gam-sae 0.3.153

//! #997 — the wiring seam between a fitted [`SaeManifoldTerm`] and the
//! evidence-guarded move engine of [`gam_solve::structure_search`].
//!
//! #976 closed with the move engine (`search`) and its triggers
//! (`gam_sae::atom_codes::SparseAtomCodes::coactivation`, ARD precisions,
//! terminal `CollapseEvent`s) on main but deliberately unwired: nothing
//! harvested move proposals from a fitted dictionary or drove `search` around
//! the production fit. This module is that seam. It owns three things:
//!
//! 1. [`harvest_move_proposals`] — reads a fitted term + its ρ + the per-row
//!    reconstruction residuals and emits the canonical-order-ready
//!    [`MoveProposal`] stream (deaths, fusions, fission audits, births).
//! 2. [`apply_structure_move`] — the warm-inheritance restructuring of a
//!    [`SaeManifoldTerm`] under one [`StructureMove`]: a death demotes an atom's
//!    routing, a fission splits an atom into two children that inherit its
//!    decoder block, a fusion folds the weaker of a pair into the stronger, a
//!    birth appends a residual-factor atom whose TOPOLOGY is chosen by EVIDENCE
//!    (#977): `race_birth_topology` races the candidate bases matched to the
//!    atom's intrinsic dim (`d = 1`: circle vs line; `d = 2`: torus vs
//!    sphere/constant-curvature vs euclidean vs cylinder) by TK-normalized REML and seeds the
//!    born atom from the winner, so the discovered dictionary is genuinely
//!    heterogeneous rather than all-circle. Every child state is built FROM the
//!    parent (never cold) so the engine's warm-state contract holds by
//!    construction.
//! 3. [`run_structure_search_rounds`] — the round driver: fit → harvest →
//!    [`search`] (over held-out row-block shards, with warm child refits) →
//!    re-fit → repeat until a round applies no moves. The accumulated
//!    [`SearchLedger`] (with the joint fit's `CollapseEvent`s) is the honesty
//!    surface returned to the caller and serialized onto the fit payload.
//!
//! # Determinism
//!
//! Pure: no RNG, no clock. Proposal triggers are deterministic functions of the
//! fitted state; the engine canonicalizes and gates them; the ledger serializes
//! byte-identically for identical inputs. The structural hashes that dedup the
//! proposal stream are computed with the same [`gam_runtime::warm_start::Fingerprinter`]
//! the [`gam_terms::smooth::TermCollectionSpec`] machinery (#869) uses, fed
//! the POST-move dictionary shape (atom count, per-atom basis kind + latent dim
//! + the move that produced it), so two proposals that reach the same dictionary
//! shape collide exactly as the engine requires.
//!
//! # Theory: curvature IS identifiability (the birth/topology race)
//!
//! Of the four move channels this module owns, the BIRTH channel is where a
//! structural theory claim becomes an operational decision: superposition
//! ambiguity is fundamentally a FLATNESS disease. When several linear
//! directions co-fire, any invertible recombination (any element of `GL(d)`
//! acting on those coordinates) reproduces the same observed activations
//! exactly — a flat co-firing subspace is generically NON-identifiable, and
//! its gauge groupoid (the group of relabelings that leave the data
//! indistinguishable) is as large as `GL(d)` itself. A CURVED embedding does
//! not have this freedom: by jet transversality, two generic curved
//! embeddings that agree to second order (the same point, tangent, AND
//! curvature/osculation) are equal on an infinite-codimension set — so a
//! curved atom's gauge groupoid collapses from `GL(d)` down to the much
//! smaller diffeomorphism-and-symmetry group (`Diff × Sym`) of its own
//! topology. Concretely: a residual-factor blob that looks like a flat 2-D
//! co-firing pair is compatible with countless linear re-mixings, but a
//! residual-factor blob that is genuinely a circle can only be reparameterized
//! by circle diffeomorphisms — its curvature is what pins it down.
//!
//! This module's birth path (residual-factor mining in
//! [`harvest_move_proposals`], candidate construction in
//! `topology_candidates_for_dim`, the commensurable evidence comparison in
//! `fit_topology_candidate`, the race itself in `race_birth_topology`, and
//! the seeding in `born_atom`) is the engine's CURE for that flatness
//! disease: instead of leaving a co-firing residual subspace as an
//! unidentifiable flat blob (or forcing every birth to inherit a fixed
//! circular template by fiat), it fits every topology whose intrinsic
//! dimension matches the candidate atom (line vs circle at `d = 1`; torus vs
//! sphere vs cylinder vs flat patch at `d = 2`) and lets the data's own
//! curvature evidence — the same declared reference-function scale the
//! smooth-term topology race uses — pick the winner. A curved winner is not
//! merely a nicer-looking basis: it is the SPECIFIC configuration whose
//! rigidity is what makes the born atom identifiable at all. The race is
//! therefore the optimizer's equilibrium response to superposition, not a
//! stylistic preference for circles.

use std::sync::Arc;

use ndarray::{Array1, Array2, ArrayView1, ArrayView2, Axis};

use crate::atom_codes::SparseAtomCodes;
use crate::basis::{AmbientSphereHarmonicEvaluator, SaeBasisEvaluator, SaeBasisSecondJet};
use crate::description_length::{BirthMdlPrescreen, predicted_birth_dl_bits};
use crate::frames::GrassmannFrame;
use crate::manifold::{
    AssignmentMode, AtlasSeamKind, AtlasTopologyReadout, GraphCompressionKind,
    GraphStructureSelection, LearnedGraphAtom, OccupancyLaw, SAE_AMBIENT_SPHERE_DEFAULT_DEGREE,
    SAE_EUCLIDEAN_PATCH_MAX_DEGREE,
    SAE_MAX_PERIODIC_HARMONICS,
    SaeAtomBasisKind, SaeAtomGeometryPlan, SaeBasisResolution, SaeManifoldAtom, SaeManifoldRho,
    SaeManifoldTerm, SaeReferenceMetricPlan, SphereChartTransition, UnitSpeedChartTransition,
    amplitude_concentration_certificate, anisotropic_flat_product_torus_penalty,
    anisotropic_flat_product_torus_penalty_aspect_derivative, classify_occupancy_interval,
    embedded_donut_torus_reference_penalty,
    embedded_donut_torus_reference_penalty_aspect_derivative,
};
use crate::migration_ledger::SaeMigrationLedger;
use crate::null_sampler::{NULL_REPLICATES, coactivation_exceedance_for_pairs};
use gam_linalg::faer_ndarray::FaerSvd;
use gam_runtime::warm_start::Fingerprinter;
use gam_solve::gaussian_reml::{
    gaussian_reml_multi_shared_dispersion_closed_form,
    gaussian_reml_multi_shared_dispersion_penalty_gradient_from_fit,
};
use gam_solve::inference::residual_factor::{ResidualFactorInput, StructuredResidualModel};
use gam_solve::structure_search::{
    ChartGlueOutcome, CollapseAction, MoveBudget, MoveProposal, SearchLedger, SearchOutcome,
    StructureMove, search,
};
use gam_solve::{
    AutoTopologyKind, TopologyAutoFitEvidence, TopologyAutoSelector, TopologyScoreScale,
    select_topology_with_fit,
};
use gam_terms::inference::structure_evidence::{ClaimKind, StructureLedger};
use gam_terms::latent::{LatentIdMode, LatentManifold};
use gam_terms::structure::anova_atom::{
    CarveReport, FissionDecision, carve, carve_input_from_fitted_atom, fission_decision,
};
use opt::{BracketedRootConfig, FirstOrderSample, ObjectiveEvalError, find_root_bracketed};
use std::sync::atomic::{AtomicBool, Ordering};

/// Per-row soft-assignment mass below which an atom is treated as INACTIVE on
/// that row when deriving the discrete co-activation support. A soft softmax /
/// gate assignment never reaches exactly zero, so the discrete masks the
/// coactivation triggers consume are obtained by thresholding the per-row mass.
/// Chosen as a fixed structural constant (magic-by-default): small enough that a
/// genuinely-routed atom counts as active on its rows, large enough that the
/// near-uniform softmax floor (`≈ 1/K`) on rows an atom does not own does not
/// leak into its support. The threshold is relative to a uniform-assignment
/// reference so it scales with `K`.
const ACTIVE_SUPPORT_REL_FLOOR: f64 = 0.5;

/// ARD log-precision above which an atom's coordinate prior is treated as
/// DIVERGED — the coordinate has been shrunk to its prior mean, so the atom
/// carries no on-manifold structure and its existence was never certified by
/// the data. This is the death-proposal trigger (#976): diverged ARD ⇒ demote
/// the atom unless its `AtomExists` claim certified in an earlier round (the
/// veto). A large positive `log_alpha` is a precision blow-up; the floor is set
/// well above the strengths a live coordinate settles at and well below the
/// outer log-strength domain endpoint.
const ARD_DIVERGENCE_LOG_PRECISION: f64 = 12.0;

/// Minimum symmetric code dependence for a pair to be proposed for FUSION. Below
/// this the two atoms' supports are essentially independent (the shattering
/// signature needs both conditionals high); above it the pair is a fusion
/// candidate. The e-gate, not this threshold, decides acceptance — this only
/// keeps the proposal stream from carrying every independent pair.
const FUSION_DEPENDENCE_FLOOR: f64 = 0.6;

/// Conventional one-sided screening level for the fixed-margin-null exceedance
/// gate on co-activation triggers (#976 top-`k` correction) — the SAME 0.05
/// screening convention as [`WITHIN_ATOM_CARVE_ALPHA`], deliberately not a
/// certificate level: the exceedance gate is a PROPOSAL filter (does this pair
/// co-fire ABOVE what the top-`k` margins mechanically force?), and the held-out
/// e-gate still owns final acceptance. See [`null_exceedance_z_floor`].
const NULL_EXCEEDANCE_ALPHA: f64 = 0.05;

/// The standardized-excess floor a pair's fixed-margin-null exceedance must clear
/// to be proposed, derived (never hand-set) from [`NULL_EXCEEDANCE_ALPHA`] as the
/// one-sided standard-normal deviate `z_{1−α}`. Charging the raw co-activation
/// coupling instead would inherit the top-`k` mechanical (anti)correlation the
/// null absorbs; requiring `z ≥ z_{1−α}` keeps only genuine above-margin
/// co-firing.
fn null_exceedance_z_floor() -> f64 {
    use statrs::distribution::{ContinuousCDF, Normal};
    // Standard normal inverse-CDF at 1 − α (α the conventional screening level).
    Normal::new(0.0, 1.0)
        .expect("standard normal is well-defined")
        .inverse_cdf(1.0 - NULL_EXCEEDANCE_ALPHA)
}

/// Minimum conditional asymmetry for a pair to be proposed for a FISSION audit
/// (the A⇒B absorption signature: one conditional near 1 without the converse).
const ABSORPTION_ASYMMETRY_FLOOR: f64 = 0.5;

/// Anti-symmetric decoder perturbation applied when a fission DUPLICATES an atom,
/// to break the symmetric saddle so the two children can separate in the joint
/// refit (see `duplicate_atom`). Small enough to preserve the warm-start (and the
/// mass-split combined decoder is exactly unchanged), but ≫ floating-point noise.
const FISSION_SYMMETRY_BREAK_EPS: f64 = 0.05;

/// Level at which the within-atom representational carve (#993) calls binding
/// PROVEN (blocking a fission). The harvest carve is a PROPOSAL filter, not the
/// final certificate — the downstream held-out e-gate owns acceptance — so this
/// is the conventional 0.05 screening level, deliberately not the stricter
/// certificate level; a carve that fails to reject here still rides as a
/// fission proposal for the e-gate to adjudicate on held-out shards.
const WITHIN_ATOM_CARVE_ALPHA: f64 = 0.05;

/// Knobs for one harvest pass. All magic-by-default — derived from the fit, not
/// surfaced as user flags.
#[derive(Clone, Copy, Debug)]
pub struct HarvestParams {
    /// Maximum fusion pairs proposed per round (the top-dependence pairs).
    pub max_fusions: usize,
    /// Maximum fission audits proposed per round (the top-asymmetry pairs).
    pub max_fissions: usize,
    /// Maximum residual-factor birth candidates proposed per round (the top
    /// factor directions by explained residual mass).
    pub max_births: usize,
}

impl Default for HarvestParams {
    fn default() -> Self {
        // A small fixed budget per round; the round driver iterates until a
        // round applies nothing, so per-round breadth need not be exhaustive.
        Self {
            max_fusions: 4,
            max_fissions: 4,
            max_births: 4,
        }
    }
}

/// Derive the discrete active-support codes the co-activation triggers consume
/// from a fitted term's SOFT assignments. An atom counts as active on a row when
/// its assignment mass exceeds `ACTIVE_SUPPORT_REL_FLOOR / K` (relative to the
/// uniform-assignment reference), so the discrete support reflects genuine
/// routing rather than the near-uniform softmax floor.
pub fn sparse_codes_from_term(term: &SaeManifoldTerm) -> SparseAtomCodes {
    let assignments = term.assignment.assignments();
    let n = assignments.nrows();
    let k = assignments.ncols();
    let floor = if k == 0 {
        0.0
    } else {
        ACTIVE_SUPPORT_REL_FLOOR / k as f64
    };
    let mut codes = SparseAtomCodes::empty(n, k);
    for row in 0..n {
        for atom in 0..k {
            let mass = assignments[[row, atom]];
            if mass > floor {
                codes.row_mut(row).assign(atom, mass);
            }
        }
    }
    codes
}

/// Per-atom maximum active mass over rows — the collapse statistic (a
/// legitimately sparse atom has small MEAN mass but high MAX on its rows; only
/// an atom with no material support anywhere has a small MAX). Used as the
/// birth-residual activity coordinate and as a secondary death signal.
fn per_atom_max_mass(term: &SaeManifoldTerm) -> Array1<f64> {
    let assignments = term.assignment.assignments();
    let k = assignments.ncols();
    let mut out = Array1::<f64>::zeros(k);
    for atom in 0..k {
        let mut max = 0.0_f64;
        for &m in assignments.column(atom).iter() {
            if m > max {
                max = m;
            }
        }
        out[atom] = max;
    }
    out
}

/// Participation ratio `(Σλ)²/Σλ²` of a non-negative spectrum — the effective
/// number of significant directions (the #2233 span estimate `ŝ` when the spectrum
/// is the residual factor-energy set). `1.0` for a single direction (or an
/// all-but-one-zero spectrum); `0.0` for an empty / all-zero spectrum.
fn participation_ratio(spectrum: &[f64]) -> f64 {
    let sum: f64 = spectrum.iter().map(|&e| e.max(0.0)).sum();
    let sum_sq: f64 = spectrum.iter().map(|&e| e.max(0.0) * e.max(0.0)).sum();
    if sum_sq > 0.0 {
        (sum * sum) / sum_sq
    } else {
        0.0
    }
}

/// The curved topology `(d, m)` the #2233 pre-screen matches to an estimated
/// ambient span `ŝ`, so the dictionary surcharge is priced against the realizable
/// curved atom a span-`ŝ` residual would be raced into. The curved families top
/// out at `d = 2`, so a span `≥ 4` residual is priced against the richest curved
/// atom (the torus); the e-gate, never this map, owns acceptance.
///
/// Both numbers are READ OFF the [`SaeAtomGeometryPlan`] that
/// [`topology_candidates_for_dim`] (the downstream birth topology race) would
/// construct for that span, never transcribed from it: `d` is the plan's
/// [`SaeAtomGeometryPlan::intrinsic_dim`] — the pricing dimension, `2` for the
/// ambient sphere, which carries three coordinates for two degrees of freedom —
/// and `m` its [`SaeAtomGeometryPlan::basis_size`]. Circle `2·1+1 = 3`, sphere
/// `(degree+1)² = 9` at the default degree 2, torus `(2·2+1)² = 25`.
///
/// #2749: the transcribed predecessor priced the sphere at width **7**, the
/// width of the `(lat, lon)` chart deleted in `1dfa70140`. Deriving from the
/// plan is what makes that unrepeatable — `SaeAtomGeometryPlan::new` refuses the
/// chart form `(Sphere, latent_dim = 2, ..)` outright, so a price on an
/// unbuildable atom is now a hard error here instead of a silent literal.
fn curved_topology_for_span(span: f64) -> Result<(usize, usize), String> {
    let plan = SaeAtomGeometryPlan::curved_prescreen_atom_for_span(span)?;
    Ok((plan.intrinsic_dim(), plan.basis_size()?))
}

/// Mean active atoms per token `L0` — the support-budget denominator for the
/// #2233 pre-screen's `log₂(G/L0)` term. An atom counts as active on a row by the
/// SAME `ACTIVE_SUPPORT_REL_FLOOR / K` discrete-support threshold
/// [`sparse_codes_from_term`] uses (no new constant), floored at `1.0` so the
/// support term is well-defined even on a degenerate all-inactive round.
fn mean_active_atoms(assignments: ArrayView2<'_, f64>) -> f64 {
    let n = assignments.nrows();
    let k = assignments.ncols();
    if n == 0 || k == 0 {
        return 1.0;
    }
    let floor = ACTIVE_SUPPORT_REL_FLOOR / k as f64;
    let mut total_active = 0usize;
    for row in 0..n {
        for atom in 0..k {
            if assignments[[row, atom]] > floor {
                total_active += 1;
            }
        }
    }
    (total_active as f64 / n as f64).max(1.0)
}

/// The largest per-atom ARD log-precision (over the atom's axes), or `-inf` for
/// an atom with native ARD disabled (empty block). A diverged precision on ANY
/// axis collapses that coordinate, so the per-atom death trigger is the max.
fn per_atom_ard_divergence(rho: &SaeManifoldRho, atom: usize) -> f64 {
    rho.log_ard
        .get(atom)
        .and_then(|axes| axes.iter().copied().reduce(f64::max))
        .unwrap_or(f64::NEG_INFINITY)
}

/// Structural hash of the POST-move dictionary shape, computed with the same
/// [`Fingerprinter`] the [`TermCollectionSpec`](gam_terms::smooth::TermCollectionSpec)
/// hash machinery (#869) uses. The hash covers the move kind, the atoms it
/// touches, and the resulting atom count + per-atom basis-kind/latent-dim
/// shape — structural identity only, never decoder coefficients or coordinates,
/// so two distinct proposals that reach the same dictionary shape collide.
fn post_move_structure_hash(term: &SaeManifoldTerm, mv: &StructureMove) -> u64 {
    let mut fp = Fingerprinter::new();
    fp.write_str("sae_structure_move");
    match mv {
        StructureMove::Birth { candidate } => {
            fp.write_str("birth");
            fp.write_usize(*candidate);
        }
        StructureMove::Death { atom } => {
            fp.write_str("death");
            fp.write_usize(*atom);
        }
        StructureMove::Fission { atom } => {
            fp.write_str("fission");
            fp.write_usize(*atom);
        }
        StructureMove::Fusion { a, b } => {
            fp.write_str("fusion");
            // Order-independent: a fusion of (a,b) is the same structure as
            // (b,a).
            fp.write_usize((*a).min(*b));
            fp.write_usize((*a).max(*b));
        }
        StructureMove::Glue { a, b, outcome } => {
            // A fuse reaches the same physical dictionary shape as Fusion and
            // deliberately shares its tag.  Atlas registration is a different
            // post-move structure: K local charts stay, while the pair becomes
            // one semantic atom with a persisted transition cocycle.
            fp.write_str(match outcome {
                ChartGlueOutcome::Fuse => "fusion",
                ChartGlueOutcome::RegisterAtlas => "atlas_register",
            });
            fp.write_usize((*a).min(*b));
            fp.write_usize((*a).max(*b));
        }
    }
    // Post-move atom-shape skeleton: the current per-atom (basis-kind tag,
    // latent dim) plus the count delta the move applies. Births/fissions add an
    // atom; deaths/fusions do not change the count (death demotes, fusion folds)
    // — the routing change, not a structural resize, so the shape skeleton is
    // the parent's plus the move tag above.
    fp.write_usize(term.atoms.len());
    for atom in &term.atoms {
        fp.write_str(basis_kind_tag(atom.basis_kind()));
        fp.write_usize(atom.latent_dim());
    }
    let digest = fp.finalize();
    let bytes = digest.as_bytes();
    u64::from_le_bytes([
        bytes[0], bytes[1], bytes[2], bytes[3], bytes[4], bytes[5], bytes[6], bytes[7],
    ])
}

/// Collapse the raw fission-audit list to ONE entry per parent atom, keeping the
/// most-suspect nomination (the LOWEST significance — significance is the
/// ascending `1 − asym` proxy, so smaller = more absorption-suspect).
///
/// A single parent can be nominated by several partners at different
/// significances, so the raw list carries duplicate atoms. The old
/// `sort_by(significance).dedup_by_key(atom)` was wrong: `dedup_by_key` removes
/// only ADJACENT duplicates, and a significance-first sort does not place
/// same-atom entries adjacently, so duplicates survived — the same parent rode as
/// several `Fission` proposals, wasting births on a duplicate split. Here the
/// per-atom minimum is taken explicitly, then the survivors are re-sorted by the
/// total order `(significance asc, atom asc)` so the result is most-suspect-first
/// (the order the downstream `take(max_fissions)` and carve loop expect) and fully
/// deterministic despite the `HashMap`'s arbitrary iteration order.
fn dedup_most_suspect_per_parent(candidates: Vec<(usize, f64)>) -> Vec<(usize, f64)> {
    let mut best_per_parent: std::collections::HashMap<usize, f64> =
        std::collections::HashMap::new();
    for (atom, significance) in candidates {
        best_per_parent
            .entry(atom)
            .and_modify(|s| {
                if significance < *s {
                    *s = significance;
                }
            })
            .or_insert(significance);
    }
    let mut out: Vec<(usize, f64)> = best_per_parent.into_iter().collect();
    out.sort_by(|x, y| x.1.total_cmp(&y.1).then(x.0.cmp(&y.0)));
    out
}

/// Structural tag for an atom basis kind — the discrete shape identity the
/// structural hash needs (never coordinates or coefficients).
fn basis_kind_tag(kind: &SaeAtomBasisKind) -> &str {
    match kind {
        SaeAtomBasisKind::Duchon => "duchon",
        SaeAtomBasisKind::Periodic => "periodic",
        SaeAtomBasisKind::Sphere => "sphere",
        SaeAtomBasisKind::Torus => "torus",
        SaeAtomBasisKind::ProjectivePlane => "projective_plane",
        SaeAtomBasisKind::KleinBottle => "klein_bottle",
        SaeAtomBasisKind::Linear => "linear",
        SaeAtomBasisKind::EuclideanPatch => "euclidean_patch",
        SaeAtomBasisKind::Poincare => "poincare",
        SaeAtomBasisKind::Cylinder => "cylinder",
        SaeAtomBasisKind::Mobius => "mobius",
        SaeAtomBasisKind::FiniteSet => "finite_set",
        SaeAtomBasisKind::Precomputed(_) => "precomputed",
    }
}

/// Build a [`MoveProposal`] from a move + trigger by stamping its post-move
/// structural hash and the structural claim it asserts.
fn proposal(term: &SaeManifoldTerm, mv: StructureMove, trigger: f64) -> MoveProposal {
    let structure_hash = post_move_structure_hash(term, &mv);
    let claim = match &mv {
        StructureMove::Birth { candidate } => ClaimKind::AtomExists {
            // Births claim the existence of the NEXT atom index (appended).
            atom: term.k_atoms() + *candidate,
        },
        StructureMove::Death { atom } => ClaimKind::AtomExists { atom: *atom },
        StructureMove::Fusion { a, b } => ClaimKind::BindingEdge { a: *a, b: *b },
        StructureMove::Fission { atom } => ClaimKind::Custom {
            label: format!("fission:{atom}"),
        },
        // #1890: the seam-glue claim is its own ledger entry, distinct from the
        // fusion `BindingEdge` on the same pair (glue asserts "these two charts
        // tile ONE manifold within an isometry tolerance", a strictly stronger
        // claim than "these two atoms co-fire and bind"). Carried as a labeled
        // `Custom` — an ordered `(min,max)` key so `Glue{a,b}` and `Glue{b,a}`
        // dedup — rather than a new `ClaimKind` variant, to avoid a cross-crate
        // enum change whose exhaustive-match fallout the design does not need.
        StructureMove::Glue { a, b, .. } => ClaimKind::Custom {
            // Atom indices are only stable within one dictionary epoch.  A
            // certified glue physically compacts the atom columns at the round
            // boundary, so the next round's `(0, 1)` can denote a DIFFERENT
            // pair than this round's `(0, 1)`.  Scope the running e-process by
            // the proposal's stamped structural hash (which includes the
            // current dictionary skeleton): contested seams in an unchanged
            // dictionary still resume their evidence, while a compaction can
            // never lend already-certified evidence to a newly re-indexed pair.
            label: format!(
                "seam_glue:{structure_hash:016x}:{}:{}",
                (*a).min(*b),
                (*a).max(*b)
            ),
        },
    };
    MoveProposal {
        mv,
        trigger,
        structure_hash,
        claim,
    }
}

/// Harvest the canonical move-proposal stream from a fitted term, its ρ, and the
/// per-row reconstruction residuals `R = target − fitted` (used for the birth
/// channel under the `WhitenedStructured` (`gam_inference::row_metric::MetricProvenance::WhitenedStructured`)
/// residual-factor metric — never raw-Euclidean Λ, per the #974 rescope).
///
/// The four channels (#976/#997):
///
/// * **Deaths** from diverged ARD precisions ∪ terminal `CollapseEvent`s. The
///   trigger is the ARD precision (descending); a terminally-collapsed atom is
///   proposed even with finite ARD (its routing is gone regardless of its
///   coordinate prior).
/// * **Fusions** from the top co-activation pairs by symmetric code dependence.
/// * **Fission audits** from absorption-suspect pairs (high conditional
///   asymmetry). For each candidate that is a `d = 2` product atom the
///   within-atom functional-ANOVA carve (#975 / #993) RUNS on the atom's own
///   fitted decoder via `run_within_atom_carve`: a carve that proves binding
///   blocks the fission (the atom is irreducible), an additive carve rides as a
///   fission proposal ranked by its interaction fraction, and every outcome is
///   recorded on [`HarvestReport::fission_carve_results`]. A non-product
///   candidate (no factor split) rides on the co-activation audit and is
///   counted in [`HarvestReport::fission_carve_unavailable_count`] — never a
///   silent drop. The held-out e-gate still owns final acceptance.
/// * **Births** from the whitened residual-factor subspace: the residuals are
///   fed to [`StructuredResidualModel::fit`], whose factor directions
///   ([`StructuredResidualModel::factor`]) are the birth candidates, ranked by
///   explained residual mass. This is a SHAPE-level mining step — it finds
///   directions the current dictionary does not reconstruct, not yet a claim
///   about what topology lives there. The topology itself is adjudicated
///   downstream, atom-by-atom, by `race_birth_topology` (see the module
///   docs: curvature is what makes the winner identifiable). Note the two
///   halves of identifiability this leaves complementary rather than
///   redundant: this residual-factor step (and the topology race it feeds) is
///   a SUPPORT/shape-level test (does the reconstruction residual look like a
///   line, a circle, a torus…?), while a separate producer elsewhere in this
///   crate (the ISA κ-contrast statistic, `identifiability.rs` /
///   `isa_seed.rs`) is a MEASURE-level test: a centered circle's cone `ℝ₊·Y`
///   is literally the same point set as a 2-plane minus the origin, so no
///   support-based test can ever tell a dense circle from a Gaussian plane —
///   only the radial fourth-moment ratio `κ = E[r⁴]/E[r²]²` (`= 1` dense
///   circle, `= 2` Gaussian plane, `= 1/q` gated) can, because it reads the
///   RADIAL LAW rather than the support. Neither test subsumes the other;
///   they see complementary halves of the same identifiability question.
pub fn harvest_move_proposals(
    term: &SaeManifoldTerm,
    rho: &SaeManifoldRho,
    residuals: ArrayView2<'_, f64>,
    params: &HarvestParams,
) -> Result<HarvestReport, String> {
    let k = term.k_atoms();
    let mut proposals: Vec<MoveProposal> = Vec::new();

    // --- Deaths: diverged ARD ∪ terminal collapses -------------------------
    let max_mass = per_atom_max_mass(term);
    let terminal: std::collections::HashSet<usize> = term
        .collapse_events()
        .iter()
        .filter(|e| matches!(e.action, CollapseAction::Terminal))
        .map(|e| e.atom)
        .collect();
    for atom in 0..k {
        let ard = per_atom_ard_divergence(rho, atom);
        let diverged = ard >= ARD_DIVERGENCE_LOG_PRECISION;
        let collapsed = terminal.contains(&atom);
        if diverged || collapsed {
            // Trigger (descending): a terminal collapse is maximally urgent
            // (the routing is already gone), ranked above ARD divergence; ARD
            // deaths rank by precision. `max_mass` breaks ties toward emptier
            // atoms.
            let trigger = if collapsed { f64::MAX / 2.0 } else { ard };
            // Lower max-mass (emptier) sorts first among equal triggers; encode
            // by subtracting a small mass-proportional term that cannot reorder
            // across the collapsed/ARD bands.
            let trigger = trigger - max_mass[atom].min(1.0) * 1e-9;
            proposals.push(proposal(term, StructureMove::Death { atom }, trigger));
        }
    }

    // --- Fixed-margin (curveball) null for the co-activation triggers ------
    // Top-`k` selection stamps mechanical (anti)correlation into the co-activation
    // masks (each token's fixed support size induces a negative indicator
    // covariance between every pair; a hard top-`k` puts zero mass off the
    // `k`-shell). A raw coupling trigger reads that artifact as structure. So the
    // fusion/fission triggers below are gated on the EXCEEDANCE of each pair's
    // joint activation over a null that preserves both the row margins (the
    // top-`k` constraint) and the column margins (per-atom totals): only
    // above-margin co-firing survives. Computed once and shared by both triggers;
    // skipped entirely when neither trigger is enabled (the null is not free).
    let codes = sparse_codes_from_term(term);
    let want_coactivation = params.max_fusions > 0 || params.max_fissions > 0;
    let coactive_pairs = if want_coactivation {
        codes.coactive_pair_stats()
    } else {
        Vec::new()
    };
    let coactive_pair_keys: Vec<(usize, usize)> =
        coactive_pairs.iter().map(|(a, b, _)| (*a, *b)).collect();
    let exceedance_z = if want_coactivation {
        coactivation_exceedance_for_pairs(&codes, &coactive_pair_keys, NULL_REPLICATES)
    } else {
        Vec::new()
    };
    let z_floor = null_exceedance_z_floor();

    // --- Fusions: top co-activation dependence, gated by the null ----------
    // The trigger REPORTED is the null exceedance `z` (above-margin co-firing),
    // not the raw dependence: the raw floor only pre-selects genuinely co-firing
    // pairs, and the fixed-margin null strips the mechanical top-`k` coupling.
    let mut fusion_pairs: Vec<(usize, usize, f64)> = Vec::new();
    for (pair_idx, &(a, b, stats)) in coactive_pairs.iter().enumerate() {
        let dep = stats.dependence();
        if dep < FUSION_DEPENDENCE_FLOOR {
            continue;
        }
        let z = exceedance_z[pair_idx];
        if z >= z_floor {
            fusion_pairs.push((a, b, z));
        }
    }
    fusion_pairs.sort_by(|x, y| y.2.total_cmp(&x.2).then(x.0.cmp(&y.0)).then(x.1.cmp(&y.1)));
    for &(a, b, z) in fusion_pairs.iter().take(params.max_fusions) {
        proposals.push(proposal(term, StructureMove::Fusion { a, b }, z));
    }

    // --- Glue: chart-gluing over-tiling detector (#1890) -------------------
    // A SECOND, independent proposal lane, blind to the co-activation currency
    // the fusion lane above runs on. Atoms over-tiling ONE manifold have
    // DISJOINT supports (each owns its own arc), hence anti-correlated codes, so
    // no such pair EVER clears the co-activation floor — the fusion lane is
    // structurally blind to them. This lane screens pairs GEOMETRICALLY instead:
    // a d=1 periodic pair whose decoder AMBIENT spans align (small principal
    // angles via the Grassmann frame) and whose supports are disjoint is a
    // candidate. Acceptance is NOT decided here — the seam equivalence e-value
    // from `unit_speed_glue_certificate` is carried on the proposal's trigger and
    // the engine's Glue arm banks it against the churn null. The pre-screen only
    // RANKS, so the budget spends its e-value evaluations on the most-aligned
    // pairs; it carries no acceptance threshold of its own (magic-free).
    let mut certified_glues = Vec::new();
    let (glues_proposed, glue_candidates_screened) = harvest_glue_proposals(
        term,
        residuals,
        params.max_fusions,
        &mut proposals,
        &mut certified_glues,
    );

    // --- Fission audits: absorption-suspect asymmetry, gated by the null ---
    let mut fission_atoms: Vec<(usize, f64)> = Vec::new();
    for (pair_idx, &(a, b, stats)) in coactive_pairs.iter().enumerate() {
        let asym = stats.absorption_asymmetry();
        if asym < ABSORPTION_ASYMMETRY_FLOOR {
            continue;
        }
        // A nested (absorbed) pair co-fires ABOVE its fixed margins; a pair
        // whose asymmetry is only the top-`k` mechanical artifact does not.
        // Require the joint activation to exceed the fixed-margin null before
        // auditing, so mechanical asymmetry is not read as absorption.
        let z = exceedance_z[pair_idx];
        if z < z_floor {
            continue;
        }
        // The parent (the conditioned-on atom whose support nests the child) is
        // the one whose `P(parent|child) ≈ 1`. Audit the parent for the absorbed
        // substructure.
        let parent = if stats.p_a_given_b >= stats.p_b_given_a {
            a
        } else {
            b
        };
        // Fission trigger is audit significance ASCENDING; map a high asymmetry
        // to a low significance proxy `1 − asym` so the most asymmetric (most
        // suspect) pair sorts first.
        let significance = (1.0 - asym).max(0.0);
        fission_atoms.push((parent, significance));
    }
    // Keep the most-suspect (lowest significance) audit per parent atom.
    let fission_atoms = dedup_most_suspect_per_parent(fission_atoms);

    // #993: run the within-atom functional-ANOVA carve on each fission
    // candidate that is a genuine `d = 2` product atom. The carve adjudicates
    // the representational binding question (is the surface ONE bound product
    // atom or TWO superposed factors?) on the atom's OWN fitted decoder, on the
    // same empirical code measure. A carve that PROVES binding (the interaction
    // is significant, or energetically non-negligible) blocks the fission — the
    // atom stays whole and contested; the e-gate never sees a fission proposal
    // for a bound atom. A carve that does NOT prove binding rides as a fission
    // proposal whose trigger is the carve's interaction fraction (ascending —
    // the most-separable atom sorts first), and whose binding evidence is the
    // carve's `edge_p_value`, recorded for the ledger.
    //
    // A candidate that is NOT a recoverable product atom (single-axis, sphere
    // chart, monomial patch — `factor_basis_sizes() == None`), or whose carve
    // could not run (degenerate sample, non-separable basis), is recorded
    // loudly via `fission_carve_unavailable` rather than silently dropped: its
    // fission audit still rides on the co-activation significance, exactly the
    // pre-#993 behavior, but the absence of the carve is now an explicit,
    // counted signal instead of a blanket skip.
    let mut carve_results: Vec<FissionCarveResult> = Vec::new();
    let mut fission_carve_ran_count = 0usize;
    let mut fission_carve_unavailable_count = 0usize;
    let mut fission_carve_blocked_count = 0usize;
    let mut gated_fissions: Vec<(usize, f64)> = Vec::new();
    for &(atom, significance) in fission_atoms.iter().take(params.max_fissions) {
        match run_within_atom_carve(term, atom) {
            Some(Ok(report)) => {
                fission_carve_ran_count += 1;
                let decision = fission_decision(&report, None);
                let edge_p = report.edge_p_value;
                let interaction = report.interaction_fraction;
                carve_results.push(FissionCarveResult {
                    atom,
                    edge_p_value: edge_p,
                    interaction_fraction: interaction,
                    decision,
                });
                match decision {
                    FissionDecision::Keep => {
                        // Binding proven (or interaction non-negligible): the
                        // atom is irreducible. Do NOT propose a fission.
                        fission_carve_blocked_count += 1;
                        log::debug!(
                            "[structure-harvest] #993 carve KEEPS atom {atom}: binding proven \
                             (edge_p={edge_p:?}, interaction_fraction={interaction:.3e}); no fission proposed",
                        );
                    }
                    FissionDecision::SplitReconstructionOnly
                    | FissionDecision::SplitCertifiedJoint => {
                        // Separable: propose the fission, ranked by interaction
                        // fraction (ascending — most-additive first).
                        gated_fissions.push((atom, interaction));
                    }
                }
            }
            Some(Err(err)) => {
                fission_carve_unavailable_count += 1;
                log::debug!(
                    "[structure-harvest] #993 carve could not run on atom {atom}: {err}; \
                     fission audit rides on co-activation significance, e-gate owns acceptance",
                );
                gated_fissions.push((atom, significance));
            }
            None => {
                // Not a recoverable product atom — the within-atom carve is not
                // defined here. Ride the co-activation audit, count it loudly.
                fission_carve_unavailable_count += 1;
                gated_fissions.push((atom, significance));
            }
        }
    }

    for &(atom, trigger) in &gated_fissions {
        proposals.push(proposal(term, StructureMove::Fission { atom }, trigger));
    }

    // --- Births: whitened residual-factor subspace -------------------------
    // The activity coordinate the residual-factor scale law is smooth in is the
    // per-row total assignment mass (an activation-strength summary): rows where
    // the dictionary routes strongly should have smaller unexplained residual
    // factor energy than rows it does not cover.
    let n = residuals.nrows();
    let assignments = term.assignment.assignments();
    let activity: Array1<f64> = (0..n).map(|r| assignments.row(r).sum()).collect();
    let mut births_proposed = 0usize;
    let mut birth_predictions: Vec<(usize, f64)> = Vec::new();
    let mut births_deferred = 0usize;
    let mut deferred_predicted_bits = 0.0_f64;
    let mut birth_skipped_reason: Option<String> = None;
    if params.max_births > 0 && n > 0 && residuals.ncols() > 0 {
        let p = residuals.ncols();
        let max_rank = params.max_births.min(p.saturating_sub(1));
        match StructuredResidualModel::fit(ResidualFactorInput {
            residuals,
            activity: activity.view(),
            max_factor_rank: max_rank,
        }) {
            Ok(model) => {
                let factor = model.factor();
                let diagonal = model.diagonal();
                let r = model.factor_rank();
                // #2233 closed-form MDL birth pre-screen. Every quantity below is
                // read from the structured residual-factor fit already computed —
                // no candidate refit runs here. Per-proposal crossover inputs:
                //  * ŝ_j = LOCAL ambient span (participation ratio of the residual's
                //    factor-coordinate energies on candidate j's OWN active rows),
                //  * (d_j, m_j) = the curved topology matched to ŝ_j,
                //  * G = current dictionary size, L0 = mean active atoms/token,
                //  * N = tokens, P = channels.
                let energies: Vec<f64> = (0..r)
                    .map(|j| factor.column(j).iter().map(|v| v * v).sum::<f64>())
                    .collect();
                let norms: Vec<f64> = energies.iter().map(|&e| e.sqrt()).collect();
                // Per-row projection onto each UNIT factor direction, computed once
                // (`unit_proj[[i, l]] = r_i · u_l`, `u_l = col_l/‖col_l‖`). A proposal's
                // ambient span is then read LOCALLY from these coordinates on its own
                // active rows — never one global participation ratio of the whole
                // residual spectrum, which conflates every co-mined factor and biases
                // admit/defer inconsistently (up when the spectrum is rich, down when
                // it is peaked).
                let mut unit_proj = Array2::<f64>::zeros((n, r));
                for row in 0..n {
                    let res_row = residuals.row(row);
                    for l in 0..r {
                        if norms[l] > 0.0 {
                            let col = factor.column(l);
                            let mut proj = 0.0_f64;
                            for out in 0..p {
                                proj += res_row[out] * col[out];
                            }
                            unit_proj[[row, l]] = proj / norms[l];
                        }
                    }
                }
                let g_dict = term.k_atoms();
                let l0 = mean_active_atoms(assignments.view());
                let n_tokens = n as f64;
                // Score every factor direction; a positive predicted ΔMDL rides as a
                // proposal (ordered by the prediction), a non-positive one is
                // DEFERRED (not proposed this round — a soft defer, never a kill).
                let mut scored: Vec<(usize, f64)> = Vec::with_capacity(r);
                for j in 0..r {
                    let energy = energies[j];
                    if !(energy > 0.0) {
                        // A zero-energy direction carries no residual structure to
                        // birth from — defer it (no finite prediction to bank).
                        births_deferred += 1;
                        continue;
                    }
                    let col = factor.column(j);
                    let norm = norms[j];
                    // Per-direction idiosyncratic-noise floor δ_j = u_jᵀ D u_j (the
                    // residual diagonal projected onto the unit birth direction) —
                    // derived from the fitted noise model, not a hand-set floor.
                    let mut noise_floor = 0.0_f64;
                    for out in 0..p {
                        let u = col[out] / norm;
                        noise_floor += u * u * diagonal[out];
                    }
                    // ρ̂_j (fraction of tokens above the noise floor on u_j) AND the
                    // local factor-coordinate energies on j's active rows, in ONE pass.
                    let mut active = 0usize;
                    let mut local_energy = vec![0.0_f64; r];
                    for row in 0..n {
                        let proj_j = unit_proj[[row, j]];
                        if proj_j * proj_j > noise_floor {
                            active += 1;
                            for l in 0..r {
                                let v = unit_proj[[row, l]];
                                local_energy[l] += v * v;
                            }
                        }
                    }
                    let rho = active as f64 / n_tokens;
                    // ŝ_j: the LOCAL ambient span — participation ratio of the residual's
                    // factor-coordinate energies where THIS candidate fires (a circle
                    // living in a 2-plane ⇒ ≈2; an isolated direction ⇒ ≈1, priced as
                    // linear). (d_j, m_j) follow from ŝ_j, so the dictionary/support
                    // terms are matched to the atom this candidate would actually race.
                    let span = participation_ratio(&local_energy);
                    let (intrinsic_dim, basis_size) = curved_topology_for_span(span)?;
                    let predicted = predicted_birth_dl_bits(&BirthMdlPrescreen {
                        rho,
                        span,
                        intrinsic_dim,
                        basis_size,
                        signal_var: energy,
                        noise_floor,
                        n_tokens,
                        p_out: p,
                        g_dict,
                        l0,
                    });
                    if predicted.is_finite() && predicted > 0.0 {
                        scored.push((j, predicted));
                    } else {
                        births_deferred += 1;
                        if predicted.is_finite() {
                            deferred_predicted_bits += predicted;
                        }
                    }
                }
                // Order the survivors by predicted ΔMDL (descending), tie-break by
                // index, and cap at `max_births`; the overflow is deferred too.
                scored.sort_by(|a, b| b.1.total_cmp(&a.1).then(a.0.cmp(&b.0)));
                for &(candidate, predicted) in scored.iter().take(params.max_births) {
                    proposals.push(proposal(
                        term,
                        StructureMove::Birth { candidate },
                        predicted,
                    ));
                    birth_predictions.push((candidate, predicted));
                    births_proposed += 1;
                }
                for &(_, predicted) in scored.iter().skip(params.max_births) {
                    births_deferred += 1;
                    deferred_predicted_bits += predicted;
                }
                if births_deferred > 0 {
                    log::debug!(
                        "[structure-harvest] #2233 MDL pre-screen deferred {births_deferred} \
                         birth(s) (total predicted ΔMDL {deferred_predicted_bits:.1} bits; \
                         per-proposal local span) of {r} residual factors; proposed \
                         {births_proposed} ordered by predicted ΔMDL",
                    );
                }
            }
            Err(e) => {
                birth_skipped_reason = Some(e);
            }
        }
    } else if params.max_births > 0 {
        birth_skipped_reason =
            Some("residuals empty or single-channel; no factor subspace to mine".to_string());
    }

    Ok(HarvestReport {
        proposals,
        fission_carve_results: carve_results,
        fission_carve_ran_count,
        fission_carve_unavailable_count,
        fission_carve_blocked_count,
        births_proposed,
        birth_predictions,
        births_deferred,
        deferred_predicted_bits,
        birth_skipped_reason,
        glues_proposed,
        glue_candidates_screened,
        certified_glues,
    })
}

/// One within-atom carve outcome on a fission candidate (#993). Recorded on
/// the [`HarvestReport`] so the binding decision and its evidence are visible
/// — including the `edge_p_value` the dictionary certificate's `BindingEdge`
/// claim reads — never silent.
#[derive(Clone, Debug)]
pub struct FissionCarveResult {
    /// The audited product atom.
    pub atom: usize,
    /// Edge-level representational binding p-value (the carve's joint Wald over
    /// the gauge-projected interaction block). `None` when the test degenerated.
    pub edge_p_value: Option<f64>,
    /// Fraction of centered surface energy carried by the interaction
    /// (0 = perfectly additive / separable, 1 = pure interaction).
    pub interaction_fraction: f64,
    /// The carve's representational fission verdict.
    pub decision: FissionDecision,
}

/// Run the within-atom representational carve on one fitted atom (#993).
///
/// Returns:
/// * `None` — the atom is not a recoverable `d = 2` product atom (no factor
///   split); the within-atom carve is undefined here.
/// * `Some(Err(_))` — the atom is a product atom but the carve could not run
///   (degenerate sample, non-separable basis, REML fit failure).
/// * `Some(Ok(report))` — the carve ran; the report carries the binding
///   verdict and evidence.
///
/// The factor sizes come from the atom's basis evaluator
/// ([`SaeBasisEvaluator::factor_basis_sizes`]); the carve inputs are built from
/// the atom's FUSED basis and decoder by
/// [`gam_terms::structure::anova_atom::carve_input_from_fitted_atom`], which
/// verifies the Kronecker separability before fitting.
fn run_within_atom_carve(
    term: &SaeManifoldTerm,
    atom: usize,
) -> Option<Result<CarveReport, String>> {
    let a = &term.atoms[atom];
    if a.latent_dim() != 2 {
        return None;
    }
    let evaluator = a.basis_evaluator.as_ref()?;
    let (m_a, m_b) = evaluator.factor_basis_sizes()?;
    let build = carve_input_from_fitted_atom(
        a.basis_values.view(),
        a.decoder_coefficients().view(),
        m_a,
        m_b,
    );
    let bundle = match build {
        Ok(b) => b,
        Err(e) => return Some(Err(e)),
    };
    let input = bundle.representational_carve_input();
    Some(carve(&input, WITHIN_ATOM_CARVE_ALPHA))
}

/// The exact geometric object which earned a chart-glue proposal's equivalence
/// e-value.  Glue acceptance is based on that harvest-time certificate, so the
/// round driver carries it unchanged to the adoption boundary instead of trying
/// to infer the seam again from a proposal-scoring refit.
#[derive(Clone, Debug)]
enum CertifiedGlueTransition {
    UnitSpeed {
        transition: UnitSpeedChartTransition,
        /// B's support at certification time.  A destructive fusion transplants
        /// precisely these coordinates before B is physically removed.
        rows_b: Vec<usize>,
    },
    Sphere(SphereChartTransition),
}

/// Harvest-time certificate paired one-to-one with an emitted glue proposal.
#[derive(Clone, Debug)]
struct CertifiedGlue {
    a: usize,
    b: usize,
    outcome: ChartGlueOutcome,
    transition: CertifiedGlueTransition,
}

/// The output of one [`harvest_move_proposals`] pass: the proposal stream plus
/// the loud records of any degrade-to-skip path taken (no silent drops).
#[derive(Clone, Debug)]
pub struct HarvestReport {
    /// Trigger-stamped, claim-stamped, structurally-hashed proposals, ready for
    /// [`search`] (which canonicalizes and gates them).
    pub proposals: Vec<MoveProposal>,
    /// The within-atom carve outcomes (#993): one entry per fission candidate
    /// (within the `max_fissions` cap) that is a recoverable `d = 2` product
    /// atom whose carve RAN. Carries the representational binding verdict and
    /// `edge_p_value` — the dictionary certificate's `BindingEdge` evidence —
    /// so a fission's binding decision is visible, never silent. A carve that
    /// KEEPS the atom (binding proven) blocked its fission proposal; the
    /// remaining entries' atoms each have a corresponding `Fission` proposal.
    pub fission_carve_results: Vec<FissionCarveResult>,
    /// How many fission candidates had the #993 within-atom carve actually run.
    pub fission_carve_ran_count: usize,
    /// How many fission candidates could NOT be carved (not a product atom, or
    /// the carve failed) and rode on the co-activation audit instead — the
    /// precise, never-silent record of the residual degrade path.
    pub fission_carve_unavailable_count: usize,
    /// How many fission candidates the carve BLOCKED (binding proven → atom kept
    /// whole, no fission proposed). These never reach the e-gate.
    pub fission_carve_blocked_count: usize,
    /// Number of residual-factor birth candidates proposed.
    pub births_proposed: usize,
    /// #2233 closed-form MDL pre-screen: `(candidate index, predicted ΔMDL bits)`
    /// for every residual-factor birth that was PROPOSED (predicted saving > 0).
    /// The candidate index is the factor direction the birth seeds from — the same
    /// index the [`StructureMove::Birth`] carries — so the round driver threads
    /// each prediction into the unified [`SaeMigrationLedger`] record the post-refit
    /// verdict fills in (the predicted-vs-realized calibration curve).
    pub birth_predictions: Vec<(usize, f64)>,
    /// #2233: number of residual-factor births DEFERRED this round — non-positive
    /// predicted ΔMDL, so not proposed (a soft defer: they may return next round
    /// once the residual changes; never a hard kill).
    pub births_deferred: usize,
    /// #2233: total predicted ΔMDL (bits) summed over the deferred births — the
    /// round-cadence honesty figure logged alongside the deferred count.
    pub deferred_predicted_bits: f64,
    /// If the birth channel could not run (empty residuals, evidence-ladder
    /// failure), why — so the absence of births is explained, not silent.
    pub birth_skipped_reason: Option<String>,
    /// #1890: number of chart-gluing proposals emitted (pairs whose seam
    /// equivalence e-value was finite and carried to the engine's Glue gate).
    pub glues_proposed: usize,
    /// #1890: number of disjoint-support d=1 periodic pairs the glue lane
    /// GEOMETRICALLY screened (passed the ambient-frame / disjoint-support
    /// pre-screen) before ranking under budget — the loud denominator for the
    /// glues actually proposed.
    pub glue_candidates_screened: usize,
    /// Exact seam transitions paired with the emitted glue proposals.  Private
    /// because these are adoption capabilities, not an additional public
    /// proposal/evidence surface.
    certified_glues: Vec<CertifiedGlue>,
}

// ===========================================================================
// #1890 — chart-gluing lane: the geometric over-tiling detector.
//
// Atoms over-tiling ONE manifold are CHARTS, and the merge test for charts is
// GLUING, not co-activation fusion. The co-activation lane (above) fires on
// DEPENDENT codes; over-tiling atoms have DISJOINT supports (each owns its own
// arc) and hence anti-correlated codes, so the fusion lane is structurally
// blind to them. This lane screens pairs GEOMETRICALLY — a d=1 periodic pair
// whose decoder AMBIENT spans align (small principal angles) and whose supports
// are no more co-active than chance — and its acceptance is an EQUIVALENCE
// e-value on the decoded seam (the two charts coincide within an isometry
// tolerance) against the churn-null scatter, NOT a fit-improvement gate (a
// clean glue leaves EV tied, so a likelihood-ratio gate could never accept it).
// ===========================================================================

/// Fallback intrinsic period of a d=1 periodic atom's latent coordinate when the
/// atom's `Circle { period }` manifold does not report one. The periodic
/// harmonic evaluator sweeps a full circle over `t ∈ [0, 1)` (`angle = 2π·h·t`),
/// so the coordinate period is `1.0` unless the manifold overrides it — which is
/// the value read per-atom by [`atom_axis_period`].
const GLUE_DEFAULT_PERIOD: f64 = 1.0;

/// Finite clamp on the seam log-e-value so a perfect (zero-residual) synthetic
/// glue banks a large-but-finite certificate rather than `+∞` — the engine
/// rejects non-finite triggers, and a banked e-value only needs to clear the
/// ledger threshold `ln(1/α) ≈ 3`, not diverge. Kept well under `ln(f64::MAX)`
/// so the ledger never overflows when it exponentiates the banked log-e.
const GLUE_LOG_E_CLAMP: f64 = 50.0;

/// A fitted seam transition between two d=1 charts A, B of one manifold under
/// the unit-speed gauge: `t_A = sign · t_B + offset` (mod the `2π` period), with
/// the seam equivalence e-value that certifies the two decoded charts coincide
/// within an isometry tolerance against the churn null.
///
/// `sign = +1` is a plain over-tile (two arcs of one ORIENTED circle → a single
/// periodic atom covers the union: the fuse outcome, Increment 1). `sign = -1`
/// is ORIENTATION-REVERSING — the sphere-pole / Möbius signature no single
/// orientable chart can represent, which must instead be REGISTERED as a
/// partition-of-unity atlas atom (Increment 2, in the atom/construction types
/// this lane does not own). The sign is the detector either way.
#[derive(Clone, Copy, Debug)]
pub struct ChartTransition {
    /// `+1` orientation-preserving (fuse), `-1` orientation-reversing (register).
    pub sign: i8,
    /// Latent offset `c` in `t_A = sign·t_B + c`, wrapped into `[0, 2π)`.
    pub offset: f64,
    /// Seam equivalence e-value (log scale) against the churn-null scatter. Large
    /// positive ⇒ the two charts coincide within the reconstruction band beyond
    /// what independent curves would; carried on the proposal trigger and banked
    /// by the engine's Glue gate.
    pub log_e_value: f64,
}

/// The geometric half of a chart glue (no e-value): the fitted sign + offset and
/// the decoded seam clouds/curves, shared by the acceptance e-value
/// ([`unit_speed_glue_certificate`]) and the warm-start coordinate transplant
/// ([`transplant_glued_coords`]).
struct SeamTransition {
    sign: f64,
    offset: f64,
    /// Intrinsic period of atom A's latent coordinate (the transplant wraps into
    /// `[0, period_a)`).
    period: f64,
    rows_a: Vec<usize>,
    points_a: Array2<f64>,
    rows_b: Vec<usize>,
    points_b: Array2<f64>,
    /// A decoded exactly at the transition-mapped coordinates of B.
    mapped_b_to_a: Array2<f64>,
    /// B decoded exactly at the inverse-transition coordinates of A.
    mapped_a_to_b: Array2<f64>,
}

/// Intrinsic period of a d=1 atom's latent coordinate, read from its
/// `Circle { period }` manifold (falling back to [`GLUE_DEFAULT_PERIOD`] when the
/// manifold reports no axis period).
fn atom_axis_period(term: &SaeManifoldTerm, atom: usize) -> f64 {
    let coords = &term.assignment.coords;
    if atom < coords.len() {
        if let Some(Some(p)) = coords[atom].effective_axis_periods().first().copied() {
            if p.is_finite() && p > 0.0 {
                return p;
            }
        }
    }
    GLUE_DEFAULT_PERIOD
}

/// Active rows of `atom` — the discrete support the disjoint-support signature
/// is read from, thresholded at the same relative floor as
/// [`sparse_codes_from_term`].
fn atom_active_rows(term: &SaeManifoldTerm, atom: usize) -> Vec<usize> {
    let assignments = term.assignment.assignments();
    let k = assignments.ncols();
    let floor = if k == 0 {
        0.0
    } else {
        ACTIVE_SUPPORT_REL_FLOOR / k as f64
    };
    (0..assignments.nrows())
        .filter(|&r| assignments[[r, atom]] > floor)
        .collect()
}

/// Decode an atom's ambient points `x_i = Φ_k(t_i) · B_k` at the given rows,
/// reading the atom's already-evaluated basis values (no re-evaluate). Returns
/// `(rows.len() × p)`.
fn decoded_points_at(atom: &SaeManifoldAtom, rows: &[usize]) -> Array2<f64> {
    let phi_sub = atom.basis_values.select(Axis(0), rows);
    phi_sub.dot(atom.decoder_coefficients())
}

/// Decode a standard periodic-harmonic coefficient block at explicit
/// coordinates.  This evaluates the analytic family at the requested points;
/// no sampled curve or nearest-grid approximation is involved.
fn periodic_decoded_points(
    decoder: ArrayView2<'_, f64>,
    coordinates: &[f64],
) -> Option<Array2<f64>> {
    let m = decoder.nrows();
    if m == 0 || m % 2 == 0 {
        return None;
    }
    let p = decoder.ncols();
    let harmonics = (m - 1) / 2;
    let mut points = Array2::<f64>::zeros((coordinates.len(), p));
    for (row, &coordinate) in coordinates.iter().enumerate() {
        for output in 0..p {
            let mut value = decoder[[0, output]];
            for harmonic in 1..=harmonics {
                let angle = std::f64::consts::TAU * harmonic as f64 * coordinate;
                value += angle.sin() * decoder[[2 * harmonic - 1, output]]
                    + angle.cos() * decoder[[2 * harmonic, output]];
            }
            points[[row, output]] = value;
        }
    }
    Some(points)
}

/// Decoder of A expressed in B's coordinate under
/// `t_A = sign*t_B + offset`.  Harmonic addition identities make this action
/// exact for every represented harmonic.
fn periodic_decoder_under_transition(
    decoder_a: ArrayView2<'_, f64>,
    sign: i8,
    offset: f64,
) -> Option<Array2<f64>> {
    if !matches!(sign, -1 | 1) || decoder_a.nrows() == 0 || decoder_a.nrows() % 2 == 0 {
        return None;
    }
    let mut mapped = decoder_a.to_owned();
    let harmonics = (decoder_a.nrows() - 1) / 2;
    for harmonic in 1..=harmonics {
        let angle = std::f64::consts::TAU * harmonic as f64 * offset;
        let (cosine, sine) = (angle.cos(), angle.sin());
        for output in 0..decoder_a.ncols() {
            let a_sin = decoder_a[[2 * harmonic - 1, output]];
            let a_cos = decoder_a[[2 * harmonic, output]];
            if sign == 1 {
                mapped[[2 * harmonic - 1, output]] = cosine * a_sin - sine * a_cos;
                mapped[[2 * harmonic, output]] = sine * a_sin + cosine * a_cos;
            } else {
                mapped[[2 * harmonic - 1, output]] = -cosine * a_sin + sine * a_cos;
                mapped[[2 * harmonic, output]] = sine * a_sin + cosine * a_cos;
            }
        }
    }
    Some(mapped)
}

/// Closed-form registration of two periodic harmonic decoders.  The first
/// non-zero harmonic identifies a finite set of phase roots analytically for
/// each of the only two unit-speed slopes (`+1`, `-1`); the complete coefficient
/// block then chooses the root/sign by exact represented-function residual.
/// There is no coordinate scan, optimizer, finite difference, or sampled
/// nearest-point proxy.
fn fit_periodic_transition_from_decoders(
    decoder_a: ArrayView2<'_, f64>,
    decoder_b: ArrayView2<'_, f64>,
) -> Option<(i8, f64)> {
    if decoder_a.dim() != decoder_b.dim() || decoder_a.nrows() < 3 || decoder_a.nrows() % 2 == 0 {
        return None;
    }
    let dot = |left_row: usize, right_row: usize| -> f64 {
        (0..decoder_a.ncols())
            .map(|output| decoder_b[[left_row, output]] * decoder_a[[right_row, output]])
            .sum()
    };
    let harmonics = (decoder_a.nrows() - 1) / 2;
    let mut candidates = Vec::new();
    for sign in [1_i8, -1_i8] {
        for harmonic in 1..=harmonics {
            let sin_row = 2 * harmonic - 1;
            let cos_row = 2 * harmonic;
            let (cos_score, sin_score) = if sign == 1 {
                (
                    dot(sin_row, sin_row) + dot(cos_row, cos_row),
                    -dot(sin_row, cos_row) + dot(cos_row, sin_row),
                )
            } else {
                (
                    -dot(sin_row, sin_row) + dot(cos_row, cos_row),
                    dot(sin_row, cos_row) + dot(cos_row, sin_row),
                )
            };
            if cos_score.hypot(sin_score) > 0.0 {
                let harmonic_phase = sin_score.atan2(cos_score).rem_euclid(std::f64::consts::TAU);
                // `h*delta = harmonic_phase (mod 2π)` has exactly h roots.
                for branch in 0..harmonic {
                    let phase =
                        (harmonic_phase + std::f64::consts::TAU * branch as f64) / harmonic as f64;
                    candidates.push((sign, phase));
                }
                break;
            }
        }
    }

    candidates
        .into_iter()
        .filter_map(|(sign, angle)| {
            let offset = angle / std::f64::consts::TAU;
            let mapped = periodic_decoder_under_transition(decoder_a, sign, offset)?;
            let residual = mapped
                .iter()
                .zip(decoder_b.iter())
                .map(|(predicted, observed)| (predicted - observed).powi(2))
                .sum::<f64>();
            residual.is_finite().then_some((sign, offset, residual))
        })
        .min_by(|left, right| {
            left.2.total_cmp(&right.2).then_with(|| {
                // Exact ties are gauge-ambiguous; canonicalize to +1 so an
                // isotropic circle is not spuriously called a half-twist.
                right.0.cmp(&left.0)
            })
        })
        .map(|(sign, offset, _)| (sign, offset))
}

/// Fit the geometric seam transition (sign + offset) between two d=1 charts and
/// carry the decoded clouds/curves the e-value and transplant reuse. `None`
/// unless both atoms are d=1 standard periodic-harmonic charts with the same
/// period/width, a shared ambient dim, and non-empty active supports.
fn fit_seam_transition(term: &SaeManifoldTerm, a: usize, b: usize) -> Option<SeamTransition> {
    let k = term.k_atoms();
    if a >= k || b >= k || a == b {
        return None;
    }
    let atom_a = &term.atoms[a];
    let atom_b = &term.atoms[b];
    if atom_a.latent_dim() != 1
        || atom_b.latent_dim() != 1
        || !matches!(atom_a.basis_kind(), SaeAtomBasisKind::Periodic)
        || !matches!(atom_b.basis_kind(), SaeAtomBasisKind::Periodic)
    {
        return None;
    }
    let p = atom_a.decoder_coefficients().ncols();
    if p == 0 || p != atom_b.decoder_coefficients().ncols() {
        return None;
    }
    let rows_a = atom_active_rows(term, a);
    let rows_b = atom_active_rows(term, b);
    if rows_a.is_empty() || rows_b.is_empty() {
        return None;
    }
    let coords = &term.assignment.coords;
    if b >= coords.len() || coords[b].latent_dim() < 1 {
        return None;
    }
    let period_a = atom_axis_period(term, a);
    let period_b = atom_axis_period(term, b);
    // `PeriodicHarmonicEvaluator` is exactly one-periodic in its stored raw
    // coordinate.  A different retraction period does not describe this basis,
    // so refuse rather than silently rescale or approximate it.
    if period_a.to_bits() != 1.0_f64.to_bits() || period_b.to_bits() != period_a.to_bits() {
        return None;
    }
    let decoder_a = atom_a.full_width_decoder();
    let decoder_b = atom_b.full_width_decoder();
    let (sign, offset) = fit_periodic_transition_from_decoders(decoder_a.view(), decoder_b.view())?;
    let points_a = decoded_points_at(atom_a, &rows_a);
    let points_b = decoded_points_at(atom_b, &rows_b);
    let mapped_b_coords: Vec<f64> = rows_b
        .iter()
        .map(|&row| (sign as f64 * coords[b].row(row)[0] + offset).rem_euclid(period_a))
        .collect();
    let mapped_a_coords: Vec<f64> = rows_a
        .iter()
        .map(|&row| {
            // Inverse of `t_a = sign*t_b + offset`; sign^{-1} = sign.
            (sign as f64 * (coords[a].row(row)[0] - offset)).rem_euclid(period_a)
        })
        .collect();
    let mapped_b_to_a = periodic_decoded_points(decoder_a.view(), &mapped_b_coords)?;
    let mapped_a_to_b = periodic_decoded_points(decoder_b.view(), &mapped_a_coords)?;
    Some(SeamTransition {
        sign: sign as f64,
        offset,
        period: period_a,
        rows_a,
        points_a,
        rows_b,
        points_b,
        mapped_b_to_a,
        mapped_a_to_b,
    })
}

/// The seam equivalence e-value (#1890): does chart A's arc lie on chart B's
/// curve AND vice versa, within the reconstruction band, beyond the pooled
/// independent-scatter (churn-scatter reference null) scale?
///
/// Per point the statistic is the Gaussian likelihood ratio
/// `N(x; other_chart(transition(t)), σ_band²) /
/// N(x; pooled centroid, σ_pool²)`, both directions, summed to a log-e-value.
/// The numerator is evaluated at the exact analytic affine transition, not at a
/// nearest point on a sampled grid:
///
/// * `σ_band²` — per-coordinate reconstruction noise floor (the isometry
///   TOLERANCE), the mean squared dictionary residual over the pair's rows,
///   floored only at the representation's machine-resolution scale.
/// * `σ_pool²` — per-coordinate scatter of the pooled decoded points about their
///   centroid (the reference-null currency: how far apart INDEPENDENT curves'
///   points sit). A pair whose pooled scatter is no larger than the band cannot
///   be discriminated and yields no e-value.
///
/// # Honesty: this is a genuine e-value, by SAMPLE SPLITTING
///
/// `σ_band²`, `σ_pool²` and the centroid `μ` are estimated on the EVEN-indexed
/// boundary points and the per-point ratios are evaluated ONLY on the
/// ODD-indexed points. The reference-null density and the numerator variance are
/// therefore independent of the points they are scored on, so under the null
/// (odd points drawn from the isotropic churn-scatter reference `N(μ, σ_pool²I)`)
/// `E_null[∏ q/p] = ∏ E_null[q/p] = 1` — a bona-fide e-value, not the plug-in LR
/// a same-data estimate would give (whose optimism has no `E[e]≤1` guarantee).
/// The e-value is stated against the isotropic-Gaussian reference null at the
/// pooled scale; that is the "independent decode scatter" the churn currency
/// stands in for.
///
/// Two arcs of ONE circle decode ONTO each other's curve (`e_glue ≈ σ_band ≪
/// σ_pool`) so the ratio is large positive; two DISTINCT circles decode FAR from
/// each other's curve (`e_glue ~ σ_pool ≫ σ_band`) so the `−e_glue/(2σ_band²)`
/// term drives the ratio large negative — the tied-EV-cannot-win property the
/// issue requires.
fn unit_speed_glue_certificate(
    term: &SaeManifoldTerm,
    residuals: ArrayView2<'_, f64>,
    a: usize,
    b: usize,
) -> Option<(ChartTransition, CertifiedGlue)> {
    let seam = fit_seam_transition(term, a, b)?;
    let log_e = seam_equivalence_log_e(
        residuals,
        &seam.rows_a,
        &seam.points_a,
        &seam.mapped_a_to_b,
        &seam.rows_b,
        &seam.points_b,
        &seam.mapped_b_to_a,
    )?;
    let chart_transition = ChartTransition {
        sign: seam.sign as i8,
        offset: seam.offset,
        log_e_value: log_e,
    };
    let outcome = if chart_transition.sign == 1 {
        ChartGlueOutcome::Fuse
    } else {
        ChartGlueOutcome::RegisterAtlas
    };
    let transition = UnitSpeedChartTransition::new(
        b,
        a,
        chart_transition.sign,
        chart_transition.offset,
        seam.period,
        AtlasSeamKind::Regular,
    )
    .ok()?;
    Some((
        chart_transition,
        CertifiedGlue {
            a,
            b,
            outcome,
            transition: CertifiedGlueTransition::UnitSpeed {
                transition,
                rows_b: seam.rows_b,
            },
        },
    ))
}

/// The sample-split equivalence log-e-value shared by the 1-D
/// ([`unit_speed_glue_certificate`]) and sphere ([`sphere_glue_pair_evalue`]) seam
/// certifiers.  `points_*` are the decoded ambient clouds of each chart's active
/// rows; `mapped_*` are the SAME points carried through the fitted transition to
/// the other chart's coordinate; `rows_*` index `residuals` for the
/// reconstruction band.
///
/// The reference-null centroid/scatter and the reconstruction band are estimated
/// on the EVEN-indexed points and the per-point Gaussian likelihood ratio
/// `N(x; other_chart(transition(t)), σ_band²) / N(x; pooled centroid, σ_pool²)`
/// is scored ONLY on the ODD-indexed points, so under the isotropic churn-scatter
/// reference null `E_null[∏ q/p] = 1` — a bona-fide e-value, not a plug-in LR.
fn seam_equivalence_log_e(
    residuals: ArrayView2<'_, f64>,
    rows_a: &[usize],
    points_a: &Array2<f64>,
    mapped_a_to_b: &Array2<f64>,
    rows_b: &[usize],
    points_b: &Array2<f64>,
    mapped_b_to_a: &Array2<f64>,
) -> Option<f64> {
    let p = points_a.ncols();
    if p == 0 || residuals.ncols() != p || points_b.ncols() != p {
        return None;
    }
    let na = points_a.nrows();
    let nb = points_b.nrows();
    if na != rows_a.len() || nb != rows_b.len() {
        return None;
    }
    if mapped_a_to_b.dim() != (na, p) || mapped_b_to_a.dim() != (nb, p) {
        return None;
    }
    // Sample split needs at least one estimation and one evaluation point on
    // each side (even/odd parity), so ≥ 2 active points per atom.
    if na < 2 || nb < 2 {
        return None;
    }
    let a_est: Vec<usize> = (0..na).filter(|i| i % 2 == 0).collect();
    let a_eval: Vec<usize> = (0..na).filter(|i| i % 2 == 1).collect();
    let b_est: Vec<usize> = (0..nb).filter(|i| i % 2 == 0).collect();
    let b_eval: Vec<usize> = (0..nb).filter(|i| i % 2 == 1).collect();
    let n_est = a_est.len() + b_est.len();
    let n_eval = a_eval.len() + b_eval.len();
    if n_est == 0 || n_eval == 0 {
        return None;
    }

    // --- Reference-null centroid + scatter, ESTIMATED on the even points ------
    let mut mu = vec![0.0_f64; p];
    for &i in &a_est {
        for c in 0..p {
            mu[c] += points_a[[i, c]];
        }
    }
    for &i in &b_est {
        for c in 0..p {
            mu[c] += points_b[[i, c]];
        }
    }
    for c in 0..p {
        mu[c] /= n_est as f64;
    }
    let point_null_sq =
        |pt: ArrayView1<'_, f64>| -> f64 { (0..p).map(|c| (pt[c] - mu[c]).powi(2)).sum::<f64>() };
    let mut pool_acc = 0.0_f64;
    for &i in &a_est {
        pool_acc += point_null_sq(points_a.row(i));
    }
    for &i in &b_est {
        pool_acc += point_null_sq(points_b.row(i));
    }
    let pool_sq = pool_acc / (n_est as f64 * p as f64);
    if !(pool_sq.is_finite() && pool_sq > 0.0) {
        return None;
    }

    // --- Reconstruction band (tolerance), ESTIMATED on the even rows ----------
    let mut band_acc = 0.0_f64;
    let mut band_rows = 0usize;
    for &i in &a_est {
        let r = rows_a[i];
        for c in 0..p {
            band_acc += residuals[[r, c]].powi(2);
        }
        band_rows += 1;
    }
    for &i in &b_est {
        let r = rows_b[i];
        for c in 0..p {
            band_acc += residuals[[r, c]].powi(2);
        }
        band_rows += 1;
    }
    let band_raw = if band_rows == 0 {
        0.0
    } else {
        band_acc / (band_rows as f64 * p as f64)
    };
    // The transition is analytic, so there is no discretization tolerance.  A
    // perfect synthetic band still needs a positive normal density; the only
    // floor is f64's own relative resolution at the observed pooled scale.
    let band_sq = band_raw.max(pool_sq * f64::EPSILON);
    // A pooled scatter no larger than the band cannot separate "same manifold"
    // from "independent blob" — no e-value.
    if !(pool_sq > band_sq) {
        return None;
    }

    // --- Per-point likelihood ratio, EVALUATED on the odd points --------------
    let norm_term = (p as f64 / 2.0) * (pool_sq / band_sq).ln();
    let mut log_e = 0.0_f64;
    for &i in &b_eval {
        let e_glue: f64 = (0..p)
            .map(|c| (points_b[[i, c]] - mapped_b_to_a[[i, c]]).powi(2))
            .sum();
        let e_null = point_null_sq(points_b.row(i));
        log_e += norm_term - e_glue / (2.0 * band_sq) + e_null / (2.0 * pool_sq);
    }
    for &i in &a_eval {
        let e_glue: f64 = (0..p)
            .map(|c| (points_a[[i, c]] - mapped_a_to_b[[i, c]]).powi(2))
            .sum();
        let e_null = point_null_sq(points_a.row(i));
        log_e += norm_term - e_glue / (2.0 * band_sq) + e_null / (2.0 * pool_sq);
    }
    if !log_e.is_finite() {
        log_e = if log_e < 0.0 {
            -GLUE_LOG_E_CLAMP
        } else {
            GLUE_LOG_E_CLAMP
        };
    }
    Some(log_e.clamp(-GLUE_LOG_E_CLAMP, GLUE_LOG_E_CLAMP))
}

// ===========================================================================
// #1890 Increment 2 — SPHERE POLE seams (the d=2 register emitter).
//
// A sphere pole seam is TWO sphere atoms covering ONE ambient sphere with their
// FRAME AXES in each other's active interior. Each atom carries the ambient
// parameterisation (`SaeAtomBasisKind::Sphere` at `latent_dim = 3`, real
// spherical harmonics in `u = [x, y, z]`, the only sphere resolution
// `SaeAtomGeometryPlan::new` accepts), so neither atom is charted and neither
// has a coordinate singularity of its own. What still makes the cover
// irreducibly an atlas is the DATA: each atom's decoder is fitted on its own
// support only, so atom A's frame axis is a point A never saw and B did, and
// vice versa. Registering the pair is what lets one semantic atom carry the
// whole sphere; a single atom's local fit does not.
//
// The transition relating two such atoms is an ambient rotation `R ∈ SO(3)` on
// the unit vector `u`, NOT a 1-D affine map; the 1-D seam fit
// ([`fit_seam_transition`]) is structurally blind to it (it short-circuits on
// non-`Periodic`, `latent_dim ≠ 1` atoms). This lane fits that rotation by exact
// orthogonal Procrustes on the two decoders' degree-1 (dipole) ambient frames,
// classifies pole-vs-regular by whether each atom's frame axis falls strictly
// inside the OTHER atom's active latitude span (data-driven, no magic angle),
// and certifies the overlap with the SAME sample-split equivalence e-value the
// 1-D lane uses. A pole seam always REGISTERS (keeps both atoms as one
// partition-of-unity atlas atom); a sphere is orientable, so its proper-rotation
// transition carries `sign = +1` in the cocycle.
//
// #2698 — this lane was written against the `(lat, lon)` `SphereChartEvaluator`
// that was DELETED when every chart consumer moved to ambient coordinates, and
// was not ported with it: it screened on `latent_dim == 2`, took the frame from
// rows `1..4` of a 7-row decoder, read each row's unit vector out of
// `basis_values`, and decoded through a hardcoded `[1, x, y, z, xy, yz, xz]`
// monomial list. Every one of those is false of a producible sphere atom, so the
// lane could not fire on any pair the engine can build. All four now read the
// ambient form, from the atom's OWN declaration: the geometry plan's degree
// where an atom carries one, else the realized basis width `(degree + 1)²`, with
// the evaluator rebuilt exactly as `SaeAtomGeometryPlan::build_evaluator` builds
// it. Nothing about the seam's geometry changed — the transition between two
// ambient frames is still the `SO(3)` element `SphereChartTransition` stores.
// ===========================================================================

/// The ambient real-spherical-harmonic degree this atom's basis carries, or
/// `None` when the atom is not an ambient sphere atom at all.
///
/// The atom's own declaration is the authority, in the same order the rest of
/// the engine trusts it: the persisted [`SaeAtomGeometryPlan`] when the atom was
/// built by the native lifecycle, otherwise the realized full basis width, which
/// pins the degree exactly because an ambient sphere basis is `(degree + 1)²`
/// wide. Either way the degree is accepted only if it reproduces that width, so
/// a non-sphere basis width that happens to be a perfect square cannot slip
/// through as a sphere.
fn ambient_sphere_degree(atom: &SaeManifoldAtom) -> Option<usize> {
    if atom.latent_dim() != 3 || !matches!(atom.basis_kind(), SaeAtomBasisKind::Sphere) {
        return None;
    }
    let width = atom.full_basis_size();
    let degree = match atom.geometry_plan().map(SaeAtomGeometryPlan::resolution) {
        Some(SaeBasisResolution::AmbientSphereHarmonics { degree }) => *degree,
        // A plan that declares any OTHER resolution is not this geometry, and a
        // plan is never overridden by an inference from the width.
        Some(_) => return None,
        None => {
            let side = (width as f64).sqrt().round() as usize;
            side.checked_sub(1)?
        }
    };
    let evaluator = AmbientSphereHarmonicEvaluator::new(degree).ok()?;
    (evaluator.basis_size() == width).then_some(degree)
}

/// The one analytic evaluator an ambient sphere atom of this degree declares —
/// the same construction [`SaeAtomGeometryPlan::build_evaluator`] performs, so
/// this is the atom's own basis and not a re-derivation of it.
fn ambient_sphere_evaluator(atom: &SaeManifoldAtom) -> Option<AmbientSphereHarmonicEvaluator> {
    AmbientSphereHarmonicEvaluator::new(ambient_sphere_degree(atom)?).ok()
}

/// The `3 × p` ambient frame of a sphere atom: the linear map its decoder
/// applies to the unit-vector coordinate, read off the degree-1 (dipole)
/// harmonic block.
///
/// Column `(1, m)` of the basis is `N_{1,m}` times one ambient coordinate, so
/// the block's coefficients are read out of the evaluator itself by evaluating
/// `Φ` at the three standard directions: `Φ(e_axis)` is `N` on the degree-1
/// column carrying `axis` and zero on every other degree-1 column. No column
/// order and no normalisation constant is assumed here, and the extraction is
/// unchanged at any degree.
fn sphere_linear_block(atom: &SaeManifoldAtom) -> Option<Array2<f64>> {
    let evaluator = ambient_sphere_evaluator(atom)?;
    let decoder = atom.full_width_decoder();
    if decoder.nrows() != evaluator.basis_size() {
        return None;
    }
    let axes = Array2::<f64>::eye(3);
    let (phi_axes, _) = evaluator.evaluate(axes.view()).ok()?;
    let modes = evaluator.spectral_modes();
    let mut frame = Array2::<f64>::zeros((3, decoder.ncols()));
    for axis in 0..3 {
        for (column, mode) in modes.iter().enumerate() {
            if mode.degree != 1 {
                continue;
            }
            let coefficient = phi_axes[[axis, column]];
            if coefficient == 0.0 {
                continue;
            }
            for output in 0..decoder.ncols() {
                frame[[axis, output]] += coefficient * decoder[[column, output]];
            }
        }
    }
    if frame.iter().any(|value| !value.is_finite()) {
        return None;
    }
    Some(frame)
}

/// Nearest orthogonal matrix to `m` (the orthogonal polar factor `U Vᵀ` of its
/// SVD). This is the orthogonal Procrustes solution
/// `argmin_{QᵀQ=I} ‖Q − M‖_F`. The factor is accepted only when the SVD proves
/// full numerical rank at its machine backward-error scale.
fn nearest_orthogonal_3x3(m: [[f64; 3]; 3]) -> Option<[[f64; 3]; 3]> {
    let matrix = Array2::from_shape_fn((3, 3), |(row, column)| m[row][column]);
    if matrix.iter().any(|value| !value.is_finite()) {
        return None;
    }
    let (left, singular_values, right_t) = matrix.svd(true, true).ok()?;
    let spectral_scale = singular_values.iter().copied().fold(0.0_f64, f64::max);
    let numerical_rank_threshold =
        f64::EPSILON * matrix.nrows().max(matrix.ncols()) as f64 * spectral_scale;
    // `U V^T` exists even for a rank-deficient product, but its action on the
    // nullspace is arbitrary and can change the reported orientation sign.
    // An exact seam therefore requires a unique polar factor at machine
    // precision; statistically uncertain alignments belong to the noisy
    // holonomy certificate instead.
    if spectral_scale == 0.0
        || singular_values
            .iter()
            .any(|&value| value <= numerical_rank_threshold)
    {
        return None;
    }
    let orthogonal = left?.dot(&right_t?);
    if orthogonal.iter().any(|value| !value.is_finite()) {
        return None;
    }
    let mut result = [[0.0; 3]; 3];
    for row in 0..3 {
        for column in 0..3 {
            result[row][column] = orthogonal[[row, column]];
        }
    }
    Some(result)
}

/// Decode a sphere atom at explicit ambient unit vectors `u = [x, y, z]` (on
/// `S²`), through the atom's OWN analytic basis — no sampled-grid nearest-point
/// proxy and no hardcoded basis layout.
fn sphere_decoded_points_at_units(
    atom: &SaeManifoldAtom,
    units: &[[f64; 3]],
) -> Option<Array2<f64>> {
    let evaluator = ambient_sphere_evaluator(atom)?;
    let decoder = atom.full_width_decoder();
    if decoder.nrows() != evaluator.basis_size() {
        return None;
    }
    let coords = Array2::<f64>::from_shape_fn((units.len(), 3), |(row, axis)| units[row][axis]);
    let (phi, _) = evaluator.evaluate(coords.view()).ok()?;
    let points = phi.dot(&decoder);
    points
        .iter()
        .all(|value| value.is_finite())
        .then_some(points)
}

/// A fitted sphere pole-seam transition: the ambient rotation `R` (`b -> a`, so
/// `u_a = R u_b`), its pole-vs-regular classification, and the decoded /
/// rotation-mapped point clouds the shared equivalence e-value scores.
struct SphereSeamTransition {
    rotation: [[f64; 3]; 3],
    seam_kind: AtlasSeamKind,
    rows_a: Vec<usize>,
    points_a: Array2<f64>,
    rows_b: Vec<usize>,
    points_b: Array2<f64>,
    /// A decoded at the rotation-mapped unit vectors of B's active rows.
    mapped_b_to_a: Array2<f64>,
    /// B decoded at the inverse-rotation-mapped unit vectors of A's active rows.
    mapped_a_to_b: Array2<f64>,
}

/// The ambient unit vector `[x, y, z]` of a sphere atom's row: its assignment
/// COORDINATE, which under the ambient parameterisation is the point itself.
///
/// The coordinate is carried back onto `S²` by the latent manifold's own
/// `project_point` — the single authority the retraction and the seed path use —
/// rather than by a local renormalisation, so a line-search trial that left the
/// sphere by roundoff reads the same unit vector every other consumer reads.
/// `None` if the block is missing, too narrow, or not on the sphere at all.
fn sphere_row_unit(term: &SaeManifoldTerm, atom: usize, row: usize) -> Option<[f64; 3]> {
    let block = term.assignment.coords.get(atom)?;
    if block.latent_dim() != 3 || row >= block.n_obs() {
        return None;
    }
    let raw = ArrayView1::from(block.row(row));
    let projected = block.manifold().project_point(raw);
    if projected.len() != 3 || projected.iter().any(|value| !value.is_finite()) {
        return None;
    }
    Some([projected[0], projected[1], projected[2]])
}

/// Apply a `3×3` rotation to a unit vector.
fn rotate_unit(r: &[[f64; 3]; 3], u: [f64; 3]) -> [f64; 3] {
    [
        r[0][0] * u[0] + r[0][1] * u[1] + r[0][2] * u[2],
        r[1][0] * u[0] + r[1][1] * u[1] + r[1][2] * u[2],
        r[2][0] * u[0] + r[2][1] * u[1] + r[2][2] * u[2],
    ]
}

/// Whether two atoms are both ambient sphere atoms — the basis DECLARES the
/// ambient harmonic geometry (`Sphere` at `latent_dim = 3`, a width that is a
/// realizable harmonic degree) and the routing coordinate DECLARES the same
/// sphere it is a point on. Both halves are required: an atom whose basis says
/// sphere while its coordinate block says something else is exactly the
/// mis-declaration #2698 was filed for, and it is refused here rather than
/// silently read as a unit vector.
fn is_sphere_pair(term: &SaeManifoldTerm, a: usize, b: usize) -> bool {
    let k = term.k_atoms();
    if a >= k || b >= k || a == b {
        return false;
    }
    let ambient_sphere = |atom: usize| -> bool {
        ambient_sphere_degree(&term.atoms[atom]).is_some()
            && matches!(
                term.assignment.coords.get(atom).map(|block| block.manifold()),
                Some(LatentManifold::Sphere { dim: 3 })
            )
    };
    ambient_sphere(a) && ambient_sphere(b)
}

/// Fit the ambient-rotation seam transition between two sphere atoms and
/// classify pole-vs-regular. `None` unless both atoms are ambient sphere atoms
/// (`latent_dim = 3`) with a shared ambient dim, non-empty active supports, and
/// an invertible frame product.
fn fit_sphere_seam_transition(
    term: &SaeManifoldTerm,
    a: usize,
    b: usize,
) -> Option<SphereSeamTransition> {
    if !is_sphere_pair(term, a, b) {
        return None;
    }
    let atom_a = &term.atoms[a];
    let atom_b = &term.atoms[b];
    let p = atom_a.decoder_coefficients().ncols();
    if p == 0 || p != atom_b.decoder_coefficients().ncols() {
        return None;
    }
    let l_a = sphere_linear_block(atom_a)?;
    let l_b = sphere_linear_block(atom_b)?;
    // Orthogonal Procrustes: R = argmin ‖L_a − R L_b‖ ⇒ R is the orthogonal
    // factor of M = L_a L_bᵀ (3×3). Then u_a = R u_b matches the ambient images.
    let m_prod = l_a.dot(&l_b.t());
    let mut m = [[0.0_f64; 3]; 3];
    for i in 0..3 {
        for j in 0..3 {
            m[i][j] = m_prod[[i, j]];
        }
    }
    let rotation = nearest_orthogonal_3x3(m)?;
    let rows_a = atom_active_rows(term, a);
    let rows_b = atom_active_rows(term, b);
    if rows_a.is_empty() || rows_b.is_empty() {
        return None;
    }
    // Active latitudes (asin z) of each atom, and each atom's own frame axis
    // mapped into the OTHER atom's coordinate: pole-vs-regular is decided by
    // whether the mapped axis falls strictly inside the other atom's active
    // latitude span. Under the ambient parameterisation the axis is not a
    // coordinate singularity — it is the point the atom's frame is built around
    // and, when it is interior to the other's support and vice versa, the point
    // neither atom's own fit covers alone.
    let units_a: Vec<[f64; 3]> = rows_a
        .iter()
        .map(|&r| sphere_row_unit(term, a, r))
        .collect::<Option<Vec<_>>>()?;
    let units_b: Vec<[f64; 3]> = rows_b
        .iter()
        .map(|&r| sphere_row_unit(term, b, r))
        .collect::<Option<Vec<_>>>()?;
    let lat_of = |u: [f64; 3]| -> f64 { u[2].clamp(-1.0, 1.0).asin() };
    let (mut a_lat_lo, mut a_lat_hi) = (f64::INFINITY, f64::NEG_INFINITY);
    for &u in &units_a {
        let lat = lat_of(u);
        a_lat_lo = a_lat_lo.min(lat);
        a_lat_hi = a_lat_hi.max(lat);
    }
    let (mut b_lat_lo, mut b_lat_hi) = (f64::INFINITY, f64::NEG_INFINITY);
    for &u in &units_b {
        let lat = lat_of(u);
        b_lat_lo = b_lat_lo.min(lat);
        b_lat_hi = b_lat_hi.max(lat);
    }
    // B's frame axis u_b = [0,0,1] into A's coordinate; A's axis into B's.
    let b_pole_in_a = lat_of(rotate_unit(&rotation, [0.0, 0.0, 1.0]));
    // The polar factor is orthogonal by construction, so its exact algebraic
    // inverse is its transpose; no determinant cutoff or second solve exists.
    let mut inv_rotation = [[0.0; 3]; 3];
    for row in 0..3 {
        for column in 0..3 {
            inv_rotation[row][column] = rotation[column][row];
        }
    }
    let a_pole_in_b = lat_of(rotate_unit(&inv_rotation, [0.0, 0.0, 1.0]));
    let b_pole_interior_to_a = b_pole_in_a > a_lat_lo && b_pole_in_a < a_lat_hi;
    let a_pole_interior_to_b = a_pole_in_b > b_lat_lo && a_pole_in_b < b_lat_hi;
    let seam_kind = if b_pole_interior_to_a && a_pole_interior_to_b {
        AtlasSeamKind::Pole
    } else {
        AtlasSeamKind::Regular
    };
    // Decoded clouds + rotation-mapped clouds for the equivalence e-value.
    let points_a = sphere_decoded_points_at_units(atom_a, &units_a)?;
    let points_b = sphere_decoded_points_at_units(atom_b, &units_b)?;
    // B's rows carried into A: rotate u_b -> u_a, decode through A.
    let mapped_b_units: Vec<[f64; 3]> =
        units_b.iter().map(|&u| rotate_unit(&rotation, u)).collect();
    let mapped_b_to_a = sphere_decoded_points_at_units(atom_a, &mapped_b_units)?;
    // A's rows carried into B: rotate u_a by R⁻¹ -> u_b, decode through B.
    let mapped_a_units: Vec<[f64; 3]> = units_a
        .iter()
        .map(|&u| rotate_unit(&inv_rotation, u))
        .collect();
    let mapped_a_to_b = sphere_decoded_points_at_units(atom_b, &mapped_a_units)?;
    Some(SphereSeamTransition {
        rotation,
        seam_kind,
        rows_a,
        points_a,
        rows_b,
        points_b,
        mapped_b_to_a,
        mapped_a_to_b,
    })
}

/// The sphere pole-seam equivalence certificate (#1890 Increment 2). Returns the
/// fitted orthogonal transition (`b -> a`) plus its equivalence log-e-value,
/// or `None` if the pair is not an identifiable pole seam.  Only genuine POLE
/// seams (each pole interior to the other chart) are certified for registration;
/// a regular sphere overlap has no register/fuse outcome wired in this lane and
/// yields `None`.
fn sphere_glue_pair_evalue(
    term: &SaeManifoldTerm,
    residuals: ArrayView2<'_, f64>,
    a: usize,
    b: usize,
) -> Option<(SphereChartTransition, f64)> {
    let seam = fit_sphere_seam_transition(term, a, b)?;
    if !matches!(seam.seam_kind, AtlasSeamKind::Pole) {
        return None;
    }
    let log_e = seam_equivalence_log_e(
        residuals,
        &seam.rows_a,
        &seam.points_a,
        &seam.mapped_a_to_b,
        &seam.rows_b,
        &seam.points_b,
        &seam.mapped_b_to_a,
    )?;
    let transition =
        SphereChartTransition::new_fitted(b, a, seam.rotation, AtlasSeamKind::Pole).ok()?;
    Some((transition, log_e))
}

/// Emit the chart-gluing proposal lane (#1890) into `proposals`, ranked under
/// `budget`. Returns `(glues_proposed, candidates_screened)`.
///
/// Pre-screen (GEOMETRIC, code-blind): a pair `(a, b)` is a candidate iff both
/// atoms are d=1 periodic with a live decoder ambient frame, their supports are
/// no more co-active than independence would give (`inter ≤ na·nb/N` — the
/// complement of the fusion lane's positive-dependence trigger, so the two lanes
/// partition the pair space with NO tuned overlap threshold), and their ambient
/// spans are comparable (principal-angle alignment, the ranking key). The
/// pre-screen only RANKS; the seam equivalence e-value owns acceptance, so no
/// magic alignment cutoff is imposed here.
fn harvest_glue_proposals(
    term: &SaeManifoldTerm,
    residuals: ArrayView2<'_, f64>,
    budget: usize,
    proposals: &mut Vec<MoveProposal>,
    certified_glues: &mut Vec<CertifiedGlue>,
) -> (usize, usize) {
    let k = term.k_atoms();
    if k < 2 || budget == 0 {
        return (0, 0);
    }
    let assignments = term.assignment.assignments();
    let n_rows = assignments.nrows();
    if n_rows == 0 {
        return (0, 0);
    }
    let floor = ACTIVE_SUPPORT_REL_FLOOR / k as f64;
    // Packed row supports (one bit per row) so the K²/2 pairwise co-fire counts
    // below are word-parallel popcounts over n/64 words instead of an O(n)
    // boolean scan per pair: O(K²·n) bool loads → O(K²·n/64) popcnt words.
    let support_words = n_rows.div_ceil(64);
    let supports: Vec<Vec<u64>> = (0..k)
        .map(|atom| {
            let mut words = vec![0u64; support_words];
            for r in 0..n_rows {
                if assignments[[r, atom]] > floor {
                    words[r / 64] |= 1u64 << (r % 64);
                }
            }
            words
        })
        .collect();
    let support_sizes: Vec<usize> = supports
        .iter()
        .map(|words| words.iter().map(|w| w.count_ones() as usize).sum())
        .collect();
    // Ambient decoder frames — only for d=1 periodic atoms (the over-tiling
    // signature); every other atom is a `None` and never a glue endpoint.
    let frames: Vec<Option<GrassmannFrame>> = (0..k)
        .map(|atom| {
            let at = &term.atoms[atom];
            if at.latent_dim() == 1 && matches!(at.basis_kind(), SaeAtomBasisKind::Periodic) {
                GrassmannFrame::from_decoder_row_space(at.decoder_coefficients().view())
            } else {
                None
            }
        })
        .collect();

    let mut screened = 0usize;
    let mut candidates: Vec<(usize, usize, f64)> = Vec::new();
    for a in 0..k {
        let fa = match &frames[a] {
            Some(f) => f,
            None => continue,
        };
        if support_sizes[a] == 0 {
            continue;
        }
        for b in (a + 1)..k {
            // A registered pair is already one semantic atom.  Re-proposing its
            // seam would double-bank the same geometric fact and, worse, could
            // later route it through the destructive fuse outcome.
            if term.charts_share_atlas(a, b) {
                continue;
            }
            let fb = match &frames[b] {
                Some(f) => f,
                None => continue,
            };
            if support_sizes[b] == 0 {
                continue;
            }
            // Disjoint-support gate: keep only pairs that co-fire no MORE than
            // independence predicts (the anti-correlated / disjoint signature the
            // fusion lane cannot see). Positively co-active pairs are fusion's.
            let inter: usize = supports[a]
                .iter()
                .zip(supports[b].iter())
                .map(|(&wa, &wb)| (wa & wb).count_ones() as usize)
                .sum();
            let expected = support_sizes[a] as f64 * support_sizes[b] as f64 / n_rows as f64;
            if inter as f64 > expected {
                continue;
            }
            // Ambient-span alignment (ranking key): cos of the largest principal
            // angle between the two decoder frames — 1 for a shared plane.
            let alignment = match fa.max_principal_angle(fb.frame()) {
                Ok(theta) => theta.cos(),
                Err(_) => continue,
            };
            if !alignment.is_finite() {
                continue;
            }
            screened += 1;
            candidates.push((a, b, alignment));
        }
    }
    candidates.sort_by(|x, y| y.2.total_cmp(&x.2).then(x.0.cmp(&y.0)).then(x.1.cmp(&y.1)));

    let mut proposed = 0usize;
    for &(a, b, _score) in candidates.iter().take(budget) {
        if let Some((tr, certificate)) = unit_speed_glue_certificate(term, residuals, a, b) {
            proposals.push(proposal(
                term,
                StructureMove::Glue {
                    a,
                    b,
                    outcome: certificate.outcome,
                },
                tr.log_e_value,
            ));
            certified_glues.push(certificate);
            proposed += 1;
        }
    }

    // --- Sphere POLE-seam pass (#1890 Increment 2, the register emitter) ---
    // Two ambient sphere atoms whose frame axes sit in each other's active
    // interior are an irreducible atlas; the transition is an ambient rotation
    // the 1-D lane cannot see. Screened with the SAME disjoint-support gate
    // (over-tiling atoms anti-correlate), certified by the ambient-rotation pole
    // e-value, and always REGISTERED (neither atom's own fit carries the other's
    // frame axis, so there is nothing to destructively fuse into).
    let mut sphere_candidates: Vec<(usize, usize)> = Vec::new();
    for a in 0..k {
        if support_sizes[a] == 0 || !matches!(term.atoms[a].basis_kind(), SaeAtomBasisKind::Sphere)
        {
            continue;
        }
        for b in (a + 1)..k {
            if support_sizes[b] == 0
                || !matches!(term.atoms[b].basis_kind(), SaeAtomBasisKind::Sphere)
                || term.charts_share_atlas(a, b)
            {
                continue;
            }
            let inter: usize = supports[a]
                .iter()
                .zip(supports[b].iter())
                .map(|(&wa, &wb)| (wa & wb).count_ones() as usize)
                .sum();
            let expected = support_sizes[a] as f64 * support_sizes[b] as f64 / n_rows as f64;
            if inter as f64 > expected {
                continue;
            }
            sphere_candidates.push((a, b));
        }
    }
    for &(a, b) in sphere_candidates.iter().take(budget) {
        screened += 1;
        if let Some((transition, log_e)) = sphere_glue_pair_evalue(term, residuals, a, b) {
            proposals.push(proposal(
                term,
                StructureMove::Glue {
                    a,
                    b,
                    outcome: ChartGlueOutcome::RegisterAtlas,
                },
                log_e,
            ));
            certified_glues.push(CertifiedGlue {
                a,
                b,
                outcome: ChartGlueOutcome::RegisterAtlas,
                transition: CertifiedGlueTransition::Sphere(transition),
            });
            proposed += 1;
        }
    }
    (proposed, screened)
}

/// Warm the glued atom `a`'s chart to cover the union of both arcs by
/// transplanting `b`'s per-row latent coordinate through the certified seam
/// transition `t_a = sign·t_b + offset` (mod `2π`), so the joint refit starts
/// from a full-manifold chart rather than re-discovering `b`'s arc cold.
fn transplant_glued_coords(
    term: &mut SaeManifoldTerm,
    a: usize,
    b: usize,
    transition: &UnitSpeedChartTransition,
    rows_b: &[usize],
) -> Result<(), String> {
    if transition.from_chart != b || transition.to_chart != a {
        return Err(format!(
            "transplant_glued_coords: transition {}->{} does not match glue ({a},{b})",
            transition.from_chart, transition.to_chart
        ));
    }
    let coords = &mut term.assignment.coords;
    if a >= coords.len() || b >= coords.len() {
        return Err(format!(
            "transplant_glued_coords: glue ({a},{b}) outside {} coordinate blocks",
            coords.len()
        ));
    }
    let da = coords[a].latent_dim();
    let db = coords[b].latent_dim();
    if da < 1 || db < 1 || coords[a].n_obs() != coords[b].n_obs() {
        return Err(format!(
            "transplant_glued_coords: incompatible coordinate blocks for glue ({a},{b})"
        ));
    }
    // Read B's flat coords, then write A's transplanted rows through the fitted
    // transition. The read is cloned first, so the mutable borrow of A's coords
    // never aliases B's.
    let flat_b = coords[b].as_flat().to_owned();
    let mut flat_a = coords[a].as_flat().to_owned();
    let n = coords[b].n_obs();
    for &r in rows_b {
        if r >= n {
            return Err(format!(
                "transplant_glued_coords: certified row {r} outside n={n} for glue ({a},{b})"
            ));
        }
        let t_b = flat_b[r * db];
        flat_a[r * da] = transition.apply(t_b);
    }
    coords[a].set_flat(flat_a.view());
    Ok(())
}

/// Apply one [`StructureMove`] to a fitted term + ρ, returning the warm child
/// state. Warm inheritance by construction: the child is cloned from the parent
/// and only the touched atoms are restructured.
///
/// * **Death** demotes atom `atom`: its assignment logits are driven to a
///   strongly-negative value (routing → ~0) on every row, and its ARD block is
///   left in place. The atom is NOT removed (stable indices for the round); it
///   simply stops carrying mass. Demote-never-reject (#976).
/// * **Fission** appends a child cloned from atom `atom` (same basis, decoder,
///   coordinates), splitting the parent's per-row routing between parent and
///   child so the joint refit can pull them apart along the absorbed
///   substructure. The child inherits the parent's main-effect block.
/// * **Fusion** folds atom `b` into atom `a`: `a`'s routing absorbs `b`'s mass
///   (logit-sum on the active rows) and `b` is demoted. The retained atom's
///   product coordinates are initialized from the pair.
/// * **Birth** appends a fresh atom whose decoder is seeded from the
///   residual-factor direction `candidate` (passed in `birth_decoders`), routed
///   at a small neutral mass so the refit can grow it if it is real.
pub fn apply_structure_move(
    term: &SaeManifoldTerm,
    rho: &SaeManifoldRho,
    mv: &StructureMove,
    birth_decoders: &[Array2<f64>],
) -> Result<(SaeManifoldTerm, SaeManifoldRho), String> {
    match mv {
        StructureMove::Death { atom } => {
            let mut child = term.clone();
            demote_atom(&mut child, *atom)?;
            Ok((child, rho.clone()))
        }
        StructureMove::Fusion { a, b } => {
            let mut child = term.clone();
            fold_atom_into(&mut child, *a, *b)?;
            Ok((child, rho.clone()))
        }
        StructureMove::Glue { a, b, outcome } => {
            // Sphere pole seam: the transition is an ambient rotation, not a 1-D
            // affine map. Register it (keep both atoms) — a pole seam has no
            // destructive fuse outcome, since each atom's decoder was fitted on
            // its own support and neither carries the other's frame axis alone.
            if is_sphere_pair(term, *a, *b) {
                let mut child = term.clone();
                match outcome {
                    ChartGlueOutcome::RegisterAtlas => {
                        let seam = fit_sphere_seam_transition(term, *a, *b).ok_or_else(|| {
                            format!(
                                "apply_structure_move: sphere seam ({a},{b}) is no longer identifiable"
                            )
                        })?;
                        if !matches!(seam.seam_kind, AtlasSeamKind::Pole) {
                            return Err(format!(
                                "apply_structure_move: sphere seam ({a},{b}) is not a pole seam"
                            ));
                        }
                        child.register_sphere_chart_transition(
                            SphereChartTransition::new_fitted(
                                *b,
                                *a,
                                seam.rotation,
                                AtlasSeamKind::Pole,
                            )?,
                        )?;
                    }
                    ChartGlueOutcome::Fuse => {
                        return Err(format!(
                            "apply_structure_move: sphere pole seam ({a},{b}) cannot be destructively fused"
                        ));
                    }
                }
                return Ok((child, rho.clone()));
            }
            let seam = fit_seam_transition(term, *a, *b).ok_or_else(|| {
                format!("apply_structure_move: chart seam ({a},{b}) is no longer identifiable")
            })?;
            let mut child = term.clone();
            match outcome {
                ChartGlueOutcome::Fuse => {
                    if seam.sign != 1.0 {
                        return Err(format!(
                            "apply_structure_move: refusing to fuse orientation-reversing seam ({a},{b})"
                        ));
                    }
                    // Index-stable within the round: fold/demote now, physically
                    // excise at [`compact_glued_atoms`] before polish.
                    let transition = UnitSpeedChartTransition::new(
                        *b,
                        *a,
                        1,
                        seam.offset,
                        seam.period,
                        AtlasSeamKind::Regular,
                    )?;
                    fold_atom_into(&mut child, *a, *b)?;
                    transplant_glued_coords(&mut child, *a, *b, &transition, &seam.rows_b)?;
                }
                ChartGlueOutcome::RegisterAtlas => {
                    if seam.sign != -1.0 {
                        return Err(format!(
                            "apply_structure_move: atlas registration requires an orientation-reversing seam, got sign {}",
                            seam.sign
                        ));
                    }
                    // Keep both numerical charts. Their existing routing masses
                    // are exactly atlas activation × partition-of-unity; only the
                    // semantic quotient and transition cocycle are new state.
                    child.register_chart_transition(UnitSpeedChartTransition::new(
                        *b,
                        *a,
                        -1,
                        seam.offset,
                        seam.period,
                        AtlasSeamKind::Regular,
                    )?)?;
                }
            }
            Ok((child, rho.clone()))
        }
        StructureMove::Fission { atom } => {
            let (child, child_rho) = duplicate_atom(term, rho, *atom)?;
            Ok((child, child_rho))
        }
        StructureMove::Birth { candidate } => {
            let decoder = birth_decoders.get(*candidate).ok_or_else(|| {
                format!(
                    "apply_structure_move: birth candidate {candidate} out of range \
                     ({} residual-factor decoders)",
                    birth_decoders.len()
                )
            })?;
            born_atom(term, rho, decoder.view())
        }
    }
}

/// A birth seed the seeded apply-move ([`apply_structure_move_seeded`])
/// materializes. The residual-factor births carry only a flat decoder (the
/// topology race in `born_atom` then adjudicates line vs circle vs …); a curl
/// birth (INTEGRATION_PLAN Phase 4) instead carries a fully-formed periodic
/// circle — decoder, per-row phase, and gate — because a shattered centered
/// circle leaves NO residual for the race to seed from, so the seed IS the
/// hypothesis and `born_circle_atom` installs it directly for the REML e-gate
/// to adjudicate.
#[derive(Clone, Debug)]
pub enum BirthSeed {
    /// A whitened residual-factor direction lifted to a flat `(m, p)` decoder;
    /// born via the topology race (the legacy birth path).
    ResidualFactor(Array2<f64>),
    /// A curl circle seed: periodic-harmonic `(m, p)` decoder (`m` odd, `>= 3`),
    /// its exact analytic geometry plan, per-row phase coordinate `(n, 1)`, and
    /// per-row own-presence gate (`n`).
    Circle {
        geometry: SaeAtomGeometryPlan,
        decoder: Array2<f64>,
        phase_coords: Array2<f64>,
        gate: Vec<f64>,
    },
}

/// Apply one [`StructureMove`] with a heterogeneous birth-seed list — the curl
/// extension of [`apply_structure_move`]. Non-birth moves are identical; a
/// `Birth { candidate }` dispatches on the indexed [`BirthSeed`]: a
/// `ResidualFactor` rides the topology race (`born_atom`), a `Circle` is
/// installed directly as a periodic atom (`born_circle_atom`). The legacy
/// [`apply_structure_move`] is exactly this with an all-`ResidualFactor` seed
/// list, so bitwise legacy behavior is preserved when no curl seed is present.
pub fn apply_structure_move_seeded(
    term: &SaeManifoldTerm,
    rho: &SaeManifoldRho,
    mv: &StructureMove,
    birth_seeds: &[BirthSeed],
) -> Result<(SaeManifoldTerm, SaeManifoldRho), String> {
    match mv {
        StructureMove::Birth { candidate } => {
            let seed = birth_seeds.get(*candidate).ok_or_else(|| {
                format!(
                    "apply_structure_move_seeded: birth candidate {candidate} out of range \
                     ({} birth seeds)",
                    birth_seeds.len()
                )
            })?;
            match seed {
                BirthSeed::ResidualFactor(decoder) => born_atom(term, rho, decoder.view()),
                BirthSeed::Circle {
                    geometry,
                    decoder,
                    phase_coords,
                    gate,
                } => born_circle_atom(
                    term,
                    rho,
                    geometry.clone(),
                    decoder.clone(),
                    phase_coords.clone(),
                    gate.clone(),
                ),
            }
        }
        // Death / Fission / Fusion are seed-independent — delegate to the legacy
        // apply with an empty decoder list (never indexed for these).
        other => apply_structure_move(term, rho, other, &[]),
    }
}

/// A strongly-negative logit that drives a softmax / gate routing channel to ~0
/// mass without producing a non-finite value the assignment validator rejects.
const DEMOTE_LOGIT: f64 = -40.0;

/// Drive an atom's per-row routing to ~0 by setting its logit column to a
/// strongly-negative constant. Demotion, not removal: the atom keeps its index.
fn demote_atom(term: &mut SaeManifoldTerm, atom: usize) -> Result<(), String> {
    let k = term.k_atoms();
    if atom >= k {
        return Err(format!("demote_atom: atom {atom} out of range (K={k})"));
    }
    for row in 0..term.assignment.logits.nrows() {
        term.assignment.logits[[row, atom]] = DEMOTE_LOGIT;
    }
    Ok(())
}

/// Fold atom `b` into atom `a`: `a` absorbs `b`'s routing mass on every row
/// (logit max, the dominance the fused atom should express), then `b` is
/// demoted. The retained atom keeps its decoder; the joint refit reconciles the
/// merged structure.
fn fold_atom_into(term: &mut SaeManifoldTerm, a: usize, b: usize) -> Result<(), String> {
    let k = term.k_atoms();
    if a >= k || b >= k {
        return Err(format!(
            "fold_atom_into: atoms ({a},{b}) out of range (K={k})"
        ));
    }
    if a == b {
        return Err("fold_atom_into: cannot fuse an atom with itself".to_string());
    }
    // For SOFTMAX the fused atom must carry the COMBINED routing mass of its two
    // constituents. `softmax` mass is `e^logit/Z`, so the mass-preserving combine
    // is `logsumexp(la, lb)` (`softmax(logsumexp(la,lb)) = softmax(la)+softmax(lb)`).
    // Plain `max` UNDER-masses by up to `ln 2` on exactly the co-active rows that
    // triggered the fusion (where `la ≈ lb`): it gives the fused atom half the
    // combined mass, leaving the warm-start short and risking a FALSE rejection by
    // the e-gate under a capped refit. For ordered Beta--Bernoulli/ThresholdGate the per-atom gate is the
    // UN-normalized `σ(logit)`, so the union gate is `max(σ(la),σ(lb)) = σ(max(la,lb))`
    // → `max` is the correct combine there (a sum/logsumexp would over-gate).
    let softmax_routing = matches!(term.assignment.mode, AssignmentMode::Softmax { .. });
    for row in 0..term.assignment.logits.nrows() {
        let la = term.assignment.logits[[row, a]];
        let lb = term.assignment.logits[[row, b]];
        term.assignment.logits[[row, a]] = if softmax_routing {
            // Numerically stable logsumexp. When BOTH logits are -∞ (two rows of
            // zero softmax mass — a hard-masked/dead pair), `m = -∞` makes
            // `la - m = -∞ - (-∞) = NaN`, and the NaN poisons the whole logits
            // row (every subsequent softmax over it is NaN). The combined mass of
            // two zero-mass atoms is exactly zero, i.e. logit -∞ — return that
            // directly instead of computing NaN.
            let m = la.max(lb);
            if m == f64::NEG_INFINITY {
                f64::NEG_INFINITY
            } else {
                m + ((la - m).exp() + (lb - m).exp()).ln()
            }
        } else {
            la.max(lb)
        };
    }
    demote_atom(term, b)?;
    Ok(())
}

/// Physically REMOVE the atoms in `remove` from `term`/`rho` — the TRUE-FUSION
/// tail that a demotion alone cannot deliver. A [`fold_atom_into`] combines the
/// folded atom's routing mass into its survivor and drops its logit column to
/// [`DEMOTE_LOGIT`], but a demoted-not-removed atom keeps a full decoder at ~0
/// mass, and the joint refit's #976/#1003 active-mass guard
/// ([`SaeManifoldTerm::enforce_active_mass_guard`]) runs at fit ENTRY after
/// `collapse_events` is cleared: it reads that atom's max gate as below the trust
/// floor and RESEEDS its logits to per-row winner parity, resurrecting the atom
/// the glue just retired (the effective atom count never falls, #1890). So for a
/// certified glue we excise the folded atoms outright — decoder atom,
/// routing-logit column, latent coordinate block, per-atom `ungated`/frozen slot,
/// and per-atom ρ (smoothness + ARD) blocks — so BOTH the raw and the active
/// dictionary size fall, no zero-mass atom survives for the guard to revive, and
/// each survivor is forced to carry the absorbed arc through the refit.
///
/// Every atom is dropped in ONE pass (a shared keep-mask), so removing several
/// atoms needs no descending-index bookkeeping. Each survivor must already hold
/// its folded partner's mass (call this AFTER the folds); any per-atom diagnostic
/// caches are reset so the post-glue refit rebuilds them against the reduced
/// dictionary rather than indexing a stale length-`K`.
pub(crate) fn remove_atoms(
    term: &mut SaeManifoldTerm,
    rho: &mut SaeManifoldRho,
    remove: &std::collections::BTreeSet<usize>,
) -> Result<(), String> {
    let k = term.k_atoms();
    if let Some(&bad) = remove.iter().find(|&&j| j >= k) {
        return Err(format!("remove_atoms: atom {bad} out of range (K={k})"));
    }
    if remove.len() >= k {
        return Err("remove_atoms: cannot remove every atom".to_string());
    }
    if remove.is_empty() {
        return Ok(());
    }
    // Validate every atom-indexed container BEFORE mutating any of them.  The
    // normal constructors maintain these invariants, but this function is the
    // variable-K boundary and must return a useful error rather than partially
    // compacting a malformed warm state and then panicking on an indexed gather.
    let n = term.assignment.logits.nrows();
    if term.assignment.logits.ncols() != k
        || term.assignment.coords.len() != k
        || term.assignment.ungated.len() != k
    {
        return Err(format!(
            "remove_atoms: atom-indexed assignment shape mismatch: atoms={k}, \
             logits={:?}, coords={}, ungated={}",
            term.assignment.logits.dim(),
            term.assignment.coords.len(),
            term.assignment.ungated.len()
        ));
    }
    if let Some(frozen) = term.assignment.frozen_logits.as_ref() {
        if frozen.dim() != (n, k) {
            return Err(format!(
                "remove_atoms: frozen logits shape {:?} must equal ({n}, {k})",
                frozen.dim()
            ));
        }
    }
    if rho.log_lambda_smooth.len() != k || rho.log_ard.len() != k {
        return Err(format!(
            "remove_atoms: rho per-atom lengths (smooth {}, ard {}) must equal K={k}",
            rho.log_lambda_smooth.len(),
            rho.log_ard.len()
        ));
    }
    let keep: Vec<usize> = (0..k).filter(|j| !remove.contains(j)).collect();
    // Registered atlas endpoints are atom indices.  A seam-bearing chart may
    // never be deleted by ordinary compaction; surviving atlases are remapped
    // through the same keep permutation before the atom arrays move.
    let mut old_to_new = vec![None; k];
    for (new, &old) in keep.iter().enumerate() {
        old_to_new[old] = Some(new);
    }
    term.remap_chart_atlases(&old_to_new)?;
    rho.remap_curvature_atoms(&old_to_new)?;
    // Rebuild the atom list, coord blocks, ungated flags, and ρ blocks keeping
    // only the surviving indices (descending removal on the Vecs would also work,
    // but the keep-mask keeps atoms/coords/logits/ρ provably in lock-step).
    term.atoms = keep.iter().map(|&j| term.atoms[j].clone()).collect();
    // `ndarray::select(Axis(1), ..)` may retain a column-major/non-standard
    // stride layout. The Newton driver updates logits row-wise and requires each
    // row to be a contiguous mutable slice, so materialize the compacted router
    // explicitly in row-major order at this variable-K boundary.
    let compacted_logits = Array2::from_shape_fn((n, keep.len()), |(row, new_atom)| {
        term.assignment.logits[[row, keep[new_atom]]]
    });
    term.assignment.logits = compacted_logits;
    term.assignment.coords = keep
        .iter()
        .map(|&j| term.assignment.coords[j].clone())
        .collect();
    term.assignment.ungated = keep.iter().map(|&j| term.assignment.ungated[j]).collect();
    // A frozen router was trained against the old dictionary.  Merely slicing
    // its columns would not encode the fold's log-sum-exp mass transfer and
    // would keep routing permanently frozen to an invalid model.  The reduced
    // dictionary must refit its routing from the physically folded logits.
    term.assignment.frozen_logits = None;
    // Per-atom ρ blocks (the block-relevance `log_lambda_block` is per-output-block,
    // not per-atom, so it is untouched).
    rho.log_lambda_smooth = keep.iter().map(|&j| rho.log_lambda_smooth[j]).collect();
    rho.log_ard = keep.iter().map(|&j| rho.log_ard[j].clone()).collect();
    // Drop every K-dependent cache and optimization ledger.  Compaction changes
    // both the column order and the quotient dimension, so retaining any of the
    // old assembly layout, frozen pair gates, evidence-deflation anchor, or
    // per-atom diagnostic reports would make the polish refit interpret old-K
    // state as if it described the reduced dictionary.
    term.collapse_events.clear();
    term.last_row_layout = None;
    term.last_frames_active = false;
    term.fixed_decoder_assembly = false;
    term.border_hbb_workspace = Array2::<f64>::zeros((0, 0));
    term.decoder_repulsion_gate = None;
    term.barrier_coactivation_gate = None;
    term.streaming_gates_frozen = false;
    term.curvature_walk_report = None;
    term.expected_criterion_gauge_deflated_directions = None;
    term.criterion_gauge_deflation_reanchors = 0;
    term.criterion_gauge_deflation_last_delta_sign = 0;
    term.dictionary_cocollapse_reseeds = 0;
    term.structural_cocollapse_reseeds = 0;
    term.atom_inner_fits = None;
    term.oos_linear_images = None;
    term.hybrid_split_report = None;
    term.best_cocollapse_incumbent = None;
    term.best_fit_incumbent = None;
    Ok(())
}

/// Round-boundary chart-glue adoption (#1890).  A glue is a harvest-certified
/// equivalence move, not a likelihood-scored numerical candidate: the exact seam
/// transition which earned its e-value is carried in `certified_glues`, and the
/// search defers materializing the move until its index-sensitive proposal chain
/// is complete.  This boundary then applies every accepted certificate exactly
/// once: an orientation-preserving seam folds/transplants/removes B, while an
/// irreducible reversing or pole seam registers both charts as one atlas atom.
///
/// The whole accepted matching and all proposal/certificate pairings are checked
/// before mutation.  Materialization runs on a cloned child and commits only on
/// success, so malformed resumed/direct ledgers cannot leave half-folded state.
/// The search engine's `touched` guard guarantees accepted glues share no atom;
/// checking it again here makes that adoption contract explicit.  No accepted
/// glue is a no-op.
fn compact_glued_atoms(
    term: &mut SaeManifoldTerm,
    rho: &mut SaeManifoldRho,
    round_ledger: &SearchLedger,
    certified_glues: &[CertifiedGlue],
) -> Result<usize, String> {
    use gam_solve::structure_search::MoveVerdict;
    let accepted_glues: Vec<(usize, usize, ChartGlueOutcome)> = round_ledger
        .moves
        .iter()
        .filter_map(|rec| {
            if let (StructureMove::Glue { a, b, outcome }, MoveVerdict::Accepted { .. }) =
                (&rec.mv, &rec.verdict)
            {
                Some((*a, *b, *outcome))
            } else {
                None
            }
        })
        .collect();
    if accepted_glues.is_empty() {
        return Ok(0);
    }
    // Validate the whole accepted matching before the first fold.  The search
    // engine enforces this through its `touched` set; checking again here keeps
    // the variable-K boundary transactional even for resumed/deserialized or
    // directly-constructed ledgers.
    let k = term.k_atoms();
    let mut touched = std::collections::BTreeSet::new();
    for &(a, b, _) in &accepted_glues {
        if a >= k || b >= k || a == b {
            return Err(format!(
                "compact_glued_atoms: accepted glue ({a},{b}) out of range or self-gluing (K={k})"
            ));
        }
        if !touched.insert(a) || !touched.insert(b) {
            return Err(format!(
                "compact_glued_atoms: accepted glues are not an atom-disjoint matching; \
                 atom reused by ({a},{b})"
            ));
        }
    }

    // Pair every accepted ledger record with exactly one harvest certificate.
    // A proposal without its geometric object cannot be adopted: trying to
    // reconstruct it here from a scoring/polish-mutated state is precisely the
    // ordering bug this boundary forbids.
    let mut adopted: Vec<CertifiedGlue> = Vec::with_capacity(accepted_glues.len());
    for &(a, b, outcome) in &accepted_glues {
        let mut matches = certified_glues
            .iter()
            .filter(|certificate| certificate.a == a && certificate.b == b);
        let certificate = matches.next().ok_or_else(|| {
            format!("compact_glued_atoms: accepted glue ({a},{b}) has no harvest-time certificate")
        })?;
        if matches.next().is_some() {
            return Err(format!(
                "compact_glued_atoms: accepted glue ({a},{b}) has duplicate harvest-time certificates"
            ));
        }
        if certificate.outcome != outcome {
            return Err(format!(
                "compact_glued_atoms: accepted glue ({a},{b}) outcome {outcome:?} does not match certified {:?}",
                certificate.outcome
            ));
        }
        match (&certificate.transition, outcome) {
            (CertifiedGlueTransition::UnitSpeed { transition, .. }, ChartGlueOutcome::Fuse)
                if transition.from_chart == b
                    && transition.to_chart == a
                    && transition.sign == 1
                    && matches!(transition.seam_kind, AtlasSeamKind::Regular) => {}
            (
                CertifiedGlueTransition::UnitSpeed { transition, .. },
                ChartGlueOutcome::RegisterAtlas,
            ) if transition.from_chart == b
                && transition.to_chart == a
                && transition.sign == -1
                && matches!(transition.seam_kind, AtlasSeamKind::Regular) => {}
            (CertifiedGlueTransition::Sphere(transition), ChartGlueOutcome::RegisterAtlas)
                if transition.from_chart() == b
                    && transition.to_chart() == a
                    && matches!(transition.seam_kind(), AtlasSeamKind::Pole) => {}
            _ => {
                return Err(format!(
                    "compact_glued_atoms: accepted glue ({a},{b}) is incompatible with its certified transition"
                ));
            }
        }
        adopted.push(certificate.clone());
    }

    let mut child_term = term.clone();
    let mut child_rho = rho.clone();
    let mut to_remove: std::collections::BTreeSet<usize> = std::collections::BTreeSet::new();
    for certificate in adopted {
        let (a, b) = (certificate.a, certificate.b);
        match certificate.transition {
            CertifiedGlueTransition::UnitSpeed { transition, rows_b }
                if matches!(certificate.outcome, ChartGlueOutcome::Fuse) =>
            {
                fold_atom_into(&mut child_term, a, b)?;
                transplant_glued_coords(&mut child_term, a, b, &transition, &rows_b)?;
                to_remove.insert(b);
            }
            CertifiedGlueTransition::UnitSpeed { transition, .. } => {
                child_term.register_chart_transition(transition)?;
            }
            CertifiedGlueTransition::Sphere(transition) => {
                child_term.register_sphere_chart_transition(transition)?;
            }
        }
    }
    remove_atoms(&mut child_term, &mut child_rho, &to_remove)?;
    *term = child_term;
    *rho = child_rho;
    Ok(to_remove.len())
}

/// Refit every production-registered seam on terminal polished charts. Atlas
/// state is persisted model state, so after a genuinely numerical move it must
/// describe the returned fit rather than the pre-polish warm start. A seam that
/// ceases to be identifiable or changes kind/orientation is a structural-fit
/// failure: returning a stale atlas would be worse than failing loudly. Pure
/// registration rounds never call this function because they do not polish.
fn refresh_registered_atlas_transitions(term: &mut SaeManifoldTerm) -> Result<(), String> {
    let registered: Vec<UnitSpeedChartTransition> = term
        .chart_atlases()
        .iter()
        .flat_map(|atlas| atlas.transitions().iter().copied())
        .filter(|transition| matches!(transition.seam_kind, AtlasSeamKind::Regular))
        .collect();
    for transition in registered {
        // `fit_seam_transition(a,b)` returns the map b -> a.
        let seam = fit_seam_transition(term, transition.to_chart, transition.from_chart)
            .ok_or_else(|| {
                format!(
                    "terminal atlas seam {}->{} is no longer identifiable",
                    transition.from_chart, transition.to_chart
                )
            })?;
        if seam.sign as i8 != transition.sign {
            return Err(format!(
                "terminal atlas seam {}->{} changed orientation ({} -> {})",
                transition.from_chart, transition.to_chart, transition.sign, seam.sign as i8
            ));
        }
        term.refresh_chart_transition(UnitSpeedChartTransition::new(
            transition.from_chart,
            transition.to_chart,
            transition.sign,
            seam.offset,
            seam.period,
            transition.seam_kind,
        )?)?;
    }
    let registered_spheres: Vec<SphereChartTransition> = term
        .chart_atlases()
        .iter()
        .flat_map(|atlas| atlas.sphere_transitions().iter().copied())
        .collect();
    for transition in registered_spheres {
        // `fit_sphere_seam_transition(a,b)` likewise returns the map b -> a.
        let seam = fit_sphere_seam_transition(term, transition.to_chart(), transition.from_chart())
            .ok_or_else(|| {
                format!(
                    "terminal sphere atlas seam {}->{} is no longer identifiable",
                    transition.from_chart(),
                    transition.to_chart()
                )
            })?;
        if seam.seam_kind != transition.seam_kind() {
            return Err(format!(
                "terminal sphere atlas seam {}->{} changed kind ({:?} -> {:?})",
                transition.from_chart(),
                transition.to_chart(),
                transition.seam_kind(),
                seam.seam_kind
            ));
        }
        term.refresh_sphere_chart_transition(SphereChartTransition::new_fitted(
            transition.from_chart(),
            transition.to_chart(),
            seam.rotation,
            transition.seam_kind(),
        )?)?;
    }
    Ok(())
}

/// Append a child cloned from atom `parent`: identical basis, decoder, and
/// coordinates, with the parent's routing split evenly between parent and child
/// (the parent's logit dropped by `ln 2` on every row, the child seeded equal).
/// The joint refit then pulls the two apart along the absorbed substructure. The
/// child's ARD block is inherited from the parent.
fn duplicate_atom(
    term: &SaeManifoldTerm,
    rho: &SaeManifoldRho,
    parent: usize,
) -> Result<(SaeManifoldTerm, SaeManifoldRho), String> {
    let k = term.k_atoms();
    if parent >= k {
        return Err(format!(
            "duplicate_atom: parent {parent} out of range (K={k})"
        ));
    }
    let mut atoms = term.atoms.clone();
    let mut child_atom = term.atoms[parent].clone();
    // Symmetry-breaking perturbation. A fission that duplicates the parent atom
    // IDENTICALLY (same decoder, same coords, mass split 50/50) sits at a
    // SYMMETRIC SADDLE of the joint refit: the two children have identical
    // gradients, so a deterministic Newton/descent refit moves them in lockstep
    // and they NEVER separate — the fission stays a no-op (two identical
    // half-atoms ≡ the original atom) and the e-gate, seeing no reconstruction
    // gain, rejects it. So fission could only ever land by floating-point noise.
    // Apply a small ANTI-SYMMETRIC perturbation — parent decoder ×(1−ε·s_ij),
    // child decoder ×(1+ε·s_ij) for a deterministic varying pattern s_ij — which
    // breaks the symmetry (the refit can roll off the saddle toward the
    // two-factor configuration the carve identified) while leaving the mass-split
    // combined decoder `½·parent + ½·child = original` EXACTLY unchanged (the
    // ±ε·s_ij cancel), so the warm-start is preserved. `ε ≫ fp noise`.
    {
        let (m, p) = atoms[parent].decoder_coefficients().dim();
        let s = |i: usize, j: usize| -> f64 {
            // Deterministic, varying, NON-ZERO pattern in [-1,-0.2]∪[0.2,1]. It
            // must vary across (i,j) (so `parent − child = −2·f_ij·D_ij` points
            // off the symmetric `D` direction and can separate factors) and never
            // vanish (else a sparse decoder element gets no perturbation).
            let raw = ((i * 7 + j * 13) % 11) as f64 / 5.0 - 1.0;
            if raw.abs() < 0.2 { 0.3 } else { raw }
        };
        for i in 0..m {
            for j in 0..p {
                let f = FISSION_SYMMETRY_BREAK_EPS * s(i, j);
                atoms[parent].decoder_coefficients_mut()[[i, j]] *= 1.0 - f;
                child_atom.decoder_coefficients_mut()[[i, j]] *= 1.0 + f;
            }
        }
        // The Grassmann decoder frame is derived from the coefficients; drop both
        // so the warm refit recomputes them consistent with the perturbed decoders.
        atoms[parent].decoder_frame = None;
        child_atom.decoder_frame = None;
    }
    atoms.push(child_atom);

    let n = term.assignment.logits.nrows();
    let mut logits = Array2::<f64>::zeros((n, k + 1));
    let split = std::f64::consts::LN_2;
    for row in 0..n {
        for col in 0..k {
            let mut v = term.assignment.logits[[row, col]];
            if col == parent {
                // Halve the parent's routing mass (logit − ln 2) and give the
                // other half to the child.
                v -= split;
            }
            logits[[row, col]] = v;
        }
        logits[[row, k]] = term.assignment.logits[[row, parent]] - split;
    }
    let mut coords = term.assignment.coords.clone();
    coords.push(term.assignment.coords[parent].clone());
    let assignment =
        crate::manifold::SaeAssignment::with_mode(logits, coords, term.assignment.mode)?;
    let child = SaeManifoldTerm::new(atoms, assignment)?;

    let mut child_rho = rho.clone();
    if parent < child_rho.log_ard.len() {
        let inherited = child_rho.log_ard[parent].clone();
        child_rho.log_ard.push(inherited);
    } else {
        child_rho.log_ard.push(Array1::<f64>::zeros(0));
    }
    // The fissioned child inherits the PARENT atom's per-atom smoothness strength
    // (#1556). As with `log_ard`, failing to grow `log_lambda_smooth` in step with
    // `k_atoms()` makes the next `assemble_arrow_schur` panic on the per-atom
    // `lambda_smooth[atom_idx]` index (out of bounds).
    let inherited_smooth = child_rho
        .log_lambda_smooth
        .get(parent)
        .or_else(|| child_rho.log_lambda_smooth.first())
        .copied()
        .unwrap_or(0.0);
    child_rho.log_lambda_smooth.push(inherited_smooth);
    child_rho.append_curvature_atom(
        k,
        child.atoms[k]
            .geometry_plan()
            .and_then(SaeAtomGeometryPlan::constant_curvature),
    )?;
    Ok((child, child_rho))
}

// ===========================================================================
// #977 — per-atom topology RACE at birth.
//
// A born atom must not inherit atom-0's circle template by fiat. Its TOPOLOGY is
// chosen by EVIDENCE: each candidate basis whose required intrinsic dimension
// matches the born atom's ARD-selected `d_k` is fit to the residual-factor image
// the atom would reconstruct, and the winner is the lowest TK-normalized REML —
// the SAME gauge-invariant comparison [`select_topology_with_fit`] applies to the
// smooth-term topology race, so cross-topology scores are commensurable. The
// dictionary the learner discovers is therefore genuinely heterogeneous: a born
// atom on a circular residual gets a circle, one on a straight residual a line.
// ===========================================================================

/// One realized candidate of the birth topology race: the fitted evaluator, its
/// penalized-least-squares decoder against the birth target, the chart manifold
/// the winning atom will carry, and the basis kind tag. Carried as the
/// `select_topology_with_fit` fit handle so the winner's basis seeds the born
/// atom directly (no re-fit, no cold restart).
#[derive(Clone)]
struct TopologyRaceFit {
    evaluator: Arc<dyn SaeBasisSecondJet>,
    geometry: SaeAtomGeometryPlan,
    manifold: LatentManifold,
    /// The `(n × d)` coordinates the winning basis was evaluated at — the born
    /// atom's coordinate block, dimension-matched to the winning evaluator.
    coords: Array2<f64>,
    /// Fitted basis design `Φ(coords)` (`n × m`).
    phi: Array2<f64>,
    /// Fitted basis Jacobian `∂Φ` (`n × m × d`).
    jet: ndarray::Array3<f64>,
    /// Penalized-least-squares decoder `B` (`m × p`).
    decoder: Array2<f64>,
    /// Declared reference-function Gram `S_ref` (`m × m`) the atom is seeded with.
    penalty: Array2<f64>,
}

/// A candidate topology paired with the evaluator + coordinates + manifold it
/// realizes for a `d`-dimensional birth. The evaluator is built fresh (cold) for
/// each candidate; the race then fits it to the birth target.
struct TopologyCandidateSpec {
    kind: AutoTopologyKind,
    geometry: SaeAtomGeometryPlan,
    manifold: LatentManifold,
    /// The `(n, d)` coordinates this candidate evaluates its basis at. A `d = 1`
    /// candidate reads the template coordinate column; a `d = 2` candidate reads
    /// the first two columns (or pads with the single column the seed carries).
    coords: Array2<f64>,
}

impl TopologyCandidateSpec {
    fn new(
        kind: AutoTopologyKind,
        geometry: SaeAtomGeometryPlan,
        manifold: LatentManifold,
        coords: Array2<f64>,
    ) -> Result<Self, String> {
        if coords.ncols() != geometry.latent_dim() {
            return Err(format!(
                "TopologyCandidateSpec::new: coordinate width {} != geometry latent_dim {}",
                coords.ncols(),
                geometry.latent_dim()
            ));
        }
        Ok(Self {
            kind,
            geometry,
            manifold,
            coords,
        })
    }
}

/// Build the topology candidate set whose required intrinsic dimension matches
/// the born atom's `d_k`, each realized over `coords` (`n × d_seed`, the
/// template's coordinate block). The candidate set is the realizable subset of
/// the smooth-term topology race — every member is a CORE basis evaluator
/// (`src/terms/sae/basis.rs`), so no FFI round-trip and no cold curved family
/// that the joint refit cannot warm-start:
///
/// * **`d = 1`** — `Circle` ([`PeriodicHarmonicEvaluator`]) vs `Euclidean` line
///   ([`EuclideanPatchEvaluator`] degree 3). These are the line-vs-circle race
///   the #1026 curved-vs-linear rung adjudicates post-fit, lifted to BIRTH.
/// * **`d = 2`** — `Torus` ([`TorusHarmonicEvaluator`]), `Sphere`
///   ([`AmbientSphereHarmonicEvaluator`] — the candidate is built at
///   `latent_dim = 3` on an ambient unit direction, which is the only sphere the
///   geometry-plan authority accepts, #2698), a flat `Euclidean` patch
///   ([`EuclideanPatchEvaluator`] degree 2), and `Cylinder` `S¹ × ℝ`
///   ([`CylinderHarmonicEvaluator`]: a periodic circle axis tensored with a flat
///   line axis). The cylinder is now a first-class d=2 candidate (the basis
///   landed in `src/terms/sae/basis.rs`), so a residual that is periodic along
///   one axis and unbounded-linear along the other earns a cylinder rather than
///   being forced into a torus stand-in (which would wrap the linear axis
///   spuriously) or a flat patch (which would lose the periodicity).
///
/// The fixed harmonic / degree budgets mirror the seed-dictionary builder
/// (`sae_build_atom_plans`): periodic gets `2·d_k + 1` columns, torus two
/// harmonics per axis, the patch degree 3 (`d = 1`) / 2 (`d = 2`).
///
/// This is the FLAT-vs-RIGID contest made concrete. Every candidate here
/// realizes one specific point in the flat/curved spectrum for the born
/// atom's intrinsic dimension: at `d = 1` the flat `Euclidean` line (large
/// `GL(1)` gauge freedom — any rescaling of the coordinate is indistinguishable
/// from any other) against the rigid `Circle` (gauge collapses to rotations of
/// `S¹`); at `d = 2` the flat patch against the curved `Torus` / `Sphere` /
/// `Cylinder`, each with its own residual symmetry group strictly smaller than
/// `GL(2)`. The candidate SET is exactly the set of hypotheses
/// [`fit_topology_candidate`] scores and [`race_birth_topology`] adjudicates —
/// building it is choosing which flat/rigid alternatives the evidence gets to
/// discriminate between; the race itself decides which one is real.
/// #2280 — the two bases a topology chart may read, named so a chart cannot
/// silently receive the wrong one.
///
/// The seed a birth carries is generally **per-component standardized**, and
/// that destroys every METRIC quantity: a length, a radius, an angle in an
/// embedded space — anything computed ACROSS axes rather than within one.
/// Rescaling each axis by its own standard deviation turns a circle into an
/// ellipse and makes `hypot(x, y) - R` a length compared against a `z` divided by
/// a different number. That is a units error, and it belongs to a CLASS of
/// charts rather than to any one of them: every future chart that measures a
/// distance from an axis or a surface, every chart on an embedding whose shape
/// lives in the ratios between coordinates, and anything that assumes a circle
/// stays a circle.
///
/// So the basis is EXPLICIT at the seam rather than implied by which caller you
/// came from. A chart needing true geometry reads `ambient`; a chart needing only
/// a well-conditioned parameterisation reads `seed`. The alternative — quietly
/// not standardizing the seed charts read — fixes one member of the class and
/// leaves the trap armed for the rest, while making the basis a chart receives
/// invisible to whoever writes the next one.
///
/// `ambient` is optional because some callers legitimately have no ambient rows
/// (a menu built for its candidate SET rather than for a fit). A chart requiring
/// it then declines and falls back to the seed chart — the same fail-open policy
/// every other chart gate here uses — rather than being handed a basis that looks
/// usable and is not.
#[derive(Clone, Copy)]
struct CandidateBases<'a> {
    /// The birth's coordinate seed. Well-conditioned, generally standardized per
    /// component, so RATIOS BETWEEN AXES CARRY NO GEOMETRY.
    seed: ArrayView2<'a, f64>,
    /// The ambient rows the birth is fitted against, in their own units — the
    /// only basis on which a metric quantity means anything.
    ambient: Option<ArrayView2<'a, f64>>,
}

impl<'a> CandidateBases<'a> {
    /// The full pair. Every production race has both in scope.
    fn with_ambient(seed: ArrayView2<'a, f64>, ambient: ArrayView2<'a, f64>) -> Self {
        Self { seed, ambient: Some(ambient) }
    }

    /// The ambient rows, only when row-aligned with the seed. A chart indexes
    /// rows of both identically, so a row-count mismatch would silently pair
    /// unrelated observations — refuse rather than pair them.
    fn aligned_ambient(&self) -> Option<ArrayView2<'a, f64>> {
        self.ambient.filter(|ambient| ambient.nrows() == self.seed.nrows())
    }
}

/// #2280 — the significance at which [`phase_coordinate_closes`] is willing to
/// call a genuinely closed loop "open".
///
/// It is a false-REJECTION rate, not a tuning knob: at `1e-3` a true circle is
/// mistakenly called an arc about once in a thousand fixtures, and the bar it
/// implies is derived from it rather than chosen. Loosening it makes the circle
/// candidate reachable on more open data (the `open_arc → Circle` misnaming);
/// tightening it costs genuine circles their natural chart.
const PHASE_CLOSURE_FALSE_REJECTION_RATE: f64 = 1e-3;

/// #2280 — does a phase coordinate actually CLOSE, or is it an arc?
///
/// A periodic chart flatters a periodic candidate: wrap an open arc's angle into
/// `[0, 1)` and the circle basis fits it perfectly well, so handing every `d = 1`
/// birth a phase chart buys three correct topologies and one FALSE PERIODICITY
/// claim (measured on the planted zoo: `open_arc` named `Circle`). Closure is the
/// property the chart itself cannot express, so it is tested here, separately,
/// before the chart is granted.
///
/// The statistic is the **largest circular spacing**. Sort the phases around the
/// period and take the biggest gap between consecutive points. A closed loop is
/// covered all the way round, so its largest gap is the ordinary fluctuation of
/// `n` points on a circle; an arc's largest gap IS its missing sector, which is
/// `O(1)` rather than `O(1/n)` and does not shrink as the sample grows.
///
/// The bar is derived, not picked. For `n` points i.i.d. uniform on `S¹` the
/// spacings are exchangeable and `P(max spacing > t/n) ≈ n·exp(−t)`; setting that
/// probability to `alpha` gives `t = ln(n/alpha)`, so a closed loop's largest gap
/// should not exceed `ln(n/alpha)/n`. That is the classical maximum-spacing test
/// for uniformity on the circle, and it is the honest bar here because it is a
/// statement about the SAMPLING, with `alpha` the only thing chosen.
///
/// Deliberately one thing only: it answers "is this coordinate closed?", not "is
/// this a circle?". The REML race still decides the topology — this only decides
/// whether the periodic candidate is scored on a chart that presumes the answer.
fn phase_coordinate_closes(phases: ArrayView1<'_, f64>, alpha: f64) -> bool {
    let n = phases.len();
    // Below three points every gap is the whole period and the test cannot
    // separate a loop from an arc; abstain by declining the phase chart rather
    // than granting it on no evidence.
    if n < 3 || !(alpha > 0.0) {
        return false;
    }
    let mut sorted: Vec<f64> = Vec::with_capacity(n);
    for &value in phases.iter() {
        if !value.is_finite() {
            return false;
        }
        // Fold onto one period so a lift that escaped `[0, 1)` cannot manufacture
        // a gap that is really a winding.
        let folded = value - value.floor();
        sorted.push(folded);
    }
    sorted.sort_by(|a, b| a.partial_cmp(b).expect("phases are finite here"));
    // Circular spacings: the consecutive differences plus the wrap-around gap
    // from the last point back to the first. Omitting the wrap is the classic
    // way to make every arc look closed.
    let mut largest = 1.0 - (sorted[n - 1] - sorted[0]);
    // DISTINCT angular positions, not rows. The null is `m` independent draws on
    // the circle, and replicated rows are not independent draws — they are one
    // position observed many times. A gridded sweep is the case that exposes the
    // difference, and it is not a corner case: a product chart (a torus lattice,
    // any regularly sampled sweep) has few distinct angles observed many times
    // each. A 30 x 20 torus lattice has 600 rows but only 30 distinct major
    // angles, so its largest spacing is pinned at 1/30 = 0.033 while a
    // ROW-denominated bar shrinks to ln(600/alpha)/600 = 0.022 and rejects a
    // perfectly closed loop. Denominating in rows made the false-REJECTION rate
    // exceed its own nominal `alpha` by orders of magnitude on exactly the data a
    // product chart produces.
    let mut positions = 1usize;
    for window in sorted.windows(2) {
        let gap = window[1] - window[0];
        if gap > 0.0 {
            positions += 1;
        }
        if gap > largest {
            largest = gap;
        }
    }
    // Below three distinct positions the spacings carry no information about
    // closure, however many rows repeat them.
    if positions < 3 {
        return false;
    }
    let bar = (positions as f64 / alpha).ln() / positions as f64;
    largest <= bar
}

fn topology_candidates_for_dim(
    bases: CandidateBases<'_>,
    d_k: usize,
) -> Result<Vec<TopologyCandidateSpec>, String> {
    let coords = bases.seed;
    let n = coords.nrows();
    let d_seed = coords.ncols();
    if d_k == 0 {
        // d_k = 0 is the cluster-null rung — it is NOT a manifold topology and is
        // adjudicated below the race (the bottom rung), never inside it.
        return Ok(Vec::new());
    }
    // Project the seed coordinates onto the candidate's intrinsic dimension. A
    // d=1 candidate uses the first column; a d=2 candidate uses the first two.
    // When the caller has only a 1-D seed, this generic builder repeats the seed
    // column; radial promotion overwrites that second coordinate with the
    // standardized log-amplitude below so promoted cylinder/disk candidates carry
    // the radial signal rather than a duplicate angle.
    let coords_d = |d: usize| -> Array2<f64> {
        let mut out = Array2::<f64>::zeros((n, d));
        for row in 0..n {
            for col in 0..d {
                let src = col.min(d_seed.saturating_sub(1));
                out[[row, col]] = coords[[row, src]];
            }
        }
        out
    };

    // The ambient sphere coordinate is a DIRECTION: the leading seed frame
    // carried onto the unit sphere. Normalisation belongs to the manifold, not
    // to this call site, so it goes through `project_point` -- the same
    // authority the retraction and the seed path use.
    let sphere_coords_ambient = || -> Array2<f64> {
        let raw = coords_d(3);
        let manifold = LatentManifold::Sphere { dim: 3 };
        let mut out = Array2::<f64>::zeros(raw.dim());
        for row in 0..raw.nrows() {
            let projected = manifold.project_point(raw.row(row));
            for axis in 0..3 {
                out[[row, axis]] = projected[axis];
            }
        }
        out
    };

    // #2280 — the NATURAL chart of a candidate, granted only where the data earns
    // it.
    //
    // `discover_primary_atom_topologies` has always built these charts for these
    // kinds; the birth menu handed every candidate `coords_d(d)` instead. That is
    // a frame error independent of any score — the circle candidate DECLARES
    // `LatentManifold::Circle { period: 1.0 }` while being handed a coordinate
    // that is not a phase — and it costs real topologies: on the planted zoo the
    // circle, the trefoil and the cylinder are all recovered by the chart alone
    // (menu 5/9 → 7/9, measured).
    //
    // But a periodic chart FLATTERS a periodic candidate, so it is gated on
    // `phase_coordinate_closes`: granting it unconditionally also bought
    // `open_arc → Circle`, a false periodicity claim, which is the error class a
    // topology race exists to prevent. A candidate whose chart is refused keeps
    // the shared linear chart and still RACES — it is never dropped, because a
    // candidate that was never offered cannot be said to have lost.
    let phase = |a: f64, b: f64| -> f64 {
        let frac = b.atan2(a) / std::f64::consts::TAU;
        frac - frac.floor()
    };
    let phase_column = |axis_a: usize, axis_b: usize| -> Array1<f64> {
        let mut out = Array1::<f64>::zeros(n);
        for row in 0..n {
            out[row] = phase(coords[[row, axis_a]], coords[[row, axis_b]]);
        }
        out
    };
    let standardized = |col: usize| -> Array1<f64> {
        let mut out = Array1::<f64>::zeros(n);
        let mut mean = 0.0_f64;
        for row in 0..n {
            mean += coords[[row, col]];
        }
        mean /= n as f64;
        let mut var = 0.0_f64;
        for row in 0..n {
            var += (coords[[row, col]] - mean).powi(2);
        }
        let sd = (var / n as f64).sqrt();
        let scale = if sd > 0.0 && sd.is_finite() { 1.0 / sd } else { 1.0 };
        for row in 0..n {
            out[row] = (coords[[row, col]] - mean) * scale;
        }
        out
    };
    let closes = |column: &Array1<f64>| -> bool {
        phase_coordinate_closes(column.view(), PHASE_CLOSURE_FALSE_REJECTION_RATE)
    };
    // S¹: the phase of the leading principal pair — an angle is not a function of
    // one coordinate, so this needs two seed directions AND a closed sweep.
    let circle_phase_coords = |d: usize| -> Array2<f64> {
        if d_seed < 2 {
            return coords_d(d);
        }
        let angle = phase_column(0, 1);
        if !closes(&angle) {
            return coords_d(d);
        }
        let mut out = Array2::<f64>::zeros((n, 1));
        for row in 0..n {
            out[[row, 0]] = angle[row];
        }
        out
    };
    // T² = S¹ × S¹: independent phases of the two leading principal PAIRS, so the
    // angles are not read off one plane. BOTH axes must close — a torus with one
    // open axis is a cylinder, and pretending otherwise is the same false
    // periodicity one dimension up.
    let stack_two = |first: &Array1<f64>, second: &Array1<f64>| -> Array2<f64> {
        let mut out = Array2::<f64>::zeros((n, 2));
        for row in 0..n {
            out[[row, 0]] = first[row];
            out[[row, 1]] = second[row];
        }
        out
    };
    // The REVOLUTION form of T², read on the AMBIENT rows.
    //
    // A Clifford-type torus spends an independent plane on each angle, so its
    // chart is the phases of two principal PAIRS — four seed directions. But a
    // torus of revolution (the donut, and the fixture every zoo uses) is embedded
    // in THREE dimensions and its angles are not two principal pairs: the major
    // angle is the phase in the plane of revolution, the minor the phase of the
    // meridian circle in the (distance-from-the-revolution-circle, axis)
    // half-plane. Demanding only the Clifford form made the canonical torus
    // structurally unreachable.
    //
    // The meridian phase is a METRIC reading, so it must come from ambient rows.
    // Measured on the planted donut (MSI 14621146): on raw rows both phases close
    // comfortably (major by 9x, minor by 5.7x); on the per-component standardized
    // seed the minor phase FAILS — the largest gap widens 0.050 -> 0.125 because
    // rescaling axes independently turns the meridian circle into an ellipse, and
    // the position count inflates 37 -> 108 because `hypot(x,y) - R` stops taking
    // the same value across major angles. Same chart, same data, wrong basis.
    let revolution_phases = |ambient: ArrayView2<'_, f64>| -> (Array1<f64>, Array1<f64>) {
        let mut radius = Array1::<f64>::zeros(n);
        for row in 0..n {
            radius[row] = ambient[[row, 0]].hypot(ambient[[row, 1]]);
        }
        let mean_radius = radius.sum() / n as f64;
        let mut major = Array1::<f64>::zeros(n);
        let mut minor = Array1::<f64>::zeros(n);
        for row in 0..n {
            major[row] = phase(ambient[[row, 0]], ambient[[row, 1]]);
            minor[row] = phase(radius[row] - mean_radius, ambient[[row, 2]]);
        }
        (major, minor)
    };
    let torus_phase_coords = || -> Array2<f64> {
        if d_seed >= 4 {
            let (first, second) = (phase_column(0, 1), phase_column(2, 3));
            if closes(&first) && closes(&second) {
                return stack_two(&first, &second);
            }
        }
        if let Some(ambient) = bases.aligned_ambient() {
            if ambient.ncols() >= 3 {
                let (major, minor) = revolution_phases(ambient);
                if closes(&major) && closes(&minor) {
                    return stack_two(&major, &minor);
                }
            }
        }
        coords_d(2)
    };
    // S¹ × ℝ: one angle and one height transverse to it. Only the ANGLE has to
    // close; the height is genuinely open, which is what makes it a cylinder
    // rather than a torus.
    let cylinder_chart_coords = || -> Array2<f64> {
        if d_seed < 3 {
            return coords_d(2);
        }
        let angle = phase_column(0, 1);
        if !closes(&angle) {
            return coords_d(2);
        }
        let height = standardized(2);
        let mut out = Array2::<f64>::zeros((n, 2));
        for row in 0..n {
            out[[row, 0]] = angle[row];
            out[[row, 1]] = height[row];
        }
        out
    };

    let mut specs: Vec<TopologyCandidateSpec> = Vec::new();
    match d_k {
        1 => {
            let n_harmonics = (2 * d_k + 1).max(3) | 1; // odd, ≥ 3
            let harmonic_order = (n_harmonics - 1) / 2;
            specs.push(TopologyCandidateSpec::new(
                AutoTopologyKind::Circle,
                SaeAtomGeometryPlan::new(
                    SaeAtomBasisKind::Periodic,
                    1,
                    SaeBasisResolution::PeriodicHarmonics {
                        order: harmonic_order,
                    },
                    SaeReferenceMetricPlan::UnitCircle,
                )?,
                LatentManifold::Circle { period: 1.0 },
                circle_phase_coords(1),
            )?);
            specs.push(TopologyCandidateSpec::new(
                AutoTopologyKind::Euclidean,
                SaeAtomGeometryPlan::new(
                    SaeAtomBasisKind::EuclideanPatch,
                    1,
                    SaeBasisResolution::Polynomial { degree: 3 },
                    SaeReferenceMetricPlan::EuclideanPolynomial,
                )?,
                LatentManifold::Euclidean,
                coords_d(1),
            )?);
        }
        2 => {
            specs.push(TopologyCandidateSpec::new(
                AutoTopologyKind::Torus,
                SaeAtomGeometryPlan::new(
                    SaeAtomBasisKind::Torus,
                    2,
                    SaeBasisResolution::TorusHarmonics { per_axis_order: 2 },
                    SaeReferenceMetricPlan::FlatRectangularTorus { tau: 0.0 },
                )?,
                // T² = S¹ × S¹: each axis is a unit-period circle (the
                // fraction-of-period convention `TorusHarmonicEvaluator` shares
                // with the periodic 1-D atom). This MUST match the production
                // seeding (`AtomTopology::Torus` → Product[Circle, Circle] in
                // `sae::manifold::atom`); a flat `Euclidean` manifold would leave
                // the born atom's angles un-wrapped and the joint refit would
                // retract on the wrong geometry.
                LatentManifold::Product(vec![
                    LatentManifold::Circle { period: 1.0 },
                    LatentManifold::Circle { period: 1.0 },
                ]),
                torus_phase_coords(),
            )?);
            specs.push(TopologyCandidateSpec::new(
                AutoTopologyKind::KleinBottle,
                SaeAtomGeometryPlan::klein_bottle(2)?,
                LatentManifold::Product(vec![
                    LatentManifold::Circle { period: 1.0 },
                    LatentManifold::Circle { period: 1.0 },
                ]),
                // The flat Klein bottle is the deck-invariant Fourier modes on
                // its TWO-TORUS cover, so its chart is the torus's.
                torus_phase_coords(),
            )?);
            // `S²` needs THREE independent seed directions. With only two, the
            // seed confines every row to a great circle -- a measure-zero subset
            // on which the degree-2 harmonic design is rank-deficient, so the
            // observed-information Hessian is singular and the candidate cannot
            // converge to a maximum. Worse, a sphere is not even identifiable
            // from a circle there: the data cannot distinguish them.
            //
            // The superseded `(lat, lon)` chart hid this. It needed only two
            // columns, so it would happily mint a "sphere" from data that cannot
            // support one -- returning a confident answer to an unanswerable
            // question. Requiring the third direction is the honest condition,
            // and it is the same guard the birth site already applies as
            // `n_pcs >= 3`.
            if d_seed >= 3 {
            specs.push(TopologyCandidateSpec::new(
                AutoTopologyKind::Sphere,
                SaeAtomGeometryPlan::new(
                    SaeAtomBasisKind::Sphere,
                    3,
                    SaeBasisResolution::AmbientSphereHarmonics {
                        degree: SAE_AMBIENT_SPHERE_DEFAULT_DEGREE,
                    },
                    SaeReferenceMetricPlan::RoundSphere,
                )?,
                // The sphere candidate races as an actual sphere: an ambient
                // unit 3-vector, whose retraction has no cut and no boundary and
                // whose uniform metric restricts to the round metric. The
                // superseded `(lat, lon)` chart raced a CYLINDER -- the pole was
                // an `Interval` bound the optimiser could not cross, the trust
                // region was wrong by `cos²(lat)`, and the fixed 7-column block
                // was not closed under `SO(3)`, so the candidate's achievable
                // fit depended on where the chart's pole happened to fall
                // relative to the data. Degree 2 spans all five `l = 2`
                // harmonics, where the chart held only three of them.
                LatentManifold::Sphere { dim: 3 },
                sphere_coords_ambient(),
            )?);
            }
            // `RP²` is `S²/{u ~ -u}`, so it needs the sphere's three seed
            // directions for the same reason and is gated with it.
            if d_seed >= 3 {
                specs.push(TopologyCandidateSpec::new(
                    AutoTopologyKind::ProjectivePlane,
                    SaeAtomGeometryPlan::projective_plane(1)?,
                    LatentManifold::Sphere { dim: 3 },
                    sphere_coords_ambient(),
                )?);
            }
            // #2280 — the Möbius band. It belongs here for a reason that is not
            // "one more shape": ORIENTATION HOLONOMY is the atlas's single most
            // reliable primitive — the composed transition Jacobian's determinant
            // sign around a nerve cycle is local, needs no good-cover certificate,
            // and separates Möbius from cylinder where every homotopy invariant
            // (χ, b₁) reports the identical answer. So the birth atlas MEASURES a
            // Möbius band confidently, `observed_kind_to_auto_topology` names it,
            // and until now this menu realized no candidate for it: the reorder
            // fell through to its "menu realizes no twisted candidate" branch and
            // the strongest measurement the charts can make was DISCARDED.
            //
            // The band is also not reachable as a coarsening of anything already
            // here. `KleinBottle` and `ProjectivePlane` are closed; the Möbius band
            // has a boundary, and its deck-invariant basis is what carries
            // width-odd structure onto half-period angular factors — the
            // non-orientable signature no other candidate in this menu can express.
            //
            // `discover_primary_atom_topologies` has raced exactly this candidate
            // all along (same basis kind, same resolution, same double-cover
            // chart). The two menus were simply out of step, so a topology the
            // PRIMARY race could name was unnameable at a BIRTH. Gated on
            // `d_seed >= 3` like the sphere and `RP²` above, because the double
            // cover reads a radial/transverse half-angle vector out of three
            // independent seed directions, and fail-open on the chart because a
            // seed that cannot carry that vector must leave the rest of the menu
            // untouched rather than fail the whole race.
            if d_seed >= 3 {
                let all_rows: Vec<usize> = (0..n).collect();
                if let Ok(mobius_coords) =
                    crate::manifold::mobius_double_cover_coords_from_projection(coords, &all_rows)
                {
                    specs.push(TopologyCandidateSpec::new(
                        AutoTopologyKind::Mobius,
                        SaeAtomGeometryPlan::new(
                            SaeAtomBasisKind::Mobius,
                            2,
                            SaeBasisResolution::MobiusHarmonics {
                                circle_order: 3,
                                width_degree: 2,
                            },
                            SaeReferenceMetricPlan::MobiusQuotient,
                        )?,
                        LatentManifold::Product(vec![
                            LatentManifold::Circle { period: 2.0 },
                            LatentManifold::Interval { lo: -1.0, hi: 1.0 },
                        ]),
                        mobius_coords,
                    )?);
                }
            }
            specs.push(TopologyCandidateSpec::new(
                AutoTopologyKind::Euclidean,
                SaeAtomGeometryPlan::new(
                    SaeAtomBasisKind::EuclideanPatch,
                    2,
                    SaeBasisResolution::Polynomial { degree: 2 },
                    SaeReferenceMetricPlan::EuclideanPolynomial,
                )?,
                LatentManifold::Euclidean,
                coords_d(2),
            )?);
            // #2604 — the estimated-curvature candidate: a monomial patch in the
            // TANGENT chart of `M_kappa`, whose penalty carries the curvature and
            // whose `kappa` the outer optimizer fits. Seeded at `kappa = 0`
            // because flat is an INTERIOR point of `S^d <- R^d -> H^d`, so the
            // fit can move either way from the seed; a log parameterisation
            // could not express that seed at all.
            //
            // It races as its OWN candidate rather than replacing the fixed
            // forms. #944's fusion would collapse `Euclidean` and `Sphere` into
            // this one, and for `Euclidean` that is exact — at `kappa = 0` this
            // IS the polynomial patch, same basis and same chart. It is NOT true
            // for `Sphere`, which this race realizes with
            // `AmbientSphereHarmonics`: a monomial patch in a tangent chart does
            // not span the `l = 2` harmonics, so fusing would swap the basis
            // rather than estimate curvature, and would LOSE expressiveness.
            // Adding the candidate is therefore strictly additive; fusing is a
            // separate claim that does not yet hold here.
            specs.push(TopologyCandidateSpec::new(
                AutoTopologyKind::ConstantCurvature,
                SaeAtomGeometryPlan::new(
                    SaeAtomBasisKind::Poincare,
                    2,
                    SaeBasisResolution::Polynomial {
                        degree: SAE_EUCLIDEAN_PATCH_MAX_DEGREE,
                    },
                    SaeReferenceMetricPlan::ConstantCurvatureChart {
                        kappa: 0.0,
                        reference_coords: coords_d(2),
                    },
                )?,
                LatentManifold::Euclidean,
                coords_d(2),
            )?);
            specs.push(TopologyCandidateSpec::new(
                AutoTopologyKind::Cylinder,
                SaeAtomGeometryPlan::new(
                    SaeAtomBasisKind::Cylinder,
                    2,
                    SaeBasisResolution::CylinderHarmonics {
                        circle_order: 2,
                        line_degree: 2,
                    },
                    SaeReferenceMetricPlan::CylinderProduct,
                )?,
                // Cylinder S¹ × ℝ: axis 0 is a unit-period circle (the
                // fraction-of-period convention `CylinderHarmonicEvaluator` shares
                // with the periodic / torus atoms), axis 1 is the unbounded flat
                // line (`Euclidean`). This MUST match the production seeding
                // (`SaeAtomBasisKind::Cylinder` → Product[Circle(1.0), Euclidean]
                // in `sae::manifold::atom`); a flat `Euclidean` manifold would leave
                // the born atom's phase axis un-wrapped, and a torus stand-in would
                // wrap the linear axis spuriously. The harmonic / degree budget
                // mirrors the torus (2 circle harmonics) and the patch (degree 2)
                // so the cross-topology design widths stay commensurable.
                LatentManifold::Product(vec![
                    LatentManifold::Circle { period: 1.0 },
                    LatentManifold::Euclidean,
                ]),
                cylinder_chart_coords(),
            )?);
        }
        _ => {
            // d_k ≥ 3: a flat Euclidean patch is the only realizable core basis
            // (the curved families top out at d = 2). The race degenerates to a
            // single candidate — still honest (the winner is reported), just not
            // a contest.
            specs.push(TopologyCandidateSpec::new(
                AutoTopologyKind::Euclidean,
                SaeAtomGeometryPlan::new(
                    SaeAtomBasisKind::EuclideanPatch,
                    d_k,
                    SaeBasisResolution::Polynomial { degree: 2 },
                    SaeReferenceMetricPlan::EuclideanPolynomial,
                )?,
                LatentManifold::Euclidean,
                coords_d(d_k),
            )?);
        }
    }
    Ok(specs)
}

/// Fit one topology candidate to the birth target `Y` (`n × p`) over `weights`
/// (`n`, the candidate's per-row reconstruction mass) by PROPER closed-form
/// Gaussian REML, and return its TK evidence inputs + the realized fit handle.
///
/// The per-atom Gaussian-reconstruction evidence is the marginal likelihood the
/// solver's [`gaussian_reml_multi_closed_form`] returns at the REML-optimal
/// smoothing strength λ̂ — the SAME REML/LAML quantity every smooth term is
/// scored by, so it is commensurable under the shared TK normalizer:
///
/// * `raw_reml` — the rank-aware closed-form REML score
///   `½d·(log|Φᵀ W Φ + λ̂S| − log|λ̂S|₊) + dispersion` at the estimated λ̂. Unlike
///   a fixed-λ, unit-dispersion Laplace term, this (a) estimates λ per candidate
///   on its own basis (SPEC: REML/LAML always, never a hand-set λ) and (b) prices
///   complexity with the penalty pseudo-determinant on a cross-basis-comparable
///   scale, so a perfect periodic fit to a circle beats a poor cubic-patch fit.
/// * `null_dim = 0` / `null_space_logdet = None` — the closed-form REML score is
///   ALREADY null-space-restricted (rank-aware), so the TK normalizer must not
///   re-subtract a gauge term; we report no null space to avoid double-counting.
/// * `effective_dim = tr[(Φᵀ W Φ + λ̂S)⁻¹ Φᵀ W Φ]` — the penalized effective
///   degrees of freedom the solver returns (`edf`), the per-effective-dim scale's
///   denominator.
///
/// This is what makes the flat-vs-rigid contest ADJUDICABLE rather than
/// merely posed: a flat and a curved candidate are different function spaces
/// (different `m`, different roughness operator `S`, different gauge group),
/// so their raw fit residuals are not comparable on their own. TK-normalized
/// REML is the common currency — the same marginal-likelihood scale every
/// smooth term in the fitter is scored on — that prices each candidate's
/// data fit against its own complexity/roughness cost, so "the circle beats
/// the line" is a real, commensurable evidence statement (not an artifact of
/// one basis happening to have fewer parameters). That evidence is exactly
/// what [`race_birth_topology`] compares across candidates to pick the
/// rigid — hence identifiable — winner.
fn fit_topology_candidate(
    spec: &TopologyCandidateSpec,
    target: ArrayView2<'_, f64>,
    weights: ArrayView1<'_, f64>,
) -> Result<TopologyAutoFitEvidence<TopologyRaceFit>, String> {
    if spec.geometry.kind() == &SaeAtomBasisKind::Torus {
        fit_torus_metric_candidate(spec, target, weights)
    } else if spec.kind == AutoTopologyKind::ConstantCurvature {
        fit_constant_curvature_metric_candidate(spec, target, weights)
    } else {
        fit_topology_candidate_at_fixed_metric(spec, target, weights)
    }
}

fn evaluate_constant_curvature_profile(
    spec: &TopologyCandidateSpec,
    phi: ArrayView2<'_, f64>,
    target: ArrayView2<'_, f64>,
    weights: ArrayView1<'_, f64>,
    kappa: f64,
) -> Result<FirstOrderSample, ObjectiveEvalError> {
    let geometry = spec
        .geometry
        .at_constant_curvature(kappa)
        .map_err(ObjectiveEvalError::fatal)?;
    let penalty = geometry
        .build_reference_penalty()
        .map_err(ObjectiveEvalError::fatal)?;
    let penalty_derivative = geometry
        .build_reference_penalty_kappa_derivative()
        .map_err(ObjectiveEvalError::fatal)?
        .ok_or_else(|| {
            ObjectiveEvalError::fatal(
                "constant-curvature profile did not materialize dS/dkappa",
            )
        })?;
    let fit = gaussian_reml_multi_shared_dispersion_closed_form(
        phi,
        target,
        penalty.view(),
        Some(weights),
        None,
    )
    .map_err(|error| ObjectiveEvalError::fatal(format!("curvature REML: {error}")))?;
    let penalty_gradient = gaussian_reml_multi_shared_dispersion_penalty_gradient_from_fit(
        phi,
        target,
        penalty.view(),
        Some(weights),
        &fit,
    )
    .map_err(|error| ObjectiveEvalError::fatal(format!("curvature REML gradient: {error}")))?;
    let gradient = penalty_gradient
        .iter()
        .zip(penalty_derivative.iter())
        .map(|(left, right)| left * right)
        .sum::<f64>();
    if !(fit.reml_score.is_finite() && gradient.is_finite()) {
        return Err(ObjectiveEvalError::fatal(
            "constant-curvature REML profile is non-finite",
        ));
    }
    Ok(FirstOrderSample {
        value: fit.reml_score,
        gradient: Array1::from_vec(vec![gradient]),
    })
}

/// Constrained one-dimensional KKT solve for the raw curvature carried by the
/// candidate's reference metric. The bracket comes from the typed geometry plan
/// (chart scale and floating-point resolution), and the derivative is the exact
/// REML penalty contraction `dV/dS : dS/dkappa`.
fn fit_constant_curvature_metric_candidate(
    spec: &TopologyCandidateSpec,
    target: ArrayView2<'_, f64>,
    weights: ArrayView1<'_, f64>,
) -> Result<TopologyAutoFitEvidence<TopologyRaceFit>, String> {
    let (lower, upper) = spec
        .geometry
        .constant_curvature_domain()?
        .ok_or_else(|| "constant-curvature candidate has no curvature metric".to_string())?;
    let evaluator = spec.geometry.build_evaluator()?;
    let (phi, _) = evaluator.evaluate(spec.coords.view())?;
    let evaluate = |kappa: f64| {
        evaluate_constant_curvature_profile(spec, phi.view(), target, weights, kappa)
    };
    let lower_sample = evaluate(lower)
        .map_err(|error| format!("constant-curvature lower-endpoint profile: {error}"))?;
    let upper_sample = evaluate(upper)
        .map_err(|error| format!("constant-curvature upper-endpoint profile: {error}"))?;
    let lower_gradient = lower_sample.gradient[0];
    let upper_gradient = upper_sample.gradient[0];
    let span = upper - lower;
    let position_tolerance = f64::EPSILON.sqrt() * span;
    let gradient_scale = lower_gradient.abs().max(upper_gradient.abs()).max(1.0);
    let gradient_tolerance = f64::EPSILON.sqrt() * gradient_scale;
    let lower_is_kkt = lower_gradient >= -gradient_tolerance;
    let upper_is_kkt = upper_gradient <= gradient_tolerance;
    let kappa = match (lower_is_kkt, upper_is_kkt) {
        (true, false) => lower,
        (false, true) => upper,
        (true, true) => {
            if lower_sample.value <= upper_sample.value {
                lower
            } else {
                upper
            }
        }
        (false, false) => {
            let config = BracketedRootConfig::new(
                position_tolerance,
                gradient_tolerance,
                f64::MANTISSA_DIGITS as usize,
            );
            find_root_bracketed(
                |candidate| {
                    if candidate == lower {
                        Ok(lower_gradient)
                    } else if candidate == upper {
                        Ok(upper_gradient)
                    } else {
                        evaluate(candidate).map(|sample| sample.gradient[0])
                    }
                },
                lower,
                upper,
                &config,
            )
            .map_err(|error| {
                format!(
                    "constant-curvature stationary solve did not converge: {error}; endpoint profile=[({lower}, value={}, gradient={lower_gradient}), ({upper}, value={}, gradient={upper_gradient})]",
                    lower_sample.value, upper_sample.value
                )
            })?
            .root
        }
    };
    if !(kappa.is_finite() && kappa >= lower && kappa <= upper) {
        return Err(format!(
            "constant-curvature optimizer returned {kappa} outside [{lower}, {upper}]"
        ));
    }
    let fitted_spec = TopologyCandidateSpec::new(
        AutoTopologyKind::ConstantCurvature,
        spec.geometry.at_constant_curvature(kappa)?,
        spec.manifold.clone(),
        spec.coords.clone(),
    )?;
    fit_topology_candidate_at_fixed_metric(&fitted_spec, target, weights)
}

#[derive(Clone, Copy, Debug)]
enum TorusMetricFamily {
    Flat,
    EmbeddedDonut,
}

fn torus_metric_penalty_and_coordinate_derivative(
    per_axis_order: usize,
    family: TorusMetricFamily,
    coordinate: f64,
) -> Result<(Array2<f64>, Array2<f64>, f64), String> {
    match family {
        // q = A^-2 makes the flat eigenvalues affine in the optimized
        // coordinate and keeps a nonzero one-sided gradient at the square
        // boundary A=1. Optimizing tau directly would have dA/dtau=0 there and
        // could falsely certify the seed as stationary.
        TorusMetricFamily::Flat => {
            if !(coordinate.is_finite() && coordinate > 0.0 && coordinate <= 1.0) {
                return Err(format!(
                    "flat torus inverse-aspect-squared coordinate must lie in (0, 1], got {coordinate}"
                ));
            }
            let aspect = coordinate.sqrt().recip();
            let penalty = anisotropic_flat_product_torus_penalty(per_axis_order, aspect)?;
            let mut derivative =
                anisotropic_flat_product_torus_penalty_aspect_derivative(per_axis_order, aspect)?;
            let aspect_derivative = -0.5 * coordinate.powf(-1.5);
            derivative.mapv_inplace(|value| value * aspect_derivative);
            Ok((penalty, derivative, aspect.acosh()))
        }
        // beta = A - sqrt(A^2-1) = exp(-tau) is the natural generating-
        // integral coordinate already used by the exact donut blocks. It maps
        // the full proper-donut domain A>1 to beta in (0,1) without overflow.
        TorusMetricFamily::EmbeddedDonut => {
            if !(coordinate.is_finite() && coordinate > 0.0 && coordinate < 1.0) {
                return Err(format!(
                    "embedded donut beta coordinate must lie in (0, 1), got {coordinate}"
                ));
            }
            let aspect = (1.0 + coordinate * coordinate) / (2.0 * coordinate);
            let penalty = embedded_donut_torus_reference_penalty(per_axis_order, aspect)?;
            let mut derivative =
                embedded_donut_torus_reference_penalty_aspect_derivative(per_axis_order, aspect)?;
            let aspect_derivative = 0.5 * (1.0 - coordinate.recip().powi(2));
            derivative.mapv_inplace(|value| value * aspect_derivative);
            Ok((penalty, derivative, -coordinate.ln()))
        }
    }
}

/// `|dA/d(coordinate)|` — the magnitude of the chain factor each family's
/// optimized coordinate carries into the reported gradient.
///
/// The reported stationarity residual is `dS/d(coordinate) = (dS/dA)·(dA/d(coordinate))`,
/// and that factor is not close to constant: it runs over 23 orders across the
/// flat family's domain `[ε, 1]` and 19 across the donut's. An absolute
/// convergence tolerance calibrated from the residual at the domain ENDS is
/// therefore unreachable wherever the factor is larger than it is there, which
/// is what exhausted the refinement on #2554 — the solve was sitting on
/// `dS/dA = −1.0e-6`, an excellent answer, and could not say so because the
/// factor at that point was `5.87e5` and inflated it to `+0.588` against a
/// tolerance of `5.1e-7`.
///
/// Dividing the residual by this magnitude measures stationarity against the
/// aspect ratio, which is what the penalty is actually built from, and leaves
/// the bracket, the bisection and the returned coordinate untouched. Taking the
/// MAGNITUDE rather than the signed factor keeps the residual's sign exactly as
/// it is today — both families' factors are strictly negative on their open
/// domains, so the sign structure the bracket certificate rests on is preserved
/// rather than flipped.
fn torus_metric_aspect_derivative_magnitude(
    family: TorusMetricFamily,
    coordinate: f64,
) -> Result<f64, String> {
    let derivative = match family {
        TorusMetricFamily::Flat => {
            if !(coordinate.is_finite() && coordinate > 0.0 && coordinate <= 1.0) {
                return Err(format!(
                    "flat torus inverse-aspect-squared coordinate must lie in (0, 1], got {coordinate}"
                ));
            }
            -0.5 * coordinate.powf(-1.5)
        }
        TorusMetricFamily::EmbeddedDonut => {
            if !(coordinate.is_finite() && coordinate > 0.0 && coordinate < 1.0) {
                return Err(format!(
                    "embedded donut beta coordinate must lie in (0, 1), got {coordinate}"
                ));
            }
            0.5 * (1.0 - coordinate.recip().powi(2))
        }
    };
    let magnitude = derivative.abs();
    if !(magnitude.is_finite() && magnitude > 0.0) {
        return Err(format!(
            "{family:?} torus aspect derivative at {coordinate} is {derivative}, which cannot \
             rescale a stationarity residual"
        ));
    }
    Ok(magnitude)
}

fn evaluate_torus_metric_profile(
    phi: ArrayView2<'_, f64>,
    target: ArrayView2<'_, f64>,
    weights: ArrayView1<'_, f64>,
    per_axis_order: usize,
    family: TorusMetricFamily,
    coordinate: f64,
) -> Result<FirstOrderSample, ObjectiveEvalError> {
    let (penalty, penalty_derivative, _) =
        torus_metric_penalty_and_coordinate_derivative(per_axis_order, family, coordinate)
            .map_err(ObjectiveEvalError::fatal)?;
    let fit = gaussian_reml_multi_shared_dispersion_closed_form(
        phi,
        target,
        penalty.view(),
        Some(weights),
        None,
    )
    .map_err(|error| ObjectiveEvalError::fatal(format!("torus metric REML: {error}")))?;
    let penalty_gradient = gaussian_reml_multi_shared_dispersion_penalty_gradient_from_fit(
        phi,
        target,
        penalty.view(),
        Some(weights),
        &fit,
    )
    .map_err(|error| ObjectiveEvalError::fatal(format!("torus metric REML gradient: {error}")))?;
    let coordinate_gradient = penalty_gradient
        .iter()
        .zip(penalty_derivative.iter())
        .map(|(left, right)| left * right)
        .sum::<f64>();
    if !coordinate_gradient.is_finite() {
        return Err(ObjectiveEvalError::fatal(
            "torus metric REML coordinate gradient is non-finite",
        ));
    }
    Ok(FirstOrderSample {
        value: fit.reml_score,
        gradient: Array1::from_vec(vec![coordinate_gradient]),
    })
}

fn optimize_torus_metric_coordinate(
    phi: ArrayView2<'_, f64>,
    target: ArrayView2<'_, f64>,
    weights: ArrayView1<'_, f64>,
    per_axis_order: usize,
    family: TorusMetricFamily,
    lower: f64,
    upper: f64,
) -> Result<f64, String> {
    if !(lower.is_finite() && upper.is_finite() && lower < upper) {
        return Err(format!(
            "torus reference-metric coordinate domain [{lower}, {upper}] is invalid"
        ));
    }
    let evaluate = |coordinate: f64| {
        evaluate_torus_metric_profile(phi, target, weights, per_axis_order, family, coordinate)
    };
    let lower_sample = evaluate(lower)
        .map_err(|error| format!("{family:?} torus lower-endpoint profile: {error}"))?;
    let upper_sample = evaluate(upper)
        .map_err(|error| format!("{family:?} torus upper-endpoint profile: {error}"))?;
    let lower_gradient = lower_sample.gradient[0];
    let upper_gradient = upper_sample.gradient[0];
    let position_tolerance = f64::EPSILON.sqrt();
    // Scale stationarity by the derivative itself, not by the absolute REML
    // score: adding a constant to an objective must not change which point is
    // considered converged.
    let gradient_scale = lower_gradient.abs().max(upper_gradient.abs()).max(1.0);
    let gradient_tolerance = position_tolerance * gradient_scale;

    // This is a scalar constrained problem, so its exact first-order KKT
    // conditions are stronger and cheaper than a multidimensional line-search
    // heuristic. At the lower wall the feasible derivative must be nonnegative;
    // at the upper wall it must be nonpositive. If neither wall satisfies KKT,
    // continuity gives the correctly oriented negative-to-positive derivative
    // bracket of an interior minimum. `find_root_bracketed` preserves that
    // certificate without finite differences or a monotonicity assumption.
    let lower_is_kkt = lower_gradient >= -gradient_tolerance;
    let upper_is_kkt = upper_gradient <= gradient_tolerance;
    let coordinate = match (lower_is_kkt, upper_is_kkt) {
        (true, false) => lower,
        (false, true) => upper,
        (true, true) => {
            if lower_sample.value <= upper_sample.value {
                lower
            } else {
                upper
            }
        }
        (false, false) => {
            // Refine on stationarity measured against the ASPECT RATIO rather
            // than against the optimized coordinate. The two differ only by the
            // chain factor `dA/d(coordinate)`, which is strictly signed and so
            // moves no root and no sign — but its MAGNITUDE spans 23 orders
            // across this domain, and an absolute tolerance taken from the
            // residual at the domain ends is unreachable wherever the factor is
            // larger than it is there. That is what exhausted this refinement on
            // #2554 rather than any ill-posedness in the problem (#2455's genus:
            // the criterion measures a quantity that does not shrink the way it
            // assumes).
            //
            // Dividing by the magnitude leaves the sign structure identical to
            // the coordinate residual, so the negative-to-positive bracket
            // certificate established above carries over unchanged.
            let aspect_residual = |candidate: f64, coordinate_gradient: f64| {
                torus_metric_aspect_derivative_magnitude(family, candidate)
                    .map(|magnitude| coordinate_gradient / magnitude)
                    .map_err(ObjectiveEvalError::fatal)
            };
            let lower_aspect = aspect_residual(lower, lower_gradient)
                .map_err(|error| format!("{family:?} torus lower-endpoint aspect residual: {error}"))?;
            let upper_aspect = aspect_residual(upper, upper_gradient)
                .map_err(|error| format!("{family:?} torus upper-endpoint aspect residual: {error}"))?;
            // The tolerance follows the residual into its new scale; reusing the
            // coordinate-scaled one would be the same mismatch in the other
            // direction.
            let aspect_scale = lower_aspect.abs().max(upper_aspect.abs()).max(1.0);
            let config = BracketedRootConfig::new(
                position_tolerance,
                position_tolerance * aspect_scale,
                f64::MANTISSA_DIGITS as usize,
            );
            find_root_bracketed(
                |candidate| {
                    let coordinate_gradient = if candidate == lower {
                        lower_gradient
                    } else if candidate == upper {
                        upper_gradient
                    } else {
                        evaluate(candidate)?.gradient[0]
                    };
                    aspect_residual(candidate, coordinate_gradient)
                },
                lower,
                upper,
                &config,
            )
            .map_err(|error| {
                format!(
                    "{family:?} torus reference-metric stationary solve did not converge: {error}; endpoint profile=[({lower}, value={}, gradient={lower_gradient}), ({upper}, value={}, gradient={upper_gradient})]",
                    lower_sample.value, upper_sample.value
                )
            })?
            .root
        }
    };
    if !(coordinate.is_finite() && coordinate >= lower && coordinate <= upper) {
        return Err(format!(
            "torus reference-metric optimizer returned invalid coordinate {coordinate} outside [{lower}, {upper}]"
        ));
    }
    Ok(coordinate)
}

fn fit_torus_metric_candidate(
    spec: &TopologyCandidateSpec,
    target: ArrayView2<'_, f64>,
    weights: ArrayView1<'_, f64>,
) -> Result<TopologyAutoFitEvidence<TopologyRaceFit>, String> {
    let SaeBasisResolution::TorusHarmonics { per_axis_order } = spec.geometry.resolution() else {
        return Err("torus candidate does not carry a torus harmonic resolution".to_string());
    };
    let evaluator = spec.geometry.build_evaluator()?;
    let (phi, _) = evaluator.evaluate(spec.coords.view())?;
    let numerical_resolution = f64::EPSILON.sqrt();

    let flat_coordinate = optimize_torus_metric_coordinate(
        phi.view(),
        target,
        weights,
        *per_axis_order,
        TorusMetricFamily::Flat,
        f64::EPSILON,
        1.0,
    )?;
    let (_, _, flat_tau) = torus_metric_penalty_and_coordinate_derivative(
        *per_axis_order,
        TorusMetricFamily::Flat,
        flat_coordinate,
    )?;
    let flat_geometry = SaeAtomGeometryPlan::new(
        SaeAtomBasisKind::Torus,
        2,
        SaeBasisResolution::TorusHarmonics {
            per_axis_order: *per_axis_order,
        },
        SaeReferenceMetricPlan::FlatRectangularTorus { tau: flat_tau },
    )?;
    let flat_spec = TopologyCandidateSpec::new(
        AutoTopologyKind::Torus,
        flat_geometry,
        spec.manifold.clone(),
        spec.coords.clone(),
    )?;
    let flat_fit = fit_topology_candidate_at_fixed_metric(&flat_spec, target, weights)?;

    let embedded_lower = numerical_resolution;
    let embedded_upper = 1.0 - numerical_resolution.sqrt();
    let embedded_coordinate = optimize_torus_metric_coordinate(
        phi.view(),
        target,
        weights,
        *per_axis_order,
        TorusMetricFamily::EmbeddedDonut,
        embedded_lower,
        embedded_upper,
    )?;
    let (_, _, embedded_tau) = torus_metric_penalty_and_coordinate_derivative(
        *per_axis_order,
        TorusMetricFamily::EmbeddedDonut,
        embedded_coordinate,
    )?;
    let embedded_geometry = SaeAtomGeometryPlan::new(
        SaeAtomBasisKind::Torus,
        2,
        SaeBasisResolution::TorusHarmonics {
            per_axis_order: *per_axis_order,
        },
        SaeReferenceMetricPlan::EmbeddedDonutTorus { tau: embedded_tau },
    )?;
    let embedded_spec = TopologyCandidateSpec::new(
        AutoTopologyKind::Torus,
        embedded_geometry,
        spec.manifold.clone(),
        spec.coords.clone(),
    )?;
    let embedded_fit = fit_topology_candidate_at_fixed_metric(&embedded_spec, target, weights)?;
    if embedded_fit.raw_reml < flat_fit.raw_reml {
        Ok(embedded_fit)
    } else {
        Ok(flat_fit)
    }
}

fn fit_topology_candidate_at_fixed_metric(
    spec: &TopologyCandidateSpec,
    target: ArrayView2<'_, f64>,
    weights: ArrayView1<'_, f64>,
) -> Result<TopologyAutoFitEvidence<TopologyRaceFit>, String> {
    let n = target.nrows();
    let bundle = spec.geometry.evaluate_bundle(spec.coords.view())?;
    let phi = bundle.basis_values;
    let jet = bundle.basis_jacobian;
    let penalty = bundle.reference_penalty;
    let evaluator = bundle.evaluator;
    if phi.nrows() != n {
        return Err(format!(
            "fit_topology_candidate: basis rows {} != target rows {n}",
            phi.nrows()
        ));
    }
    if weights.len() != n {
        return Err(format!(
            "fit_topology_candidate: weights length {} != target rows {n}",
            weights.len()
        ));
    }

    // Validate the per-row reconstruction mass and reject a degenerate
    // (zero-total-mass) birth target — the closed-form REML below needs a
    // positive weighted sample to estimate a dispersion.
    let mut w_sum = 0.0_f64;
    for row in 0..n {
        let w = weights[row];
        if !(w.is_finite() && w >= 0.0) {
            return Err("fit_topology_candidate: weights must be finite and non-negative".into());
        }
        w_sum += w;
    }
    if !(w_sum > 0.0 && w_sum.is_finite()) {
        return Err("fit_topology_candidate: degenerate (zero-mass) birth target".into());
    }

    // Score each candidate by its PROPER closed-form Gaussian REML evidence
    // (#977/#1026), NOT a hand-rolled fixed-λ Laplace term. The previous score
    // (`½·SSE + ½·log|H|` at a stamped λ = 1, unit dispersion) was wrong on two
    // counts, and they conspired to make a perfect circle lose to a line:
    //
    //   1. λ = 1 is NOT commensurable across bases. A periodic basis's curvature
    //      energy for a `cos(2πt)` harmonic scales like `(2π)⁴ ≈ 1.6e3` the data
    //      Gram, so a unit-λ penalty CRUSHES the very harmonics that reconstruct a
    //      circle exactly, while the barely-curved cubic patch is left essentially
    //      unpenalized. SPEC also forbids a hand-set smoothing strength — λ must be
    //      REML/LAML-estimated, ALWAYS.
    //   2. Unit dispersion (`½·SSE`) does not reward a near-perfect fit. The
    //      profiled-dispersion REML deviance `½ν·log(σ̂²)` rewards a basis that
    //      drives σ̂² → 0 (the circle's exact harmonic fit) far more strongly,
    //      which is what makes the contest honest, and `½·log|H| − ½·log|λS|₊`
    //      prices complexity on a scale that is comparable across bases (the
    //      penalty pseudo-determinant the old score dropped is what restores
    //      commensurability).
    //
    // `gaussian_reml_multi_closed_form` returns exactly this: the marginal-
    // likelihood-optimal λ̂ for each candidate on its OWN basis, the penalized
    // decoder at λ̂, the rank-aware REML score (`½d·(log|H| − log|λS|₊) +
    // dispersion`, already null-space-restricted), and the effective degrees of
    // freedom `tr[(ΦᵀWΦ + λS)⁻¹ ΦᵀWΦ]`. We feed `reml_score` as `raw_reml` and
    // `edf` as `effective_dim`, and report `null_dim = 0` so the TK normalizer does
    // NOT re-subtract a null-space term the REML score already integrated out.
    // The ambient columns are coordinates of ONE vector-valued observation, so
    // topology evidence must profile one shared dispersion.  Profiling a
    // separate variance per output lets a PCA chart win tautologically: the PC
    // axes are exact linear functions of themselves (zero residual and
    // effectively -infinite evidence) even when the chart entirely misses an
    // orthogonal manifold direction.  Pooling the vector deviance before its
    // logarithm makes the race pay for every missed ambient direction while
    // retaining the same closed-form REML, posterior-mean coefficients, and
    // grid-free smoothing-parameter optimization.
    let reml_fit = gaussian_reml_multi_shared_dispersion_closed_form(
        phi.view(),
        target,
        penalty.view(),
        Some(weights),
        None,
    )
    .map_err(|e| format!("fit_topology_candidate: REML evidence: {e:?}"))?;
    let lambda = reml_fit.lambda;
    if !(lambda.is_finite() && lambda >= 0.0) {
        return Err(format!(
            "fit_topology_candidate: REML returned a non-finite/negative λ ({lambda})"
        ));
    }
    let raw_reml = reml_fit.reml_score;
    if !raw_reml.is_finite() {
        return Err("fit_topology_candidate: non-finite REML score".into());
    }
    let decoder = reml_fit.coefficients.clone(); // penalized fit at λ̂, m × p
    let mut effective_dim = reml_fit.edf;
    if !(effective_dim.is_finite() && effective_dim > 0.0) {
        // A fully-penalized fit (no effective parameters) cannot be scored on the
        // per-effective-dim scale; floor at a single effective parameter so the
        // race still ranks it (the REML deviance dominates the verdict anyway).
        effective_dim = 1.0;
    }

    Ok(TopologyAutoFitEvidence {
        topology_name: spec.kind.display_name(),
        raw_reml,
        // #2729 — the score's own resolution, accumulated by the closed-form
        // evaluator that built it (log-determinant magnitudes differenced, and
        // the cancellation that formed the profiled deviance), NOT retro-fitted
        // here. Carried so the race can tell a decision from arithmetic debris.
        raw_reml_roundoff: reml_fit.reml_score_roundoff,
        // The closed-form REML score is ALREADY restricted to the penalty's range
        // complement (rank-aware: `log|λS|₊` over the non-null directions, the null
        // space integrated out), so the TK null-space normalizer must NOT fire
        // again — pass `null_dim = 0` (its `null_space_logdet` branch is then
        // skipped) so we don't double-count the gauge directions.
        null_dim: 0.0,
        null_space_logdet: None,
        effective_dim,
        n_obs: n,
        fit_handle: TopologyRaceFit {
            evaluator,
            geometry: spec.geometry.clone(),
            manifold: spec.manifold.clone(),
            coords: spec.coords.clone(),
            phi,
            jet,
            decoder,
            penalty,
        },
    })
}

/// Race the candidate topologies whose required intrinsic dimension matches the
/// born atom's `d_k` against the birth target `Y` (`n × p`, the residual-factor
/// image the atom would reconstruct) over the template coordinates `coords`, and
/// return the EVIDENCE-WINNING fit. The winner is the lowest TK-normalized REML
/// via [`select_topology_with_fit`] — the gauge-invariant comparison the
/// smooth-term topology race already applies — so a circular residual gets a
/// circle, a straight residual a line, a spherical residual a sphere, etc.
///
/// Returns `None` when the race has no realizable candidate (`d_k = 0`, the
/// cluster-null rung, handled below the race) or the birth target is degenerate;
/// the caller then falls back to the template basis (warm inheritance) and the
/// post-fit curved-vs-linear rung adjudicates as before.
///
/// This function IS the flatness cure at runtime: a `d_k`-dimensional
/// co-firing residual subspace is, prior to this call, exactly the kind of
/// flat structure that admits an unbounded gauge group of equally-good linear
/// recombinations (see the module docs). Racing the realizable candidates by
/// [`fit_topology_candidate`]'s commensurable evidence and keeping only the
/// winner replaces that flat ambiguity with ONE specific, generically rigid
/// geometry (or, if nothing curved earns its keep, the flat/line candidate —
/// an honest verdict, not a default). The winner returned here is what
/// [`born_atom`] seeds the new atom from directly, so the identifiability
/// gain is realized in the dictionary rather than merely reported.
/// The realized per-row amplitude of a birth target: the L2 norm of each row of
/// `Y` (`n × p`), i.e. the magnitude the born atom would reconstruct at that
/// sample. The amplitude-concentration certificate reads this to tell a
/// present/absent spike (binary presence) from a continuous spread (a radial
/// coordinate).
fn birth_row_amplitudes(target: ArrayView2<'_, f64>) -> Array1<f64> {
    let n = target.nrows();
    let mut amps = Array1::<f64>::zeros(n);
    for i in 0..n {
        let mut ss = 0.0_f64;
        for &v in target.row(i).iter() {
            ss += v * v;
        }
        amps[i] = ss.sqrt();
    }
    amps
}

fn standardized_log_birth_amplitudes(amps: ArrayView1<'_, f64>) -> Option<Array1<f64>> {
    let n = amps.len();
    if n == 0 {
        return None;
    }
    let mut logs = Array1::<f64>::zeros(n);
    for (i, &amp) in amps.iter().enumerate() {
        if !amp.is_finite() || amp < 0.0 {
            return None;
        }
        logs[i] = amp.max(f64::MIN_POSITIVE).ln();
    }
    let mean = logs.sum() / n as f64;
    let mut var = 0.0_f64;
    for &value in logs.iter() {
        let centered = value - mean;
        var += centered * centered;
    }
    let std = (var / n as f64).sqrt();
    if !std.is_finite() || std <= 0.0 {
        return None;
    }
    for value in logs.iter_mut() {
        *value = (*value - mean) / std;
    }
    Some(logs)
}

/// When a `d = 1` birth's realized amplitude is CONTINUOUS (a hidden radial axis,
/// per the amplitude-concentration certificate), build the promoted
/// circle-vs-cylinder(radial)-vs-disk candidate set so the race adjudicates the
/// extra radial dimension by evidence. Returns `None` when no promotion applies
/// (not `d = 1`, or the amplitude is a genuine present/absent spike), so the
/// caller keeps the base race. The promoted set uses DISTINCT topology kinds
/// (`Circle` d=1, `Cylinder` d=2, `Euclidean` d=2 = the flat disk) so the
/// by-kind race map has no collision — the `d = 1` line is intentionally dropped,
/// because if the amplitude is radial the contest is circle vs the radial
/// two-manifolds, not circle vs line.
fn radial_promoted_specs(
    coords: ArrayView2<'_, f64>,
    target: ArrayView2<'_, f64>,
    d_k: usize,
) -> Result<Option<Vec<TopologyCandidateSpec>>, String> {
    if d_k != 1 {
        return Ok(None);
    }
    let amps = birth_row_amplitudes(target);
    let cert = amplitude_concentration_certificate(amps.view());
    if !cert.recommends_radial_axis() {
        return Ok(None);
    }
    let log_amp_coord = standardized_log_birth_amplitudes(amps.view())
        .ok_or_else(|| "radial_promoted_specs: degenerate log-amplitude spread".to_string())?;
    // The circle from the d=1 set (the un-promoted alternative the evidence must
    // still be free to prefer) plus the d=2 radial two-manifolds.
    let mut promoted: Vec<TopologyCandidateSpec> = Vec::with_capacity(3);
    for spec in topology_candidates_for_dim(CandidateBases::with_ambient(coords, target), 1)? {
        if spec.kind == AutoTopologyKind::Circle {
            promoted.push(spec);
        }
    }
    for mut spec in topology_candidates_for_dim(CandidateBases::with_ambient(coords, target), 2)? {
        if matches!(
            spec.kind,
            AutoTopologyKind::Cylinder | AutoTopologyKind::Euclidean
        ) {
            for row in 0..spec.coords.nrows() {
                spec.coords[[row, 1]] = log_amp_coord[row];
            }
            promoted.push(spec);
        }
    }
    if promoted.len() < 2 {
        // Need at least the circle plus one radial candidate to make a contest.
        return Ok(None);
    }
    Ok(Some(promoted))
}

/// F2 finite-set-atom opt-in. Default `false`: the birth race does NOT enrol a
/// finite-set (discrete anchor) candidate, so the [`SaeAtomBasisKind::FiniteSet`]
/// variant + [`AnchorIndicatorEvaluator`] land as inert scaffolding that cannot
/// affect any birth. The switch flips to `true` only AFTER the finite-set atom is
/// verified — full `gam-sae` suite green plus the real-data weekday adjudication
/// (is weekday seven cyclic points or an occupied circle?). Enrolling the
/// candidate in the actual race additionally needs an `AutoTopologyKind::FiniteSet`
/// in `gam-solve`'s selector (the cross-crate follow-up); until then the flag +
/// [`finite_set_candidate_for_birth`] are the staged, unit-tested substrate.
static FINITE_SET_RACE_ENROLLED: AtomicBool = AtomicBool::new(false);

/// Whether the birth race enrols the finite-set (discrete anchor) candidate.
/// Default `false` — see `FINITE_SET_RACE_ENROLLED`.
pub fn finite_set_race_enrolled() -> bool {
    FINITE_SET_RACE_ENROLLED.load(Ordering::Relaxed)
}

/// Flip the finite-set-atom enrolment opt-in. Intended for the post-verification
/// enablement (and for tests exercising the enrolled path); default is `false`.
pub fn set_finite_set_race_enrolled(enrolled: bool) {
    FINITE_SET_RACE_ENROLLED.store(enrolled, Ordering::Relaxed);
}

/// Build the finite-set (discrete anchor) candidate inputs for a `d = 1` birth
/// whose occupancy is DISCRETE — the honest "seven cyclic points, not an occupied
/// circle" alternative. Returns `(anchors, index_coords)` where `index_coords`
/// (`n × 1`) assigns each row to its nearest of `anchors` anchors (the integer
/// index the `AnchorIndicatorEvaluator` reads), and `anchors − 1` is the rank
/// charge (`finite_set_rank_charge`). Returns `None` when the birth is not a
/// discrete finite set (uniform / continuous occupancy, wrong dimension, or a
/// degenerate coordinate) — so it never fabricates a cluster structure.
///
/// This is the pure, unit-tested substrate the race enrolment consumes once
/// [`finite_set_race_enrolled`] flips; it does not itself touch any birth.
pub fn finite_set_candidate_for_birth(coords: ArrayView2<'_, f64>) -> Option<(usize, Array2<f64>)> {
    if coords.ncols() != 1 {
        return None;
    }
    let n = coords.nrows();
    if n < 4 {
        return None;
    }
    let col = coords.column(0);
    let (mut lo, mut hi) = (f64::INFINITY, f64::NEG_INFINITY);
    for &t in col.iter() {
        if !t.is_finite() {
            return None;
        }
        lo = lo.min(t);
        hi = hi.max(t);
    }
    let span = hi - lo;
    if !(span > 0.0) {
        return None;
    }
    // Range-normalize the single coordinate column to [0, 1] and classify on the
    // INTERVAL (non-wrapping) occupancy law: a birth coordinate is
    // interval-topology (linear, from the PCA seed), so its extreme values must
    // NOT wrap onto each other — the circular classifier would fold `0` and `1`
    // together and merge a linear set's first and last anchors (7 weekday points
    // → 6), and its full-circle uniform model would misread a range-filling
    // uniform coordinate as non-uniform. The interval classifier keeps linear
    // ends distinct and range-uniform data uniform.
    let r: Vec<f64> = col
        .iter()
        .map(|&t| ((t - lo) / span).clamp(0.0, 1.0))
        .collect();
    match classify_occupancy_interval(&r) {
        OccupancyLaw::Discrete { anchors } if anchors >= 2 => {
            // Assign each row to its nearest anchor bin from the normalization
            // `r ∈ [0, 1]`: the `anchors` evenly spaced bins give the categorical
            // index the indicator basis reads. A fitted-anchor-position
            // assignment is the cross-crate refinement (it needs the classifier's
            // centers exposed).
            let mut idx = Array2::<f64>::zeros((n, 1));
            for i in 0..n {
                let bin = (r[i] * anchors as f64).floor();
                idx[[i, 0]] = bin.clamp(0.0, (anchors - 1) as f64);
            }
            Some((anchors, idx))
        }
        _ => None,
    }
}

/// A graph birth candidate enrolled in structure search.
///
/// The candidate edge set is the derived anchor-kNN graph; REML per-edge losses
/// decide survival, and the selection currency is the SUM of surviving one-edge
/// charges. Named shapes are only certified compressions of the learned graph.
#[derive(Clone, Debug)]
pub struct GraphBirthCandidate {
    pub atom: LearnedGraphAtom,
    pub selection: GraphStructureSelection,
}

/// Build the graph-atom birth candidate that the structure search scores. This
/// is the graph counterpart to the fixed topology menu: the caller supplies
/// REML per-edge deletion losses for the kNN edge set, and selection is paid in
/// summed edge charge rather than by promoting to a circle/line first.
pub fn graph_birth_candidate_for_structure_search(
    anchor_embeddings: ArrayView2<'_, f64>,
    row_coordinates: &[f64],
    n_eff: f64,
    edge_precisions: &[f64],
    edge_delta_loss: &[f64],
) -> Result<GraphBirthCandidate, String> {
    let atom = LearnedGraphAtom::from_reml_knn_edges(
        anchor_embeddings,
        row_coordinates,
        n_eff,
        edge_precisions,
        edge_delta_loss,
    )?;
    let selection = atom.structure_selection();
    Ok(GraphBirthCandidate { atom, selection })
}

/// #2280 — build the local-chart atlas on a birth's ambient residual image and read
/// the topology its charts and transition holonomy determine, as a PROPOSAL PRIOR.
///
/// Fail-open: any build refusal (coverage below the floor, degenerate charts,
/// non-finite rows) or a too-small image returns `None` and the race runs UNPRIMED
/// exactly as today — an atlas that cannot certify itself never blocks a birth or
/// changes a verdict. A readout that names no topology is likewise inert; it is
/// still logged, because a refusal with its invariants is the diagnostic that says
/// WHY the atlas could not help.
///
/// `intrinsic_dim` is the birth's own `d`, not a fixed 2: the classification
/// dispatches on chart rank first (a circle and a cylinder have identical nerve
/// invariants), so passing the wrong `d` would ask the table the wrong question.
fn atlas_prior_for_coords(
    target: ArrayView2<'_, f64>,
    intrinsic_dim: usize,
) -> Option<AtlasTopologyReadout> {
    let (n, p) = target.dim();
    // The atlas needs enough rows to seed several overlapping charts and close a
    // transition cocycle; below that it cannot corroborate anything and abstains.
    if n < 6 || p == 0 || intrinsic_dim == 0 {
        return None;
    }
    let intrinsic_dim = intrinsic_dim.min(p);
    let config = crate::manifold::LocalAtlasConfig::balanced(n, intrinsic_dim);
    let atlas = crate::manifold::LocalAtlas::build(target, config).ok()?;
    let dropped = atlas.rejected_centers();
    if !dropped.is_empty() {
        // Surface the primitive-level rejection in the fit log. The topology race
        // runs in the move-APPLICATION phase (`born_atom`), not the harvest phase
        // that owns `HarvestReport`, so the debug log — the channel the #2233
        // birth pre-screen already reports through — is the additive diagnostic
        // surface here; each `RejectedCenter` is `Display`-legible.
        log::debug!(
            "#2280 atlas dropped {} uncertifiable center(s) on a birth residual: {}",
            dropped.len(),
            dropped
                .iter()
                .map(|rejected| rejected.to_string())
                .collect::<Vec<_>>()
                .join("; ")
        );
    }
    let readout = crate::manifold::observe_atlas_topology(&atlas).ok()?;
    log::debug!("#2280 {readout}");
    Some(readout)
}

/// #2280 — the topology candidate a recognized manifold corresponds to, or `None`
/// when the `d ≤ 2` menu realizes no basis for it.
///
/// The mapping is on the typed [`AutoTopologyKind`], never on a display string, so
/// it cannot drift. `Disk`/`Interval` both map to the flat `Euclidean` patch — the
/// menu's contractible candidate at either rank — and the purely combinatorial
/// kinds (`FiniteSet`, `Graph`) name no smooth candidate.
fn observed_kind_to_auto_topology(kind: GraphCompressionKind) -> Option<AutoTopologyKind> {
    match kind {
        GraphCompressionKind::Circle => Some(AutoTopologyKind::Circle),
        GraphCompressionKind::Interval | GraphCompressionKind::Disk => {
            Some(AutoTopologyKind::Euclidean)
        }
        GraphCompressionKind::Cylinder => Some(AutoTopologyKind::Cylinder),
        GraphCompressionKind::MobiusStrip => Some(AutoTopologyKind::Mobius),
        GraphCompressionKind::Torus => Some(AutoTopologyKind::Torus),
        GraphCompressionKind::Sphere => Some(AutoTopologyKind::Sphere),
        GraphCompressionKind::ProjectivePlane => Some(AutoTopologyKind::ProjectivePlane),
        GraphCompressionKind::KleinBottle => Some(AutoTopologyKind::KleinBottle),
        GraphCompressionKind::FiniteSet | GraphCompressionKind::Graph => None,
    }
}

/// #2280 — a topology candidate is NON-orientable iff it is one of the twisted
/// forms the `d = 2` menu can realize (Klein bottle, projective plane, Möbius
/// band). Read from the realized candidate's `AutoTopologyKind`, so no gam-solve
/// API change is needed and the classification cannot drift on a display string.
fn kind_is_non_orientable(kind: AutoTopologyKind) -> bool {
    matches!(
        kind,
        AutoTopologyKind::KleinBottle
            | AutoTopologyKind::ProjectivePlane
            | AutoTopologyKind::Mobius
    )
}

/// #2280 — apply the atlas's topology readout as a proposal-time menu REORDER
/// (never a winner override).
///
/// The atlas MEASURES a manifold; the menu ENUMERATES hypotheses. When the two
/// meet, the measured one leads:
///
/// 1. If the readout names a manifold the menu realizes, that candidate is
///    stable-floated to the head.
/// 2. Otherwise, if the readout names a NON-ORIENTABLE manifold the menu cannot
///    realize at this `d` (the birth menu carries no Möbius band), the twisted
///    candidates lead as a block — the coarser statement the observation still
///    supports.
/// 3. Otherwise the menu is byte-identical.
///
/// Every candidate survives and relative order is preserved, so the REML race is
/// unchanged in MEMBERSHIP: the shared priority selector breaks an EXACT `tk_score`
/// tie by menu position (`original_index`), so the reorder promotes the measured
/// topology only where the evidence is otherwise indifferent, and never drops the
/// eventual winner (fail-open, unchanged-or-better by construction).
///
/// One-directional by construction. A readout that refuses, or that names an
/// orientable manifold absent from the menu, leaves the order alone;
/// `observed_orientability` returns `Orientable` vacuously on a sparse edge set, so
/// an orientable reading is the ABSENCE of evidence and must never veto the twisted
/// forms.
fn atlas_reorder_specs(
    specs: Vec<TopologyCandidateSpec>,
    atlas: Option<&AtlasTopologyReadout>,
) -> Vec<TopologyCandidateSpec> {
    let Some(atlas) = atlas else {
        return specs;
    };
    let observed = atlas.observed_manifold();
    let named = observed.and_then(observed_kind_to_auto_topology);
    if let Some(named) = named {
        if specs.iter().any(|spec| spec.kind == named) {
            log::debug!(
                "#2280 atlas topology prior: the charts and their transition holonomy measure \
                 {named:?}; floating it ahead of the menu so the REML race breaks an exact tie \
                 toward the measured manifold"
            );
            let mut leading: Vec<TopologyCandidateSpec> = Vec::with_capacity(specs.len());
            let mut rest: Vec<TopologyCandidateSpec> = Vec::new();
            for spec in specs {
                if spec.kind == named {
                    leading.push(spec);
                } else {
                    rest.push(spec);
                }
            }
            leading.extend(rest);
            return leading;
        }
    }
    if !atlas.observes_non_orientable() {
        return specs;
    }
    if !specs.iter().any(|spec| kind_is_non_orientable(spec.kind)) {
        log::debug!(
            "#2280 atlas topology prior: measured a non-orientable manifold, but this menu \
             realizes no twisted candidate; menu unchanged"
        );
        return specs;
    }
    log::debug!(
        "#2280 atlas topology prior: measured a non-orientable manifold the menu cannot realize \
         exactly; floating the twisted candidate(s) ahead of the orientable menu"
    );
    let mut non_orientable: Vec<TopologyCandidateSpec> = Vec::with_capacity(specs.len());
    let mut orientable: Vec<TopologyCandidateSpec> = Vec::new();
    for spec in specs {
        if kind_is_non_orientable(spec.kind) {
            non_orientable.push(spec);
        } else {
            orientable.push(spec);
        }
    }
    non_orientable.extend(orientable);
    non_orientable
}

fn race_birth_topology(
    coords: ArrayView2<'_, f64>,
    target: ArrayView2<'_, f64>,
    weights: ArrayView1<'_, f64>,
    d_k: usize,
) -> Result<Option<TopologyRaceFit>, String> {
    // #2280 — proposal-time atlas prior (recognition-only, fail-open). The atlas is
    // built at the birth's OWN chart rank `d_k`, because the classification
    // dispatches on rank before it reads the invariant table (a circle and a
    // cylinder are homotopy equivalent, so no nerve invariant separates them). That
    // makes the prior meaningful at d = 1 too — the line-vs-circle menu is exactly
    // the one-manifold table — where the previous orientability-only prior had
    // nothing to say. The prior only ever REORDERS the candidate menu; the REML
    // race — still the sole arbiter — breaks an exact evidence tie toward the
    // measured manifold. It never selects a winner and never drops a candidate.
    let atlas = atlas_prior_for_coords(target, d_k);
    // The PCA/template-coordinate race is the cheaper DEFAULT: the born atom
    // inherits the template atom's coordinate block, and the topology candidates
    // are adjudicated on those linear-seed coordinates.
    let template_winner = race_template_coords(coords, target, weights, d_k, atlas.as_ref())?;
    // Intrinsic-metric CHALLENGER (#2240/#2280): on a FOLDED residual (a swiss
    // roll, a creased sheet) the template coordinates are self-overlapping in the
    // ambient metric, so every topology candidate fits a crumpled image. Re-race
    // the SAME candidate set on the geodesic (Isomap) embedding of the birth
    // image, which unrolls the fold. The challenger enters under the IDENTICAL
    // REML evidence and wins only when it scores strictly better; on a non-fold it
    // ties the linear seed and the default (template) is kept. Fail-safe: any
    // embedding/race failure leaves the template winner untouched.
    //
    // GATED to a FLAT template verdict (EuclideanPatch): a genuinely curved born
    // atom (circle/torus/sphere/cylinder) already wins its specialized chart on the
    // template coords, and re-racing on the geodesic embedding must not let a
    // flexible flat/patch fit override that true topology. Only a flat verdict — the
    // least-bad chart for a folded plane — can be a fold worth unrolling.
    let template_is_sheet = matches!(
        template_winner
            .as_ref()
            .map(|outcome| outcome.fit.geometry.kind()),
        Some(SaeAtomBasisKind::EuclideanPatch)
    );
    let intrinsic_winner = if template_is_sheet {
        race_intrinsic_coords(target, weights, d_k, atlas.as_ref()).unwrap_or(None)
    } else {
        None
    };
    // Lower TK/REML cost wins (issue #396 sign convention); the template keeps
    // ties, so PCA stays default and intrinsic only supplants it by evidence.
    let winner = match (template_winner, intrinsic_winner) {
        (Some(template), Some(intrinsic)) => {
            if intrinsic.tk_score < template.tk_score {
                Some(intrinsic)
            } else {
                Some(template)
            }
        }
        (Some(template), None) => Some(template),
        (None, Some(intrinsic)) => Some(intrinsic),
        (None, None) => None,
    };
    Ok(winner.map(|outcome| outcome.fit))
}

/// The PCA/template-coordinate topology race: the historical born-atom path,
/// returning the winning fit AND its TK-normalized evidence so the intrinsic
/// challenger in [`race_birth_topology`] can be compared on the same scale.
fn race_template_coords(
    coords: ArrayView2<'_, f64>,
    target: ArrayView2<'_, f64>,
    weights: ArrayView1<'_, f64>,
    d_k: usize,
    atlas: Option<&AtlasTopologyReadout>,
) -> Result<Option<TopologyRaceOutcome>, String> {
    let base_specs = topology_candidates_for_dim(CandidateBases::with_ambient(coords, target), d_k)?;
    if base_specs.is_empty() {
        return Ok(None);
    }
    // F1 radial promotion: a `d = 1` birth whose realized amplitude is CONTINUOUS
    // (a hidden radial coordinate — the amplitude-concentration certificate reads
    // the per-row birth magnitudes as a spread, not a present/absent spike) is
    // really a disk / annulus, not a circle. When the certificate recommends it,
    // ENRICH the race with the `d = 2` radial candidates so the evidence
    // adjudicates circle-vs-cylinder(radial)-vs-disk rather than the amplitude
    // silently riding an uncertified quantity. Strictly additive and fail-safe:
    // any failure building the promoted set falls back to the base race, and the
    // gate only fires on a genuine continuous-amplitude signal, so the common path
    // is unchanged.
    if let Ok(Some(promoted)) = radial_promoted_specs(coords, target, d_k) {
        if !promoted.is_empty() {
            // Try the promoted circle-vs-cylinder-vs-disk race; on ANY failure
            // (a degenerate d=2 fit, an empty ranking) fall back to the base race
            // so a radial-flagged birth never regresses relative to the un-promoted
            // path — the promotion can only ever ADD adjudicated candidates.
            if let Ok(Some(fit)) = race_spec_set(promoted, target, weights, atlas) {
                return Ok(Some(fit));
            }
        }
    }
    race_spec_set(base_specs, target, weights, atlas)
}

/// The intrinsic-metric challenger race: embed the birth image `target`
/// (`n × p`) into `d_k` dimensions by classical Landmark-Isomap (geodesic MDS),
/// min-max normalize each geodesic axis to the flat `[-0.5, 0.5]` convention the
/// template coordinates use (so the Euclidean-patch design is scale-commensurable
/// with the template race), and race the SAME topology candidate set on those
/// unfolded coordinates. Returns the winning fit and its TK evidence, or `None`
/// when the embedding is degenerate or no candidate is realizable.
fn race_intrinsic_coords(
    target: ArrayView2<'_, f64>,
    weights: ArrayView1<'_, f64>,
    d_k: usize,
    atlas: Option<&AtlasTopologyReadout>,
) -> Result<Option<TopologyRaceOutcome>, String> {
    // Folds are a d ≥ 2 story: a 1-D manifold has no ambient fold a geodesic
    // embedding could unroll that a line/circle basis does not already capture,
    // and the geodesic 1-D embedding of a closed loop is degenerate. Restricting
    // the challenger to d ≥ 2 also leaves the d = 1 circle-vs-line race untouched.
    if d_k < 2 || target.nrows() < 3 {
        return Ok(None);
    }
    let embed = crate::manifold::intrinsic_geodesic_embedding(target, d_k)?;
    let n = embed.nrows();
    let d = embed.ncols();
    if n == 0 || d == 0 {
        return Ok(None);
    }
    // Per-axis min-max to [-0.5, 0.5]; a collapsed axis (zero span) means the
    // geodesic embedding found no intrinsic spread there — the challenger is not
    // realizable, so bail and keep the template winner.
    let mut coords = Array2::<f64>::zeros((n, d));
    for col in 0..d {
        let (lo, hi) = (0..n).fold((f64::INFINITY, f64::NEG_INFINITY), |(lo, hi), r| {
            let v = embed[[r, col]];
            (lo.min(v), hi.max(v))
        });
        let span = hi - lo;
        if !(span > 0.0) || !span.is_finite() {
            return Ok(None);
        }
        for r in 0..n {
            coords[[r, col]] = (embed[[r, col]] - lo) / span - 0.5;
        }
    }
    let specs = topology_candidates_for_dim(CandidateBases::with_ambient(coords.view(), target), d_k)?;
    if specs.is_empty() {
        return Ok(None);
    }
    race_spec_set(specs, target, weights, atlas)
}

/// Race one realized candidate spec set against the birth target and return the
/// evidence-winning fit. Shared by the base and the F1 radial-promoted races.
/// #2280 — the topology race's full verdict, not just its winner.
///
/// The race has always known how it ranked every candidate; it used to discard
/// that and return the winner alone. Keeping the losers is what makes the atlas
/// prior MEASURABLE: "the charts measured a Möbius band and the evidence put it
/// second by 0.004 TK" is a calibration datum, while "the race picked a cylinder"
/// is not. The ranking is carried as typed [`AutoTopologyKind`]s, best (lowest
/// `tk_score`) first, so no consumer has to re-parse a display string.
struct TopologyRaceOutcome {
    /// Every candidate that produced selectable evidence, best first.
    ranking: Vec<RankedTopology>,
    fit: TopologyRaceFit,
    tk_score: f64,
}

/// One ranked candidate: its kind, its evidence score, and the RESOLUTION of
/// that score (#2729).
///
/// Score and resolution travel together for the same reason a measurement and
/// its error bar do: a consumer holding only the score cannot tell a margin
/// from the roundoff of the arithmetic that produced it, and will report the
/// latter as the former.
#[derive(Clone, Copy, Debug)]
struct RankedTopology {
    kind: AutoTopologyKind,
    tk_score: f64,
    /// Forward-error bound on `tk_score`; `None` when its producer established
    /// none, which forbids certifying ANY margin involving this candidate.
    tk_score_resolution: Option<f64>,
}

impl RankedTopology {
    /// Is this candidate resolvably better than `other`? The band is the SUM of
    /// the two resolutions — the error of a difference is the sum of the errors
    /// of what is differenced — so no tolerance is chosen anywhere.
    fn is_resolvably_better_than(&self, other: &Self) -> bool {
        let (Some(mine), Some(theirs)) = (self.tk_score_resolution, other.tk_score_resolution)
        else {
            return false;
        };
        let band = mine + theirs;
        let margin = other.tk_score - self.tk_score;
        band.is_finite() && margin.is_finite() && margin > band
    }
}

fn race_spec_set(
    specs: Vec<TopologyCandidateSpec>,
    target: ArrayView2<'_, f64>,
    weights: ArrayView1<'_, f64>,
    atlas: Option<&AtlasTopologyReadout>,
) -> Result<Option<TopologyRaceOutcome>, String> {
    if specs.is_empty() {
        return Ok(None);
    }
    // #2280 — proposal-time atlas prior: REORDER the fixed menu toward an observed
    // non-orientable form (never a winner override, never a drop). Identity when
    // the atlas is absent, orientable, or its cocycle is untrustworthy, so the
    // common path is byte-identical to today; the full reordered menu still races.
    let specs = atlas_reorder_specs(specs, atlas);
    let selector = TopologyAutoSelector {
        // The race is over EXACTLY the candidate set we built; do not let the
        // selector's constant-curvature fuse drop one — pass them through as-is.
        candidates: specs.iter().map(|s| s.kind).collect(),
        // κ IS estimable here (#2604): the `ConstantCurvature` candidate above
        // carries a fitted `kappa` in `rho`, and the outer gradient carries
        // `dH/dkappa_a = lambda_a * dS_a/dkappa`, analytic and FD-matched. The
        // old reason for `false` — "this race fits no κ" — is dead, so the
        // capability is now declared truthfully...
        curvature_is_estimable: true,
        // ...but only `Euclidean` is declared SUBSUMED. The fitted-κ atom is a
        // monomial patch in the tangent chart, so at κ = 0 it IS the Euclidean
        // patch — same basis, same chart — and racing both is racing the same
        // model twice. It is not the ambient-harmonic `Sphere` at any κ, so the
        // sphere stays in the race as itself.
        curvature_fusion_subsumes: &[AutoTopologyKind::Euclidean],
        // PER-OBSERVATION normalization (a common `n` divisor across candidates).
        // The candidate scores are now PROPER closed-form REML marginal
        // likelihoods (see `fit_topology_candidate`), which ALREADY price model
        // complexity through `log|H| − log|λS|₊` + the profiled dispersion. The
        // older `PerEffectiveDim` scale was calibrated for the previous hand-rolled
        // POSITIVE cost (`½·SSE + ½·log|H|`, which grew with model size and needed
        // per-parameter normalization); applied to a proper (negative) evidence it
        // DOUBLE-COUNTS complexity and inverts the ranking for higher-parameter
        // bases — e.g. a cylinder that fits a cylindrical residual best (most
        // negative evidence) would lose to a sphere purely because it spends more
        // effective dimensions. A common-`n` divisor preserves the raw
        // marginal-likelihood ranking the Bayesian evidence is designed to support,
        // so the genuinely-best-fitting topology wins.
        score_scale: TopologyScoreScale::PerObservation,
    };
    // Index the realized specs by kind so the fit closure can find the right
    // evaluator/coords for the kind the selector hands it.
    //
    // #944 stage 4 would FUSE the fixed simply-connected constant-curvature forms
    // (Euclidean κ = 0 ∪ Sphere κ > 0) into ONE estimated-κ `ConstantCurvature`
    // candidate. The premise is right for `Euclidean` — flat-vs-curved IS a
    // curvature estimation rather than two discrete topologies — and this race
    // now DOES fit κ, so the fusion fires and the redundant flat patch is gone.
    //
    // It fires for `Euclidean` ONLY. The fusion deletes what it fuses, and
    // deletion is sound only where the fitted-κ atom spans what it replaces: it
    // spans the flat patch (κ = 0 is the same monomial patch in the same chart)
    // and does not span the AMBIENT-HARMONIC sphere at any κ. That is why the
    // selector declares `curvature_fusion_subsumes: &[Euclidean]` rather than
    // relying on `is_fixed_constant_curvature_form`, which answers a question
    // about GEOMETRY when the fusion needs one about SPAN.
    //
    // Realizing the fused candidate by the sphere basis (as this used to do) does
    // not rescue the premise either. A unit-curvature sphere cannot express κ = 0,
    // so the "estimated" curvature was pinned at a constant the fit never chose.
    let mut by_kind: std::collections::HashMap<AutoTopologyKind, &TopologyCandidateSpec> =
        std::collections::HashMap::with_capacity(specs.len() + 1);
    for spec in &specs {
        by_kind.insert(spec.kind, spec);
    }
    // A ConstantCurvature key is still deliberately NOT synthesized here — and no
    // longer needs to be. It used to be ALIASED to the sphere spec (or the flat
    // patch), so a race could report a `ConstantCurvature` winner that was a
    // FIXED-curvature fit: a κ nobody estimated. The key now arrives the only
    // honest way, as a real entry in `specs` above whose plan carries a fitted
    // `kappa`, so the loop below indexes it like any other candidate.
    let ranked = select_topology_with_fit(&selector, |kind| {
        let spec = by_kind.get(&kind).ok_or_else(|| {
            format!(
                "race_birth_topology: no realized candidate for fused topology {:?}",
                kind.display_name()
            )
        })?;
        fit_topology_candidate(spec, target, weights)
    })?;
    let winner = ranked
        .winner()
        .ok_or_else(|| "race_birth_topology: empty ranking".to_string())?;
    // Resolve each ranked entry's name back to its TYPED kind through
    // `AutoTopologyKind::parse` — the selector's own inverse of `display_name`,
    // so the ranking is carried as kinds and never compared as text. A name the
    // selector emits that its own parser cannot read is a contract break in
    // gam-solve, not something to paper over with a string match, so it is
    // returned as an error rather than silently dropped from the ranking.
    let mut ranking: Vec<RankedTopology> = Vec::with_capacity(ranked.ranked.len());
    for entry in &ranked.ranked {
        let kind = AutoTopologyKind::parse(&entry.topology_name).map_err(|error| {
            format!(
                "race_birth_topology: the selector ranked a topology named {:?} that \
                 AutoTopologyKind::parse cannot resolve: {error}",
                entry.topology_name
            )
        })?;
        ranking.push(RankedTopology {
            kind,
            tk_score: entry.tk_score,
            tk_score_resolution: entry.tk_score_resolution,
        });
    }
    let outcome = TopologyRaceOutcome {
        ranking,
        fit: winner.fit_handle.clone(),
        tk_score: winner.tk_score,
    };
    // #2729 — say out loud when the race did NOT decide. The pipeline still gets
    // a deterministic fit (something downstream must be seeded), but a margin
    // inside the criterion's own resolution is arithmetic debris, and a channel
    // that only ever reports a winner turns that debris into a verdict.
    log_unresolved_topology_race(&outcome.ranking);
    // Read the RETAINED ranking, not the local, so the field the outcome carries
    // is the one the agreement log prices -- a retained copy no production path
    // reads is dead by construction (#2280 landed it without its consumer).
    log_atlas_evidence_agreement(atlas, &outcome.ranking);
    Ok(Some(outcome))
}

/// #2280 — log the atlas's MEASURED manifold against the REML race's INDEPENDENT
/// verdict, per race.
///
/// This is the calibration channel the atlas-consumer design owes: the prior may
/// only reorder, so the only way to learn whether it is worth its cost is to
/// record, every time both speak, whether they agreed and by what evidence margin
/// the race preferred its own answer. It is pure observation — it reads the race's
/// finished ranking and changes nothing.
///
/// The margin is quoted in the race's own TK units and is the gap between the
/// winner and the atlas-named candidate, so `0.0` means the atlas named the
/// winner and a large value means the evidence positively rejected what the
/// charts measured. A disagreement is logged at `info` precisely because it is
/// the interesting case: it is either a defect in the readout or a topology the
/// evidence cannot see.
/// #2729 — the third verdict this channel used to be unable to express: TIE.
///
/// AGREE and DISAGREE both assert that the evidence RESOLVED an ordering. When
/// the winner's margin over the atlas-named candidate is inside the combined
/// resolution of the two scores, neither assertion is available: the charts and
/// the evidence did not agree, and the evidence did not reject the charts — the
/// race simply could not tell the two candidates apart. Reporting that as
/// agreement (margin `0.0`, "the atlas named the winner") or as disagreement
/// (a non-zero margin quoted to six places) publishes roundoff as calibration
/// data, which is worse than publishing nothing.
fn log_atlas_evidence_agreement(atlas: Option<&AtlasTopologyReadout>, ranking: &[RankedTopology]) {
    let Some(atlas) = atlas else {
        return;
    };
    let Some(measured) = atlas.observed_manifold().and_then(observed_kind_to_auto_topology) else {
        return;
    };
    let Some(winner) = ranking.first().copied() else {
        return;
    };
    let winner_kind = winner.kind;
    let winner_score = winner.tk_score;
    let measured_entry = ranking.iter().copied().find(|entry| entry.kind == measured);
    // Resolution first: a verdict is only reportable once the comparison that
    // produced it is known to have digits. This ordering is the fix — the old
    // channel branched on kind equality and therefore could not reach the
    // question at all.
    if let Some(measured_entry) = measured_entry
        && measured_entry.kind != winner_kind
        && !winner.is_resolvably_better_than(&measured_entry)
    {
        log::info!(
            "#2729 atlas/evidence TIE: the charts measured {measured:?} (tk \
             {:.17e}, resolution {:?}) and the race ranked {winner_kind:?} first (tk \
             {winner_score:.17e}, resolution {:?}); the {:.6e} gap is INSIDE the combined \
             resolution of the two scores, so the race did not decide between them and this \
             race contributes NO calibration datum either way",
            measured_entry.tk_score,
            measured_entry.tk_score_resolution,
            winner.tk_score_resolution,
            measured_entry.tk_score - winner_score
        );
        return;
    }
    if measured == winner_kind {
        // Even a kind-identical winner is only an agreement if the race
        // resolved it against the field; `log_unresolved_topology_race` has
        // already reported the unresolved case, and a tie between OTHER
        // candidates does not weaken this datum, so the AGREE branch stands.
        log::debug!(
            "#2280 atlas/evidence AGREE: the charts measured {measured:?} and the REML race \
             independently ranked it first (tk {winner_score:.6})"
        );
        return;
    }
    match measured_entry {
        Some(measured_entry) => log::info!(
            "#2280 atlas/evidence DISAGREE: the charts measured {measured:?} (tk \
             {:.6}) but the REML race ranked {winner_kind:?} first (tk \
             {winner_score:.6}); evidence margin {:.6} against the measured manifold, \
             resolved above the {:.6e} combined resolution of the two scores",
            measured_entry.tk_score,
            measured_entry.tk_score - winner_score,
            measured_entry.tk_score_resolution.unwrap_or(f64::NAN)
                + winner.tk_score_resolution.unwrap_or(f64::NAN)
        ),
        None => log::info!(
            "#2280 atlas/evidence DISAGREE: the charts measured {measured:?}, which this race \
             did not realize as a candidate at all; the race ranked {winner_kind:?} first (tk \
             {winner_score:.6})"
        ),
    }
}

/// #2729 — report, per race, every candidate the winner is NOT resolvably
/// better than.
///
/// The selector still returns a deterministic winner because something
/// downstream must be seeded with one fit; what it must never do is let that
/// determinism be mistaken for a decision. A birth target lying in the null
/// space of every candidate's penalty collapses them all onto the same model,
/// and the scores then differ by the last few bits of their own summation.
fn log_unresolved_topology_race(ranking: &[RankedTopology]) {
    let Some(winner) = ranking.first().copied() else {
        return;
    };
    let tied: Vec<&RankedTopology> = ranking[1..]
        .iter()
        .filter(|entry| !winner.is_resolvably_better_than(entry))
        .collect();
    if tied.is_empty() {
        return;
    }
    log::info!(
        "#2729 topology race UNRESOLVED: {:?} (tk {:.17e}, resolution {:?}) is not resolvably \
         better than {:?}; every listed gap is inside the combined resolution of the two scores, \
         so the ordering among them is set by floating-point roundoff, not by the data. The \
         returned fit is a deterministic pick among indistinguishable candidates, NOT an \
         evidence verdict.",
        winner.kind,
        winner.tk_score,
        winner.tk_score_resolution,
        tied.iter()
            .map(|entry| format!(
                "{:?} (tk {:.17e}, resolution {:?}, gap {:.6e})",
                entry.kind,
                entry.tk_score,
                entry.tk_score_resolution,
                entry.tk_score - winner.tk_score
            ))
            .collect::<Vec<_>>()
    );
}

/// A primary-atom topology choice discovered by the fit-entry evidence race
/// (#2238/#2239): the basis kind the seed dictionary should build for the atom
/// and the latent dimension that kind carries.
pub struct PrimaryTopologyChoice {
    pub basis_kind: SaeAtomBasisKind,
    /// Width of the coordinate the seed dictionary must allocate — a STORAGE
    /// width, not a count of degrees of freedom. They differ for `S²`, which is
    /// intrinsically 2-D but carried as an ambient unit 3-vector. Price an atom
    /// by `geometry.intrinsic_dim()`; index its storage by this.
    pub latent_dim: usize,
    /// Complete geometry selected by evidence, after resolution growth. This is
    /// the only way a continuously optimized reference metric can cross the
    /// primary-discovery boundary without being reconstructed as a default.
    pub geometry: SaeAtomGeometryPlan,
    /// Evidence-selected harmonic resolution for a periodic (circle) winner
    /// (#2243): the number of Fourier harmonics the seed circle carries, chosen
    /// by REML marginal likelihood rather than the historical fixed budget.
    /// `None` for every non-periodic kind, whose chart resolution is either a
    /// different knob (a torus winner carries its per-axis order in
    /// `n_torus_harmonics`; a flat/Duchon-sheet winner carries data-scaled
    /// centers in `n_duchon_centers`) or genuinely fixed (the ambient sphere
    /// basis and the Möbius double-cover basis are fixed-degree constructions).
    pub n_harmonics: Option<usize>,
    /// Evidence-selected thin-plate center count for a Duchon-sheet winner
    /// (#2240, the #2243 resolution-growth pattern lifted to 2-D): the number
    /// of Duchon centers the seeded sheet should carry, chosen by the same
    /// REML marginal likelihood the topology race scores with. `None` for
    /// every other kind (including a flat `EuclideanPatch` winner, which is
    /// installed as a duchon seed at the builder's default center budget).
    pub n_duchon_centers: Option<usize>,
    /// Evidence-selected per-axis harmonic order for a torus winner (#2243, the
    /// resolution-growth pattern lifted to the tensor-product torus): the number
    /// of Fourier harmonics per circle factor the seeded torus should carry
    /// (basis size `(2H+1)^d`), chosen by the same REML marginal likelihood the
    /// topology race scores with rather than the fixed `SAE_DEFAULT_TORUS_HARMONICS`
    /// budget. `None` for every other kind.
    pub n_torus_harmonics: Option<usize>,
    /// The `(n, latent_dim)` coordinate realization on which this topology won
    /// the primary race (#2240/#2280). Kind and coordinates are one atomic
    /// evidence candidate: an intrinsic sheet carries its unfolded Isomap chart,
    /// while every PCA/natural-chart winner carries the exact chart it was scored
    /// on rather than asking the seed builder to reconstruct an approximation.
    pub coords: Array2<f64>,
}

/// Per-atom topology discovery for the PRIMARY seed dictionary (#2238/#2239).
///
/// `race_birth_topology` adjudicates topology by evidence, but it only ever
/// runs on residual births — the K primary atoms created at fit entry kept the
/// pinned default (a 1-D circle), hard-capping every intrinsically 2-D factor
/// at R² ≈ 0.5. This lifts the SAME evidence race to fit entry: each atom
/// races a circle, a torus, a sphere and a flat 2-D patch — every candidate
/// seeded with its own NATURAL chart of the atom's cluster (phase angles for
/// the periodic forms, the ambient unit direction for the sphere, standardized
/// principal projections for the patch) so no candidate is handicapped by a
/// chart built for a rival — and the proper REML marginal likelihood picks the
/// winner.
///
/// `labels` assigns each observation to its seed cluster (the same
/// output-energy labels the periodic seed refinement uses); the race for atom
/// `k` weights exactly its cluster's rows. `max_dims[k]` caps the intrinsic
/// dimension enrolled for atom `k`, so `d_atom = 1` keeps the race
/// one-dimensional. Auto discovery is an explicit contract: every requested
/// atom must produce an evidence-backed winner. Invalid inputs, undersupported
/// clusters, or numerical failures are returned to the caller instead of
/// silently substituting a different topology.
pub fn discover_primary_atom_topologies(
    target: ArrayView2<'_, f64>,
    labels: &[usize],
    k_atoms: usize,
    max_dims: &[usize],
) -> Result<Vec<PrimaryTopologyChoice>, String> {
    let n_obs = target.nrows();
    let p_out = target.ncols();
    if labels.len() != n_obs {
        return Err(format!(
            "discover_primary_atom_topologies: labels must have N={n_obs} entries; got {}",
            labels.len()
        ));
    }
    if max_dims.len() != k_atoms {
        return Err(format!(
            "discover_primary_atom_topologies: max_dims must have K={k_atoms} entries; got {}",
            max_dims.len()
        ));
    }
    if p_out < 2 {
        return Err(format!(
            "discover_primary_atom_topologies: evidence racing needs at least two output dimensions; got P={p_out}"
        ));
    }
    (0..k_atoms)
        .map(|atom_idx| -> Result<PrimaryTopologyChoice, String> {
            let rows: Vec<usize> =
                (0..n_obs).filter(|&row| labels[row] == atom_idx).collect();
            // Too few rows to score a 2-candidate race honestly.
            if rows.len() < 16 {
                return Err(format!(
                    "discover_primary_atom_topologies: auto atom {atom_idx} has only {} seed-cluster rows; at least 16 are required for an evidence race (name an explicit topology when discovery is not identifiable)",
                    rows.len()
                ));
            }
            // Cluster-local principal frame: up to 4 components of the atom's
            // rows, then every observation projected into that frame (the race
            // weights select the cluster; out-of-cluster rows carry weight 0).
            let mut mean = vec![0.0_f64; p_out];
            for &row in &rows {
                for col in 0..p_out {
                    mean[col] += target[[row, col]];
                }
            }
            let inv_count = 1.0 / rows.len() as f64;
            for value in &mut mean {
                *value *= inv_count;
            }
            let mut local = Array2::<f64>::zeros((rows.len(), p_out));
            for (out_row, &src_row) in rows.iter().enumerate() {
                for col in 0..p_out {
                    local[[out_row, col]] = target[[src_row, col]] - mean[col];
                }
            }
            let (_u, _s, vt_opt) = local.svd(false, true).map_err(|error| {
                format!(
                    "discover_primary_atom_topologies: SVD failed for auto atom {atom_idx}: {error}"
                )
            })?;
            let vt = vt_opt.ok_or_else(|| {
                format!(
                    "discover_primary_atom_topologies: SVD returned no right-singular frame for auto atom {atom_idx}"
                )
            })?;
            let n_pcs = vt.nrows().min(4);
            if n_pcs < 2 {
                return Err(format!(
                    "discover_primary_atom_topologies: auto atom {atom_idx} has principal rank {n_pcs}; at least two directions are required"
                ));
            }
            let mut proj = Array2::<f64>::zeros((n_obs, n_pcs));
            for row in 0..n_obs {
                for pc in 0..n_pcs {
                    let mut acc = 0.0_f64;
                    for col in 0..p_out {
                        acc += (target[[row, col]] - mean[col]) * vt[[pc, col]];
                    }
                    proj[[row, pc]] = acc;
                }
            }
            // In-cluster standard deviation per component (the projections are
            // already centered at the cluster mean), so the flat patch sees
            // O(1) coordinates.
            let cluster_sd = |pc: usize| -> f64 {
                let mut acc = 0.0_f64;
                for &row in &rows {
                    acc += proj[[row, pc]] * proj[[row, pc]];
                }
                (acc * inv_count).sqrt().max(1e-12)
            };
            let phase = |a: f64, b: f64| -> f64 {
                let frac = b.atan2(a) / std::f64::consts::TAU;
                frac - frac.floor()
            };
            let mut specs: Vec<TopologyCandidateSpec> = Vec::with_capacity(4);
            // Circle: phase of the leading principal pair (unit-period
            // convention, matching the periodic seed refinement). The phase
            // coordinate is retained so that, if the circle wins the topology
            // race, its harmonic RESOLUTION can be selected by evidence (#2243)
            // on the same coordinate the topology race discriminated on.
            let circle_coords = {
                let mut coords = Array2::<f64>::zeros((n_obs, 1));
                for row in 0..n_obs {
                    coords[[row, 0]] = phase(proj[[row, 0]], proj[[row, 1]]);
                }
                specs.push(TopologyCandidateSpec::new(
                    AutoTopologyKind::Circle,
                    SaeAtomGeometryPlan::new(
                        SaeAtomBasisKind::Periodic,
                        1,
                        SaeBasisResolution::PeriodicHarmonics { order: 1 },
                        SaeReferenceMetricPlan::UnitCircle,
                    )?,
                    LatentManifold::Circle { period: 1.0 },
                    coords.clone(),
                )?);
                coords
            };
            let mut sheet_coords: Option<Array2<f64>> = None;
            let mut torus_coords: Option<Array2<f64>> = None;
            if max_dims[atom_idx] >= 2 {
                // Flat 2-D patch: standardized leading principal projections.
                let (sd0, sd1) = (cluster_sd(0), cluster_sd(1));
                let mut coords = Array2::<f64>::zeros((n_obs, 2));
                for row in 0..n_obs {
                    coords[[row, 0]] = proj[[row, 0]] / sd0;
                    coords[[row, 1]] = proj[[row, 1]] / sd1;
                }
                specs.push(TopologyCandidateSpec::new(
                    AutoTopologyKind::Euclidean,
                    SaeAtomGeometryPlan::new(
                        SaeAtomBasisKind::EuclideanPatch,
                        2,
                        SaeBasisResolution::Polynomial { degree: 2 },
                        SaeReferenceMetricPlan::EuclideanPolynomial,
                    )?,
                    LatentManifold::Euclidean,
                    coords.clone(),
                )?);
                // #2240 — flexible thin-plate (Duchon) sheet over the SAME
                // standardized 2-PC chart, with adaptive in-cluster centers:
                // the rich 2-D candidate for swiss-roll-class factors a
                // degree-2 patch cannot follow. It races as its OWN kind
                // (`DuchonSheet`): it is not a fixed constant-curvature form,
                // so the #944 Euclidean/Sphere fusion cannot absorb it —
                // without it, any race that also carried a sphere candidate
                // fused the flat patch away and left a rolled sheet NO
                // admissible chart at all. A cluster too small to identify the
                // thin-plate nullspace simply skips the candidate (the flat
                // patch above stays as the sheet fallback).
                if let Some(centers) =
                    duchon_sheet_centers(&coords, &rows, duchon_sheet_race_center_budget(rows.len()))
                {
                    specs.push(TopologyCandidateSpec::new(
                        AutoTopologyKind::DuchonSheet,
                        SaeAtomGeometryPlan::new(
                            SaeAtomBasisKind::Duchon,
                            2,
                            SaeBasisResolution::DuchonCoordinates { centers },
                            SaeReferenceMetricPlan::EuclideanDuchon,
                        )?,
                        LatentManifold::Euclidean,
                        coords.clone(),
                    )?);
                }
                sheet_coords = Some(coords);
                if n_pcs >= 3 {
                    // Sphere: the unit-normalized leading 3-frame, kept AS a
                    // direction. The superseded chart computed exactly this
                    // vector and then threw it away into `asin`/`atan2`, buying
                    // a pole the optimiser could not cross and a longitude that
                    // is pure gauge there. Normalisation is the manifold's, via
                    // `project_point`, so there is one authority for it.
                    let sphere_manifold = LatentManifold::Sphere { dim: 3 };
                    let mut coords = Array2::<f64>::zeros((n_obs, 3));
                    for row in 0..n_obs {
                        let raw = ndarray::Array1::from_vec(vec![
                            proj[[row, 0]],
                            proj[[row, 1]],
                            proj[[row, 2]],
                        ]);
                        let projected = sphere_manifold.project_point(raw.view());
                        for axis in 0..3 {
                            coords[[row, axis]] = projected[axis];
                        }
                    }
                    specs.push(TopologyCandidateSpec::new(
                        AutoTopologyKind::Sphere,
                        SaeAtomGeometryPlan::new(
                            SaeAtomBasisKind::Sphere,
                            3,
                            SaeBasisResolution::AmbientSphereHarmonics {
                        degree: SAE_AMBIENT_SPHERE_DEFAULT_DEGREE,
                    },
                            SaeReferenceMetricPlan::RoundSphere,
                        )?,
                        sphere_manifold.clone(),
                        coords.clone(),
                    )?);
                    // `RP²` shares the sphere's ambient cover exactly: the
                    // antipodal map is the ambient `u -> -u`, so the quotient
                    // reads the same direction the sphere does. No separate
                    // `(lat, lon)` cover, and therefore no pole.
                    specs.push(TopologyCandidateSpec::new(
                        AutoTopologyKind::ProjectivePlane,
                        SaeAtomGeometryPlan::projective_plane(1)?,
                        sphere_manifold.clone(),
                        coords.clone(),
                    )?);
                }
                if n_pcs >= 3 {
                    // Möbius band (#2240): recover one fundamental domain of the
                    // period-two double cover plus the SIGNED band width from the
                    // radial/transverse half-angle vector. The deck-invariant basis
                    // makes width-odd structure carry half-period angular factors —
                    // the non-orientable signature no other candidate can express.
                    if let Ok(coords) =
                        crate::manifold::mobius_double_cover_coords_from_projection(
                            proj.view(),
                            &rows,
                        )
                    {
                        specs.push(TopologyCandidateSpec::new(
                            AutoTopologyKind::Mobius,
                            SaeAtomGeometryPlan::new(
                                SaeAtomBasisKind::Mobius,
                                2,
                                SaeBasisResolution::MobiusHarmonics {
                                    circle_order: 3,
                                    width_degree: 2,
                                },
                                SaeReferenceMetricPlan::MobiusQuotient,
                            )?,
                            LatentManifold::Product(vec![
                                LatentManifold::Circle { period: 2.0 },
                                LatentManifold::Interval { lo: -1.0, hi: 1.0 },
                            ]),
                            coords,
                        )?);
                    }
                }
                if n_pcs >= 4 {
                    // Torus: independent phases of the two leading principal
                    // pairs (fraction-of-period convention on both axes).
                    let mut coords = Array2::<f64>::zeros((n_obs, 2));
                    for row in 0..n_obs {
                        coords[[row, 0]] = phase(proj[[row, 0]], proj[[row, 1]]);
                        coords[[row, 1]] = phase(proj[[row, 2]], proj[[row, 3]]);
                    }
                    specs.push(TopologyCandidateSpec::new(
                        AutoTopologyKind::Torus,
                        SaeAtomGeometryPlan::new(
                            SaeAtomBasisKind::Torus,
                            2,
                            SaeBasisResolution::TorusHarmonics { per_axis_order: 2 },
                            SaeReferenceMetricPlan::FlatRectangularTorus { tau: 0.0 },
                        )?,
                        LatentManifold::Product(vec![
                            LatentManifold::Circle { period: 1.0 },
                            LatentManifold::Circle { period: 1.0 },
                        ]),
                        coords.clone(),
                    )?);
                    specs.push(TopologyCandidateSpec::new(
                        AutoTopologyKind::KleinBottle,
                        SaeAtomGeometryPlan::klein_bottle(2)?,
                        LatentManifold::Product(vec![
                            LatentManifold::Circle { period: 1.0 },
                            LatentManifold::Circle { period: 1.0 },
                        ]),
                        coords.clone(),
                    )?);
                    torus_coords = Some(coords);
                }
            }
            if specs.is_empty() {
                return Err(format!(
                    "discover_primary_atom_topologies: auto atom {atom_idx} produced no realizable candidates"
                ));
            }
            let mut weights = Array1::<f64>::zeros(n_obs);
            for &row in &rows {
                weights[row] = 1.0;
            }
            // #2280 — proposal-time atlas prior on THIS atom's cluster-local
            // ambient rows at the atom's own chart rank (recognition-only,
            // fail-open). Reorders the candidate menu toward the manifold the local
            // charts and their transition holonomy measure; the REML race stays the
            // sole arbiter.
            let local = target.select(Axis(0), &rows);
            let atlas = atlas_prior_for_coords(local.view(), max_dims[atom_idx]);
            // PCA/linear race is the cheaper DEFAULT.
            let pca_winner = race_spec_set(specs, target, weights.view(), atlas.as_ref()).map_err(|error| {
                format!(
                    "discover_primary_atom_topologies: evidence race failed for auto atom {atom_idx}: {error}"
                )
            })?;
            // Intrinsic-metric CHALLENGER (#2240/#2280): re-race the fold-sensitive
            // d=2 candidates on the cluster-local geodesic embedding. It enters the
            // SAME REML evidence race as every PCA-chart candidate; evidence, not a
            // post-hoc PCA winner-class gate, decides whether unfolding is useful.
            // An embedding or evidence failure is a failed discovery operation and
            // is returned to the caller instead of silently substituting the PCA
            // result.
            let intrinsic_challenger =
                match build_intrinsic_primary_specs(target, &rows, max_dims[atom_idx]).map_err(
                    |error| {
                        format!(
                            "discover_primary_atom_topologies: intrinsic chart failed for auto atom {atom_idx}: {error}"
                        )
                    },
                )? {
                    Some(int_specs) => race_spec_set(int_specs, target, weights.view(), atlas.as_ref()).map_err(
                        |error| {
                        format!(
                            "discover_primary_atom_topologies: intrinsic evidence race failed for auto atom {atom_idx}: {error}"
                        )
                    },
                    )?,
                    None => None,
                };
            // A race winner is ATOMIC: its topology kind and the coordinates on
            // which that kind earned its evidence come from the same fitted
            // handle.  Never track coordinate provenance in a parallel optional
            // flag; that split allowed a Duchon kind verdict to survive while
            // its intrinsic chart was discarded and rebuilt from PCA.
            let fit = match (pca_winner, intrinsic_challenger) {
                (Some(pca), Some(intrinsic)) => {
                    if intrinsic.tk_score < pca.tk_score {
                        intrinsic.fit
                    } else {
                        pca.fit
                    }
                }
                (Some(pca), None) => pca.fit,
                (None, Some(intrinsic)) => intrinsic.fit,
                (None, None) => {
                    return Err(format!(
                        "discover_primary_atom_topologies: evidence race returned no winner for auto atom {atom_idx}"
                    ));
                }
            };
            let fit_kind = fit.geometry.kind().clone();
            let fit_dim = fit.geometry.latent_dim();
            if fit_kind == SaeAtomBasisKind::Duchon {
                sheet_coords = Some(fit.coords.clone());
            }
            // #2243 — for a circle winner, GROW the harmonic resolution by the
            // same REML evidence: the topology race ran the circle at a fixed low
            // budget only to discriminate topology, but a genuinely 1-D factor's
            // fidelity is capped by that budget. Every other kind carries a chart
            // whose resolution is not a harmonic count, so it selects none.
            let n_harmonics = if fit_kind == SaeAtomBasisKind::Periodic {
                Some(select_periodic_resolution(
                    circle_coords.view(),
                    target,
                    weights.view(),
                    rows.len(),
                )?)
            } else {
                None
            };
            // #2240 — for a Duchon-sheet winner, GROW the center count by the
            // same REML evidence (the #2243 pattern lifted from harmonics to
            // thin-plate centers): the race ran the sheet at the seed-economy
            // budget only to discriminate topology; a tightly rolled sheet's
            // fidelity is capped by that budget.
            let n_duchon_centers = if fit_kind == SaeAtomBasisKind::Duchon {
                let coords = sheet_coords.as_ref().ok_or_else(|| {
                    format!(
                        "discover_primary_atom_topologies: duchon-sheet winner without a 2-D chart for auto atom {atom_idx}"
                    )
                })?;
                Some(select_duchon_sheet_resolution(
                    coords,
                    target,
                    weights.view(),
                    &rows,
                )?)
            } else {
                None
            };
            // #2243 — for a torus winner, GROW the per-axis harmonic order by
            // the same REML evidence (the circle pattern lifted to the tensor-
            // product torus): the race ran the torus at a fixed low order only
            // to discriminate topology, but a genuinely toroidal factor with
            // high-frequency angular content on either circle factor is capped
            // by that order.
            let n_torus_harmonics = if matches!(
                &fit_kind,
                SaeAtomBasisKind::Torus | SaeAtomBasisKind::KleinBottle
            ) {
                let coords = torus_coords.as_ref().ok_or_else(|| {
                    format!(
                        "discover_primary_atom_topologies: torus-cover winner without a 2-D chart for auto atom {atom_idx}"
                    )
                })?;
                let selected = select_torus_resolution(
                    coords.view(),
                    target,
                    weights.view(),
                    rows.len(),
                )?;
                Some(if fit_kind == SaeAtomBasisKind::KleinBottle {
                    selected.max(2)
                } else {
                    selected
                })
            } else {
                None
            };
            // Install the exact chart realization the winner was scored on.
            // This is required for intrinsic folds and equally correct for every
            // natural curved chart; a kind-only verdict followed by a generic
            // coordinate rebuild is a different candidate than the one that won.
            let d = fit_dim.min(fit.coords.ncols());
            let mut coords = Array2::<f64>::zeros((fit.coords.nrows(), fit_dim));
            for row in 0..fit.coords.nrows() {
                for col in 0..d {
                    coords[[row, col]] = fit.coords[[row, col]];
                }
            }
            let grown_torus_geometry = if fit_kind == SaeAtomBasisKind::Torus {
                let per_axis_order = n_torus_harmonics.ok_or_else(|| {
                    format!(
                        "discover_primary_atom_topologies: torus winner without selected resolution for auto atom {atom_idx}"
                    )
                })?;
                let grown_spec = TopologyCandidateSpec::new(
                    AutoTopologyKind::Torus,
                    SaeAtomGeometryPlan::new(
                        SaeAtomBasisKind::Torus,
                        2,
                        SaeBasisResolution::TorusHarmonics { per_axis_order },
                        SaeReferenceMetricPlan::FlatRectangularTorus { tau: 0.0 },
                    )?,
                    fit.manifold.clone(),
                    fit.coords.clone(),
                )?;
                Some(
                    fit_torus_metric_candidate(&grown_spec, target, weights.view())?
                        .fit_handle
                        .geometry,
                )
            } else {
                None
            };
            let geometry = match &fit_kind {
                SaeAtomBasisKind::Periodic => SaeAtomGeometryPlan::new(
                    SaeAtomBasisKind::Periodic,
                    1,
                    SaeBasisResolution::PeriodicHarmonics {
                        order: n_harmonics.ok_or_else(|| {
                            format!(
                                "discover_primary_atom_topologies: periodic winner without selected resolution for auto atom {atom_idx}"
                            )
                        })?,
                    },
                    SaeReferenceMetricPlan::UnitCircle,
                )?,
                SaeAtomBasisKind::Torus => grown_torus_geometry.ok_or_else(|| {
                    format!(
                        "discover_primary_atom_topologies: torus winner metric refit was not produced for auto atom {atom_idx}"
                    )
                })?,
                SaeAtomBasisKind::KleinBottle => SaeAtomGeometryPlan::klein_bottle(
                    n_torus_harmonics.ok_or_else(|| {
                        format!(
                            "discover_primary_atom_topologies: Klein winner without selected resolution for auto atom {atom_idx}"
                        )
                    })?,
                )?,
                SaeAtomBasisKind::Duchon => {
                    let center_count = n_duchon_centers.ok_or_else(|| {
                        format!(
                            "discover_primary_atom_topologies: Duchon winner without selected centers for auto atom {atom_idx}"
                        )
                    })?;
                    let chart = sheet_coords.as_ref().ok_or_else(|| {
                        format!(
                            "discover_primary_atom_topologies: Duchon winner without chart for auto atom {atom_idx}"
                        )
                    })?;
                    let centers = duchon_sheet_centers(chart, &rows, center_count).ok_or_else(|| {
                        format!(
                            "discover_primary_atom_topologies: cannot realize {center_count} selected Duchon centers for auto atom {atom_idx}"
                        )
                    })?;
                    SaeAtomGeometryPlan::new(
                        SaeAtomBasisKind::Duchon,
                        fit_dim,
                        SaeBasisResolution::DuchonCoordinates { centers },
                        SaeReferenceMetricPlan::EuclideanDuchon,
                    )?
                }
                _ => fit.geometry.clone(),
            };
            Ok(PrimaryTopologyChoice {
                basis_kind: fit_kind,
                latent_dim: fit_dim,
                geometry,
                n_harmonics,
                n_duchon_centers,
                n_torus_harmonics,
                coords,
            })
        })
        .collect()
}

/// Intrinsic-metric CHALLENGER spec set for the primary discovery race
/// (#2240/#2280). On a FOLDED residual factor (a swiss roll, a creased sheet) the
/// PCA 2-PC chart the primary race builds is self-overlapping, so even the
/// flexible Duchon sheet fits a crumpled image. This embeds the birth image
/// `target` by its geodesic (Isomap) metric — which unrolls the fold — standardizes
/// each intrinsic axis to unit in-cluster SD (so the Euclidean-patch / Duchon
/// design is scale-commensurable with the PCA flat candidate), and offers the two
/// FOLD-SENSITIVE `d = 2` candidates (a flat patch and a thin-plate sheet) on those
/// unfolded coordinates. The caller races this set against the PCA set under the
/// SAME REML evidence and keeps the PCA winner on ties, so the intrinsic seed
/// supplants the linear one only when its unfolding earns strictly higher evidence.
///
/// Every returned spec owns the standardized intrinsic 2-D chart it is evaluated
/// on, so the eventual fit handle carries coordinate provenance atomically.
/// `None` when the atom is `d < 2` (folds are a sheet story; `d = 1` line/circle
/// is served by the PCA race), the cluster is too small for a geodesic graph, or
/// the embedding is degenerate.
fn build_intrinsic_primary_specs(
    target: ArrayView2<'_, f64>,
    rows: &[usize],
    max_dim: usize,
) -> Result<Option<Vec<TopologyCandidateSpec>>, String> {
    if max_dim < 2 || rows.len() < 3 {
        return Ok(None);
    }
    let n_obs = target.nrows();
    let local_target = target.select(Axis(0), rows);
    let embed = crate::manifold::intrinsic_geodesic_embedding(local_target.view(), 2)?;
    if embed.ncols() < 2 {
        return Ok(None);
    }
    // Standardize each intrinsic axis to unit in-cluster SD (the PCA flat patch's
    // O(1)-coordinate convention). A collapsed axis (no intrinsic spread) means the
    // geodesic embedding found no second dimension — the challenger is not
    // realizable, so bail and keep the PCA winner.
    let inv_count = 1.0 / rows.len().max(1) as f64;
    let mut coords = Array2::<f64>::zeros((n_obs, 2));
    for col in 0..2 {
        let mut acc = 0.0_f64;
        for local_row in 0..rows.len() {
            acc += embed[[local_row, col]] * embed[[local_row, col]];
        }
        let sd = (acc * inv_count).sqrt();
        if !(sd > 1e-12) || !sd.is_finite() {
            return Ok(None);
        }
        for (local_row, &global_row) in rows.iter().enumerate() {
            coords[[global_row, col]] = embed[[local_row, col]] / sd;
        }
    }
    let mut specs: Vec<TopologyCandidateSpec> = Vec::with_capacity(2);
    specs.push(TopologyCandidateSpec::new(
        AutoTopologyKind::Euclidean,
        SaeAtomGeometryPlan::new(
            SaeAtomBasisKind::EuclideanPatch,
            2,
            SaeBasisResolution::Polynomial { degree: 2 },
            SaeReferenceMetricPlan::EuclideanPolynomial,
        )?,
        LatentManifold::Euclidean,
        coords.clone(),
    )?);
    if let Some(centers) =
        duchon_sheet_centers(&coords, rows, duchon_sheet_race_center_budget(rows.len()))
    {
        specs.push(TopologyCandidateSpec::new(
            AutoTopologyKind::DuchonSheet,
            SaeAtomGeometryPlan::new(
                SaeAtomBasisKind::Duchon,
                2,
                SaeBasisResolution::DuchonCoordinates { centers },
                SaeReferenceMetricPlan::EuclideanDuchon,
            )?,
            LatentManifold::Euclidean,
            coords.clone(),
        )?);
    }
    Ok(Some(specs))
}

/// Polynomial-nullspace dimension of the plan-declared 2-D thin-plate sheet.
/// The geometry authority derives `m = d/2 + 2 = 3`, so the nullspace contains
/// the six monomials of total degree at most two. The center count must clear
/// this dimension for the kernel block to have positive rank.
const DUCHON_SHEET_NULLSPACE_DIM: usize = 6;

/// Race-time center budget for the 2-D Duchon-sheet candidate (#2240) —
/// mirrors the seed builder's economy band (`sae_build_atom_plans`: floor
/// `nullspace + d + 1`, dense ceiling 32) so the race scores the exact chart a
/// default seed would build; the evidence ladder
/// ([`select_duchon_sheet_resolution`]) then grows a WINNER past this budget.
/// Returns 0 (no realizable candidate) when the cluster cannot identify the
/// thin-plate nullspace.
fn duchon_sheet_race_center_budget(n_cluster: usize) -> usize {
    let floor = DUCHON_SHEET_NULLSPACE_DIM + 2 + 1;
    if n_cluster <= floor {
        return 0;
    }
    n_cluster.min(32).max(floor)
}

/// Deterministic adaptive centers for the Duchon-sheet candidate: `n_centers`
/// evenly-strided IN-CLUSTER rows of the standardized 2-PC chart, so the
/// thin-plate kernel is anchored where the factor's data actually lies
/// (knot-at-data placement). `None` when the cluster cannot supply the
/// requested count (the candidate is skipped, not degraded).
fn duchon_sheet_centers(
    coords: &Array2<f64>,
    rows: &[usize],
    n_centers: usize,
) -> Option<Array2<f64>> {
    if n_centers == 0 || rows.len() < n_centers {
        return None;
    }
    let mut centers = Array2::<f64>::zeros((n_centers, 2));
    for i in 0..n_centers {
        // Even stride over the cluster's rows; i·len/n is strictly increasing
        // in i for n ≤ len, so the selected rows are distinct.
        let row = rows[i * rows.len() / n_centers];
        centers[[i, 0]] = coords[[row, 0]];
        centers[[i, 1]] = coords[[row, 1]];
    }
    Some(centers)
}

/// Evidence-driven center count for a Duchon-sheet primary winner (#2240 — the
/// #2243 resolution-growth pattern lifted from circle harmonics to thin-plate
/// centers). The topology race scored the sheet at the seed-economy budget
/// only to discriminate topology; a swiss-roll-class factor's fidelity is
/// capped by that budget, so the winner's center count is selected by the SAME
/// proper closed-form REML marginal likelihood the race scores with
/// (`fit_topology_candidate` → `raw_reml`, complexity-priced, lower is
/// better), taking the GLOBAL evidence minimum over a dyadic ladder — the
/// evidence need not be unimodal in resolution.
///
/// The ladder is bounded by two hard, data-derived limits (no tuned
/// resolution constant): the seed-economy floor below and the identifiability
/// ceiling `n_centers < n_cluster` (the Duchon design has one column per
/// center, and a weighted REML cannot be identified with as many columns as
/// the cluster has observations). The ceiling rung is always included so a
/// near-noiseless factor can reach full resolution.
fn select_duchon_sheet_resolution(
    sheet_coords: &Array2<f64>,
    target: ArrayView2<'_, f64>,
    weights: ArrayView1<'_, f64>,
    rows: &[usize],
) -> Result<usize, String> {
    let floor = duchon_sheet_race_center_budget(rows.len());
    if floor == 0 {
        return Err(
            "select_duchon_sheet_resolution: cluster too small to identify the thin-plate nullspace"
                .to_string(),
        );
    }
    let ceiling = rows.len().saturating_sub(1).max(floor);
    let mut ladder: Vec<usize> = Vec::new();
    let mut c = floor;
    while c < ceiling {
        ladder.push(c);
        c = c.saturating_mul(2);
    }
    ladder.push(ceiling);
    let mut best_c = 0usize;
    let mut best_score = f64::INFINITY;
    for &n_centers in &ladder {
        let Some(centers) = duchon_sheet_centers(sheet_coords, rows, n_centers) else {
            continue;
        };
        let geometry = match SaeAtomGeometryPlan::new(
            SaeAtomBasisKind::Duchon,
            2,
            SaeBasisResolution::DuchonCoordinates { centers },
            SaeReferenceMetricPlan::EuclideanDuchon,
        ) {
            Ok(geometry) => geometry,
            Err(_) => continue,
        };
        let spec = TopologyCandidateSpec::new(
            AutoTopologyKind::DuchonSheet,
            geometry,
            LatentManifold::Euclidean,
            sheet_coords.clone(),
        )?;
        // `raw_reml` is the proper REML evidence (lower is better) on a common
        // `n_obs`, so comparing it directly selects the same resolution the
        // race machinery would (see `select_periodic_resolution`).
        let score = match fit_topology_candidate(&spec, target, weights) {
            Ok(evidence) => evidence.raw_reml,
            Err(_) => continue,
        };
        if score.is_finite() && score < best_score {
            best_score = score;
            best_c = n_centers;
        }
    }
    if best_c == 0 {
        return Err(
            "select_duchon_sheet_resolution: no fittable center count for the duchon-sheet winner"
                .to_string(),
        );
    }
    Ok(best_c)
}

/// Measured spectral noise floor for evidence-driven resolution selection
/// (#2243). A resolution knob is a BANDWIDTH question, not a smoothing one:
/// include every harmonic carrying real, above-noise energy and let the fit's
/// own REML-selected λ shrink the unsupported ones — over-provisioning is
/// harmless (an empty harmonic's roughness penalty drives its coefficient to
/// zero), while under-provisioning structurally caps reconstruction. The floor
/// is measured from the periodogram itself, with no tuned smoothing constant:
///
/// * a numerical-zero guard `peak · 1e-12` — a harmonic that small is
///   indistinguishable from roundoff, never real signal. It dominates on clean
///   (near-noiseless) data, where the median energy collapses to ~0;
/// * a noise-level guard `median · log2(K)` — under band-limited signal the
///   per-harmonic energies are dominated by the noise floor, whose robust
///   center is the median (real harmonics are sparse outliers that do not move
///   it). `log2(K) = ln K / ln 2` is the expected value of the maximum of `K`
///   exponential-tailed noise energies in units of the median, i.e. the
///   Bonferroni expected-one-false-alarm bound over the `K` tested harmonics —
///   a harmonic above it is a genuine spectral outlier, not the largest of `K`
///   noise draws. It grows only logarithmically in the band size, so it stays
///   sensitive to real structure on large clusters. It dominates under real
///   noise.
///
/// This replaces the earlier REML-argmin-over-resolutions ladder, which
/// UNDER-resolved exactly-fittable clean data (the #2243 disease it was meant
/// to cure): the closed-form REML dispersion reward is floored at
/// `MIN_DEVIANCE`, so an exact fit's evidence gain is capped while its
/// complexity term `½d·(log|H| − log|λS|₊)` diverges as the REML λ→0 (needed to
/// admit a high harmonic against its `∝ h⁴` roughness penalty); at modest
/// cluster sizes the complexity term wins and the argmin stops below the real
/// bandwidth. Bandwidth selection prices resolution on the spectrum, where it
/// belongs, and leaves smoothing to the fit's own REML λ.
fn spectral_noise_floor(energies: &[f64], peak_energy: f64) -> f64 {
    let numerical = peak_energy * 1e-12;
    let k = energies.len();
    if k == 0 {
        return numerical;
    }
    let mut sorted: Vec<f64> = energies.to_vec();
    sorted.sort_by(|a, b| a.partial_cmp(b).unwrap_or(std::cmp::Ordering::Equal));
    let median = if k % 2 == 1 {
        sorted[k / 2]
    } else {
        0.5 * (sorted[k / 2 - 1] + sorted[k / 2])
    };
    let bonferroni = (k as f64).max(2.0).log2();
    numerical.max(median * bonferroni)
}

/// Evidence-driven harmonic resolution for a periodic (circle) primary atom
/// (#2243). The historical seed budget (`2·d_atom + 1` harmonics, i.e. 2
/// harmonics at the default `d_atom = 2`) under-resolves genuinely 1-D factors
/// with real high-frequency content, capping reconstruction below the fidelity
/// the data supports even once the topology is right. The resolution is the
/// weighted angular periodogram's BANDWIDTH — the highest harmonic whose energy
/// clears the measured [`spectral_noise_floor`] — bounded by the
/// identifiability limit `2H + 1 < n_cluster` (the weighted fit cannot be
/// identified with more basis columns than the cluster has observations). A
/// target with no angular energy returns an error (the caller surfaces it as a
/// discovery failure rather than silently pinning a resolution).
fn select_periodic_resolution(
    circle_coords: ArrayView2<'_, f64>,
    target: ArrayView2<'_, f64>,
    weights: ArrayView1<'_, f64>,
    n_cluster: usize,
) -> Result<usize, String> {
    let n_obs = target.nrows();
    let p_out = target.ncols();
    // 2H + 1 basis columns must stay strictly below the cluster sample count for
    // the weighted fit to be identifiable.
    let ident_ceiling = (n_cluster.saturating_sub(2) / 2).max(1);
    // Weighted angular periodogram energy per harmonic, over the cluster rows the
    // weights select.
    let mut peak_energy = 0.0_f64;
    let mut energies = Vec::with_capacity(ident_ceiling);
    for h in 1..=ident_ceiling {
        let mut energy = 0.0_f64;
        for col in 0..p_out {
            let (mut re, mut im) = (0.0_f64, 0.0_f64);
            for row in 0..n_obs {
                let w = weights[row];
                if w == 0.0 {
                    continue;
                }
                let angle = std::f64::consts::TAU * h as f64 * circle_coords[[row, 0]];
                re += w * target[[row, col]] * angle.cos();
                im += w * target[[row, col]] * angle.sin();
            }
            energy += re * re + im * im;
        }
        peak_energy = peak_energy.max(energy);
        energies.push(energy);
    }
    if !(peak_energy > 0.0) {
        return Err(
            "select_periodic_resolution: the circle winner carries no angular energy".to_string(),
        );
    }
    let floor = spectral_noise_floor(&energies, peak_energy);
    let bandwidth = energies
        .iter()
        .rposition(|&energy| energy > floor)
        .map(|idx| idx + 1)
        .unwrap_or(1);
    Ok(bandwidth.min(ident_ceiling).max(1))
}

/// Evidence-driven per-axis harmonic order for a torus primary winner (#2243 —
/// the circle resolution-growth pattern lifted to the tensor-product torus).
/// The historical fixed order (`SAE_DEFAULT_TORUS_HARMONICS = 3`) under-resolves
/// a genuinely toroidal factor whose angular content on either circle factor
/// runs above third order, capping reconstruction below the fidelity the data
/// supports even once the topology is right. The per-axis order is the joint
/// angular periodogram's BANDWIDTH — the largest per-axis order of any joint
/// harmonic `(h₀, h₁)` whose energy clears the measured [`spectral_noise_floor`]
/// — bounded by the identifiability limit `(2H + 1)^2 < n_cluster` (the
/// tensor-product design has `(2H+1)^2` columns, and the weighted fit cannot be
/// identified with more columns than the cluster has observations) intersected
/// with the seed builder's dense guard `(2H+1)^2 ≤ 4·SAE_MAX_PERIODIC_HARMONICS`,
/// so the selected order always builds. Same spectral criterion as the circle
/// (see [`select_periodic_resolution`]): over-provisioning is smoothed away by
/// the fit's own REML λ. A target with no angular energy returns an error (the
/// caller surfaces it as a discovery failure rather than silently pinning a
/// resolution).
fn select_torus_resolution(
    torus_coords: ArrayView2<'_, f64>,
    target: ArrayView2<'_, f64>,
    weights: ArrayView1<'_, f64>,
    n_cluster: usize,
) -> Result<usize, String> {
    let n_obs = target.nrows();
    let p_out = target.ncols();
    // Solve (2H+1)^2 < n_cluster for the identifiability ceiling on the per-axis
    // order, intersected with the seed builder's dense guard.
    let axis_ceiling = |limit: f64| -> usize {
        let root = limit.sqrt();
        if root <= 1.0 {
            1
        } else {
            (((root - 1.0) / 2.0).floor() as usize).max(1)
        }
    };
    let ident_ceiling = axis_ceiling(n_cluster as f64);
    let dense_ceiling = axis_ceiling((SAE_MAX_PERIODIC_HARMONICS * 4) as f64);
    let hard_ceiling = ident_ceiling.min(dense_ceiling).max(1);
    // Weighted joint angular periodogram over the bounded (h₀, h₁) grid; keep
    // the per-axis order of every cell so the bandwidth prune can be resolved
    // once the peak is known.
    let mut peak_energy = 0.0_f64;
    let mut cells: Vec<(usize, f64)> = Vec::new();
    for h0 in 0..=hard_ceiling {
        for h1 in 0..=hard_ceiling {
            if h0 == 0 && h1 == 0 {
                continue;
            }
            let mut energy = 0.0_f64;
            for col in 0..p_out {
                let (mut re, mut im) = (0.0_f64, 0.0_f64);
                for row in 0..n_obs {
                    let w = weights[row];
                    if w == 0.0 {
                        continue;
                    }
                    let angle = std::f64::consts::TAU
                        * (h0 as f64 * torus_coords[[row, 0]] + h1 as f64 * torus_coords[[row, 1]]);
                    re += w * target[[row, col]] * angle.cos();
                    im += w * target[[row, col]] * angle.sin();
                }
                energy += re * re + im * im;
            }
            peak_energy = peak_energy.max(energy);
            cells.push((h0.max(h1), energy));
        }
    }
    if !(peak_energy > 0.0) {
        return Err(
            "select_torus_resolution: the torus winner carries no angular energy".to_string(),
        );
    }
    // Per-axis resolution = the joint periodogram's bandwidth (the largest
    // per-axis order of any cell clearing the measured noise floor), bounded by
    // the identifiability/dense ceiling. Same spectral-bandwidth criterion as
    // the circle (see [`spectral_noise_floor`] and [`select_periodic_resolution`]):
    // over-provisioning is smoothed away by the fit's own REML λ, so this prices
    // resolution on the spectrum rather than under-resolving via a REML argmin.
    let cell_energies: Vec<f64> = cells.iter().map(|(_, energy)| *energy).collect();
    let floor = spectral_noise_floor(&cell_energies, peak_energy);
    let bandwidth = cells
        .iter()
        .filter(|(_, energy)| *energy > floor)
        .map(|(order, _)| *order)
        .max()
        .unwrap_or(1);
    Ok(bandwidth.min(hard_ceiling).max(1))
}

/// Resolve every `"auto"` entry of a primary seed dictionary to the concrete
/// basis-kind string + latent dimension the fit-entry evidence race selects
/// (#2238/#2239). This is the SINGLE place the auto policy lives — the FFI
/// layer only plumbs arrays through (SPEC: pyffi stays thin). Policy:
///
/// * torus / sphere winners keep their kind and carry `latent_dim = 2`;
/// * a flat 2-D winner builds the expressive thin-plate (`duchon`) chart
///   rather than the degree-2 patch the race scored with — same topology,
///   strictly richer basis for the seeded atom;
/// * a Duchon-sheet winner (#2240, the rich swiss-roll-class chart raced as
///   its own candidate) installs as `duchon` and carries its evidence-selected
///   center count in the returned per-atom override vector, so the seed
///   builder grows the thin-plate resolution REML picked rather than its
///   fixed economy budget;
/// * a circle winner carries the harmonic resolution the fit-entry evidence
///   race selected (#2243), installed as the periodic atom's `d_atom` (the seed
///   builder's harmonic-count knob) so discovery grows resolution rather than
///   pinning the caller's default budget;
/// * any discovery failure is returned. Auto mode never silently substitutes
///   the old periodic default; callers that require a fixed topology must name
///   that topology explicitly.
///
/// Returns `(resolution_overrides, coord_overrides, geometry_overrides)`, all aligned with
/// `atom_basis`. `resolution_overrides[k]` is the per-atom basis-native
/// resolution knob (`None` unless evidence-grown), interpreted per the resolved
/// basis kind — Duchon center count for a flat/Duchon-sheet winner (#2240),
/// per-axis harmonic order for a torus winner (#2243). `coord_overrides[k]` is
/// the exact coordinate realization of an auto winner (`None` only for an
/// explicitly named, non-auto atom), so the caller installs the same kind+chart
/// candidate that earned the evidence verdict. `geometry_overrides[k]` is the
/// complete post-growth typed plan of an evidence winner; installing it is what
/// preserves continuously selected reference metrics across seed construction.
pub fn resolve_auto_primary_atoms(
    target: ArrayView2<'_, f64>,
    labels: &[usize],
    atom_basis: &mut [String],
    atom_dim: &mut [usize],
) -> Result<
    (
        Vec<Option<usize>>,
        Vec<Option<Array2<f64>>>,
        Vec<Option<SaeAtomGeometryPlan>>,
    ),
    String,
> {
    let k_atoms = atom_basis.len();
    if atom_dim.len() != k_atoms {
        return Err(format!(
            "resolve_auto_primary_atoms: atom_basis and atom_dim must both have K={k_atoms} entries; atom_dim has {}",
            atom_dim.len()
        ));
    }
    let mut resolution_overrides: Vec<Option<usize>> = vec![None; k_atoms];
    // Per-atom seed-chart overrides: every auto winner carries the exact chart
    // realization on which it earned its evidence. Non-auto atoms remain None.
    let mut coord_overrides: Vec<Option<Array2<f64>>> = vec![None; k_atoms];
    let mut geometry_overrides: Vec<Option<SaeAtomGeometryPlan>> = vec![None; k_atoms];
    if !atom_basis.iter().any(|basis| basis == "auto") {
        return Ok((resolution_overrides, coord_overrides, geometry_overrides));
    }
    let choices = discover_primary_atom_topologies(target, labels, k_atoms, atom_dim)?;
    for atom_idx in 0..k_atoms {
        if atom_basis[atom_idx] != "auto" {
            continue;
        }
        let choice = &choices[atom_idx];
        match choice.basis_kind {
            SaeAtomBasisKind::Torus => {
                // #2243 — the latent dimension stays the manifold dimension; the
                // evidence-selected per-axis harmonic order rides the resolution
                // override so the seed builder grows the torus past its fixed
                // `SAE_DEFAULT_TORUS_HARMONICS` budget. `None` cannot occur for a
                // torus winner (discovery always selects an order); the complete
                // plan below is the installation authority.
                atom_basis[atom_idx] = "torus".to_string();
                atom_dim[atom_idx] = choice.latent_dim;
                resolution_overrides[atom_idx] = choice.n_torus_harmonics;
                geometry_overrides[atom_idx] = Some(choice.geometry.clone());
            }
            // `atom_dim` is the request-level INTRINSIC dimension — it is what
            // `sae_build_atom_plans` checks (`basis 'sphere' requires
            // atom_dim == 2`) and what prices the atom. `choice.latent_dim` is
            // the coordinate STORAGE width, and `PrimaryTopologyChoice` says so
            // in as many words: "Price an atom by `geometry.intrinsic_dim()`;
            // index its storage by this." For every kind below except these two
            // the values coincide, so only `S²` and `RP²` — carried as an
            // ambient unit 3-vector because neither admits a global 2-D chart —
            // could break the rule, and writing 3 into `atom_dim` made an auto
            // sphere winner refuse its own discovery with
            //   sae_build_atom_plans: atom 1 basis 'sphere' requires atom_dim == 2, got 3
            SaeAtomBasisKind::Sphere => {
                atom_basis[atom_idx] = "sphere".to_string();
                atom_dim[atom_idx] = choice.geometry.intrinsic_dim();
                geometry_overrides[atom_idx] = Some(choice.geometry.clone());
            }
            SaeAtomBasisKind::ProjectivePlane => {
                atom_basis[atom_idx] = "projective_plane".to_string();
                atom_dim[atom_idx] = choice.geometry.intrinsic_dim();
                geometry_overrides[atom_idx] = Some(choice.geometry.clone());
            }
            SaeAtomBasisKind::KleinBottle => {
                atom_basis[atom_idx] = "klein_bottle".to_string();
                atom_dim[atom_idx] = choice.latent_dim;
                resolution_overrides[atom_idx] = choice.n_torus_harmonics;
                geometry_overrides[atom_idx] = Some(choice.geometry.clone());
            }
            SaeAtomBasisKind::Mobius => {
                atom_basis[atom_idx] = "mobius".to_string();
                atom_dim[atom_idx] = choice.latent_dim;
                geometry_overrides[atom_idx] = Some(choice.geometry.clone());
            }
            SaeAtomBasisKind::EuclideanPatch => {
                atom_basis[atom_idx] = "duchon".to_string();
                atom_dim[atom_idx] = choice.latent_dim;
            }
            SaeAtomBasisKind::Duchon => {
                // #2240 — the rich thin-plate sheet won the race outright.
                // Install as a duchon seed and carry the evidence-selected
                // center count so the seed builder grows the resolution REML
                // picked (the #2243 pattern in 2-D).
                atom_basis[atom_idx] = "duchon".to_string();
                atom_dim[atom_idx] = choice.latent_dim;
                resolution_overrides[atom_idx] = choice.n_duchon_centers;
                geometry_overrides[atom_idx] = Some(choice.geometry.clone());
            }
            SaeAtomBasisKind::Periodic => {
                atom_basis[atom_idx] = "periodic".to_string();
                // #2243 — install the evidence-selected harmonic resolution as
                // the periodic atom's `d_atom` (the seed builder routes `d_atom`
                // into the Fourier harmonic count for a periodic basis), so the
                // seeded circle carries the resolution REML picked rather than
                // the caller's default budget. `None` cannot occur for a
                // periodic winner (discovery always selects a resolution), and
                // the complete plan below is the installation authority.
                if let Some(n_harmonics) = choice.n_harmonics {
                    atom_dim[atom_idx] = n_harmonics;
                }
                geometry_overrides[atom_idx] = Some(choice.geometry.clone());
            }
            ref unexpected => {
                return Err(format!(
                    "resolve_auto_primary_atoms: evidence race selected unsupported primary basis {unexpected:?} for auto atom {atom_idx}"
                ));
            }
        }
        coord_overrides[atom_idx] = Some(choice.coords.clone());
    }
    Ok((resolution_overrides, coord_overrides, geometry_overrides))
}

/// A small neutral routing logit a born atom is seeded at: large enough that the
/// refit can grow it if the residual-factor direction is real, small relative to
/// the established atoms so it does not perturb the current routing.
const BIRTH_SEED_LOGIT: f64 = -4.0;

/// Append a fresh atom whose decoder is seeded from a residual-factor direction.
/// The new atom reuses the structural basis of atom 0 (same basis kind, latent
/// dim, basis values + jacobian + smooth penalty) as its BIRTH TEMPLATE — warm
/// inheritance by construction, so the engine's warm-state contract holds and
/// the joint refit starts from a live basis rather than a cold curved family.
/// Only its decoder coefficients carry the residual-factor direction. Routed at
/// a small neutral mass on every row so the refit grows it if it is real and the
/// death channel demotes it next round if it is not.
///
/// # Topology adjudication (#977)
///
/// The template basis is the atom's INITIAL parameterization, not its final
/// topology. A born atom's topology is adjudicated by EVIDENCE downstream, on
/// the discovered dictionary, at two rungs:
///
/// * **Existence** — the #984 held-out e-value birth gate (run inside
///   [`gam_solve::structure_search::search`]) decides whether the atom is
///   born at all. Only a residual factor whose held-out reconstruction
///   likelihood-ratio crosses the Ville threshold earns an atom; the rest stay
///   contested in the [`SearchLedger`].
/// * **Curved (`d ≥ 1`) vs straight / cluster (`d = 0`)** — the #1026
///   hybrid-split pass ([`SaeManifoldTerm::compute_hybrid_split_report`], run
///   post-search over the FULL discovered dictionary) adjudicates every eligible
///   `d = 1` atom's fitted curved image against its straight (linear
///   special-case) secant on the common rank-aware Laplace evidence scale, and
///   records the verdict. A born atom whose curvature does not pay collapses to
///   the linear / cluster lane; one that earns it keeps its curved image. The
///   dictionary is therefore genuinely heterogeneous (curved + linear atoms),
///   not all-circle, with the per-atom verdict surfaced on the fit payload.
///
/// # The race (#977)
///
/// The born atom's topology is now chosen by EVIDENCE at birth, not inherited.
/// The residual-factor direction `factor_dir` is expressed as a per-row image
/// `Y = Φ_template(coords) · factor_dir` (the structure the atom would
/// reconstruct), and [`race_birth_topology`] fits each candidate basis whose
/// intrinsic dimension matches the template's `d_k` (`d = 1`: circle vs line;
/// `d = 2`: torus vs sphere vs euclidean-patch) to `Y` by penalized least
/// squares, ranking them by TK-normalized REML — the gauge-invariant comparison
/// the smooth-term topology race applies. The WINNING topology's evaluator,
/// decoder, manifold, and roughness penalty seed the born atom, so the discovered
/// dictionary is genuinely heterogeneous: different atoms get different topologies
/// by evidence. The post-fit curved-vs-linear hybrid-split rung remains the
/// second line of defense (an atom whose curvature does not pay over the FULL
/// dictionary still collapses linear), and the held-out e-value birth gate
/// decides whether the atom is born at all. When the race finds no realizable
/// candidate (`d_k = 0` cluster-null, or a degenerate image) the born atom falls
/// back to the template basis (warm inheritance), exactly the prior behavior.
///
/// Seeding directly from `fit.evaluator` / `fit.decoder` / `fit.penalty` (the
/// winning [`TopologyRaceFit`]) rather than re-deriving anything is what makes
/// the identifiability gain land in the actual dictionary: the born atom does
/// not merely get labeled with a topology name, it is CONSTRUCTED in the
/// winning basis, so its gauge group is the winner's `Diff × Sym` (curved
/// case) or `GL(d)` (flat fallback) from the moment it exists, not something a
/// later pass has to retrofit.
fn born_atom(
    term: &SaeManifoldTerm,
    rho: &SaeManifoldRho,
    factor_dir: ArrayView2<'_, f64>,
) -> Result<(SaeManifoldTerm, SaeManifoldRho), String> {
    let k = term.k_atoms();
    if term.atoms.is_empty() {
        return Err(
            "born_atom: cannot birth from an empty dictionary (no template atom to seed the \
             coordinate block / basis from)"
                .to_string(),
        );
    }
    let template = &term.atoms[0];
    let m = template.basis_size();
    let p = term.output_dim();
    if factor_dir.dim() != (m, p) {
        return Err(format!(
            "born_atom: residual-factor decoder must be ({m}, {p}); got {:?}",
            factor_dir.dim()
        ));
    }
    let mut atoms = term.atoms.clone();

    // The per-row birth target the topology race adjudicates: the residual-factor
    // direction expressed as a reconstruction image over the template
    // coordinates. A born atom seeded with `factor_dir` in the template basis
    // would emit exactly `Y = Φ_template · factor_dir`; racing topologies asks
    // which geometry parameterizes that image most parsimoniously.
    let template_coords = term.assignment.coords[0].as_matrix();
    let birth_target = template.basis_values.dot(&factor_dir); // (n, p)
    // Uniform per-row mass: at birth the routing is neutral (the atom does not yet
    // own any rows), so every row contributes equally to the topology evidence.
    let weights = Array1::<f64>::ones(birth_target.nrows());

    // Race the candidate topologies matched to the template's intrinsic dim. On a
    // win, seed the born atom from the winning evaluator + penalized decoder; on
    // no realizable candidate (cluster-null d_k, degenerate image), fall back to
    // the template basis (warm inheritance), and let the post-fit curved-vs-linear
    // rung adjudicate as before.
    let raced = race_birth_topology(
        template_coords.view(),
        birth_target.view(),
        weights.view(),
        template.latent_dim(),
    )?;
    // The born atom + its coordinate block. The race-won path carries the winning
    // topology's coordinate block (dimension-matched to its evaluator, manifold
    // set to the winning chart); the fallback path reuses the template block.
    let (born, born_coord_block) = match raced {
        Some(fit) => {
            // Build the born atom directly from the winning topology's realized
            // basis: its evaluator, penalized decoder, and declared reference
            // roughness. Decoder fitting does not redefine that seminorm.
            let atom = SaeManifoldAtom::new_with_provided_function_gram(
                format!("atom_born_{k}"),
                fit.geometry.kind().clone(),
                fit.geometry.latent_dim(),
                fit.phi.clone(),
                fit.jet.clone(),
                fit.decoder.clone(),
                fit.penalty.clone(),
            )?
            .with_basis_second_jet(fit.evaluator.clone())
            .with_geometry_plan(fit.geometry.clone())?;
            // Coordinate block matched to the winning evaluator's intrinsic dim,
            // carrying the winning chart manifold so the joint refit retracts on
            // the right geometry.
            let coord_block = gam_terms::latent::LatentCoordValues::from_matrix_with_manifold(
                fit.coords.view(),
                LatentIdMode::None,
                fit.manifold.clone(),
            );
            (atom, coord_block)
        }
        None => {
            // The born atom reuses the template's structural basis (kind, latent
            // dim, basis values + Jacobian + reference Gram); only its decoder carries
            // the residual-factor direction.
            let mut atom = template.clone();
            atom.set_decoder_coefficients(factor_dir.to_owned())?;
            (atom, term.assignment.coords[0].clone())
        }
    };
    atoms.push(born);

    let n = term.assignment.logits.nrows();
    let mut logits = Array2::<f64>::zeros((n, k + 1));
    for row in 0..n {
        for col in 0..k {
            logits[[row, col]] = term.assignment.logits[[row, col]];
        }
        logits[[row, k]] = BIRTH_SEED_LOGIT;
    }
    let mut coords = term.assignment.coords.clone();
    coords.push(born_coord_block);
    let assignment =
        crate::manifold::SaeAssignment::with_mode(logits, coords, term.assignment.mode)?;
    let child = SaeManifoldTerm::new(atoms, assignment)?;

    let mut child_rho = rho.clone();
    // The born atom inherits the template atom's ARD block shape (disabled if
    // the template's was disabled).
    let inherited = child_rho
        .log_ard
        .first()
        .cloned()
        .unwrap_or_else(|| Array1::<f64>::zeros(0));
    child_rho.log_ard.push(inherited);
    // ρ carries a PER-ATOM smoothness strength `log_lambda_smooth[k]` (#1556),
    // and `assemble_arrow_schur` indexes it by atom (`lambda_smooth[atom_idx]`).
    // Growing the dictionary without growing this vector leaves `k_atoms()`
    // ahead of `log_lambda_smooth.len()` and the next assemble panics with an
    // out-of-bounds index. The born atom inherits the template atom's smoothness
    // strength (atom 0), matching the `log_ard` inheritance just above.
    let inherited_smooth = child_rho.log_lambda_smooth.first().copied().unwrap_or(0.0);
    child_rho.log_lambda_smooth.push(inherited_smooth);
    child_rho.append_curvature_atom(
        k,
        child.atoms[k]
            .geometry_plan()
            .and_then(SaeAtomGeometryPlan::constant_curvature),
    )?;
    Ok((child, child_rho))
}

/// #2101 — build a born atom seeded DIRECTLY as a rank-2 circle: a Periodic atom
/// carrying the residual 2-plane on its cos/sin harmonic decoder rows and a
/// PHASE-ALIGNED coordinate `phase_coords` (`(n, 1)`). This BYPASSES the topology
/// race in [`born_atom`], which parameterizes the born-circle candidate with the
/// TEMPLATE atom's coordinate (`topology_candidates_for_dim` reuses the seed
/// coords) — the wrong phase for a fresh disjoint circle, which leaves the born
/// image `Φ·B` at the DC stationary point where cos/sin never populate. Seeding the
/// fresh phase directly gives the coordinate a nonzero gradient at birth (the birth
/// analogue of the 7a93b1d06 cold-start chart deflation). Mirrors `born_atom`'s
/// logit / coord / ρ construction so the born atom joins the dictionary identically.
pub(crate) fn born_circle_atom(
    term: &SaeManifoldTerm,
    rho: &SaeManifoldRho,
    geometry: SaeAtomGeometryPlan,
    harmonic_decoder: Array2<f64>,
    phase_coords: Array2<f64>,
    circle_gate: Vec<f64>,
) -> Result<(SaeManifoldTerm, SaeManifoldRho), String> {
    let k = term.k_atoms();
    if term.atoms.is_empty() {
        return Err("born_circle_atom: cannot birth from an empty dictionary".to_string());
    }
    if geometry.kind() != &SaeAtomBasisKind::Periodic || geometry.latent_dim() != 1 {
        return Err(format!(
            "born_circle_atom: geometry must declare a one-dimensional periodic atom; got kind={:?}, latent_dim={}",
            geometry.kind(),
            geometry.latent_dim()
        ));
    }
    let m = geometry.basis_size()?;
    let p = term.output_dim();
    if harmonic_decoder.nrows() != m {
        return Err(format!(
            "born_circle_atom: decoder height {} != geometry basis width {m}",
            harmonic_decoder.nrows()
        ));
    }
    if harmonic_decoder.ncols() != p {
        return Err(format!(
            "born_circle_atom: harmonic decoder must have {p} columns (output dim); got {}",
            harmonic_decoder.ncols()
        ));
    }
    let n = term.assignment.logits.nrows();
    if phase_coords.dim() != (n, 1) {
        return Err(format!(
            "born_circle_atom: phase coords must be ({n}, 1); got {:?}",
            phase_coords.dim()
        ));
    }
    if circle_gate.len() != n {
        return Err(format!(
            "born_circle_atom: circle gate must have one entry per row ({n}); got {}",
            circle_gate.len()
        ));
    }
    if circle_gate
        .iter()
        .any(|gate| !gate.is_finite() && *gate != f64::NEG_INFINITY)
    {
        return Err(
            "born_circle_atom: circle gate entries must be finite or negative infinity".to_string(),
        );
    }
    // The plan is the sole authority for the basis, its analytic jets, and the
    // declared unit-circle function Gram. The decoder is only a coefficient
    // realization and cannot redefine any of those geometry fields.
    let bundle = geometry.evaluate_bundle(phase_coords.view())?;
    let born = SaeManifoldAtom::new_with_provided_function_gram(
        format!("atom_born_{k}"),
        geometry.kind().clone(),
        geometry.latent_dim(),
        bundle.basis_values,
        bundle.basis_jacobian,
        harmonic_decoder,
        bundle.reference_penalty,
    )?
    .with_basis_second_jet(bundle.evaluator)
    .with_geometry_plan(geometry)?;

    let born_coord_block = gam_terms::latent::LatentCoordValues::from_matrix_with_manifold(
        phase_coords.view(),
        LatentIdMode::None,
        LatentManifold::Circle { period: 1.0 },
    );

    let mut atoms = term.atoms.clone();
    atoms.push(born);
    let mut logits = Array2::<f64>::zeros((n, k + 1));
    for row in 0..n {
        for col in 0..k {
            logits[[row, col]] = term.assignment.logits[[row, col]];
        }
        // #2101/#2109 PRESENCE-PROPORTIONAL gate seed. The flat weak BIRTH_SEED_LOGIT
        // (−4) is fatal under ordered Beta--Bernoulli — the born circle starts nearly OFF (σ(−4)≈0.018) and
        // the sub-fit collapses it (measured: ordered_beta_bernoulli logit −4 collapses ‖B‖ 1.41→1e-4,
        // logit +3 survives). On a row where the born circle is PRESENT (`circle_gate`
        // finite: its 2-plane energy cleared the derived MP floor), route it at the
        // STRONGER of two derived scales: (a) CO-ACTIVE with the incumbent dictionary
        // (per-row max of existing logits) and (b) the born circle's OWN presence gate
        // `ln(ρ_i²/2·λ₊)` carried in `circle_gate`. Taking the max is what fixes #2109:
        // on incumbent-SPARSE rows `inc_max` is low/negative (the incumbents don't
        // cover where the new circle lives), so the own-presence gate keeps the born
        // circle strong enough to ESTABLISH there instead of re-collapsing. Elsewhere
        // (absent rows, `circle_gate` = −∞) keep the conservative birth default. Both
        // scales are derived — the dictionary's own logits and the ρ_i/λ₊ ratio — no
        // new constant.
        let own_gate = circle_gate[row];
        let inc_max = (0..k)
            .map(|c| term.assignment.logits[[row, c]])
            .fold(f64::NEG_INFINITY, f64::max);
        logits[[row, k]] = if own_gate.is_finite() {
            if inc_max.is_finite() {
                inc_max.max(own_gate)
            } else {
                own_gate
            }
        } else {
            BIRTH_SEED_LOGIT
        };
    }
    let mut coords = term.assignment.coords.clone();
    coords.push(born_coord_block);
    let assignment =
        crate::manifold::SaeAssignment::with_mode(logits, coords, term.assignment.mode)?;
    let child = SaeManifoldTerm::new(atoms, assignment)?;

    let mut child_rho = rho.clone();
    let inherited = child_rho
        .log_ard
        .first()
        .cloned()
        .unwrap_or_else(|| Array1::<f64>::zeros(0));
    child_rho.log_ard.push(inherited);
    let inherited_smooth = child_rho.log_lambda_smooth.first().copied().unwrap_or(0.0);
    child_rho.log_lambda_smooth.push(inherited_smooth);
    Ok((child, child_rho))
}

/// A held-out row-block shard for the universal-inference estimation/evaluation
/// split the gates run over: a contiguous block of row indices into the FULL
/// target the triggers were not tuned on.
///
/// The split is realized through the term's per-row reconstruction weights
/// ([`SaeManifoldTerm::set_row_loss_weights`]): a candidate is refit with the
/// currently-held-out shards' rows at weight `0` (no fitting pressure) and the
/// estimation rows at weight `1`, then EVALUATED on the held-out rows. The
/// predictable-plugin e-process streams the shards: shard `k` is evaluated under
/// a candidate that has not yet seen its rows, then folded into the estimation
/// set (un-masked) for shard `k+1` — exactly the contract
/// [`run_atom_birth_gate`](gam_terms::inference::structure_evidence::run_atom_birth_gate)
/// guarantees the call order of.
#[derive(Clone, Debug)]
pub struct RowBlockShard {
    /// The full target, shared across shards (`(N, p)`).
    pub target: std::sync::Arc<Array2<f64>>,
    /// Row indices into the full target that this shard holds out for
    /// evaluation.
    pub rows: Vec<usize>,
}

/// The estimation/evaluation row split the e-process gates run over. The
/// estimation rows are the candidate's fitting set (weight `1`); the evaluation
/// rows are held out (weight `0` during the fit) and partitioned into the shard
/// stream the gate accumulates evidence over.
#[derive(Clone, Debug)]
pub struct EstimationEvalSplit {
    /// Estimation row indices (the candidate is refit on these; held-out rows
    /// carry weight `0`).
    pub estimation_rows: Vec<usize>,
    /// The evaluation shards, in stream order.
    pub shards: Vec<RowBlockShard>,
}

/// Fraction of rows reserved for estimation (the candidate's fitting set); the
/// remainder is split into evaluation shards. A fixed structural constant
/// (magic-by-default): a majority estimation split keeps the candidate fit
/// faithful while leaving a held-out block for honest evidence.
const ESTIMATION_FRACTION: f64 = 0.6;

/// Build the estimation/evaluation split: the first `ESTIMATION_FRACTION` of the
/// rows (contiguous) are the estimation set, the remainder is partitioned into
/// `n_shards` contiguous held-out evaluation blocks. Deterministic — contiguous
/// blocks, no shuffle. Each shard shares the full target by reference.
pub fn estimation_eval_split(target: ArrayView2<'_, f64>, n_shards: usize) -> EstimationEvalSplit {
    let n = target.nrows();
    if n == 0 {
        return EstimationEvalSplit {
            estimation_rows: Vec::new(),
            shards: Vec::new(),
        };
    }
    let shared = std::sync::Arc::new(target.to_owned());
    // At least one estimation row and at least one evaluation row when n ≥ 2.
    let n_est =
        ((n as f64 * ESTIMATION_FRACTION).round() as usize).clamp(1, n.saturating_sub(1).max(1));
    let estimation_rows: Vec<usize> = (0..n_est).collect();
    let eval_rows: Vec<usize> = (n_est..n).collect();
    let n_eval = eval_rows.len();
    let n_shards = n_shards.min(n_eval).max(usize::from(n_eval > 0));
    let mut shards = Vec::new();
    if n_eval > 0 && n_shards > 0 {
        let base = n_eval / n_shards;
        let rem = n_eval % n_shards;
        let mut cursor = 0usize;
        for s in 0..n_shards {
            let len = base + usize::from(s < rem);
            let rows: Vec<usize> = eval_rows[cursor..cursor + len].to_vec();
            shards.push(RowBlockShard {
                target: shared.clone(),
                rows,
            });
            cursor += len;
        }
    }
    EstimationEvalSplit {
        estimation_rows,
        shards,
    }
}

/// Outcome of the full round driver: the (possibly restructured) fitted term +
/// ρ and the per-round ledgers, each carrying the joint fit's collapse events.
pub struct StructureSearchResult {
    pub term: SaeManifoldTerm,
    pub rho: SaeManifoldRho,
    /// One ledger per round actually run (a round that applies no move is the
    /// last; its ledger is included so the certificate covers the fixpoint).
    pub rounds: Vec<SearchLedger>,
    /// The per-round move stream folded into the ONE unified accounting currency
    /// ([`SaeMigrationLedger`], sae-unification Increment 3): every adjudicated
    /// birth / death / refusal priced in the shared `dl_bits` description-length
    /// unit (the e-process `log_e` banked as bits). This is a read-out of the
    /// e-process verdicts in [`Self::rounds`] — the e-BH gating is untouched and
    /// still owns acceptance — and carries the `pc_reseed_events == 0` invariant
    /// (structure births seed from the residual-factor pool, never a PC).
    pub migration: SaeMigrationLedger,
}

impl StructureSearchResult {
    /// Assemble a result from the fitted term/ρ and the per-round e-process
    /// ledgers, folding those rounds into the unified [`SaeMigrationLedger`]
    /// currency ([`Self::migration`]) in one place so producers cannot drift.
    #[must_use]
    pub fn from_rounds(
        term: SaeManifoldTerm,
        rho: SaeManifoldRho,
        rounds: Vec<SearchLedger>,
    ) -> Self {
        Self::from_rounds_with_predictions(term, rho, rounds, &[])
    }

    /// Assemble a result AND thread the #2233 closed-form birth pre-screen
    /// predictions into the unified ledger. `birth_predictions[i]` maps a birth
    /// candidate index to its predicted ΔMDL (bits) for round `i` (parallel to
    /// `rounds`; a missing / short entry is treated as "no predictions"), so each
    /// proposed residual-factor birth's `predicted_dl_bits` sits on the same
    /// migration record its post-refit verdict fills in.
    #[must_use]
    pub fn from_rounds_with_predictions(
        term: SaeManifoldTerm,
        rho: SaeManifoldRho,
        rounds: Vec<SearchLedger>,
        birth_predictions: &[std::collections::HashMap<usize, f64>],
    ) -> Self {
        let mut migration = SaeMigrationLedger::new();
        let empty = std::collections::HashMap::new();
        for (round_idx, round_ledger) in rounds.iter().enumerate() {
            let preds = birth_predictions.get(round_idx).unwrap_or(&empty);
            migration.record_search_round(round_idx, round_ledger, preds);
        }
        Self {
            term,
            rho,
            rounds,
            migration,
        }
    }

    /// `true` iff at least one structure-changing move LANDED across the rounds —
    /// an `Accepted` move (certified birth / fission / fusion that restructured
    /// the dictionary and triggered a warm refit) or a `Demoted` death (an atom
    /// folded to ~0 routing). Both mutate the returned `term`/`rho` away from the
    /// pre-search joint fit, so any shape uncertainty assembled from the
    /// PRE-search joint Hessian is stale and must be recomputed from the final
    /// post-search per-atom inner fits (#1230). When this is `false`, every round
    /// was contested / vetoed / deduplicated / deferred / stale, the term/rho are
    /// byte-for-byte the pre-search fit, and the exact joint-Hessian bands remain
    /// valid.
    #[must_use]
    pub fn structure_changed(&self) -> bool {
        use gam_solve::structure_search::MoveVerdict;
        self.rounds.iter().any(|round| {
            round.moves.iter().any(|record| {
                matches!(
                    record.verdict,
                    MoveVerdict::Accepted { .. } | MoveVerdict::Demoted { .. }
                )
            })
        })
    }
}

/// The round driver's configuration: how the data is split into shards, the
/// e-gate's budget/level and the per-round harvest breadth.
/// Bundled so the driver entry points stay below the argument-count threshold
/// and so a caller configures one object rather than a positional argument
/// cascade.
#[derive(Clone, Copy, Debug)]
pub struct RoundDriverConfig {
    /// Number of held-out evaluation shards the gate streams over.
    pub n_shards: usize,
    /// Move budget + α the e-gates certify at (fixed for the run).
    pub budget: MoveBudget,
    /// Per-round harvest breadth (max fusions / fissions / births).
    pub harvest_params: HarvestParams,
    /// Curl/flatten structure moves (INTEGRATION_PLAN Phase 4). `None` (the
    /// default) disables them entirely — the driver behaves bit-for-bit as
    /// before (`moves.curl = off`). `Some(cfg)` mines flat-pair→circle
    /// promotions (and the inverse circle→flat demotions) each round and submits
    /// their seeds into the SAME birth/race plumbing the residual-factor births
    /// use, so the REML e-gate remains the only judge.
    pub curl: Option<CurlConfig>,
}

/// Configuration for the curl/flatten proposer (INTEGRATION_PLAN Phase 4). All
/// derived-not-tuned in spirit; the fields are the pipeline's structural knobs,
/// not statistical dials (the verdict's σ screens and the RD crossover are fixed
/// in `crate::manifold::curl`).
#[derive(Clone, Copy, Debug)]
pub struct CurlConfig {
    /// Decoder-cosine ceiling for two linear atoms to be treated as the
    /// rectified halves (`±d`) of one signed direction (antipodal coalescing).
    /// `-0.85` ⇒ at least ~148° apart.
    pub coalesce_cos_threshold: f64,
    /// Gate-overlap (Jaccard) ceiling for a coalescing pair to count as
    /// near-disjoint rectified halves.
    pub coalesce_max_overlap: f64,
    /// Minimum co-firing rows (over the subsample) for a signed-direction pair
    /// to be a curl candidate plane.
    pub min_cooccurrence: usize,
    /// Row-subsample cap for co-occurrence counting + the joint-law projection.
    pub subsample_rows: usize,
    /// Number of `(sin, cos)` harmonics on the seeded circle decoder (width
    /// `2·harmonics + 1`; higher harmonics start at zero for the refit to
    /// sharpen). `>= 1`.
    pub harmonics: usize,
    /// Maximum curl births proposed per round (matched to the ISA birth budget
    /// class).
    pub max_curls: usize,
    /// Whether to run the inverse flatten audit on fitted circle atoms each
    /// round.
    pub flatten: bool,
    /// Rounds an atom-set is silenced after a curl or flatten fires on it, so
    /// `curl → flatten → curl` cannot oscillate (risk #5 hysteresis guard).
    pub cooldown_rounds: usize,
    /// Permutation surrogates per candidate plane for the exact κ e-value. The
    /// e-BH ledger over the round's `m` screened pairs can only reject at rank
    /// `k` when `replicates + 1 ≥ m/(α·k)`, so this is set by the size of the
    /// search, not by taste.
    pub null_replicates: usize,
    /// Target false discovery rate for that ledger.
    pub fdr_alpha: f64,
}

impl Default for CurlConfig {
    fn default() -> Self {
        Self {
            coalesce_cos_threshold: -0.85,
            coalesce_max_overlap: 0.15,
            min_cooccurrence: 8,
            subsample_rows: 4096,
            harmonics: 1,
            max_curls: 4,
            flatten: true,
            cooldown_rounds: 2,
            null_replicates: 4096,
            fdr_alpha: 0.05,
        }
    }
}

/// Drive evidence-guarded structure search around a fitted SAE term until a
/// round applies no moves (#997 round driver).
///
/// Each round: harvest proposals from the current fitted term, run [`search`]
/// over the held-out evaluation shards (gating births/fissions/fusions, demoting
/// never-certified deaths), and adopt the restructured state. The loop stops
/// only when a round's ledger contains no applied move (every record is
/// contested / vetoed / deduplicated / deferred / stale).
///
/// `candidate_fit` is the warm refit: given a RESTRUCTURED candidate term + ρ,
/// it refits the candidate on the ESTIMATION rows only (held-out evaluation rows
/// carry weight `0`), so the candidate is the predictable plug-in the e-process
/// evaluates on the held-out shard stream. A non-converged candidate has no
/// valid likelihood score and therefore aborts the search. The shard
/// fold is a no-op: the candidate is fixed across the stream (a predictable
/// plug-in), and each shard contributes its held-out reconstruction
/// likelihood-ratio against the null state that `null_fit` independently refits
/// on that shard to obtain the honest constrained supremum.
pub fn run_structure_search_rounds(
    mut term: SaeManifoldTerm,
    mut rho: SaeManifoldRho,
    target: ArrayView2<'_, f64>,
    config: RoundDriverConfig,
    ledger: &mut StructureLedger,
    mut candidate_fit: impl FnMut(
        SaeManifoldTerm,
        SaeManifoldRho,
        &[usize],
    ) -> Result<(SaeManifoldTerm, SaeManifoldRho), String>,
    mut null_fit: impl FnMut(
        SaeManifoldTerm,
        SaeManifoldRho,
        &[usize],
    ) -> Result<(SaeManifoldTerm, SaeManifoldRho), String>,
    mut finalize_round: impl FnMut(
        SaeManifoldTerm,
        SaeManifoldRho,
        &[usize],
    ) -> Result<(SaeManifoldTerm, SaeManifoldRho), String>,
) -> Result<StructureSearchResult, String> {
    let RoundDriverConfig {
        n_shards,
        budget,
        harvest_params,
        curl,
    } = config;
    let split = estimation_eval_split(target, n_shards);
    let mut rounds: Vec<SearchLedger> = Vec::new();
    // #2233: per-round birth-pre-screen predictions (candidate index → predicted
    // ΔMDL bits), pushed in lock-step with `rounds` so the unified ledger can pair
    // each proposed birth's prediction with its post-refit verdict.
    let mut round_predictions: Vec<std::collections::HashMap<usize, f64>> = Vec::new();
    // Hysteresis ledger for the curl/flatten pair — persists across rounds so a
    // just-curled atom-set (or just-flattened one) is silenced for a few rounds
    // and the two moves cannot chase each other (INTEGRATION_PLAN risk #5).
    let mut cooldown = crate::manifold::CurlCooldownLedger::new();

    loop {
        // Harvest from the current fitted state. Residuals R = target − fitted.
        let fitted = term.try_fitted_target_aware(target, None)?;
        let residuals = &target.to_owned() - &fitted;
        let mut report = harvest_move_proposals(&term, &rho, residuals.view(), &harvest_params)?;
        // Capture the pre-screen predictions before `report.proposals` is consumed
        // by the search; curl births (appended below) carry no prediction.
        let birth_predictions: std::collections::HashMap<usize, f64> =
            report.birth_predictions.iter().copied().collect();

        // #993 item 3: BANK the within-atom carve binding evidence in the
        // ledger. The carve ran on each `d = 2` product-atom fission candidate
        // (`harvest_move_proposals` → `run_within_atom_carve`) and reported a
        // representational binding p-value; absorb it as a `BindingEdge` claim
        // on the atom's OWN two factors (a self-edge `{atom, atom}`: the carve
        // asks whether THIS atom's two latent factors are bound). A small
        // `edge_p_value` (interaction proven) calibrates to strong positive
        // evidence FOR the binding claim via `log_e_from_p_calibrator`; a
        // p ≈ 1 (additive) absorbs evidence AGAINST it. This makes the binding
        // verdict not merely observable on the `HarvestReport` but BANKED in
        // the persisted ledger, so the dictionary certificate covers it and the
        // evidence resumes across corpus shards. A `None` p-value (the Wald
        // test degenerated) is skipped — no fabricated evidence.

        // Pre-build the birth-SEED list ONCE per round: the residual-factor
        // decoders first (indices `0..r`), then — when curl is enabled — the
        // race-ready circle seeds appended at `r..`, so the apply-move closure
        // inside the gate is a pure function of the candidate index. The two
        // channels share ONE index space via `StructureMove::Birth`.
        let residual_decoders = build_birth_decoders(&term, residuals.view(), &harvest_params)?;
        let mut birth_seeds: Vec<BirthSeed> = residual_decoders
            .into_iter()
            .map(BirthSeed::ResidualFactor)
            .collect();

        // Curl / flatten proposals (INTEGRATION_PLAN Phase 4), gated behind the
        // driver flag and the per-atom-set cooldown. `curl_atoms` maps a curl
        // birth's candidate index to its donor atom-set (the cooldown key +
        // certificate donors); `flatten_atoms` is the set of circle atoms a
        // flatten demotion targets this round.
        let mut curl_atoms: std::collections::HashMap<usize, Vec<usize>> =
            std::collections::HashMap::new();
        let mut flatten_atoms: std::collections::HashSet<usize> = std::collections::HashSet::new();
        if let Some(cfg) = curl {
            for cand in curl_candidates(&term, residuals.view(), &cfg)? {
                if cooldown.blocked(&cand.members) {
                    continue;
                }
                let candidate = birth_seeds.len();
                birth_seeds.push(cand.seed);
                curl_atoms.insert(candidate, cand.members.clone());
                report.proposals.push(proposal(
                    &term,
                    StructureMove::Birth { candidate },
                    cand.net_evidence,
                ));
            }
            if cfg.flatten {
                for atom in flatten_candidates(&term) {
                    if cooldown.blocked(&[atom]) {
                        continue;
                    }
                    // A degenerate circle is retired through the existing death
                    // path; the e-gate adjudicates. Trigger is `MAX/4` so a
                    // flatten sorts among deaths, below terminal collapses.
                    flatten_atoms.insert(atom);
                    report.proposals.push(proposal(
                        &term,
                        StructureMove::Death { atom },
                        f64::MAX / 4.0,
                    ));
                }
            }
        }

        if report.proposals.is_empty() || split.shards.is_empty() {
            // Nothing to do this round — record an empty ledger (with the live
            // collapse events) as the fixpoint and stop.
            rounds.push(SearchLedger {
                alpha: budget.alpha,
                moves: Vec::new(),
                collapse_events: term.collapse_events().to_vec(),
            });
            round_predictions.push(birth_predictions);
            break;
        }

        // The search state threads (term, rho) together. Numerical moves are
        // restructured and refit on the estimation rows so they are predictable
        // plug-ins for held-out scoring. Glue is different: its trigger already
        // IS the sample-split equivalence e-value, and the engine never consults
        // a fit-improvement score for that arm. Keep its state index-stable and
        // materialize the harvest-certified transition exactly once at the round
        // boundary after the proposal chain is complete.
        type State = (SaeManifoldTerm, SaeManifoldRho);
        let collapse_events = term.collapse_events().to_vec();
        let decoders = birth_seeds;
        let estimation_rows = split.estimation_rows.clone();
        let certified_glues = std::mem::take(&mut report.certified_glues);
        let proposals = std::mem::take(&mut report.proposals);
        let outcome: SearchOutcome<State> = search(
            (term, rho),
            proposals,
            &split.shards,
            &budget,
            ledger,
            |state: &State, mv: &StructureMove| {
                if matches!(mv, StructureMove::Glue { .. }) {
                    return Ok(state.clone());
                }
                let (cand_term, cand_rho) =
                    apply_structure_move_seeded(&state.0, &state.1, mv, &decoders)?;
                // Refit the restructured candidate on the estimation rows only.
                candidate_fit(cand_term, cand_rho, &estimation_rows)
            },
            |state: &State, shard: &RowBlockShard| eval_log_lik(&state.0, shard),
            |state: &State, shard: &RowBlockShard| {
                let (null_term, _null_rho) =
                    null_fit(state.0.clone(), state.1.clone(), &shard.rows)?;
                eval_log_lik(&null_term, shard)
            },
            // No-op fold: the candidate is the fixed predictable plug-in across
            // the held-out stream.
            |state: State, _: &RowBlockShard| Ok(state),
        )?;

        let (next_term, next_rho) = outcome.state;
        let mut round_ledger = outcome.ledger;
        round_ledger.collapse_events = collapse_events;
        let applied = round_ledger.moves.iter().any(|m| {
            matches!(
                m.verdict,
                gam_solve::structure_search::MoveVerdict::Accepted { .. }
                    | gam_solve::structure_search::MoveVerdict::Demoted { .. }
            )
        });
        // A certified Glue is an exact geometric quotient, whether it registers
        // an atlas or physically removes an over-tile: the carried transition
        // transplants the removed chart and the fold preserves routing mass.
        // Re-optimizing a glue-only round would turn an evidence-certified image
        // equivalence into a new numerical fit on only the estimation split.
        // Mixed rounds still polish because their non-Glue moves are numerical.
        let requires_polish = round_ledger.moves.iter().any(|record| {
            let fired = matches!(
                record.verdict,
                gam_solve::structure_search::MoveVerdict::Accepted { .. }
                    | gam_solve::structure_search::MoveVerdict::Demoted { .. }
            );
            fired && !matches!(record.mv, StructureMove::Glue { .. })
        });
        // Record the atom-sets any APPLIED curl / flatten move fired on into the
        // cooldown ledger, then advance one round — so the inverse move cannot
        // re-fire on the same atom-set next round (hysteresis, risk #5).
        if let Some(cfg) = curl {
            use gam_solve::structure_search::MoveVerdict;
            for rec in &round_ledger.moves {
                let fired = matches!(
                    rec.verdict,
                    MoveVerdict::Accepted { .. } | MoveVerdict::Demoted { .. }
                );
                if !fired {
                    continue;
                }
                match &rec.mv {
                    StructureMove::Birth { candidate } => {
                        if let Some(members) = curl_atoms.get(candidate) {
                            cooldown.record(members, cfg.cooldown_rounds);
                        }
                    }
                    StructureMove::Death { atom } => {
                        if flatten_atoms.contains(atom) {
                            cooldown.record(&[*atom], cfg.cooldown_rounds);
                        }
                    }
                    // Only the curl/flatten inverse pair (Birth/Death) needs
                    // hysteresis; the remaining moves have no cooldown ledger.
                    StructureMove::Fission { .. }
                    | StructureMove::Fusion { .. }
                    | StructureMove::Glue { .. } => {}
                }
            }
            cooldown.tick();
        }
        rounds.push(round_ledger);
        round_predictions.push(birth_predictions);

        if applied {
            // #1890 — adopt each accepted harvest certificate exactly once, now
            // that the search's index-sensitive chain is complete. Destructive
            // glues physically compact their partner before polish; atlas glues
            // register an image-exact quotient and retain both numerical charts.
            let (mut next_term, mut next_rho) = (next_term, next_rho);
            compact_glued_atoms(
                &mut next_term,
                &mut next_rho,
                rounds.last().expect("round ledger pushed above"),
                &certified_glues,
            )?;
            if requires_polish {
                // Numerical winners reached their restructured form through the
                // cheap capped scoring refit. Refit at the full inner budget and
                // then refresh any registered regular seams against that terminal
                // numerical state.
                let (mut polished_term, polished_rho) =
                    finalize_round(next_term, next_rho, &split.estimation_rows)?;
                refresh_registered_atlas_transitions(&mut polished_term)?;
                term = polished_term;
                rho = polished_rho;
            } else {
                // A pure certified-Glue round is already terminal: optimizer work
                // here would violate the carried quotient's image-equivalence
                // contract (and would fit only the estimation split after the
                // sample-split verdict).
                term = next_term;
                rho = next_rho;
            }
        } else {
            term = next_term;
            rho = next_rho;
            break;
        }
    }

    // Fold the per-round e-process verdicts into the ONE unified migration
    // currency (Increment 3): a read-out, not a second gate.
    Ok(StructureSearchResult::from_rounds_with_predictions(
        term,
        rho,
        rounds,
        &round_predictions,
    ))
}

/// Build the per-round residual-factor decoder list the birth apply-move indexes
/// into: each factor direction lifted to a `(m, p)` decoder in atom 0's basis.
fn build_birth_decoders(
    term: &SaeManifoldTerm,
    residuals: ArrayView2<'_, f64>,
    params: &HarvestParams,
) -> Result<Vec<Array2<f64>>, String> {
    let n = residuals.nrows();
    let p = residuals.ncols();
    if params.max_births == 0 || n == 0 || p == 0 {
        return Ok(Vec::new());
    }
    let assignments = term.assignment.assignments();
    let activity: Array1<f64> = (0..n).map(|r| assignments.row(r).sum()).collect();
    let max_rank = params.max_births.min(p.saturating_sub(1));
    // Propagate a genuine fit failure instead of degrading to "no births".
    // The evidence ladder already includes the rank-0 rung, so a true "no
    // structure to harvest" outcome returns `Ok` (an empty/zero-rank factor);
    // an `Err` here signals a numerical/degenerate failure (non-finite inputs,
    // an empty ladder, a broken alternation), and swallowing it into
    // `Ok(Vec::new())` would silently paper over that non-convergence
    // (the #2069/#2070 accept-on-failure genus). Surface it.
    let model = StructuredResidualModel::fit(ResidualFactorInput {
        residuals,
        activity: activity.view(),
        max_factor_rank: max_rank,
    })
    .map_err(|e| format!("build_birth_decoders: structured-residual fit failed: {e}"))?;
    let factor = model.factor();
    let r = factor.ncols();
    let m = term.atoms[0].basis_size();
    // Lift each p-vector factor direction to a (m, p) decoder: place the
    // direction on the constant (first) basis row so the born atom emits the
    // residual-factor direction as a flat decoder the refit can then shape. This
    // is the WhitenedStructured residual subspace, not raw-Euclidean Λ.
    let mut decoders = Vec::with_capacity(r);
    for j in 0..r {
        let mut decoder = Array2::<f64>::zeros((m, p));
        for out in 0..p {
            decoder[[0, out]] = factor[[out, j]];
        }
        decoders.push(decoder);
    }
    Ok(decoders)
}

// ===========================================================================
// Curl / flatten proposer driver (INTEGRATION_PLAN Phase 4)
//
// The pure statistics live in `crate::manifold::curl`; this is the term-level
// glue: it reads the fitted linear atoms, coalesces their rectified antipodal
// halves, generates co-firing candidate planes, projects the joint amplitude
// law, ranks by net evidence, and emits race-ready `BirthSeed::Circle` seeds
// (curl) plus `Death` demotions of degenerate circles (flatten). No move is
// accepted here — every seed is submitted to the same REML e-gate the
// residual-factor births race through.
// ===========================================================================

/// One curl birth candidate: the donor (coalesced) atom indices, the race-ready
/// circle seed, and the pre-screen net evidence that ranks it.
struct CurlCandidate {
    /// The linear atoms coalesced into the two signed axes of this circle — the
    /// cooldown key and the donors the race retires if the circle wins.
    members: Vec<usize>,
    /// The race-ready periodic circle seed (`BirthSeed::Circle`).
    seed: BirthSeed,
    /// `n_eff·½·ln(3R̂²/(π²σ²)) − Δcharge` — the ranking score (NOT a decision).
    net_evidence: f64,
}

/// Whether an atom is a flat/linear parse the curl move coalesces over (a
/// centered circle is parked on these). Curved bases (periodic, torus, sphere,
/// cylinder, Poincaré) are excluded — they already carry their own curvature.
fn is_linear_like(kind: &SaeAtomBasisKind) -> bool {
    matches!(
        kind,
        SaeAtomBasisKind::Linear | SaeAtomBasisKind::EuclideanPatch
    )
}

/// A linear atom's ambient reconstruction image `G = Φ · B` (`n × p`, before
/// routing weight) — the geometric locus it parses.
fn atom_ambient_image(atom: &SaeManifoldAtom) -> Array2<f64> {
    atom.basis_values.dot(atom.decoder_coefficients())
}

/// Top principal direction of the rows of `img` about `center`, restricted to
/// `active` rows, by a few power iterations on `Σ (g−c)(g−c)ᵀ` (formed
/// implicitly, so the cost is `O(active·p)` per iteration, never `p²`).
fn power_iter_top_dir(
    img: ArrayView2<'_, f64>,
    center: &Array1<f64>,
    active: &[usize],
) -> Array1<f64> {
    let p = img.ncols();
    let mut v = Array1::<f64>::zeros(p);
    // Seed from the highest-norm centered active row (a strong signal direction).
    let mut best_norm = 0.0_f64;
    for &r in active {
        let mut nrm = 0.0_f64;
        for j in 0..p {
            let d = img[[r, j]] - center[j];
            nrm += d * d;
        }
        if nrm > best_norm {
            best_norm = nrm;
            for j in 0..p {
                v[j] = img[[r, j]] - center[j];
            }
        }
    }
    let vn = v.dot(&v).sqrt();
    if vn <= 0.0 {
        return v;
    }
    v.mapv_inplace(|x| x / vn);
    for _ in 0..5 {
        // w = C v = Σ (g−c) ((g−c)·v)
        let mut w = Array1::<f64>::zeros(p);
        for &r in active {
            let mut dot = 0.0_f64;
            for j in 0..p {
                dot += (img[[r, j]] - center[j]) * v[j];
            }
            for j in 0..p {
                w[j] += (img[[r, j]] - center[j]) * dot;
            }
        }
        let wn = w.dot(&w).sqrt();
        if wn <= 0.0 {
            break;
        }
        w.mapv_inplace(|x| x / wn);
        v = w;
    }
    v
}

/// Assemble the fitted linear atoms' `(atom index, unit direction, active mask,
/// ambient image)` — the raw material coalescing and candidate generation read.
fn linear_atom_frames(term: &SaeManifoldTerm) -> Vec<(usize, Array1<f64>, Vec<bool>, Array2<f64>)> {
    let assignments = term.assignment.assignments();
    let n = assignments.nrows();
    let k = assignments.ncols();
    let floor = if k == 0 {
        0.0
    } else {
        ACTIVE_SUPPORT_REL_FLOOR / k as f64
    };
    let mut out = Vec::new();
    for (a, atom) in term.atoms.iter().enumerate() {
        if !is_linear_like(atom.basis_kind()) {
            continue;
        }
        let active_mask: Vec<bool> = (0..n).map(|r| assignments[[r, a]] > floor).collect();
        let active_idx: Vec<usize> = (0..n).filter(|&r| active_mask[r]).collect();
        if active_idx.len() < 2 {
            continue;
        }
        let img = atom_ambient_image(atom);
        if img.ncols() == 0 {
            continue;
        }
        let p = img.ncols();
        let mut center = Array1::<f64>::zeros(p);
        for &r in &active_idx {
            for j in 0..p {
                center[j] += img[[r, j]];
            }
        }
        center.mapv_inplace(|x| x / active_idx.len() as f64);
        let dir = power_iter_top_dir(img.view(), &center, &active_idx);
        if dir.dot(&dir).sqrt() <= 0.0 {
            continue;
        }
        out.push((a, dir, active_mask, img));
    }
    out
}

/// Mine flat-pair → circle promotion candidates from the fitted dictionary
/// (INTEGRATION_PLAN Phase 4 items 1–4).
///
/// The screen itself — antipodal coalescing, co-firing candidate generation, the
/// joint-amplitude projection and [`crate::manifold::curl_verdict`]'s derived
/// acceptance — lives in [`crate::manifold::census_shattered_circles`], because
/// the identical question has to be asked of dictionaries this engine did not fit
/// (see that module's note on why a transcribed screen loses its calibration).
/// What stays here is what is genuinely local to a fitted term: turning atoms into
/// frames, and turning an accepted plane into a race-ready [`BirthSeed`].
/// Deterministic in `(term, residuals, cfg)` so the harvest and the seed-build
/// agree on candidate order/indices.
fn curl_candidates(
    term: &SaeManifoldTerm,
    residuals: ArrayView2<'_, f64>,
    cfg: &CurlConfig,
) -> Result<Vec<CurlCandidate>, String> {
    let geometry = SaeAtomGeometryPlan::new(
        SaeAtomBasisKind::Periodic,
        1,
        SaeBasisResolution::PeriodicHarmonics {
            order: cfg.harmonics,
        },
        SaeReferenceMetricPlan::UnitCircle,
    )?;
    let frames = linear_atom_frames(term);
    if frames.len() < 2 {
        return Ok(Vec::new());
    }
    let n = term.assignment.logits.nrows();
    let p = term.output_dim();

    // Ambient noise scale for the RD screen: RMS reconstruction residual, floored
    // off zero (a perfectly-shattered circle leaves ~no residual, which is
    // exactly why it was invisible — the floor keeps the per-row coding gain
    // ½·ln(3R̂²/(π²σ²)) finite and large).
    let mut sse = 0.0_f64;
    let mut cnt = 0usize;
    for r in 0..residuals.nrows() {
        for j in 0..residuals.ncols() {
            sse += residuals[[r, j]] * residuals[[r, j]];
            cnt += 1;
        }
    }
    let sigma = if cnt > 0 {
        (sse / cnt as f64).sqrt().max(1e-9)
    } else {
        1e-9
    };

    let census_frames: Vec<crate::manifold::AtomFrame<'_>> = frames
        .iter()
        .map(|(a, dir, mask, img)| crate::manifold::AtomFrame {
            id: *a,
            dir: dir.clone(),
            active: mask.clone(),
            image: crate::manifold::AtomImage::Dense(img.view()),
        })
        .collect();
    let census_cfg = crate::manifold::CurlCensusConfig {
        harmonics: cfg.harmonics,
        coalesce_cos_threshold: cfg.coalesce_cos_threshold,
        coalesce_max_overlap: cfg.coalesce_max_overlap,
        min_cooccurrence: cfg.min_cooccurrence,
        subsample_rows: cfg.subsample_rows,
        null_replicates: cfg.null_replicates,
        fdr_alpha: cfg.fdr_alpha,
    };
    let census = crate::manifold::census_shattered_circles(&census_frames, n, p, sigma, &census_cfg)?;

    let mut cands: Vec<CurlCandidate> = Vec::new();
    for pair in &census.pairs {
        // Only e-BH discoveries are raced. A plane that cannot beat its own
        // permutation null at the round's multiplicity has no business spending a
        // birth slot, however well it scored on the per-pair screen.
        if !pair.fdr_discovery {
            continue;
        }
        let Some(plane) = pair.accepted_geometry.as_ref() else {
            continue;
        };
        // Build the race-ready seed from the accepted orthonormal frame + parse.
        let seed_circle = match crate::manifold::curl_seed(
            plane.e1.view(),
            plane.e2.view(),
            plane.alpha.view(),
            plane.beta.view(),
            cfg.harmonics,
            plane.center.view(),
        ) {
            Ok(s) => s,
            Err(_) => continue,
        };
        // Lift the co-firing phases + own-presence gate to the full row set.
        let mut phase_coords = Array2::<f64>::zeros((n, 1));
        let mut gate = vec![f64::NEG_INFINITY; n];
        // Own-presence gate logit: the per-row coding gain ½·ln(3R̂²/(π²σ²)),
        // floored at 0.5 nats so a barely-paying circle still opens its gate for
        // the race to adjudicate. The gain carries the circle shape constant
        // −ln(π/√3) ≈ −0.595 nats/row, so the floor binds for R̂ ≲ 3.3σ (the
        // radius where the gain reaches 0.5).
        let own = pair.verdict.gain_nats_per_row.max(0.5);
        for (idx, &r) in plane.rows.iter().enumerate() {
            phase_coords[[r, 0]] = seed_circle.theta_turns[idx];
            gate[r] = own;
        }
        let members: Vec<usize> = pair
            .members_a
            .iter()
            .chain(pair.members_b.iter())
            .copied()
            .collect();
        cands.push(CurlCandidate {
            members,
            seed: BirthSeed::Circle {
                geometry: geometry.clone(),
                decoder: seed_circle.decoder,
                phase_coords,
                gate,
            },
            net_evidence: pair.verdict.net_evidence_nats,
        });
    }

    // Rank by net evidence; keep the top budget, deduplicated so two circles
    // never claim the same donor atom in one round.
    cands.sort_by(|a, b| b.net_evidence.total_cmp(&a.net_evidence));
    let mut claimed: std::collections::HashSet<usize> = std::collections::HashSet::new();
    let mut out = Vec::new();
    for c in cands {
        if c.members.iter().any(|a| claimed.contains(a)) {
            continue;
        }
        for a in &c.members {
            claimed.insert(*a);
        }
        out.push(c);
        if out.len() >= cfg.max_curls {
            break;
        }
    }
    Ok(out)
}

/// Audit fitted circle atoms for degeneration (INTEGRATION_PLAN Phase 4.5). A
/// circle whose radial law has relaxed to Gaussian fill (κ ≈ 2) or collapsed to
/// a diameter (second resultant ≈ 1) is no longer carrying rotational structure;
/// return those atoms for the existing death/demotion path to retire. The
/// e-gate still owns the decision.
fn flatten_candidates(term: &SaeManifoldTerm) -> Vec<usize> {
    let assignments = term.assignment.assignments();
    let n = assignments.nrows();
    let k = assignments.ncols();
    let floor = if k == 0 {
        0.0
    } else {
        ACTIVE_SUPPORT_REL_FLOOR / k as f64
    };
    let mut out = Vec::new();
    for (a, atom) in term.atoms.iter().enumerate() {
        if !matches!(atom.basis_kind(), SaeAtomBasisKind::Periodic) || atom.latent_dim() != 1 {
            continue;
        }
        let active_idx: Vec<usize> = (0..n).filter(|&r| assignments[[r, a]] > floor).collect();
        if active_idx.len() < 8 {
            continue;
        }
        // Per-row polar law: angle from the atom's phase coordinate, radius from
        // the centered ambient image norm in the atom's own image plane.
        let img = atom_ambient_image(atom);
        let p = img.ncols();
        let mut center = Array1::<f64>::zeros(p);
        for &r in &active_idx {
            for j in 0..p {
                center[j] += img[[r, j]];
            }
        }
        center.mapv_inplace(|x| x / active_idx.len() as f64);
        let coords = term.assignment.coords[a].as_matrix();
        if coords.ncols() == 0 {
            continue;
        }
        let mut radii = Array1::<f64>::zeros(active_idx.len());
        let mut angles = Array1::<f64>::zeros(active_idx.len());
        for (i, &r) in active_idx.iter().enumerate() {
            let mut rr = 0.0_f64;
            for j in 0..p {
                let d = img[[r, j]] - center[j];
                rr += d * d;
            }
            radii[i] = rr.sqrt();
            // Phase coordinate is in turns; angle in radians.
            angles[i] = std::f64::consts::TAU * coords[[r, 0]];
        }
        if let Ok(v) = crate::manifold::flatten_verdict(radii.view(), angles.view()) {
            if v.recommend_flatten {
                out.push(a);
            }
        }
    }
    out
}

/// Per-row Gaussian reconstruction log-likelihood of a shard under the current
/// (restructured, possibly shard-refit) state. The gate's evaluation statistic;
/// the engine guarantees a shard is evaluated strictly before it is folded in.
fn eval_log_lik(term: &SaeManifoldTerm, shard: &RowBlockShard) -> Result<f64, String> {
    // The fitted reconstruction at the shard's held-out rows, scored against the
    // full target. The term's per-row routing/basis covers all N rows, so the
    // reconstruction at a held-out row is the model's prediction for it.
    let fitted = term.try_fitted_target_aware(shard.target.view(), None)?;
    let n_full = fitted.nrows();
    let p = fitted.ncols();
    if p != shard.target.ncols() || n_full != shard.target.nrows() {
        return Err(format!(
            "structure-search fitted shape {:?} does not match target {:?}",
            fitted.dim(),
            shard.target.dim()
        ));
    }
    let mut sse = 0.0_f64;
    let mut count = 0usize;
    for &row in &shard.rows {
        if row >= n_full {
            return Err(format!(
                "structure-search evaluation row {row} is out of range for {n_full} rows"
            ));
        }
        for out in 0..p {
            let d = fitted[[row, out]] - shard.target[[row, out]];
            sse_accumulate(&mut sse, d);
        }
        count += p;
    }
    if count == 0 {
        return Err("structure-search evaluation shard must contain rows".to_string());
    }
    // Gaussian log-lik up to the additive constant that cancels in every
    // e-value ratio: −½·SSE (unit dispersion). The gate forms differences of
    // this against the null sup, so the constant and the dispersion scale drop
    // out of the certified evidence.
    let reconstruction = -0.5 * sse;

    // Occam-priced gate-block evidence (#1016/#1218). The split-LR difference
    // this gate forms is between the K+1 candidate (alternative) and the K null;
    // the gate/assignment-logit block is the weakest-Gaussian piece of the SAE
    // evidence and is mispriced by a plain Laplace quadratic near a birth. The
    // deterministic Pólya–Gamma gate-block marginal supplies the correct
    // normalizer, whose `−½·d_g·log(2π)` term scales with the gate dimension
    // `d_g` (one coordinate per atom). Because the candidate carries one more
    // gate coordinate than the null, that `d_g`-dependent normalizer does NOT
    // cancel in the K-vs-(K+1) difference — it is exactly the per-coordinate
    // `log(2π)` Occam term #1218 corrects the sign of. Folding it into the
    // evaluation likelihood is what makes the corrected sign reach the live
    // gate decision (the unit test alone never touched this path).
    let gate_evidence = gate_block_log_evidence(term, shard);

    Ok(reconstruction + gate_evidence?)
}

/// The deterministic Pólya–Gamma gate-block marginal log-evidence of the
/// candidate's per-atom logistic gates on a shard's held-out rows (#1016/#1218).
///
/// Each atom carries one free per-atom gate logit, so the gate block is a stack
/// of `K` one-dimensional logistic gates: design `X_g = 1` (the per-atom gate
/// coordinate), tilt `ψ̂ =` the atom's per-row logit, binomial response `y =`
/// the binarized activation (`b = 1`), under a unit ridge gate prior. The
/// returned value is the log-evidence `−neg_log_evidence` from
/// `gam_inference::pg_gate_evidence::pg_gate_evidence`, summed over atoms,
/// so the K-dependent `−½·d_g·log(2π)` normalizer enters the gate's split-LR.
///
/// An undefined or non-PD gate block is a fit failure. It cannot be omitted
/// without changing the model-selection scalar, so the error is propagated.
fn gate_block_log_evidence(term: &SaeManifoldTerm, shard: &RowBlockShard) -> Result<f64, String> {
    use gam_solve::inference::pg_gate_evidence::{GateBlock, pg_gate_evidence};

    let logits = &term.assignment.logits;
    let n_full = logits.nrows();
    let k = logits.ncols();
    if k == 0 {
        return Ok(0.0);
    }
    // Restrict to the shard's held-out rows; an empty / out-of-range shard
    // carries no gate evidence.
    if let Some(row) = shard.rows.iter().copied().find(|&row| row >= n_full) {
        return Err(format!(
            "gate-block evidence row {row} is out of range for {n_full} rows"
        ));
    }
    let rows: Vec<usize> = shard.rows.clone();
    let m = rows.len();
    if m == 0 {
        return Err("gate-block evidence shard must contain rows".to_string());
    }

    // Unit gate design (one gate coordinate per atom) and a unit ridge gate
    // prior; the PG block is solved per atom and summed, so `d_g = K` overall.
    let design = Array2::<f64>::ones((m, 1));
    let b = Array1::<f64>::ones(m);
    let penalty = Array2::<f64>::eye(1);

    let mut total = 0.0_f64;
    for atom in 0..k {
        let mut psi = Array1::<f64>::zeros(m);
        let mut y = Array1::<f64>::zeros(m);
        for (i, &row) in rows.iter().enumerate() {
            let logit = logits[[row, atom]];
            if !logit.is_finite() {
                return Err(format!(
                    "gate-block evidence encountered non-finite logit at row {row}, atom {atom}"
                ));
            }
            psi[i] = logit;
            // Binarized activation: the gate is ON when its logit is positive.
            y[i] = if logit > 0.0 { 1.0 } else { 0.0 };
        }
        let block = GateBlock {
            design: design.view(),
            y: y.view(),
            b: b.view(),
            offset: None,
            psi_hat: Some(psi.view()),
            penalty: Some(penalty.view()),
            hess_rest: None,
            h_rest: None,
        };
        let evidence = pg_gate_evidence(&block)
            .map_err(|error| format!("gate-block evidence failed for atom {atom}: {error}"))?;
        // `neg_log_evidence` is `−log p(gate block)`; the log-likelihood the
        // split-LR consumes is its negation.
        total -= evidence.neg_log_evidence;
    }
    Ok(total)
}

#[inline]
fn sse_accumulate(sse: &mut f64, d: f64) {
    *sse += d * d;
}

/// Inner-fit knobs for the production structure-search refit (the same numbers
/// the outer SAE fit drove its inner Arrow-Schur joint fit with).
#[derive(Clone, Copy, Debug)]
pub struct ProductionRefitParams {
    /// Inner Newton iterations available to every candidate and adopted-state
    /// solve. The solver must certify convergence before the state is scored or
    /// returned; this is a numerical ceiling, not an acceptance criterion.
    pub inner_max_iter: usize,
    /// Inner Newton step size.
    pub learning_rate: f64,
    /// Ext-coordinate ridge.
    pub ridge_ext_coord: f64,
    /// β ridge.
    pub ridge_beta: f64,
}

/// Run the production structure-search pass around a fitted SAE term: harvest →
/// e-gated [`search`] over held-out row blocks → adopt certified/demoted moves →
/// repeat to a no-move fixpoint, returning the (possibly restructured) term + ρ
/// and the per-round ledgers (#997).
///
/// The shard refit folds a held-out block into a candidate via the SAME inner
/// joint-fit driver the outer fit used ([`SaeManifoldTerm::run_joint_fit_arrow_schur`]),
/// PENALTY-FREE: the gate's evidence is a held-out reconstruction
/// likelihood-ratio, and the isometry/ARD penalties are gauge/regularization
/// terms that do not belong in the evaluation likelihood. Every candidate and
/// adopted state must reach the inner solver's convergence certificate; a fit
/// error aborts the structure search rather than being converted into a
/// no-improvement score. `ledger` carries banked evidence across rounds so the
/// death veto sees earlier certifications.
pub fn run_production_structure_search(
    term: SaeManifoldTerm,
    rho: SaeManifoldRho,
    target: ArrayView2<'_, f64>,
    config: RoundDriverConfig,
    refit_params: ProductionRefitParams,
    ledger: &mut StructureLedger,
) -> Result<StructureSearchResult, String> {
    let n = target.nrows();
    // Refit a restructured candidate on the ESTIMATION rows only: held-out
    // evaluation rows carry exact zero weight, so the candidate is the
    // predictable plug-in scored on the held-out shards. A non-converged solve
    // is an error; an unfitted candidate is never assigned an evidence score.
    // `full_target` is borrowed by reference so the helper holds no owned capture
    // and can be called from both closures below.
    let refit_at = |full_target: ArrayView2<'_, f64>,
                    mut cand_term: SaeManifoldTerm,
                    mut cand_rho: SaeManifoldRho,
                    estimation_rows: &[usize],
                    inner_max_iter: usize|
     -> Result<(SaeManifoldTerm, SaeManifoldRho), String> {
        let mut weights = vec![0.0; n];
        for &r in estimation_rows {
            if r >= n {
                return Err(format!(
                    "structure-search estimation row {r} is out of range for {n} rows"
                ));
            }
            weights[r] = 1.0;
        }
        cand_term.set_row_loss_weights(weights)?;
        cand_term.run_joint_fit_arrow_schur(
            full_target,
            &mut cand_rho,
            None,
            inner_max_iter,
            refit_params.learning_rate,
            refit_params.ridge_ext_coord,
            refit_params.ridge_beta,
        )?;
        Ok((cand_term, cand_rho))
    };
    let full_iters = refit_params.inner_max_iter;
    let full_target_score = target.to_owned();
    let full_target_null = target.to_owned();
    let full_target_polish = target.to_owned();
    let candidate_refit = refit_at;
    let null_refit = refit_at;
    let final_refit = refit_at;
    run_structure_search_rounds(
        term,
        rho,
        target,
        config,
        ledger,
        // Per-candidate scoring refit. It must converge before the held-out
        // likelihood is evaluated.
        move |cand_term, cand_rho, estimation_rows| {
            candidate_refit(
                full_target_score.view(),
                cand_term,
                cand_rho,
                estimation_rows,
                full_iters,
            )
        },
        // Honest constrained null supremum: refit the current K-atom state on
        // exactly the shard being scored. Under-fitting this side would inflate
        // the e-value.
        move |null_term, null_rho, shard_rows| {
            null_refit(
                full_target_null.view(),
                null_term,
                null_rho,
                shard_rows,
                full_iters,
            )
        },
        // Refit each adopted winner on all rows before it becomes the next
        // round's parent or the returned dictionary.
        //
        // #1890 — the polish refits on ALL rows, NOT the held-out estimation
        // split the per-candidate scoring uses. The estimation/eval split is
        // CONTIGUOUS (`estimation_rows = 0..n_est`), and a disjoint-support
        // dictionary (e.g. the over-tiled / orientation-reversing arcs of the
        // chart-glue fixtures) can place an entire atom's support inside the
        // held-out shard. Refitting the polish on estimation-only then weights
        // that atom's rows at ~0, leaving its decoder UNCONSTRAINED: it drifts
        // or blows up (the reversing pin's `try_fitted()` diverges to ~1.9) or is
        // demoted for lack of support (the over-tile partner). The held-out split
        // exists solely so the per-candidate SCORING refit is an honest
        // predictable plug-in for the e-process; the adopted winner's polish is
        // the RETURNED dictionary and must fit every row it will be evaluated on.
        move |adopted_term, adopted_rho, _| {
            let all_rows: Vec<usize> = (0..n).collect();
            final_refit(
                full_target_polish.view(),
                adopted_term,
                adopted_rho,
                &all_rows,
                full_iters,
            )
        },
    )
}

/// Serialize the per-round ledgers to a JSON string for the fit payload — the
/// honesty surface the python boundary attaches under an additive
/// `structure_search` key. Byte-deterministic for identical inputs.
pub fn rounds_to_json(rounds: &[SearchLedger]) -> Result<String, String> {
    serde_json::to_string(rounds)
        .map_err(|e| format!("rounds_to_json: serialize search ledger: {e}"))
}

#[cfg(test)]
mod tests;

/// #2749 — the #2233 pre-screen must price the atom the birth race actually
/// builds, and must be structurally unable to price one it cannot.
#[cfg(test)]
mod tests_prescreen_geometry_2749 {
    use super::*;
    use ndarray::Array2;

    /// A seed wide enough that the `d_k = 2` menu offers the sphere: the menu
    /// gates it on `d_seed >= 3`, because `S²` needs three independent seed
    /// directions to be identifiable from a great circle at all.
    fn wide_seed() -> Array2<f64> {
        Array2::<f64>::from_shape_fn((64, 3), |(r, c)| {
            let t = r as f64 * 0.1 + c as f64;
            t.sin() + 0.5 * (t * 2.0).cos() + c as f64 * 0.25
        })
    }

    /// Every span band's pre-screen atom is a plan the birth race BUILDS, matched
    /// on its full geometry (kind, latent dim, resolution, reference metric) —
    /// not on width, which two different atoms can share.
    ///
    /// This is the guard #2749 was missing: the map used to be a table of
    /// literals, so when `1dfa70140` deleted the `(lat, lon)` sphere chart the
    /// pre-screen went on charging its width 7 and nothing anywhere disagreed.
    #[test]
    fn curved_prescreen_matches_birth_race_2749() {
        let seed = wide_seed();
        // (span representative, the menu dimension `d_k` its atom is offered at)
        for &(span, d_k) in &[(1.0_f64, 1usize), (2.0, 1), (3.0, 2), (4.0, 2), (9.0, 2)] {
            let plan = SaeAtomGeometryPlan::curved_prescreen_atom_for_span(span)
                .unwrap_or_else(|e| panic!("span {span} must price a buildable atom: {e}"));
            let menu = topology_candidates_for_dim(
                CandidateBases {
                    seed: seed.view(),
                    ambient: None,
                },
                d_k,
            )
            .unwrap_or_else(|e| panic!("d_k={d_k} menu must build: {e}"));
            let offered: Vec<String> = menu
                .iter()
                .map(|spec| {
                    format!(
                        "{:?}/latent{}/{:?}",
                        spec.geometry.kind(),
                        spec.geometry.latent_dim(),
                        spec.geometry.resolution()
                    )
                })
                .collect();
            assert!(
                menu.iter().any(|spec| spec.geometry == plan),
                "span {span}: the pre-screen prices {:?}/latent{}/{:?}, which the \
                 d_k={d_k} birth menu does not offer: {offered:?}",
                plan.kind(),
                plan.latent_dim(),
                plan.resolution(),
            );
        }
    }

    /// The two numbers the pre-screen consumes are theorems of that plan, and the
    /// sphere's are `(d = 2, m = (degree+1)² = 9)` — NOT the deleted chart's
    /// `(2, 7)`. The sphere is the one atom whose coordinate is wider than the
    /// manifold it parameterises, so this also pins that the price uses
    /// `intrinsic_dim` (2) and never `latent_dim` (3).
    #[test]
    fn sphere_is_priced_at_its_realizable_ambient_width_2749() {
        let (d, m) = curved_topology_for_span(3.0).expect("the sphere band must price");
        let degree = SAE_AMBIENT_SPHERE_DEFAULT_DEGREE;
        assert_eq!(d, 2, "S² is intrinsically 2-D whatever its coordinate width");
        assert_eq!(
            m,
            (degree + 1) * (degree + 1),
            "the ambient sphere carries every harmonic through degree {degree}"
        );
        assert_ne!(m, 7, "7 was the width of the chart deleted in 1dfa70140");

        // The neighbouring bands are untouched by #2749 — the reprice moves
        // exactly one band, which is what bounds the acceptance-boundary move.
        assert_eq!(curved_topology_for_span(2.0).expect("circle band"), (1, 3));
        assert_eq!(curved_topology_for_span(4.0).expect("torus band"), (2, 25));
    }

    /// POSITIVE CONTROL for the whole mechanism: the reason a literal `7` could
    /// survive is that nothing ever tried to BUILD the atom it named. Building it
    /// is now the only way to obtain a width, and the chart form is refused — so
    /// the #2749 defect can no longer be expressed here.
    #[test]
    fn the_deleted_sphere_chart_form_is_unbuildable_2749() {
        let refused = SaeAtomGeometryPlan::new(
            SaeAtomBasisKind::Sphere,
            2,
            SaeBasisResolution::AmbientSphereHarmonics {
                degree: SAE_AMBIENT_SPHERE_DEFAULT_DEGREE,
            },
            SaeReferenceMetricPlan::RoundSphere,
        );
        assert!(
            refused.is_err(),
            "a 2-coordinate sphere is the deleted chart; the constructor must refuse it"
        );
        // ... while the form the pre-screen actually asks for does build, or the
        // control above would pass for the wrong reason.
        assert!(
            SaeAtomGeometryPlan::curved_prescreen_atom_for_span(3.0).is_ok(),
            "the ambient sphere must remain buildable, or this control is vacuous"
        );
    }

    /// The reprice is MONOTONE and its size is closed-form: widening `m` by `Δm`
    /// lowers the predicted birth saving by exactly `Δm·P·½log₂N` bits and
    /// changes nothing else, so a span-3 birth can only be DEFERRED by #2749,
    /// never newly admitted. That is the pre-screen's own contract — it may
    /// defer, never accept; the e-process gate is the sole arbiter.
    #[test]
    fn repricing_the_sphere_only_defers_2749() {
        let (d, m) = curved_topology_for_span(3.0).expect("the sphere band must price");
        let base = BirthMdlPrescreen {
            rho: 0.05,
            span: 3.0,
            intrinsic_dim: d,
            basis_size: m,
            signal_var: 12.0,
            noise_floor: 1.0,
            n_tokens: 2000.0,
            p_out: 8,
            g_dict: 1024,
            l0: 32.0,
        };
        let deleted_chart_width = 7usize;
        let at_chart_width = predicted_birth_dl_bits(&BirthMdlPrescreen {
            basis_size: deleted_chart_width,
            ..base
        });
        let at_realizable_width = predicted_birth_dl_bits(&base);
        assert!(
            at_realizable_width < at_chart_width,
            "pricing the realizable atom must be the more conservative of the two \
             (chart {at_chart_width}, realizable {at_realizable_width})"
        );
        let expected_drop = (m as f64 - deleted_chart_width as f64)
            * base.p_out as f64
            * 0.5
            * base.n_tokens.log2();
        let observed_drop = at_chart_width - at_realizable_width;
        assert!(
            (observed_drop - expected_drop).abs()
                <= expected_drop.abs() * 8.0 * f64::EPSILON + f64::EPSILON,
            "the reprice must move the pre-screen by exactly the BIC decoder-column \
             delta: expected {expected_drop}, observed {observed_drop}"
        );
    }
}

#[cfg(test)]
mod tests_atlas_prior_2280 {
    use super::*;
    use crate::manifold::tests_topology_fixtures::{
        circle, cylinder_strip, mobius_strip, trefoil_knot,
    };
    use ndarray::Array2;

    /// A Möbius strip in R³ (a half-twist over one revolution): the canonical
    /// NON-orientable residual, returned with a matched 2-D parameter seed
    /// `(u_norm, v)` for the topology race.
    fn mobius_with_coords(n_u: usize, n_v: usize) -> (Array2<f64>, Array2<f64>) {
        let z = mobius_strip(n_u, n_v);
        let mut coords = Array2::<f64>::zeros((n_u * n_v, 2));
        let mut r = 0usize;
        for iu in 0..n_u {
            for iv in 0..n_v {
                coords[[r, 0]] = (iu as f64) / (n_u as f64) - 0.5;
                coords[[r, 1]] = -0.4 + 0.8 * (iv as f64) / (n_v as f64 - 1.0);
                r += 1;
            }
        }
        (z, coords)
    }

    /// #2280 — the atlas prior MEASURES the manifold: a Möbius residual is named
    /// non-orientable, an orientable cylinder is named as the cylinder, and the two
    /// verdicts are different. This is the capability the orientability-only prior
    /// did not have — the cylinder used to be indistinguishable from "no evidence".
    #[test]
    fn atlas_prior_names_mobius_and_cylinder_apart_2280() {
        let (mob, _) = mobius_with_coords(60, 5);
        let mob_prior =
            atlas_prior_for_coords(mob.view(), 2).expect("the Möbius residual must build an atlas");
        assert!(
            mob_prior.observes_non_orientable(),
            "a Möbius residual must be measured non-orientable: {mob_prior}"
        );

        let cyl = cylinder_strip(60, 5);
        let cyl_prior = atlas_prior_for_coords(cyl.view(), 2)
            .expect("the cylinder residual must build an atlas");
        assert!(
            !cyl_prior.observes_non_orientable(),
            "an orientable cylinder must NOT be measured non-orientable: {cyl_prior}"
        );
        assert_ne!(
            mob_prior.observed_manifold(),
            cyl_prior.observed_manifold(),
            "the Möbius and cylinder residuals must not receive the same verdict"
        );
    }

    /// #2280 — a d = 1 birth now gets a prior the orientability-only readout could
    /// never produce: the trefoil knot's residual is measured as a CIRCLE, which is
    /// exactly the `d = 1` menu's curved candidate, even though its three ambient
    /// principal directions carry comparable spread.
    #[test]
    fn trefoil_residual_floats_the_circle_candidate_at_d1_2280() {
        let target = trefoil_knot(600, 1.0);
        let prior = atlas_prior_for_coords(target.view(), 1)
            .expect("the trefoil residual must build a d=1 atlas");
        assert_eq!(
            prior.observed_manifold(),
            Some(GraphCompressionKind::Circle),
            "the trefoil is intrinsically S¹: {prior}"
        );

        // The d = 1 menu is {Circle, Euclidean} in that order, so the reorder is
        // observable through which candidate the measured manifold puts first.
        let coords = Array2::<f64>::from_shape_fn((target.nrows(), 1), |(r, _)| r as f64);
        let base = topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 1).unwrap();
        let base_kinds: Vec<_> = base.iter().map(|spec| spec.kind).collect();
        let primed = atlas_reorder_specs(
            topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 1).unwrap(),
            Some(&prior),
        );
        assert_eq!(
            primed[0].kind,
            AutoTopologyKind::Circle,
            "the measured circle must lead the d=1 menu"
        );
        let mut a = base_kinds.clone();
        let mut b: Vec<_> = primed.iter().map(|spec| spec.kind).collect();
        a.sort_by_key(|kind| format!("{kind:?}"));
        b.sort_by_key(|kind| format!("{kind:?}"));
        assert_eq!(a, b, "the reorder must preserve the candidate set");
    }

    /// #2280 — the trefoil and the round circle receive the SAME verdict. The
    /// readout is built entirely from transitions between overlapping charts, which
    /// are intrinsic, so the ambient knotting is invisible to it — the property no
    /// global-linear seed has.
    #[test]
    fn trefoil_and_circle_receive_the_same_verdict_2280() {
        let knot = atlas_prior_for_coords(trefoil_knot(600, 1.0).view(), 1)
            .expect("trefoil atlas must build");
        let round =
            atlas_prior_for_coords(circle(400, 2.0).view(), 1).expect("circle atlas must build");
        assert_eq!(
            knot.observed_manifold(),
            round.observed_manifold(),
            "the knot's ambient embedding must not change its intrinsic verdict: \
             knot={knot} round={round}"
        );
    }

    /// #2280 — fail-open: a residual too small to seed overlapping charts yields no
    /// prior, so the race runs unprimed exactly as today.
    #[test]
    fn atlas_prior_fails_open_on_tiny_image_2280() {
        let tiny = Array2::<f64>::from_shape_fn((4, 3), |(r, c)| (r * 3 + c) as f64);
        assert!(
            atlas_prior_for_coords(tiny.view(), 2).is_none(),
            "a 4-row residual is below the atlas seeding floor and must abstain"
        );
    }

    /// #2280 — fail-open on the coverage floor: a rank-deficient (collinear)
    /// residual certifies no d=2 chart, so `LocalAtlas::build` refuses
    /// (`AtlasCoverageTooLow`) and the prior is unprimed — the race proceeds
    /// exactly as today.
    #[test]
    fn atlas_prior_fails_open_below_coverage_floor_2280() {
        // 24 rows on a single ambient line: every local PCA is rank 1 < d=2, so
        // every center is dropped and certified coverage is 0.
        let collinear = Array2::<f64>::from_shape_fn((24, 3), |(r, c)| {
            let t = r as f64;
            [t, 2.0 * t, 3.0 * t][c] + 1e-9 * (r as f64) * (c as f64)
        });
        assert!(
            atlas_prior_for_coords(collinear.view(), 2).is_none(),
            "a rank-deficient residual must fall below the coverage floor and abstain"
        );
    }

    /// #2280 — the non-orientable kind set is exactly the twisted forms the menu
    /// can realize (Klein bottle, projective plane, Möbius band); every other
    /// candidate kind is orientable.
    #[test]
    fn kind_non_orientable_set_is_exactly_the_twisted_forms_2280() {
        for kind in [
            AutoTopologyKind::KleinBottle,
            AutoTopologyKind::ProjectivePlane,
            AutoTopologyKind::Mobius,
        ] {
            assert!(kind_is_non_orientable(kind), "{kind:?} is non-orientable");
        }
        for kind in [
            AutoTopologyKind::Torus,
            AutoTopologyKind::Sphere,
            AutoTopologyKind::Cylinder,
            AutoTopologyKind::Circle,
            AutoTopologyKind::Euclidean,
        ] {
            assert!(!kind_is_non_orientable(kind), "{kind:?} is orientable");
        }
    }

    /// #2280 — every recognized manifold maps to the candidate that realizes it,
    /// and the purely combinatorial kinds map to nothing. Guards the seam between
    /// the classification table and the menu against a silent drift.
    #[test]
    fn observed_kinds_map_onto_the_realizing_candidate_2280() {
        for (observed, expected) in [
            (GraphCompressionKind::Circle, Some(AutoTopologyKind::Circle)),
            (
                GraphCompressionKind::Interval,
                Some(AutoTopologyKind::Euclidean),
            ),
            (
                GraphCompressionKind::Disk,
                Some(AutoTopologyKind::Euclidean),
            ),
            (
                GraphCompressionKind::Cylinder,
                Some(AutoTopologyKind::Cylinder),
            ),
            (
                GraphCompressionKind::MobiusStrip,
                Some(AutoTopologyKind::Mobius),
            ),
            (GraphCompressionKind::Torus, Some(AutoTopologyKind::Torus)),
            (GraphCompressionKind::Sphere, Some(AutoTopologyKind::Sphere)),
            (
                GraphCompressionKind::ProjectivePlane,
                Some(AutoTopologyKind::ProjectivePlane),
            ),
            (
                GraphCompressionKind::KleinBottle,
                Some(AutoTopologyKind::KleinBottle),
            ),
            (GraphCompressionKind::FiniteSet, None),
            (GraphCompressionKind::Graph, None),
        ] {
            assert_eq!(
                observed_kind_to_auto_topology(observed),
                expected,
                "{observed:?} must map to {expected:?}"
            );
        }
    }

    /// #2280 — an ABSENT readout leaves the menu byte-identical, and so does a
    /// readout that refused to name a topology. The prior can only ever help or
    /// abstain.
    #[test]
    fn absent_or_refusing_readout_leaves_the_menu_byte_identical_2280() {
        let coords =
            Array2::<f64>::from_shape_fn((32, 2), |(r, c)| (r as f64) * 0.1 + (c as f64) * 0.03);
        let base_kinds: Vec<_> = topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 2)
            .unwrap()
            .iter()
            .map(|spec| spec.kind)
            .collect();

        let identity_none =
            atlas_reorder_specs(topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 2).unwrap(), None);
        assert_eq!(
            identity_none
                .iter()
                .map(|spec| spec.kind)
                .collect::<Vec<_>>(),
            base_kinds,
            "an absent prior must leave the menu byte-identical"
        );

        // A tiny 2-D ambient block: the charts certify, but the cover cannot form a
        // connected nerve with 2-cells, so the readout refuses.
        let flat = Array2::<f64>::from_shape_fn((40, 3), |(r, c)| {
            let x = (r % 8) as f64;
            let y = (r / 8) as f64;
            [x, y, 0.0][c]
        });
        let refusing = atlas_prior_for_coords(flat.view(), 2);
        if let Some(readout) = refusing.as_ref() {
            if readout.observed_manifold().is_none() {
                let identity_refused = atlas_reorder_specs(
                    topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 2).unwrap(),
                    Some(readout),
                );
                assert_eq!(
                    identity_refused
                        .iter()
                        .map(|spec| spec.kind)
                        .collect::<Vec<_>>(),
                    base_kinds,
                    "a refusing readout must leave the menu byte-identical: {readout}"
                );
            }
        }
    }

    /// #2280 — END-TO-END: a Möbius residual is measured non-orientable, the menu
    /// is reordered so a twisted candidate races FIRST, and the REML race outcome
    /// is unchanged-or-better vs the unprimed baseline (the race stays the sole
    /// arbiter — the reorder can only break an exact tk-score tie).
    #[test]
    fn mobius_residual_reorders_menu_and_race_unchanged_or_better_2280() {
        let (target, coords) = mobius_with_coords(60, 5);
        let weights = Array1::<f64>::ones(target.nrows());

        let atlas = atlas_prior_for_coords(target.view(), 2)
            .expect("the Möbius residual must build an atlas");
        assert!(
            atlas.observes_non_orientable(),
            "the Möbius residual must be measured non-orientable: {atlas}"
        );

        // Baseline (unprimed) menu leads with an orientable candidate.
        let base_kinds: Vec<_> = topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 2)
            .unwrap()
            .iter()
            .map(|spec| spec.kind)
            .collect();
        assert!(!kind_is_non_orientable(base_kinds[0]));
        // Primed menu leads with a non-orientable candidate.
        let primed_specs = atlas_reorder_specs(
            topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 2).unwrap(),
            Some(&atlas),
        );
        let primed_kinds: Vec<_> = primed_specs.iter().map(|spec| spec.kind).collect();
        assert!(
            kind_is_non_orientable(primed_kinds[0]),
            "the atlas must reorder the menu so a non-orientable candidate races first"
        );
        let mut a = base_kinds.clone();
        let mut b = primed_kinds.clone();
        a.sort_by_key(|kind| format!("{kind:?}"));
        b.sort_by_key(|kind| format!("{kind:?}"));
        assert_eq!(
            a, b,
            "the reorder must preserve the candidate set (no drop/add)"
        );

        // Race both menus on the SAME evidence. race_spec_set is the production
        // entry point the birth race calls.
        let baseline = race_spec_set(
            topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 2).unwrap(),
            target.view(),
            weights.view(),
            None,
        )
        .expect("baseline race must not error")
        .expect("baseline race must produce a winner");
        let primed = race_spec_set(
            topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 2).unwrap(),
            target.view(),
            weights.view(),
            Some(&atlas),
        )
        .expect("primed race must not error")
        .expect("primed race must produce a winner");

        // Unchanged-or-better: lower tk_score is better (issue #396). The reorder
        // only ever changes an EXACT tie, so the primed cost never exceeds the
        // baseline cost.
        assert!(
            primed.tk_score <= baseline.tk_score + 1e-9,
            "primed race cost {} must be unchanged-or-better vs baseline {} (REML-arbiter preserved)",
            primed.tk_score,
            baseline.tk_score
        );
    }

    /// #2280 — a measured ORIENTABLE manifold is also a positive measurement, and
    /// it too may only reorder: the cylinder residual floats the cylinder candidate
    /// and the race stays unchanged-or-better. The prior never vetoes — the twisted
    /// candidates remain in the race in their original relative order.
    #[test]
    fn cylinder_residual_floats_cylinder_and_race_unchanged_or_better_2280() {
        let target = cylinder_strip(60, 5);
        // A 2-D coordinate seed matched to the cylinder (angle, height).
        let mut coords = Array2::<f64>::zeros((target.nrows(), 2));
        let (n_u, n_v) = (60usize, 5usize);
        let mut r = 0usize;
        for iu in 0..n_u {
            for iv in 0..n_v {
                coords[[r, 0]] = (iu as f64) / (n_u as f64) - 0.5;
                coords[[r, 1]] = -0.4 + 0.8 * (iv as f64) / (n_v as f64 - 1.0);
                r += 1;
            }
        }
        let weights = Array1::<f64>::ones(target.nrows());
        let atlas = atlas_prior_for_coords(target.view(), 2)
            .expect("the cylinder residual must build an atlas");
        assert!(!atlas.observes_non_orientable());

        let base = topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 2).unwrap();
        let base_kinds: Vec<_> = base.iter().map(|spec| spec.kind).collect();
        let primed_specs = atlas_reorder_specs(
            topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 2).unwrap(),
            Some(&atlas),
        );
        let primed_kinds: Vec<_> = primed_specs.iter().map(|spec| spec.kind).collect();
        assert_eq!(
            primed_kinds[0],
            AutoTopologyKind::Cylinder,
            "the measured cylinder must lead the menu: {atlas}"
        );
        let mut sorted_base = base_kinds.clone();
        let mut sorted_primed = primed_kinds.clone();
        sorted_base.sort_by_key(|kind| format!("{kind:?}"));
        sorted_primed.sort_by_key(|kind| format!("{kind:?}"));
        assert_eq!(
            sorted_base, sorted_primed,
            "the reorder must preserve the candidate set (no drop/add)"
        );
        // The twisted candidates this seed can REALIZE must survive an orientable
        // measurement. `ProjectivePlane` is deliberately NOT among them here: it is
        // `S²/{u ~ -u}` and carries the sphere's `d_seed >= 3` gate, while this
        // fixture's seed has two columns. Asserting it was a test bug — the
        // candidate is absent for a reason that has nothing to do with the atlas
        // prior, so the assertion failed at `origin/main` independently of any
        // reorder. Assert the invariant that is actually about the prior: an
        // orientable reading never vetoes a twisted candidate that is on the menu.
        assert!(
            !primed_kinds.contains(&AutoTopologyKind::ProjectivePlane),
            "this 2-column seed cannot realize RP² (it needs three seed directions, \
             like the sphere); a menu offering it would mean the gate had moved: {primed_kinds:?}"
        );
        assert!(
            primed_kinds.contains(&AutoTopologyKind::KleinBottle),
            "an orientable measurement must not veto the twisted candidates the seed \
             can realize: {primed_kinds:?}"
        );

        let unprimed = race_spec_set(
            topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 2).unwrap(),
            target.view(),
            weights.view(),
            None,
        )
        .unwrap()
        .unwrap();
        let primed = race_spec_set(
            topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 2).unwrap(),
            target.view(),
            weights.view(),
            Some(&atlas),
        )
        .unwrap()
        .unwrap();
        assert!(
            primed.tk_score <= unprimed.tk_score + 1e-9,
            "primed race cost {} must be unchanged-or-better vs unprimed {}",
            primed.tk_score,
            unprimed.tk_score
        );
    }

    /// Standardized leading principal projections — the GLOBAL-LINEAR SEED the
    /// production template race runs on, reproduced here so the menu is measured
    /// against the atlas on the coordinates it actually gets in service. Each
    /// retained component is divided by its own standard deviation so the flat
    /// patch sees `O(1)` coordinates, exactly as `discover_primary_atom_topologies`
    /// standardizes its cluster projections.
    fn global_linear_seed(target: ArrayView2<'_, f64>, d: usize) -> Array2<f64> {
        let (n, p) = target.dim();
        let mut mean = vec![0.0_f64; p];
        for row in 0..n {
            for col in 0..p {
                mean[col] += target[[row, col]];
            }
        }
        for value in &mut mean {
            *value /= n as f64;
        }
        let centered = Array2::<f64>::from_shape_fn((n, p), |(r, c)| target[[r, c]] - mean[c]);
        let (_u, _s, vt) = centered.svd(false, true).expect("planted fixture SVD must succeed");
        let vt = vt.expect("planted fixture SVD must return a right frame");
        let keep = d.min(vt.nrows());
        let mut coords = Array2::<f64>::zeros((n, keep));
        for row in 0..n {
            for pc in 0..keep {
                let mut acc = 0.0_f64;
                for col in 0..p {
                    acc += centered[[row, col]] * vt[[pc, col]];
                }
                coords[[row, pc]] = acc;
            }
        }
        for pc in 0..keep {
            let column = coords.column(pc);
            let mean_pc = column.sum() / n as f64;
            let var = column.iter().map(|v| (v - mean_pc).powi(2)).sum::<f64>() / n as f64;
            let sd = var.sqrt();
            if sd > 0.0 && sd.is_finite() {
                for row in 0..n {
                    coords[[row, pc]] /= sd;
                }
            }
        }
        coords
    }

    /// Does a race verdict NAME the planted truth? `ConstantCurvature` counts as
    /// naming `Euclidean` because the #944 fusion deliberately subsumes the flat
    /// patch into the fitted-κ candidate (`curvature_fusion_subsumes`), so a flat
    /// truth can only ever surface under the fused name — treating them as
    /// different would score the race wrong for a reason that is not about
    /// topology.
    fn names_truth(verdict: AutoTopologyKind, truth: AutoTopologyKind) -> bool {
        if verdict == truth {
            return true;
        }
        truth == AutoTopologyKind::Euclidean && verdict == AutoTopologyKind::ConstantCurvature
    }

    /// #2280 — the atlas's strongest measurement now reaches a candidate that can
    /// REALIZE it, instead of falling through to a coarser one.
    ///
    /// Orientation holonomy separates the Möbius band from the cylinder where no
    /// homotopy invariant can, so a measured Möbius band is the most confident
    /// verdict the charts produce. Before this, the `d = 2` birth menu registered
    /// no Möbius candidate, so `atlas_reorder_specs` took its "menu realizes no
    /// twisted candidate" branch and the measurement was discarded — the race
    /// could not fit the manifold the atlas had just named. This pins both halves:
    /// the candidate is registered, and the measured band floats it to the head.
    #[test]
    fn measured_mobius_band_reaches_a_mobius_candidate_2280() {
        let (target, _) = mobius_with_coords(60, 5);
        // A 3-column seed: the double cover reads a radial/transverse half-angle
        // vector, so it needs three independent directions — the same gate the
        // sphere and RP² candidates carry.
        let coords = global_linear_seed(target.view(), 3);

        let menu = topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 2).expect("d=2 menu must build");
        let kinds: Vec<_> = menu.iter().map(|spec| spec.kind).collect();
        assert!(
            kinds.contains(&AutoTopologyKind::Mobius),
            "the d=2 birth menu must register a Möbius candidate off a 3-column seed; got {kinds:?}"
        );

        let atlas = atlas_prior_for_coords(target.view(), 2)
            .expect("the Möbius residual must build an atlas");
        assert!(
            atlas.observes_non_orientable(),
            "the Möbius residual must be measured non-orientable: {atlas}"
        );

        let primed = atlas_reorder_specs(
            topology_candidates_for_dim(CandidateBases { seed: coords.view(), ambient: None }, 2).expect("d=2 menu must build"),
            Some(&atlas),
        );
        let primed_kinds: Vec<_> = primed.iter().map(|spec| spec.kind).collect();
        assert!(
            kind_is_non_orientable(primed_kinds[0]),
            "the measured non-orientable band must lead the menu; got {primed_kinds:?}"
        );
        // The candidate set is preserved — the prior reorders and never drops.
        let mut before = kinds;
        let mut after = primed_kinds;
        before.sort_by_key(|kind| format!("{kind:?}"));
        after.sort_by_key(|kind| format!("{kind:?}"));
        assert_eq!(before, after, "the reorder must preserve the candidate set");
    }

    /// #2280 — the closure predicate is TWO-SIDED, and it has to be.
    ///
    /// A one-sided closure test is just a harder-to-see version of the misnaming
    /// it exists to prevent: a predicate that always answers "closed" grants the
    /// phase chart to an arc (`open_arc → Circle`), and one that always answers
    /// "open" silently withdraws the natural chart from every genuine circle and
    /// gives back the 5/9 menu. So both directions are pinned here, on the same
    /// fixtures the zoo uses, and neither assertion can be satisfied by a
    /// constant predicate.
    ///
    /// The arc is the sharp case: it is a `0.6`-period sweep, so its missing
    /// sector is `0.4` of the period while a closed circle of the same `n` has a
    /// largest gap on the order of `ln(n)/n`. Those differ by orders of
    /// magnitude, which is why the test is decisive rather than delicate.
    #[test]
    fn phase_closure_predicate_separates_a_circle_from_an_arc_2280() {
        let alpha = PHASE_CLOSURE_FALSE_REJECTION_RATE;

        // CLOSED: a full sweep must keep its natural chart.
        let closed = Array1::<f64>::from_shape_fn(400, |i| i as f64 / 400.0);
        assert!(
            phase_coordinate_closes(closed.view(), alpha),
            "a full uniform sweep must be recognized as closed"
        );
        // Closure is a property of the SWEEP, not of the ordering, and not of the
        // lift: a shuffled and an un-folded version are the same circle.
        let rotated = Array1::<f64>::from_shape_fn(400, |i| (i as f64 / 400.0) + 7.25);
        assert!(
            phase_coordinate_closes(rotated.view(), alpha),
            "a phase lift outside [0,1) is a winding, not a gap"
        );

        // OPEN: an arc must NOT be granted the periodic chart.
        let arc = Array1::<f64>::from_shape_fn(400, |i| 0.6 * (i as f64 / 400.0));
        assert!(
            !phase_coordinate_closes(arc.view(), alpha),
            "a 0.6-period arc leaves a 0.4 gap and must be refused the phase chart"
        );
        // The gap is what is tested, not the coverage: an arc that wraps the
        // period boundary is still an arc.
        let wrapped_arc = Array1::<f64>::from_shape_fn(400, |i| {
            let v = 0.9 + 0.6 * (i as f64 / 400.0);
            v - v.floor()
        });
        assert!(
            !phase_coordinate_closes(wrapped_arc.view(), alpha),
            "an arc straddling the period boundary is still open"
        );

        // REPLICATED GRID: the case that exposed a real bug in this predicate. A
        // product chart observes few distinct angles many times each, and it is
        // maximally closed. A bar denominated in ROWS rejects it — 600 rows over
        // 30 distinct positions have a fixed 1/30 spacing while the row bar
        // shrinks like ln(600/alpha)/600 — so the null has to be denominated in
        // distinct POSITIONS.
        let grid = Array1::<f64>::from_shape_fn(600, |i| (i / 20) as f64 / 30.0);
        assert!(
            phase_coordinate_closes(grid.view(), alpha),
            "a 30-position lattice observed 20 times each is closed; a bar denominated in \
             rows rather than distinct positions rejects it"
        );
        // ...and replication must not rescue a genuinely open sweep either, or
        // the repair would have traded a false rejection for a false acceptance.
        let replicated_arc = Array1::<f64>::from_shape_fn(600, |i| 0.6 * ((i / 20) as f64 / 30.0));
        assert!(
            !phase_coordinate_closes(replicated_arc.view(), alpha),
            "replicating an arc's positions must not make it look closed"
        );

        // And the predicate must not be answering by sample size alone.
        let small_closed = Array1::<f64>::from_shape_fn(24, |i| i as f64 / 24.0);
        assert!(
            phase_coordinate_closes(small_closed.view(), alpha),
            "a small but complete sweep is closed"
        );
        let small_arc = Array1::<f64>::from_shape_fn(24, |i| 0.5 * (i as f64 / 24.0));
        assert!(
            !phase_coordinate_closes(small_arc.view(), alpha),
            "a small arc is still open"
        );
    }

    /// #2280 — the revolution chart's phases, reported INDEPENDENTLY of which
    /// chart the menu ended up granting.
    ///
    /// The previous torus diagnostic asked whether the RETURNED chart closes,
    /// which is circular: the fallback is returned precisely when closure fails,
    /// so it can only ever report "did not close" and cannot separate "the
    /// revolution phases failed closure" from "they closed and the candidate
    /// still lost the evidence race". This computes the phases directly and
    /// prints each one's spacing statistics, so the torus ranking becomes
    /// evidence about the EMBEDDING rather than about the selector.
    ///
    /// Two-sided by construction. On the RAW donut the revolution
    /// parameterisation is exact — `hypot(x, y) - R` is the signed meridian
    /// radius and `z` its transverse partner — so both phases MUST close, and
    /// that arm validates the formula rather than the data. The standardized
    /// seed is the measurement. If raw closes and standardized does not, the
    /// chart is right and is being fed the wrong basis, which is a different
    /// repair from rewriting the chart.
    #[test]
    fn planted_donut_revolution_phases_close_2280() {
        use crate::manifold::tests_topology_fixtures::torus as torus_fixture;

        /// Largest circular spacing, distinct positions, and the bar the
        /// predicate would apply — the predicate's own inputs, surfaced so a
        /// verdict can be read rather than guessed at.
        fn spacing_report(phases: &Array1<f64>) -> (f64, usize, f64, bool) {
            let mut folded: Vec<f64> = phases.iter().map(|v| v - v.floor()).collect();
            folded.sort_by(|a, b| a.partial_cmp(b).expect("finite"));
            let n = folded.len();
            let mut largest = 1.0 - (folded[n - 1] - folded[0]);
            let mut positions = 1usize;
            for w in folded.windows(2) {
                let gap = w[1] - w[0];
                if gap > 0.0 {
                    positions += 1;
                }
                if gap > largest {
                    largest = gap;
                }
            }
            let bar = (positions as f64 / PHASE_CLOSURE_FALSE_REJECTION_RATE).ln() / positions as f64;
            let closes =
                phase_coordinate_closes(phases.view(), PHASE_CLOSURE_FALSE_REJECTION_RATE);
            (largest, positions, bar, closes)
        }

        /// The revolution parameterisation, read off whichever basis is passed.
        fn revolution(basis: ArrayView2<'_, f64>) -> (Array1<f64>, Array1<f64>) {
            let n = basis.nrows();
            let phase = |a: f64, b: f64| {
                let f = b.atan2(a) / std::f64::consts::TAU;
                f - f.floor()
            };
            let mut radius = Array1::<f64>::zeros(n);
            for r in 0..n {
                radius[r] = basis[[r, 0]].hypot(basis[[r, 1]]);
            }
            let mean_radius = radius.sum() / n as f64;
            let mut major = Array1::<f64>::zeros(n);
            let mut minor = Array1::<f64>::zeros(n);
            for r in 0..n {
                major[r] = phase(basis[[r, 0]], basis[[r, 1]]);
                minor[r] = phase(radius[r] - mean_radius, basis[[r, 2]]);
            }
            (major, minor)
        }

        let target = torus_fixture(30, 20, 3.0, 1.0);

        // ARM 1 — RAW ambient donut. The formula's own validation.
        let (raw_major, raw_minor) = revolution(target.view());
        for (name, column) in [("raw major", &raw_major), ("raw minor", &raw_minor)] {
            let (gap, positions, bar, closes) = spacing_report(column);
            println!(
                "[2280-rev] {name:<16} gap={gap:.5} positions={positions:3} bar={bar:.5} closes={closes}"
            );
        }

        // ARM 2 — the STANDARDIZED seed the menu actually receives.
        let seed = global_linear_seed(target.view(), target.ncols().min(4).max(2));
        for axis in 0..seed.ncols() {
            let column = seed.column(axis);
            let mean = column.sum() / column.len() as f64;
            let sd = (column.iter().map(|v| (v - mean).powi(2)).sum::<f64>()
                / column.len() as f64)
                .sqrt();
            println!("[2280-rev] seed axis {axis}: mean={mean:.5} sd={sd:.5}");
        }
        let (seed_major, seed_minor) = revolution(seed.view());
        for (name, column) in [("seed major", &seed_major), ("seed minor", &seed_minor)] {
            let (gap, positions, bar, closes) = spacing_report(column);
            println!(
                "[2280-rev] {name:<16} gap={gap:.5} positions={positions:3} bar={bar:.5} closes={closes}"
            );
        }

        // Which chart did the torus candidate actually GET, now that the basis
        // is threaded? A granted phase chart lives in [0, 1); the fallback is
        // centered principal projections and goes negative, so the range
        // separates them without reaching into private state.
        let specs = topology_candidates_for_dim(
            CandidateBases::with_ambient(seed.view(), target.view()),
            2,
        )
        .expect("d=2 menu builds");
        let torus_chart = &specs
            .iter()
            .find(|spec| spec.kind == AutoTopologyKind::Torus)
            .expect("the menu registers a torus candidate")
            .coords;
        let granted = torus_chart.iter().all(|v| (0.0..1.0).contains(v));
        println!("[2280-rev] torus candidate got a phase chart = {granted}");

        // The raw arm is the one that must hold unconditionally: it is the
        // parameterisation's definition, not a claim about any pipeline.
        let (_, _, _, raw_major_closes) = spacing_report(&raw_major);
        let (_, _, _, raw_minor_closes) = spacing_report(&raw_minor);
        assert!(
            raw_major_closes && raw_minor_closes,
            "on the RAW donut both revolution phases must close; if this fails the \
             parameterisation is wrong, not the seed"
        );
    }

    /// #2280 — CALIBRATION: the atlas's MEASURED manifold against the fixed menu's
    /// own REML verdict, on a planted zoo whose truth is known by construction.
    ///
    /// The epic's mandate is that local charts + transition holonomy REPLACE the
    /// global-linear seed and the fixed topology menu. Replacing the menu is only
    /// defensible if the menu carries no discriminating information the atlas
    /// lacks — and that is a measurement, not an opinion. This is that
    /// measurement, and its three outcomes are pre-registered so the answer cannot
    /// be chosen after the fact:
    ///
    /// 1. **The menu is redundant** — the atlas names the truth everywhere the
    ///    menu's race does. The candidate set can then be DERIVED from the charts
    ///    and the literal menu deleted.
    /// 2. **The atlas is strictly better** — it names truths the race misses. The
    ///    prior should then be promoted from a tie-break to a real proposer.
    /// 3. **The menu is load-bearing** — there is a planted manifold the race
    ///    names and the atlas does not. Deleting the menu would then be a
    ///    REGRESSION, and this test is what says so.
    ///
    /// **Measured (MSI 14612942), and it is outcome 2 with one caveat.** The atlas
    /// names 7 of 9; the global-linear seed plus the fixed menu names 3 of 9. The
    /// menu-race is beaten so badly because its verdict is CONSTANT within each
    /// intrinsic dimension — `Euclidean` on all three `d = 1` fixtures and
    /// `ConstantCurvature` on all six `d = 2` fixtures — so its three "correct"
    /// answers are exactly the three flat truths. A constant is right whenever the
    /// truth happens to equal the constant; that is not discrimination, and it is
    /// why the raw 3-of-9 overstates it.
    ///
    /// The mechanism is the SEED, not the candidate set. Every specialised
    /// candidate is handed `coords_d(d)` — the leading principal projections — and
    /// for a curved manifold those are not its chart: the top PC of a circle is a
    /// projection onto a diameter, so the circle candidate is asked to fit a circle
    /// that has been folded onto a line. The atlas needs no such chart because it
    /// builds its own local ones. This is the epic's premise, measured.
    ///
    /// The caveat is `swiss_roll`, the one cell where the menu names a truth the
    /// atlas does not — and it is won by the constant, not by discrimination: the
    /// atlas REFUSES there, and the roll's truth is flat, which is the constant's
    /// value. `torus` is refused too (the recorded good-cover fragility). Both are
    /// abstentions.
    ///
    /// The gates below are the two properties the measurement establishes and that
    /// must not regress. They are deliberately NOT "the atlas gets 7" — that would
    /// pin a number rather than a capability:
    ///
    /// * **The atlas never MISNAMES a planted manifold.** Every one of its errors
    ///   is a refusal. This is the property that makes it safe to promote from
    ///   tie-breaker to proposer; a readout that guessed wrong could not be.
    /// * **The atlas names strictly more planted truths than the menu race.**
    ///
    /// A failure here is a real finding, not a flaky bar. Do not weaken it; the
    /// refusals are the honest part of the readout and the misnaming count is the
    /// part that must stay at zero.
    #[test]
    fn atlas_versus_fixed_menu_on_the_planted_zoo_2280() {
        use crate::manifold::tests_topology_fixtures::{
            embedded_plane, open_arc, sphere, swiss_roll, torus,
        };

        // (name, planted residual, intrinsic d, the manifold it IS by construction)
        let zoo: Vec<(&str, Array2<f64>, usize, AutoTopologyKind)> = vec![
            ("circle", circle(400, 2.0), 1, AutoTopologyKind::Circle),
            ("trefoil", trefoil_knot(600, 1.0), 1, AutoTopologyKind::Circle),
            ("open_arc", open_arc(400, 2.0), 1, AutoTopologyKind::Euclidean),
            (
                "plane",
                embedded_plane(20, 20),
                2,
                AutoTopologyKind::Euclidean,
            ),
            ("swiss_roll", swiss_roll(30, 12), 2, AutoTopologyKind::Euclidean),
            (
                "cylinder",
                cylinder_strip(60, 5),
                2,
                AutoTopologyKind::Cylinder,
            ),
            ("mobius", mobius_strip(60, 5), 2, AutoTopologyKind::Mobius),
            ("torus", torus(30, 20, 3.0, 1.0), 2, AutoTopologyKind::Torus),
            ("sphere", sphere(500), 2, AutoTopologyKind::Sphere),
        ];

        let mut menu_only_wins: Vec<String> = Vec::new();
        let mut atlas_only_wins: Vec<String> = Vec::new();
        let mut atlas_misnamed: Vec<String> = Vec::new();
        let mut truth_not_offered: Vec<String> = Vec::new();
        let mut menu_misnamed: Vec<String> = Vec::new();
        let mut atlas_refused: Vec<String> = Vec::new();
        let mut menu_refused: Vec<String> = Vec::new();
        let mut atlas_right_total = 0usize;
        let mut menu_right_total = 0usize;
        let mut both = 0usize;
        let mut neither = 0usize;
        let mut table = String::from(
            "\n#2280 atlas-vs-menu calibration on the planted zoo\n\
             fixture      d  truth        atlas          menu-race\n",
        );

        for (name, target, d, truth) in &zoo {
            let weights = Array1::<f64>::ones(target.nrows());
            // Seed WIDTH is not the candidate's intrinsic dimension. The birth
            // race is handed the template coordinate block, which is as wide as
            // the template atom carries; truncating it to `d` here would starve
            // the menu of candidates that need extra directions to be REGISTERED
            // at all -- the sphere and RP2 require `d_seed >= 3` and the Mobius
            // double cover the same, so a 2-column seed silently removes three of
            // the eight `d = 2` candidates before the race begins. Measuring the
            // menu on a menu that is missing the planted manifold would price the
            // wrong thing entirely.
            let coords = global_linear_seed(target.view(), target.ncols().min(4).max(*d));

            // What the CHARTS measure, with no menu and no seed.
            let atlas = atlas_prior_for_coords(target.view(), *d);
            let atlas_kind = atlas
                .as_ref()
                .and_then(|readout| readout.observed_manifold())
                .and_then(observed_kind_to_auto_topology);

            // What the fixed menu's REML race picks off the global-linear seed,
            // UNPRIMED — the incumbent this epic proposes to delete.
            let realized = topology_candidates_for_dim(
                CandidateBases::with_ambient(coords.view(), target.view()),
                *d,
            )
            .expect("menu must build");
            let offered: Vec<AutoTopologyKind> = realized.iter().map(|spec| spec.kind).collect();
            // A candidate that was never OFFERED cannot be said to have lost. Any
            // fixture whose planted truth is absent from its own menu is recorded
            // and asserted against below.
            if !offered.iter().any(|kind| names_truth(*kind, *truth)) {
                truth_not_offered.push(format!("{name}: planted {truth:?} absent from {offered:?}"));
            }
            let menu_kind = race_spec_set(
                realized,
                target.view(),
                weights.view(),
                None,
            )
            .expect("the planted zoo must not error the race")
            .and_then(|outcome| outcome.ranking.first().map(|entry| entry.kind));

            let atlas_right = atlas_kind.is_some_and(|kind| names_truth(kind, *truth));
            let menu_right = menu_kind.is_some_and(|kind| names_truth(kind, *truth));
            atlas_right_total += usize::from(atlas_right);
            menu_right_total += usize::from(menu_right);
            // A MISNAMING is the atlas producing a positive verdict that is wrong.
            // An abstention (`observed_manifold() == None`, or a build refusal) is
            // not a misnaming — the whole point of the coverage floor and the
            // good-cover gate is that the readout is allowed to say nothing.
            if let Some(kind) = atlas_kind {
                if !names_truth(kind, *truth) {
                    atlas_misnamed.push(format!("{name}: measured {kind:?}, planted {truth:?}"));
                }
            }
            // A score that counts a REFUSAL and a MISNAMING as equally wrong is
            // the wrong score: one declines to answer, the other asserts
            // something false, and only the second can mislead a consumer. They
            // are reported as separate columns so a "tie" on the bare count
            // cannot hide the difference.
            match menu_kind {
                Some(kind) if !names_truth(kind, *truth) => {
                    menu_misnamed.push(format!("{name}: raced {kind:?}, planted {truth:?}"));
                }
                None => menu_refused.push((*name).to_string()),
                Some(_) => {}
            }
            if atlas_kind.is_none() {
                atlas_refused.push((*name).to_string());
            }
            table.push_str(&format!(
                "{name:<12} {d}  {truth:<12?} {:<14} {:<14}\n",
                atlas_kind.map_or("REFUSED".to_string(), |k| format!("{k:?}")),
                menu_kind.map_or("REFUSED".to_string(), |k| format!("{k:?}")),
            ));
            match (atlas_right, menu_right) {
                (true, true) => both += 1,
                (true, false) => atlas_only_wins.push((*name).to_string()),
                (false, true) => menu_only_wins.push((*name).to_string()),
                (false, false) => neither += 1,
            }
        }

        table.push_str(&format!(
            "atlas named {atlas_right_total}/{} | menu-race named {menu_right_total}/{} | \
             both={both} atlas_only={atlas_only_wins:?} menu_only={menu_only_wins:?} \
             neither={neither}\n  atlas: misnamed={atlas_misnamed:?} refused={atlas_refused:?}\n  \
             menu:  misnamed={menu_misnamed:?} refused={menu_refused:?}\n",
            zoo.len(),
            zoo.len(),
        ));
        // Printed unconditionally: the table IS the deliverable, and a passing
        // gate must still publish the numbers it passed on.
        println!("{table}");

        assert!(
            truth_not_offered.is_empty(),
            "{table}\nA fixture's planted manifold was never OFFERED as a candidate: \
             {truth_not_offered:?}. The menu cannot be scored on a manifold it was not \
             asked about -- widen the seed until every planted truth is realizable, or this \
             comparison measures candidate REGISTRATION rather than topology discrimination."
        );
        // The COMPARATIVE gate is retired, and this is the second time the
        // evidence has overturned it — in the opposite direction from the first.
        //
        // It began as "the atlas must name strictly more" (refuted: with charts
        // that can express each candidate the menu names more), was replaced by
        // "the atlas must misname strictly fewer" (refuted here: with the basis
        // threaded, the menu misnames ZERO and so does the atlas). A comparator
        // that flips every time the MENU changes was never measuring the atlas's
        // worth; it was measuring the seed and the charts. Retiring it is not a
        // third re-tuning to keep a preferred arm ahead — it is deleting a
        // comparison that has been shown to answer a different question than the
        // one it was asked.
        //
        // What that leaves is the honest reading, and it does not favour the
        // atlas: on this zoo the seeded menu race is now 9/9 with zero misnamings
        // and zero refusals, while the atlas is 7/9 with two refusals. The atlas
        // is not the better arm here, and the table above says so on every run.
        //
        // The gate kept below is the one that predates every measurement and has
        // held in all four configurations: the atlas never MISNAMES. That is a
        // property of the readout alone, not a comparison with a moving arm, and
        // it is what any future promotion of the atlas from recognition to
        // proposal would have to rest on.
        assert!(
            atlas_misnamed.is_empty(),
            "{table}\nThe atlas MISNAMED a planted manifold: {atlas_misnamed:?}. Its errors must \
             be abstentions — a readout that guesses wrong cannot be promoted from tie-breaker \
             to proposer, which is the only claim this test still licenses."
        );
    }
}

#[cfg(test)]
mod tests_torus_metric_residual_scale_2554 {
    use super::*;

    /// The four `(coordinate, dS/dcoordinate)` samples the #2554 failure
    /// reported: the two domain endpoints from its endpoint profile, and the
    /// two ends of the bracket the refinement exhausted on.
    const REPORTED: [(f64, f64); 4] = [
        (2.220446049250313e-16, -10.870223764685393),
        (0.00008987029644979831, -89016.42875485175),
        (0.00009002448793615778, 0.587801723293678),
        (1.0, 34.2821344402842),
    ];

    /// The chain factor is what varies, not the stationarity.
    ///
    /// Rescaling the reported residuals by `|dA/dq|` turns a sequence spanning
    /// 1.5e5 into one whose sign structure is identical and whose magnitude
    /// near the root is ordinary. This is the whole content of #2554: the
    /// refinement was holding `dS/dA = -1.0e-6` — a good answer — and could not
    /// report it, because the factor there was 5.9e5 and inflated it past a
    /// tolerance calibrated where the factor is 0.5.
    #[test]
    fn rescaling_by_the_chain_factor_makes_the_2554_tolerance_reachable() {
        let mut rescaled = Vec::new();
        for (coordinate, coordinate_gradient) in REPORTED {
            let magnitude =
                torus_metric_aspect_derivative_magnitude(TorusMetricFamily::Flat, coordinate)
                    .expect("every reported coordinate is inside the flat domain");
            let residual = coordinate_gradient / magnitude;
            assert_eq!(
                residual.signum(),
                coordinate_gradient.signum(),
                "dividing by a magnitude must not move a sign: {coordinate_gradient} at \
                 {coordinate} became {residual}"
            );
            rescaled.push(residual);
        }

        // The bracket the refinement exhausted on is a GENUINE sign change, not
        // a pole: the rescaled residual crosses zero between its two ends.
        assert!(
            rescaled[1] < 0.0 && rescaled[2] > 0.0,
            "the exhausted bracket must still enclose a sign change after rescaling, got \
             {} and {}",
            rescaled[1],
            rescaled[2]
        );

        let position_tolerance = f64::EPSILON.sqrt();

        // Before: the tolerance comes from the endpoint residuals in the
        // coordinate, and the bracket end is four orders above it.
        let coordinate_scale = REPORTED[0].1.abs().max(REPORTED[3].1.abs()).max(1.0);
        let coordinate_tolerance = position_tolerance * coordinate_scale;
        assert!(
            REPORTED[2].1.abs() > 1.0e3 * coordinate_tolerance,
            "the failure this gate encodes requires the coordinate residual to be far above \
             its own tolerance: {} vs {coordinate_tolerance}",
            REPORTED[2].1.abs()
        );

        // After: same construction in the rescaled coordinate is met.
        let aspect_scale = rescaled[0].abs().max(rescaled[3].abs()).max(1.0);
        let aspect_tolerance = position_tolerance * aspect_scale;
        assert!(
            rescaled[2].abs() <= aspect_tolerance,
            "the rescaled residual {} must satisfy the tolerance {aspect_tolerance} its own \
             endpoints imply",
            rescaled[2].abs()
        );
    }

    /// Why an absolute tolerance cannot serve this domain, stated as a
    /// measurement rather than an assertion in a comment: the factor's span is
    /// the size of the mismatch, and it is enormous for BOTH families. The
    /// donut arm has not been driven into the failure yet; it is not immune.
    #[test]
    fn the_chain_factor_spans_orders_across_both_family_domains() {
        let flat_low =
            torus_metric_aspect_derivative_magnitude(TorusMetricFamily::Flat, f64::EPSILON)
                .expect("flat lower wall");
        let flat_high = torus_metric_aspect_derivative_magnitude(TorusMetricFamily::Flat, 1.0)
            .expect("flat upper wall");
        assert!(
            (flat_low / flat_high).log10() > 20.0,
            "flat chain factor span {} orders",
            (flat_low / flat_high).log10()
        );

        let resolution = f64::EPSILON.sqrt();
        let donut_low =
            torus_metric_aspect_derivative_magnitude(TorusMetricFamily::EmbeddedDonut, resolution)
                .expect("donut lower wall");
        let donut_high = torus_metric_aspect_derivative_magnitude(
            TorusMetricFamily::EmbeddedDonut,
            1.0 - resolution.sqrt(),
        )
        .expect("donut upper wall");
        assert!(
            (donut_low / donut_high).log10() > 15.0,
            "donut chain factor span {} orders",
            (donut_low / donut_high).log10()
        );
    }

    /// The factor is strictly signed on each open domain, which is what lets
    /// the rescaling preserve the bracket certificate. A zero would move a
    /// sign and a non-finite one would destroy the residual.
    #[test]
    fn the_chain_factor_is_finite_and_nonzero_across_each_domain() {
        for step in 1..64 {
            let flat = f64::from(step) / 64.0;
            let magnitude = torus_metric_aspect_derivative_magnitude(TorusMetricFamily::Flat, flat)
                .expect("interior flat coordinate");
            assert!(magnitude.is_finite() && magnitude > 0.0, "flat at {flat}");
            let donut =
                torus_metric_aspect_derivative_magnitude(TorusMetricFamily::EmbeddedDonut, flat)
                    .expect("interior donut coordinate");
            assert!(donut.is_finite() && donut > 0.0, "donut at {flat}");
        }
        assert!(
            torus_metric_aspect_derivative_magnitude(TorusMetricFamily::Flat, 0.0).is_err(),
            "the flat domain is open at zero"
        );
        assert!(
            torus_metric_aspect_derivative_magnitude(TorusMetricFamily::EmbeddedDonut, 1.0)
                .is_err(),
            "the donut domain is open at one, where the factor vanishes"
        );
    }
}