gam_terms/basis/

types.rs

use super::*;

/// Wrapper to send a raw pointer across thread boundaries for parallel buffer fills.
/// SAFETY: every `SendPtr` value must be built from live, properly aligned `f64`
/// storage whose mutable borrow is held until all worker threads finish; callers
/// may only dereference offsets that are in-bounds and disjoint across workers.
#[derive(Clone, Copy)]
pub(crate) struct SendPtr(pub(crate) *mut f64);

// SAFETY: SendPtr only grants raw-pointer transport. Actual dereferences occur
// at call sites after row-chunk partitioning proves each thread writes a
// distinct in-bounds element of the backing Array/Vec allocation.
unsafe impl Send for SendPtr {}

// SAFETY: shared references to SendPtr are sound because the pointee is never
// accessed through the wrapper without the call-site disjoint-offset proof.
unsafe impl Sync for SendPtr {}

impl SendPtr {
    #[inline(always)]
    pub(crate) fn add(self, offset: usize) -> *mut f64 {
        // SAFETY: callers pass offsets within the backing allocation and only
        // dereference the returned pointer after proving the target element is
        // uniquely owned by that worker's chunk for the whole parallel region.
        unsafe { self.0.add(offset) }
    }
}

/// Re-export of the neutral basis-error contract. #1521: `BasisError` lives
/// in `gam-problem` so `EstimationError` can wrap it (`#[from]`) without a
/// back-edge; gam-terms re-exports it to preserve `gam_terms::basis::BasisError`.
pub use gam_problem::BasisError;

// ============================================================================
// Unified Basis Generation API
// ============================================================================

/// Options for basis generation, controlling derivative order.
#[derive(Clone, Copy, Debug, Default)]
pub struct BasisOptions {
    /// Derivative order: 0 = value (default), 1 = first derivative, 2 = second derivative
    pub derivative_order: usize,
    /// Basis family to evaluate.
    pub basis_family: BasisFamily,
}

impl BasisOptions {
    /// Create options for evaluating basis functions (no derivative).
    pub const fn value() -> Self {
        Self {
            derivative_order: 0,
            basis_family: BasisFamily::BSpline,
        }
    }

    /// Create options for evaluating first derivatives of basis functions.
    pub const fn first_derivative() -> Self {
        Self {
            derivative_order: 1,
            basis_family: BasisFamily::BSpline,
        }
    }

    /// Create options for evaluating second derivatives of basis functions.
    pub const fn second_derivative() -> Self {
        Self {
            derivative_order: 2,
            basis_family: BasisFamily::BSpline,
        }
    }

    /// Create options for evaluating I-spline basis values.
    pub const fn i_spline() -> Self {
        Self {
            derivative_order: 0,
            basis_family: BasisFamily::ISpline,
        }
    }
}

/// Basis-family selector for 1D spline evaluation.
#[derive(Clone, Copy, Debug, Default, PartialEq, Eq)]
pub enum BasisFamily {
    /// Standard B-splines.
    #[default]
    BSpline,
    /// M-splines: normalized B-splines, M_i = ((k+1)/(t_{i+k+1}-t_i)) B_i.
    MSpline,
    /// I-splines: integrated M-splines, implemented by right-cumulative
    /// sums of B-splines at degree k+1.
    ISpline,
}

/// Specifies the source of knots for basis generation.
#[derive(Clone, Debug)]
pub enum KnotSource<'a> {
    /// Use a pre-computed knot vector.
    Provided(ArrayView1<'a, f64>),
    /// Generate uniformly spaced knots based on data range.
    Generate {
        /// Data range (min, max) for knot placement.
        data_range: (f64, f64),
        /// Number of internal knots to place between boundaries.
        num_internal_knots: usize,
    },
}
/// Thin-plate regression spline basis and penalty (order m=2).
///
/// The returned basis has columns `[K_c | P]` where:
/// - `K_c` is the constrained radial basis block (`K * Z`) with
///   `P(knots)^T * α = 0` enforced via nullspace projection
/// - `P` is the TPS polynomial null-space block containing all monomials of
///   total degree `< m`, where `m = thin_plate_penalty_order(d)` (so `P` is
///   just `[1, x_1, ..., x_d]` for `d <= 3`)
///
/// The returned penalty matrix is block-diagonal with:
/// - upper-left `Omega_c = Z^T Omega Z` for the constrained radial block
/// - zero lower-right block for unpenalized polynomial terms.
///
/// For double-penalty GAMs, a second ridge penalty `I` is also returned so the
/// caller can optimize `(lambda_bending, lambdaridge)` jointly.
#[derive(Debug, Clone)]
pub struct ThinPlateSplineBasis {
    pub basis: Array2<f64>,
    pub penalty_bending: Array2<f64>,
    pub penalty_ridge: Array2<f64>,
    pub num_kernel_basis: usize,
    pub num_polynomial_basis: usize,
    pub dimension: usize,
    /// Wood-TPRS radial reparameterization matrix `V`.
    ///
    /// Rows live in the side-constrained radial coefficient space. Columns are
    /// the retained positive bending eigendirections of `Z' Ω Z`; numerically
    /// near-null radial directions are dropped before the basis is exposed.
    /// Therefore `V` can be rectangular: design columns are `Φ Z V`, and the
    /// radial penalty is `diag(Λ_retained)`.
    pub radial_reparam: Array2<f64>,
}

/// Matérn smoothness parameter `nu` (half-integer variants with closed forms).
#[derive(Debug, Clone, Copy, Serialize, Deserialize)]
pub enum MaternNu {
    Half,
    ThreeHalves,
    FiveHalves,
    SevenHalves,
    NineHalves,
}

impl MaternNu {
    /// The half-integer smoothness value ν as an `f64` (0.5, 1.5, …).
    pub const fn half_integer_value(self) -> f64 {
        match self {
            MaternNu::Half => 0.5,
            MaternNu::ThreeHalves => 1.5,
            MaternNu::FiveHalves => 2.5,
            MaternNu::SevenHalves => 3.5,
            MaternNu::NineHalves => 4.5,
        }
    }
}

/// Matérn radial basis and penalties.
#[derive(Debug, Clone)]
pub struct MaternSplineBasis {
    pub basis: Array2<f64>,
    pub penalty_kernel: Array2<f64>,
    pub penalty_ridge: Array2<f64>,
    pub num_kernel_basis: usize,
    pub num_polynomial_basis: usize,
    pub dimension: usize,
}

#[derive(Debug, Clone)]
pub(crate) struct DuchonBasisDesign {
    pub(crate) basis: Array2<f64>,
}

/// Boundary-condition policy for one-dimensional smooth bases.
#[derive(Debug, Clone, Serialize, Deserialize, Default)]
pub enum OneDimensionalBoundary {
    /// Ordinary open interval basis with clamped endpoint behavior.
    #[default]
    Open,
    /// Periodic/cyclic basis over the half-open interval `[start, end)`.
    ///
    /// Values are evaluated modulo `period = end - start`; the basis and its
    /// first `degree - 1` derivatives agree at the two endpoints for B-splines.
    Cyclic { start: f64, end: f64 },
}

impl OneDimensionalBoundary {
    pub(crate) fn period(&self) -> Option<(f64, f64, f64)> {
        match *self {
            OneDimensionalBoundary::Open => None,
            OneDimensionalBoundary::Cyclic { start, end } if end > start => {
                Some((start, end, end - start))
            }
            OneDimensionalBoundary::Cyclic { .. } => None,
        }
    }
}

/// Which knot strategy to use for 1D B-spline bases.
#[derive(Debug, Clone, Serialize, Deserialize)]
pub enum BSplineKnotSpec {
    Generate {
        data_range: (f64, f64),
        num_internal_knots: usize,
    },
    /// Uniform cyclic B-spline basis on `[data_range.0, data_range.1)`.
    ///
    /// The first and last endpoints are identified, so evaluating at `x` and
    /// `x + m * period` gives identical rows. `num_basis` is the number of
    /// periodic control sites around the loop and must be at least
    /// `degree + 1` for an unaliased local support stencil.
    PeriodicUniform {
        data_range: (f64, f64),
        num_basis: usize,
    },
    Automatic {
        num_internal_knots: Option<usize>,
        placement: BSplineKnotPlacement,
    },
    Provided(Array1<f64>),
    /// Natural cubic regression spline (`bs="cr"`/`"cs"`) knot set (#1074).
    ///
    /// Unlike the open-spline variants above, these `knots` are the `k`
    /// Lancaster–Salkauskas knots `x*_1 < … < x*_k` that *directly* index the
    /// basis values `β_i = f(x*_i)` — the basis dimension equals `knots.len()`
    /// (not `knots.len() - degree - 1`). The 1-D builder routes this variant to
    /// the cubic-regression builder; the cr identity therefore round-trips
    /// through freeze/reload by virtue of the variant itself (no separate
    /// metadata marker is required), and tensor margins inherit cr by carrying
    /// this knotspec into `build_bspline_basis_1d`.
    NaturalCubicRegression {
        knots: Array1<f64>,
    },
}

/// Internal-knot placement strategy when knots are automatically inferred.
#[derive(Debug, Clone, Copy, PartialEq, Eq, Serialize, Deserialize)]
pub enum BSplineKnotPlacement {
    Uniform,
    Quantile,
}

/// 1D B-spline basis configuration.
#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct BSplineBasisSpec {
    pub degree: usize,
    pub penalty_order: usize,
    pub knotspec: BSplineKnotSpec,
    pub double_penalty: bool,
    pub identifiability: BSplineIdentifiability,
    #[serde(default)]
    pub boundary: OneDimensionalBoundary,
    /// Optional endpoint boundary constraints (Hermite-style pin of value and/or
    /// derivative at the left/right knot extents). Default = `Free` on both
    /// sides which is a no-op.
    #[serde(default)]
    pub boundary_conditions: BSplineBoundaryConditions,
}

/// Per-endpoint boundary constraint policy for B-spline 1D bases.
#[derive(Debug, Clone, Copy, PartialEq, Serialize, Deserialize, Default)]
pub enum BSplineEndpointBoundaryCondition {
    /// No endpoint constraint.
    #[default]
    Free,
    /// Pin the first derivative to zero at this endpoint.
    Clamped,
    /// Hermite pin: fix the endpoint value to `value` and its first derivative
    /// to zero.
    Anchored { value: f64 },
}

/// Left/right pair of B-spline endpoint constraints.
#[derive(Debug, Clone, Copy, PartialEq, Serialize, Deserialize, Default)]
pub struct BSplineBoundaryConditions {
    #[serde(default)]
    pub left: BSplineEndpointBoundaryCondition,
    #[serde(default)]
    pub right: BSplineEndpointBoundaryCondition,
}

impl BSplineBoundaryConditions {
    pub const fn is_free(&self) -> bool {
        matches!(self.left, BSplineEndpointBoundaryCondition::Free)
            && matches!(self.right, BSplineEndpointBoundaryCondition::Free)
    }

    /// Whether either endpoint fixes the function's absolute level.
    ///
    /// An anchored endpoint (one *or* both sides) replaces the global intercept
    /// as the level-setting constraint: the fitted function itself, not only a
    /// centered deviation, must obey the endpoint pin. Centering that same
    /// smooth to zero would impose a second, incompatible level constraint and
    /// exclude every non-zero-mean anchored function from the model space, and a
    /// free global intercept would float the whole curve off its pin. A
    /// *two*-sided anchor fixes the level even more strongly than a one-sided
    /// one, so it must be treated identically here — the earlier XOR (exactly
    /// one endpoint) silently dropped both pins for the two-sided case (#2297).
    pub const fn has_anchor(&self) -> bool {
        matches!(self.left, BSplineEndpointBoundaryCondition::Anchored { .. })
            || matches!(
                self.right,
                BSplineEndpointBoundaryCondition::Anchored { .. }
            )
    }

    /// Whether either endpoint carries an inhomogeneous value constraint.
    pub fn has_nonzero_anchor(&self) -> bool {
        let nonzero = |condition: BSplineEndpointBoundaryCondition| {
            matches!(
                condition,
                BSplineEndpointBoundaryCondition::Anchored { value } if value != 0.0
            )
        };
        nonzero(self.left) || nonzero(self.right)
    }
}

/// Per-smooth identifiability policy for 1D B-spline bases.
///
/// These constraints are applied directly in the builder via a reparameterization
/// `B_constrained = B * Z`, and every penalty matrix is projected as
/// `S_constrained = Z' S Z`, so solver geometry stays consistent.
#[derive(Debug, Clone, Serialize, Deserialize)]
pub enum BSplineIdentifiability {
    /// Keep unconstrained basis columns.
    None,
    /// Enforce weighted sum-to-zero: `B' w = 0` (or unweighted when `weights=None`).
    // Smooth terms are centered by default to avoid intercept confounding.
    WeightedSumToZero { weights: Option<Array1<f64>> },
    /// Remove intercept + linear trend in coefficient space using Greville geometry.
    RemoveLinearTrend,
    /// Enforce orthogonality to supplied design columns `C` (n x q):
    /// `B_c' W C = 0` (or unweighted when `weights=None`).
    ///
    /// To enforce `[intercept, x, ...]`, provide `columns` with those columns.
    OrthogonalToDesignColumns {
        columns: Array2<f64>,
        weights: Option<Array1<f64>>,
    },
    /// Apply an explicit coefficient-space transform `Z` learned at fit time.
    ///
    /// This freezes identifiability behavior so prediction cannot drift based on
    /// new-data distribution. The constrained basis is `B * Z`.
    FrozenTransform { transform: Array2<f64> },
}

impl Default for BSplineIdentifiability {
    fn default() -> Self {
        BSplineIdentifiability::WeightedSumToZero { weights: None }
    }
}

/// Spatial center selection strategy.
///
/// `num_centers` is the exact number of knot/center rows selected by the
/// strategy. Polynomial nullspace columns are added separately by each basis
/// builder and must never be folded into this count.
#[derive(Debug, Clone, Serialize, Deserialize)]
pub enum CenterStrategy {
    Auto(Box<CenterStrategy>),
    /// Select a potentially rich knot cloud, then retain an explicit
    /// low-dimensional spectral subspace of its kernel. The knot strategy and
    /// retained rank are independent by construction.
    DuchonSpectral {
        knots: Box<CenterStrategy>,
        basis: DuchonSpectralBasis,
    },
    UserProvided(Array2<f64>),
    /// Joint multidimensional equal-mass partitioning in the full smooth space.
    EqualMass {
        num_centers: usize,
    },
    /// Covariate-representative equal-mass partitioning along one selected axis.
    EqualMassCovarRepresentative {
        num_centers: usize,
    },
    FarthestPoint {
        num_centers: usize,
    },
    KMeans {
        num_centers: usize,
        max_iter: usize,
    },
    UniformGrid {
        points_per_dim: usize,
    },
}

impl CenterStrategy {
    /// The number of centers this strategy will select, computed from the
    /// strategy alone (no data pass). `d` is the smooth's covariate
    /// dimensionality, needed only by `UniformGrid` whose count is
    /// `points_per_dim^d`. Adaptive-fit provenance consults this before freeze,
    /// because the frozen center matrix can contain periodic image expansion
    /// and therefore is not the requested resolution for the next refit.
    pub fn planned_num_centers(&self, d: usize) -> usize {
        match self {
            Self::Auto(inner) => inner.planned_num_centers(d),
            Self::DuchonSpectral { knots, .. } => knots.planned_num_centers(d),
            Self::UserProvided(centers) => centers.nrows(),
            Self::EqualMass { num_centers }
            | Self::EqualMassCovarRepresentative { num_centers }
            | Self::FarthestPoint { num_centers }
            | Self::KMeans { num_centers, .. } => *num_centers,
            Self::UniformGrid { points_per_dim } => {
                points_per_dim.saturating_pow(d.clamp(1, u32::MAX as usize) as u32)
            }
        }
    }
}

#[derive(Debug, Clone, Copy, PartialEq, Eq, PartialOrd, Ord)]
pub enum CenterStrategyKind {
    UserProvided,
    EqualMass,
    EqualMassCovarRepresentative,
    FarthestPoint,
    KMeans,
    UniformGrid,
}

/// Adaptive default center count for spatial smooths (TPS, Duchon, Matérn).
///
/// Use this when the user has not explicitly specified a knot/center count.
/// The basis size is the sub-linear `ceil(8 * d_factor * n^0.4)`, clamped above
/// at `K_MAX = 2000` and below at a *data-proportional* floor `min(200, n/8)` so
/// the floor only engages once there are enough observations to support a rich
/// basis. The result is additionally capped at `n/4` so the penalty matrices
/// stay well-conditioned relative to the data:
///
/// | n      | d=1  | d=2  | d=5  |
/// |--------|------|------|------|
/// | 800    | 116  | 134  | 186  |
/// | 1 000  | 127  | 146  | 200  |
/// | 2 000  | 200  | 200  | 268  |
/// | 10 000 | 319  | 367  | 510  |
/// | 100 000| 801  | 921  | 1281 |
/// | 400 000| 1393 | 1602 | 2000 |
/// | 1 000 000| 2000 | 2000 | 2000 |
///
/// The flat `200` floor used to inflate moderate-`n` spatial smooths (a few
/// hundred to ~2000 rows) up to a dense 200-column design even though the raw
/// sub-linear count — and the mesh/knot density that mgcv and R-INLA use on the
/// same data — is far smaller. On ~800 rows that turned a single 2-D thin-plate
/// REML fit into an `O(n·p² + p³)` grind at `p ≈ 200` (#718). Smoothness is
/// already controlled by REML's penalty weight λ, not by the center count, so a
/// data-proportional floor recovers the same surface at a fraction of the cost.
///
/// # Arguments
/// * `n` - sample size (number of observations)
/// * `d` - covariate dimensionality (number of input variables in the smooth)
pub fn default_num_centers(n: usize, d: usize) -> usize {
    const K_MIN: usize = 200;
    const K_MAX: usize = 2000;
    const ALPHA: f64 = 0.4;
    const C: f64 = 8.0;
    /// Per-extra-dimension growth in the center count: each covariate axis
    /// beyond the first widens the basis by 15% to keep the per-axis mesh
    /// density roughly constant as the smooth's domain dimensionality grows.
    const PER_DIM_GROWTH: f64 = 0.15;
    /// Divisor for the data-proportional floor: the `K_MIN` floor only engages
    /// once `n` exceeds `K_MIN * FLOOR_N_DIVISOR`, so small samples are not
    /// forced up to a dense `K_MIN`-column design.
    const FLOOR_N_DIVISOR: usize = 8;
    /// Divisor for the conditioning cap: the center count never exceeds `n /
    /// COND_N_DIVISOR`, keeping the penalty matrices well-conditioned relative
    /// to the data.
    const COND_N_DIVISOR: usize = 4;

    let d_factor = 1.0 + PER_DIM_GROWTH * (d.max(1) - 1) as f64;
    let raw = (C * d_factor * (n as f64).powf(ALPHA)).ceil() as usize;

    // Data-proportional floor: never inflate beyond n/FLOOR_N_DIVISOR, so the
    // K_MIN-center floor only takes effect once n is large enough (~1600) to
    // genuinely support that many basis columns.
    let floor = K_MIN.min(n / FLOOR_N_DIVISOR);
    let k = raw.clamp(floor, K_MAX);

    // Never exceed n itself; cap at n/COND_N_DIVISOR to keep the penalty
    // matrices well-conditioned relative to the data.
    k.min(n).min(n / COND_N_DIVISOR)
}

/// Conservative center count for a *secondary* (distributional) predictor's
/// spatial smooth — e.g. the log-σ scale model in a Gaussian location-scale
/// fit.
///
/// The mean is identified directly by the response, so it warrants the
/// generous [`default_num_centers`] basis. A scale/shape predictor is
/// identified only through (noisy) squared residuals: handing it a basis sized
/// for the mean lets REML/LAML smoothing selection over-fit it, because where
/// the fitted scale is driven small the *observed* information collapses and
/// the determinant penalty stops holding the wiggle down (#501). This mirrors
/// standard GAMLSS/mgcv practice of giving distribution parameters a modest
/// default (mgcv's modest default basis for a 1-D `s()`), grown gently with
/// dimensionality and never exceeding the generous primary-predictor default.
pub fn conservative_secondary_centers(n: usize, d: usize) -> usize {
    const BASE_1D_CENTERS: usize = 15;
    let modest = BASE_1D_CENTERS.saturating_mul(d.max(1));
    default_num_centers(n, d).min(modest).max(1)
}

/// Low-rank starting center count for saturation-driven spatial fitting.
///
/// The structural minimum (`d + 1` polynomial directions plus one radial
/// direction) is only enough to make the algebra identifiable. It is not an
/// adequate pilot function space: structure orthogonal to that single radial
/// direction is absorbed into the residual, so REML can legitimately shrink
/// the direction and report EDF below its ceiling even when the surface is
/// badly under-resolved (#1689). Start from the project's established
/// thin-plate-style low-rank resolution `10 * 3^(d - 1)` instead. This is the
/// same dimension rule already used by the automatic Duchon builder, capped by
/// [`default_num_centers`] so the pilot never exceeds the validated production
/// basis at small sample sizes.
pub fn starting_num_centers(n: usize, d: usize) -> usize {
    let low_rank_resolution = 10usize
        .saturating_mul(3usize.saturating_pow(d.saturating_sub(1).min(u32::MAX as usize) as u32));
    low_rank_resolution
        .min(default_num_centers(n, d))
        .min(n)
        .max(1)
}

/// Next evidence-backed center count for a saturated spatial basis, bounded by
/// the already validated production-default resolution.
///
/// Growth is geometric so the number of certified refits is logarithmic. The
/// ceiling is supplied by the owning workflow because it depends on the
/// spatial family/dimension and resource plan; the standard formula workflow
/// uses [`default_num_centers`]. Adaptive resolution may therefore avoid work
/// below the previous default, but can never turn an ordinary fit into an
/// unvalidated row-rank dense basis. `None` means the validated function-space
/// ceiling has been reached.
pub fn expanded_num_centers(current: usize, ceiling: usize) -> Option<usize> {
    if current >= ceiling {
        return None;
    }
    let expanded = current.saturating_mul(2).min(ceiling);
    (expanded > current).then_some(expanded)
}

/// Is a fitted spatial smooth's basis SATURATED — i.e. does its own evidence say
/// the data wants more resolution than its realized coefficient span provides (#1689)?
///
/// The penalizable capacity is `realized_width − nullspace_dim`: the unpenalized
/// polynomial null space is always fully used, so it is excluded from the "is the
/// PENALIZED part maxed out?" test. The supplied `edf` is the total term EDF;
/// subtracting `nullspace_dim` yields its penalized contribution, which rises
/// toward that capacity exactly as REML drives the penalty
/// λ toward its floor to chase structure the basis cannot resolve. Saturated ⟺
/// `edf ≥ capacity − ε`, with the margin `ε` DERIVED from the outer REML
/// numerical resolution (`ε = capacity · resolution_tol`, floored at
/// `resolution_tol` so a tiny-capacity block still has a positive margin) rather
/// than a tuned knob. The workflow derives `resolution_tol` from the maximum of
/// its outer convergence tolerance and any rho-independent penalty shrinkage
/// floor, because that floor bounds how closely EDF can approach the algebraic
/// ceiling even as lambda tends to zero. Non-positive capacity (a block whose
/// null space already exhausts its columns) is never saturated. The absolute
/// scale of `ε` is what the MSI truth-recovery sweep
/// (sin8/kappa/large_scale + #1074) validates — the criterion SHAPE
/// (edf-vs-capacity, nullspace excluded, tol-tied margin) is the load-bearing
/// contract this function pins.
pub fn basis_is_saturated(
    edf: f64,
    realized_width: usize,
    nullspace_dim: usize,
    resolution_tol: f64,
) -> bool {
    let capacity = realized_width.saturating_sub(nullspace_dim) as f64;
    if !(capacity > 0.0) || !edf.is_finite() {
        return false;
    }
    let penalized_edf = (edf - nullspace_dim as f64).clamp(0.0, capacity);
    let margin = (capacity * resolution_tol).max(resolution_tol);
    penalized_edf >= capacity - margin
}

/// Resource-aware plan for a spatial smooth (Duchon / Matérn / TPS).
///
/// Returned by [`plan_spatial_basis`]. Captures the resolved center count,
/// final basis dimension `p`, the dense byte cost for the value matrix and
/// each derivative tier, and a recommended storage mode that is consistent
/// with the supplied [`gam_runtime::resource::ResourcePolicy`].
#[derive(Clone, Debug)]
pub struct SpatialBasisPlan {
    pub n: usize,
    pub d: usize,
    pub centers: usize,
    pub p_final_estimate: usize,
    pub dense_design_bytes: usize,
    pub first_derivative_dense_bytes: usize,
    pub second_derivative_dense_bytes: usize,
    pub recommended_storage: SpatialStorageMode,
}

/// Storage mode recommended by [`plan_spatial_basis`].
///
/// * `DenseValueDenseDerivatives` — both the value design and its derivative
///   matrices fit under the policy's single-materialization budget.
/// * `LazyValueImplicitDerivatives` — the value design fits dense but the
///   derivative matrices do not; switch derivatives to the implicit operator.
/// * `OperatorOnly` — neither the design nor its derivatives fit; everything
///   must be operator-backed.
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub enum SpatialStorageMode {
    DenseValueDenseDerivatives,
    LazyValueImplicitDerivatives,
    OperatorOnly,
}

/// How [`plan_spatial_basis`] should pick the spatial center count.
#[derive(Clone, Copy, Debug)]
pub enum CenterCountRequest {
    /// Use the heuristic [`default_num_centers`].
    Default,
    /// Use the caller-supplied count exactly.
    Explicit(usize),
    /// Use [`default_num_centers`] but cap at `cap` to bound dense cost.
    HeuristicCapped { cap: usize },
}

/// Build a resource-aware plan for a spatial smooth basis.
///
/// Computes the resolved center count, final basis dimension, dense byte
/// estimates for the value design and first/second derivative tiers, and a
/// recommended [`SpatialStorageMode`] derived from `policy`. This is the
/// resource-aware replacement for ad-hoc calls to [`default_num_centers`] /
/// `heuristic_centers`.
pub fn plan_spatial_basis(
    n: usize,
    d: usize,
    requested_centers: CenterCountRequest,
    nullspace_order: DuchonNullspaceOrder,
    scale_dims: bool,
    policy: &gam_runtime::resource::ResourcePolicy,
) -> Result<SpatialBasisPlan, BasisError> {
    if n == 0 {
        crate::bail_invalid_basis!("plan_spatial_basis: n must be >= 1");
    }
    if d == 0 {
        crate::bail_invalid_basis!("plan_spatial_basis: d must be >= 1");
    }

    // 1. Resolve center count.
    let centers = match requested_centers {
        CenterCountRequest::Default => default_num_centers(n, d),
        CenterCountRequest::Explicit(k) => k,
        CenterCountRequest::HeuristicCapped { cap } => default_num_centers(n, d).min(cap),
    };

    // 2. Nullspace dimension (Duchon polynomial null space of degree p-1).
    //    `duchon_p_from_nullspace_order` returns m such that the null space is
    //    polynomials of total degree < m, matching `duchon_nullspace_dimension`'s
    //    `max_total_degree = m - 1` argument.
    let m = duchon_p_from_nullspace_order(nullspace_order);
    let nullspace_dim = if m == 0 {
        0
    } else {
        duchon_nullspace_dimension(d, m - 1)
    };

    let p = centers.saturating_add(nullspace_dim);

    // 3. Dense byte estimates.
    let derivative_axes = if scale_dims { d } else { 0 };
    let bytes_per_f64 = std::mem::size_of::<f64>();
    let dense_design_bytes = bytes_per_f64.saturating_mul(n).saturating_mul(p);
    let first_derivative_dense_bytes = dense_design_bytes.saturating_mul(derivative_axes);
    // Diagonal second derivatives are also (D × n × p); off-diagonal cross terms
    // would scale as D^2 but the planner reports the diagonal tier here.
    let second_derivative_dense_bytes = first_derivative_dense_bytes;

    // 4. Pick storage mode based on policy.
    let recommended_storage = match policy.derivative_storage_mode {
        gam_runtime::resource::DerivativeStorageMode::AnalyticOperatorRequired => {
            SpatialStorageMode::OperatorOnly
        }
        gam_runtime::resource::DerivativeStorageMode::MaterializeIfSmall => {
            let budget = policy.max_single_materialization_bytes;
            if derivative_axes == 0 {
                if dense_design_bytes <= budget {
                    SpatialStorageMode::DenseValueDenseDerivatives
                } else {
                    SpatialStorageMode::LazyValueImplicitDerivatives
                }
            } else {
                let total = dense_design_bytes
                    .saturating_add(first_derivative_dense_bytes)
                    .saturating_add(second_derivative_dense_bytes);
                if total <= budget {
                    SpatialStorageMode::DenseValueDenseDerivatives
                } else if dense_design_bytes <= budget {
                    SpatialStorageMode::LazyValueImplicitDerivatives
                } else {
                    SpatialStorageMode::OperatorOnly
                }
            }
        }
        gam_runtime::resource::DerivativeStorageMode::DiagnosticsOnly => {
            // Diagnostic mode still prefers analytic storage for correctness.
            SpatialStorageMode::OperatorOnly
        }
    };

    Ok(SpatialBasisPlan {
        n,
        d,
        centers,
        p_final_estimate: p,
        dense_design_bytes,
        first_derivative_dense_bytes,
        second_derivative_dense_bytes,
        recommended_storage,
    })
}

pub const fn default_spatial_center_strategy(num_centers: usize, d: usize) -> CenterStrategy {
    if d <= 3 {
        CenterStrategy::FarthestPoint { num_centers }
    } else {
        CenterStrategy::EqualMassCovarRepresentative { num_centers }
    }
}

pub fn auto_spatial_center_strategy(num_centers: usize, d: usize) -> CenterStrategy {
    let strategy = if d == 1 {
        // In one dimension, farthest-point selection is the deterministic
        // maximin grid over the observed domain. Equal-mass midpoints leave the
        // low-frequency Duchon radial block slightly under-resolved at the
        // boundaries, and REML then compensates with an over-smooth λ on
        // low-noise signals (#504). The maximin grid matches the native
        // reproducing-kernel interpolation geometry. The default strategy below
        // extends the same space-filling contract to low-dimensional spatial
        // GP bases, where kriging accuracy is governed by fill distance rather
        // than marginal quantile balance.
        CenterStrategy::FarthestPoint { num_centers }
    } else {
        default_spatial_center_strategy(num_centers, d)
    };
    CenterStrategy::Auto(Box::new(strategy))
}

pub const fn center_strategy_is_auto(strategy: &CenterStrategy) -> bool {
    match strategy {
        CenterStrategy::Auto(_) => true,
        CenterStrategy::DuchonSpectral { knots, .. } => center_strategy_is_auto(knots),
        _ => false,
    }
}

pub(crate) fn realized_center_strategy(strategy: &CenterStrategy) -> &CenterStrategy {
    match strategy {
        CenterStrategy::Auto(inner) => inner.as_ref(),
        CenterStrategy::DuchonSpectral { knots, .. } => realized_center_strategy(knots),
        other => other,
    }
}

pub(crate) fn center_strategy_spectral_basis(
    strategy: &CenterStrategy,
) -> Option<&DuchonSpectralBasis> {
    match strategy {
        CenterStrategy::Auto(inner) => center_strategy_spectral_basis(inner),
        CenterStrategy::DuchonSpectral { basis, .. } => Some(basis),
        _ => None,
    }
}

/// Whether a Duchon center plan is fully fit-time-resolved.
///
/// A spectral plan is frozen only when both pieces of fit-time state are
/// explicit: the selected knots and the learned kernel-to-basis transform.
/// Keeping this predicate beside the state types prevents model validation
/// from accidentally treating the spectral wrapper itself as an unresolved
/// center-selection strategy.
pub(crate) fn duchon_center_strategy_is_frozen(strategy: &CenterStrategy) -> bool {
    match strategy {
        CenterStrategy::UserProvided(_) => true,
        CenterStrategy::DuchonSpectral {
            knots,
            basis: DuchonSpectralBasis::Frozen { .. },
        } => matches!(knots.as_ref(), CenterStrategy::UserProvided(_)),
        _ => false,
    }
}

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

    #[test]
    fn spectral_state_is_frozen_only_when_knots_and_transform_are_resolved() {
        let centers = Array2::zeros((3, 2));
        let unresolved = CenterStrategy::DuchonSpectral {
            knots: Box::new(CenterStrategy::UserProvided(centers.clone())),
            basis: DuchonSpectralBasis::Fresh { rank: 2 },
        };
        assert!(!duchon_center_strategy_is_frozen(&unresolved));

        let resolved = CenterStrategy::DuchonSpectral {
            knots: Box::new(CenterStrategy::UserProvided(centers)),
            basis: DuchonSpectralBasis::Frozen {
                rank: 2,
                kernel_transform: Array2::zeros((3, 1)),
                bending_penalty: Array2::zeros((1, 1)),
            },
        };
        assert!(duchon_center_strategy_is_frozen(&resolved));
    }
}

pub fn center_strategy_kind(strategy: &CenterStrategy) -> CenterStrategyKind {
    match strategy {
        CenterStrategy::Auto(inner) => center_strategy_kind(inner.as_ref()),
        CenterStrategy::DuchonSpectral { knots, .. } => center_strategy_kind(knots),
        CenterStrategy::UserProvided(_) => CenterStrategyKind::UserProvided,
        CenterStrategy::EqualMass { .. } => CenterStrategyKind::EqualMass,
        CenterStrategy::EqualMassCovarRepresentative { .. } => {
            CenterStrategyKind::EqualMassCovarRepresentative
        }
        CenterStrategy::FarthestPoint { .. } => CenterStrategyKind::FarthestPoint,
        CenterStrategy::KMeans { .. } => CenterStrategyKind::KMeans,
        CenterStrategy::UniformGrid { .. } => CenterStrategyKind::UniformGrid,
    }
}

pub fn center_strategy_num_centers(strategy: &CenterStrategy) -> Option<usize> {
    match strategy {
        CenterStrategy::Auto(inner) => center_strategy_num_centers(inner.as_ref()),
        CenterStrategy::DuchonSpectral { knots, .. } => center_strategy_num_centers(knots),
        CenterStrategy::UserProvided(centers) => Some(centers.nrows()),
        CenterStrategy::EqualMass { num_centers }
        | CenterStrategy::EqualMassCovarRepresentative { num_centers }
        | CenterStrategy::FarthestPoint { num_centers }
        | CenterStrategy::KMeans { num_centers, .. } => Some(*num_centers),
        CenterStrategy::UniformGrid { .. } => None,
    }
}

pub fn center_strategy_with_num_centers(
    strategy: &CenterStrategy,
    num_centers: usize,
    d: usize,
) -> Result<CenterStrategy, BasisError> {
    validate_center_count(num_centers)?;
    fn rebuild_inner(
        strategy: &CenterStrategy,
        num_centers: usize,
        d: usize,
    ) -> Result<CenterStrategy, BasisError> {
        match strategy {
            CenterStrategy::Auto(inner) => rebuild_inner(inner.as_ref(), num_centers, d),
            CenterStrategy::DuchonSpectral { knots, basis } => Ok(CenterStrategy::DuchonSpectral {
                knots: Box::new(rebuild_inner(knots, num_centers, d)?),
                basis: basis.clone(),
            }),
            CenterStrategy::EqualMass { .. } => Ok(CenterStrategy::EqualMass { num_centers }),
            CenterStrategy::EqualMassCovarRepresentative { .. } => {
                Ok(CenterStrategy::EqualMassCovarRepresentative { num_centers })
            }
            CenterStrategy::FarthestPoint { .. } => {
                Ok(CenterStrategy::FarthestPoint { num_centers })
            }
            CenterStrategy::KMeans { max_iter, .. } => Ok(CenterStrategy::KMeans {
                num_centers,
                max_iter: *max_iter,
            }),
            CenterStrategy::UniformGrid { .. } if d == 1 => Ok(CenterStrategy::UniformGrid {
                points_per_dim: num_centers,
            }),
            CenterStrategy::UserProvided(_) | CenterStrategy::UniformGrid { .. } => {
                Err(BasisError::InvalidInput(format!(
                    "cannot replace center count for {:?} strategy",
                    center_strategy_kind(strategy)
                )))
            }
        }
    }
    let rebuilt = rebuild_inner(strategy, num_centers, d)?;
    Ok(match strategy {
        CenterStrategy::Auto(_) => CenterStrategy::Auto(Box::new(rebuilt)),
        _ => rebuilt,
    })
}

/// Thin-plate basis configuration.
#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct ThinPlateBasisSpec {
    pub center_strategy: CenterStrategy,
    #[serde(default)]
    pub periodic: Option<Vec<Option<f64>>>,
    pub length_scale: f64,
    pub double_penalty: bool,
    #[serde(default)]
    pub identifiability: SpatialIdentifiability,
    /// Frozen Wood-TPRS radial reparameterization. When `Some`, the builder
    /// reuses this `(raw_radial_cols) × (kept_radial_cols)` matrix instead of
    /// recomputing it from the constrained kernel penalty eigensystem. The
    /// rectangular case is the truncated regression-spline path; carrying it
    /// into prediction guarantees identical radial modes to fit-time.
    #[serde(default)]
    pub radial_reparam: Option<Array2<f64>>,
}

/// Per-smooth identifiability policy for spatial (TPS / Duchon) bases.
///
/// For a raw local basis `B` and parametric design block `C`, the orthogonalized
/// basis is `B_c = B Z` where columns of `Z` span `null((B^T C)^T)`. This enforces:
///   `B_c^T C = 0`
/// in the unweighted inner product, so spatial effects cannot absorb parametric
/// directions that actually exist in the model. The standalone basis builder has
/// only an implicit intercept available, so it centers smooths against that
/// intercept. The term-collection builder augments `C` with explicit linear
/// terms when those terms are present in the formula.
#[derive(Debug, Default, Clone, Serialize, Deserialize)]
pub enum SpatialIdentifiability {
    /// Keep unconstrained basis columns.
    None,
    /// Orthogonalize the smooth against model-owned parametric columns.
    // "Magic" default for modular GAMs with explicit parametric block:
    // keep spatial smooth orthogonal to intercept/linear terms.
    // ApproxKind: Exact (orthogonalization is an exact projection).
    #[default]
    OrthogonalToParametric,
    /// Freeze a fit-time transform `Z`; prediction uses `B_new * Z` unchanged.
    FrozenTransform { transform: Array2<f64> },
}

pub(crate) use sphere_half_angle::{
    SphereTrig, ambient_half_angle_separation, half_angle_partials, half_angle_separation,
    half_angle_separation_scalar,
};

pub(crate) use sphere_kernels::{
    wahba_sphere_kernel_derivative_dhav_kind, wahba_sphere_kernel_kind,
    wahba_sphere_kernel_simd_kind, wahba_sphere_kernel_sobolev_derivative_dhav,
};

pub use sphere_spectral::{
    pseudo_s2_truncated_coefficients, sobolev_s2_truncated_coefficients,
    sphere_truncated_spectral_eval,
};

/// User intent and resolved numeric state for a Matérn kernel length scale.
///
/// `Auto` remains auto-owned after the planner resolves its data-dependent
/// numeric seed.  This is deliberately not represented by a magic floating
/// point value: callers can distinguish an omitted `length_scale` from an
/// explicit value before and after center planning, and subsequent κ updates
/// preserve that provenance.
#[derive(Debug, Clone, Copy, Serialize, Deserialize, PartialEq)]
pub enum MaternLengthScale {
    Auto { resolved: Option<f64> },
    Fixed(f64),
}

impl MaternLengthScale {
    pub const fn auto() -> Self {
        Self::Auto { resolved: None }
    }

    pub const fn fixed(value: f64) -> Self {
        Self::Fixed(value)
    }

    pub const fn is_fixed(self) -> bool {
        matches!(self, Self::Fixed(_))
    }

    pub const fn resolved(self) -> Option<f64> {
        match self {
            Self::Auto { resolved } => resolved,
            Self::Fixed(value) => Some(value),
        }
    }

    /// Install a numeric value without changing who owns the scale.
    pub fn set_resolved(&mut self, value: f64) {
        match self {
            Self::Auto { resolved } => *resolved = Some(value),
            Self::Fixed(fixed) => *fixed = value,
        }
    }

    /// Resolve an omitted scale exactly once.  Replanning a frozen or
    /// κ-updated Auto scale must retain its current numeric value.
    pub fn resolve_auto_once(&mut self, value: f64) {
        if let Self::Auto { resolved } = self
            && resolved.is_none()
        {
            *resolved = Some(value);
        }
    }
}

/// Matérn basis configuration.
#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct MaternBasisSpec {
    pub center_strategy: CenterStrategy,
    #[serde(default)]
    pub periodic: Option<Vec<Option<f64>>>,
    pub length_scale: MaternLengthScale,
    pub nu: MaternNu,
    #[serde(default)]
    pub include_intercept: bool,
    pub double_penalty: bool,
    #[serde(default)]
    pub identifiability: MaternIdentifiability,
    /// Per-axis anisotropy log-scales η_a (contrasts with Ση_a = 0).
    ///
    /// This implements geometric anisotropy: Λ = κA where A = diag(exp(η_a)),
    /// det(A) = 1. The kernel is evaluated at r = κ|Ah| instead of r = κ|h|.
    /// The decomposition preserves the isotropic scaling law for global κ
    /// and adds d−1 shape parameters for directional relevance.
    ///
    /// Conditional positive definiteness is preserved under any invertible
    /// linear coordinate transform (Schoenberg), so the kernel remains valid.
    ///
    /// When Some, the distance is r = √(Σ_a exp(2η_a) · (x_a - c_a)²).
    /// When None, isotropic distance r = ‖x - c‖ is used.
    #[serde(default)]
    pub aniso_log_scales: Option<Vec<f64>>,
}

/// Per-smooth identifiability policy for Matérn kernel coefficients.
///
/// These constraints are geometric (center-based), so they are stable across
/// train/predict and do not depend on response weights.
#[derive(Debug, Default, Clone, Serialize, Deserialize)]
pub enum MaternIdentifiability {
    /// Keep the unconstrained kernel coefficient space.
    None,
    /// Enforce `1^T alpha = 0` at center locations (removes constant drift).
    // Safe default with model intercepts: prevent kernel block from absorbing
    // a global mean level.
    #[default]
    CenterSumToZero,
    /// Enforce orthogonality to `[1, c_1, ..., c_d]` at centers.
    /// Use this when explicit linear terms should own global trends.
    CenterLinearOrthogonal,
    /// Freeze a fit-time transform `Z` so prediction cannot drift.
    FrozenTransform { transform: Array2<f64> },
}

/// Duchon null-space polynomial degree.
///
/// Controls the polynomial null space of the Duchon / polyharmonic spline. The
/// Duchon seminorm `‖D^m f‖²` annihilates all polynomials of total degree
/// `< m`, so those polynomials must be handled as explicit unpenalized columns.
///
/// The user-facing `order` knob selects the polynomial degree cutoff `r`, and
/// the resulting polynomial null space has dimension `C(d + r, r)` where `d`
/// is the covariate dimension.  In the `duchon(...)` formula DSL:
///
/// | `order=` | Variant         | max total degree | null-space dim  |
/// |----------|-----------------|------------------|-----------------|
/// | `0`      | `Zero`          | 0                | `C(d+0,0) = 1`  |
/// | `1`      | `Linear`        | 1                | `C(d+1,1) = d+1`|
/// | `k≥2`    | `Degree(k)`     | k                | `C(d+k,k)`      |
///
/// **How the polynomial null space is consumed during basis construction:**
///
/// 1. `polynomial_block_from_order` materialises an `(n, C(d+r,r))` block `P`
///    of monomials up to total degree `r` at the selected `centers`.
/// 2. `kernel_constraint_nullspace` computes `Z = null(P_centers^T)`, a
///    `(k, k − C(d+r,r))` matrix. Reparameterising the radial kernel
///    coefficients as `α = Z γ` enforces the side condition `P_centers^T α = 0`
///    and yields `k − C(d+r,r)` free kernel parameters.
/// 3. The polynomial block `P_data` evaluated at the data rows is appended to
///    the kernel block `Φ Z`, giving a total of
///    `(k − C(d+r,r)) + C(d+r,r) = k` columns before the spatial
///    identifiability transform.  Crucially, the total width equals the
///    requested center count `k`, **not** `k + C(d+r,r)`.
///
/// **Example — `duchon(PC1, PC2, PC3, centers=10, order=1)` (d=3):**
///
/// - Polynomial null space: `C(3+1,1) = 4` monomials `{1, x₁, x₂, x₃}`.
/// - Kernel columns after constraint: `10 − 4 = 6`.
/// - Appended polynomial block: 4 columns.
/// - Pre-identifiability total: `6 + 4 = 10` columns, i.e. exactly `centers`.
///
/// The variant naming matches the Duchon `m` parameter:
/// `Zero` → `m=1`, `Linear` → `m=2`, `Degree(k)` → `m=k+1`.
#[derive(Debug, Clone, Copy, PartialEq, Eq, Hash, Serialize, Deserialize)]
pub enum DuchonNullspaceOrder {
    Zero,
    Linear,
    Degree(usize),
}

/// Explicit low-rank spectral construction for a Duchon kernel.
///
/// `rank` is the total retained spline dimension, including the polynomial
/// null space. `Fresh` asks the basis builder to compute the dominant
/// center-kernel eigenspace once. `Frozen` carries both the resulting direct
/// center-to-radial transform and the reduced bending operator into prediction
/// and derivative rebuilds. Keeping both pieces in the state is essential:
/// recomputing `Vᵀ K V` after a residual-certified Ritz solve silently replaces
/// the tiny tridiagonal's Galerkin operator by a numerically different one.
#[derive(Debug, Clone, Serialize, Deserialize)]
#[serde(tag = "state", rename_all = "snake_case", deny_unknown_fields)]
pub enum DuchonSpectralBasis {
    Fresh {
        rank: usize,
    },
    Frozen {
        rank: usize,
        kernel_transform: Array2<f64>,
        bending_penalty: Array2<f64>,
    },
}

impl DuchonSpectralBasis {
    pub fn rank(&self) -> usize {
        match self {
            Self::Fresh { rank } | Self::Frozen { rank, .. } => *rank,
        }
    }

    pub fn kernel_transform(&self) -> Option<&Array2<f64>> {
        match self {
            Self::Fresh { .. } => None,
            Self::Frozen {
                kernel_transform, ..
            } => Some(kernel_transform),
        }
    }

    pub fn bending_penalty(&self) -> Option<&Array2<f64>> {
        match self {
            Self::Fresh { .. } => None,
            Self::Frozen {
                bending_penalty, ..
            } => Some(bending_penalty),
        }
    }
}

/// Duchon-like basis configuration with explicit low-frequency null-space
/// control and explicit spectral power.
#[derive(Debug, Clone, Serialize, Deserialize)]
#[serde(deny_unknown_fields)]
pub struct DuchonBasisSpec {
    pub center_strategy: CenterStrategy,
    #[serde(default)]
    pub periodic: Option<Vec<Option<f64>>>,
    /// Optional hybrid Matérn width. `None` means pure scale-free Duchon with
    /// spectrum `||w||^(2p + 2s)`. `Some(length_scale)` enables the hybrid
    /// spectrum `||w||^(2p) * (kappa^2 + ||w||^2)^s`, `kappa = 1/length_scale`.
    pub length_scale: Option<f64>,
    /// Literal Duchon spectral power `s` (`f64`, fractional values fully
    /// threaded end-to-end). The pure-Duchon kernel exponent is `2(p + s) − d`,
    /// so this is the knob that sets `φ(r)`: `s = 0` is the integer-order Duchon
    /// kernel `r^{2p−d}` (its `r²·log r` log case in even `d`, ≡ the thin-plate
    /// kernel); `s = (d − 1)/2` gives the cubic `r³` in every dimension.
    ///
    /// This field is taken LITERALLY by the basis builder — `power = 0` means
    /// `s = 0`, NOT "use a default". The magic cubic default (applied when the
    /// user gives no explicit power) is a request-layer choice resolved by the
    /// formula / CLI / pyffi front-ends via [`duchon_cubic_default`]; by the time
    /// a spec reaches the builder this value is the final intended `s`. The
    /// hybrid Duchon–Matérn path (`length_scale = Some`) still requires an
    /// integer `s` (read via `spec.power_as_usize()`).
    pub power: f64,
    pub nullspace_order: DuchonNullspaceOrder,
    #[serde(default)]
    pub identifiability: SpatialIdentifiability,
    /// Per-axis anisotropy log-scales η_a.
    ///
    /// For hybrid Duchon (`length_scale=Some`), these are centered contrasts in
    /// the decomposition Λ = κA with det(A)=1. For pure Duchon
    /// (`length_scale=None`), they parameterize shape-only axis warping on the
    /// public path and are centered before basis evaluation/writeback so no
    /// global length scale is introduced.
    ///
    /// When Some, the distance is r = √(Σ_a exp(2η_a) · (x_a - c_a)²).
    /// When None, isotropic distance r = ‖x - c‖ is used.
    #[serde(default)]
    pub aniso_log_scales: Option<Vec<f64>>,
    #[serde(default)]
    pub operator_penalties: DuchonOperatorPenaltySpec,
    #[serde(default)]
    pub boundary: OneDimensionalBoundary,
    /// Data-metric radial reparameterization `V` (#1355), mirroring the
    /// thin-plate Wood-TPRS reparam. When `Some`, the constrained kernel
    /// transform is folded to `Z·V` so the realized design columns rotate into
    /// the `G_c`-orthonormal generalized eigenbasis of `Ω_c v = μ G_c v` and the
    /// native penalty becomes the diagonal curvature-per-unit-data-variance
    /// spectrum (mgcv's cliff), preventing the REML over-smoothing collapse to
    /// EDF = 1. Frozen at the cold dense build and replayed verbatim by the
    /// predict / κ-trial / ψ-derivative paths so they stay bit-consistent with
    /// the fit-time design. `None` on the lazy/streaming path (huge `n`), which
    /// retains the original constrained basis.
    #[serde(default)]
    pub radial_reparam: Option<Array2<f64>>,
}

impl DuchonBasisSpec {
    /// Integer view of `power` for the existing integer-only downstream chain.
    /// Non-finite or non-integer values fall back to `0` (the integer-only
    /// validators downstream already reject this case with a clear message).
    pub fn power_as_usize(&self) -> usize {
        duchon_power_to_usize(self.power)
    }
}

/// Convert a Duchon spectral-power `f64` into the integer view used by the
/// closed-form code paths. Non-finite, negative, or fractional values clamp to
/// `0` so the validator downstream emits the canonical error.
pub fn duchon_power_to_usize(power: f64) -> usize {
    if !power.is_finite() || power < 0.0 {
        return 0;
    }
    let rounded = power.round();
    if (rounded - power).abs() > 1e-9 {
        return 0;
    }
    rounded as usize
}

#[derive(Clone, Debug, Serialize, Deserialize)]
pub struct DuchonOperatorPenaltySpec {
    pub mass: OperatorPenaltySpec,
    pub tension: OperatorPenaltySpec,
    pub stiffness: OperatorPenaltySpec,
}

#[derive(Clone, Debug, Serialize, Deserialize)]
pub enum OperatorPenaltySpec {
    Active {
        initial_log_lambda: f64,
        prior: Option<RhoPrior>,
    },
    Disabled,
}

impl Default for DuchonOperatorPenaltySpec {
    fn default() -> Self {
        // ALL ON. The Duchon penalty is a Hilbert scale: curvature is the
        // always-on exact RKHS `Primary` Gram and the trend ridge is always on;
        // the lower orders — mass (amplitude `Σ(f−f̄)²`) and tension (first-order
        // roughness `Σ‖∇f‖²`) — are active here, collocated on a density-blind
        // data-support sample. REML deselects any the data don't support (SPEC:
        // recover the null by default, opt INTO overfitting). Stiffness (`D2`)
        // stays off — `Primary` is the exact, superior curvature. (The Matérn
        // collocation overlay builds its own `all_active()`; SAE atoms, which
        // ship only `Primary`, use `all_disabled()`.)
        Self {
            mass: OperatorPenaltySpec::Active {
                initial_log_lambda: 0.0,
                prior: None,
            },
            tension: OperatorPenaltySpec::Active {
                initial_log_lambda: 0.0,
                prior: None,
            },
            stiffness: OperatorPenaltySpec::Disabled,
        }
    }
}

impl DuchonOperatorPenaltySpec {

    pub fn all_disabled() -> Self {
        Self {
            mass: OperatorPenaltySpec::Disabled,
            tension: OperatorPenaltySpec::Disabled,
            stiffness: OperatorPenaltySpec::Disabled,
        }
    }

    /// All three operator dials active — used by the Matérn collocation overlay.
    pub fn all_active() -> Self {
        let active = || OperatorPenaltySpec::Active {
            initial_log_lambda: 0.0,
            prior: None,
        };
        Self {
            mass: active(),
            tension: active(),
            stiffness: active(),
        }
    }

    /// Operator-penalty dials appropriate for a Matérn-ν kernel in dimension `d`.
    ///
    /// The Matérn-ν RKHS is the Sobolev space `H^m` with `m = ν + d/2`: its
    /// squared norm controls the order-`j` derivative in L2 exactly when
    /// `j ≤ m`. The collocation overlay penalizes the squared L2 norms of the
    /// value (mass, `D0`, j=0), gradient (tension, `D1`, j=1) and Hessian
    /// (stiffness, `D2`, j=2). Activating a penalty whose derivative order
    /// exceeds the RKHS smoothness (`j > m`) imposes a roughness constraint the
    /// true kernel does NOT — it over-smooths the reduced-rank fit relative to
    /// the exact GP (mgcv `bs="gp"`, GpGp).
    ///
    /// The ν=1/2 Ornstein–Uhlenbeck kernel is the sole exception: its cusp at a
    /// center makes the collocated gradient/Hessian undefined, so it retains
    /// mass only (#707). Every differentiable order uses the inclusive Sobolev
    /// boundary. In particular ν=3/2 in d=1 has `m=2`, and its finite `D2`
    /// stiffness energy belongs to H². Omitting that block leaves the rough
    /// kernel with only mass+tension, inflates EDF, and changes the REML model
    /// class relative to its stated RKHS.
    pub fn matern_for_smoothness(nu: MaternNu, d: usize) -> Self {
        let m = nu.half_integer_value() + 0.5 * d as f64;
        // Tolerance keeps the mathematically inclusive `j ≤ m` boundary stable
        // under floating-point representation of half-integer orders.
        const ORDER_EPS: f64 = 1e-9;
        let active = || OperatorPenaltySpec::Active {
            initial_log_lambda: 0.0,
            prior: None,
        };
        let gate = |order: f64| {
            if !matches!(nu, MaternNu::Half) && m + ORDER_EPS >= order {
                active()
            } else {
                OperatorPenaltySpec::Disabled
            }
        };
        Self {
            mass: active(),
            tension: gate(1.0),
            stiffness: gate(2.0),
        }
    }
}

/// Resolve a fully admissible Duchon `(nullspace_order, power)` pair.
///
/// Three constraints fold into one resolution:
///   (a) operator collocation up to `max_op`:        `2(p + s) > d + max_op`
///   (b) pure-mode CPD vs polynomial nullspace P_p:  `2s < d`
///       (Wendland Thm 8.17: pure polyharmonic kernel of order m = p+s in
///        R^d is CPD of order `m − ⌊d/2⌋ + 1[d even, log] / m − (d−1)/2
///        [d odd]`, and Duchon interpolation against P_p is well-posed iff
///        CPD order ≤ p, which collapses to `2s < d` since 2s, d are
///        integers and 2s is even.)
///   (a) implies the kernel-existence condition `2(p + s) > d`.
///   (b) is dropped when `length_scale` is `Some` (hybrid Matérn-blended
///       kernel is strictly PD, CPD order 0).
///
/// Strategy: at the requested `nullspace_order`, take the smallest `s`
/// satisfying (a). If that `s` violates (b) in pure mode, escalate the
/// nullspace order by one and retry. Termination: at `p ≥ ⌈(d+max_op)/2⌉ + 1`
/// the operator constraint (a) admits `s = 0`, and `0 < d` satisfies (b)
/// for any `d ≥ 1`, so escalation always converges.
///
/// The returned nullspace order is monotone in the request: it never
/// decreases the user's requested order — only strengthens it when pure-mode
/// CPD requires a richer polynomial absorption space.
pub fn resolve_duchon_orders(
    dim: usize,
    requested_nullspace_order: DuchonNullspaceOrder,
    max_operator_derivative_order: usize,
    length_scale: Option<f64>,
) -> (DuchonNullspaceOrder, usize) {
    assert!(dim >= 1, "Duchon basis requires dim >= 1");
    let pure = length_scale.is_none();
    let mut nullspace = requested_nullspace_order;
    // Bounded loop: escalation terminates by the argument above.
    for _ in 0..=(dim + max_operator_derivative_order + 1) {
        let p = duchon_p_from_nullspace_order(nullspace);
        // Smallest s with 2(p + s) > d + max_op:
        //   2p > d + max_op            ⇒ s = 0
        //   else s = ⌈(d + max_op + 1 − 2p) / 2⌉ = (d + max_op + 2 − 2p) / 2
        let s_op = if 2 * p > dim + max_operator_derivative_order {
            0
        } else {
            (dim + max_operator_derivative_order + 2 - 2 * p) / 2
        };
        if !pure || 2 * s_op < dim {
            return (nullspace, s_op);
        }
        nullspace = duchon_next_nullspace_order(nullspace);
    }
    // Bounded-loop fallback: by the analysis in the docstring, for
    // `p >= ceil((dim + max_op) / 2) + 1` the operator constraint admits
    // `s = 0` and (in pure mode) `0 < dim` satisfies the kernel-existence
    // condition. The loop above always reaches that regime within the bound,
    // so returning the last `nullspace` with `s = 0` is a valid answer.
    (nullspace, 0)
}

#[inline]
pub(crate) fn duchon_next_nullspace_order(order: DuchonNullspaceOrder) -> DuchonNullspaceOrder {
    match order {
        DuchonNullspaceOrder::Zero => DuchonNullspaceOrder::Linear,
        DuchonNullspaceOrder::Linear => DuchonNullspaceOrder::Degree(2),
        DuchonNullspaceOrder::Degree(k) => DuchonNullspaceOrder::Degree(k + 1),
    }
}

pub(crate) fn duchon_previous_nullspace_order(order: DuchonNullspaceOrder) -> DuchonNullspaceOrder {
    match order {
        DuchonNullspaceOrder::Zero => DuchonNullspaceOrder::Zero,
        DuchonNullspaceOrder::Linear => DuchonNullspaceOrder::Zero,
        DuchonNullspaceOrder::Degree(2) => DuchonNullspaceOrder::Linear,
        DuchonNullspaceOrder::Degree(k) => DuchonNullspaceOrder::Degree(k - 1),
    }
}

/// Returns the maximum derivative order required by the *active* operator
/// penalties: 2 if stiffness is Active, else 1 if tension is Active, else 0.
/// Mass-only (or no active operator) penalties only require kernel validity
/// (`2(p+s) > d`), tension requires D1 collocation (`2(p+s) > d+1`), and
/// stiffness requires D2 collocation (`2(p+s) > d+2`).
pub fn duchon_max_active_operator_derivative_order(
    operator_penalties: &DuchonOperatorPenaltySpec,
) -> usize {
    if matches!(
        operator_penalties.stiffness,
        OperatorPenaltySpec::Active { .. }
    ) {
        2
    } else if matches!(
        operator_penalties.tension,
        OperatorPenaltySpec::Active { .. }
    ) {
        1
    } else {
        0
    }
}

/// Metadata returned by generic basis builders.
#[derive(Debug, Clone)]
pub enum BasisMetadata {
    BSpline1D {
        knots: Array1<f64>,
        identifiability_transform: Option<Array2<f64>>,
        periodic: Option<(f64, f64, usize)>,
        /// Effective B-spline polynomial degree carried by `knots`.
        ///
        /// Persisted alongside `knots` so prediction can reconstruct an
        /// evaluator that matches fit-time geometry, even when the fit-time
        /// auto-shrink (issue #340) reduced the user's requested degree to
        /// fit the available data (`n` too small for cubic ⇒ quadratic ⇒
        /// linear). When `None` the consumer should fall back to the
        /// upstream `BSplineBasisSpec.degree` (legacy / non-shrunk path).
        degree: Option<usize>,
        /// Human-readable description of an automatic basis shrink (issue #340)
        /// when the user's requested `(degree, num_internal_knots)` exceeded the
        /// available evaluation count `n`. `Some(note)` records the before→after
        /// configuration; `None` means no auto-shrink occurred for this basis.
        auto_shrink_note: Option<String>,
        /// Raw-basis particular-solution coefficients `β_p` for a *non-zero*
        /// endpoint anchor (#2297), if any. The term carries a fixed affine
        /// offset function `B_raw(x) · β_p` in addition to its constrained
        /// design `B_raw(x) · Z`; the design assembler realizes that offset into
        /// the model's linear predictor at both fit and predict time. `None`
        /// for free / clamped / zero-anchor bases (the ordinary pure-linear
        /// chart). Recomputed deterministically from the frozen `knots`,
        /// `degree` and boundary conditions on every rebuild, so a saved model
        /// replays the identical offset; it is serialized here so the assembler
        /// need not re-derive it from the spec. This metadata is transient
        /// (rebuilt at predict from the serialized frozen spec), not persisted.
        anchor_offset_coeffs: Option<Array1<f64>>,
    },
    /// Natural cubic regression spline (`bs="cr"`/`"cs"`) metadata (#1074).
    ///
    /// `knots` are the `k` Lancaster–Salkauskas knots that index the basis
    /// values directly (basis dim = `knots.len()`). Predict-time rebuilds
    /// reconstruct the cr geometry from `knots` and replay the captured
    /// `identifiability_transform` exactly, mirroring `BSpline1D`.
    CubicRegression1D {
        knots: Array1<f64>,
        identifiability_transform: Option<Array2<f64>>,
    },
    ThinPlate {
        /// Kernel centers in the STANDARDIZED frame (`x / input_scale`).
        centers: Array2<f64>,
        /// Kernel range in the user's ORIGINAL units — a different frame from
        /// `centers`.  Any consumer that evaluates the kernel against
        /// `centers` must first go through
        /// [`crate::IsotropicScale::to_standardized_units`]; the frame tag is
        /// what makes forgetting that a compile error (#2636).
        length_scale: crate::OriginalUnits,
        periodic: Option<Vec<Option<f64>>>,
        identifiability_transform: Option<Array2<f64>>,
        /// Uniform coordinate scale used for isotropic input standardization.
        input_scale: crate::IsotropicScale,
        /// Wood-TPRS radial reparameterization carried into prediction so the
        /// rotated radial basis at predict-time matches fit-time exactly. `None`
        /// in the lazy/streaming path which retains the original basis.
        radial_reparam: Option<Array2<f64>>,
    },
    Sphere {
        centers: Array2<f64>,
        penalty_order: usize,
        method: SphereMethod,
        max_degree: Option<usize>,
        wahba_kernel: SphereWahbaKernel,
        constraint_transform: Option<Array2<f64>>,
    },
    /// Constant-curvature (`M_κ`) geodesic-kernel smooth (#944). `kappa` and
    /// the realized `length_scale` are persisted so predict-time (and the
    /// future ψ-channel per-trial) rebuilds replay the exact fit-time
    /// geometry; `constraint_transform` is the composed `z · z_parametric`
    /// frozen by the global identifiability pipeline (#532 pattern).
    ConstantCurvature {
        centers: Array2<f64>,
        kappa: f64,
        length_scale: f64,
        constraint_transform: Option<Array2<f64>>,
    },
    /// Measure-jet spline smooth: multiscale local-jet-residual energy of the
    /// empirical measure, quadratured on the center set. `centers` are the
    /// REALIZED barycenter nodes; `order_s` stores the spec's order sentinel
    /// verbatim as the mode marker (0.0 = per-level/spectral, > 0 = fused
    /// pin — persisting a realized default would flip the rebuilt mode). The
    /// penalty depends on the FIT data through `masses`, the realized
    /// `eps_band`, the support anchors, and the normalization scales, so all
    /// are persisted and replayed verbatim by
    /// predict-time (and per-ψ-trial) rebuilds — recomputing either from
    /// predict rows would change the penalty the coefficients were estimated
    /// under. `constraint_transform` is the composed `z · z_parametric`
    /// frozen by the global identifiability pipeline (#532 pattern).
    MeasureJet {
        centers: Array2<f64>,
        input_scale: crate::IsotropicScale,
        /// Kernel range in the STANDARDIZED frame — the SAME frame as
        /// `centers`, and the odd one out among the four Euclidean spatial
        /// families (ThinPlate/Matern/Duchon all store original units).  The
        /// asymmetry is deliberate: a frozen MeasureJet range replays
        /// verbatim.  It used to live only in a private policy enum
        /// (`InputFrameNormalization::AutoStandardizedFreshOriginalReplayRealized`)
        /// that the value could not carry; the tag now carries it (#2636).
        length_scale: crate::StandardizedUnits,
        eps_band: Vec<f64>,
        order_s: f64,
        alpha: f64,
        tau0: f64,
        masses: Array1<f64>,
        support_means: Vec<f64>,
        penalty_normalization_scales: Vec<f64>,
        raw_penalty_normalization_scales: Vec<f64>,
        fused_penalty_normalization_scale: Option<f64>,
        constraint_transform: Option<Array2<f64>>,
        /// Ambient input-measurement-error scale `σ_coord` (issue #2225): the
        /// perpendicular off-manifold residual spread of the fit rows, in the
        /// centers' (standardized) frame. `None` when it could not be estimated.
        /// Carried into `MeasureJetFrozenQuadrature::sigma_coord` at freeze time.
        sigma_coord: Option<f64>,
    },
    Matern {
        /// Kernel centers in the STANDARDIZED frame (`x / input_scale`).
        centers: Array2<f64>,
        /// Kernel range in the user's ORIGINAL units; see
        /// [`BasisMetadata::ThinPlate::length_scale`] for the frame contract.
        length_scale: crate::OriginalUnits,
        periodic: Option<Vec<Option<f64>>>,
        nu: MaternNu,
        include_intercept: bool,
        identifiability_transform: Option<Array2<f64>>,
        /// Uniform coordinate scale used for isotropic input standardization.
        input_scale: crate::IsotropicScale,
        /// Per-axis anisotropy log-scales η_a for geometric anisotropy.
        /// When Some, distance is r = √(Σ_a exp(2η_a) · (x_a - c_a)²).
        aniso_log_scales: Option<Vec<f64>>,
    },
    Duchon {
        /// Kernel centers in the STANDARDIZED frame (`x / input_scale`).
        centers: Array2<f64>,
        /// Hybrid Duchon–Matérn range in the user's ORIGINAL units, or `None`
        /// for the pure scale-free spectrum; see
        /// [`BasisMetadata::ThinPlate::length_scale`] for the frame contract.
        length_scale: Option<crate::OriginalUnits>,
        periodic: Option<Vec<Option<f64>>>,
        power: f64,
        nullspace_order: DuchonNullspaceOrder,
        identifiability_transform: Option<Array2<f64>>,
        /// Uniform coordinate scale used for isotropic input standardization.
        input_scale: crate::IsotropicScale,
        /// Per-axis anisotropy log-scales η_a, stored for prediction.
        aniso_log_scales: Option<Vec<f64>>,
        /// Support points used to build the active lower-order operator
        /// penalties (mass/tension/stiffness). Stored so runtime adaptive
        /// caches can rebuild the exact same operator rows instead of guessing
        /// from centers.
        operator_collocation_points: Option<Array2<f64>>,
        /// Data-metric radial reparameterization `V` (#1355). When `Some`, the
        /// constrained kernel transform is folded to `Z·V` so predict-time and
        /// κ-trial rebuilds replay the exact fit-time rotated radial basis.
        /// `None` on the lazy/streaming path (original constrained basis).
        radial_reparam: Option<Array2<f64>>,
        /// Frozen direct center-to-radial transform for an explicitly spectral
        /// basis. Unlike `radial_reparam`, this already includes the polynomial
        /// side-condition projection and therefore has `centers.nrows()` rows.
        spectral_basis: Option<DuchonSpectralBasis>,
    },
    Pca {
        feature_cols: Vec<usize>,
        basis_matrix: Array2<f64>,
        centered: bool,
        smooth_penalty: f64,
        center_mean: Option<Array1<f64>>,
        pca_basis_path: Option<std::path::PathBuf>,
        chunk_size: usize,
    },
    TensorBSpline {
        feature_cols: Vec<usize>,
        knots: Vec<Array1<f64>>,
        degrees: Vec<usize>,
        periods: Vec<Option<f64>>,
        /// Per-margin flag: `true` when that margin is a natural cubic
        /// regression spline (`NaturalCubicRegression` knotspec) rather than an
        /// open/periodic B-spline (#1074). Persisted so the tensor freeze
        /// rebuilds the cr marginal knotspec (value-at-knot) instead of an open
        /// `Provided(knots)` B-spline, keeping predict-time marginals identical
        /// to the fit-time cr margins. Defaults to all-`false` (legacy B-spline
        /// tensors) when deserialized from an older persisted model (the
        /// older-model default is applied on the persisted `SmoothBasisSpec`
        /// side; `BasisMetadata` itself is transient builder output and is not
        /// serde-serialized, so it carries no `#[serde]` attributes).
        is_cr: Vec<bool>,
        identifiability_transform: Option<Array2<f64>>,
    },
    SphereHarmonics {
        max_degree: usize,
        radians: bool,
    },
    /// Wrap an inner basis metadata to record a multiplicative `by` (continuous or
    /// factor) along a column of the dataset.
    BySmooth {
        inner: Box<BasisMetadata>,
        by_col: usize,
        levels: Option<Vec<u64>>,
        ordered: bool,
    },
    /// Factor-by-smooth (mgcv-style `s(x, by=g, bs="fs"|"sz"|"re")`).
    FactorSmooth {
        continuous_cols: Vec<usize>,
        group_col: usize,
        knots: Array1<f64>,
        degree: usize,
        periodic: Option<(f64, f64, usize)>,
        group_levels: Vec<u64>,
        flavour: String,
        /// `true` when the per-level marginal is a cubic regression spline
        /// (`NaturalCubicRegression` knotspec, mgcv's `bs="sz"` default marginal,
        /// #1074). Predict-time freeze must then restore a cr knotspec from the
        /// stored value-knots rather than treating them as a B-spline knot
        /// vector. Defaults to `false` (B-spline marginal) for backward compat.
        marginal_is_cr: bool,
    },
}

/// Standardized basis build result for engine-level composition.
#[derive(Clone)]
pub struct BasisBuildResult {
    pub design: DesignMatrix,
    /// Fixed row-wise contribution carried by an affine basis chart.
    ///
    /// Ordinary bases are linear in their fitted coefficients and leave this
    /// as `None`. An inhomogeneous boundary condition, such as a non-zero
    /// B-spline endpoint anchor, realizes the basis as
    /// `offset(x) + design(x) * beta`; the known `offset(x)` belongs here, not
    /// in a fake coefficient column. Term-collection assembly sums these
    /// channels and routes the result through the model's ordinary likelihood
    /// offset at fit and prediction time.
    pub affine_offset: Option<Array1<f64>>,
    /// Canonical active penalties. Matrix, spectral metadata, operator form,
    /// and semantic identity are one record so dropping an earlier candidate
    /// cannot shift one channel without shifting all of them.
    pub active_penalties: Vec<ActivePenalty>,
    /// Candidate diagnostics excluded from the active smoothing-parameter
    /// layout. Dropped candidates never share a positional container with
    /// active matrices.
    pub dropped_penalties: Vec<DroppedPenaltyInfo>,
    pub metadata: BasisMetadata,
    /// Optional factored rowwise-Kronecker representation for tensor-product
    /// bases. When present, downstream code can keep the design operator-backed
    /// instead of forcing a fully materialized `n x prod(q_j)` block.
    pub kronecker_factored: Option<KroneckerFactoredBasis>,
    /// Joint-null absorption rotation for this basis, when the basis carries
    /// any penalties with a non-trivial joint null space.
    ///
    /// `Some(rotation)` records `Q = [U_range | U_null]` where `U_null` spans
    /// the joint null space `null(Σ_k S_k)` over this basis's active
    /// penalties (unscaled — the structural joint null is independent of
    /// `λ`). After the basis pipeline applies this rotation, the design
    /// becomes `X · Q` and each penalty becomes `Qᵀ S_k Q`, block-diagonal
    /// with a guaranteed-zero null tail. The same `Q` must be replayed at
    /// prediction time, so it is persisted in the fitted model. `None`
    /// indicates either no penalties on this basis, or a full-rank joint
    /// penalty (joint nullity = 0). A `Some` value is never recorded with
    /// `joint_nullity == 0` — the `None` discriminant is canonical for
    /// "nothing to absorb".
    ///
    /// Stage-2 commit A: this field is plumbed into the struct but neither
    /// computed nor applied yet. Stage-2 commit B populates it; Stage-2
    /// commit D applies the rotation to `design` and `penalties`.
    pub joint_null_rotation: Option<JointNullRotation>,
}

/// Joint-null absorption rotation, attached to a smooth's basis when the
/// basis's joint penalty `Σ_k S_k` has a non-trivial null space.
///
/// The `rotation` field stores the orthonormal eigenvector matrix
/// `Q = [U_range | U_null]` of the symmetric joint penalty: the first
/// `range_dim = rotation.ncols() - joint_nullity` columns span
/// `range(Σ_k S_k)`; the remaining `joint_nullity` columns span
/// `null(Σ_k S_k)`. After the pipeline applies the rotation, the smooth's
/// coefficient vector satisfies `β = Q · γ`, the design becomes `X · Q`,
/// and each per-block penalty `S_k` becomes `Qᵀ S_k Q`, which is guaranteed
/// block-diagonal with a zero `(joint_nullity × joint_nullity)` tail
/// (because the joint null annihilates every active `S_k`).
#[derive(Clone, Serialize, Deserialize)]
pub struct JointNullRotation {
    /// `(p_smooth × p_smooth)` orthonormal matrix; range columns first,
    /// joint-null columns last.
    pub rotation: Array2<f64>,
    /// Number of columns at the tail of `rotation` that span the joint
    /// null space. Always `> 0` when wrapped in `Some` — the value `0`
    /// is encoded as `None`.
    pub joint_nullity: usize,
}

impl std::fmt::Debug for JointNullRotation {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        f.debug_struct("JointNullRotation")
            .field(
                "rotation",
                &format_args!("{}×{}", self.rotation.nrows(), self.rotation.ncols()),
            )
            .field("joint_nullity", &self.joint_nullity)
            .finish()
    }
}

impl std::fmt::Debug for BasisBuildResult {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        f.debug_struct("BasisBuildResult")
            .field("design", &self.design)
            .field(
                "affine_offset_len",
                &self.affine_offset.as_ref().map(|offset| offset.len()),
            )
            .field("active_penalties", &self.active_penalties)
            .field("dropped_penalties", &self.dropped_penalties)
            .field("metadata", &self.metadata)
            .field("kronecker_factored", &self.kronecker_factored)
            .field("joint_null_rotation", &self.joint_null_rotation)
            .finish()
    }
}

/// Factored tensor-product basis metadata for operator-backed downstream use.
#[derive(Debug)]
pub struct KroneckerFactoredBasis {
    /// Marginal design matrices: `marginal_designs[j]` is `(n, q_j)`.
    pub marginal_designs: Vec<Array2<f64>>,
    /// Marginal penalty matrices: `marginal_penalties[k]` is `(q_k, q_k)`.
    pub marginal_penalties: Vec<Array2<f64>>,
    /// Marginal basis dimensions: `[q_0, ..., q_{d-1}]`.
    pub marginal_dims: Vec<usize>,
    /// Whether the system includes a global ridge (double) penalty.
    pub has_double_penalty: bool,
    /// λ-invariant tensor structure (marginal eigensystems, reparameterized
    /// marginals, shrinkage scale), memoized once per fit. The marginal
    /// designs/penalties are fixed for the whole fit, so the expensive marginal
    /// `eigh()` and `B_k·U_k` GEMMs only need to run once — every outer REML
    /// iterate (50+ on the #1082 tensor cases) then reuses this. Filled lazily
    /// on first use via [`Self::invariant_structure`]. NOT serialized and reset
    /// to empty on `Clone` (it is purely a within-fit performance cache; a fresh
    /// owner recomputes on first demand, keeping every result bit-identical).
    invariant: std::sync::OnceLock<std::sync::Arc<crate::kronecker::KroneckerInvariantStructure>>,
}

impl Clone for KroneckerFactoredBasis {
    fn clone(&self) -> Self {
        Self {
            marginal_designs: self.marginal_designs.clone(),
            marginal_penalties: self.marginal_penalties.clone(),
            marginal_dims: self.marginal_dims.clone(),
            has_double_penalty: self.has_double_penalty,
            // Propagate the memoized structure when present so a clone made
            // mid-fit keeps the hoist; otherwise start empty (recomputed on
            // first demand, identical result).
            invariant: match self.invariant.get() {
                Some(s) => {
                    let cell = std::sync::OnceLock::new();
                    cell.get_or_init(|| std::sync::Arc::clone(s));
                    cell
                }
                None => std::sync::OnceLock::new(),
            },
        }
    }
}

impl KroneckerFactoredBasis {
    /// Construct from the fixed marginal data with an empty invariant cache.
    pub fn new(
        marginal_designs: Vec<Array2<f64>>,
        marginal_penalties: Vec<Array2<f64>>,
        marginal_dims: Vec<usize>,
        has_double_penalty: bool,
    ) -> Self {
        Self {
            marginal_designs,
            marginal_penalties,
            marginal_dims,
            has_double_penalty,
            invariant: std::sync::OnceLock::new(),
        }
    }

    /// Lazily compute (once) and return the λ-invariant tensor structure
    /// (marginal eigensystems, reparameterized marginals, shrinkage scale).
    ///
    /// Computed from the fixed marginal designs/penalties, so the result is the
    /// same on every call within a fit; the first call pays the `eigh()` cost
    /// and every later call is a pointer load. Because the cache is keyed on the
    /// fixed marginal data and `marginal_penalties`/`marginal_designs` are
    /// immutable for the fit's lifetime, no invalidation is needed.
    pub fn invariant_structure(
        &self,
    ) -> Result<std::sync::Arc<crate::kronecker::KroneckerInvariantStructure>, gam_problem::EstimationError> {
        // Fast path: already memoized.
        if let Some(s) = self.invariant.get() {
            return Ok(std::sync::Arc::clone(s));
        }
        // Compute outside the cell (fallible) and install via `get_or_init`. If a
        // concurrent racer already won, `get_or_init` drops our `computed` and
        // returns the stored one; either way the value is the unique function of
        // the fixed marginal data, so the returned Arc is correct.
        let computed = std::sync::Arc::new(crate::kronecker::KroneckerInvariantStructure::compute(
            &self.marginal_designs,
            &self.marginal_penalties,
            &self.marginal_dims,
        )?);
        let installed = self.invariant.get_or_init(|| computed);
        Ok(std::sync::Arc::clone(installed))
    }
}

#[derive(Debug, Clone, PartialEq, Eq, Serialize, Deserialize)]
pub enum PenaltySource {
    Primary,
    DoublePenaltyNullspace,
    OperatorMass,
    OperatorTension,
    OperatorStiffness,
    /// One per input axis `a` of a multivariate Duchon smooth: the gradient
    /// energy along axis `a`, `Σ(∂f/∂x_a)²`, each with its own REML λ_a. REML
    /// shrinks an axis's contribution toward flat only when it does not earn
    /// its keep — penalty-based ARD / variable relevance, the replacement for
    /// brittle kernel-η optimization. Emitted when `scale_dims` is on.
    OperatorRelevance {
        axis: usize,
    },
    TensorMarginal {
        dim: usize,
    },
    TensorSeparable {
        penalized_margins: Vec<usize>,
    },
    TensorGlobalRidge,
    Other(String),
}

#[derive(Debug, Clone, PartialEq, Eq, Serialize, Deserialize)]
pub enum PenaltyDropReason {
    ZeroMatrix,
    NumericalRankZero,
}

fn default_normalization_scale() -> f64 {
    1.0
}

/// Metadata for one retained penalty coordinate.
///
/// This type is active-only by construction. In particular it has no
/// `active` flag or optional drop reason: those fields allowed a metadata
/// position to exist without a corresponding matrix and made positional
/// indexing silently wrong after an earlier candidate was dropped.
#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct ActivePenaltyInfo {
    pub source: PenaltySource,
    pub original_index: usize,
    pub effective_rank: usize,
    #[serde(default = "default_normalization_scale")]
    pub normalization_scale: f64,
    /// Kronecker factors preserved from tensor penalty construction.
    /// When present, spectral decomposition can use per-factor eigendecomposition.
    #[serde(skip)]
    pub kronecker_factors: Option<Vec<Array2<f64>>>,
    /// Structural null frame carried from the candidate's
    /// [`ConstructiveQuadratic`] (see
    /// [`ConstructiveQuadratic::with_structural_null_frame`]): the declared
    /// null space of the seminorm this penalty represents, in the penalty's
    /// own coefficient chart. Downstream rebuilds
    /// (`rebuild_metric_consistent_ridge` at the term-collection chokepoint)
    /// re-attach it so the double-penalty topology stays a carried theorem
    /// through every chart instead of a per-chart rank measurement (#2445).
    /// Runtime-only, like `kronecker_factors`: a frozen replay rebuilds it
    /// from the basis factory, which is the single source.
    #[serde(skip)]
    pub structural_null_frame: Option<Array2<f64>>,
}

/// Diagnostic for one penalty candidate excluded from the optimizer layout.
/// It is intentionally a different type from [`ActivePenaltyInfo`] so a
/// dropped record cannot be used as an active matrix index.
#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct DroppedPenaltyInfo {
    pub source: PenaltySource,
    pub original_index: usize,
    pub reason: PenaltyDropReason,
    #[serde(default = "default_normalization_scale")]
    pub normalization_scale: f64,
}

/// One atomic active penalty identity.
///
/// Every field describes the same retained candidate. Consumers may reorder,
/// transform, or remove a penalty only by moving the whole record, which makes
/// matrix/role/nullity/operator skew unrepresentable.
#[derive(Clone)]
pub struct ActivePenalty {
    pub matrix: Array2<f64>,
    pub nullity: usize,
    pub null_eigenvectors: Option<Array2<f64>>,
    pub op: Option<std::sync::Arc<dyn crate::analytic_penalties::PenaltyOp>>,
    pub info: ActivePenaltyInfo,
}

impl std::fmt::Debug for ActivePenalty {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        f.debug_struct("ActivePenalty")
            .field(
                "matrix",
                &format_args!("{}×{}", self.matrix.nrows(), self.matrix.ncols()),
            )
            .field("nullity", &self.nullity)
            .field(
                "null_eigenvectors",
                &self
                    .null_eigenvectors
                    .as_ref()
                    .map(|basis| format!("{}×{}", basis.nrows(), basis.ncols())),
            )
            .field("op_dim", &self.op.as_ref().map(|op| op.dim()))
            .field("info", &self.info)
            .finish()
    }
}

#[derive(Debug, Clone)]
pub struct FilteredPenalties {
    pub active: Vec<ActivePenalty>,
    pub dropped: Vec<DroppedPenaltyInfo>,
}

/// A positive-semidefinite quadratic with a construction witness.
///
/// `factor` is the authoritative representation: for coefficients `β`, the
/// penalty is `‖factor · β‖²`, hence its dense matrix is
/// `factorᵀ factor` by construction.  The cached dense matrix exists only for
/// consumers that require it; rank/null-space logic must use the factor and
/// must never attempt to recover PSD provenance from signed eigenvalues of a
/// rounded dense congruence (#2318).
#[derive(Clone)]
pub struct ConstructiveQuadratic {
    factor: Array2<f64>,
    matrix: Array2<f64>,
    /// Orthonormal basis, in this quadratic's own coefficient chart, for the
    /// null space of the seminorm this matrix REPRESENTS — as opposed to the
    /// numerical null space of the matrix itself, which can differ by a
    /// deliberate conditioning term (#2445: the Duchon affine native ridge is
    /// `√ε`-relative and sits within a decade of the spectral rank cutoff, so
    /// a rank test on the shipped matrix decides penalty TOPOLOGY by the
    /// Gram's conditioning). `None` means "no structural declaration; a
    /// consumer that needs the null space must measure it". `Some` with zero
    /// columns is a declaration that the seminorm is structurally full rank.
    structural_null_frame: Option<Array2<f64>>,
}

impl ConstructiveQuadratic {
    /// Construct directly from an energy factor `A`, representing `AᵀA`.
    pub fn from_energy_factor(factor: Array2<f64>, context: &str) -> Result<Self, BasisError> {
        if factor.iter().any(|value| !value.is_finite()) {
            crate::bail_invalid_basis!(
                "{context}: constructive penalty factor contains a non-finite value"
            );
        }
        let matrix = fast_ata(&factor);
        if matrix.iter().any(|value| !value.is_finite()) {
            crate::bail_invalid_basis!("{context}: constructive penalty Gram is not representable");
        }
        Ok(Self {
            factor,
            matrix,
            structural_null_frame: None,
        })
    }

    /// Declare the structural null frame of the represented seminorm (see the
    /// field doc). The frame is a carried certificate (#2427): the factory
    /// that BUILT the seminorm knows its null space as a theorem (Duchon's
    /// polynomial block), and consumers that decide topology
    /// (`crate::basis::rebuild_metric_consistent_ridge`) consume the
    /// declaration instead of re-deriving it from a rank test on a matrix
    /// that deliberately contains a conditioning term.
    pub fn with_structural_null_frame(
        mut self,
        frame: Array2<f64>,
        context: &str,
    ) -> Result<Self, BasisError> {
        if frame.nrows() != self.matrix.nrows() {
            crate::bail_dim_basis!(
                "{context}: structural null frame has {} rows but the quadratic chart has {}",
                frame.nrows(),
                self.matrix.nrows()
            );
        }
        if frame.iter().any(|value| !value.is_finite()) {
            crate::bail_invalid_basis!("{context}: structural null frame is not finite");
        }
        // Orthonormality is what makes the congruence transport below exact.
        let gram = fast_ata(&frame);
        for row in 0..gram.nrows() {
            for col in 0..gram.ncols() {
                let expected = if row == col { 1.0 } else { 0.0 };
                if (gram[[row, col]] - expected).abs() > 1e-8 {
                    crate::bail_invalid_basis!(
                        "{context}: structural null frame is not orthonormal \
                         (FᵀF deviates by {:.3e} at [{row},{col}])",
                        (gram[[row, col]] - expected).abs()
                    );
                }
            }
        }
        self.structural_null_frame = Some(frame);
        Ok(self)
    }

    /// The declared structural null frame, if any (see
    /// [`Self::with_structural_null_frame`]).
    pub fn structural_null_frame(&self) -> Option<&Array2<f64>> {
        self.structural_null_frame.as_ref()
    }

    /// The declared frame restricted to the coefficient block `[lo, hi)`,
    /// or `None` when no frame is declared or the frame has support outside
    /// the block (in which case the block does not own the null space and a
    /// consumer must fall back to measuring).
    pub fn structural_null_frame_block(&self, lo: usize, hi: usize) -> Option<Array2<f64>> {
        let frame = self.structural_null_frame.as_ref()?;
        if lo >= hi || hi > frame.nrows() {
            return None;
        }
        let outside = frame
            .rows()
            .into_iter()
            .enumerate()
            .filter(|(row, _)| *row < lo || *row >= hi)
            .flat_map(|(_, row)| row.to_vec())
            .fold(0.0_f64, |acc, value| acc.max(value.abs()));
        if outside > 1e-12 {
            return None;
        }
        Some(frame.slice(s![lo..hi, ..]).to_owned())
    }

    /// Checked bridge for legacy dense factories that already produce a PSD
    /// function quadratic but do not yet expose their native energy factor.
    ///
    /// This is deliberately fallible and reconstructs a factor from the
    /// canonical range spectrum. Material negative curvature is rejected; a
    /// caller can no longer place an unchecked `Array2` in a
    /// [`PenaltyCandidate`]. New factories should use
    /// [`Self::from_energy_factor`] so PSD is true by construction rather than
    /// inferred after dense assembly.
    pub fn try_from_dense_psd(dense: Array2<f64>, context: &str) -> Result<Self, BasisError> {
        if dense.nrows() != dense.ncols() {
            crate::bail_dim_basis!(
                "{context}: dense penalty must be square, got {}x{}",
                dense.nrows(),
                dense.ncols()
            );
        }
        if dense.iter().any(|value| !value.is_finite()) {
            crate::bail_invalid_basis!("{context}: dense penalty contains a non-finite value");
        }
        if dense.nrows() == 0 {
            return Self::from_energy_factor(Array2::zeros((0, 0)), context);
        }
        let sym = symmetrize_penalty(&dense);
        let (evals, evecs) = FaerEigh::eigh(&sym, Side::Lower).map_err(BasisError::LinalgError)?;
        let tolerance = spectral_tolerance(&evals);
        if let Some(&negative) = evals.iter().find(|&&value| value < -tolerance) {
            return Err(BasisError::IndefinitePenalty {
                context: context.to_string(),
                min_eigenvalue: negative,
                tolerance,
                guidance: "supply the native energy factor for a PSD function penalty; negative curvature is not a penalty null direction".to_string(),
            });
        }
        let positive: Vec<usize> = evals
            .iter()
            .enumerate()
            .filter_map(|(index, &value)| (value > tolerance).then_some(index))
            .collect();
        let mut factor = Array2::<f64>::zeros((positive.len(), dense.nrows()));
        for (row, index) in positive.into_iter().enumerate() {
            let scale = evals[index].sqrt();
            for column in 0..dense.nrows() {
                factor[[row, column]] = scale * evecs[[column, index]];
            }
        }
        Self::from_energy_factor(factor, context)
    }

    /// The authoritative rectangular energy factor.
    pub fn factor(&self) -> &Array2<f64> {
        &self.factor
    }

    /// Dense materialization `AᵀA` for consumers that require a matrix.
    pub fn dense(&self) -> &Array2<f64> {
        &self.matrix
    }

    /// Apply a coefficient gauge to the factor, preserving PSD by
    /// construction instead of multiplying the rounded dense Gram twice.
    ///
    /// A declared structural null frame is transported through the same
    /// congruence: for the (single-block) transform `T`, the structural null
    /// space of `TᵀST` is `{γ : Tγ ∈ span(F)} = null((I − FFᵀ)T)`, which is a
    /// well-conditioned computation on orthonormal inputs — the rank decision
    /// has O(1) principal-angle gaps, never the conditioning of `S` (#2445).
    pub fn restricted(
        &self,
        gauge: &gam_problem::Gauge,
        context: &str,
    ) -> Result<Self, BasisError> {
        let mut out =
            Self::from_energy_factor(gauge.restrict_quadratic_factor(&self.factor), context)?;
        if let Some(frame) = self.structural_null_frame.as_ref() {
            if gauge.n_blocks() == 1 {
                let transform = gauge.block_transform(0);
                out.structural_null_frame = transport_structural_null_frame(frame, &transform);
            }
            // Multi-block gauges do not arise on the paths that declare
            // frames; dropping the declaration is always safe (consumers
            // fall back to measuring), inventing one is not.
        }
        Ok(out)
    }

    /// Multiply the represented quadratic by a finite non-negative scalar.
    pub fn scaled(&self, scale: f64, context: &str) -> Result<Self, BasisError> {
        if !scale.is_finite() || scale < 0.0 {
            crate::bail_invalid_basis!(
                "{context}: constructive penalty scale must be finite and non-negative, got {scale}"
            );
        }
        let root = scale.sqrt();
        let mut out = Self::from_energy_factor(self.factor.mapv(|value| value * root), context)?;
        // A positive rescale does not move the null space; scaling to exactly
        // zero collapses the seminorm and voids the declaration.
        if scale > 0.0 {
            out.structural_null_frame = self.structural_null_frame.clone();
        }
        Ok(out)
    }

    /// Sum PSD quadratics by vertically concatenating their energy factors.
    pub fn sum(terms: &[Self], context: &str) -> Result<Self, BasisError> {
        let coefficient_dim = terms.first().map(|term| term.factor.ncols()).unwrap_or(0);
        if terms
            .iter()
            .any(|term| term.factor.ncols() != coefficient_dim)
        {
            crate::bail_dim_basis!(
                "{context}: constructive penalty sum has inconsistent coefficient dimensions"
            );
        }
        let rows = terms.iter().map(|term| term.factor.nrows()).sum();
        let mut factor = Array2::<f64>::zeros((rows, coefficient_dim));
        let mut start = 0usize;
        for term in terms {
            let end = start + term.factor.nrows();
            factor.slice_mut(s![start..end, ..]).assign(&term.factor);
            start = end;
        }
        Self::from_energy_factor(factor, context)
    }

    /// The exact zero quadratic on a coefficient chart of `dimension`.
    pub fn zero(dimension: usize) -> Self {
        Self {
            factor: Array2::zeros((0, dimension)),
            matrix: Array2::zeros((dimension, dimension)),
            structural_null_frame: None,
        }
    }
}

/// Transport a structural null frame through an injective coefficient
/// transform `T` (raw → reduced): the structural null space of `TᵀST` is
/// `{γ : Tγ ∈ span(F)} = null((I − FFᵀ)T)`, computed with a rank-revealing QR
/// on orthonormal inputs. Returns `Some` with possibly zero columns (a
/// structural "no null space survives the chart" is a valid declaration);
/// `None` only when the shapes cannot compose or the factorization fails, in
/// which case the declaration is dropped rather than guessed.
fn transport_structural_null_frame(
    frame: &Array2<f64>,
    transform: &Array2<f64>,
) -> Option<Array2<f64>> {
    if transform.nrows() != frame.nrows() {
        return None;
    }
    let projected = transform - &frame.dot(&frame.t().dot(transform));
    // `rrqr_nullspace_basis(a)` returns an orthonormal basis of `null(aᵀ)`,
    // so pass `projectedᵀ` to obtain `null(projected)` over the reduced
    // coordinates. Machine-precision cutoff: the singular values here are
    // sines of principal angles between orthonormal frames, so the rank gap
    // is O(1) unless the chart genuinely grazes the subspace.
    gam_linalg::faer_ndarray::rrqr_nullspace_basis(&projected.t().to_owned(), 1.0)
        .ok()
        .map(|(null, _)| null)
}

impl std::fmt::Debug for ConstructiveQuadratic {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        f.debug_struct("ConstructiveQuadratic")
            .field(
                "factor",
                &format_args!("{}×{}", self.factor.nrows(), self.factor.ncols()),
            )
            .field(
                "matrix",
                &format_args!("{}×{}", self.matrix.nrows(), self.matrix.ncols()),
            )
            .field(
                "structural_null_frame",
                &self
                    .structural_null_frame
                    .as_ref()
                    .map(|frame| format!("{}×{}", frame.nrows(), frame.ncols())),
            )
            .finish()
    }
}

impl std::ops::Deref for ConstructiveQuadratic {
    type Target = Array2<f64>;

    fn deref(&self) -> &Self::Target {
        &self.matrix
    }
}

#[derive(Clone)]
pub struct PenaltyCandidate {
    /// Constructive PSD quadratic. Raw dense matrices cannot inhabit a
    /// candidate without passing through a checked constructor.
    pub matrix: ConstructiveQuadratic,
    pub source: PenaltySource,
    pub normalization_scale: f64,
    /// Optional Kronecker factors whose product equals `matrix`.
    /// When present, spectral decomposition can be done per-factor
    /// (O(Σ q_j³) instead of O((Π q_j)³)).
    pub kronecker_factors: Option<Vec<Array2<f64>>>,
    /// Optional operator-form handle whose `as_dense()` matches `matrix`. When
    /// populated by the closed-form factories, this is propagated through to
    /// `CanonicalPenaltyBlock` so downstream consumers can use exact matvec
    /// algebra without rebuilding the dense Gram. When `None`, only the dense
    /// `matrix` path is available.
    pub op: Option<std::sync::Arc<dyn crate::analytic_penalties::PenaltyOp>>,
}

impl std::fmt::Debug for PenaltyCandidate {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        f.debug_struct("PenaltyCandidate")
            .field(
                "matrix",
                &format_args!("{}×{}", self.matrix.nrows(), self.matrix.ncols()),
            )
            .field("source", &self.source)
            .field("normalization_scale", &self.normalization_scale)
            .field(
                "kronecker_factors",
                &self.kronecker_factors.as_ref().map(|v| v.len()),
            )
            .field("op", &self.op.as_ref().map(|o| o.dim()))
            .finish()
    }
}

#[derive(Clone)]
pub struct CanonicalPenaltyBlock {
    pub sym_penalty: Array2<f64>,
    /// Eigenvalues from spectral decomposition (retained to avoid recomputation).
    pub eigenvalues: Array1<f64>,
    /// Eigenvectors from spectral decomposition (retained to avoid recomputation).
    pub eigenvectors: Array2<f64>,
    pub rank: usize,
    pub nullity: usize,
    /// Number of genuine negative-curvature eigendirections (`ev < -tol`).
    /// A non-PSD penalty has `negative_dim > 0`; these directions are
    /// neither range nor null and are never absorbed as unpenalized (#1425).
    pub negative_dim: usize,
    pub rank_tol: f64,
    pub noise_tol: f64,
    pub iszero: bool,
    /// Optional operator-form handle that is bit-equivalent to `sym_penalty`.
    /// Propagated from `PenaltyCandidate.op` when present so downstream
    /// consumers can use matvec without rebuilding the dense Gram.
    pub op: Option<std::sync::Arc<dyn crate::analytic_penalties::PenaltyOp>>,
}

impl std::fmt::Debug for CanonicalPenaltyBlock {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        f.debug_struct("CanonicalPenaltyBlock")
            .field(
                "sym_penalty",
                &format_args!("{}×{}", self.sym_penalty.nrows(), self.sym_penalty.ncols()),
            )
            .field("eigenvalues", &self.eigenvalues)
            .field(
                "eigenvectors",
                &format_args!(
                    "{}×{}",
                    self.eigenvectors.nrows(),
                    self.eigenvectors.ncols()
                ),
            )
            .field("rank", &self.rank)
            .field("nullity", &self.nullity)
            .field("negative_dim", &self.negative_dim)
            .field("rank_tol", &self.rank_tol)
            .field("noise_tol", &self.noise_tol)
            .field("iszero", &self.iszero)
            .field("op", &self.op.as_ref().map(|o| o.dim()))
            .finish()
    }
}

#[derive(Debug)]
pub struct BasisPsiDerivativeResult {
    pub design_derivative: Array2<f64>,
    pub penalties_derivative: Vec<Array2<f64>>,
    /// Operator-backed design derivative for standalone first-derivative
    /// callers. Bundled first+second callers receive the shared operator on
    /// `BasisPsiDerivativeBundle` instead.
    pub implicit_operator: Option<ImplicitDesignPsiDerivative>,
}

#[derive(Debug)]
pub struct BasisPsiSecondDerivativeResult {
    pub designsecond_derivative: Array2<f64>,
    pub penaltiessecond_derivative: Vec<Array2<f64>>,
    /// Operator-backed design derivative for standalone second-derivative
    /// callers. Bundled first+second callers receive the shared operator on
    /// `BasisPsiDerivativeBundle` instead.
    pub implicit_operator: Option<ImplicitDesignPsiDerivative>,
}

#[derive(Debug)]
pub struct BasisPsiDerivativeBundle {
    pub first: BasisPsiDerivativeResult,
    pub second: BasisPsiSecondDerivativeResult,
    /// Shared operator-backed design derivative for the first and second
    /// psi derivatives. Bundled callers consume this once instead of storing
    /// duplicate materialized/streaming operators in both derivative payloads.
    pub implicit_operator: Option<ImplicitDesignPsiDerivative>,
}

/// Per-axis psi_a derivative package for anisotropic spatial terms.
///
/// For a d-dimensional anisotropic term, the kernel phi(r) depends on
/// the anisotropic distance r = |Lambda h| where Lambda = diag(kappa_a). Each axis a
/// has its own log-scale psi_a = log(kappa_a), yielding d first derivatives,
/// d diagonal second derivatives, and d*(d-1)/2 cross second derivatives.
///
/// The cross second derivative d2 phi/(d psi_a d psi_b) = t * s_a * s_b (a != b)
/// is rank-1, so we store the t_values and s_components vectors rather
/// than materializing d^2 matrices.
#[derive(Clone)]
pub struct AnisoBasisPsiDerivatives {
    /// d matrices, each (n x p_smooth): dX/d psi_a.
    pub design_first: Vec<Array2<f64>>,
    /// d matrices, each (n x p_smooth): d2X/d psi_a^2 (diagonal second derivatives).
    pub design_second_diag: Vec<Array2<f64>>,
    /// Cross second derivatives d2X/(d psi_a d psi_b) for a < b.
    pub design_second_cross: Vec<Array2<f64>>,
    /// Axis-pair indices corresponding to `design_second_cross`.
    pub design_second_cross_pairs: Vec<(usize, usize)>,
    /// d x num_penalties: dS_m/d psi_a for each axis a and penalty m.
    pub penalties_first: Vec<Vec<Array2<f64>>>,
    /// d x num_penalties: d2S_m/d psi_a^2 for each axis a and penalty m.
    pub penalties_second_diag: Vec<Vec<Array2<f64>>>,
    /// The (a, b) axis pairs supported by the on-demand cross-penalty
    /// provider. Only the upper triangle (a < b) is stored.
    pub penalties_cross_pairs: Vec<(usize, usize)>,
    /// On-demand cross-penalty second-derivative provider. Exact anisotropic
    /// cross-axis penalty seconds are streamed one pair at a time rather than
    /// stored as a dense upper triangle of blocks.
    pub penalties_cross_provider: Option<AnisoPenaltyCrossProvider>,
    /// Shared operator-backed representation of the anisotropic kernel-side
    /// design derivatives. When `design_first` / `design_second_diag` are empty,
    /// callers must use this operator directly; when they are present, this
    /// operator still provides exact cross-axis second derivatives without
    /// duplicating separate `t` / `s_a` storage layouts.
    pub implicit_operator: Option<ImplicitDesignPsiDerivative>,
}

#[derive(Clone)]
pub struct AnisoPenaltyCrossProvider(
    std::sync::Arc<
        dyn Fn(usize, usize) -> Result<Vec<Array2<f64>>, BasisError> + Send + Sync + 'static,
    >,
);

impl AnisoPenaltyCrossProvider {
    pub(crate) fn new<F>(f: F) -> Self
    where
        F: Fn(usize, usize) -> Result<Vec<Array2<f64>>, BasisError> + Send + Sync + 'static,
    {
        Self(std::sync::Arc::new(f))
    }

    pub fn evaluate(&self, axis_a: usize, axis_b: usize) -> Result<Vec<Array2<f64>>, BasisError> {
        (self.0)(axis_a, axis_b)
    }
}

/// `Qᵀ S Q` for one penalty-derivative block, or `None` when `S` is not square
/// in `Q`'s dimension.
fn rotate_psi_penalty_block(s_local: &Array2<f64>, q: &Array2<f64>) -> Option<Array2<f64>> {
    if s_local.nrows() != q.nrows() || s_local.ncols() != q.nrows() {
        return None;
    }
    Some(fast_ab(&fast_atb(q, s_local), q))
}

impl AnisoBasisPsiDerivatives {
    /// Carry a per-axis ψ-derivative bundle through the stage-2 joint-null
    /// absorption rotation `Q` the realized design was built in (gam#2760).
    ///
    /// `Q` acts on the smooth's COEFFICIENTS: the design becomes `X·Q` and each
    /// penalty block `S_k` becomes `Qᵀ S_k Q`, so every ψ-derivative of either
    /// object transforms the same way — differentiation and the (ψ-independent)
    /// rotation commute. The isotropic arm of the spatial ψ route has always
    /// done this; a per-axis bundle handed to a rotated design without it is an
    /// unrotated derivative of a rotated model, which is why the enrollment
    /// predicate used to exclude rotated terms from per-axis ψ altogether.
    ///
    /// Returns `Ok(None)` — an honest decline, matching the isotropic arm —
    /// when a block's shape does not admit `Q`, so the caller can fall back
    /// rather than assert on a mismatch it cannot repair.
    ///
    /// MUST be applied before any fixed row-space projector: `Q` acts on
    /// coefficients and the projector on rows, and the implicit operator
    /// refuses a coefficient transform once its projector is installed.
    pub fn rotated_by_joint_null(
        mut self,
        rotation: &JointNullRotation,
    ) -> Result<Option<Self>, BasisError> {
        let q = &rotation.rotation;
        if q.nrows() != q.ncols() {
            return Ok(None);
        }
        for matrix in self
            .design_first
            .iter_mut()
            .chain(self.design_second_diag.iter_mut())
            .chain(self.design_second_cross.iter_mut())
        {
            if matrix.ncols() != q.nrows() {
                return Ok(None);
            }
            *matrix = fast_ab(&*matrix, q);
        }
        for blocks in self
            .penalties_first
            .iter_mut()
            .chain(self.penalties_second_diag.iter_mut())
        {
            for block in blocks.iter_mut() {
                let Some(rotated) = rotate_psi_penalty_block(block, q) else {
                    return Ok(None);
                };
                *block = rotated;
            }
        }
        if let Some(provider) = self.penalties_cross_provider.take() {
            let q_owned = q.clone();
            self.penalties_cross_provider = Some(AnisoPenaltyCrossProvider::new(
                move |axis_a, axis_b| {
                    provider
                        .evaluate(axis_a, axis_b)?
                        .into_iter()
                        .map(|block| {
                            rotate_psi_penalty_block(&block, &q_owned).ok_or_else(|| {
                                BasisError::InvalidInput(format!(
                                    "anisotropic cross-penalty block for axes ({axis_a}, {axis_b}) \
                                     is {}x{}, which the joint-null rotation's {} coefficients \
                                     cannot rotate",
                                    block.nrows(),
                                    block.ncols(),
                                    q_owned.nrows()
                                ))
                            })
                        })
                        .collect()
                },
            ));
        }
        if let Some(operator) = self.implicit_operator.take() {
            if operator.p_out() != q.nrows() {
                return Ok(None);
            }
            self.implicit_operator = Some(operator.append_full_transform(q)?);
        }
        Ok(Some(self))
    }
}

// ═══════════════════════════════════════════════════════════════════════════
//  Implicit derivative operator for scalable anisotropic REML gradients
// ═══════════════════════════════════════════════════════════════════════════

pub(crate) const SPATIAL_CENTER_CENTER_MAX_BYTES: usize = 512 * 1024 * 1024; // 512 MiB
pub(crate) const DESIGN_CROSS_CHUNK_SIZE: usize = 1024;

/// Determine whether implicit operators should be used based on problem size
/// and the supplied `ResourcePolicy`.
///
/// Returns `true` when the dense materialization of D first-derivative
/// matrices would exceed `policy.max_single_materialization_bytes`.
///
/// For D axes with n data points and p_smooth basis columns, the dense path
/// allocates D * n * p_smooth * 8 bytes for first-derivative matrices alone
/// (plus a similar amount for second derivatives). The implicit path stores
/// only the compact (n * n_knots) radial jets plus (n * n_knots * D) axis
/// fractions, which is O(n * k * D) instead of O(n * p * D).
pub fn should_use_implicit_operators_with_policy(
    n: usize,
    p: usize,
    d: usize,
    policy: &gam_runtime::resource::ResourcePolicy,
) -> bool {
    // Each first-derivative matrix is (n x p) f64 → n*p*8 bytes.
    // We need D of them for first derivatives, D for second diag, plus
    // the cross-t matrix and s_components. Conservative estimate: 3*D matrices.
    let dense_bytes = 3usize
        .saturating_mul(n)
        .saturating_mul(p)
        .saturating_mul(d)
        .saturating_mul(8);
    dense_bytes > policy.max_single_materialization_bytes
}

pub(crate) fn implicit_radial_cache_bytes(n: usize, k: usize, n_axes: usize) -> usize {
    n.saturating_mul(k)
        .saturating_mul(n_axes.saturating_add(3))
        .saturating_mul(8)
}

pub(crate) fn should_cache_implicit_radial_components(
    n: usize,
    k: usize,
    n_axes: usize,
    policy: &gam_runtime::resource::ResourcePolicy,
) -> bool {
    implicit_radial_cache_bytes(n, k, n_axes) <= policy.max_operator_cache_bytes
}

pub fn assert_no_dense_derivative_materialization(n: usize, p: usize, d_pc: usize) {
    let first = dense_design_bytes(n, p).saturating_mul(d_pc);
    let second = dense_design_bytes(n, p).saturating_mul(d_pc.saturating_mul(d_pc));
    // Consult the library default ResourcePolicy. Production large-scale runs
    // configure `AnalyticOperatorRequired`, which still refuses every dense
    // materialization here. The default `MaterializeIfSmall` mode lets tiny
    // problems (and small-data/test usage) materialize as long as the combined
    // first- and second-order dense bytes fit under the single-materialization
    // byte budget. `DiagnosticsOnly` is treated like `MaterializeIfSmall` for
    // this guard: it permits dense materialization under the same byte cap.
    let policy = gam_runtime::resource::ResourcePolicy::default_library();
    let budget = policy.max_single_materialization_bytes;
    let needed = first.saturating_add(second);
    match policy.derivative_storage_mode {
        gam_runtime::resource::DerivativeStorageMode::AnalyticOperatorRequired => {
            // SAFETY: this assertion helper exists specifically to enforce
            // the large-scale invariant that spatial-PC Duchon derivative
            // designs never persist as dense `Array2<f64>` storage. When the
            // resource policy is `AnalyticOperatorRequired`, any caller that
            // reached this point has materialized something the strict
            // operator contract forbids.
            // SAFETY: AnalyticOperatorRequired forbids dense derivative materialization.
            panic!(
                "spatial PC Duchon derivative designs must remain operator-backed; refused persistent dense derivative materialization (n={n}, p={p}, d_pc={d_pc}, first_order={:.1} MiB, second_order={:.1} MiB)",
                first as f64 / (1024.0 * 1024.0),
                second as f64 / (1024.0 * 1024.0),
            );
        }
        gam_runtime::resource::DerivativeStorageMode::MaterializeIfSmall
        | gam_runtime::resource::DerivativeStorageMode::DiagnosticsOnly => {
            // SAFETY: exceeding the single-materialization budget here is a
            // contract violation by an upstream caller that must route through
            // the operator-backed path; failing loudly surfaces it rather than
            // silently materializing an oversized dense derivative design.
            assert!(
                needed <= budget,
                "spatial PC Duchon derivative designs would exceed the single-materialization budget; refused persistent dense derivative materialization (n={n}, p={p}, d_pc={d_pc}, first_order={:.1} MiB, second_order={:.1} MiB, budget={:.1} MiB)",
                first as f64 / (1024.0 * 1024.0),
                second as f64 / (1024.0 * 1024.0),
                budget as f64 / (1024.0 * 1024.0),
            );
        }
    }
}

pub fn assert_spatial_centers_below_large_scale_cap(
    d_pc: usize,
    centers: ArrayView2<'_, f64>,
) -> Result<(), BasisError> {
    if centers.ncols() != d_pc {
        crate::bail_dim_basis!(
            "spatial PC center dimension mismatch: centers have {} columns, expected {d_pc}",
            centers.ncols()
        );
    }
    let k = centers.nrows();
    let centers_bytes = dense_design_bytes(k, d_pc);
    let center_center_bytes = dense_design_bytes(k, k);
    if centers_bytes > SPATIAL_CENTER_CENTER_MAX_BYTES {
        crate::bail_invalid_basis!(
            "spatial PC centers exceed center storage cap: K={k}, d_pc={d_pc}, centers={:.1} MiB, cap={:.1} MiB",
            centers_bytes as f64 / (1024.0 * 1024.0),
            SPATIAL_CENTER_CENTER_MAX_BYTES as f64 / (1024.0 * 1024.0),
        );
    }
    if center_center_bytes > SPATIAL_CENTER_CENTER_MAX_BYTES {
        crate::bail_invalid_basis!(
            "spatial PC centers exceed center-center large-scale cap: K={k}, d_pc={d_pc}, KxK={:.1} MiB, cap={:.1} MiB",
            center_center_bytes as f64 / (1024.0 * 1024.0),
            SPATIAL_CENTER_CENTER_MAX_BYTES as f64 / (1024.0 * 1024.0),
        );
    }
    Ok(())
}

pub(crate) fn dense_design_bytes(n: usize, p: usize) -> usize {
    n.saturating_mul(p)
        .saturating_mul(std::mem::size_of::<f64>())
}

pub(crate) fn should_use_lazy_spatial_design(
    n: usize,
    p: usize,
    policy: &gam_runtime::resource::ResourcePolicy,
) -> bool {
    matches!(
        policy.derivative_storage_mode,
        gam_runtime::resource::DerivativeStorageMode::AnalyticOperatorRequired
    ) || dense_design_bytes(n, p) > policy.max_single_materialization_bytes
}

pub(crate) fn wrap_dense_design_with_transform(
    design: DesignMatrix,
    transform: &Array2<f64>,
    label: &str,
) -> Result<DesignMatrix, BasisError> {
    match design {
        DesignMatrix::Dense(inner) => {
            let op = CoefficientTransformOperator::new(inner, transform.clone()).map_err(|e| {
                BasisError::InvalidInput(format!("{label} coefficient transform failed: {e}"))
            })?;
            Ok(DesignMatrix::Dense(
                gam_linalg::matrix::DenseDesignMatrix::from(Arc::new(op)),
            ))
        }
        DesignMatrix::Sparse(_) => Err(BasisError::InvalidInput(format!(
            "{label} coefficient transform requires a dense/operator-backed design"
        ))),
    }
}

/// Single-pass `(Bᵀ(W·C), BᵀB)` accumulation over the streamed design.
///
/// Materialises each row chunk of the design **once** and reuses it for both
/// the constraint cross `Bᵀ(W·C)` and the Gram `BᵀB`. On the lazy chunked
/// spatial path each `try_row_chunk` re-evaluates all kernel columns for the
/// chunk, so accumulating both products in a single sweep halves the per-build
/// kernel re-evaluation work (the dominant cost at large scale) versus two
/// independent streaming passes — without changing the result beyond
/// floating-point reassociation. The cross is masked off (`q == 0`) by the
/// caller, which never invokes this when there is no constraint block.
pub(crate) fn design_cross_and_gram(
    design: &DesignMatrix,
    constraint_matrix: ArrayView2<'_, f64>,
    weights: Option<ArrayView1<'_, f64>>,
) -> Result<(Array2<f64>, Array2<f64>), BasisError> {
    let n = design.nrows();
    let k = design.ncols();
    if constraint_matrix.nrows() != n {
        return Err(BasisError::ConstraintMatrixRowMismatch {
            basisrows: n,
            constraintrows: constraint_matrix.nrows(),
        });
    }
    if let Some(w) = weights
        && w.len() != n
    {
        return Err(BasisError::WeightsDimensionMismatch {
            expected: n,
            found: w.len(),
        });
    }
    let q = constraint_matrix.ncols();
    let mut cross = Array2::<f64>::zeros((k, q));
    let mut gram = Array2::<f64>::zeros((k, k));
    for start in (0..n).step_by(DESIGN_CROSS_CHUNK_SIZE) {
        let end = (start + DESIGN_CROSS_CHUNK_SIZE).min(n);
        let basis_chunk = design
            .try_row_chunk(start..end)
            .map_err(|e| BasisError::InvalidInput(e.to_string()))?;
        let mut constraint_chunk = constraint_matrix.slice(s![start..end, ..]).to_owned();
        if let Some(w) = weights {
            for (mut row, &weight) in constraint_chunk
                .axis_iter_mut(Axis(0))
                .zip(w.slice(s![start..end]).iter())
            {
                row *= weight;
            }
        }
        cross += &fast_atb(&basis_chunk, &constraint_chunk);
        gram += &fast_atb(&basis_chunk, &basis_chunk);
    }
    Ok((cross, gram))
}

pub(crate) fn positive_spectral_whitener_from_gram(
    gram: &Array2<f64>,
) -> Result<Array2<f64>, BasisError> {
    // Inverse-square-root for the positive part of `gram`. Eigenvalues at or
    // below the relative rank tolerance `α·ε·n·max_eval` are *dropped*: the
    // returned whitener has shape `(n × keep)` where `keep` counts strictly
    // positive eigendirections of `gram`.
    //
    // Dropping (rather than ridging) is what makes the result a true
    // square-root inverse on the column space of `gram`. This whitener is
    // used by `stabilized_orthogonality_transform_from_gram` to make a
    // pre-existing transform `K_raw` orthonormal under the W-inner product:
    // when some columns of `K_raw` map to zero (or near-zero) under `B`, the
    // constrained Gram `K_raw^T G K_raw` is rank-deficient. Ridging those
    // tail directions with `1/sqrt(ε)` produced spurious basis columns
    // whose coefficient norms blew up to `~1/sqrt(ε)` while their image in
    // `B` was floating-point zero, contaminating downstream linear algebra
    // (in particular it forced `smooth.rs` to widen the post-transform
    // orthogonality residual tolerance to absorb a `cond ≈ 1/sqrt(ε)`
    // rounding floor). Dropping these directions is the right behavior:
    // they contribute nothing to `B`'s column space, and removing them
    // tightens the orthogonality residual back down to the genuine
    // floating-point limit.
    let (eigenvalues, eigenvectors) = gram.eigh(Side::Lower).map_err(BasisError::LinalgError)?;
    let n = gram.nrows();
    let max_eval = eigenvalues.iter().copied().fold(0.0_f64, f64::max);
    // Scale-invariant rank tolerance: the cutoff must track the Gram's own
    // spectrum (`α·ε·n·max_eval`), not an absolute floor. An earlier `max_eval
    // .max(1.0)` clamped the reference scale to 1.0, which is only harmless when
    // `max_eval ≥ 1`; for a genuinely well-conditioned but small-magnitude Gram
    // (e.g. a Duchon hybrid whose evaluated kernel sits far below unit scale in
    // moderate-to-high d) it inflated the tolerance to an absolute `α·ε·n` floor
    // that swallows the entire — perfectly valid — spectrum, spuriously reporting
    // `keep == 0`. Using the true `max_eval` makes `keep` invariant to a uniform
    // rescaling of the Gram (which scales every eigenvalue and the cutoff
    // identically). The residual `.max(f64::EPSILON)` only guards the degenerate
    // all-zero Gram so that numerical-zero roundoff directions are still dropped.
    let tol =
        (default_rrqr_rank_alpha() * f64::EPSILON * (n.max(1) as f64) * max_eval).max(f64::EPSILON);
    let keep = eigenvalues.iter().filter(|&&ev| ev > tol).count();
    if keep == 0 {
        let min_ev = eigenvalues.iter().copied().fold(f64::INFINITY, f64::min);
        return Err(BasisError::ConstraintNullspaceCollapsed {
            site: "positive_spectral_whitener_from_gram",
            cross_rank: 0,
            coeff_dim: gram.nrows(),
            cross_frobenius: gram.iter().map(|v| v * v).sum::<f64>().sqrt(),
            gram_spectrum: format!(
                "max eigenvalue {max_eval:.3e} (min {min_ev:.3e}, spectral tolerance {tol:.3e})"
            ),
        });
    }
    // `eigh` returns eigenvalues in ascending order, so the largest `keep`
    // eigenvalues live at the tail.
    let eig_start = eigenvalues.len() - keep;
    let kept_vectors = eigenvectors.slice(s![.., eig_start..]).to_owned();
    let mut inv_sqrt = Array2::<f64>::zeros((keep, keep));
    for (out_i, eig_i) in (eig_start..eigenvalues.len()).enumerate() {
        inv_sqrt[[out_i, out_i]] = 1.0 / eigenvalues[eig_i].sqrt();
    }
    Ok(fast_ab(&kept_vectors, &inv_sqrt))
}

pub(crate) fn stabilized_orthogonality_transform_from_gram(
    gram: &Array2<f64>,
    transform: &Array2<f64>,
) -> Result<Array2<f64>, BasisError> {
    let constrained_gram = {
        let gt = fast_ab(gram, transform);
        fast_atb(transform, &gt)
    };
    let whitening = positive_spectral_whitener_from_gram(&constrained_gram)?;
    Ok(fast_ab(transform, &whitening))
}

pub(crate) fn orthogonality_transform_from_cross_and_gram(
    constraint_cross: &Array2<f64>,
    gram: &Array2<f64>,
) -> Result<Array2<f64>, BasisError> {
    // Compute null(M^T) directly on M = B^T W C (k × q) via column-pivoted QR.
    // Working in the original k-dim coefficient space rather than first
    // whitening by B^T B avoids a fundamental failure mode: when B is heavily
    // collinear, `positive_spectral_whitener_from_gram` truncates the design
    // column-space to a `keep`-dim subspace, and if `keep <= q` the subsequent
    // nullspace search has no room — even though dim null(M^T) = k - rank(M)
    // ≥ k - q is always positive when k > q. The constraint nullspace is a
    // property of M alone; conditioning of the design only matters for the
    // downstream stabilization of B*K_raw.
    let k = constraint_cross.nrows();
    if k == 0 {
        return Err(BasisError::InsufficientColumnsForConstraint { found: 0 });
    }
    let (transform_raw, rank) = rrqr_nullspace_basis(constraint_cross, default_rrqr_rank_alpha())
        .map_err(BasisError::LinalgError)?;
    if rank >= k || transform_raw.ncols() == 0 {
        return Err(BasisError::ConstraintNullspaceCollapsed {
            site: "orthogonality_transform_from_cross_and_gram",
            cross_rank: rank,
            coeff_dim: k,
            cross_frobenius: constraint_cross.iter().map(|v| v * v).sum::<f64>().sqrt(),
            gram_spectrum: "not computed (structural cross-rank collapse: null(Mᵀ) is empty, \
                            so no constrained design exists to eigendecompose)"
                .to_string(),
        });
    }

    // Make the constrained design B*K_raw orthonormal under the W-inner product.
    // If the constrained Gram K_raw^T G K_raw is rank-deficient (because some
    // directions in null(M^T) collapse under B), the spectral whitener drops
    // them — that is the right behavior: a degenerate column never contributes
    // to B's column space and shouldn't appear in the reparameterized basis.
    stabilized_orthogonality_transform_from_gram(gram, &transform_raw)
}

/// The part of `constraint_matrix` that `design`'s realized column span actually
/// **CONTAINS**, as vectors in row space (an `n × r` block, `r ≤ q`).
///
/// # Overlap is not containment, and only containment licenses a deletion
///
/// [`orthogonality_transform_for_design`] removes `rank(BᵀWC)` coefficient
/// directions from the smooth — one for every parametric direction the design
/// has any measurable overlap with. That is the wrong predicate. A direction may
/// be deleted **without loss** only when it is contained in the design's span:
/// then the deleted function IS the parametric column, the parametric block
/// keeps it, and the model span is unchanged. When the design merely
/// *correlates* with the parametric column, the deleted direction is a genuine
/// function the model can no longer represent at all.
///
/// `smooth_requires_parametric_orthogonality` asserts containment for the whole
/// kernel/radial class — *"their realized column span contains the constant …
/// a structural rank-1 collision"* — and for the constant-curvature geodesic
/// kernel that is false. Measured on the `kappa_one_...` fixture (400 rows, 30
/// centers), the orthogonal projection of the planted truth onto the span — the
/// best R² any fit could reach at any smoothing parameter — falls from **0.9984
/// to 0.8957** when the constraint is applied, and the loss grows with the
/// kernel range (0.9440 → 0.8915 from `ℓ = 0.2` to `ℓ = 3`) because the columns
/// grow more collinear and the deleted mean-carrying direction carries more.
/// The shipped pipeline's ceiling matches the hand-applied constraint to six
/// decimals at every range, and the fitted R² sits AT that ceiling: the fit is
/// not failing, a model dimension is missing.
///
/// # The test
///
/// Principal angles between `span(B)` and `span(C)`: with `G = BᵀWB`,
/// `M = BᵀWC` and `N = CᵀWC`, the generalized problem `MᵀG⁻M v = cos²θ · N v`
/// gives `cos θ_i` per direction, and `θ = 0` is containment. The decision is
/// made on `sin²θ = 1 − cos²θ`, and its threshold is DERIVED from that
/// expression's own resolution rather than chosen: it is a difference of two
/// `O(1)` quantities accumulated over `k + q` terms, so anything below
/// `(k + q)·ε` is indistinguishable from zero and anything above it is real.
/// The two populations are nowhere near that boundary — a contained constant
/// sits at `κ·ε`, a merely-correlated one at `10⁻¹`–`10⁰` — so the floor has six
/// orders of margin on both sides and no fixture rides it.
///
/// When EVERY direction is contained the original `constraint_matrix` is
/// returned unchanged rather than a rotation of it, so a basis whose span really
/// does contain the constant keeps its transform bit-for-bit.
pub fn contained_constraint_directions(
    design: &DesignMatrix,
    constraint_matrix: ArrayView2<'_, f64>,
    weights: Option<ArrayView1<'_, f64>>,
) -> Result<Array2<f64>, BasisError> {
    let n = design.nrows();
    let k = design.ncols();
    let q = constraint_matrix.ncols();
    if q == 0 || k == 0 {
        return Ok(Array2::zeros((n, 0)));
    }
    let normalized = unit_normalize_constraint_columns(constraint_matrix, weights);
    let (cross, gram) = design_cross_and_gram(design, normalized.view(), weights)?;
    // `N = CᵀWC` on the unit-normalized block: unit diagonal, off-diagonal
    // cosines between constraint columns.
    let mut constraint_gram = Array2::<f64>::zeros((q, q));
    for i in 0..q {
        for j in i..q {
            let mut acc = 0.0_f64;
            for row in 0..n {
                let w = weights.map_or(1.0, |ws| ws[row]);
                acc += w * normalized[[row, i]] * normalized[[row, j]];
            }
            constraint_gram[[i, j]] = acc;
            constraint_gram[[j, i]] = acc;
        }
    }
    // `MᵀG⁻M`, with `G⁻` truncated at the design Gram's own spectral floor.
    let (design_evals, design_evecs) =
        FaerEigh::eigh(&gram, Side::Lower).map_err(BasisError::LinalgError)?;
    let design_top = design_evals.iter().cloned().fold(0.0_f64, f64::max);
    let mut whitened_cross = design_evecs.t().dot(&cross);
    for i in 0..k {
        let scale = if design_evals[i] > design_top * (k as f64) * f64::EPSILON {
            1.0 / design_evals[i].sqrt()
        } else {
            0.0
        };
        for j in 0..q {
            whitened_cross[[i, j]] *= scale;
        }
    }
    let cos2 = whitened_cross.t().dot(&whitened_cross);
    // Whiten the constraint side so the problem is an ordinary symmetric
    // eigenproblem; a constraint block with dependent columns simply loses those
    // directions, which cannot be contained in anything as separate directions.
    let (constraint_evals, constraint_evecs) =
        FaerEigh::eigh(&constraint_gram, Side::Lower).map_err(BasisError::LinalgError)?;
    let constraint_top = constraint_evals.iter().cloned().fold(0.0_f64, f64::max);
    let keep: Vec<usize> = (0..q)
        .filter(|&i| constraint_evals[i] > constraint_top * (q as f64) * f64::EPSILON)
        .collect();
    if keep.is_empty() {
        return Ok(Array2::zeros((n, 0)));
    }
    let mut inverse_root = Array2::<f64>::zeros((q, keep.len()));
    for (slot, &i) in keep.iter().enumerate() {
        let scale = 1.0 / constraint_evals[i].sqrt();
        for row in 0..q {
            inverse_root[[row, slot]] = constraint_evecs[[row, i]] * scale;
        }
    }
    let reduced = inverse_root.t().dot(&cos2).dot(&inverse_root);
    let (_, angle_evecs) =
        FaerEigh::eigh(&reduced, Side::Lower).map_err(BasisError::LinalgError)?;
    // The principal directions themselves, in row space. The DECISION is not
    // taken on the eigenvalues: `sin²θ = 1 − cos²θ` is a difference of two O(1)
    // quantities and cannot resolve a small angle at all. Forming the residual
    // `c − B(BᵀWB)⁻BᵀWc` explicitly costs one `n × k` pass per direction and is
    // accurate to `ε‖c‖` however small the angle is, which is what makes `√ε` a
    // usable bar rather than a hopeful one.
    let directions = normalized.dot(&inverse_root.dot(&angle_evecs));
    let mut direction_cross = Array2::<f64>::zeros((k, directions.ncols()));
    for start in (0..n).step_by(DESIGN_CROSS_CHUNK_SIZE) {
        let end = (start + DESIGN_CROSS_CHUNK_SIZE).min(n);
        let basis_chunk = design
            .try_row_chunk(start..end)
            .map_err(|e| BasisError::InvalidInput(e.to_string()))?;
        let mut direction_chunk = directions.slice(s![start..end, ..]).to_owned();
        if let Some(ws) = weights {
            for (mut row, &weight) in direction_chunk
                .axis_iter_mut(Axis(0))
                .zip(ws.slice(s![start..end]).iter())
            {
                row *= weight;
            }
        }
        direction_cross += &fast_atb(&basis_chunk, &direction_chunk);
    }
    let mut design_pinv_cross = design_evecs.t().dot(&direction_cross);
    for i in 0..k {
        let scale = if design_evals[i] > design_top * (k as f64) * f64::EPSILON {
            1.0 / design_evals[i]
        } else {
            0.0
        };
        for j in 0..directions.ncols() {
            design_pinv_cross[[i, j]] *= scale;
        }
    }
    let coefficients = design_evecs.dot(&design_pinv_cross);
    let mut residual_sq = vec![0.0_f64; directions.ncols()];
    let mut direction_sq = vec![0.0_f64; directions.ncols()];
    for start in (0..n).step_by(DESIGN_CROSS_CHUNK_SIZE) {
        let end = (start + DESIGN_CROSS_CHUNK_SIZE).min(n);
        let basis_chunk = design
            .try_row_chunk(start..end)
            .map_err(|e| BasisError::InvalidInput(e.to_string()))?;
        let approximation = basis_chunk.dot(&coefficients);
        for j in 0..directions.ncols() {
            for row in start..end {
                let w = weights.map_or(1.0, |ws| ws[row]);
                let target = directions[[row, j]];
                let gap = target - approximation[[row - start, j]];
                residual_sq[j] += w * gap * gap;
                direction_sq[j] += w * target * target;
            }
        }
    }
    // `√ε`: the direction is reproduced by the design to within the square root
    // of machine precision, i.e. it IS in the span as far as this arithmetic can
    // tell. The two populations sit seven orders either side of it — a contained
    // constant reproduces at `κ·ε`, a merely-correlated one at `10⁻¹`–`10⁰`.
    let containment_bar = f64::EPSILON.sqrt();
    let contained: Vec<usize> = (0..directions.ncols())
        .filter(|&i| {
            direction_sq[i] > 0.0 && (residual_sq[i] / direction_sq[i]).sqrt() <= containment_bar
        })
        .collect();
    if contained.is_empty() {
        return Ok(Array2::zeros((n, 0)));
    }
    if contained.len() == keep.len() {
        // Every resolvable parametric direction is inside the span: this is the
        // case the shipped transform was written for, and it keeps it verbatim
        // rather than a rotation of it, so those bases do not move at all.
        return Ok(constraint_matrix.to_owned());
    }
    let mut out = Array2::<f64>::zeros((n, contained.len()));
    for (slot, &i) in contained.iter().enumerate() {
        for row in 0..n {
            out[[row, slot]] = directions[[row, i]];
        }
    }
    Ok(out)
}

/// The span-preserving orthogonalization of a smooth design against a
/// constraint block: the realized block becomes `X·T − C·R`.
///
/// See [`parametric_residualization_for_design`] for the derivation. `T` is the
/// ordinary coefficient-space transform every basis already carries (it goes
/// into the basis metadata and restricts the penalties); `R` is the part that is
/// new, and it is what makes the construction cost no model dimension.
#[derive(Clone, Debug)]
pub struct ParametricResidualization {
    /// `T` — the coefficient-space transform, `p × k`.
    pub coefficient_transform: Array2<f64>,
    /// `R = B·T` in the RAW constraint block's own columns, `q × k`. The
    /// realized block is `X·T − C·R`, so a predict-time rebuild needs `C` at the
    /// new rows and this matrix, and nothing else.
    pub row_space_correction: Array2<f64>,
}

/// Orthogonalize `design` against `constraint_matrix` **without deleting a model
/// dimension**, by projecting in row space rather than restricting in
/// coefficient space.
///
/// # Why this and not [`orthogonality_transform_for_design`]
///
/// That function returns a `Z` spanning `null((XᵀWC)ᵀ)`, so the realized block
/// becomes `X·Z` with span `col(X) ∩ col(C)^⊥` — it drops one coefficient
/// direction per parametric direction the cross resolves, whatever the geometry.
/// `76a520c45` established that such a deletion is free only under CONTAINMENT
/// (see [`contained_constraint_directions`]): when `C`'s direction is inside
/// `col(X)`, the deleted function IS the parametric column and the parametric
/// block keeps it. When it is not, the deleted direction is a genuine function
/// nothing else carries. That fix withheld the deletion, and left nothing in its
/// place — measured (gam#2747),
/// the shipped smooth block then sits at `‖XᵀC‖/(‖X‖‖C‖) = 1.6e-1 … 4.9e-1`
/// against the `1e-8` bar the same step asserts whenever a transform IS applied,
/// and `analyze_smooth_ownership`'s hierarchy is inert for every dependent
/// smooth, because an owner's realized columns are contained in no other basis's
/// span.
///
/// Residualization is the operation that is licensed unconditionally:
///
/// ```text
///     X̃ = X − C(CᵀWC)⁻CᵀWX          span([C | X̃]) = span([C | X])   ALWAYS
/// ```
///
/// — column operations on a block whose partner is in the model — so it makes
/// `X̃ᵀWC = 0` exactly while the joint span is untouched. The rank of `X̃` falls
/// by `dim(col X ∩ col C)` and by nothing else, so the whitener below drops
/// precisely the directions the deletion is entitled to drop and no others.
///
/// It is also CONTINUOUS in the containment residual, which the delete/don't
/// dichotomy is not: the direction the classical constraint removes has
/// residualized norm exactly `sin θ = ‖1 − P_X 1‖/‖1‖`, so as a basis approaches
/// containment its extra direction shrinks to zero and the two constructions
/// meet, instead of the model dimension stepping by one when a fit walks its own
/// range across a threshold.
///
/// # Numerics
///
/// `G̃ = X̃ᵀWX̃` is formed from `X̃` STREAMED chunk by chunk, not as
/// `G − M N⁻ Mᵀ`. The two are equal in exact arithmetic and the second is a
/// difference of near-equal `O(‖G‖)` quantities precisely in the contained case
/// this has to resolve — the same argument `contained_constraint_directions`
/// makes for forming its residual explicitly rather than reading `sin²θ` off
/// `1 − cos²θ`.
///
/// The constraint columns are unit-normalized internally, for the scale reason
/// [`orthogonality_transform_for_design`] documents at length; the normalization
/// is folded back into `row_space_correction` so the returned matrix is stated
/// against the RAW block a predict-time rebuild will reconstruct.
pub fn parametric_residualization_for_design(
    design: &DesignMatrix,
    constraint_matrix: ArrayView2<'_, f64>,
    weights: Option<ArrayView1<'_, f64>>,
) -> Result<ParametricResidualization, BasisError> {
    let n = design.nrows();
    let p = design.ncols();
    let q = constraint_matrix.ncols();
    if p == 0 {
        return Err(BasisError::InsufficientColumnsForConstraint { found: 0 });
    }
    if q == 0 {
        return Ok(ParametricResidualization {
            coefficient_transform: Array2::eye(p),
            row_space_correction: Array2::zeros((0, p)),
        });
    }
    if constraint_matrix.nrows() != n {
        return Err(BasisError::ConstraintMatrixRowMismatch {
            basisrows: n,
            constraintrows: constraint_matrix.nrows(),
        });
    }
    // Column norms are needed twice: to condition the cross/Gram below, and to
    // restate the correction against the raw block at the end.
    let mut column_norms = vec![0.0_f64; q];
    for (col, norm) in column_norms.iter_mut().enumerate() {
        let mut norm_sq = 0.0_f64;
        for row in 0..n {
            let value = constraint_matrix[[row, col]];
            let weight = weights.map_or(1.0, |ws| ws[row]);
            norm_sq += weight * value * value;
        }
        *norm = norm_sq.sqrt();
    }
    let normalized = unit_normalize_constraint_columns(constraint_matrix, weights);

    // `N = ĈᵀWĈ`, and its pseudo-inverse truncated at its own spectral floor so
    // a constraint block with dependent columns simply loses those directions:
    // a direction that is a combination of the others is already projected out
    // by them.
    let mut constraint_gram = Array2::<f64>::zeros((q, q));
    for i in 0..q {
        for j in i..q {
            let mut acc = 0.0_f64;
            for row in 0..n {
                let weight = weights.map_or(1.0, |ws| ws[row]);
                acc += weight * normalized[[row, i]] * normalized[[row, j]];
            }
            constraint_gram[[i, j]] = acc;
            constraint_gram[[j, i]] = acc;
        }
    }
    let (constraint_evals, constraint_evecs) =
        FaerEigh::eigh(&constraint_gram, Side::Lower).map_err(BasisError::LinalgError)?;
    let constraint_top = constraint_evals.iter().cloned().fold(0.0_f64, f64::max);
    let constraint_floor = constraint_top * (q as f64) * f64::EPSILON;
    let mut constraint_pinv = Array2::<f64>::zeros((q, q));
    for slot in 0..q {
        if constraint_evals[slot] <= constraint_floor {
            continue;
        }
        let scale = 1.0 / constraint_evals[slot];
        for i in 0..q {
            for j in 0..q {
                constraint_pinv[[i, j]] +=
                    scale * constraint_evecs[[i, slot]] * constraint_evecs[[j, slot]];
            }
        }
    }

    // `B̂ = N⁻ ĈᵀWX` (q × p): the regression of the design on the normalized
    // constraint block.
    let (cross, _gram) = design_cross_and_gram(design, normalized.view(), weights)?;
    let regression = constraint_pinv.dot(&cross.t());

    // `G̃ = X̃ᵀWX̃`, streamed from the explicit residual.
    let mut residual_gram = Array2::<f64>::zeros((p, p));
    for start in (0..n).step_by(DESIGN_CROSS_CHUNK_SIZE) {
        let end = (start + DESIGN_CROSS_CHUNK_SIZE).min(n);
        let basis_chunk = design
            .try_row_chunk(start..end)
            .map_err(|e| BasisError::InvalidInput(e.to_string()))?;
        let residual_chunk = &basis_chunk - &normalized.slice(s![start..end, ..]).dot(&regression);
        let weighted = match weights {
            Some(ws) => {
                let mut scaled = residual_chunk.clone();
                for (mut row, &weight) in scaled
                    .axis_iter_mut(Axis(0))
                    .zip(ws.slice(s![start..end]).iter())
                {
                    row *= weight;
                }
                scaled
            }
            None => residual_chunk.clone(),
        };
        residual_gram += &fast_atb(&residual_chunk, &weighted);
    }
    // The streamed accumulation is symmetric in exact arithmetic; make it so in
    // floating point before the eigensolver is asked to assume it.
    for i in 0..p {
        for j in (i + 1)..p {
            let averaged = 0.5 * (residual_gram[[i, j]] + residual_gram[[j, i]]);
            residual_gram[[i, j]] = averaged;
            residual_gram[[j, i]] = averaged;
        }
    }
    let coefficient_transform = positive_spectral_whitener_from_gram(&residual_gram)?;

    // Restate the correction against the RAW constraint columns: with
    // `Ĉ = C·diag(1/‖c_j‖)`, `Ĉ·B̂·T = C·(diag(1/‖c_j‖)·B̂·T)`.
    let mut row_space_correction = regression.dot(&coefficient_transform);
    for (row, norm) in column_norms.iter().enumerate() {
        let scale = if *norm > 0.0 && norm.is_finite() {
            1.0 / norm
        } else {
            0.0
        };
        for col in 0..row_space_correction.ncols() {
            row_space_correction[[row, col]] *= scale;
        }
    }
    Ok(ParametricResidualization {
        coefficient_transform,
        row_space_correction,
    })
}

pub fn orthogonality_transform_for_design(
    design: &DesignMatrix,
    constraint_matrix: ArrayView2<'_, f64>,
    weights: Option<ArrayView1<'_, f64>>,
) -> Result<Array2<f64>, BasisError> {
    let k = design.ncols();
    if k == 0 {
        return Err(BasisError::InsufficientColumnsForConstraint { found: 0 });
    }
    let q = constraint_matrix.ncols();
    if q == 0 {
        return Ok(Array2::eye(k));
    }
    // Scale every constraint column to unit (weighted) L2 norm before forming the
    // design/constraint cross `M = Bᵀ W C`. The downstream rank detection
    // (`rrqr_nullspace_basis`) decides HOW MANY parametric directions the smooth
    // genuinely spans by testing the pivoted magnitudes of `M` against an
    // essentially absolute floor `α·ε·max(k,q)` (the `max(|R₀₀|, 1)` reference
    // clamps to 1 whenever the cross is sub-unit). With a RAW constraint column
    // that floor is scale-wrong: the all-ones intercept has norm √n, so `M`
    // carries a √n factor while the tolerance is referenced to 1. For a
    // kernel/radial smooth whose realized design is already (numerically)
    // orthogonal to the constant, `‖Bᵀ1‖` is pure floating-point roundoff
    // (~ε·‖B‖·√n); the √n inflation lands it right at the floor, so a rigid
    // rotation of the covariates — which only perturbs that roundoff — flips the
    // detected rank between 0 and 1. A spurious rank 1 then removes an ARBITRARY
    // real smooth direction (the pivot of a noise vector), and the fitted
    // surface, its EDF, and the REML-selected λ all drift under rotation
    // (gam#1818). Measuring the design's overlap with UNIT constraint directions
    // turns the test into a genuine rotation-invariant cosine: a real overlap is
    // O(1) and always detected, while roundoff-level overlap stays consistently
    // below the floor (rank 0). Column scaling of `C` leaves `null(Mᵀ)` — hence
    // the constrained-design span and the emitted transform — unchanged wherever
    // the rank is unchanged; it only removes the roundoff-driven rank flip.
    let normalized_constraint = unit_normalize_constraint_columns(constraint_matrix, weights);
    let (constraint_cross, gram) =
        design_cross_and_gram(design, normalized_constraint.view(), weights)?;
    orthogonality_transform_from_cross_and_gram(&constraint_cross, &gram)
}

/// Scale each column of a constraint block to unit L2 norm under the inner
/// product used to form the identifiability cross — the `weights`-weighted
/// product when `Some`, the plain product otherwise. A column that is already
/// numerically zero (norm 0 or non-finite) is left untouched: its cross entries
/// are zero regardless, so it contributes no rank. The returned owned copy is
/// used only to build the cross; the emitted transform and the realized
/// constrained design are unaffected (column scaling of `C` preserves
/// `null(Mᵀ)`).
fn unit_normalize_constraint_columns(
    constraint_matrix: ArrayView2<'_, f64>,
    weights: Option<ArrayView1<'_, f64>>,
) -> Array2<f64> {
    let mut c = constraint_matrix.to_owned();
    let (n, q) = c.dim();
    for col in 0..q {
        let mut norm_sq = 0.0_f64;
        for row in 0..n {
            let v = c[[row, col]];
            let w = weights.map_or(1.0, |ws| ws[row]);
            norm_sq += w * v * v;
        }
        let norm = norm_sq.sqrt();
        if norm > 0.0 && norm.is_finite() {
            let inv = 1.0 / norm;
            for row in 0..n {
                c[[row, col]] *= inv;
            }
        }
    }
    c
}

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

    #[test]
    fn starting_count_is_a_supported_low_rank_pilot_capped_by_default() {
        assert_eq!(starting_num_centers(800, 2), 30);
        assert_eq!(starting_num_centers(100_000, 1), 10);
        // The generic conditioning ceiling is `n / 4` and therefore reports
        // zero below four rows; the pilot retains the basis-wide one-center
        // degenerate minimum, which materialization subsequently raises to the
        // exact polynomial floor for the requested family.
        assert_eq!(starting_num_centers(3, 5), 1);
        assert_eq!(starting_num_centers(1, 2), 1);
    }

    #[test]
    fn saturated_expansion_doubles_then_pins_at_validated_ceiling() {
        assert_eq!(expanded_num_centers(30, 157), Some(60));
        assert_eq!(expanded_num_centers(120, 157), Some(157));
        assert_eq!(expanded_num_centers(157, 157), None);
        assert_eq!(
            expanded_num_centers(usize::MAX - 1, usize::MAX),
            Some(usize::MAX)
        );
    }

    #[test]
    fn saturation_excludes_the_nullspace_and_tracks_edf() {
        let tol = 1e-4;
        // Total term EDF includes the three-dimensional nullspace. Saturation
        // means its penalized component spends all 97 remaining directions.
        assert!(basis_is_saturated(100.0, 100, 3, tol));
        // Half-used basis is NOT saturated.
        assert!(!basis_is_saturated(48.5, 100, 3, tol));
        // Just below capacity by more than the derived margin: not saturated.
        assert!(!basis_is_saturated(90.0, 100, 3, tol));
        // A block whose null space already exhausts its columns has no penalizable
        // capacity and is never saturated.
        assert!(!basis_is_saturated(3.0, 3, 3, tol));
        assert!(!basis_is_saturated(f64::NAN, 100, 3, tol));
    }

    #[test]
    fn saturation_is_monotone_in_edf() {
        let tol = 1e-3;
        let (k, null) = (60usize, 3usize);
        let full_width = k as f64;
        // Once saturated at some edf, any larger edf stays saturated.
        let mut first_true: Option<f64> = None;
        let mut e = full_width - 5.0;
        while e <= full_width {
            let sat = basis_is_saturated(e, k, null, tol);
            if sat && first_true.is_none() {
                first_true = Some(e);
            }
            if let Some(t) = first_true {
                assert!(
                    basis_is_saturated(e.max(t), k, null, tol),
                    "saturation must not flip back to false as edf grows"
                );
            }
            e += 0.25;
        }
        assert!(first_true.is_some(), "edf reaching capacity must saturate");
    }
}

#[cfg(test)]
mod containment_tests {
    use super::*;
    fn dense(m: Array2<f64>) -> DesignMatrix {
        DesignMatrix::Dense(gam_linalg::matrix::DenseDesignMatrix::from(m))
    }

    /// A design whose span CONTAINS the constant keeps its constraint block
    /// verbatim — the shipped transform is right for it and must not move.
    #[test]
    fn a_span_that_contains_the_constant_keeps_the_whole_constraint_block() {
        let n = 40usize;
        let mut basis = Array2::<f64>::zeros((n, 3));
        for i in 0..n {
            let x = i as f64 / (n as f64 - 1.0);
            basis[[i, 0]] = 1.0;
            basis[[i, 1]] = x;
            basis[[i, 2]] = x * x;
        }
        let intercept = Array2::<f64>::ones((n, 1));
        let contained =
            contained_constraint_directions(&dense(basis), intercept.view(), None).expect("test");
        assert_eq!(
            contained.dim(),
            (n, 1),
            "the constant IS in this span, so the whole block is contained"
        );
        assert!(
            contained.iter().all(|&v| (v - 1.0).abs() < 1e-14),
            "an all-contained block must come back verbatim, not rotated"
        );
    }

    /// A design that merely CORRELATES with the constant contributes nothing to
    /// residualize against — deleting a coefficient direction there removes a
    /// function the parametric block does not carry.
    #[test]
    fn a_span_that_only_correlates_with_the_constant_contains_nothing() {
        let n = 40usize;
        // Two strictly positive, non-constant columns: heavily correlated with
        // the constant (cosines ~0.99) and containing it in neither.
        let mut basis = Array2::<f64>::zeros((n, 2));
        for i in 0..n {
            let x = i as f64 / (n as f64 - 1.0);
            basis[[i, 0]] = (-0.3 * x).exp();
            basis[[i, 1]] = (-0.9 * x).exp();
        }
        let intercept = Array2::<f64>::ones((n, 1));
        let contained =
            contained_constraint_directions(&dense(basis.clone()), intercept.view(), None)
                .expect("test");
        assert_eq!(
            contained.ncols(),
            0,
            "a merely-correlated constant is not contained and licenses no deletion"
        );
        // And the correlation really is high, so this is not passing by the
        // directions being unrelated: the test would be vacuous if it were.
        let ones = Array1::<f64>::ones(n);
        let cross = basis.t().dot(&ones);
        let gram = basis.t().dot(&basis);
        let (evals, evecs) = FaerEigh::eigh(&gram, Side::Lower).expect("gram");
        let projected = evecs.t().dot(&cross);
        let mut solved = Array1::<f64>::zeros(projected.len());
        for i in 0..projected.len() {
            solved[i] = projected[i] / evals[i];
        }
        let fitted = basis.dot(&evecs.dot(&solved));
        let residual = &ones - &fitted;
        let sine = residual.dot(&residual).sqrt() / ones.dot(&ones).sqrt();
        assert!(
            sine > 1.0e-3 && sine < 0.5,
            "the fixture must be genuinely correlated-but-not-containing; sin θ = {sine}"
        );
    }

    /// A block with one contained direction and one merely-correlated one keeps
    /// exactly the contained one, and it comes back orthogonal to nothing in
    /// particular — only its span matters downstream.
    #[test]
    fn a_mixed_block_keeps_exactly_the_contained_direction() {
        let n = 40usize;
        let mut basis = Array2::<f64>::zeros((n, 2));
        for i in 0..n {
            let x = i as f64 / (n as f64 - 1.0);
            basis[[i, 0]] = 1.0;
            basis[[i, 1]] = (-0.9 * x).exp();
        }
        let mut block = Array2::<f64>::zeros((n, 2));
        for i in 0..n {
            let x = i as f64 / (n as f64 - 1.0);
            block[[i, 0]] = 1.0;
            block[[i, 1]] = x;
        }
        let contained =
            contained_constraint_directions(&dense(basis), block.view(), None).expect("test");
        assert_eq!(
            contained.ncols(),
            1,
            "one of the two block directions is in the span and the other is not"
        );
        // The kept direction must BE the constant, up to scale and sign.
        let column = contained.column(0).to_owned();
        let first = column[0];
        assert!(
            first.abs() > 1e-8,
            "the kept direction must be non-degenerate"
        );
        assert!(
            column.iter().all(|&v| (v / first - 1.0).abs() < 1e-8),
            "the kept direction must be the constant, got {column:?}"
        );
    }
    /// The span-preserving construction, on the fixture that made the deletion
    /// wrong: two decaying exponentials that CORRELATE with the constant.
    ///
    /// Three properties, and they are the whole argument for residualizing
    /// rather than deleting:
    ///
    /// 1. the realized block comes out orthogonal to the constraint at roundoff
    ///    — the invariant the step exists for;
    /// 2. it costs NO coefficient direction — the deletion costs one;
    /// 3. `span([C | X·T − C·R]) == span([C | X])` exactly, which is what makes
    ///    (2) a fact rather than a preference.
    #[test]
    fn residualizing_a_correlated_block_is_orthogonal_and_costs_no_dimension() {
        let n = 40usize;
        let mut basis = Array2::<f64>::zeros((n, 2));
        for i in 0..n {
            let x = i as f64 / (n as f64 - 1.0);
            basis[[i, 0]] = (-0.3 * x).exp();
            basis[[i, 1]] = (-0.9 * x).exp();
        }
        let intercept = Array2::<f64>::ones((n, 1));
        let plan =
            parametric_residualization_for_design(&dense(basis.clone()), intercept.view(), None)
                .expect("test");
        assert_eq!(
            plan.coefficient_transform.ncols(),
            2,
            "a non-contained constraint costs no coefficient direction"
        );
        assert_eq!(plan.row_space_correction.dim(), (1, 2));
        let realized =
            basis.dot(&plan.coefficient_transform) - intercept.dot(&plan.row_space_correction);
        let cross = realized.t().dot(&intercept);
        let relative = cross.iter().map(|v| v * v).sum::<f64>().sqrt()
            / (realized.iter().map(|v| v * v).sum::<f64>().sqrt()
                * intercept.iter().map(|v| v * v).sum::<f64>().sqrt());
        // The bar is DERIVED rather than chosen. `X̃ᵀC` is accumulated over `n`
        // products of size `‖x‖‖c‖`, so relative to `‖X̃‖‖C‖` its floating-point
        // floor carries the amplification `‖X‖/‖X̃‖` — which on a fixture built
        // to be nearly collinear with its constraint is exactly `1/sin θ` and is
        // the reason a fixed `1e-14` would be a statement about this fixture
        // rather than about the arithmetic.
        let amplification = basis.iter().map(|v| v * v).sum::<f64>().sqrt()
            / realized.iter().map(|v| v * v).sum::<f64>().sqrt();
        let floor = (n as f64) * f64::EPSILON * amplification;
        assert!(
            floor < 1.0e-8,
            "the derived floor must stay far below the shipped ORTHOGONALITY_REL_RESIDUAL_TOL \
             or this assertion is vacuous; got {floor:e} at amplification {amplification:e}"
        );
        assert!(
            relative <= floor,
            "residualized block must be orthogonal to its constraint at the accumulation's own \
             floor; got {relative:e} against {floor:e}"
        );
        // Span preservation, stated as a measurement: the orthogonal projection
        // of an arbitrary vector onto `[C | X]` and onto `[C | X·T − C·R]` must
        // agree. The classical deletion FAILS this, and the same statistic on it
        // is asserted below so the test cannot pass by the fixture being easy.
        let mut target = Array1::<f64>::zeros(n);
        for i in 0..n {
            let x = i as f64 / (n as f64 - 1.0);
            target[i] = (-0.6 * x).exp();
        }
        let mut original = Array2::<f64>::zeros((n, 3));
        original.slice_mut(s![.., ..1]).assign(&intercept);
        original.slice_mut(s![.., 1..]).assign(&basis);
        let mut residualized = Array2::<f64>::zeros((n, 3));
        residualized.slice_mut(s![.., ..1]).assign(&intercept);
        residualized.slice_mut(s![.., 1..]).assign(&realized);
        let gap = |design: &Array2<f64>| -> f64 {
            let gram = design.t().dot(design);
            let rhs = design.t().dot(&target);
            let (evals, evecs) = FaerEigh::eigh(&gram, Side::Lower).expect("gram");
            let top = evals.iter().cloned().fold(0.0_f64, f64::max);
            let projected = evecs.t().dot(&rhs);
            let mut solved = Array1::<f64>::zeros(projected.len());
            for i in 0..projected.len() {
                if evals[i] > top * 1.0e-12 {
                    solved[i] = projected[i] / evals[i];
                }
            }
            let fitted = design.dot(&evecs.dot(&solved));
            let residual = &target - &fitted;
            residual.dot(&residual).sqrt() / target.dot(&target).sqrt()
        };
        let original_gap = gap(&original);
        let residualized_gap = gap(&residualized);
        assert!(
            (original_gap - residualized_gap).abs() <= 1.0e-10 * (1.0 + original_gap),
            "residualization must preserve the model span: {original_gap:e} vs {residualized_gap:e}"
        );
        // The negative control: the deletion this replaces LOSES span here.
        let deletion =
            orthogonality_transform_for_design(&dense(basis.clone()), intercept.view(), None)
                .expect("test");
        assert_eq!(
            deletion.ncols(),
            1,
            "the deletion costs exactly the dimension this test is about"
        );
        let mut deleted = Array2::<f64>::zeros((n, 2));
        deleted.slice_mut(s![.., ..1]).assign(&intercept);
        deleted.slice_mut(s![.., 1..]).assign(&basis.dot(&deletion));
        let deleted_gap = gap(&deleted);
        assert!(
            deleted_gap > 10.0 * original_gap.max(1.0e-14),
            "the fixture must be one where the deletion actually loses something: \
             {original_gap:e} -> {deleted_gap:e}"
        );
    }

    /// On a span that CONTAINS the constant, residualization reproduces the
    /// classical constrained basis: it drops exactly one coefficient direction,
    /// by the rank test rather than by a predicate, and lands on the same span.
    ///
    /// This is the continuity claim made concrete — the two constructions are
    /// not alternatives that meet at a threshold, they agree at containment.
    #[test]
    fn residualizing_a_contained_block_reproduces_the_classical_deletion() {
        let n = 40usize;
        let mut basis = Array2::<f64>::zeros((n, 3));
        for i in 0..n {
            let x = i as f64 / (n as f64 - 1.0);
            basis[[i, 0]] = 1.0;
            basis[[i, 1]] = x;
            basis[[i, 2]] = x * x;
        }
        let intercept = Array2::<f64>::ones((n, 1));
        let plan =
            parametric_residualization_for_design(&dense(basis.clone()), intercept.view(), None)
                .expect("test");
        assert_eq!(
            plan.coefficient_transform.ncols(),
            2,
            "the constant IS in this span, so the rank test drops exactly one direction"
        );
        let realized =
            basis.dot(&plan.coefficient_transform) - intercept.dot(&plan.row_space_correction);
        let deletion =
            orthogonality_transform_for_design(&dense(basis.clone()), intercept.view(), None)
                .expect("test");
        let deleted = basis.dot(&deletion);
        assert_eq!(deleted.ncols(), realized.ncols());
        // Same span: each block's columns are reproduced by the other to
        // roundoff.
        let reproduces = |from: &Array2<f64>, to: &Array2<f64>| -> f64 {
            let gram = from.t().dot(from);
            let rhs = from.t().dot(to);
            let (evals, evecs) = FaerEigh::eigh(&gram, Side::Lower).expect("gram");
            let top = evals.iter().cloned().fold(0.0_f64, f64::max);
            let mut solved = evecs.t().dot(&rhs);
            for i in 0..evals.len() {
                let scale = if evals[i] > top * 1.0e-12 {
                    1.0 / evals[i]
                } else {
                    0.0
                };
                for j in 0..solved.ncols() {
                    solved[[i, j]] *= scale;
                }
            }
            let approximation = from.dot(&evecs.dot(&solved));
            let gap = to - &approximation;
            gap.iter().map(|v| v * v).sum::<f64>().sqrt()
                / to.iter().map(|v| v * v).sum::<f64>().sqrt().max(1.0e-300)
        };
        assert!(
            reproduces(&realized, &deleted) < 1.0e-12,
            "the deletion's span must be inside the residualization's"
        );
        assert!(
            reproduces(&deleted, &realized) < 1.0e-12,
            "the residualization's span must be inside the deletion's"
        );
        // And the correction is genuinely inert here: with the constant in the
        // span, the whitener already produced a block orthogonal to it.
        let cross = realized.t().dot(&intercept);
        let relative = cross.iter().map(|v| v * v).sum::<f64>().sqrt()
            / (realized.iter().map(|v| v * v).sum::<f64>().sqrt()
                * intercept.iter().map(|v| v * v).sum::<f64>().sqrt());
        assert!(relative < 1.0e-14, "got {relative:e}");
    }
}