gam-sae 0.3.153

use super::*;

#[derive(Debug, Clone)]
pub struct SaeArrowVector {
    pub t: Array1<f64>,
    pub beta: Array1<f64>,
}

pub(crate) struct DeflatedArrowSolver<'a> {
    pub(crate) cache: &'a ArrowFactorCache,
    pub(crate) gauge_basis: Vec<Array1<f64>>,
    pub(crate) gauge_response_physical: Vec<Array1<f64>>,
    pub(crate) woodbury_factor: Option<FaerCholeskyFactor>,
    pub(crate) gauge_stiffness: f64,
}

impl<'a> DeflatedArrowSolver<'a> {
    pub(crate) fn plain(cache: &'a ArrowFactorCache) -> Self {
        Self {
            cache,
            gauge_basis: Vec::new(),
            gauge_response_physical: Vec::new(),
            woodbury_factor: None,
            gauge_stiffness: 0.0,
        }
    }

    pub(crate) fn from_orthonormal_gauges(
        cache: &'a ArrowFactorCache,
        gauge_basis: Vec<Array1<f64>>,
        stiffness: f64,
    ) -> Result<Self, String> {
        if gauge_basis.is_empty() {
            return Ok(Self::plain(cache));
        }
        if !(stiffness.is_finite() && stiffness > 0.0) {
            return Err(format!(
                "DeflatedArrowSolver: gauge stiffness must be finite and positive; got {stiffness}"
            ));
        }
        let full_len = cache.delta_t_len() + cache.k;
        let mut gauge_responses = Vec::with_capacity(gauge_basis.len());
        for gauge in &gauge_basis {
            if gauge.len() != full_len {
                return Err(format!(
                    "DeflatedArrowSolver: gauge length {} != cache full length {full_len}",
                    gauge.len()
                ));
            }
            let (sol_t, sol_beta) = cache
                .full_inverse_apply(
                    gauge.slice(s![..cache.delta_t_len()]),
                    gauge.slice(s![cache.delta_t_len()..]),
                )
                .map_err(|err| format!("DeflatedArrowSolver: gauge back-solve: {err}"))?;
            gauge_responses.push(flatten_arrow_parts(sol_t.view(), sol_beta.view()));
        }

        let rank = gauge_basis.len();
        let stiffness_recip = stiffness.recip();
        let mut gauge_metric = Array2::<f64>::zeros((rank, rank));
        let mut woodbury = Array2::<f64>::eye(rank);
        for i in 0..rank {
            woodbury[[i, i]] *= stiffness_recip;
            for j in 0..rank {
                let value = gauge_basis[i].dot(&gauge_responses[j]);
                gauge_metric[[i, j]] = value;
                woodbury[[i, j]] += value;
            }
        }
        let woodbury_factor = woodbury
            .cholesky(Side::Lower)
            .map_err(|err| format!("DeflatedArrowSolver: gauge Woodbury factor failed: {err}"))?;
        let mut gauge_response_physical = gauge_responses;
        for j in 0..rank {
            for i in 0..rank {
                let coeff = gauge_metric[[i, j]];
                for row in 0..full_len {
                    gauge_response_physical[j][row] -= coeff * gauge_basis[i][row];
                }
            }
        }
        Ok(Self {
            cache,
            gauge_basis,
            gauge_response_physical,
            woodbury_factor: Some(woodbury_factor),
            gauge_stiffness: stiffness,
        })
    }

    pub(crate) fn solve(
        &self,
        rhs_t: ArrayView1<'_, f64>,
        rhs_beta: ArrayView1<'_, f64>,
    ) -> Result<SaeArrowVector, String> {
        let (sol_t, sol_beta) = self
            .cache
            .full_inverse_apply(rhs_t, rhs_beta)
            .map_err(|err| format!("DeflatedArrowSolver: full inverse: {err}"))?;
        let Some(factor) = self.woodbury_factor.as_ref() else {
            return Ok(SaeArrowVector {
                t: sol_t,
                beta: sol_beta,
            });
        };

        let full_len = self.cache.delta_t_len() + self.cache.k;
        let mut flat = flatten_arrow_parts(sol_t.view(), sol_beta.view());
        if flat.len() != full_len {
            return Err(format!(
                "DeflatedArrowSolver: solution length {} != cache full length {full_len}",
                flat.len()
            ));
        }
        let mut gauge_coeffs = Array1::<f64>::zeros(self.gauge_basis.len());
        for (idx, gauge) in self.gauge_basis.iter().enumerate() {
            gauge_coeffs[idx] = gauge.dot(&flat);
        }
        let weights = factor.solvevec(&gauge_coeffs);
        for (gauge, &coeff) in self.gauge_basis.iter().zip(gauge_coeffs.iter()) {
            for i in 0..flat.len() {
                flat[i] -= gauge[i] * coeff;
            }
        }
        for (response, &weight) in self.gauge_response_physical.iter().zip(weights.iter()) {
            for i in 0..flat.len() {
                flat[i] -= response[i] * weight;
            }
        }
        for (gauge, &weight) in self.gauge_basis.iter().zip(weights.iter()) {
            let coeff = self.gauge_stiffness.recip() * weight;
            for i in 0..flat.len() {
                flat[i] += gauge[i] * coeff;
            }
        }
        Ok(SaeArrowVector {
            t: flat.slice(s![..self.cache.delta_t_len()]).to_owned(),
            beta: flat.slice(s![self.cache.delta_t_len()..]).to_owned(),
        })
    }

    /// Per-row latent-block inverse diagonal with the UNIT-stiffness deflated
    /// subspace REMOVED — the kept-subspace selected inverse the outer ρ/θ
    /// gradient diagonal traces must contract against.
    ///
    /// [`Self::latent_inverse_diagonal`] returns the diagonal of the DEFLATED
    /// inverse, which assigns `1/λ̃ = 1` to every per-row direction `vᵢ` that the
    /// undamped criterion factor stiffened to unit curvature; a `½ tr(H⁻¹ ∂H/∂ρ)`
    /// diagonal contraction against it therefore spuriously includes
    /// `Σ_i vᵢ[s]²` at slot `s`, a ρ/θ-independent contribution that must be 0.
    /// This variant subtracts the per-row deflated outer-product diagonal
    /// `Σ_i vᵢ[s]²` so the diagonal traces (ARD precision, ordered Beta--Bernoulli/softmax assignment
    /// log-strength) see only the kept subspace. The deflated subspace's β-Schur
    /// coupling is higher order and left to the per-block subtraction the
    /// off-diagonal (`solve`-based) traces apply directly.
    pub(crate) fn latent_inverse_diagonal_kept(&self) -> Result<Array1<f64>, String> {
        let mut out = self.latent_inverse_diagonal()?;
        let cache = self.cache;
        for (row, dirs) in cache.deflated_row_directions.iter().enumerate() {
            if dirs.is_empty() {
                continue;
            }
            let base = cache.row_offsets[row];
            for v in dirs {
                for s in 0..v.len() {
                    if base + s < out.len() {
                        out[base + s] -= v[s] * v[s];
                    }
                }
            }
        }
        Ok(out)
    }

    /// #932 FRONT C — whether the cheap row-local Takahashi selected inverse
    /// ([`Self::beta_inv`] / [`Self::selected_inverse_row_blocks`]) reproduces
    /// `solve`'s selected entries EXACTLY. It does so only on the plain bordered
    /// arrow: when a gauge Woodbury deflation is active (`woodbury_factor`) the
    /// `solve` output carries the rank-`R` gauge correction the row-local blocks
    /// omit. Callers must then fall back to the per-row `solve` loop.
    pub(crate) fn plain_selected_inverse_available(&self) -> bool {
        self.woodbury_factor.is_none()
    }

    /// #932 FRONT C — the full `(H⁻¹)_ββ = S⁻¹` block (`K×K`), formed ONCE per
    /// outer step from the cached dense Schur factor (no per-column full-system
    /// `solve`). On the plain arrow this equals the `beta_inv` the logdet /
    /// α-trace consumers used to build with `K` calls to [`Self::solve`] with
    /// unit β-RHS. ONLY valid when [`Self::plain_selected_inverse_available`].
    pub(crate) fn beta_inv(&self) -> Result<Array2<f64>, String> {
        let k = self.cache.k;
        if k == 0 {
            return Ok(Array2::<f64>::zeros((0, 0)));
        }
        self.cache
            .schur_inverse_block(0..k)
            .map_err(|err| format!("DeflatedArrowSolver::beta_inv: {err}"))
    }

    /// #932 FRONT C — row-local Takahashi selected inverse of the PLAIN bordered
    /// arrow: returns this row's own `(H⁻¹)_tt` block (`q×q`) and its `(H⁻¹)_tβ`
    /// block (`q×K`) WITHOUT the O(n) full-system sweep that one
    /// [`Self::solve`] per unit RHS performs. Mirrors
    /// `ArrowFactorCache::latent_block_inverse_diagonal` (system.rs) but returns
    /// the full blocks rather than only the diagonal. With `A_i =
    /// undamped_factor(i)`, `B_i = H_tβ^(i)`, `G_i = A_i⁻¹ B_i`, `S⁻¹ = beta_inv`:
    ///
    /// ```text
    ///   (H⁻¹)_tt[i,i] = A_i⁻¹ + G_i S⁻¹ G_iᵀ
    ///   (H⁻¹)_tβ[i]   = −G_i S⁻¹
    /// ```
    ///
    /// Touches ONLY row `i`'s own factor, its `H_tβ^(i)` coupling, and the shared
    /// `S⁻¹` — O(q·(q+K)) per row, no `n`-sweep. ONLY valid when
    /// [`Self::plain_selected_inverse_available`]; pass the `S⁻¹` from
    /// [`Self::beta_inv`].
    pub(crate) fn selected_inverse_row_blocks(
        &self,
        row: usize,
        beta_inv: &Array2<f64>,
    ) -> Result<(Array2<f64>, Array2<f64>), String> {
        let cache = self.cache;
        let q = cache.row_dims[row];
        let k = cache.k;
        let factor = cache.undamped_factor(row);

        // A_i⁻¹ (q×q): solve A_i x = e_j per column.
        let mut a_inv = Array2::<f64>::zeros((q, q));
        let mut e_j = Array1::<f64>::zeros(q);
        for j in 0..q {
            e_j.fill(0.0);
            e_j[j] = 1.0;
            let col = cholesky_solve_vector(factor, e_j.view());
            for r in 0..q {
                a_inv[[r, j]] = col[r];
            }
        }

        if k == 0 {
            return Ok((a_inv, Array2::<f64>::zeros((q, 0))));
        }

        // G_i = A_i⁻¹ B_i (q×K): column c is A_i⁻¹ (B_i e_c), where B_i e_c is the
        // c-th column of H_tβ^(i) recovered via `apply_htbeta_row`.
        let mut g = Array2::<f64>::zeros((q, k));
        let mut e_c = Array1::<f64>::zeros(k);
        let mut b_col = Array1::<f64>::zeros(q);
        for c in 0..k {
            e_c.fill(0.0);
            e_c[c] = 1.0;
            b_col.fill(0.0);
            if !cache.apply_htbeta_row(row, e_c.view(), &mut b_col) {
                return Err(format!(
                    "DeflatedArrowSolver::selected_inverse_row_blocks: H_tβ^({row}) apply failed"
                ));
            }
            let g_col = cholesky_solve_vector(factor, b_col.view());
            for r in 0..q {
                g[[r, c]] = g_col[r];
            }
        }

        // GS = G_i S⁻¹ (q×K), via the cache-blocked ndarray/matrixmultiply gemm
        // instead of an O(q·K²) scalar triple loop (K up to 32k).
        let gs = g.dot(beta_inv);

        // (H⁻¹)_tβ[i] = −G_i S⁻¹ = −GS, layout [col, b].
        let inv_vbeta = -&gs;

        // (H⁻¹)_tt[i,i] = A_i⁻¹ + G_i S⁻¹ G_iᵀ = A_i⁻¹ + GS·Gᵀ, layout [r, col].
        // `GS·Gᵀ` is another gemm (q×K · K×q); accumulate onto A_i⁻¹ in place.
        let mut inv_vv = a_inv;
        inv_vv += &gs.dot(&g.t());

        Ok((inv_vv, inv_vbeta))
    }

    pub(crate) fn latent_inverse_diagonal(&self) -> Result<Array1<f64>, String> {
        if self.woodbury_factor.is_none() {
            return self
                .cache
                .latent_block_inverse_diagonal()
                .map_err(|err| format!("DeflatedArrowSolver: latent inverse diagonal: {err}"));
        }
        let total_t = self.cache.delta_t_len();
        let mut out = Array1::<f64>::zeros(total_t);
        let rhs_beta = Array1::<f64>::zeros(self.cache.k);
        // Reuse one unit-vector buffer: set/clear a single entry per index rather
        // than allocating and zeroing a total_t-sized RHS on every iteration.
        let mut rhs_t = Array1::<f64>::zeros(total_t);
        for idx in 0..total_t {
            rhs_t[idx] = 1.0;
            let solved = self.solve(rhs_t.view(), rhs_beta.view())?;
            rhs_t[idx] = 0.0;
            out[idx] = solved.t[idx];
        }
        Ok(out)
    }
}

/// #2712 — the matrix-free sibling of
/// [`DeflatedArrowSolver::selected_inverse_row_blocks`]: row `i`'s own
/// `(H⁻¹)_tt` (`q×q`) and `(H⁻¹)_tβ` (`q×K`) blocks of the bordered arrow,
/// reconstructed from a shared reduced-Schur probe bundle `(z_l, S⁻¹ z_l)`
/// instead of a materialized `K×K` `S⁻¹`. Single source of truth for the three
/// from-probes selected-inverse channels
/// (`logdet_theta_adjoint_from_probes`, `ard_log_precision_hessian_trace_from_probes`,
/// `assignment_log_strength_hessian_trace_from_probes`), which previously each
/// carried their own copy of this reconstruction.
///
/// With `A_i = cache.undamped_factor(i)`, `B_i = H_tβ^(i)`, `G_i = A_i⁻¹ B_i`,
/// and the Rademacher probe identity `E[z zᵀ] = I` (EXACT at the full-basis
/// probe set `z_j = √k·e_j`):
///
/// ```text
///   (H⁻¹)_tt[i] = A_i⁻¹ + G_i S⁻¹ G_iᵀ ,  (G_i S⁻¹ G_iᵀ)[a,b] ≈ (1/m)Σ_l w_l[a] s_l[b]
///   (H⁻¹)_tβ[i] = −G_i S⁻¹             ,  (G_i S⁻¹)[a,c]      ≈ (1/m)Σ_l w_l[a] (S⁻¹z_l)[c]
/// ```
///
/// with `w_l = A_i⁻¹ B_i z_l` and `s_l = A_i⁻¹ B_i (S⁻¹ z_l)`. The `t–t` outer
/// product is symmetrized, which is a no-op at full-basis probes and removes the
/// asymmetry a finite Rademacher set would otherwise introduce into a block the
/// callers contract as symmetric. Its DIAGONAL is unchanged by the
/// symmetrization (`½(w[a]s[a] + s[a]w[a]) = w[a]s[a]`), so diagonal-only
/// consumers get bit-identical values.
///
/// # These blocks are the DEFLATED selected inverse (#2712)
///
/// `cache.undamped_factor(i)` is the Cholesky of the SPECTRALLY CONDITIONED
/// `Φ(H_tt^(i))` — the block that pinned `λ̃ = 1` on every direction recorded in
/// `cache.deflated_row_directions[i]` — not of the raw `H_tt^(i)`, and the
/// reduced Schur `S` behind the bundle is that same conditioned arrow's Schur
/// complement. So `A_i⁻¹ + G_i S⁻¹ G_iᵀ` is the DEFLATED per-row inverse block,
/// exactly the object [`DeflatedArrowSolver::selected_inverse_row_blocks`]
/// returns from the dense route and exactly the object the Daleckii–Krein
/// deflation correction `tr(inv_vv·(D − DΦ[D]))` must be contracted against.
///
/// The from-probes channels used to hard-refuse deflated rows on the stated
/// grounds that "the plain-`S⁻¹` bundle cannot reconstruct the DEFLATED block".
/// That was a misreading of which operand was missing: the block is
/// reconstructed here, and what the probe routes actually lacked was the
/// correction TERM, whose remaining operands (`cache.deflated_row_directions`,
/// `cache.deflation_row_spectra`, and the raw per-row derivative `D` each
/// channel already assembles locally) never involved `S⁻¹` at all.
///
/// `want_tbeta = false` skips the `q×K` `t–β` block — the only part of the
/// reconstruction that costs `O(q·K)` per row — for the trace channels that
/// contract the `t–t` block alone.
///
/// # Edge cases the deflated regime raises, and why they are handled
///
/// * **Stochastic probes (`m < k`).** The reconstruction is then a Hutchinson
///   ESTIMATE, but the Daleckii–Krein correction
///   `Σ_{a,b} W[a,b]·M[a,b]·(1 − F[a,b])`, `W = Uᵀ inv_vv U`, is LINEAR in
///   `inv_vv` — exactly like the `Σ inv_vv[b,a]·dh` contraction it corrects. An
///   unbiased block gives an unbiased correction, so deflation introduces no new
///   approximation class beyond the one the route already accepts.
/// * **Extreme magnitudes.** A KEPT near-null eigendirection makes `inv_vv`
///   legitimately enormous (measured `1.7e7` against a `6e-8` kept eigenvalue on
///   the #2712 fixture). Those entries do NOT amplify the correction: `1 − F` is
///   identically zero on kept×kept pairs, so the huge `W[a,a]` terms drop out and
///   only pairs touching a deflated index survive, each weighted by the
///   deflated direction's own unit stiffness.
/// * **Degenerate / near-degenerate raw eigenvalues.** Handled once, upstream, in
///   `row_deflation_frechet_coefficients`' `eigen_gap_threshold` branch, which
///   both routes call — this reconstruction never re-derives the branch.
/// * **Non-deflating solver paths.** `deflated_row_directions` is empty on the
///   streaming / cross-row-CG / device paths by construction, so the correction
///   is skipped there exactly as before.
pub(crate) fn row_selected_inverse_from_probes(
    cache: &ArrowFactorCache,
    row: usize,
    probes: &[Array1<f64>],
    sinv_probes: &[Array1<f64>],
    want_tbeta: bool,
    context: &str,
) -> Result<(Array2<f64>, Array2<f64>), String> {
    let q = cache.row_dims[row];
    let k = cache.k;
    let factor = cache.undamped_factor(row);

    // A_i⁻¹ (q×q) via the row-local undamped Cholesky.
    let mut inv_vv = Array2::<f64>::zeros((q, q));
    let mut unit = Array1::<f64>::zeros(q);
    for j in 0..q {
        unit.fill(0.0);
        unit[j] = 1.0;
        let col = cholesky_solve_vector(factor, unit.view());
        for r in 0..q {
            inv_vv[[r, j]] = col[r];
        }
    }

    let border_cols = if want_tbeta { k } else { 0 };
    let mut inv_vbeta = Array2::<f64>::zeros((q, border_cols));
    let m = probes.len();
    // A borderless arrow (`k == 0`) is block diagonal: the row block inverse IS
    // `A_i⁻¹` and there is no bundle to fold.
    if k == 0 || m == 0 {
        return Ok((inv_vv, inv_vbeta));
    }
    let inv_m = 1.0 / m as f64;
    let mut b_tmp = Array1::<f64>::zeros(q);
    for l in 0..m {
        b_tmp.fill(0.0);
        if !cache.apply_htbeta_row(row, probes[l].view(), &mut b_tmp) {
            return Err(format!("{context}: H_tβ^({row}) probe apply failed"));
        }
        let w = cholesky_solve_vector(factor, b_tmp.view());
        b_tmp.fill(0.0);
        if !cache.apply_htbeta_row(row, sinv_probes[l].view(), &mut b_tmp) {
            return Err(format!("{context}: H_tβ^({row}) solve apply failed"));
        }
        let s = cholesky_solve_vector(factor, b_tmp.view());
        for a in 0..q {
            for b in 0..q {
                inv_vv[[a, b]] += 0.5 * inv_m * (w[a] * s[b] + s[a] * w[b]);
            }
        }
        if want_tbeta {
            for a in 0..q {
                inv_vbeta.row_mut(a).scaled_add(-inv_m * w[a], &sinv_probes[l]);
            }
        }
    }
    Ok((inv_vv, inv_vbeta))
}

#[cfg(test)]
mod selected_inverse_row_blocks_oracle_tests {
    //! #932 FRONT C oracle: the row-local Takahashi selected-inverse blocks
    //! ([`DeflatedArrowSolver::selected_inverse_row_blocks`] / [`beta_inv`])
    //! MUST reproduce the per-row full-system `solve` loop they replace, to
    //! ≤1e-9, on the plain bordered arrow. This is the gate the logdet /
    //! α-trace consumers rely on when they take the fast path.
    use super::*;
    use gam_solve::arrow_schur::{
        ArrowFactorSlab, ArrowHtbetaCache, ArrowPcgDiagnostics, ArrowSolverMode,
        ArrowUndampedFactors,
    };
    use ndarray::array;
    use std::sync::Arc;

    /// A plain bordered-arrow cache with a NONZERO `H_tβ` coupling and a PD
    /// dense Schur factor, so the β-Schur back-substitution genuinely exercises
    /// the `G S⁻¹ Gᵀ` / `−G S⁻¹` terms (not just the block-diagonal `A⁻¹`). The
    /// stored factors are lower-Cholesky factors `L` (the represented block is
    /// `L Lᵀ`); the row-local identity holds for any PD `A`/`S` and any `B`.
    fn coupled_arrow_cache() -> ArrowFactorCache {
        let htt = ArrowFactorSlab::from_blocks(vec![
            array![[1.3_f64, 0.0], [0.4, 1.1]],
            array![[0.9_f64]],
        ]);
        let schur = array![[1.2_f64, 0.0], [0.25, 0.95]];
        ArrowFactorCache {
            htt_factors: htt,
            htt_factors_undamped: ArrowUndampedFactors::SameAsDamped,
            schur_factor: Some(schur),
            schur_factor_is_undamped: true,
            beta_schur_deflation: None,
            joint_hessian_log_det: None,
            solver_mode: ArrowSolverMode::Direct,
            ridge_t: 0.0,
            ridge_beta: 0.0,
            htbeta: ArrowHtbetaCache::Dense {
                blocks: Arc::from(
                    vec![array![[0.5_f64, -0.2], [0.1, 0.4]], array![[0.3_f64, 0.7]]]
                        .into_boxed_slice(),
                ),
                estimated_bytes: 0,
            },
            d: 2,
            row_dims: Arc::from(vec![2usize, 1usize].into_boxed_slice()),
            row_offsets: Arc::from(vec![0usize, 2usize, 3usize].into_boxed_slice()),
            k: 2,
            manifold_mode_fingerprint: 0,
            row_hessian_fingerprint: 0,
            pcg_diagnostics: ArrowPcgDiagnostics::default(),
            gauge_deflated_directions: 0,
            deflated_row_directions: Arc::from(Vec::new()),
            deflation_row_spectra: Arc::from(Vec::new()),
            beta_gauge_quotient: None,
        }
    }

    #[test]
    fn row_local_blocks_match_per_row_solve() {
        let cache = coupled_arrow_cache();
        let solver = DeflatedArrowSolver::plain(&cache);
        assert!(
            solver.plain_selected_inverse_available(),
            "plain cache must take the fast selected-inverse path"
        );
        let total_t = cache.delta_t_len();
        let k = cache.k;

        // β-block `(H⁻¹)_ββ = S⁻¹`: beta_inv() vs the per-column unit-β solve.
        let beta_inv = solver.beta_inv().expect("beta_inv");
        let rhs_t_zero = Array1::<f64>::zeros(total_t);
        for col in 0..k {
            let mut rhs_beta = Array1::<f64>::zeros(k);
            rhs_beta[col] = 1.0;
            let solved = solver
                .solve(rhs_t_zero.view(), rhs_beta.view())
                .expect("β solve");
            for r in 0..k {
                assert!(
                    (beta_inv[[r, col]] - solved.beta[r]).abs() <= 1e-9,
                    "beta_inv[{r},{col}] {} != solve {}",
                    beta_inv[[r, col]],
                    solved.beta[r]
                );
            }
        }

        // Per-row `(H⁻¹)_tt` (q×q) and `(H⁻¹)_tβ` (q×K) blocks.
        let rhs_beta_zero = Array1::<f64>::zeros(k);
        for row in 0..cache.n_rows() {
            let q = cache.row_dims[row];
            let base = cache.row_offsets[row];
            let (inv_vv, inv_vbeta) = solver
                .selected_inverse_row_blocks(row, &beta_inv)
                .expect("row blocks");
            for col in 0..q {
                let mut rhs_t = Array1::<f64>::zeros(total_t);
                rhs_t[base + col] = 1.0;
                let solved = solver
                    .solve(rhs_t.view(), rhs_beta_zero.view())
                    .expect("t solve");
                for r in 0..q {
                    assert!(
                        (inv_vv[[r, col]] - solved.t[base + r]).abs() <= 1e-9,
                        "inv_vv[{r},{col}] {} != solve {}",
                        inv_vv[[r, col]],
                        solved.t[base + r]
                    );
                }
                for b in 0..k {
                    assert!(
                        (inv_vbeta[[col, b]] - solved.beta[b]).abs() <= 1e-9,
                        "inv_vbeta[{col},{b}] {} != solve {}",
                        inv_vbeta[[col, b]],
                        solved.beta[b]
                    );
                }
            }
        }
    }
}

pub(crate) fn flatten_arrow_parts(
    t: ArrayView1<'_, f64>,
    beta: ArrayView1<'_, f64>,
) -> Array1<f64> {
    let mut out = Array1::<f64>::zeros(t.len() + beta.len());
    for i in 0..t.len() {
        out[i] = t[i];
    }
    for i in 0..beta.len() {
        out[t.len() + i] = beta[i];
    }
    out
}

pub(crate) fn apply_cached_arrow_hessian(
    cache: &ArrowFactorCache,
    v_t: ArrayView1<'_, f64>,
    v_beta: ArrayView1<'_, f64>,
) -> Result<SaeArrowVector, String> {
    let total_t = cache.delta_t_len();
    if v_t.len() != total_t || v_beta.len() != cache.k {
        return Err(format!(
            "apply_cached_arrow_hessian: vector shapes (t={}, beta={}) != cache shapes \
             (t={total_t}, beta={})",
            v_t.len(),
            v_beta.len(),
            cache.k
        ));
    }

    let mut out_t = Array1::<f64>::zeros(total_t);
    let mut out_beta = Array1::<f64>::zeros(cache.k);
    for row in 0..cache.n_rows() {
        let di = cache.row_dims[row];
        let base = cache.row_offsets[row];
        let row_v = v_t.slice(s![base..base + di]);
        let factor = cache.undamped_factor(row);
        let av = cholesky_factor_apply(factor, row_v);
        for j in 0..di {
            out_t[base + j] += av[j];
        }
        if cache.k > 0 {
            let mut b_vbeta = Array1::<f64>::zeros(di);
            if !cache.apply_htbeta_row(row, v_beta, &mut b_vbeta) {
                return Err(format!(
                    "apply_cached_arrow_hessian: H_tβ^({row}) apply failed"
                ));
            }
            for j in 0..di {
                out_t[base + j] += b_vbeta[j];
            }
            if !cache.apply_htbeta_row_transpose(row, row_v, &mut out_beta, None) {
                return Err(format!(
                    "apply_cached_arrow_hessian: H_βt^({row}) apply failed"
                ));
            }
        }
    }

    if cache.k > 0 {
        let Some(schur_factor) = cache.schur_factor.as_ref() else {
            return Err(
                "apply_cached_arrow_hessian: dense Schur factor is required for gauge probing"
                    .to_string(),
            );
        };
        if !cache.schur_factor_is_undamped {
            return Err(
                "apply_cached_arrow_hessian: Schur factor was not built from the undamped evidence row factors"
                    .to_string(),
            );
        }
        let schur_v = cholesky_factor_apply(schur_factor.view(), v_beta);
        for i in 0..cache.k {
            out_beta[i] += schur_v[i];
        }
        for row in 0..cache.n_rows() {
            let di = cache.row_dims[row];
            let mut b_vbeta = Array1::<f64>::zeros(di);
            if !cache.apply_htbeta_row(row, v_beta, &mut b_vbeta) {
                return Err(format!(
                    "apply_cached_arrow_hessian: H_tβ^({row}) Schur correction apply failed"
                ));
            }
            let a_inv_b_vbeta = cholesky_solve_vector(cache.undamped_factor(row), b_vbeta.view());
            if !cache.apply_htbeta_row_transpose(row, a_inv_b_vbeta.view(), &mut out_beta, None) {
                return Err(format!(
                    "apply_cached_arrow_hessian: H_βt^({row}) Schur correction apply failed"
                ));
            }
        }
    }

    Ok(SaeArrowVector {
        t: out_t,
        beta: out_beta,
    })
}

pub(crate) fn cholesky_factor_apply(
    factor: ArrayView2<'_, f64>,
    vector: ArrayView1<'_, f64>,
) -> Array1<f64> {
    let n = factor.nrows();
    // `factor` is a lower-triangular Cholesky factor `L` stored row-major; the
    // represented action is `out = L (Lᵀ v)`.
    //
    // Phase 1 — `lt_v = Lᵀ v`. The natural inner-product form reads
    // `factor[[col, row]]` down a COLUMN (stride `n`), which thrashes cache for
    // the K×K Schur factor (K up to 32 000). Instead iterate over ROWS `j` of
    // `L` — contiguous in memory — and scatter `L[j, 0..=j]·v[j]` into `lt_v`,
    // touching each `L` row once in row-major order (summation order preserved
    // ⇒ bit-identical).
    let mut lt_v = Array1::<f64>::zeros(n);
    for j in 0..n {
        let vj = vector[j];
        if vj == 0.0 {
            continue;
        }
        for (i, &lji) in factor.row(j).iter().enumerate().take(j + 1) {
            lt_v[i] += lji * vj;
        }
    }
    // Phase 2 — `out = L lt_v`. Already contiguous row-major (`L[row, 0..=row]`).
    let mut out = Array1::<f64>::zeros(n);
    for row in 0..n {
        let mut acc = 0.0_f64;
        for (col, &lrc) in factor.row(row).iter().enumerate().take(row + 1) {
            acc += lrc * lt_v[col];
        }
        out[row] = acc;
    }
    out
}

/// The local coordinates a row jet differentiates against.
///
/// #2330 — OPEN QUESTION, recorded here because the measurement is unambiguous even
/// though the attribution is not. The ordered Beta--Bernoulli concentration `α` is a free
/// parameter whenever `learnable_alpha` is set and no override pins it (see
/// `SaeAssignment::effective_alpha_is_learnable`), and it has no variant here. Note
/// that `α` varies with `ρ`, so its absence from this INNER-`θ` basis may well be
/// deliberate — the open question is whether the θ-adjoint under a learnable `α` is
/// then being assembled against a gate the basis cannot see vary.
/// Every derivative assembly keyed on this enum skips `α` — the `dz` / `d2z`
/// loops in `construction_row_jet_logdet_channels.rs` open with
/// `let SaeLocalRowVar::Logit { .. } = *var else { continue; }`, and the border-channel
/// match falls through to `_ => 0.0`. The result is not an inaccurate `∂/∂α`; it is an
/// identically zero one, in a direction the outer optimizer is free to move.
///
/// Measured: with `learnable_alpha`, `fd = -1.428192e0` against `analytic = 5.500943e-3`,
/// relative error FROZEN at `5.904e-1` across a 10x reduction in `h` (a missing term, not
/// a precision limit), while the coordinate channel in the same dump converges
/// quadratically. The same fixture WITHOUT the learnable concentration passes.
///
/// Adding a variant is the real repair, but this enum is indexed positionally by several
/// assemblies (`vars.iter().enumerate()`), so widening it requires auditing each consumer.
#[derive(Debug, Clone, Copy)]
pub(crate) enum SaeLocalRowVar {
    Logit { atom: usize },
    Coord { atom: usize, axis: usize },
}

#[derive(Debug, Clone)]
pub(crate) struct SaeBorderChannel {
    pub(crate) atom: usize,
    pub(crate) basis_col: usize,
    pub(crate) index: usize,
    pub(crate) output: Vec<f64>,
}

#[derive(Debug, Clone)]
pub(crate) struct SaeRowJets {
    pub(crate) vars: Vec<SaeLocalRowVar>,
    pub(crate) channels: crate::row_jet_program::SaeScheduledRowJets,
}

impl SaeRowJets {
    #[inline]
    pub(crate) fn first(&self, primary: usize) -> &[f64] {
        self.channels.first(primary)
    }

    #[inline]
    pub(crate) fn second(&self, a: usize, b: usize) -> &[f64] {
        self.channels.second(a, b)
    }

    #[inline]
    pub(crate) fn beta(&self, border: usize) -> &[f64] {
        self.channels.beta(border)
    }

    #[inline]
    pub(crate) fn beta_deriv(&self, primary: usize, border: usize) -> &[f64] {
        self.channels.beta_deriv(primary, border)
    }

    #[inline]
    pub(crate) fn beta_l_deriv(&self, primary: usize, border: usize) -> &[f64] {
        self.channels.beta_l_deriv(primary, border)
    }
}

pub(crate) fn sae_dot(a: &[f64], b: &[f64]) -> f64 {
    a.iter().zip(b.iter()).map(|(&x, &y)| x * y).sum()
}

/// Euclidean inner product `⟨a, b⟩` over the concatenated `(t, β)` blocks of two
/// arrow vectors. Used by the exact-stationarity Krylov solve and its residual
/// verification.
pub(crate) fn sae_inner(a: &SaeArrowVector, b: &SaeArrowVector) -> f64 {
    sae_dot(a.t.as_slice().unwrap_or(&[]), b.t.as_slice().unwrap_or(&[]))
        + sae_dot(
            a.beta.as_slice().unwrap_or(&[]),
            b.beta.as_slice().unwrap_or(&[]),
        )
}

/// Euclidean norm `‖a‖` over the concatenated `(t, β)` blocks of an arrow vector.
pub(crate) fn sae_norm(a: &SaeArrowVector) -> f64 {
    sae_inner(a, a).max(0.0).sqrt()
}

/// Largest Arnoldi basis admitted by the live cgroup-aware host budget.
///
/// A flexible-GMRES cycle owns `(m+1)` length-`dim` residual-basis vectors,
/// `m` length-`dim` preconditioned directions, an `(m+1)×m` Hessenberg matrix,
/// and the fixed work vectors. Choosing `m` from the memory ledger makes
/// small/medium arrow systems full-memory (and hence exact in at most `dim`
/// Krylov directions) without imposing a dimension or iteration constant;
/// genuinely large systems use the largest restart that can be allocated
/// without violating the process budget.
/// Largest Arnoldi restart whose storage fits `budget`, as a pure function of
/// the shape and the budget.
///
/// #2486 — separated from the host probe so the two regimes that actually
/// depend on the budget are testable at a chosen value rather than only on a
/// memory-starved machine. The search is capped at `dim`, so it returns `dim`
/// — full, unrestarted, exact-in-`dim`-directions GMRES — whenever
/// `24·dim² + 96·dim <= budget`. That threshold is `dim ≈ 9457` even at the
/// 2 GiB in-core floor, so for every shape this repository fits the result is
/// the shape's own `dim` and the budget does not enter. The budget binds only
/// in the large-border regime, which is also the regime with no test coverage;
/// `gmres_restart_saturates_below_the_binding_dimension` pins both halves.
fn admitted_gmres_restart_with_budget(dim: usize, budget: usize) -> Result<usize, String> {
    let storage_bytes = |m: usize| -> Option<usize> {
        let basis = m.checked_add(1)?.checked_mul(dim)?;
        let preconditioned_directions = m.checked_mul(dim)?;
        let hessenberg = m.checked_add(1)?.checked_mul(m)?;
        let fixed = dim.checked_mul(6)?.checked_add(m.checked_mul(4)?)?;
        basis
            .checked_add(preconditioned_directions)?
            .checked_add(hessenberg)?
            .checked_add(fixed)?
            .checked_mul(std::mem::size_of::<f64>())
    };
    let minimum = storage_bytes(1).ok_or_else(|| {
        format!("solve_b_preconditioned_gmres: storage size overflow for dimension {dim}")
    })?;
    if minimum > budget {
        return Err(format!(
            "solve_b_preconditioned_gmres: even one Arnoldi direction needs {minimum} bytes, \
             exceeding the cgroup-aware Krylov budget {budget}"
        ));
    }
    let mut low = 1usize;
    let mut high = dim;
    while low < high {
        let mid = low + (high - low).div_ceil(2);
        if storage_bytes(mid).is_some_and(|bytes| bytes <= budget) {
            low = mid;
        } else {
            high = mid - 1;
        }
    }
    Ok(low)
}

fn admitted_gmres_restart(dim: usize) -> Result<usize, String> {
    let (budget, available) = sae_host_in_core_budget_bytes();
    admitted_gmres_restart_with_budget(dim, budget)
        .map_err(|reason| format!("{reason} (available {available})"))
}

/// Orthogonalize one Arnoldi column with two modified Gram--Schmidt passes.
///
/// A single pass can leave a component parallel to the existing basis when
/// `A z_j` is nearly in the Krylov span. Normalizing that small remainder then
/// magnifies the component, so the Hessenberg/Givens residual is no longer the
/// physical residual. The second pass removes that round-off component while
/// its correction is accumulated into the same Hessenberg column, preserving
/// the Arnoldi decomposition (#2653).
fn reorthogonalize_arnoldi_column(
    basis: &[Array1<f64>],
    w: &mut Array1<f64>,
    h: &mut Array2<f64>,
    column: usize,
) {
    for _ in 0..2 {
        for i in 0..=column {
            let correction = basis[i].dot(w);
            h[[i, column]] += correction;
            for slot in 0..w.len() {
                w[slot] -= correction * basis[i][slot];
            }
        }
    }
}

/// Minimize the physical cycle residual over the stored flexible-GMRES images.
///
/// The Arnoldi Hessenberg problem is a cheap residual predictor, but in finite
/// precision it can become optimistic when the Krylov map is ill-conditioned.
/// Solving `min_y ||r - [A z_0 ... A z_j] y||` by rank-revealing SVD uses the
/// operator images that define the physical equation itself. The resulting
/// update is therefore the minimum-residual member of the generated flexible
/// Krylov space even when the Hessenberg proxy has developed a residual gap
/// (#2653).
fn physical_krylov_least_squares(
    residual: &Array1<f64>,
    operator_images: &[Array1<f64>],
) -> Result<Array1<f64>, String> {
    if operator_images.is_empty() {
        return Err("solve_b_preconditioned_gmres: empty Krylov image space".to_string());
    }
    let dim = residual.len();
    let directions = operator_images.len();
    let mut design = Array2::<f64>::zeros((dim, directions));
    for (column, image) in operator_images.iter().enumerate() {
        if image.len() != dim || image.iter().any(|value| !value.is_finite()) {
            return Err(format!(
                "solve_b_preconditioned_gmres: invalid physical operator image {column}"
            ));
        }
        design.column_mut(column).assign(image);
    }
    let (u_opt, singular_values, vt_opt) = design
        .svd(true, true)
        .map_err(|error| format!("solve_b_preconditioned_gmres: physical SVD failed: {error}"))?;
    let u = u_opt.ok_or_else(|| {
        "solve_b_preconditioned_gmres: physical SVD omitted left vectors".to_string()
    })?;
    let vt = vt_opt.ok_or_else(|| {
        "solve_b_preconditioned_gmres: physical SVD omitted right vectors".to_string()
    })?;
    let largest = singular_values
        .iter()
        .fold(0.0_f64, |current, &value| current.max(value));
    if !(largest.is_finite() && largest > 0.0) {
        return Err(
            "solve_b_preconditioned_gmres: physical Krylov image has zero numerical rank"
                .to_string(),
        );
    }
    let rank_floor =
        largest * f64::EPSILON * (design.nrows().max(design.ncols()) as f64);
    let projected = u.t().dot(residual);
    let mut scaled = Array1::<f64>::zeros(singular_values.len());
    for index in 0..singular_values.len() {
        if singular_values[index] > rank_floor {
            scaled[index] = projected[index] / singular_values[index];
        }
    }
    let coefficients = vt.t().dot(&scaled);
    if coefficients.iter().any(|value| !value.is_finite()) {
        return Err(
            "solve_b_preconditioned_gmres: physical Krylov coefficients are non-finite"
                .to_string(),
        );
    }
    Ok(coefficients)
}

#[cfg(test)]
mod gmres_restart_budget_tests {
    use super::admitted_gmres_restart_with_budget;

    /// Saturated storage at `m == dim`, in bytes: the basis, the preconditioned
    /// directions, the Hessenberg block and the fixed workspace.
    fn saturated_bytes(dim: usize) -> usize {
        24 * dim * dim + 96 * dim
    }

    #[test]
    fn gmres_restart_saturates_below_the_binding_dimension() {
        // A budget that admits the whole Krylov space: the restart is the
        // shape's own dim, so the budget does not enter the answer at all.
        for dim in [26usize, 200, 2_048] {
            let budget = saturated_bytes(dim);
            assert_eq!(
                admitted_gmres_restart_with_budget(dim, budget),
                Ok(dim),
                "dim={dim} must take full unrestarted GMRES when its whole basis fits"
            );
            // One byte short of saturation must NOT return dim, otherwise the
            // assertion above would pass for a rule that ignores the budget.
            let restart = admitted_gmres_restart_with_budget(dim, budget - 1)
                .expect("one Arnoldi direction still fits");
            assert!(
                restart < dim,
                "dim={dim} must restart below dim once its full basis no longer fits, got {restart}"
            );
        }
    }

    #[test]
    fn gmres_restart_refuses_when_a_single_direction_does_not_fit() {
        let error = admitted_gmres_restart_with_budget(1_000, 8)
            .expect_err("8 bytes cannot hold one Arnoldi direction at dim=1000");
        assert!(
            error.contains("even one Arnoldi direction"),
            "the refusal must name the storage it could not afford, got {error}"
        );
    }
}

/// Solve `A x = rhs` by flexible right-preconditioned restarted GMRES.
///
/// The exact stationarity Jacobian `A` contains residual and prior curvature and
/// can be indefinite. Conjugate gradients is therefore not admissible: its SPD
/// recurrence can stop at negative curvature and silently return a non-solution.
/// Flexible GMRES instead minimizes the original residual without an SPD or
/// linear-preconditioner assumption. At Arnoldi direction `v_j` it stores
/// `z_j = P_j(v_j)`, applies `A z_j`, and updates the physical solution as
/// `x += Σ_j z_j y_j`. This is essential for the matrix-free arrow inverse:
/// its tolerance-driven adaptive CG is not one fixed linear `B⁻¹`, so recovering
/// `P(Σ_j v_j y_j)` after the least-squares solve is not equal to the Arnoldi
/// combination `Σ_j P_j(v_j)y_j` (#2258). Because preconditioning remains on the
/// right, the Arnoldi least-squares norm is the physical residual that certifies
/// `A x = rhs`; an ill-scaled or adaptive inverse cannot certify only a
/// preconditioned proxy. Exhaustion or Arnoldi breakdown is a typed error, never
/// a last-iterate fallback. The same implementation serves the fixed dense
/// factor cache and the adaptive matrix-free reduced-Schur inverse.
pub(crate) fn solve_b_preconditioned_gmres_with<F, P>(
    rhs: &SaeArrowVector,
    apply_a: F,
    precondition: P,
) -> Result<SaeArrowVector, String>
where
    F: Fn(&SaeArrowVector) -> Result<SaeArrowVector, String>,
    P: Fn(&SaeArrowVector) -> Result<SaeArrowVector, String>,
{
    let initial = SaeArrowVector {
        t: Array1::zeros(rhs.t.len()),
        beta: Array1::zeros(rhs.beta.len()),
    };
    solve_b_preconditioned_gmres_from(rhs, &initial, apply_a, precondition)
        .map(|(solution, _iterations)| solution)
}

/// Warm-started sibling of solve_b_preconditioned_gmres_with.
///
/// The physical initial vector is checked against the original residual before
/// any Arnoldi step. An exact warm start therefore returns with zero iterations;
/// every inexact start follows the same flexible right-preconditioned recurrence
/// and the same original-residual certificate as the zero-start entry. Returning
/// the actual iteration count makes descending shifted systems auditable and
/// allows their neighboring solutions to be reused without changing the solve.
pub(crate) fn solve_b_preconditioned_gmres_from<F, P>(
    rhs: &SaeArrowVector,
    initial: &SaeArrowVector,
    apply_a: F,
    precondition: P,
) -> Result<(SaeArrowVector, usize), String>
where
    F: Fn(&SaeArrowVector) -> Result<SaeArrowVector, String>,
    P: Fn(&SaeArrowVector) -> Result<SaeArrowVector, String>,
{
    let t_len = rhs.t.len();
    let beta_len = rhs.beta.len();
    if initial.t.len() != t_len || initial.beta.len() != beta_len {
        return Err(format!(
            "solve_b_preconditioned_gmres: initial dimensions ({}, {}) do not match rhs ({t_len}, {beta_len})",
            initial.t.len(),
            initial.beta.len(),
        ));
    }
    let dim = t_len + beta_len;
    if dim == 0 {
        return Ok((
            SaeArrowVector {
                t: Array1::zeros(0),
                beta: Array1::zeros(0),
            },
            0,
        ));
    }
    let rhs_flat = flatten_arrow_parts(rhs.t.view(), rhs.beta.view());
    let rhs_norm = rhs_flat.dot(&rhs_flat).sqrt();
    if rhs_norm == 0.0 {
        return Ok((
            SaeArrowVector {
                t: Array1::zeros(t_len),
                beta: Array1::zeros(beta_len),
            },
            0,
        ));
    }
    if !rhs_norm.is_finite() {
        return Err("solve_b_preconditioned_gmres: non-finite right-hand side".to_string());
    }
    let b = rhs_flat;
    let b_norm = rhs_norm;
    let relative_floor = f64::EPSILON.sqrt();
    // Full-memory whenever the live memory ledger admits it. Each restarted
    // cycle must make a strictly representable reduction in the original
    // residual, and inability to do so is the typed numerical-stagnation
    // certificate.
    //
    // #2627 — that strict test alone was not a termination argument. `next_norm
    // < residual_norm` is satisfied by a reduction of one ulp, so on a
    // near-singular `A = B + ΔC` the restart loop contracted by `1 - O(eps)` per
    // cycle and ran forever, each cycle costing `restart` operator applies. The
    // two additions below are an anti-runaway ceiling on restarted cycles and a
    // contraction-margin projection; neither is a convergence bound (see their
    // notes at the bottom of the loop).
    const RESTART_CYCLE_CEILING: usize = 256;
    let restart = admitted_gmres_restart(dim)?;
    let started = std::time::Instant::now();
    let mut iterations = 0usize;
    let mut cycles = 0usize;
    // Smallest per-cycle contraction observed so far. Projecting with the BEST
    // rate is the most optimistic estimate available, so a refusal derived from
    // it cannot cut off a solve that any observed rate could still finish.
    let mut best_contraction = 1.0_f64;
    let mut solution = flatten_arrow_parts(initial.t.view(), initial.beta.view());
    if solution.iter().any(|value| !value.is_finite()) {
        return Err("solve_b_preconditioned_gmres: non-finite initial solution".to_string());
    }

    let as_arrow = |flat: &Array1<f64>| SaeArrowVector {
        t: flat.slice(s![..t_len]).to_owned(),
        beta: flat.slice(s![t_len..]).to_owned(),
    };
    let apply_preconditioner = |flat: &Array1<f64>| -> Result<Array1<f64>, String> {
        let direction = as_arrow(flat);
        let preconditioned = precondition(&direction)
            .map_err(|err| format!("solve_b_preconditioned_gmres: B inverse: {err}"))?;
        Ok(flatten_arrow_parts(
            preconditioned.t.view(),
            preconditioned.beta.view(),
        ))
    };
    let apply_operator = |flat: &Array1<f64>| -> Result<Array1<f64>, String> {
        let physical = as_arrow(flat);
        let applied = apply_a(&physical)?;
        Ok(flatten_arrow_parts(applied.t.view(), applied.beta.view()))
    };

    loop {
        let ax = apply_operator(&solution)?;
        let mut residual = &b - &ax;
        let residual_norm = residual.dot(&residual).sqrt();
        if residual_norm <= relative_floor * b_norm {
            let candidate = as_arrow(&solution);
            let ax = apply_a(&candidate)?;
            let original = SaeArrowVector {
                t: &rhs.t - &ax.t,
                beta: &rhs.beta - &ax.beta,
            };
            let original_norm = sae_norm(&original);
            if original_norm <= relative_floor * rhs_norm {
                return Ok((candidate, iterations));
            }
        }
        if !(residual_norm.is_finite() && residual_norm > 0.0) {
            return Err("solve_b_preconditioned_gmres: non-finite original residual".to_string());
        }

        let cycle_residual = residual.clone();
        residual.mapv_inplace(|value| value / residual_norm);
        let cycle = restart;
        let mut basis: Vec<Array1<f64>> = Vec::with_capacity(cycle + 1);
        basis.push(residual);
        let mut preconditioned_basis: Vec<Array1<f64>> = Vec::with_capacity(cycle);
        let mut operator_images: Vec<Array1<f64>> = Vec::with_capacity(cycle);
        let mut h = Array2::<f64>::zeros((cycle + 1, cycle));
        let mut cosines = vec![0.0_f64; cycle];
        let mut sines = vec![0.0_f64; cycle];
        let mut g = Array1::<f64>::zeros(cycle + 1);
        g[0] = residual_norm;
        let mut used = 0usize;

        for j in 0..cycle {
            let preconditioned_direction = apply_preconditioner(&basis[j])?;
            if !preconditioned_direction
                .iter()
                .all(|value| value.is_finite())
            {
                return Err(format!(
                    "solve_b_preconditioned_gmres: non-finite preconditioned direction at \
                     iteration {}",
                    iterations + j
                ));
            }
            let mut w = apply_operator(&preconditioned_direction)?;
            operator_images.push(w.clone());
            preconditioned_basis.push(preconditioned_direction);
            reorthogonalize_arnoldi_column(&basis, &mut w, &mut h, j);
            let next_norm = w.dot(&w).sqrt();
            let arnoldi_space_closed = next_norm <= f64::EPSILON;
            h[[j + 1, j]] = next_norm;
            if !arnoldi_space_closed {
                w.mapv_inplace(|value| value / next_norm);
                basis.push(w);
            } else {
                basis.push(Array1::zeros(dim));
            }

            for i in 0..j {
                let upper = cosines[i] * h[[i, j]] + sines[i] * h[[i + 1, j]];
                let lower = -sines[i] * h[[i, j]] + cosines[i] * h[[i + 1, j]];
                h[[i, j]] = upper;
                h[[i + 1, j]] = lower;
            }
            let diagonal = h[[j, j]];
            let below = h[[j + 1, j]];
            let radius = diagonal.hypot(below);
            if !(radius.is_finite() && radius > f64::EPSILON) {
                return Err(format!(
                    "solve_b_preconditioned_gmres: Arnoldi breakdown after {} iterations",
                    iterations + j
                ));
            }
            cosines[j] = diagonal / radius;
            sines[j] = below / radius;
            h[[j, j]] = radius;
            h[[j + 1, j]] = 0.0;
            let gj = g[j];
            g[j] = cosines[j] * gj;
            g[j + 1] = -sines[j] * gj;
            used = j + 1;
            iterations = iterations.checked_add(1).ok_or_else(|| {
                "solve_b_preconditioned_gmres: iteration counter overflow".to_string()
            })?;
            // #2472 — the only progress signal this solve has ever emitted is
            // its return value, so a solve that is converging slowly and one
            // that has stalled look identical from outside: both are silence.
            // The cadence is powers of two, so the line count is logarithmic in
            // the iteration count no matter how long a cycle runs, and the
            // interval between lines grows with the cost already sunk.
            if iterations.is_power_of_two() {
                log::info!(
                    "[SAE-GMRES] dim={dim} restart={restart} iter={iterations} \
                     rel_residual={:.3e} target={:.3e} elapsed={:.1}s",
                    g[j + 1].abs() / b_norm,
                    relative_floor,
                    started.elapsed().as_secs_f64(),
                );
            }
            // The recursive Hessenberg residual is only a predictor. A false
            // convergence signal was the residual-gap defect in #2653, so it
            // cannot truncate a still-growing physical Krylov space. Continue
            // to the shape/memory-derived restart and let the stored `A z_j`
            // least-squares decide the cycle, stopping early only when Arnoldi
            // proves the generated space itself is closed.
            if arnoldi_space_closed {
                break;
            }
        }

        let y = physical_krylov_least_squares(&cycle_residual, &operator_images[..used])?;
        let mut represented_residual = cycle_residual;
        for i in 0..used {
            for slot in 0..dim {
                solution[slot] += y[i] * preconditioned_basis[i][slot];
                represented_residual[slot] -= y[i] * operator_images[i][slot];
            }
        }
        let represented_norm = represented_residual.dot(&represented_residual).sqrt();

        let candidate = as_arrow(&solution);
        let ax = apply_a(&candidate)?;
        let original = SaeArrowVector {
            t: &rhs.t - &ax.t,
            beta: &rhs.beta - &ax.beta,
        };
        let original_norm = sae_norm(&original);
        // Full-memory GMRES terminates within `dim` directions up to round-off.
        // The attainable residual of a moderately conditioned system is
        // O(kappa*eps), so sqrt(eps) is the scalar-type-derived certification
        // floor rather than a last-iterate fallback.
        let roundoff_floor = relative_floor * rhs_norm;
        cycles += 1;
        log::info!(
            "[SAE-GMRES] cycle {cycles} closed: dim={dim} restart={restart} iters={iterations} \
             rel_original_residual={:.3e} floor={:.3e} elapsed={:.1}s",
            original_norm / rhs_norm,
            roundoff_floor / rhs_norm,
            started.elapsed().as_secs_f64(),
        );
        if original_norm <= roundoff_floor {
            return Ok((candidate, iterations));
        }
        let next_ax = apply_operator(&solution)?;
        let next_residual = &b - &next_ax;
        let next_norm = next_residual.dot(&next_residual).sqrt();
        if !(next_norm.is_finite() && next_norm < residual_norm) {
            return Err(format!(
                "solve_b_preconditioned_gmres: no representable residual reduction after \
                 {iterations} iterations (restart {restart}, dimension {dim}); original \
                residual {residual_norm:.3e} -> {next_norm:.3e}, relative residual \
                 {:.3e}, stored-operator residual {represented_norm:.3e}, round-off \
                 certification floor {:.3e}",
                original_norm / rhs_norm,
                roundoff_floor / rhs_norm,
            ));
        }
        // #2627 — anti-runaway ceiling on restarted cycles.
        //
        // It must not impersonate a convergence bound, so it is placed where no
        // solve that converges in a practical count can reach it. Full-memory
        // GMRES (`restart == dim`) is exact within `dim` directions, so it
        // certifies in ONE cycle up to round-off; further cycles are round-off
        // recovery only. A memory-forced restart `m < dim` has no finite-
        // termination theorem at all, which is precisely why an unbounded loop
        // was never a termination argument here. `256` is more than two orders
        // of magnitude above the single cycle a well-posed system needs, so it
        // decides nothing about convergence — it only bounds a crawl.
        best_contraction = best_contraction.min(next_norm / residual_norm);
        if cycles >= RESTART_CYCLE_CEILING {
            return Err(format!(
                "solve_b_preconditioned_gmres: restart anti-runaway ceiling \
                 {RESTART_CYCLE_CEILING} cycles reached after {iterations} iterations \
                 (restart {restart}, dimension {dim}); relative original residual \
                 {:.3e} against round-off certification floor {:.3e}, best per-cycle \
                 contraction {best_contraction:.9}",
                original_norm / rhs_norm,
                roundoff_floor / rhs_norm,
            ));
        }
        // Contraction MARGIN. The strict test above accepts a one-ulp reduction,
        // which is why it could not terminate: the loop is allowed to spend the
        // whole ceiling proving a rate that will never reach the floor. Project
        // the cycles remaining to the round-off floor from the BEST contraction
        // measured so far — `log(floor / current) / log(rate)` — and refuse now,
        // naming the measured rate, when even that optimistic projection cannot
        // land inside the ceiling. The refusal is a statement about the observed
        // rate, not about where convergence lies: any rate that CAN finish inside
        // the ceiling passes untouched.
        if best_contraction > 0.0 && best_contraction < 1.0 && next_norm > roundoff_floor {
            let projected = (roundoff_floor / next_norm).ln() / best_contraction.ln();
            if projected.is_finite() && projected > 0.0 {
                let remaining = (RESTART_CYCLE_CEILING - cycles) as f64;
                if projected.ceil() > remaining {
                    return Err(format!(
                        "solve_b_preconditioned_gmres: restarted residual contracts too \
                         slowly to certify — best measured per-cycle contraction \
                         {best_contraction:.9} projects {projected:.1} further cycles to \
                         reach the round-off floor, against {remaining} left under the \
                         {RESTART_CYCLE_CEILING}-cycle anti-runaway ceiling (restart \
                         {restart}, dimension {dim}, {iterations} iterations); relative \
                         original residual {:.3e}, floor {:.3e}",
                        original_norm / rhs_norm,
                        roundoff_floor / rhs_norm,
                    ));
                }
            }
        }
    }
}

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

    #[test]
    fn reorthogonalizes_nearly_dependent_arnoldi_column_before_residual_projection_2653() {
        // A Krylov image that is almost in the existing span is exactly the
        // regime in which one-pass MGS lost enough orthogonality for the
        // Hessenberg residual to disagree with the recomputed physical one in
        // #2653. The symmetric direction is analytically orthogonal to `q`;
        // its tiny magnitude makes the one-pass round-off component observable
        // after normalization without relying on a large or random fixture.
        let dim = 7usize;
        let q = Array1::from_elem(dim, (dim as f64).sqrt().recip());
        let mut transverse = Array1::from_iter((-3..=3).map(f64::from));
        let transverse_norm = transverse.dot(&transverse).sqrt();
        transverse.mapv_inplace(|value| value / transverse_norm);
        let original = &q + &(1.0e-12 * &transverse);

        let mut once = original.clone();
        let once_projection = q.dot(&once);
        for slot in 0..dim {
            once[slot] -= once_projection * q[slot];
        }
        let once_norm = once.dot(&once).sqrt();
        once.mapv_inplace(|value| value / once_norm);
        let one_pass_overlap = q.dot(&once).abs();
        assert!(
            one_pass_overlap > 1.0e-5,
            "fixture must expose the one-pass Arnoldi residual gap, overlap={one_pass_overlap:.3e}"
        );

        let basis = vec![q];
        let mut twice = original;
        let mut h = Array2::<f64>::zeros((2, 1));
        reorthogonalize_arnoldi_column(&basis, &mut twice, &mut h, 0);
        let twice_norm = twice.dot(&twice).sqrt();
        twice.mapv_inplace(|value| value / twice_norm);
        let two_pass_overlap = basis[0].dot(&twice).abs();
        assert!(
            two_pass_overlap <= 64.0 * f64::EPSILON,
            "reorthogonalized Arnoldi residual must remain physical, overlap={two_pass_overlap:.3e}"
        );
    }

    #[test]
    fn exact_physical_warm_start_returns_without_an_arnoldi_step_2515() {
        let rhs = SaeArrowVector {
            t: array![5.0_f64, 5.0],
            beta: Array1::zeros(0),
        };
        let exact = SaeArrowVector {
            t: array![1.0_f64, 1.0],
            beta: Array1::zeros(0),
        };
        let apply_a = |value: &SaeArrowVector| -> Result<SaeArrowVector, String> {
            Ok(SaeArrowVector {
                t: array![
                    3.0 * value.t[0] + 2.0 * value.t[1],
                    value.t[0] + 4.0 * value.t[1],
                ],
                beta: Array1::zeros(0),
            })
        };
        let identity =
            |value: &SaeArrowVector| -> Result<SaeArrowVector, String> { Ok(value.clone()) };

        let (solved, iterations) =
            solve_b_preconditioned_gmres_from(&rhs, &exact, apply_a, identity)
                .expect("exact warm start");
        assert_eq!(iterations, 0, "an exact warm start must do no Arnoldi work");
        assert_eq!(solved.t, exact.t);
        assert_eq!(solved.beta, exact.beta);
    }

    #[test]
    fn certifies_original_residual_under_ill_scaled_preconditioner_2258() {
        // A x = rhs has x=(1,1). B^-1 scales the two physical coordinates in
        // opposite directions, so a left-preconditioned residual is a poor
        // proxy for the physical equation. Right GMRES must still solve and
        // certify rhs-Ax in the original norm.
        let rhs = SaeArrowVector {
            t: array![5.0_f64, 5.0],
            beta: Array1::zeros(0),
        };
        let apply_a = |value: &SaeArrowVector| -> Result<SaeArrowVector, String> {
            Ok(SaeArrowVector {
                t: array![
                    3.0 * value.t[0] + 2.0 * value.t[1],
                    value.t[0] + 4.0 * value.t[1],
                ],
                beta: Array1::zeros(0),
            })
        };
        let precondition = |value: &SaeArrowVector| -> Result<SaeArrowVector, String> {
            Ok(SaeArrowVector {
                t: array![1.0e-3 * value.t[0], 1.0e3 * value.t[1]],
                beta: Array1::zeros(0),
            })
        };

        let solved = solve_b_preconditioned_gmres_with(&rhs, apply_a, precondition)
            .expect("right-preconditioned solve");
        let applied = apply_a(&solved).expect("physical operator");
        let residual = SaeArrowVector {
            t: &rhs.t - &applied.t,
            beta: Array1::zeros(0),
        };
        assert!(
            sae_norm(&residual) <= f64::EPSILON.sqrt() * sae_norm(&rhs),
            "physical residual was not certified: relative={:.3e}",
            sae_norm(&residual) / sae_norm(&rhs),
        );
        assert!((solved.t[0] - 1.0).abs() <= f64::EPSILON.sqrt());
        assert!((solved.t[1] - 1.0).abs() <= f64::EPSILON.sqrt());
    }

    #[test]
    fn supports_non_linear_adaptive_preconditioner_2258() {
        // The adaptive inverse normalizes each requested direction. It is
        // deliberately nonlinear: for rhs=5v, P(rhs)=v while 5P(v)=5v. A
        // standard right-GMRES implementation that recovers P(Vy) therefore
        // returns v after its one-direction Arnoldi solve and cannot reduce the
        // physical residual. Flexible GMRES stores z=P(v) and updates x=zy=5v,
        // which exactly solves the identity equation.
        let rhs = SaeArrowVector {
            t: array![3.0_f64, 4.0],
            beta: Array1::zeros(0),
        };
        let apply_a =
            |value: &SaeArrowVector| -> Result<SaeArrowVector, String> { Ok(value.clone()) };
        let precondition = |value: &SaeArrowVector| -> Result<SaeArrowVector, String> {
            let norm = sae_norm(value);
            if norm == 0.0 {
                return Ok(value.clone());
            }
            Ok(SaeArrowVector {
                t: &value.t / norm,
                beta: &value.beta / norm,
            })
        };

        let solved = solve_b_preconditioned_gmres_with(&rhs, apply_a, precondition)
            .expect("flexible right-preconditioned solve");
        let residual = SaeArrowVector {
            t: &rhs.t - &solved.t,
            beta: &rhs.beta - &solved.beta,
        };
        assert!(
            sae_norm(&residual) <= f64::EPSILON.sqrt() * sae_norm(&rhs),
            "adaptive inverse must certify the physical equation: relative={:.3e}",
            sae_norm(&residual) / sae_norm(&rhs),
        );
    }
}