gam_problem/

block_spec.rs

//! Data model for the blockwise carrier (subset moved down to `gam-problem`):
//! parameter-block specs, the effective-Jacobian and channel-Hessian
//! abstractions, per-block working sets and states, and the block geometry
//! directional derivative.
//!
//! The coefficient-group/label/prior types, `custom_family_block_role`, and the
//! blockspec validators remain in the family crate because they depend on
//! `CoefficientGroupPrior`/`RhoPrior`/`BlockRole`/`CustomFamilyError`.

use std::ops::Range;
use std::sync::Arc;

use ndarray::{Array1, Array2};

use crate::PenaltyMatrix;
use gam_linalg::matrix::{DesignMatrix, SymmetricMatrix};

/// Per-subject channel Hessian provider for multi-output families.
///
/// The Fisher information decomposition for multi-output families is
///
/// ```text
/// I(β) = Σ_i  J_iᵀ W_i J_i
/// ```
///
/// where `J_i` is the channel-stacked Jacobian (shape `n_outputs × p` for
/// subject `i`) and `W_i` is the `n_outputs × n_outputs` per-subject channel
/// Hessian of the row negative log-likelihood (the second-derivative block of
/// `−log L_i(u_i)` at a pilot β, PSD-clamped).
///
/// For single-output families this is the scalar IRLS weight; for multi-output
/// families (survival marginal-slope: `n_outputs = 4`; location-scale:
/// `n_outputs = 2`) it carries full cross-channel curvature.
///
/// The identifiability canonicalisation step uses the `n_outputs`-channel
/// weighted joint design `W_joint = Σ_i sqrt(W_i) ⊗ J_i` to detect
/// block-against-block aliasing.  When this trait is present on
/// `ParameterBlockSpec::channel_hessian`, `canonicalize_for_identifiability`
/// routes through `audit_identifiability_channel_aware`; when absent it falls
/// back to the scalar-weight flat audit.
///
/// # W-metric rank theorem
///
/// The canonicalisation computes `rank(J^T W J)` where `W_blkdiag =
/// block-diagonal of per-subject W_i`.  This rank equals
///
/// ```text
/// rank(J) − dim(range(J) ∩ ker(W_blkdiag))
/// ```
///
/// i.e. columns of `J` that lie in the kernel of `W_blkdiag` (flat directions
/// in the curvature landscape at the pilot β) are correctly identified as
/// curvature-redundant and may be dropped.
pub trait FamilyChannelHessian: Send + Sync {
    /// Number of output channels `n_outputs` (= K in the row Jacobian).
    fn n_outputs(&self) -> usize;

    /// Number of subjects (rows).
    fn n_subjects(&self) -> usize;

    /// Fill the `n_outputs × n_outputs` per-subject channel Hessian `W_i`
    /// into `out` (row-major, length `n_outputs * n_outputs`) for subject `i`.
    /// Negative eigenvalues must be clamped to zero (PSD projection) before
    /// or inside this call.
    fn fill_subject(&self, i: usize, out: &mut [f64]);

    /// Materialise the full `(n_subjects × n_outputs × n_outputs)` tensor.
    /// Default implementation calls `fill_subject` for each row.
    fn evaluate_full(&self) -> ndarray::Array3<f64> {
        let n = self.n_subjects();
        let k = self.n_outputs();
        let mut out = ndarray::Array3::<f64>::zeros((n, k, k));
        let mut buf = vec![0.0_f64; k * k];
        for i in 0..n {
            self.fill_subject(i, &mut buf);
            for a in 0..k {
                for b in 0..k {
                    out[[i, a, b]] = buf[a * k + b];
                }
            }
        }
        out
    }
}

/// β-linearization state passed to [`BlockEffectiveJacobian::effective_jacobian_at`].
///
/// At pre-fit initialization, pass `beta = &[]` / zeros and `family_scalars = None`.
/// Families that need β-dependent scalars (e.g. survival marginal-slope's q0, q1,
/// g, c, z) store them in `family_scalars` as a concrete type behind
/// `Arc<dyn Any + Send + Sync>` and downcast inside their impl.
pub struct FamilyLinearizationState<'a> {
    pub beta: &'a [f64],
    /// Optional family-shared scalars at this β linearization.
    /// Downcast via `state.family_scalars.as_ref().and_then(|a| a.downcast_ref::<T>())`.
    pub family_scalars: Option<Arc<dyn std::any::Any + Send + Sync>>,
    /// Optional per-subject channel Hessian for multi-output families.
    /// When `Some`, the identifiability canonicalisation step and the Gram
    /// builder use the channel-stacked Fisher information instead of the
    /// scalar-weight approximation.  Single-output families leave this `None`.
    pub channel_hessian: Option<Arc<dyn FamilyChannelHessian>>,
    /// Probit frailty scale factor `s_f = 1/√(1+σ²)`.
    ///
    /// For survival marginal-slope families the logslope η contribution is
    /// `s_f · g · z`, so any Jacobian callback that depends on g or z must
    /// read `s_f` from here rather than from a captured-at-construction value.
    /// When σ = 0 (no frailty) or for non-frailty families, set this to 1.0.
    ///
    /// Since σ is always **fixed** (not jointly optimised with β) in the
    /// survival family, `s_f` is a static scalar for the entire inner fit;
    /// `∂s_f/∂σ` never appears in the β-Jacobian.  The field is nonetheless
    /// carried through state so that Jacobian callbacks are not required to
    /// capture `s_f` at spec-construction time — they can read it at
    /// evaluation time and thus stay correct across outer-loop σ updates.
    pub probit_frailty_scale: f64,
}

/// β-dependent Jacobian callback for a parameter block.
///
/// Principled long-term contract for expressing how a block contributes to
/// the stacked linear predictor at a given β:
///
/// ```text
/// J(β) ∈ ℝ^{n_rows · n_outputs × p_block}
/// ```
///
/// - Single-output linear block: returns `design.clone()`.
/// - Row-scaled block (`RowScaledJacobian`): returns `diag(eta_scaling) · design` (still linear in β).
/// - Multi-output block (e.g. survival marginal-slope with η0, η1, ad1):
///   stacks `∂eta_r/∂β_k` for `r ∈ 0..n_outputs`, row-major ordering.
///
/// The default impl on [`ParameterBlockSpec::effective_jacobian_at`] is:
/// - `jacobian_callback = None` → `design.clone()`.
/// - `jacobian_callback = Some(cb)` → delegates to `cb.effective_jacobian_at`.
pub trait BlockEffectiveJacobian: Send + Sync {
    /// Stacked multi-output Jacobian for a contiguous observation row range.
    ///
    /// Shape: `(rows.len() * n_outputs, p_block)`, with the same channel-major
    /// layout as [`Self::effective_jacobian_at`]: row
    /// `channel * rows.len() + local_row` is `rows.start + local_row` in that
    /// output channel. Implementations should keep this as the single source of
    /// row math so large construction-time audits can stream chunks instead of
    /// materialising all `n * p * K` entries at once.
    fn effective_jacobian_rows(
        &self,
        state: &FamilyLinearizationState<'_>,
        rows: Range<usize>,
    ) -> Result<Array2<f64>, String>;

    /// Stacked multi-output Jacobian at the current β.
    ///
    /// Shape: `(n_rows * n_outputs, p_block)`, **channel-major**: rows
    /// `r * n_rows .. (r + 1) * n_rows` carry output channel `r`'s row
    /// Jacobian, so `stacked[r * n_rows + i, j]` is observation `i`'s row at
    /// output `r` and coefficient column `j`.  Every consumer that destacks
    /// this matrix (audit, canonicaliser, fit) relies on this layout — see
    /// `BlockJacobianAsRowOp::from_callback` for the destacking transpose.
    /// For `n_outputs = 1` this is identical to the `(n_rows, p_block)` effective
    /// design used by the flat identifiability audit.
    fn effective_jacobian_at(
        &self,
        state: &FamilyLinearizationState<'_>,
    ) -> Result<Array2<f64>, String> {
        let full = self.effective_jacobian_rows(state, 0..usize::MAX)?;
        Ok(full)
    }

    /// Number of stacked output channels. 1 for most blocks.
    fn n_outputs(&self) -> usize {
        1
    }

    /// Returns the per-row scaling vector when this callback is a simple
    /// diagonal-scaling block (`RowScaledJacobian`).  Used by the
    /// identifiability audit's skewness-aware bias correction (T25).
    ///
    /// Returns `None` for all blocks except `RowScaledJacobian`.
    fn eta_row_scaling_for_skewness(&self) -> Option<Arc<[f64]>> {
        None
    }

    /// Whether the identifiability canonicaliser must keep this block at its
    /// full raw column width instead of column-reducing it.
    ///
    /// The `#933` reduction path wraps a `jacobian_callback` block in a
    /// gauge-composed Jacobian so the family fits in a reduced section — sound
    /// only when the family's effective geometry is DERIVED from the callback
    /// (marginal-slope logslope). It is NOT sound for a callback whose
    /// effective Jacobian is a **fixed nonlinear functional basis** recomputed
    /// at the raw coefficient width on every evaluation (the survival
    /// marginal-slope monotone time-wiggle time block), nor for one that merely
    /// DECLARES an output channel for a family that materialises its own
    /// raw-width design (the multinomial softmax, whose every assembly reads the
    /// `X` it captured at construction — #2744): such a family's downstream
    /// likelihood reads raw-width internal designs and asserts
    /// `beta.len() == p_raw`, so a reduced β desynchronises that layout — the
    /// same failure mode the competing-risks dead-column veto already guards.
    /// Such a block returns `true` so the canonicaliser keeps it at raw width
    /// (its own penalty nullspace regularises the weak directions instead).
    ///
    /// Defaults to `false`: every existing callback reduces safely.
    fn locks_raw_width_reduction(&self) -> bool {
        false
    }
}

/// A [`BlockEffectiveJacobian`] for any block that contributes linearly to
/// exactly one output of a multi-output family.
///
/// `own_output` is the zero-based output index that this block drives.
/// `n_family_outputs` is the total number of outputs (e.g. 2 for location-scale).
/// `design` is the block's effective design matrix (n × p_block).
///
/// The returned Jacobian has shape `(n_family_outputs * n, p_block)`:
/// rows `own_output * n .. (own_output + 1) * n` contain `design`,
/// all other rows are zero.
pub struct AdditiveBlockJacobian {
    pub design: Array2<f64>,
    pub own_output: usize,
    pub n_family_outputs: usize,
}

impl BlockEffectiveJacobian for AdditiveBlockJacobian {
    fn effective_jacobian_rows(
        &self,
        state: &FamilyLinearizationState<'_>,
        rows: Range<usize>,
    ) -> Result<Array2<f64>, String> {
        let n = self.design.nrows();
        let p = self.design.ncols();
        let rows = clamp_jacobian_rows(rows, n);
        // Additive (linear) block: Jacobian is β-independent — design does
        // not depend on state.beta. Verify beta contains no NaN when provided.
        if !state.beta.is_empty() && state.beta.iter().any(|v| v.is_nan()) {
            return Err(
                "AdditiveBlockJacobian::effective_jacobian_at: beta contains NaN".to_string(),
            );
        }
        let chunk = rows.end - rows.start;
        let total_rows = self.n_family_outputs * chunk;
        let mut jac = Array2::<f64>::zeros((total_rows, p));
        let row_start = self.own_output * chunk;
        jac.slice_mut(ndarray::s![row_start..row_start + chunk, ..])
            .assign(&self.design.slice(ndarray::s![rows.start..rows.end, ..]));
        Ok(jac)
    }

    fn n_outputs(&self) -> usize {
        self.n_family_outputs
    }
}

/// A [`BlockEffectiveJacobian`] for a single-output block whose contribution
/// to the linear predictor is `diag(eta_scaling) · design` (row-wise scaling).
///
/// This is the canonical replacement for the former `eta_row_scaling` field on
/// [`ParameterBlockSpec`].  The identifiability audit's skewness-aware bias
/// correction can recover the scaling vector via
/// [`BlockEffectiveJacobian::eta_row_scaling_for_skewness`].
pub struct RowScaledJacobian {
    pub design: Arc<Array2<f64>>,
    pub eta_scaling: Arc<[f64]>,
}

impl BlockEffectiveJacobian for RowScaledJacobian {
    fn effective_jacobian_rows(
        &self,
        state: &FamilyLinearizationState<'_>,
        rows: Range<usize>,
    ) -> Result<Array2<f64>, String> {
        let n = self.design.nrows();
        let rows = clamp_jacobian_rows(rows, n);
        if self.eta_scaling.len() != n {
            return Err(format!(
                "RowScaledJacobian: eta_scaling length {} != design nrows {}",
                self.eta_scaling.len(),
                n,
            ));
        }
        // Row-scaled blocks are β-linear; verify the linearization point
        // contains no NaN when β is provided (sanity check on caller state).
        if !state.beta.is_empty() && state.beta.iter().any(|v| v.is_nan()) {
            return Err(
                "RowScaledJacobian::effective_jacobian_at: state.beta contains NaN".to_string(),
            );
        }
        let mut scaled = self
            .design
            .slice(ndarray::s![rows.start..rows.end, ..])
            .to_owned();
        for local_i in 0..scaled.nrows() {
            let s = self.eta_scaling[rows.start + local_i];
            for j in 0..scaled.ncols() {
                scaled[[local_i, j]] *= s;
            }
        }
        Ok(scaled)
    }

    fn eta_row_scaling_for_skewness(&self) -> Option<Arc<[f64]>> {
        Some(Arc::clone(&self.eta_scaling))
    }
}

pub(crate) fn clamp_jacobian_rows(rows: Range<usize>, n: usize) -> Range<usize> {
    let start = rows.start.min(n);
    let end = rows.end.min(n);
    start..end.max(start)
}

/// A [`BlockEffectiveJacobian`] that composes an inner callback's raw-width
/// effective Jacobian with a fixed reduced→raw block transform `T_b`
/// (`p_raw × r_reduced`), so the family sees the **reduced** coordinates by
/// construction (#933).
///
/// The inner callback emits its row Jacobian in the raw coordinate system
/// (`(rows · k) × p_raw`), the layout every `BlockEffectiveJacobian` impl
/// produces — channel-major rows, raw columns. Post-multiplying each row by
/// `T_b` rotates those raw columns into the reduced section: the effective
/// reduced Jacobian is `J_raw · T_b`, with `r_reduced` columns. On the model
/// `η = J_raw · β_raw = (J_raw · T_b) · θ` this is the exact reduced operator
/// for the reduced coefficient θ, and the family lifts θ back to β_raw through
/// the SAME `T_b` via the one [`Gauge`](crate::Gauge).
///
/// This is the inversion #933 calls for: instead of forwarding a raw-width
/// callback alongside a column-selection `T_i` (which leaves the family
/// asserting raw column counts on a reduced spec and panicking), the callback
/// is wrapped so its output already has the reduced width — the family captures
/// the reduced design and its row-Hessian column-count assertions hold by
/// construction. A column-selection `T_b` (zero/one entries) makes this exactly
/// the audit's drop; a general orthonormal `T_b` makes it any gauge section.
pub struct GaugeComposedJacobian {
    inner: Arc<dyn BlockEffectiveJacobian>,
    /// Reduced→raw block transform `T_b`, shape `(p_raw × r_reduced)`.
    t_block: Arc<Array2<f64>>,
}

impl GaugeComposedJacobian {
    /// Wrap `inner` so its effective Jacobian is post-multiplied by `t_block`
    /// (`p_raw × r_reduced`). `t_block.nrows()` must equal the inner callback's
    /// raw column count.
    pub fn new(inner: Arc<dyn BlockEffectiveJacobian>, t_block: Arc<Array2<f64>>) -> Self {
        Self { inner, t_block }
    }
}

impl BlockEffectiveJacobian for GaugeComposedJacobian {
    fn effective_jacobian_rows(
        &self,
        state: &FamilyLinearizationState<'_>,
        rows: Range<usize>,
    ) -> Result<Array2<f64>, String> {
        let raw_width = self.t_block.nrows();
        let reduced_width = self.t_block.ncols();
        let lifted_beta;
        let lifted_state;
        let zero_raw_beta;
        let delegate_state = if state.beta.len() == raw_width {
            state
        } else if state.beta.len() == reduced_width {
            lifted_beta = self.t_block.dot(&ndarray::ArrayView1::from(state.beta));
            lifted_state = FamilyLinearizationState {
                beta: lifted_beta
                    .as_slice()
                    .expect("GaugeComposedJacobian lifted beta is contiguous"),
                family_scalars: state.family_scalars.clone(),
                channel_hessian: state.channel_hessian.clone(),
                probit_frailty_scale: state.probit_frailty_scale,
            };
            &lifted_state
        } else if state.beta.is_empty() {
            zero_raw_beta = ndarray::Array1::<f64>::zeros(raw_width);
            lifted_state = FamilyLinearizationState {
                beta: zero_raw_beta
                    .as_slice()
                    .expect("GaugeComposedJacobian zero raw beta is contiguous"),
                family_scalars: state.family_scalars.clone(),
                channel_hessian: state.channel_hessian.clone(),
                probit_frailty_scale: state.probit_frailty_scale,
            };
            &lifted_state
        } else {
            return Err(format!(
                "GaugeComposedJacobian: beta has length {}, expected raw width {} \
                 or reduced width {}; this wrapper cannot infer a block slice from a joint \
                 coefficient vector",
                state.beta.len(),
                raw_width,
                reduced_width,
            ));
        };
        let j_raw = self.inner.effective_jacobian_rows(delegate_state, rows)?;
        if j_raw.ncols() != self.t_block.nrows() {
            return Err(format!(
                "GaugeComposedJacobian: inner Jacobian has {} columns but T_b has {} rows",
                j_raw.ncols(),
                self.t_block.nrows(),
            ));
        }
        // (rows·k × p_raw) · (p_raw × r_reduced) = (rows·k × r_reduced).
        Ok(j_raw.dot(self.t_block.as_ref()))
    }

    fn n_outputs(&self) -> usize {
        self.inner.n_outputs()
    }

    // Skewness scaling is a raw-row property; reducing the column space does not
    // change the per-row scaling, so it is forwarded unchanged when present.
    fn eta_row_scaling_for_skewness(&self) -> Option<Arc<[f64]>> {
        self.inner.eta_row_scaling_for_skewness()
    }
}

#[cfg(test)]
mod gauge_composed_jacobian_tests {
    use super::*;
    use ndarray::array;

    struct BetaScaledJacobian {
        design: Array2<f64>,
    }

    impl BlockEffectiveJacobian for BetaScaledJacobian {
        fn effective_jacobian_rows(
            &self,
            state: &FamilyLinearizationState<'_>,
            rows: Range<usize>,
        ) -> Result<Array2<f64>, String> {
            let n = self.design.nrows();
            let rows = rows.start.min(n)..rows.end.min(n);
            let mut out = self.design.slice(ndarray::s![rows, ..]).to_owned();
            for col in 0..out.ncols() {
                let scale = 1.0 + state.beta.get(col).copied().unwrap_or(0.0);
                out.column_mut(col).mapv_inplace(|v| v * scale);
            }
            Ok(out)
        }

        fn n_outputs(&self) -> usize {
            1
        }
    }

    #[test]
    fn gauge_composed_jacobian_lifts_reduced_block_beta_before_delegating() {
        let inner: Arc<dyn BlockEffectiveJacobian> = Arc::new(BetaScaledJacobian {
            design: array![[2.0, 3.0], [5.0, 7.0]],
        });
        let t_block = Arc::new(array![[0.0], [1.0]]);
        let wrapped = GaugeComposedJacobian::new(inner, Arc::clone(&t_block));

        let theta = [4.0];
        let reduced_state = FamilyLinearizationState {
            beta: &theta,
            family_scalars: None,
            channel_hessian: None,
            probit_frailty_scale: 1.0,
        };
        let reduced = wrapped
            .effective_jacobian_rows(&reduced_state, 0..2)
            .expect("reduced beta should be lifted through T before inner callback");

        let raw_beta = [0.0, 4.0];
        let raw_state = FamilyLinearizationState {
            beta: &raw_beta,
            family_scalars: None,
            channel_hessian: None,
            probit_frailty_scale: 1.0,
        };
        let raw = wrapped
            .effective_jacobian_rows(&raw_state, 0..2)
            .expect("raw beta state remains valid");

        assert_eq!(reduced, raw);
        assert_eq!(reduced, array![[15.0], [35.0]]);
    }

    struct StrictRawWidthJacobian {
        design: Array2<f64>,
    }

    impl BlockEffectiveJacobian for StrictRawWidthJacobian {
        fn effective_jacobian_rows(
            &self,
            state: &FamilyLinearizationState<'_>,
            rows: Range<usize>,
        ) -> Result<Array2<f64>, String> {
            if state.beta.len() != self.design.ncols() {
                return Err(format!(
                    "StrictRawWidthJacobian expected raw beta len {}, got {}",
                    self.design.ncols(),
                    state.beta.len(),
                ));
            }
            Ok(self.design.slice(ndarray::s![rows, ..]).to_owned())
        }
    }

    #[test]
    fn gauge_composed_jacobian_lifts_zero_reduced_beta_before_delegating() {
        let inner: Arc<dyn BlockEffectiveJacobian> = Arc::new(StrictRawWidthJacobian {
            design: array![[2.0, 3.0], [5.0, 7.0]],
        });
        let wrapped = GaugeComposedJacobian::new(inner, Arc::new(array![[0.0], [1.0]]));

        let theta = [0.0];
        let reduced_state = FamilyLinearizationState {
            beta: &theta,
            family_scalars: None,
            channel_hessian: None,
            probit_frailty_scale: 1.0,
        };

        let reduced = wrapped
            .effective_jacobian_rows(&reduced_state, 0..2)
            .expect("zero reduced beta must still be lifted to raw width");

        assert_eq!(reduced, array![[3.0], [7.0]]);
    }

    #[test]
    fn gauge_composed_jacobian_rejects_nonzero_unknown_beta_layout() {
        let inner: Arc<dyn BlockEffectiveJacobian> = Arc::new(BetaScaledJacobian {
            design: array![[2.0, 3.0]],
        });
        let wrapped = GaugeComposedJacobian::new(inner, Arc::new(array![[0.0], [1.0]]));
        let joint_like_beta = [1.0, 0.0, 0.0];
        let state = FamilyLinearizationState {
            beta: &joint_like_beta,
            family_scalars: None,
            channel_hessian: None,
            probit_frailty_scale: 1.0,
        };

        let err = wrapped
            .effective_jacobian_rows(&state, 0..1)
            .expect_err("nonzero joint-layout beta cannot be inferred from one block T");
        assert!(
            err.contains("cannot infer a block slice"),
            "unexpected error: {err}"
        );
    }
}

/// Static specification for one parameter block in a custom family.
///
/// `design` and `stacked_design` are two structurally distinct operators:
///
/// * `design` is the **canonical, single-channel, n-observation operator**.
///   `design.nrows()` ALWAYS equals `n_obs` (one row per training
///   observation).  This is the matrix the identifiability audit, the
///   shape policy, and every "what shape is this block?" reader inspect.
///   For most blocks `design` is also the eta-producing operator used by
///   the solver — see [`Self::solver_design`].
/// * `stacked_design`, when `Some`, is the **multi-channel eta-producing
///   operator** used by the solver.  Survival time-varying blocks stack
///   `[exit; entry; deriv]` into a `(3·n × p)` operator here so the
///   solver can produce a `3·n`-long `eta` in one mat-vec; the audit
///   never sees this matrix.  When `None`, the solver uses `design` (the
///   single-channel default).
///
/// The single contract that downstream code can rely on:
/// `design.nrows() == n_obs`.  No more dual semantics on `design`.
///
/// Read access:
/// * Audit / canonicalize / "n_obs is the row count" code → `&spec.design`.
/// * Eta-producing solver code → [`Self::solver_design`].
#[derive(Clone)]
pub struct ParameterBlockSpec {
    pub name: String,
    pub design: DesignMatrix,
    pub offset: Array1<f64>,
    /// Block-local penalty matrices (all p_block x p_block).
    pub penalties: Vec<PenaltyMatrix>,
    /// Structural nullspace dimension of each penalty matrix (same length as `penalties`).
    /// Used by the penalty pseudo-logdet to determine rank without numerical thresholds.
    /// If empty, falls back to eigenvalue-based rank detection.
    pub nullspace_dims: Vec<usize>,
    /// Initial log-smoothing parameters for this block (same length as `penalties`).
    pub initial_log_lambdas: Array1<f64>,
    /// Optional initial coefficients (defaults to zeros if omitted).
    pub initial_beta: Option<Array1<f64>>,
    /// Gauge ownership priority. Higher = more likely to retain a
    /// redundant direction during canonical-gauge reparameterisation.
    /// Defaults to 100. Set higher for blocks that should "own" shared
    /// affine/null-space directions (e.g. baseline time in survival).
    pub gauge_priority: u8,
    /// Full β-dependent Jacobian callback.  When `Some`, this is the
    /// authoritative source for `effective_jacobian_at`.  For simple
    /// single-output row-scaled blocks use [`RowScaledJacobian`].
    pub jacobian_callback: Option<Arc<dyn BlockEffectiveJacobian>>,
    /// Optional multi-channel eta-producing operator used by the solver.
    ///
    /// When `Some`, the solver consumes this matrix (typically
    /// `(k·n × p)` for `k` stacked channels — e.g. survival
    /// `[exit; entry; deriv]` with `k = 3`) to evaluate `eta = stacked · β + stacked_offset`.
    /// The audit and shape policy NEVER read this field; they only ever
    /// inspect `design` (which always has `n_obs` rows).
    ///
    /// When `None`, the solver falls back to `design` — the correct
    /// behavior for every single-channel block (i.e. all non-survival
    /// time-varying blocks).
    ///
    /// Read this field via [`Self::solver_design`], never directly.
    ///
    /// Invariant: when `stacked_design = Some(_)`, `stacked_offset` MUST
    /// also be `Some(_)` and its length MUST equal `stacked_design.nrows()`.
    pub stacked_design: Option<DesignMatrix>,
    /// Optional offset paired with [`Self::stacked_design`]. Same Option
    /// state as `stacked_design` (both `Some` or both `None`).
    /// Read via [`Self::solver_offset`].
    pub stacked_offset: Option<Array1<f64>>,
}

impl std::fmt::Debug for ParameterBlockSpec {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        f.debug_struct("ParameterBlockSpec")
            .field("name", &self.name)
            .field("design", &self.design)
            .field("offset", &self.offset)
            .field("penalties", &self.penalties)
            .field("nullspace_dims", &self.nullspace_dims)
            .field("initial_log_lambdas", &self.initial_log_lambdas)
            .field("initial_beta", &self.initial_beta)
            .field("gauge_priority", &self.gauge_priority)
            .field(
                "jacobian_callback",
                &self
                    .jacobian_callback
                    .as_ref()
                    .map(|_| "<BlockEffectiveJacobian>"),
            )
            .finish()
    }
}

impl ParameterBlockSpec {
    /// Returns a ParameterBlockSpec with sensible defaults for all optional
    /// fields. Callers using struct literal syntax can use
    /// `..ParameterBlockSpec::defaults()` to fill in any fields added after
    /// the literal was written.
    pub fn defaults() -> Self {
        Self {
            name: String::new(),
            design: DesignMatrix::Dense(gam_linalg::matrix::DenseDesignMatrix::from(
                ndarray::Array2::<f64>::zeros((0, 0)),
            )),
            offset: ndarray::Array1::<f64>::zeros(0),
            penalties: Vec::new(),
            nullspace_dims: Vec::new(),
            initial_log_lambdas: ndarray::Array1::<f64>::zeros(0),
            initial_beta: None,
            gauge_priority: 100,
            jacobian_callback: None,
            stacked_design: None,
            stacked_offset: None,
        }
    }

    /// Returns the eta-producing operator used by the solver.
    ///
    /// Resolution order:
    ///   1. `stacked_design = Some(d)` → return `d` (multi-channel
    ///      operator, e.g. `(3n × p)` for survival time-varying blocks).
    ///   2. otherwise → return `&self.design` (the single-channel default).
    ///
    /// Solver code that needs `eta = D · β` MUST call this accessor;
    /// reading `&self.design` directly silently breaks multi-channel
    /// (survival LS time-varying) blocks because `self.design.nrows()`
    /// always equals `n_obs`, never `3·n_obs`.
    pub fn solver_design(&self) -> &DesignMatrix {
        self.stacked_design.as_ref().unwrap_or(&self.design)
    }

    /// Returns the offset paired with [`Self::solver_design`]. When
    /// `stacked_offset = Some(o)` this returns `&o`; otherwise it falls
    /// back to `&self.offset`.
    pub fn solver_offset(&self) -> &Array1<f64> {
        self.stacked_offset.as_ref().unwrap_or(&self.offset)
    }

    /// Returns the effective design `D_eff` for this block at β = 0 with no
    /// family scalars — a convenience wrapper around [`Self::effective_jacobian_at`]
    /// for the single-output (n_outputs = 1) case.
    ///
    /// Callers that need multi-output Jacobians or β-dependent scalars should
    /// call `effective_jacobian_at` directly with the appropriate state.
    ///
    /// Returns `Err` if the design cannot be densified.
    pub fn effective_design(&self, caller: &str) -> Result<ndarray::Array2<f64>, String> {
        let p = self.design.ncols();
        let zeros = vec![0.0f64; p];
        let state = FamilyLinearizationState {
            beta: &zeros,
            family_scalars: None,
            channel_hessian: None,
            probit_frailty_scale: 1.0,
        };
        self.effective_jacobian_at(caller, &state)
    }

    /// Returns the β-dependent stacked Jacobian `J(β)` for this block.
    ///
    /// Shape: `(n_rows * n_outputs, p_block)`.  For most blocks `n_outputs = 1`
    /// and the result is the familiar `(n_rows, p_block)` effective design.
    ///
    /// Dispatch order:
    ///   1. `jacobian_callback = Some(cb)` → `cb.effective_jacobian_at(state)`.
    ///   2. `jacobian_callback = None` → `design.clone()` (ignores `beta` and `family_scalars`).
    ///
    /// Returns `Err` if the design cannot be densified.
    pub fn effective_jacobian_at(
        &self,
        caller: &str,
        state: &FamilyLinearizationState<'_>,
    ) -> Result<ndarray::Array2<f64>, String> {
        if let Some(cb) = self.jacobian_callback.as_ref() {
            return cb.effective_jacobian_at(state);
        }
        self.design
            .try_to_dense_arc(&format!(
                "{caller}::effective_jacobian_at block '{}'",
                self.name
            ))
            .map(|arc| arc.as_ref().clone())
    }
}

/// Current state for a parameter block.
#[derive(Clone, Debug)]
pub struct ParameterBlockState {
    pub beta: Array1<f64>,
    pub eta: Array1<f64>,
}

#[derive(Clone)]
pub struct BlockGeometryDirectionalDerivative {
    /// Directional derivative of the block design matrix along a coefficient-space direction.
    pub d_design: Option<Array2<f64>>,
    /// Directional derivative of the block offset along the same direction.
    pub d_offset: Array1<f64>,
}

/// Working quantities supplied by a custom family for one block.
///
/// # Observed vs expected information (see response.md Section 3)
///
/// For the outer REML/LAML criterion, the Hessian used in log|H| and trace terms
/// must be the **observed** (actual) Hessian at the mode, not the expected Fisher.
///
/// - `ExactNewton`: provides -nabla^2 log L directly, which is the observed Hessian
///   by construction. This is always correct.
///
/// - `Diagonal`: provides IRLS working weights W such that the per-block Hessian
///   is X'WX. For canonical links (logit-Binomial, log-Poisson), W_obs = W_Fisher.
///   For supported non-canonical diagonal links, W must be the observed weight
///   W_obs = W_Fisher - (y-mu)*B so the outer REML uses the exact Laplace
///   Hessian. The matching `CustomFamily::diagonalworking_weights_directional_derivative`
///   callback must differentiate the same observed W surface; silently using Fisher
///   weights or zero `dW` would change the criterion into a PQL-type surrogate.
#[derive(Clone, Debug)]
pub enum BlockWorkingSet {
    /// Standard IRLS/GLM-style diagonal working set for eta-space updates.
    Diagonal {
        /// IRLS pseudo-response for this block's linear predictor.
        working_response: Array1<f64>,
        /// IRLS working curvature for this block (finite signed values, length n).
        ///
        /// For the inner solver, Fisher or observed weights both find the same mode.
        /// For the outer REML/LAML log|H| term, observed weights are the correct
        /// Laplace choice (see response.md Section 3). Canonical-link families need
        /// no correction since observed = Fisher.
        working_weights: Array1<f64>,
    },
    /// Exact Newton block update in coefficient space.
    ///
    /// `gradient` is nabla log L wrt block coefficients.
    /// `hessian` is -nabla^2 log L wrt block coefficients (positive semidefinite near optimum).
    ///
    /// This is the observed Hessian by construction (actual second derivative of the
    /// log-likelihood), which is the correct quantity for the outer REML Laplace
    /// approximation.
    ExactNewton {
        gradient: Array1<f64>,
        hessian: SymmetricMatrix,
    },
}

impl BlockWorkingSet {
    /// Construct a `Diagonal` working set with its length and finite-value
    /// invariants enforced at the type boundary. Signed observed curvature is
    /// preserved exactly; stabilization belongs to the assembled matrix.
    #[inline]
    pub fn diagonal_checked(
        working_response: Array1<f64>,
        working_weights: Array1<f64>,
    ) -> Result<Self, String> {
        if working_response.len() != working_weights.len() {
            return Err(format!(
                "BlockWorkingSet::Diagonal length mismatch: working_response={}, working_weights={}",
                working_response.len(),
                working_weights.len(),
            ));
        }
        if let Some((row, value)) = working_response
            .iter()
            .chain(working_weights.iter())
            .enumerate()
            .find(|(_, value)| !value.is_finite())
        {
            return Err(format!(
                "BlockWorkingSet::Diagonal contains a non-finite value at flattened index {row}: {value}"
            ));
        }
        Ok(Self::Diagonal {
            working_response,
            working_weights,
        })
    }
}

/// What a parameter block's COEFFICIENT COORDINATE is, as opposed to what its
/// column space is (#2748).
///
/// # The distinction, and why one bit is needed to state it
///
/// For most blocks the coefficients are an arbitrary basis: any `V` with
/// `X V` spanning `range(X)` gives the same model, so the identifiability
/// canonicaliser is free to reparameterise `β ↦ Vᵀβ`, pull the penalties back
/// as `VᵀSV`, and let [`crate::Gauge`] lift the answer home. That freedom is
/// what lets it remove a cross-block structural confound EXACTLY instead of
/// ridging it away.
///
/// Some blocks are not like that. The monotone link-wiggle warp is the
/// canonical case: its family imposes `β_w ≥ 0` **componentwise on those very
/// coefficients**, because an I-spline with non-negative coefficients is what
/// makes the learned link monotone. `β ↦ Vᵀβ` maps that cone to
/// `{A V β̃ ≥ 0}`, and the hook that produces it
/// (`CustomFamily::block_linear_constraints`) is a function of the block's
/// WIDTH — it cannot express `A V`, so after a reparameterisation it would
/// return a cone in rotated coordinates that means nothing, and the
/// coordinatewise projection beside it (`post_update_block_beta`) would enforce
/// that nothing. The model would silently stop being monotone.
///
/// So "may this block be reparameterised?" is a property of the block's
/// coordinate, and it is NOT the question [`ParameterBlockSpec::gauge_priority`]
/// answers. A priority answers "if a shared direction must be given up, whose
/// is it?" — an ordering AMONG blocks. Reading an ordering as a licence to
/// rotate is how gam#2748 broke: giving the warp a strictly lower priority
/// (correctly, so a cross-block alias stops being *unfittable*) also lifted the
/// canonicaliser's equal-priority guard and authorised rotating the one block
/// that cannot be rotated.
///
/// # Fail-closed
///
/// [`Spanning`](Self::Spanning) is the default because it is the common case,
/// but every DERIVATION of this value must resolve doubt toward
/// [`Structural`](Self::Structural): declining a reparameterisation always
/// preserves the model (the canonicaliser falls through to the audit gate,
/// which is where it lived before the orthogonalisation pass existed), while
/// performing one on a structural coordinate silently changes it.
#[derive(Clone, Copy, Debug, PartialEq, Eq, Default)]
pub enum CoefficientCoordinate {
    /// Only the block's column SPACE is model content. Any basis of it is the
    /// same model, so a reparameterisation `β ↦ Vᵀβ` with the penalties and the
    /// warm start pulled back is exact.
    #[default]
    Spanning,
    /// The coordinate ITSELF is model content — a componentwise sign cone, a
    /// monotonicity ordering, a box the family projects onto, or a geometry the
    /// family rebuilds at this exact width. No change of basis preserves the
    /// model, and no change of width preserves the family's own rebuild.
    Structural,
}

impl CoefficientCoordinate {
    /// Is this coordinate free to be reparameterised?
    pub fn is_spanning(self) -> bool {
        matches!(self, Self::Spanning)
    }

    /// Does this coordinate carry model structure a reparameterisation would
    /// destroy?
    pub fn is_structural(self) -> bool {
        matches!(self, Self::Structural)
    }
}