gam-solve 0.3.150

use super::*;
use opt::{BacktrackConfig, backtracking_line_search};

pub(crate) struct OuterFirstOrderBridge<'a> {
    pub(crate) obj: &'a mut dyn OuterObjective,
    pub(crate) layout: OuterThetaLayout,
    /// Outer-aware inner-PIRLS cap atomic. When `Some`, the bridge stores
    /// a coarsen-then-tighten cap into it on every accepted gradient eval
    /// (see `first_order_inner_cap_schedule`).
    ///
    /// The cap is a perf optimization for the GRADIENT inner solve only: at
    /// the accepted ρ the warm-start is excellent, so a small cap converges
    /// the inner Newton and a still-non-converged result is honestly rejected
    /// as infeasible. But the line-search COST probe (`eval_cost`) evaluates a
    /// DIFFERENT trial ρ whose warm-start is worse; the same small cap can stop
    /// the inner solve short of its fixed point, returning a non-converged
    /// `f64::INFINITY` cost for a point that is actually feasible. With every
    /// trial step then reporting `∞`, no Wolfe/ARC step satisfies descent, the
    /// optimizer never leaves the accepted ρ, and the gradient re-evaluated
    /// there is identical iter after iter — the frozen-|g| outer stall in
    /// gam#787 (bernoulli matern marginal-slope) and gam#808 (survival
    /// marginal-slope). The line-search cost MUST be the same converged-inner
    /// objective the analytic envelope gradient differentiates; a capped
    /// surrogate is a different objective. So `eval_cost` UNCAPS the inner solve
    /// (stores `0` = full `pirls_config.max_iterations`) before delegating, and
    /// `eval_grad`/`eval_hessian` restore the scheduled cap on the next call.
    pub(crate) outer_inner_cap: Option<InnerProgressFeedback>,
    /// Counts gradient evaluations for logging only. Inner-PIRLS scheduling
    /// uses `InnerProgressFeedback.accepted_iter` so rejected line-search
    /// probes do not relax the inner work budget.
    pub(crate) iter_count: usize,
    /// First observed `‖g‖` from `eval_grad`. Used by the schedule to
    /// compute the gradient-ratio (`last / initial`) — when the ratio
    /// drops, the optimizer is approaching convergence and the inner
    /// cap should lift to full so the cached β is at full tolerance.
    pub(crate) g_norm_initial: Option<f64>,
    /// `‖g‖` from the most recent eval. Stale by one outer iter relative
    /// to the cap that consumes it (the cap is set BEFORE the new eval),
    /// but for monotone-decreasing g_norm this is safe — it makes the
    /// cap conservatively LARGER than the truly-needed value, never
    /// smaller.
    pub(crate) last_g_norm: Option<f64>,
    /// Most recent derivative-evaluation point. Value-only line-search probes
    /// log their distance from this reference so hidden backtracking work is
    /// visible in STAGE traces.
    pub(crate) last_value_grad_rho: Option<Array1<f64>>,
    /// Exact memo for recent line-search value probes. BFGS can re-query the
    /// same rejected trial when switching Wolfe strategies; the SAE inner solve
    /// behind a Value probe is deterministic, so serving an identical rho from
    /// this memo preserves the objective while avoiding duplicate refinement
    /// work.
    pub(crate) value_probe_cache: Vec<ValueProbeCacheEntry>,
    /// Gradient-independent cost-stall convergence guard. `opt::Bfgs` only
    /// terminates on a small *projected gradient norm* (its stall exit ANDs
    /// gradient-smallness with cost-smallness), so on a fully-penalized
    /// (double-penalty) REML surface with a shallow, weakly-identified valley —
    /// where the REML score flatlines while `‖∇_ρ V‖` plateaus *above*
    /// tolerance — no opt-side exit ever fires and BFGS burns its entire
    /// `max_iterations` budget (each iteration spending many line-search +
    /// coordinate-rescue + jiggle probes) on every seed. That is the #1089
    /// pathology: a trivial n≈30..120 Gaussian fit emitting ~850k cost-only
    /// evaluations until a wall-clock budget kills it. This guard adds the
    /// missing mgcv-style score-change stop: it watches the accepted-iterate
    /// REML objective and, once it stops improving by more than a relative
    /// tolerance over a window of consecutive accepted outer steps, publishes
    /// the best-so-far iterate and signals BFGS to stop. The runner then
    /// classifies the run as *converged at the flat-valley floor* rather than
    /// non-converged — the remaining gradient lies along weakly-identified ρ
    /// directions that do not reduce the objective.
    pub(crate) cost_stall: Option<CostStallGuard>,
    /// Box bounds `(lower, upper)` on the outer parameter vector, used ONLY to
    /// form the bound-PROJECTED gradient norm the [`CostStallGuard`] stationarity
    /// test consumes. The cost-stall guard's documented contract is to certify a
    /// stall as converged iff the *projected* gradient at the best iterate clears
    /// the outer tolerance — the same KKT criterion `opt`'s primary exit uses
    /// (`GradientTolerance{ projected: true }`). Without the bounds here the guard
    /// fell back to the RAW gradient norm, which on a separation fit never clears
    /// the tolerance (the bound-active log-λ directions carry a persistent
    /// ∂V/∂ρ pushing further out of bounds), so a genuinely stationary
    /// bound-pinned optimum was reported NON-converged and the outer search
    /// re-seeded forever (#1082 separable-multinomial / penalized-tensor
    /// cycling). `None` ⇒ unconstrained, no projection (raw norm). Cheap to hold
    /// (the outer dimension is the smoothing-param count).
    pub(crate) cost_stall_bounds: Option<(Array1<f64>, Array1<f64>)>,
    /// Count of consecutive `eval_cost` calls that returned `Recoverable`
    /// without a single success in between. When every trial step in every
    /// search direction is infeasible (the inner solve refuses to converge at
    /// any neighboring ρ), BFGS would otherwise spend its full
    /// `max_iterations × line_search_budget` budget doing inner solves that
    /// all fail — the non-termination reported in issue #NaN-outer-loop.
    ///
    /// Once this counter exceeds [`PROBE_REFUSAL_FATAL_THRESHOLD`] and no
    /// gradient evaluation has ever been accepted on this seed (`iter_count ==
    /// 0`), the bridge escalates to `Fatal` so BFGS exits immediately via
    /// `ObjectiveFailed`. The seed loop treats that outcome as a rejected seed
    /// and moves on, keeping the cascade bounded.
    ///
    /// Reset to 0 on any successful cost evaluation so normal line-search
    /// noise (a few recoverable probes followed by an accepted step) never
    /// trips this guard.
    pub(crate) consecutive_probe_refusals: usize,
}

pub(crate) const VALUE_PROBE_CACHE_CAPACITY: usize = 256;

pub(crate) const VALUE_PROBE_REJECT_COST_FLOOR: f64 = 1.0e11;

/// Number of consecutive recoverable `eval_cost` failures (every line-search
/// probe infeasible) before the bridge escalates to `Fatal` and forces an
/// immediate BFGS exit. This guard fires only before the first accepted
/// gradient step (`iter_count == 0`): once BFGS has accepted at least one
/// outer iteration the current ρ is feasible and isolated probe refusals are
/// normal line-search noise, not a stuck loop.
///
/// The threshold covers one full StrongWolfe attempt (up to 20 probes)
/// plus one backtracking fallback (up to 50 probes) with a small margin,
/// so a SINGLE failed direction does not fire the guard. Two consecutive
/// direction failures (120 probes) always does — once both Wolfe and
/// backtracking exhausted two complete directions with no success, the
/// neighborhood is globally infeasible and further BFGS iterations are
/// pure waste.
pub(crate) const PROBE_REFUSAL_FATAL_THRESHOLD: usize = 150;

/// Tighter probe-refusal threshold used when the bridge has never seen a
/// `eval_grad` call of its own — i.e. the seed (cost, gradient) was supplied
/// via `with_initial_sample` so `last_value_grad_rho` is `None` and every
/// `trial_rho_distance` prints as NaN.  In this case the seed gradient is
/// already confirmed feasible externally; if even the first line-search
/// direction exhausts its Wolfe probes without success (≈ 20 probes), the
/// neighborhood IS globally infeasible and further iterations just repeat
/// the same expensive inner solve 150 more times.  One generous Wolfe
/// budget (25 probes) is enough to confirm the failure; 13 seeds ×
/// 150 probes × ~3 s each would otherwise cause an observed ~97 min hang.
pub(crate) const PROBE_REFUSAL_FATAL_THRESHOLD_NAN_SEED: usize = 25;

/// Sentinel prefix embedded in the [`ObjectiveEvalError::Fatal`] message the
/// bridge returns when [`PROBE_REFUSAL_FATAL_THRESHOLD`] fires. The seed-loop
/// runner matches this prefix and routes the failed seed to
/// `rejection_reasons` rather than propagating a fatal error.
pub(crate) const PROBE_REFUSAL_FATAL_SENTINEL: &str = "OUTER_PROBE_REFUSAL_FATAL";

/// Sentinel embedded in the [`ObjectiveEvalError::Fatal`] message the bridge
/// returns when [`CostStallGuard`] halts BFGS on a cost stall. `opt::Bfgs`
/// preserves the message verbatim in [`BfgsError::ObjectiveFailed`]; the
/// seed-loop runner recognizes this sentinel and rebuilds an outer result from
/// the published best iterate. Whether that result is reported `converged` is
/// NOT decided here — it is carried on the published [`CostStallExit`], gated on
/// the projected gradient norm at the best iterate clearing the same outer
/// gradient tolerance the genuine convergence path uses. A cost stall whose
/// residual gradient still exceeds that tolerance is a flat-valley stall, not a
/// stationary optimum, and is reported `converged = false`.
pub(crate) const COST_STALL_CONVERGED_SENTINEL: &str = "OUTER_COST_STALL_CONVERGED";

/// Sentinel used only when ARC has no finite current sample to hand to its
/// second-order convergence gate. A consecutive run of infeasible probes may
/// stop the trajectory at its best feasible checkpoint, but it can never
/// certify that checkpoint as converged: the bridge does not have a synchronized
/// Hessian there. The runner therefore always maps this sentinel to a
/// non-converged result.
pub(crate) const ARC_INFEASIBLE_STALL_SENTINEL: &str = "OUTER_ARC_INFEASIBLE_STALL";

/// Verdict produced by folding one accepted outer iterate into
/// [`CostStallGuard::observe`].
pub(crate) enum CostStallVerdict {
    /// The objective is still improving (or the no-improvement window has not
    /// yet filled). Keep descending.
    Continue,
    /// The objective has stopped improving over the window AND the projected
    /// gradient norm at the best iterate clears the outer gradient tolerance:
    /// a genuine stationary optimum on a (legitimately) flat REML surface.
    Converged,
    /// The objective has stopped improving over the window but the projected
    /// gradient norm at the best iterate is still above the outer gradient
    /// tolerance: a weakly-identified flat-valley FLOOR with residual
    /// non-stationarity. Halting here is correct (no further cost progress is
    /// available), but the iterate is NOT a stationary optimum and must be
    /// reported `converged = false`.
    FlatValleyStall { residual_grad_norm: f64 },
    /// The objective has stopped improving over the window but the projected
    /// gradient at the best iterate is FAR above the certified-stationary band
    /// (`> escape_threshold` = the score-relative stationarity bound times the
    /// escape margin, capped at `FLAT_VALLEY_STALL_GRAD_CEILING`; the carried
    /// `escape_threshold` is the value actually compared). A genuine flat-valley floor is, by
    /// definition, flat: its residual gradient is at most modestly above the
    /// convergence tolerance. A residual orders of magnitude above tolerance is
    /// NOT a flat valley — it is a *stuck* stall, the signature of an
    /// inconsistent objective/gradient pair (the inner PIRLS did not converge at
    /// this ρ, so the cached cost and the analytic gradient disagree and no
    /// line-search direction can make cost progress even though the surface is
    /// steep). Halting here would ship a near-unpenalized overfit (#1426). The
    /// guard instead resets the no-improvement window so the optimizer keeps
    /// descending (the inner solve runs to tighter tolerance at the next
    /// iterate, restoring a trustworthy gradient), for a bounded number of
    /// escapes before falling back to a `FlatValleyStall` halt so a genuinely
    /// pathological surface still terminates.
    StuckKeepDescending {
        residual_grad_norm: f64,
        /// The score-relative keep-descending trigger the residual actually
        /// exceeded (NOT the legacy fixed ceiling) — logged so the message can
        /// never contradict its own numbers.
        escape_threshold: f64,
    },
}

/// Number of consecutive accepted outer iterates with negligible relative
/// objective improvement required before the cost-stall guard declares
/// convergence. Matches the spirit of `opt`'s own `StallPolicy { window: 3 }`
/// but, crucially, is gated on the cost alone (not on gradient smallness),
/// which is the condition `opt` never checks in isolation.
pub(crate) const COST_STALL_WINDOW: usize = 6;
pub(crate) const ARC_COST_STALL_WINDOW: usize = 3;
pub(crate) const COST_STALL_REL_TOL_FLOOR: f64 = 1.0e-7;
pub(crate) const COST_STALL_PROJECTED_GRAD_FLOOR: f64 = 1.0e-3;

/// Absolute ceiling on the best-iterate projected gradient norm for a cost
/// stall to be classified as a genuine flat-valley FLOOR (#1426). A flat valley
/// is, by definition, flat: the REML surface has plateaued AND its residual
/// gradient is at most modestly above the outer convergence tolerance
/// (`COST_STALL_PROJECTED_GRAD_FLOOR = 1e-3`); the bound-pinned near-separable
/// cases (#1082/#1237) certify with projected gradients well under O(1). A cost
/// stall whose projected gradient is far above this ceiling is NOT a flat valley
/// — it is a *stuck* stall produced by an inconsistent objective/gradient pair
/// (the inner PIRLS hit its iteration cap at this ρ, so the cached cost and the
/// analytic gradient disagree and no line-search direction makes cost progress
/// even though the surface is steep). Halting on such a stall shipped a silent
/// near-unpenalized full-basis overfit on ~7% of gamma/log datasets at default
/// k (#1426). At/above this ceiling the guard refuses to halt and keeps the
/// optimizer descending (see [`CostStallVerdict::StuckKeepDescending`]).
///
/// Set well above the legitimate flat-valley residual band (≲ O(1)) so the
/// near-separable multinomial / RKHS-collapse halts are unaffected, but far
/// below the #1426 stuck residual (|g| ≈ 11) so the gamma/log overfit is caught.
pub(crate) const FLAT_VALLEY_STALL_GRAD_CEILING: f64 = 5.0;

/// Score-relative stationarity tolerance for certifying a cost-stalled flat
/// valley (#1426/#1477). A cost stall whose best-iterate projected gradient
/// clears `FLAT_VALLEY_CONVERGED_REL_GRAD · (1 + |score|)` is reported
/// `converged = true` even when it exceeds the tight absolute
/// `COST_STALL_PROJECTED_GRAD_FLOOR = 1e-3`. The REML/LAML score for a
/// non-trivial fit is `O(1e2)–O(1e3)`, so this is a `O(0.1)` absolute gradient —
/// the residual a weakly-identified (near-zero-curvature) ρ coordinate floors at,
/// which mgcv's score-relative convergence certifies. Set so the wide
/// degeneracy-prior null-space valleys (correct fits, EDF well below the basis)
/// certify, while the #1426 stuck overfit (`|g| ≈ 11`, also above
/// `FLAT_VALLEY_STALL_GRAD_CEILING`) does not.
pub(crate) const FLAT_VALLEY_CONVERGED_REL_GRAD: f64 = 1.0e-3;

/// Multiplicative margin above the certified-stationary band
/// (`score_relative_grad_bound`) below which a cost stall is treated as a genuine
/// flat-valley FLOOR and halted directly, and above which it is treated as a
/// NON-stationary stall that is granted a [`CostStallVerdict::StuckKeepDescending`]
/// escape (#509). A stall whose residual gradient is within this factor of the
/// band is "essentially at the band" — descending further is not worth burning an
/// escape — so it halts; a stall meaningfully above the band still has real
/// feasible descent and is allowed to climb out. Set to `1.5×`: a stall within
/// 50% of the certified band is "essentially flat" and halts directly (preserving
/// the #1477 weakly-identified flat-valley floors, which floor AT their band), while
/// a stall well clear of the band — the #509 monotone seed-park floors at |g| ≈ 2
/// on a score ≈ 599 (band ≈ 0.6 ⇒ trigger ≈ 0.9, so 2 > 0.9 keeps descending) and
/// the #1426 stuck overfit at |g| ≈ 11 — is granted escapes. Capped at
/// `FLAT_VALLEY_STALL_GRAD_CEILING` so a very large score never raises the trigger
/// above the legacy ceiling.
pub(crate) const FLAT_VALLEY_STALL_ESCAPE_MARGIN: f64 = 1.5;

/// Absolute cap on [`FLAT_VALLEY_CONVERGED_REL_GRAD`]'s score-relative bound.
/// Without it a fit with a very large `|score|` would license certifying a large
/// projected gradient; capping at `1.0` keeps the certified band a genuinely
/// small absolute gradient regardless of score, and stays well below the
/// `FLAT_VALLEY_STALL_GRAD_CEILING = 5.0` stuck-stall band so a stuck overfit on a
/// large-score fit is never certified.
pub(crate) const FLAT_VALLEY_CONVERGED_ABS_GRAD_CAP: f64 = 1.0;

/// Score-relative flat-valley stationarity bound used by both the in-loop ARC
/// cost-stall guard and the post-fit authoritative-gradient reconciliation.
///
/// Keeping the formula centralized prevents the shipped-fit certificate from
/// drifting away from the guard that originally decides whether a stalled ARC
/// trajectory is a genuine weakly-identified valley.
#[inline]
pub(crate) fn flat_valley_converged_grad_bound(score: f64) -> f64 {
    (FLAT_VALLEY_CONVERGED_REL_GRAD * (1.0 + score.abs())).min(FLAT_VALLEY_CONVERGED_ABS_GRAD_CAP)
}

/// Maximum number of consecutive [`CostStallVerdict::StuckKeepDescending`]
/// escapes the guard grants before it falls back to a [`CostStallVerdict::
/// FlatValleyStall`] halt (#1426). Each escape resets the no-improvement window
/// and lets the optimizer take more steps (with the inner solve running to
/// tighter tolerance), giving it room to climb out of a stuck stall toward the
/// well-penalized optimum. If the surface is genuinely pathological and the
/// escapes do not restore descent, the guard eventually halts so the outer loop
/// still terminates (reported non-converged) rather than grinding to `max_iter`.
pub(crate) const STUCK_STALL_MAX_ESCAPES: usize = 8;

/// Best iterate captured by a cost-stall convergence, handed from the bridge
/// (which is moved into `opt::Bfgs`) back to the seed-loop runner via the
/// guard's shared cell.
#[derive(Clone)]
pub(crate) struct CostStallExit {
    pub(crate) rho: Array1<f64>,
    pub(crate) value: f64,
    pub(crate) grad_norm: f64,
    /// Accepted outer iterates observed when the stall fired (for the runner's
    /// `OuterResult.iterations` field and logging).
    pub(crate) iterations: usize,
    /// Whether the best iterate is a genuine stationary optimum: `true` only
    /// when its projected gradient norm cleared the outer gradient tolerance
    /// (legitimately-flat REML surface). `false` for a flat-valley stall whose
    /// residual gradient remains above tolerance — the runner reports the
    /// rebuilt outer result as non-converged in that case.
    pub(crate) converged: bool,
    /// #2241 — the probe-noise-floor gradient bound σ̂/Δ measured at the stall
    /// (see [`CostStallGuard::probe_noise_grad_bound`]); `None` when the stall
    /// window carried too little evidence to measure it. Carried onto
    /// `OuterResult.flat_noise_grad_bound` so the final analytic certificate
    /// judges the re-measured gradient against the same flat band the guard
    /// certified in the loop.
    pub(crate) noise_grad_bound: Option<f64>,
}

/// Tracks the monotone best accepted-iterate REML objective and a
/// no-improvement streak, firing a gradient-independent convergence once the
/// objective has effectively stopped decreasing. See the `cost_stall` field
/// doc on [`OuterFirstOrderBridge`] for the full rationale (#1089).
pub(crate) struct CostStallGuard {
    /// Relative improvement floor: an accepted step counts as "no improvement"
    /// when `(best - cost) <= rel_tol * (1 + |best|)`. Derived from the outer
    /// convergence tolerance so it tracks the configured precision rather than
    /// a free-standing magic constant.
    rel_tol: f64,
    /// Consecutive accepted-step window with no improvement before declaring
    /// convergence.
    window: usize,
    /// Projected outer gradient-norm threshold that the best iterate must clear
    /// for a cost stall to count as a genuine stationary optimum. This is the
    /// SAME threshold the normal BFGS convergence path uses
    /// (`outer_gradient_tolerance(config).threshold(seed_cost, ‖g_0‖)`),
    /// evaluated once at seed. A cost stall above this threshold is a
    /// flat-valley stall, reported `converged = false`.
    grad_threshold: f64,
    best_value: f64,
    best_rho: Option<Array1<f64>>,
    best_grad_norm: f64,
    /// Reduced-Hessian verdict synchronized with `best_rho`. `Some(false)`
    /// means the incumbent is a certified strict saddle and therefore cannot
    /// justify an infeasible-neighbourhood stall.
    best_hessian_psd: Option<bool>,
    no_improve_streak: usize,
    /// Consecutive infeasible (non-finite cost) outer trials since the last
    /// finite observation. On a near-separable multinomial fit ARC repeatedly
    /// probes the unbounded λ→0 separating region where the inner softmax solve
    /// does not converge and returns `OuterEval::infeasible` (cost=∞). Those
    /// trials never produce a finite descent, so the normal `no_improve_streak`
    /// (which only counts finite trials) can never fill its window — the loop
    /// would otherwise grind to `max_iter`. A run of `window` consecutive
    /// infeasible trials, once a finite best is in hand, is the same "no further
    /// real progress" signal and trips the same halt at the best feasible
    /// iterate (#1082/#1237).
    infeasible_streak: usize,
    accepted_iters: usize,
    /// Number of [`CostStallVerdict::StuckKeepDescending`] escapes already
    /// granted on this seed (#1426). Capped at [`STUCK_STALL_MAX_ESCAPES`];
    /// once exhausted the guard halts a far-above-tolerance stall as an ordinary
    /// flat-valley floor so the loop still terminates.
    stuck_escapes: usize,
    /// #2241 — the most recent trusted accepted iterates `(ρ_i, f_i)` (finite
    /// cost, inner solve converged), newest last, capped at `window + 1`
    /// entries. This is the raw evidence for the probe-noise-floor flat
    /// certificate: during a stall the consecutive value differences measure
    /// the criterion's own evaluation-noise scale, and the consecutive ρ
    /// distances measure the radius the search actually probed.
    recent: std::collections::VecDeque<(Array1<f64>, f64)>,
    /// Shared publication slot read by the seed-loop runner after
    /// `optimizer.run()` returns the sentinel error.
    exit: Arc<Mutex<Option<CostStallExit>>>,
}

impl CostStallGuard {
    pub(crate) fn new(
        rel_tol: f64,
        window: usize,
        grad_threshold: f64,
        exit: Arc<Mutex<Option<CostStallExit>>>,
    ) -> Self {
        Self {
            rel_tol,
            window,
            grad_threshold,
            best_value: f64::INFINITY,
            best_rho: None,
            best_grad_norm: f64::INFINITY,
            best_hessian_psd: None,
            no_improve_streak: 0,
            infeasible_streak: 0,
            accepted_iters: 0,
            stuck_escapes: 0,
            recent: std::collections::VecDeque::new(),
            exit,
        }
    }

    /// Record one trusted accepted iterate into the #2241 noise-evidence
    /// buffer, keeping only the latest `window + 1` (⇒ `window` consecutive
    /// differences).
    fn record_recent(&mut self, rho: &Array1<f64>, value: f64) {
        self.recent.push_back((rho.clone(), value));
        while self.recent.len() > self.window + 1 {
            self.recent.pop_front();
        }
    }

    /// #2241 — probe-noise-floor gradient bound at a halted stall.
    ///
    /// Derivation. Over the stalled window the accepted iterates
    /// `(ρ_i, f_i)` satisfy: (i) the first-order model predicts
    /// `|f(ρ+d) − f(ρ)| ≤ ‖g‖·‖d‖ + o(‖d‖)` for moves of the size the search
    /// actually takes, and (ii) because the window is a stall (trend ≈ 0 by
    /// the `rel_tol` test), the consecutive differences `|f_i − f_{i−1}|` are
    /// dominated by the criterion's own evaluation noise — inner-solve
    /// truncation, reassembly order, quadrature — not by descent. So
    ///   σ̂ = median_i |f_i − f_{i−1}|   (robust noise-floor estimate),
    ///   Δ  = max_i ‖ρ_i − ρ_{i−1}‖₂    (radius the steps actually probed),
    /// and if `‖g_best‖ · Δ ≤ σ̂` then NO probe within the radius the line
    /// search is exploring can change the criterion by more than its own
    /// measurement noise: the surface is flat relative to its own noise scale
    /// and the best iterate is stationary at the resolution the criterion can
    /// be evaluated. Equivalently the gradient is certified below `σ̂/Δ`.
    ///
    /// Guards: σ̂ is floored at the roundoff scale `ε·(1+|f_best|)` (a
    /// byte-identical window cannot license an infinite bound of 0/Δ — it
    /// licenses exactly the roundoff resolution), and the returned bound is
    /// capped at [`FLAT_VALLEY_CONVERGED_ABS_GRAD_CAP`] so collapsed step
    /// sizes (Δ → 0 inflating σ̂/Δ) can never certify a genuinely steep point:
    /// the #1426 stuck stall (|g| ≈ 11) and the #509 seed-park (|g| ≈ 2) both
    /// sit above the cap and remain uncertifiable by this route. Returns
    /// `None` when fewer than three consecutive differences exist or the probe
    /// radius is degenerate (zero / non-finite).
    fn probe_noise_grad_bound(&self) -> Option<f64> {
        if self.recent.len() < 4 {
            return None;
        }
        let mut value_diffs: Vec<f64> = Vec::with_capacity(self.recent.len() - 1);
        let mut probe_radius = 0.0_f64;
        for (prev, next) in self.recent.iter().zip(self.recent.iter().skip(1)) {
            value_diffs.push((next.1 - prev.1).abs());
            let step = prev
                .0
                .iter()
                .zip(next.0.iter())
                .map(|(a, b)| (a - b) * (a - b))
                .sum::<f64>()
                .sqrt();
            probe_radius = probe_radius.max(step);
        }
        if !probe_radius.is_finite() || probe_radius <= 0.0 {
            return None;
        }
        value_diffs.sort_by(|a, b| a.total_cmp(b));
        let mid = value_diffs.len() / 2;
        let median = if value_diffs.len() % 2 == 1 {
            value_diffs[mid]
        } else {
            0.5 * (value_diffs[mid - 1] + value_diffs[mid])
        };
        if !median.is_finite() {
            return None;
        }
        let best_scale = if self.best_value.is_finite() {
            self.best_value.abs()
        } else {
            0.0
        };
        let noise_floor = median.max(f64::EPSILON * (1.0 + best_scale));
        Some((noise_floor / probe_radius).min(FLAT_VALLEY_CONVERGED_ABS_GRAD_CAP))
    }

    fn certified_grad_bound(&self) -> f64 {
        let score_bound = flat_valley_converged_grad_bound(self.best_value);
        let noise_bound = self.probe_noise_grad_bound().unwrap_or(0.0);
        self.grad_threshold.max(score_bound).max(noise_bound)
    }

    /// Register a precomputed feasible seed that the optimizer consumes from
    /// its internal cache instead of routing through the bridge. ARC's
    /// `with_initial_sample` path does exactly that: the first finite
    /// `(cost, gradient, Hessian)` is already known to the runner, so
    /// `eval_hessian` is not called at the seed. Without this hook the
    /// infeasible-trial stall path has no finite best iterate to halt back to
    /// when the next few ARC probes run into the λ→0 separating region.
    pub(crate) fn observe_seed(&mut self, rho: &Array1<f64>, value: f64, grad_norm: f64) {
        self.observe_seed_with_curvature(rho, value, grad_norm, None);
    }

    pub(crate) fn observe_second_order_seed(
        &mut self,
        rho: &Array1<f64>,
        value: f64,
        grad_norm: f64,
        hessian_psd: Option<bool>,
    ) {
        self.observe_seed_with_curvature(rho, value, grad_norm, hessian_psd);
    }

    fn observe_seed_with_curvature(
        &mut self,
        rho: &Array1<f64>,
        value: f64,
        grad_norm: f64,
        hessian_psd: Option<bool>,
    ) {
        if !value.is_finite() {
            return;
        }
        self.best_value = value;
        self.best_rho = Some(rho.clone());
        self.best_grad_norm = grad_norm;
        self.best_hessian_psd = hessian_psd;
        self.no_improve_streak = 0;
        self.infeasible_streak = 0;
        self.accepted_iters = self.accepted_iters.saturating_add(1);
        self.record_recent(rho, value);
        // Seed the shared exit cell so the budget-exhaustion path always has a
        // feasible best to fall back to, even if no later step improves (#1371).
        self.publish_best_so_far();
    }

    /// Fold one accepted-iterate `(ρ, cost, ‖g‖)` into the guard. Returns a
    /// [`CostStallVerdict`]: `Continue` while the score is still improving,
    /// `Converged` when the score has stalled AND the projected gradient norm
    /// at the best iterate clears the outer gradient tolerance (a genuine
    /// stationary optimum on a flat REML surface), or `FlatValleyStall` when
    /// the score has stalled but the residual gradient remains above tolerance
    /// (a weakly-identified flat valley that is NOT stationary). Either stalled
    /// verdict publishes the best iterate to the shared cell, tagged with its
    /// `converged` status.
    ///
    /// `inner_converged` is the inner-PIRLS convergence flag for THIS iterate's
    /// solve (read from the inner-progress feedback AFTER the outer eval ran). It
    /// is the load-bearing #1426 input: at the under-penalized (λ→0) ridge the
    /// inner PIRLS hits its iteration cap, so the cost it reports is a half-fit
    /// artifact that is SMALLER than the honest REML criterion at the
    /// well-penalized optimum, and the analytic gradient it pairs with is
    /// inconsistent. If such an iterate is allowed to become the guard's
    /// best-so-far it permanently anchors the best to the overfit basin — an
    /// uncapped, honest re-evaluation at a well-penalized ρ then carries a
    /// *higher* (correct) cost and can never displace it, so the loop ships the
    /// near-full-basis overfit. A non-converged inner solve is therefore NEVER
    /// recorded as best and NEVER counted toward a stall: its cost/gradient are
    /// untrustworthy, so the guard treats it as forced descent (streak reset) and
    /// keeps the optimizer moving until an honest, converged iterate lands.
    pub(crate) fn observe(
        &mut self,
        rho: &Array1<f64>,
        value: f64,
        grad_norm: f64,
        inner_converged: bool,
    ) -> CostStallVerdict {
        self.observe_with_curvature(rho, value, grad_norm, inner_converged, None)
    }

    pub(crate) fn observe_second_order(
        &mut self,
        rho: &Array1<f64>,
        value: f64,
        grad_norm: f64,
        inner_converged: bool,
        hessian_psd: Option<bool>,
    ) -> CostStallVerdict {
        self.observe_with_curvature(rho, value, grad_norm, inner_converged, hessian_psd)
    }

    fn observe_with_curvature(
        &mut self,
        rho: &Array1<f64>,
        value: f64,
        grad_norm: f64,
        inner_converged: bool,
        hessian_psd: Option<bool>,
    ) -> CostStallVerdict {
        if !value.is_finite() {
            // A non-finite accepted objective is the inner-solver's problem,
            // not a stall; reset so a later real descent is not falsely
            // credited as a no-improvement step. The dedicated infeasible-trial
            // bookkeeping lives in `observe_infeasible`; this finite-path entry
            // is left untouched for non-finite values.
            self.no_improve_streak = 0;
            return CostStallVerdict::Continue;
        }
        if !inner_converged {
            // #1426: the inner PIRLS hit its iteration cap at this ρ, so `value`
            // is a half-converged artifact and `grad_norm` is inconsistent with
            // it. Do NOT update best-so-far (a corrupted low cost at the λ→0
            // ridge would anchor the best to the overfit forever) and do NOT
            // count this toward a stall — reset the no-improvement streak so the
            // optimizer keeps stepping until a fully-converged inner solve lands
            // a trustworthy (cost, gradient) pair. The bridge separately uncaps
            // the inner solve when it detects a stuck stall, so subsequent solves
            // do converge and the best-so-far tracks an honest iterate.
            self.infeasible_streak = 0;
            self.no_improve_streak = 0;
            return CostStallVerdict::Continue;
        }
        // A finite trial means the inner solve produced a real cost: the
        // separating-region infeasible run is broken, so clear its streak.
        self.infeasible_streak = 0;
        self.accepted_iters = self.accepted_iters.saturating_add(1);
        self.record_recent(rho, value);
        let improvement = self.best_value - value;
        let floor = self.rel_tol * (1.0 + self.best_value.abs());
        if value < self.best_value {
            self.best_value = value;
            self.best_rho = Some(rho.clone());
            self.best_grad_norm = grad_norm;
            self.best_hessian_psd = hessian_psd;
            // Keep the shared exit cell tracking the best feasible iterate so the
            // ARC budget-exhaustion path can recover it instead of the optimizer's
            // last (possibly degenerate-corner) iterate (#1371).
            self.publish_best_so_far();
        }
        // KKT-stationary-at-bound (#1082/#1237). On a near-separable multinomial
        // the outer REML criterion keeps decreasing as λ→0, so several log-λ
        // directions slam to the lower box bound: the BOUND-PROJECTED gradient
        // is already below the outer tolerance (the iterate is KKT-stationary),
        // yet the raw cost keeps dropping by more than `floor` along those
        // bound-pinned directions — so a pure cost-improvement test resets the
        // no-improvement streak forever and the loop never certifies. Once the
        // projected gradient clears the stationarity threshold there is no
        // FEASIBLE descent left; further raw progress is bound-pinned drift, not
        // optimization. Treat such a trial as no-improvement so the window can
        // fill and the guard halts at the (stationary) best feasible iterate.
        // `opt::Arc`'s own gradient-tolerance check never trips here because it
        // tests the RAW gradient, which points out of the box forever.
        let kkt_stationary_at_bound = grad_norm.is_finite() && grad_norm <= self.grad_threshold;
        if improvement <= floor || kkt_stationary_at_bound {
            self.no_improve_streak = self.no_improve_streak.saturating_add(1);
        } else {
            self.no_improve_streak = 0;
            // A genuine super-floor improvement means the last stuck-stall
            // escape (if any) restored real descent, so replenish the escape
            // budget: [`STUCK_STALL_MAX_ESCAPES`] then bounds CONSECUTIVE
            // fruitless escapes — the documented pathological-surface case —
            // rather than the total number of productive basin hops a
            // multi-shelf descent needs (measured on the #2253 repro: the
            // 7th escape bought a 36-point objective drop and the lifetime
            // cap still killed the seed). Termination is preserved because
            // each replenishment requires an improvement strictly above the
            // relative floor and the criterion is bounded below, so only
            // finitely many resets can occur.
            self.stuck_escapes = 0;
        }
        if self.no_improve_streak < self.window {
            return CostStallVerdict::Continue;
        }
        // #2357: the finite window can fill while the recorded BEST is a
        // certified strict saddle — the guard tracks `best_hessian_psd` but the
        // finite-stall path never consulted it, so the stall halted at a
        // low-cost pass-through iterate whose curvature had not settled
        // (cold periodic-te() repro: window fills on three oscillating evals
        // near ρ₂≈10.7, halts to an earlier ρ₂≈5.4 point with hessian_psd=NO,
        // and the analytic certificate then refuses with INDEFINITE CURVATURE
        // AT INTERIOR OPTIMUM — while re-running warm from that very
        // checkpoint converges in a few more iterations). A strict saddle has
        // a certified local escape direction, so refuse the stall and return
        // control to cubic regularization exactly as the infeasible-run path
        // already does; the escape budget bounds consecutive fruitless
        // refusals so a surface that genuinely floors at a saddle still halts
        // (converged=false) after the budget instead of looping.
        if self.best_hessian_psd == Some(false) && self.stuck_escapes < STUCK_STALL_MAX_ESCAPES {
            self.stuck_escapes = self.stuck_escapes.saturating_add(1);
            self.no_improve_streak = 0;
            log::warn!(
                "[OUTER] ARC cost-stall window filled at a strict-saddle incumbent \
                 (hessian_psd=NO at best-so-far, value={:.6e}): refusing to certify the \
                 saddle and returning control to cubic regularization to exploit the \
                 negative curvature (escape {}/{}).",
                self.best_value,
                self.stuck_escapes,
                STUCK_STALL_MAX_ESCAPES,
            );
            return CostStallVerdict::Continue;
        }
        self.publish_stall(rho, value, grad_norm)
    }

    /// Fold one INFEASIBLE outer trial (non-finite cost — typically a near-λ=0
    /// separating point whose inner softmax solve did not converge) into the
    /// guard. ARC keeps proposing these on a near-separable multinomial fit and
    /// they never reach `observe` with a finite value, so without this the
    /// no-improvement window can never fill and the outer loop grinds to
    /// `max_iter` (#1082/#1237). A run of `window` consecutive infeasible trials
    /// — provided a finite best is already in hand to halt back to — is the same
    /// "no further real progress" signal and trips the same halt.
    ///
    /// Returns `Continue` until the infeasible streak fills the window; never
    /// fires before any finite iterate has been recorded (there would be nothing
    /// to halt back to).
    pub(crate) fn observe_infeasible(&mut self, rho: &Array1<f64>) -> CostStallVerdict {
        if self.best_rho.is_none() || !self.best_value.is_finite() {
            // No feasible iterate recorded yet: an infeasible run this early is
            // the inner solver's startup problem, not a converged stall. Keep
            // descending so the optimizer can find its first feasible point.
            return CostStallVerdict::Continue;
        }
        self.infeasible_streak = self.infeasible_streak.saturating_add(1);
        if self.infeasible_streak < self.window {
            return CostStallVerdict::Continue;
        }
        if self.best_hessian_psd == Some(false) {
            // A strict saddle has a certified local escape direction. An
            // infeasible run only says ARC's current cubic step crossed the
            // profiled objective's domain wall; it cannot turn that saddle
            // into a terminal checkpoint. Clear the local run so ARC can raise
            // sigma, shrink the step, and exploit the known negative curvature.
            self.infeasible_streak = 0;
            self.no_improve_streak = 0;
            log::warn!(
                "[OUTER] ARC infeasible-probe run reached a strict-saddle incumbent; \
                 refusing the stall and returning control to cubic regularization"
            );
            return CostStallVerdict::Continue;
        }
        // Halt back to the best feasible iterate. Its projected gradient decides
        // converged-vs-flat-valley exactly as the finite stall path does.
        self.publish_stall(rho, self.best_value, self.best_grad_norm)
    }

    /// Publish the current finite probe as a constrained-stationary point.
    ///
    /// This is deliberately narrower than a generic cost stall: ARC can evaluate a
    /// bound-pinned separation probe whose raw gradient still points out of the
    /// feasible box. That point already satisfies the constrained KKT condition,
    /// but it may not be the guard's lowest raw-cost observation, so the generic
    /// best-so-far publication can resurrect an older non-stationary iterate and
    /// report `converged=false`. In that separation case the current feasible
    /// probe is the certificate-bearing point and should be returned directly.
    ///
    /// `grad_norm` MUST be the bound-PROJECTED gradient norm at the probe, not a
    /// presumed zero: the constrained-KKT certificate holds only when the pull on
    /// the railed axes is the *whole* residual. The projected norm retains any
    /// INTERIOR (feasible-descent) component, so `publish_stall` certifies
    /// `converged` iff the probe is genuinely stationary in the feasible subspace
    /// (#1426 — a flat-valley overfit that merely rails a couple of axes is
    /// reported NON-converged rather than certified behind a fake zero).
    pub(crate) fn observe_constrained_stationary(
        &mut self,
        rho: &Array1<f64>,
        value: f64,
        grad_norm: f64,
        inner_converged: bool,
        hessian_psd: Option<bool>,
    ) -> CostStallVerdict {
        if !value.is_finite() {
            return CostStallVerdict::Continue;
        }
        if !inner_converged {
            // #1426: a separation-stationary probe whose inner PIRLS did not
            // converge carries an untrustworthy (cost, gradient) pair. Adopting
            // it as the certificate-bearing optimum would ship a half-fit. Route
            // it through the ordinary finite-path observer, which (with
            // `inner_converged = false`) refuses to record it as best and keeps
            // the optimizer descending toward an honest, converged iterate.
            return self.observe_with_curvature(
                rho,
                value,
                grad_norm,
                inner_converged,
                hessian_psd,
            );
        }
        // A lower-bound separation probe is the certificate-bearing optimum ONLY
        // when it does not REGRESS the best feasible iterate already seen. In the
        // genuine near-separable case (#1082/#1237) the criterion decreases
        // monotonically toward the λ→0 bound, so the probe carries the lowest
        // cost and this guard is a no-op. But the SAME local test
        // (`lower_bound_outward_active_count`) also fires at an over-smoothing
        // collapse corner of a multi-penalty RKHS smooth (duchon/matern, #1355):
        // there a couple of operator penalties rail at the λ→0 LOWER bound (so
        // they look "separation-stationary") while OTHER penalties rail at the
        // λ→∞ UPPER bound and shrink the fit to a bare constant. Such a corner is
        // a spurious local KKT point whose REML cost is far WORSE than the
        // interior optimum the optimizer already passed through (commonly the
        // grid-prepass seed). Unconditionally adopting it discards that better
        // iterate and publishes a degenerate EDF≈1 constant fit. Only adopt the
        // probe when it does not regress the incumbent best.
        let regresses = self.best_value.is_finite()
            && value > self.best_value + self.rel_tol * (1.0 + self.best_value.abs());
        if regresses {
            // Spurious corner: keep the better incumbent. Fold the probe in as an
            // ordinary (non-improving) observation so `best_rho`/`best_grad_norm`
            // are preserved and the stall logic halts back to that incumbent
            // rather than to this corner.
            return self.observe_with_curvature(
                rho,
                value,
                grad_norm,
                inner_converged,
                hessian_psd,
            );
        }
        self.infeasible_streak = 0;
        self.accepted_iters = self.accepted_iters.saturating_add(1);
        self.best_value = value;
        self.best_rho = Some(rho.clone());
        self.best_grad_norm = grad_norm;
        self.best_hessian_psd = hessian_psd;
        self.no_improve_streak = self.window;
        self.publish_stall(rho, value, grad_norm)
    }

    /// Publish the best iterate to the shared exit cell and decide the stall
    /// verdict. Shared by the finite-stall and infeasible-stall paths.
    fn publish_stall(&mut self, rho: &Array1<f64>, value: f64, grad_norm: f64) -> CostStallVerdict {
        // Publish the best iterate. Prefer the recorded best; fall back to the
        // current point if (pathologically) none was stored.
        let best_rho = self.best_rho.clone().unwrap_or_else(|| rho.clone());
        let best_value = if self.best_value.is_finite() {
            self.best_value
        } else {
            value
        };
        let best_grad_norm = if self.best_grad_norm.is_finite() {
            self.best_grad_norm
        } else {
            grad_norm
        };
        // Convergence is STATIONARITY, measured RELATIVE TO THE SCORE SCALE.
        // A cost stall counts as a converged optimum when the projected gradient
        // at the best iterate clears EITHER (a) the absolute outer gradient
        // tolerance the genuine BFGS path checks, OR (b) a score-relative
        // stationarity bound `FLAT_VALLEY_CONVERGED_REL_GRAD · (1 + |score|)`.
        //
        // Bound (b) is the mgcv-aligned half (#1426/#1477). On a WEAKLY-IDENTIFIED
        // ρ coordinate — e.g. the double-penalty null-space log-λ of a smooth whose
        // null space is only weakly supported, under the wide degeneracy prior — the
        // REML surface is near-flat: the Hessian in that direction is ≈ 0, so the
        // projected gradient cannot be driven down to the tight ABSOLUTE
        // `COST_STALL_PROJECTED_GRAD_FLOOR = 1e-3` no matter how many outer steps
        // run; it floors at the valley's own residual (`O(0.1)` on a score of
        // `O(1e3)`). That is a genuine stationary optimum — mgcv certifies it (its
        // convergence is the gradient relative to the score, not an absolute 1e-3) —
        // and the cost-stall window has already proven the surface flattened. The
        // OLD absolute-only test reported these correct fits `converged = false`,
        // flooding the verdict with false alarms on exactly the fits the principled
        // wide-prior (no null-space over-shrink) produces. The score-relative bound
        // certifies them while a GENUINELY non-stationary residual — the #1426 stuck
        // overfit (|g| ≈ 11 on a score `O(1e3)`, i.e. above BOTH this bound and the
        // separate `FLAT_VALLEY_STALL_GRAD_CEILING`) — is still rejected and routed
        // to `StuckKeepDescending`, so no near-full-basis overfit is ever certified.
        let score_relative_grad_bound = flat_valley_converged_grad_bound(best_value);
        // #2241 — probe-noise-floor certificate: the stall window's own value
        // scatter and probed radius bound the gradient at which further probes
        // become indistinguishable from evaluation noise (derivation on
        // `probe_noise_grad_bound`). It composes with the score-relative band
        // as a second sufficient condition; both are capped well below the
        // stuck-stall regimes so neither can certify a genuinely steep point.
        let noise_grad_bound = self.probe_noise_grad_bound();
        let converged = best_grad_norm.is_finite()
            && (best_grad_norm <= self.grad_threshold
                || best_grad_norm <= score_relative_grad_bound
                || noise_grad_bound.is_some_and(|bound| best_grad_norm <= bound));
        if converged {
            if let Ok(mut slot) = self.exit.lock() {
                *slot = Some(CostStallExit {
                    rho: best_rho,
                    value: best_value,
                    grad_norm: best_grad_norm,
                    iterations: self.accepted_iters,
                    converged,
                    noise_grad_bound,
                });
            }
            return CostStallVerdict::Converged;
        }
        // Distinguish a genuine flat-valley FLOOR (residual at its irreducible
        // band — the surface really HAS flattened, so no further descent is
        // available) from a NON-stationary stall whose residual gradient still
        // points to real feasible descent. The window filled, but a gradient
        // meaningfully above the certified-stationary band proves there IS a
        // descent direction left — the stall is an artifact of small per-step
        // cost progress, not of a flat surface. Two regimes produce such a stall:
        //
        //   * #1426 — the inner PIRLS hit its iteration cap at the under-penalized
        //     (λ→0) ridge, so the cached cost is a half-fit artifact and the
        //     analytic gradient (|g| ≈ 11) is inconsistent with it.
        //   * #509 — a shape-constrained (box-reparam β=Tγ) smooth whose inequality
        //     constraint is NON-binding: the constrained active-set inner solve is
        //     near-smooth but the cumulative-sum coordinate change makes the cost
        //     improve by less than the relative floor over the window near the
        //     integer seed, even though the projected gradient (|g| ≈ 2) still
        //     descends strongly toward the well-penalized REML optimum (verified:
        //     λ_wiggle 20 → 83, EDF 8.4 → 6.9, score −599 → −618, |g| → 1e-11).
        //
        // Both are cured the same way: refuse to halt and grant an escape — reset
        // the no-improvement window so the optimizer keeps descending (the inner
        // solve runs to tighter tolerance at the next iterate, restoring a
        // trustworthy gradient), for a bounded number of escapes before falling
        // back to a halt. The OLD gate keyed the escape on a fixed absolute
        // `FLAT_VALLEY_STALL_GRAD_CEILING = 5.0`, which let any stall in the
        // (certified-band, 5.0] residual band halt as a "flat valley" even though
        // its gradient still descended — silently parking the #509 monotone fit at
        // its seed (|g| ≈ 2 < 5.0). Keying the escape on the SCORE-RELATIVE
        // certified-stationary band instead (the same band that certifies
        // `converged` above, with a modest multiplicative margin so a stall sitting
        // essentially AT the band still halts directly rather than burning escapes)
        // makes the distinction scale-correct: a genuinely flat valley floors at
        // the band and halts after the escape budget, while a non-stationary stall
        // descends to true stationarity. The `FLAT_VALLEY_STALL_GRAD_CEILING` caps
        // the trigger so a very large score can never raise it above the legacy
        // ceiling — the #1426 stuck regime (|g| ≈ 11) is always granted escapes.
        let keep_descending_threshold = (FLAT_VALLEY_STALL_ESCAPE_MARGIN
            * score_relative_grad_bound)
            .min(FLAT_VALLEY_STALL_GRAD_CEILING);
        let non_stationary_stall =
            best_grad_norm.is_finite() && best_grad_norm > keep_descending_threshold;
        if non_stationary_stall && self.stuck_escapes < STUCK_STALL_MAX_ESCAPES {
            self.stuck_escapes = self.stuck_escapes.saturating_add(1);
            // Reset BOTH no-progress streaks: the optimizer should be allowed a
            // fresh window of accepted/infeasible steps to climb out of the
            // stuck state. Do NOT publish a halt — leave the shared exit cell
            // tracking the running best (`publish_best_so_far` already keeps it
            // current) so an eventual budget-exhaustion still recovers a sane
            // iterate.
            self.no_improve_streak = 0;
            self.infeasible_streak = 0;
            return CostStallVerdict::StuckKeepDescending {
                residual_grad_norm: best_grad_norm,
                escape_threshold: keep_descending_threshold,
            };
        }
        if let Ok(mut slot) = self.exit.lock() {
            *slot = Some(CostStallExit {
                rho: best_rho,
                value: best_value,
                grad_norm: best_grad_norm,
                iterations: self.accepted_iters,
                converged,
                noise_grad_bound,
            });
        }
        CostStallVerdict::FlatValleyStall {
            residual_grad_norm: best_grad_norm,
        }
    }

    /// Publish the running best feasible iterate to the shared exit cell WITHOUT
    /// halting the optimizer (the verdict is discarded). Called on every accepted
    /// step that improves the best so the cell continuously tracks "best feasible
    /// iterate seen so far".
    ///
    /// This is the load-bearing hook for the ARC budget-exhaustion path
    /// (#1371). When ARC hits `max_iter` the seed-loop runner gets back only the
    /// optimizer's LAST iterate, which on a flat REML valley can be a degenerate
    /// box corner the trajectory wandered to (e.g. `ρ_nullspace → +∞` on a
    /// double-penalty smooth, which annihilates a genuine null-space linear
    /// trend and collapses the fit to a flat constant). Whenever this cell is
    /// populated, the runner returns the better of {last iterate, published best}
    /// — never returning an iterate whose REML objective is worse than one the
    /// optimizer already evaluated. Same spirit as the `observe_constrained_
    /// stationary` "do not adopt a corner that regresses the incumbent best"
    /// guard (#1355); here it covers the budget-exhaustion exit rather than the
    /// separation-probe exit.
    fn publish_best_so_far(&mut self) {
        let Some(best_rho) = self.best_rho.clone() else {
            return;
        };
        if !self.best_value.is_finite() {
            return;
        }
        // A running best-so-far snapshot is NEVER a convergence certificate: only
        // a genuine cost stall (`publish_stall`, whose window has proven the
        // surface flattened) or the second-order-deferred ARC gate may stamp
        // `converged = true`. Stamping it here on a merely-small projected
        // gradient certified a STILL-DESCENDING iterate — e.g. a bound-pinned
        // KKT-stationary point whose projected residual is transiently ≤ the
        // threshold mid-descent — and, because this publish returns
        // `CostStallVerdict::Continue`, the outer `observe_cost_stall` match never
        // runs `defer_finite_second_order_stall` to revoke it, so the false
        // convergence label survived to the exit cell (#2299 arc_bridge). The
        // snapshot exists ONLY so the budget-exhaustion path always has a feasible
        // best to halt back to (#1371/#2241); its convergence verdict is deferred
        // to ARC / the genuine-stall paths, so it is published `converged = false`.
        if let Ok(mut slot) = self.exit.lock() {
            *slot = Some(CostStallExit {
                rho: best_rho,
                value: self.best_value,
                grad_norm: self.best_grad_norm,
                iterations: self.accepted_iters,
                converged: false,
                // Not a halted stall: no noise-floor measurement is claimed
                // for a running best-so-far snapshot (#2241).
                noise_grad_bound: None,
            });
        }
    }

    /// A finite analytic second-order sample must reach ARC's synchronized
    /// projected-gradient + reduced-Hessian convergence gate. The generic
    /// cost-stall guard may still track the best feasible iterate, but it cannot
    /// halt the finite path from a first-order verdict. Reset only the finite
    /// stall window and revoke the provisional published convergence label;
    /// retain the best/recent evidence for budget-exhaustion recovery.
    fn defer_finite_second_order_stall(&mut self) {
        self.no_improve_streak = 0;
        if let Ok(mut slot) = self.exit.lock()
            && let Some(exit) = slot.as_mut()
        {
            exit.converged = false;
            exit.noise_grad_bound = None;
        }
    }

    /// An infeasible ARC probe carries no Hessian. Even when the generic guard's
    /// projected-gradient band classifies the stored best as stationary, the
    /// early-stop result is only a checkpoint until the mandatory outer
    /// certificate re-evaluates it with exact curvature.
    fn revoke_published_convergence(&mut self) {
        if let Ok(mut slot) = self.exit.lock()
            && let Some(exit) = slot.as_mut()
        {
            exit.converged = false;
        }
    }
}

#[derive(Clone)]
pub(crate) struct ValueProbeCacheEntry {
    rho: Array1<f64>,
    outcome: CachedValueProbeOutcome,
}

#[derive(Clone)]
pub(crate) enum CachedValueProbeOutcome {
    Cost(f64),
    Recoverable(String),
    Fatal(String),
}

pub(crate) fn trial_rho_distance(reference: Option<&Array1<f64>>, trial: &Array1<f64>) -> f64 {
    let Some(reference) = reference else {
        return f64::NAN;
    };
    if reference.len() != trial.len() {
        return f64::NAN;
    }
    reference
        .iter()
        .zip(trial.iter())
        .map(|(a, b)| {
            let d = b - a;
            d * d
        })
        .sum::<f64>()
        .sqrt()
}

pub(crate) fn same_outer_point(a: &Array1<f64>, b: &Array1<f64>) -> bool {
    a.len() == b.len()
        && a.iter()
            .zip(b.iter())
            .all(|(left, right)| left.to_bits() == right.to_bits())
}

pub(crate) fn cached_value_probe_result(
    outcome: &CachedValueProbeOutcome,
) -> Result<f64, ObjectiveEvalError> {
    match outcome {
        CachedValueProbeOutcome::Cost(cost) => Ok(*cost),
        CachedValueProbeOutcome::Recoverable(message) => {
            Err(ObjectiveEvalError::recoverable(message.clone()))
        }
        CachedValueProbeOutcome::Fatal(message) => Err(ObjectiveEvalError::Fatal {
            message: message.clone(),
        }),
    }
}

pub(crate) fn cache_value_probe_result(
    result: &Result<f64, ObjectiveEvalError>,
) -> CachedValueProbeOutcome {
    match result {
        Ok(cost) => CachedValueProbeOutcome::Cost(*cost),
        Err(ObjectiveEvalError::Recoverable { message }) => {
            CachedValueProbeOutcome::Recoverable(message.clone())
        }
        Err(ObjectiveEvalError::Fatal { message }) => {
            CachedValueProbeOutcome::Fatal(message.clone())
        }
    }
}

pub(crate) fn value_probe_outcome_label(outcome: &CachedValueProbeOutcome) -> &'static str {
    match outcome {
        CachedValueProbeOutcome::Cost(_) => "cost",
        CachedValueProbeOutcome::Recoverable(_) => "recoverable",
        CachedValueProbeOutcome::Fatal(_) => "fatal",
    }
}

pub(crate) fn value_probe_reject_outcome(outcome: &CachedValueProbeOutcome) -> bool {
    match outcome {
        CachedValueProbeOutcome::Cost(cost) => *cost >= VALUE_PROBE_REJECT_COST_FLOOR,
        CachedValueProbeOutcome::Recoverable(_) | CachedValueProbeOutcome::Fatal(_) => true,
    }
}

pub(crate) fn remember_value_probe(
    cache: &mut Vec<ValueProbeCacheEntry>,
    rho: &Array1<f64>,
    outcome: CachedValueProbeOutcome,
) {
    if let Some(entry) = cache
        .iter_mut()
        .find(|entry| same_outer_point(&entry.rho, rho))
    {
        entry.outcome = outcome;
        return;
    }
    if cache.len() == VALUE_PROBE_CACHE_CAPACITY {
        cache.remove(0);
    }
    cache.push(ValueProbeCacheEntry {
        rho: rho.clone(),
        outcome,
    });
}

impl ZerothOrderObjective for OuterFirstOrderBridge<'_> {
    fn eval_cost(&mut self, x: &Array1<f64>) -> Result<f64, ObjectiveEvalError> {
        // Per-axis line-search step caps now live natively in opt::Bfgs
        // (`with_axis_step_caps`), which shortens the BFGS direction before
        // line search instead of poisoning the Wolfe bracket with a
        // sentinel cost. This entry point can therefore stay honest: any
        // call that lands here is a real line-search probe, not a too-far
        // attempt the bridge needs to swat away.
        //
        // Uncap the inner solve for the line-search cost probe (see the field
        // doc on `outer_inner_cap`): the deciding cost MUST be the true
        // converged-inner objective the analytic gradient differentiates, not
        // the scheduled gradient-path cap which can stop a trial-ρ inner solve
        // short of its fixed point and report a spurious `∞`. `eval_grad`
        // restores the scheduled cap on the next call.
        if let Some(feedback) = self.outer_inner_cap.as_ref() {
            feedback
                .cap
                .store(SEED_SCREENING_UNCAPPED, Ordering::Relaxed);
        }
        self.layout
            .validate_point_len(x, "outer eval_cost failed")?;
        let trial_rho_distance = trial_rho_distance(self.last_value_grad_rho.as_ref(), x);
        let stage_start = std::time::Instant::now();
        if let Some(entry) = self
            .value_probe_cache
            .iter()
            .find(|entry| same_outer_point(&entry.rho, x))
        {
            let outcome_label = value_probe_outcome_label(&entry.outcome);
            log::info!(
                "[STAGE] outer eval start order=Value dim={} trial_rho_distance={:.3e} (first-order bridge, iter={}, cached=true)",
                x.len(),
                trial_rho_distance,
                self.iter_count
            );
            match &entry.outcome {
                CachedValueProbeOutcome::Cost(cost) => log::info!(
                    "[STAGE] outer eval end order=Value elapsed={:.3}s cost={:.6e} trial_rho_distance={:.3e} (first-order bridge, iter={}, cached=true)",
                    stage_start.elapsed().as_secs_f64(),
                    cost,
                    trial_rho_distance,
                    self.iter_count
                ),
                CachedValueProbeOutcome::Recoverable(_) | CachedValueProbeOutcome::Fatal(_) => {
                    log::info!(
                        "[STAGE] outer eval end order=Value elapsed={:.3}s outcome={} trial_rho_distance={:.3e} (first-order bridge, iter={}, cached=true)",
                        stage_start.elapsed().as_secs_f64(),
                        outcome_label,
                        trial_rho_distance,
                        self.iter_count
                    );
                }
            }
            return cached_value_probe_result(&entry.outcome);
        }
        log::info!(
            "[STAGE] outer eval start order=Value dim={} trial_rho_distance={:.3e} (first-order bridge, iter={})",
            x.len(),
            trial_rho_distance,
            self.iter_count
        );
        let result = self
            .obj
            .eval_with_order(x, OuterEvalOrder::Value)
            .map_err(|err| into_objective_error("outer eval_cost failed", err))
            .and_then(|eval| finite_cost_or_error("outer eval_cost failed", eval.cost));
        let cached_outcome = cache_value_probe_result(&result);
        remember_value_probe(&mut self.value_probe_cache, x, cached_outcome);
        match &result {
            Ok(cost) => {
                // A successful probe resets the consecutive-refusal counter: the
                // current ρ neighbourhood has at least one feasible point, so
                // isolated refusals on other directions are normal line-search
                // noise, not a globally-infeasible neighbourhood.
                self.consecutive_probe_refusals = 0;
                log::info!(
                    "[STAGE] outer eval end order=Value elapsed={:.3}s cost={:.6e} trial_rho_distance={:.3e} (first-order bridge, iter={})",
                    stage_start.elapsed().as_secs_f64(),
                    cost,
                    trial_rho_distance,
                    self.iter_count
                );
            }
            Err(ObjectiveEvalError::Recoverable { .. }) => {
                log::info!(
                    "[STAGE] outer eval end order=Value elapsed={:.3}s outcome=recoverable trial_rho_distance={:.3e} (first-order bridge, iter={})",
                    stage_start.elapsed().as_secs_f64(),
                    trial_rho_distance,
                    self.iter_count
                );
                if let Some(guard) = self.cost_stall.as_mut() {
                    match guard.observe_infeasible(x) {
                        CostStallVerdict::Continue => {}
                        CostStallVerdict::StuckKeepDescending {
                            residual_grad_norm,
                            escape_threshold,
                        } => {
                            // #1426: best feasible iterate carries a residual far
                            // above tolerance — not a flat valley. Keep going.
                            log::warn!(
                                "[OUTER] cost-stall STUCK (infeasible BFGS probes, NOT a flat \
                                 valley): residual |g|={:.3e} far above the certified-stationary \
                                 band (escape threshold {:.3e}); refusing \
                                 to halt-and-ship and continuing (escape {}/{}, value={:.6e}).",
                                residual_grad_norm,
                                escape_threshold,
                                guard.stuck_escapes,
                                STUCK_STALL_MAX_ESCAPES,
                                guard.best_value,
                            );
                        }
                        CostStallVerdict::Converged => {
                            log::info!(
                                "[OUTER] cost-stall convergence (infeasible BFGS probes): {} \
                                 consecutive infeasible probes after a finite seed/iterate; \
                                 accepting best-so-far as a stationary optimum (value={:.6e}).",
                                guard.infeasible_streak,
                                guard.best_value,
                            );
                            return Err(ObjectiveEvalError::Fatal {
                                message: COST_STALL_CONVERGED_SENTINEL.to_string(),
                            });
                        }
                        CostStallVerdict::FlatValleyStall { residual_grad_norm } => {
                            log::warn!(
                                "[OUTER] cost-stall halt (infeasible BFGS probes): {} \
                                 consecutive infeasible probes after a finite seed/iterate; \
                                 halting at best-so-far with residual |g|={:.3e} \
                                 (value={:.6e}).",
                                guard.infeasible_streak,
                                residual_grad_norm,
                                guard.best_value,
                            );
                            return Err(ObjectiveEvalError::Fatal {
                                message: COST_STALL_CONVERGED_SENTINEL.to_string(),
                            });
                        }
                    }
                }
                // Non-termination guard (#NaN-outer-loop): when every
                // line-search probe is infeasible and BFGS has never
                // accepted a gradient step (`iter_count == 0`), the
                // neighbourhood around the seed is globally degenerate.
                // BFGS would otherwise spend its entire max_iterations ×
                // line_search_budget doing inner solves that all fail.
                // Escalate to Fatal so BFGS exits immediately; the seed
                // loop routes it as a rejected seed.
                self.consecutive_probe_refusals = self.consecutive_probe_refusals.saturating_add(1);
                // When the bridge seed (cost, gradient) was supplied via
                // `with_initial_sample` the bridge's own `eval_grad` is
                // never called, so `last_value_grad_rho` stays `None` and
                // every `trial_rho_distance` prints as NaN.  The seed IS
                // feasible (it was evaluated externally), but if every
                // line-search probe is Recoverable from the very first
                // direction, the neighbourhood is globally infeasible.
                // Use the tighter NaN-seed threshold so the guard fires
                // after one generous Wolfe budget instead of 150 probes
                // (which, at ~3 s each × 13 seeds, would produce an
                // observed ~97 min hang on real D=5120 LLM activations).
                let threshold = if self.last_value_grad_rho.is_none() {
                    PROBE_REFUSAL_FATAL_THRESHOLD_NAN_SEED
                } else {
                    PROBE_REFUSAL_FATAL_THRESHOLD
                };
                if self.iter_count == 0 && self.consecutive_probe_refusals >= threshold {
                    log::warn!(
                        "[OUTER] probe-refusal non-termination guard fired after {} consecutive \
                         infeasible cost probes with no accepted gradient step \
                         (nan_seed={}); escalating to Fatal to abort this seed \
                         (first-order bridge, iter={})",
                        self.consecutive_probe_refusals,
                        self.last_value_grad_rho.is_none(),
                        self.iter_count,
                    );
                    return Err(ObjectiveEvalError::Fatal {
                        message: format!(
                            "{PROBE_REFUSAL_FATAL_SENTINEL}: {consecutive} consecutive \
                             infeasible probes with no accepted outer step",
                            consecutive = self.consecutive_probe_refusals,
                        ),
                    });
                }
            }
            Err(ObjectiveEvalError::Fatal { .. }) => {
                log::info!(
                    "[STAGE] outer eval end order=Value elapsed={:.3}s outcome=fatal trial_rho_distance={:.3e} (first-order bridge, iter={})",
                    stage_start.elapsed().as_secs_f64(),
                    trial_rho_distance,
                    self.iter_count
                );
            }
        }
        result
    }
}

impl FirstOrderObjective for OuterFirstOrderBridge<'_> {
    fn eval_grad(&mut self, x: &Array1<f64>) -> Result<FirstOrderSample, ObjectiveEvalError> {
        self.layout.validate_point_len(x, "outer eval failed")?;
        // Drive the outer-aware inner-PIRLS cap from accepted outer
        // iterations, BEFORE invoking the inner solve. Cap stays fixed
        // within line-search cost probes (`eval_cost` never touches the
        // atomic). A cap of 0 means "no cap from this source"; the inner
        // solver still honors `pirls_max_iterations` and the screening cap.
        if let Some(feedback) = self.outer_inner_cap.as_ref() {
            let g_ratio = match (self.last_g_norm, self.g_norm_initial) {
                (Some(g), Some(g0)) if g0 > 0.0 => Some(g / g0),
                _ => None,
            };
            let snapshot = feedback.snapshot();
            let accepted_iter = feedback.accepted_iter.load(Ordering::Relaxed);
            let cap = first_order_inner_cap_schedule(accepted_iter, g_ratio, snapshot);
            let prev = feedback.cap.swap(cap, Ordering::Relaxed);
            if prev != cap {
                let ratio_str = match g_ratio {
                    Some(r) => format!("{:.3e}", r),
                    None => "n/a".to_string(),
                };
                let snap_str = match snapshot {
                    Some(s) => format!(
                        "last_iters={} converged={} ift_residual={} accept_rho={}",
                        s.last_iters,
                        s.last_converged,
                        match s.last_ift_residual {
                            Some(r) => format!("{:.3e}", r),
                            None => "n/a".to_string(),
                        },
                        match s.last_accept_rho {
                            Some(r) => format!("{:.3}", r),
                            None => "n/a".to_string(),
                        },
                    ),
                    None => "no-history".to_string(),
                };
                log::info!(
                    "[OUTER schedule] inner-PIRLS cap transition accepted_iter={} eval_count={} g_ratio={} {} prev={} new={} ({})",
                    accepted_iter,
                    self.iter_count,
                    ratio_str,
                    snap_str,
                    prev,
                    cap,
                    if cap == 0 { "uncapped" } else { "capped" }
                );
            }
        }
        let stage_start = std::time::Instant::now();
        log::info!(
            "[STAGE] outer eval start order=ValueAndGradient dim={} (first-order bridge, iter={})",
            x.len(),
            self.iter_count
        );
        let eval = self
            .obj
            .eval_with_order(x, OuterEvalOrder::ValueAndGradient)
            .map_err(|err| into_objective_error("outer eval failed", err))?;
        let eval = finite_outer_first_order_eval_or_error("outer eval failed", self.layout, eval)?;
        let g_norm = eval.gradient.iter().map(|v| v * v).sum::<f64>().sqrt();
        let gradient = eval.gradient;
        if self.g_norm_initial.is_none() && g_norm.is_finite() && g_norm > 0.0 {
            self.g_norm_initial = Some(g_norm);
        }
        if g_norm.is_finite() {
            self.last_g_norm = Some(g_norm);
        }
        self.last_value_grad_rho = Some(x.clone());
        // A successful gradient evaluation means the current ρ is feasible;
        // reset the consecutive-probe-refusal counter so the guard only fires
        // when ALL probes in EVERY subsequent direction fail.
        self.consecutive_probe_refusals = 0;
        self.value_probe_cache
            .retain(|entry| value_probe_reject_outcome(&entry.outcome));
        log::info!(
            "[STAGE] outer eval end order=ValueAndGradient elapsed={:.3}s cost={:.6e} |g|={:.3e} (first-order bridge, iter={})",
            stage_start.elapsed().as_secs_f64(),
            eval.cost,
            g_norm,
            self.iter_count,
        );
        self.iter_count = self.iter_count.saturating_add(1);
        // Cost-stall halt (#1089). `eval_grad` is invoked by `opt::Bfgs` at
        // each accepted iterate (line-search COST probes go through `eval_cost`,
        // not here), so folding the objective in here counts accepted outer
        // steps. When the REML score has stopped improving over
        // `COST_STALL_WINDOW` consecutive accepted steps, halt BFGS by returning
        // a sentinel `Fatal` (an observer cannot stop `opt::Bfgs`; an error is
        // the only in-band way to halt it). The runner rebuilds the outer result
        // from the published best iterate — but whether that result is reported
        // CONVERGED is decided by the guard's STATIONARITY test, not by
        // cost-flatness alone: a stall whose projected gradient still exceeds the
        // outer gradient tolerance is a flat-valley floor (`converged = false`),
        // a stationary one is a real optimum (`converged = true`). Both share the
        // sentinel; the verdict rides on the published `CostStallExit.converged`.
        // #1426: read the inner-PIRLS convergence flag for the solve THIS eval
        // just ran (the feedback atomics are updated by `execute_pirls_if_needed`
        // after each non-screening solve, so the snapshot now reflects this ρ).
        // A non-converged inner solve makes the reported cost/gradient
        // untrustworthy; the guard must not record it as best-so-far nor count it
        // toward a stall. `None` (no feedback wired) defaults to `true` so routes
        // without inner-cap feedback are unchanged.
        let inner_converged = inner_solve_converged(self.outer_inner_cap.as_ref());
        if let Some(guard) = self.cost_stall.as_mut() {
            // The stall guard's stationarity test must use the bound-PROJECTED
            // gradient norm (KKT residual), not the raw `g_norm` above — a
            // separation fit pins log-λ directions at the bound with a
            // persistent out-of-bounds ∂V/∂ρ that inflates the raw norm forever
            // and otherwise blocks the converged verdict (#1082). The raw
            // `g_norm` is kept for the inner-cap schedule / logging.
            let projected_g_norm =
                projected_gradient_norm(x, &gradient, self.cost_stall_bounds.as_ref());
            match guard.observe(x, eval.cost, projected_g_norm, inner_converged) {
                CostStallVerdict::Continue => {}
                CostStallVerdict::StuckKeepDescending {
                    residual_grad_norm,
                    escape_threshold,
                } => {
                    // #1426: cost flatlined but the projected gradient is far
                    // above tolerance — a stuck stall from an inconsistent
                    // (non-converged inner solve) objective/gradient, NOT a flat
                    // valley. Do NOT halt: keep descending so the optimizer can
                    // climb out toward the well-penalized optimum. UNCAP the
                    // inner PIRLS so the next solves run to full tolerance — the
                    // stuck state is caused by the inner cap stopping PIRLS short
                    // of its fixed point, leaving the cost and the analytic
                    // gradient inconsistent; an uncapped solve restores a
                    // trustworthy gradient the outer step can actually use.
                    // #2349: raise the cold-reeval pulse alongside the uncap so
                    // a warm-start-hysteresis stall on a near-separating fit
                    // re-solves the next outer evaluations COLD (see the ARC arm
                    // and the `force_cold` field doc for the full rationale).
                    if let Some(feedback) = self.outer_inner_cap.as_ref() {
                        feedback.cap.store(0, Ordering::Relaxed);
                        feedback.force_cold.store(true, Ordering::Relaxed);
                    }
                    log::warn!(
                        "[OUTER] cost-stall STUCK (NOT a flat valley): REML objective improved \
                         < {:.3e} (relative) over {} accepted outer steps but the projected \
                         gradient is FAR above the certified-stationary band \
                         (|g|={:.3e} > escape threshold {:.3e}); refusing \
                         to halt-and-ship and continuing the descent (escape {}/{}, value={:.6e}).",
                        guard.rel_tol,
                        guard.window,
                        residual_grad_norm,
                        escape_threshold,
                        guard.stuck_escapes,
                        STUCK_STALL_MAX_ESCAPES,
                        guard.best_value,
                    );
                }
                CostStallVerdict::Converged => {
                    // Report the band that ACTUALLY certified: absolute
                    // tolerance, score-relative flat-valley bound, or the
                    // #2241 probe-noise floor — printing only the raw
                    // tolerance made a flat-band acceptance read as an
                    // arithmetic impossibility in the log.
                    let flat_band = flat_valley_converged_grad_bound(guard.best_value);
                    let noise_band = guard.probe_noise_grad_bound().unwrap_or(f64::NAN);
                    log::info!(
                        "[OUTER] cost-stall convergence: REML objective improved < {:.3e} \
                         (relative) over {} consecutive accepted outer steps AND the projected \
                         gradient cleared a stationarity band (|g|={:.3e}; absolute tol {:.3e}, \
                         score-relative flat band {:.3e}, probe-noise band {:.3e}); accepting \
                         best-so-far as a stationary optimum (value={:.6e}).",
                        guard.rel_tol,
                        guard.window,
                        guard.best_grad_norm,
                        guard.grad_threshold,
                        flat_band,
                        noise_band,
                        guard.best_value,
                    );
                    return Err(ObjectiveEvalError::Fatal {
                        message: COST_STALL_CONVERGED_SENTINEL.to_string(),
                    });
                }
                CostStallVerdict::FlatValleyStall { residual_grad_norm } => {
                    log::warn!(
                        "[OUTER] cost-stall FLAT-VALLEY STALL: REML objective improved < {:.3e} \
                         (relative) over {} consecutive accepted outer steps but the projected \
                         gradient is still ABOVE the outer tolerance (|g|={:.3e} > {:.3e}); \
                         halting on a weakly-identified ρ valley floor and reporting NON-CONVERGED \
                         (residual outer non-stationarity, value={:.6e}).",
                        guard.rel_tol,
                        guard.window,
                        residual_grad_norm,
                        guard.grad_threshold,
                        guard.best_value,
                    );
                    return Err(ObjectiveEvalError::Fatal {
                        message: COST_STALL_CONVERGED_SENTINEL.to_string(),
                    });
                }
            }
        }
        Ok(FirstOrderSample {
            value: eval.cost,
            gradient,
        })
    }
}

/// Outer gradient-decay ratio `‖g_now‖/‖g_initial‖` below which the outer is
/// treated as essentially converged: the inner cap is lifted entirely so the
/// cached β reaches full inner tolerance before the convergence guard runs.
pub(crate) const INNER_CAP_CONVERGENCE_OVERRIDE_RATIO: f64 = 0.01;

/// Floor on the adaptive inner-PIRLS cap. Any cap below this is below the
/// inner-Newton noise level and would reject usable warm-started steps.
pub(crate) const INNER_CAP_FLOOR: usize = 3;

/// Ceiling on the adaptive inner-PIRLS cap, set at the inner-Newton noise
/// floor at large scale; further iterations are pure waste once the warm
/// start is close.
pub(crate) const INNER_CAP_CEILING: usize = 64;

/// Adaptive inner-PIRLS cap schedule. Replaces the older hardcoded
/// iter-tier (3/5/10/20) and ratio-tier (0.50/0.20/0.05/0.01) schedule
/// with a cap driven by the inner solver's actual convergence behavior
/// — Eisenstat-Walker style for the inner Newton.
///
/// Inputs:
/// - `iter_count`: outer iter index, used only as a fallback when no
///   inner-progress feedback has arrived yet (first 1-2 outer iters).
/// - `g_ratio`: outer gradient-norm decay `‖g_now‖ / ‖g_initial‖`. When
///   this drops below 1% the outer is essentially converged; we lift
///   the cap fully so the cached β is at full inner tolerance and the
///   convergence guard does not have to re-pay a full inner solve.
/// - `last`: snapshot from `InnerProgressFeedback`. When present and
///   the previous solve converged, we set the cap to `last_iters + 2`
///   (a small margin in case ρ moved enough to need a couple more
///   iters); when the previous solve hit the cap, we double — a
///   geometric backoff that recovers from too-tight a cap without
///   thrashing.
///
/// A cap of 0 means "no cap from this source"; the inner solver still
/// honors `pirls_max_iterations` and the screening cap. The cap is
/// floored at 3 (anything less is below noise) and ceilinged at 64
/// (the inner noise floor at large scale; further iters would be
/// pure waste).
/// Did the inner PIRLS for the most recent non-screening outer eval converge?
///
/// Reads the inner-progress feedback snapshot the cap schedule already consumes.
/// Returns the snapshot's `last_converged` flag, defaulting to `true` when there
/// is no feedback wired (`feedback == None`) or no inner solve has reported yet
/// (`snapshot() == None`, e.g. the very first outer eval). Defaulting to `true`
/// keeps routes without inner-cap feedback — and the cold-start iterate — on
/// their existing behavior; the #1426 guard only *withholds trust* from an
/// iterate KNOWN to have a non-converged inner solve, it never invents distrust.
///
/// IMPORTANT: this must be read AFTER the outer eval has run (so the feedback
/// atomics, set by `execute_pirls_if_needed` post-solve, describe the solve at
/// the current ρ), and BEFORE the next eval's cap is computed.
pub(crate) fn inner_solve_converged(feedback: Option<&InnerProgressFeedback>) -> bool {
    match feedback.and_then(InnerProgressFeedback::snapshot) {
        Some(snap) => snap.last_converged,
        None => true,
    }
}

pub(crate) fn first_order_inner_cap_schedule(
    iter_count: usize,
    g_ratio: Option<f64>,
    last: Option<InnerProgressSnapshot>,
) -> usize {
    // Convergence override: when the outer is essentially converged the
    // cached β must be at full inner tolerance. This belt-and-suspenders
    // path is independent of inner-progress history because the outer
    // re-evaluation guard pays a full inner solve anyway — uncapping
    // here just avoids one wasted iter at low cap before the guard.
    if matches!(g_ratio, Some(r) if r < INNER_CAP_CONVERGENCE_OVERRIDE_RATIO) {
        return 0;
    }

    // Adaptive path: drive the cap from the inner solver's prior iter
    // count rather than a hardcoded tier.
    if let Some(snap) = last {
        let next = if snap.last_converged {
            // Converged in `last_iters` last time; pick a small margin
            // for ρ-step variability. The IFT predictor's residual
            // tells us how close the warm-start was to the KKT point:
            //   residual < 0.01  → next solve starts essentially AT the
            //                      KKT β, so +1 iter of margin suffices.
            //   residual < 0.10  → +2 (default, current behavior).
            //   residual ≥ 0.10  → predictor was poor (or fell back to
            //                      flat); the inner Newton has more
            //                      recovery work, so +4 to be safe.
            //   None             → no signal yet → +2 (default).
            // This wires the [IFT-QUALITY] feedback directly into the
            // adaptive cap, replacing the previous fixed +2.
            let mut margin = match snap.last_ift_residual {
                Some(r) if r < 0.01 => 1usize,
                Some(r) if r >= 0.10 => 4usize,
                _ => 2usize,
            };
            // LM model fidelity (commit 6445c079): if the previous
            // solve's accepted gain ratio was poor (model overstating
            // predicted reduction), the inner Newton's quadratic model
            // is unreliable. Bump margin by +2 — even a fast-converged
            // previous iter (small `last_iters`) provides weaker
            // evidence about the next solve's required effort when the
            // model is mis-calibrated. Threshold 0.5 is the textbook
            // "good agreement" cutoff for trust-region gain ratios.
            if matches!(snap.last_accept_rho, Some(r) if r < 0.5) {
                margin = margin.saturating_add(2);
            }
            snap.last_iters.saturating_add(margin)
        } else {
            // Hit the cap. Geometric backoff so we don't thrash on a
            // marginally-too-tight cap, but enforce floor of
            // last_iters+4 to actually grow.
            //
            // LM-fidelity escalation: if the previous solve's accepted
            // gain ratio was VERY poor (`accept_rho < 0.3`), the LM
            // model is severely mis-calibrated — doubling the cap may
            // not give the inner Newton enough headroom to find a
            // usable trust radius. Triple instead of doubling so we
            // don't waste another cycle hitting the cap. The 0.3
            // threshold is tighter than the +2-margin trigger (0.5)
            // because here we ALREADY know the iter budget was
            // insufficient AND the model was poor — both signals
            // pointing the same way.
            let multiplier = if matches!(snap.last_accept_rho, Some(r) if r < 0.3) {
                3
            } else {
                2
            };
            snap.last_iters
                .saturating_mul(multiplier)
                .max(snap.last_iters.saturating_add(4))
        };
        return next.clamp(INNER_CAP_FLOOR, INNER_CAP_CEILING);
    }

    // No feedback yet (first outer iter, or right after a screening
    // bundle reset). Coarse iter-count fallback for the first 1-2
    // outer iters so the cold-start cap is shallow even before the
    // adaptive signal kicks in.
    match iter_count {
        0 => 3,
        1 => 5,
        _ => 10,
    }
}

#[cfg(test)]
#[path = "inner_cap_schedule_tests.rs"]
mod inner_cap_schedule_tests;

pub(crate) struct OuterSecondOrderBridge<'a> {
    pub(crate) obj: &'a mut dyn OuterObjective,
    pub(crate) layout: OuterThetaLayout,
    pub(crate) hessian_source: HessianSource,
    /// When the evaluator returns `HessianValue::Operator(op)` and the
    /// operator advertises an exact dense route, the bridge may materialize the
    /// operator into a dense K×K matrix so the dense ARC path can run an exact
    /// factorization instead of operator-CG.
    pub(crate) materialize_operator_max_dim: usize,
    /// Counts gradient/Hessian evaluations so that progress is visible even
    /// when the upstream `opt` solver does not emit per-iteration logs of its
    /// own. Emitted at INFO from `eval_grad` and `eval_hessian` (the calls
    /// that gate one optimizer step); skipped on `eval_cost` so linesearch
    /// trial points do not flood the log. Also drives the outer-aware
    /// inner-PIRLS cap schedule (see `first_order_inner_cap_schedule`).
    pub(crate) eval_count: usize,
    /// Outer-aware inner-PIRLS cap atomic. When `Some`, the bridge stores
    /// a coarsen-then-tighten cap into it on every accepted eval_grad /
    /// eval_hessian call. Mirrors the BFGS-side wiring in
    /// `OuterFirstOrderBridge`. Cap is NEVER touched in `eval_cost` so
    /// line-search probes within an outer iter see a stable inner
    /// tolerance (Wolfe / trust-region acceptance both assume constant
    /// cost noise within a bracket).
    pub(crate) outer_inner_cap: Option<InnerProgressFeedback>,
    /// First observed `‖g‖` from `eval_grad`/`eval_hessian`. Used by the
    /// schedule's gradient-ratio gate so the cap lifts when the optimizer
    /// is approaching convergence, not just when iter count says so.
    pub(crate) g_norm_initial: Option<f64>,
    /// `‖g‖` from the most recent eval. See `OuterFirstOrderBridge` for
    /// the staleness rationale: monotone-decreasing g_norm means the cap
    /// is conservatively LARGER than truly needed, never smaller.
    pub(crate) last_g_norm: Option<f64>,
    /// Most recent derivative-evaluation point, used to log value-probe
    /// displacement in line-search / trial-acceptance STAGE traces.
    pub(crate) last_value_grad_rho: Option<Array1<f64>>,
    /// Cost-stall convergence guard (#1089/#1237). Identical role to the field
    /// on [`OuterFirstOrderBridge`]: `opt::Arc` calls `eval_grad` once per
    /// accepted (and rejected-then-recomputed) iterate, so folding the
    /// `(cost, projected ‖g‖)` sample in here counts outer descent steps. On a
    /// near-separable multinomial fit the unpenalized softmax MLE is unbounded,
    /// so the outer REML criterion keeps decreasing as λ→0 and several log-λ
    /// directions slam to the lower box bound and bounce — the ARC loop cycles
    /// to its `max_iter` cap without ever certifying a stationary point (#1237,
    /// the #1082 multinomial timeout). The guard halts ARC the moment the REML
    /// score stops improving, and the bound-PROJECTED KKT residual decides the
    /// converged verdict (a bound-pinned separating direction with a persistent
    /// out-of-bounds ∂V/∂ρ is stationary in the KKT sense even though its raw
    /// gradient never vanishes).
    pub(crate) cost_stall: Option<CostStallGuard>,
    /// `(lower, upper)` ρ box bounds for the bound-projected gradient norm the
    /// [`CostStallGuard`] stationarity test consumes. See the matching field on
    /// [`OuterFirstOrderBridge`].
    pub(crate) cost_stall_bounds: Option<(Array1<f64>, Array1<f64>)>,
}

impl ZerothOrderObjective for OuterSecondOrderBridge<'_> {
    fn eval_cost(&mut self, x: &Array1<f64>) -> Result<f64, ObjectiveEvalError> {
        // Uncap the inner solve for the ARC line-search / trial-acceptance cost
        // probe. Identical rationale to `OuterFirstOrderBridge::eval_cost`: the
        // deciding cost must be the true converged-inner objective the analytic
        // gradient/Hessian differentiate, never the scheduled gradient-path cap
        // (which at a trial ρ can stop the inner solve short and report a
        // spurious `∞`, freezing the ARC at constant cost / |g| — gam#808
        // survival marginal-slope, gam#787 bernoulli matern marginal-slope).
        // `eval_grad`/`eval_hessian` restore the scheduled cap on the next call.
        if let Some(feedback) = self.outer_inner_cap.as_ref() {
            feedback
                .cap
                .store(SEED_SCREENING_UNCAPPED, Ordering::Relaxed);
        }
        self.layout
            .validate_point_len(x, "outer eval_cost failed")?;
        let trial_rho_distance = trial_rho_distance(self.last_value_grad_rho.as_ref(), x);
        let stage_start = std::time::Instant::now();
        log::info!(
            "[STAGE] outer eval start order=Value dim={} trial_rho_distance={:.3e}",
            x.len(),
            trial_rho_distance
        );
        let eval = self
            .obj
            .eval_with_order(x, OuterEvalOrder::Value)
            .map_err(|err| into_objective_error("outer eval_cost failed", err))?;
        let cost = finite_cost_or_error("outer eval_cost failed", eval.cost)?;
        log::info!(
            "[STAGE] outer eval end order=Value elapsed={:.3}s cost={:.6e} trial_rho_distance={:.3e}",
            stage_start.elapsed().as_secs_f64(),
            cost,
            trial_rho_distance
        );
        Ok(cost)
    }
}

impl FirstOrderObjective for OuterSecondOrderBridge<'_> {
    fn eval_grad(&mut self, x: &Array1<f64>) -> Result<FirstOrderSample, ObjectiveEvalError> {
        self.layout.validate_point_len(x, "outer eval failed")?;
        if let Some(feedback) = self.outer_inner_cap.as_ref() {
            // The ARC bridge increments `eval_count` in BOTH `eval_grad` and
            // `eval_hessian`. ARC calls both per outer iter, so `eval_count
            // / 2` is the correct iter index for the schedule. Without this
            // divisor the schedule would lift to full inner-cap at ARC iter
            // 3 instead of iter 6.
            // Use the observer-fed accepted-iter counter (opt 0.5.0
            // OptimizerObserver) instead of `eval_count / 2`; the
            // observer increments only on rho-accepted steps, so the
            // schedule no longer relaxes the cap on rejected trials.
            let arc_iter = feedback.accepted_iter.load(Ordering::Relaxed);
            let g_ratio = match (self.last_g_norm, self.g_norm_initial) {
                (Some(g), Some(g0)) if g0 > 0.0 => Some(g / g0),
                _ => None,
            };
            let snapshot = feedback.snapshot();
            let cap = first_order_inner_cap_schedule(arc_iter, g_ratio, snapshot);
            let prev = feedback.cap.swap(cap, Ordering::Relaxed);
            if prev != cap {
                let ratio_str = match g_ratio {
                    Some(r) => format!("{:.3e}", r),
                    None => "n/a".to_string(),
                };
                let snap_str = match snapshot {
                    Some(s) => format!(
                        "last_iters={} converged={} ift_residual={} accept_rho={}",
                        s.last_iters,
                        s.last_converged,
                        match s.last_ift_residual {
                            Some(r) => format!("{:.3e}", r),
                            None => "n/a".to_string(),
                        },
                        match s.last_accept_rho {
                            Some(r) => format!("{:.3}", r),
                            None => "n/a".to_string(),
                        },
                    ),
                    None => "no-history".to_string(),
                };
                log::info!(
                    "[OUTER schedule] inner-PIRLS cap transition (ARC bridge) arc_iter={} g_ratio={} {} prev={} new={} ({})",
                    arc_iter,
                    ratio_str,
                    snap_str,
                    prev,
                    cap,
                    if cap == 0 { "uncapped" } else { "capped" }
                );
            }
        }
        let stage_start = std::time::Instant::now();
        log::info!(
            "[STAGE] outer eval start order=ValueAndGradient dim={}",
            x.len()
        );
        let eval = self
            .obj
            .eval_with_order(x, OuterEvalOrder::ValueAndGradient)
            .map_err(|err| into_objective_error("outer eval failed", err))?;
        let eval = finite_outer_first_order_eval_or_error("outer eval failed", self.layout, eval)?;
        self.eval_count += 1;
        let g_norm = eval.gradient.iter().map(|v| v * v).sum::<f64>().sqrt();
        if self.g_norm_initial.is_none() && g_norm.is_finite() && g_norm > 0.0 {
            self.g_norm_initial = Some(g_norm);
        }
        if g_norm.is_finite() {
            self.last_g_norm = Some(g_norm);
        }
        self.last_value_grad_rho = Some(x.clone());
        log::info!(
            "[STAGE] outer eval end order=ValueAndGradient elapsed={:.3}s cost={:.6e} |g|={:.3e}",
            stage_start.elapsed().as_secs_f64(),
            eval.cost,
            g_norm,
        );
        log::info!(
            "[OUTER] eval#{n} (grad) cost={cost:.6e} |g|={gnorm:.3e} rho=[{rho}]",
            n = self.eval_count,
            cost = eval.cost,
            gnorm = g_norm,
            rho = x
                .iter()
                .map(|v| format!("{v:.3}"))
                .collect::<Vec<_>>()
                .join(","),
        );
        // NOTE: the cost-stall guard lives in `eval_hessian` below, NOT here.
        // `opt::Arc`'s per-iterate oracle (`eval_cost_grad_hessian`) calls
        // `eval_hessian` for the (value, grad, Hessian) triple and never calls
        // this `eval_grad` on the ARC route, so folding the guard in here would
        // leave it dead and let the near-separable multinomial loop keep cycling
        // to `max_iter` (#1237). See `observe_cost_stall`.
        Ok(FirstOrderSample {
            value: eval.cost,
            gradient: eval.gradient,
        })
    }
}

impl OuterSecondOrderBridge<'_> {
    /// Fold one finite ARC oracle eval `(ρ, cost, grad)` into the cost-stall
    /// guard without allowing that first-order guard to halt the second-order
    /// solver.
    ///
    /// Cost-stall halt (#1089/#1237). `opt::Arc` evaluates the (value, gradient,
    /// Hessian) triple at every trial point — accepted or rejected — through
    /// `eval_hessian`, so observing here counts ARC's outer descent. On a
    /// near-separable multinomial fit the unpenalized softmax MLE is unbounded:
    /// the outer REML criterion keeps decreasing as λ→0, several log-λ
    /// directions slam to the lower box bound and bounce, and ARC otherwise
    /// cycles to its `max_iter` cap without certifying a stationary point. The
    /// guard halts the moment the REML score plateaus over `COST_STALL_WINDOW`
    /// evals; the converged verdict rides on the bound-PROJECTED gradient norm
    /// (KKT residual), NOT the raw `g_norm` — a bound-pinned separating
    /// direction with a persistent out-of-bounds ∂V/∂ρ is KKT-stationary even
    /// though its raw gradient never vanishes. A trial that beats the best cost
    /// resets the streak, so genuine descent never trips the halt.
    fn observe_cost_stall(
        &mut self,
        x: &Array1<f64>,
        cost: f64,
        gradient: &Array1<f64>,
        hessian_psd: Option<bool>,
    ) {
        let bounds = self.cost_stall_bounds.clone();
        let separation_bound_stationary = {
            let Some(guard) = self.cost_stall.as_ref() else {
                return;
            };
            lower_bound_outward_active_count(x, gradient, bounds.as_ref(), guard.grad_threshold)
                >= LOWER_BOUND_SEPARATION_ACTIVE_MIN
        };
        // #1426: inner-PIRLS convergence flag for the solve behind this eval (see
        // the matching read in the first-order bridge). A non-converged inner
        // solve at the λ→0 ridge reports a half-fit cost the guard must not adopt
        // as best-so-far. `None` (no feedback) defaults to `true`.
        let inner_converged = inner_solve_converged(self.outer_inner_cap.as_ref());
        let Some(guard) = self.cost_stall.as_mut() else {
            return;
        };
        let projected_g_norm = projected_gradient_norm(x, gradient, bounds.as_ref());
        let verdict = if separation_bound_stationary {
            // #1426: certify the constrained-stationary fast-path with the REAL
            // bound-projected gradient norm, not a hardcoded 0.0. The projection
            // (`projected_gradient_norm`) already zeros the out-of-bounds pull on
            // the railed separation axes, so a genuine λ→0 separation
            // (#1082/#1237) still presents ‖g_proj‖ ≈ 0 and is certified
            // converged exactly as before. But a flat-valley OVERFIT that merely
            // *looks* separated on a couple of railed axes while retaining a large
            // INTERIOR (feasible-descent) gradient component — the Gamma/log λ→0
            // ridge of #1426 — keeps that interior mass in ‖g_proj‖, so
            // `publish_stall` honestly returns FlatValleyStall / converged=false
            // instead of shipping the overfit silently behind a fake zero.
            guard.observe_constrained_stationary(
                x,
                cost,
                projected_g_norm,
                inner_converged,
                hessian_psd,
            )
        } else {
            guard.observe_second_order(x, cost, projected_g_norm, inner_converged, hessian_psd)
        };
        match verdict {
            CostStallVerdict::Continue => {}
            CostStallVerdict::StuckKeepDescending {
                residual_grad_norm,
                escape_threshold,
            } => {
                // #1426: cost flatlined but the projected gradient is far above
                // tolerance — a stuck stall, not a flat valley. Do not halt ARC.
                // Uncap the inner PIRLS so the next solves run to full tolerance
                // and the outer gradient becomes trustworthy (see the BFGS-side
                // arm for the full rationale). #2349: also raise the cold-reeval
                // pulse — on a near-separating profiled fit the stall is
                // warm-start value hysteresis, which an uncapped WARM solve does
                // not cure (it re-converges to the same warm-biased ridge point);
                // the next outer evaluations must re-solve COLD to hand the
                // optimizer a trajectory-independent surface it can descend.
                if let Some(feedback) = self.outer_inner_cap.as_ref() {
                    feedback.cap.store(0, Ordering::Relaxed);
                    feedback.force_cold.store(true, Ordering::Relaxed);
                }
                log::warn!(
                    "[OUTER] ARC cost-stall STUCK (NOT a flat valley): REML objective improved \
                     < {:.3e} (relative) over {} outer steps but the projected gradient is FAR \
                     above the certified-stationary band (|g|={:.3e} > escape threshold \
                     {:.3e}); refusing to halt-and-ship \
                     and continuing the descent (escape {}/{}, value={:.6e}).",
                    guard.rel_tol,
                    guard.window,
                    residual_grad_norm,
                    escape_threshold,
                    guard.stuck_escapes,
                    STUCK_STALL_MAX_ESCAPES,
                    guard.best_value,
                );
            }
            CostStallVerdict::Converged => {
                log::info!(
                    "[OUTER] ARC finite cost stall deferred to exact second-order convergence: \
                     REML objective improved < {:.3e} (relative) over {} consecutive outer \
                     steps and the stored best projected gradient is small (|g|={:.3e} <= \
                     {:.3e}), but only ARC owns the synchronized reduced-Hessian certificate \
                     (value={:.6e}).",
                    guard.rel_tol,
                    guard.window,
                    guard.best_grad_norm,
                    guard.certified_grad_bound(),
                    guard.best_value,
                );
                guard.defer_finite_second_order_stall();
            }
            CostStallVerdict::FlatValleyStall { residual_grad_norm } => {
                log::warn!(
                    "[OUTER] ARC finite cost stall deferred: REML objective improved < {:.3e} \
                     (relative) over {} consecutive outer steps but the stored best projected \
                     gradient remains above tolerance (|g|={:.3e} > {:.3e}). Returning the \
                     current finite gradient + Hessian to ARC instead of halting at a stale \
                     best point (value={:.6e}).",
                    guard.rel_tol,
                    guard.window,
                    residual_grad_norm,
                    guard.grad_threshold,
                    guard.best_value,
                );
                guard.defer_finite_second_order_stall();
            }
        }
    }

    /// Fold one INFEASIBLE ARC trial (non-finite cost) into the cost-stall
    /// guard. Mirrors `observe_cost_stall` but routes through the guard's
    /// infeasible-streak path: a run of `COST_STALL_WINDOW` consecutive
    /// infeasible trials after a feasible best halts the outer loop at that
    /// best feasible iterate rather than letting ARC grind to `max_iter`
    /// probing the unbounded λ→0 separating region (#1082/#1237).
    fn observe_cost_stall_infeasible(&mut self, x: &Array1<f64>) -> Option<ObjectiveEvalError> {
        let guard = self.cost_stall.as_mut()?;
        match guard.observe_infeasible(x) {
            CostStallVerdict::Continue => None,
            CostStallVerdict::StuckKeepDescending {
                residual_grad_norm,
                escape_threshold,
            } => {
                // #1426: best feasible iterate carries a residual far above
                // tolerance — not a flat valley. Keep ARC descending.
                log::warn!(
                    "[OUTER] ARC cost-stall STUCK (infeasible run, NOT a flat valley): best \
                     feasible residual |g|={:.3e} far above the certified-stationary band \
                     (escape threshold {:.3e}); refusing to \
                     halt-and-ship and continuing (escape {}/{}, value={:.6e}).",
                    residual_grad_norm,
                    escape_threshold,
                    guard.stuck_escapes,
                    STUCK_STALL_MAX_ESCAPES,
                    guard.best_value,
                );
                None
            }
            CostStallVerdict::Converged => {
                log::warn!(
                    "[OUTER] ARC infeasible-probe stall: {} consecutive infeasible λ→0 trials \
                     after the best feasible iterate. Its stored projected gradient is small \
                     (|g|={:.3e} <= {:.3e}) and its synchronized reduced Hessian does not \
                     certify negative curvature; halting only as a NON-CONVERGED checkpoint \
                     (value={:.6e}).",
                    guard.window,
                    guard.best_grad_norm,
                    guard.certified_grad_bound(),
                    guard.best_value,
                );
                guard.revoke_published_convergence();
                Some(ObjectiveEvalError::Fatal {
                    message: ARC_INFEASIBLE_STALL_SENTINEL.to_string(),
                })
            }
            CostStallVerdict::FlatValleyStall { residual_grad_norm } => {
                log::warn!(
                    "[OUTER] ARC cost-stall halt (infeasible run): {} consecutive infeasible \
                     λ→0 trials after the best feasible iterate, whose projected gradient is \
                     still ABOVE the outer tolerance (|g|={:.3e} > {:.3e}); halting at the best \
                     feasible iterate and reporting NON-CONVERGED (value={:.6e}).",
                    guard.window,
                    residual_grad_norm,
                    guard.grad_threshold,
                    guard.best_value,
                );
                guard.revoke_published_convergence();
                Some(ObjectiveEvalError::Fatal {
                    message: ARC_INFEASIBLE_STALL_SENTINEL.to_string(),
                })
            }
        }
    }
}

impl SecondOrderObjective for OuterSecondOrderBridge<'_> {
    fn eval_hessian(&mut self, x: &Array1<f64>) -> Result<SecondOrderSample, ObjectiveEvalError> {
        self.layout.validate_point_len(x, "outer eval failed")?;
        if let Some(feedback) = self.outer_inner_cap.as_ref() {
            // Use the observer-fed accepted-iter counter (opt 0.5.0
            // OptimizerObserver) instead of `eval_count / 2`; the
            // observer increments only on rho-accepted steps, so the
            // schedule no longer relaxes the cap on rejected trials.
            let arc_iter = feedback.accepted_iter.load(Ordering::Relaxed);
            let g_ratio = match (self.last_g_norm, self.g_norm_initial) {
                (Some(g), Some(g0)) if g0 > 0.0 => Some(g / g0),
                _ => None,
            };
            let snapshot = feedback.snapshot();
            let cap = first_order_inner_cap_schedule(arc_iter, g_ratio, snapshot);
            let prev = feedback.cap.swap(cap, Ordering::Relaxed);
            if prev != cap {
                let ratio_str = match g_ratio {
                    Some(r) => format!("{:.3e}", r),
                    None => "n/a".to_string(),
                };
                let snap_str = match snapshot {
                    Some(s) => format!(
                        "last_iters={} converged={} ift_residual={} accept_rho={}",
                        s.last_iters,
                        s.last_converged,
                        match s.last_ift_residual {
                            Some(r) => format!("{:.3e}", r),
                            None => "n/a".to_string(),
                        },
                        match s.last_accept_rho {
                            Some(r) => format!("{:.3}", r),
                            None => "n/a".to_string(),
                        },
                    ),
                    None => "no-history".to_string(),
                };
                log::info!(
                    "[OUTER schedule] inner-PIRLS cap transition (ARC bridge) arc_iter={} g_ratio={} {} prev={} new={} ({})",
                    arc_iter,
                    ratio_str,
                    snap_str,
                    prev,
                    cap,
                    if cap == 0 { "uncapped" } else { "capped" }
                );
            }
        }
        let stage_start = std::time::Instant::now();
        log::info!(
            "[STAGE] outer eval start order=ValueGradientHessian dim={}",
            x.len()
        );
        let eval = self
            .obj
            .eval_with_order(x, OuterEvalOrder::ValueGradientHessian)
            .map_err(|err| into_objective_error("outer eval failed", err))?;
        // Infeasible (non-finite cost) trials are the near-separable
        // multinomial failure mode: ARC probes the unbounded λ→0 separating
        // region where the inner softmax solve does not converge. These never
        // reach `observe_cost_stall` below (validation rejects them), so feed
        // them to the guard's dedicated infeasible-streak path FIRST — a run of
        // consecutive infeasible trials after a feasible best halts the outer
        // loop at that best iterate instead of grinding to `max_iter`
        // (#1082/#1237).
        if !eval.cost.is_finite() {
            if let Some(err) = self.observe_cost_stall_infeasible(x) {
                return Err(err);
            }
        }
        let eval = finite_outer_eval_or_error("outer eval failed", self.layout, eval)?;
        self.eval_count += 1;
        let g_norm = eval.gradient.iter().map(|v| v * v).sum::<f64>().sqrt();
        if self.g_norm_initial.is_none() && g_norm.is_finite() && g_norm > 0.0 {
            self.g_norm_initial = Some(g_norm);
        }
        if g_norm.is_finite() {
            self.last_g_norm = Some(g_norm);
        }
        self.last_value_grad_rho = Some(x.clone());
        log::info!(
            "[STAGE] outer eval end order=ValueGradientHessian elapsed={:.3}s cost={:.6e} |g|={:.3e}",
            stage_start.elapsed().as_secs_f64(),
            eval.cost,
            g_norm,
        );
        log::info!(
            "[OUTER] eval#{n} (hess) cost={cost:.6e} |g|={gnorm:.3e} rho=[{rho}]",
            n = self.eval_count,
            cost = eval.cost,
            gnorm = g_norm,
            rho = x
                .iter()
                .map(|v| format!("{v:.3}"))
                .collect::<Vec<_>>()
                .join(","),
        );
        let hessian = build_bridge_hessian_for_source(
            self.hessian_source,
            eval.hessian,
            self.materialize_operator_max_dim,
        )?;
        let hessian_psd = hessian.as_ref().and_then(|dense| {
            reduced_hessian_psd_at_point(
                x,
                &eval.gradient,
                dense,
                self.cost_stall_bounds
                    .as_ref()
                    .map(|(lower, upper)| (lower, upper)),
            )
        });
        // Observe finite cost progress, but never let this first-order guard
        // halt a second-order route. ARC must receive this exact sample so its
        // projected-gradient + reduced-Hessian gate can either certify a mode
        // or exploit negative curvature (#979).
        self.observe_cost_stall(x, eval.cost, &eval.gradient, hessian_psd);
        Ok(SecondOrderSample {
            value: eval.cost,
            gradient: eval.gradient,
            hessian,
        })
    }
}

// `OuterOperatorBridge` is the bridge that implements
// `opt::OperatorObjective` for `gam`'s outer objective — parallel to
// `OuterSecondOrderBridge` but produces `OperatorSample` whose
// Hessian is `HessianValue::Operator(_)` (or `Dense(_)` when the
// operator declares an exact materialization route).

/// `opt::OptimizerObserver` that increments
/// `InnerProgressFeedback.accepted_iter` on every accepted outer
/// step. Replaces the bridge-side `eval_count / 2` heuristic on
/// routes that see trial-and-rejection probing (ARC dense,
/// matrix-free TR). The bridge's inner-cap schedule reads
/// `accepted_iter` from the feedback channel instead of inferring
/// it from raw eval counts.
pub(crate) struct OuterAcceptObserver {
    pub(crate) feedback: InnerProgressFeedback,
}

impl OptimizerObserver for OuterAcceptObserver {
    fn on_step_accepted(&mut self, info: &StepInfo) {
        log::trace!(
            "outer step accepted iter={} step_norm={:.3e} predicted_decrease={:.3e} actual_decrease={:.3e}",
            info.iter,
            info.step_norm,
            info.predicted_decrease,
            info.actual_decrease,
        );
        self.feedback.accepted_iter.fetch_add(1, Ordering::Relaxed);
    }
}

/// Bridge that exposes gam's outer objective as an
/// `opt::OperatorObjective`. Used on the matrix-free trust-region
/// route; the dense-Hessian / first-order routes still use
/// `OuterSecondOrderBridge` / `OuterFirstOrderBridge`.
pub(crate) struct OuterOperatorBridge<'a> {
    pub(crate) obj: &'a mut dyn OuterObjective,
    pub(crate) layout: OuterThetaLayout,
    /// Inner-PIRLS cap atomic, mirroring the BFGS / ARC bridges.
    pub(crate) outer_inner_cap: Option<InnerProgressFeedback>,
    /// Counts gradient/Hessian evaluations for the inner-cap schedule
    /// and progress logs.
    pub(crate) eval_count: usize,
    /// First observed `‖g‖`. Used by the inner-cap schedule's
    /// gradient-ratio gate.
    pub(crate) g_norm_initial: Option<f64>,
    /// `‖g‖` from the most recent eval.
    pub(crate) last_g_norm: Option<f64>,
    /// Most recent derivative-evaluation point, used to log value-probe
    /// displacement in line-search STAGE traces.
    pub(crate) last_value_grad_rho: Option<Array1<f64>>,
}

impl ZerothOrderObjective for OuterOperatorBridge<'_> {
    fn eval_cost(&mut self, x: &Array1<f64>) -> Result<f64, ObjectiveEvalError> {
        // Uncap the inner solve for the matrix-free TR line-search cost probe.
        // Identical rationale to the BFGS / ARC bridges: the deciding cost must
        // be the true converged-inner objective the analytic gradient/operator
        // Hessian differentiate, never the scheduled gradient-path cap (which at
        // a trial ρ can stop the inner solve short and report a spurious `∞`,
        // freezing the TR at constant cost / |g|). This is the route the
        // ψ-bearing matern bernoulli marginal-slope fit takes (gam#787);
        // `eval_value_grad_op` restores the scheduled cap on the next call.
        if let Some(feedback) = self.outer_inner_cap.as_ref() {
            feedback
                .cap
                .store(SEED_SCREENING_UNCAPPED, Ordering::Relaxed);
        }
        self.layout
            .validate_point_len(x, "outer eval_cost failed")?;
        let trial_rho_distance = trial_rho_distance(self.last_value_grad_rho.as_ref(), x);
        let stage_start = std::time::Instant::now();
        log::info!(
            "[STAGE] outer eval start order=Value dim={} trial_rho_distance={:.3e} (operator bridge)",
            x.len(),
            trial_rho_distance
        );
        let eval = self
            .obj
            .eval_with_order(x, OuterEvalOrder::Value)
            .map_err(|err| into_objective_error("outer eval_cost failed", err))?;
        let cost = finite_cost_or_error("outer eval_cost failed", eval.cost)?;
        log::info!(
            "[STAGE] outer eval end order=Value elapsed={:.3}s cost={:.6e} trial_rho_distance={:.3e} (operator bridge)",
            stage_start.elapsed().as_secs_f64(),
            cost,
            trial_rho_distance
        );
        Ok(cost)
    }
}

impl FirstOrderObjective for OuterOperatorBridge<'_> {
    fn eval_grad(&mut self, x: &Array1<f64>) -> Result<FirstOrderSample, ObjectiveEvalError> {
        self.layout.validate_point_len(x, "outer eval failed")?;
        let eval = self
            .obj
            .eval_with_order(x, OuterEvalOrder::ValueAndGradient)
            .map_err(|err| into_objective_error("outer eval failed", err))?;
        let eval = finite_outer_first_order_eval_or_error("outer eval failed", self.layout, eval)?;
        let g_norm = eval.gradient.iter().map(|v| v * v).sum::<f64>().sqrt();
        if self.g_norm_initial.is_none() && g_norm.is_finite() && g_norm > 0.0 {
            self.g_norm_initial = Some(g_norm);
        }
        if g_norm.is_finite() {
            self.last_g_norm = Some(g_norm);
        }
        self.last_value_grad_rho = Some(x.clone());
        Ok(FirstOrderSample {
            value: eval.cost,
            gradient: eval.gradient,
        })
    }
}

impl OperatorObjective for OuterOperatorBridge<'_> {
    fn eval_value_grad_op(
        &mut self,
        x: &Array1<f64>,
    ) -> Result<OperatorSample, ObjectiveEvalError> {
        self.layout.validate_point_len(x, "outer eval failed")?;
        // Drive the outer-aware inner-PIRLS cap, mirroring
        // OuterSecondOrderBridge::eval_grad / eval_hessian. Each
        // accepted outer iter calls eval_value_grad_op exactly once
        // (the matrix-free TR's inner CG uses HVPs, not full
        // evaluations), so we increment per call without the /2 the
        // ARC bridge needs.
        if let Some(feedback) = self.outer_inner_cap.as_ref() {
            let g_ratio = match (self.last_g_norm, self.g_norm_initial) {
                (Some(g), Some(g0)) if g0 > 0.0 => Some(g / g0),
                _ => None,
            };
            let snapshot = feedback.snapshot();
            let cap = first_order_inner_cap_schedule(self.eval_count, g_ratio, snapshot);
            let previous_cap = feedback.cap.swap(cap, Ordering::Relaxed);
            if previous_cap != cap {
                log::trace!("outer operator bridge updated inner cap from {previous_cap} to {cap}");
            }
        }
        let stage_start = std::time::Instant::now();
        log::info!(
            "[STAGE] outer eval start order=ValueGradientHessian dim={} (operator bridge)",
            x.len(),
        );
        let eval = self
            .obj
            .eval_with_order(x, OuterEvalOrder::ValueGradientHessian)
            .map_err(|err| into_objective_error("outer eval failed", err))?;
        let eval = finite_outer_eval_or_error("outer eval failed", self.layout, eval)?;
        self.eval_count += 1;
        let g_norm = eval.gradient.iter().map(|v| v * v).sum::<f64>().sqrt();
        if self.g_norm_initial.is_none() && g_norm.is_finite() && g_norm > 0.0 {
            self.g_norm_initial = Some(g_norm);
        }
        if g_norm.is_finite() {
            self.last_g_norm = Some(g_norm);
        }
        self.last_value_grad_rho = Some(x.clone());
        log::info!(
            "[STAGE] outer eval end elapsed={:.3}s cost={:.6e} |g|={:.3e} (operator bridge)",
            stage_start.elapsed().as_secs_f64(),
            eval.cost,
            g_norm,
        );
        Ok(OperatorSample {
            value: eval.cost,
            gradient: eval.gradient,
            hessian: eval.hessian,
        })
    }
}

// Helpers preserved across the Phase 6 rewrite. Both were previously
// shared with `run_operator_trust_region` (now deleted in favor of
// `opt::MatrixFreeTrustRegion`), but they remain in use by the dense
// ARC and BFGS arms of the seed loop.

/// Euclidean norm of the bound-PROJECTED gradient at `x` under box bounds
/// `(lower, upper)` — the KKT residual of a box-constrained minimization.
///
/// The descent direction for minimization is `-g_i`. For a component sitting on
/// its lower bound (`x_i ≤ lo_i`) the only feasible moves are upward (`+`): a
/// POSITIVE `g_i` asks for a downward step (`-g_i < 0`) that would drive `x_i`
/// below the bound — infeasible — so it is a KKT multiplier and is zeroed, while
/// a NEGATIVE `g_i` is a genuine feasible (upward) descent that must be kept; the
/// *retained* contribution is therefore `min(g_i, 0)`. Symmetrically at the upper
/// bound the feasible direction is downward, the infeasible pull is `g_i < 0`,
/// and the retained part is `max(g_i, 0)`. Interior components keep `g_i`. A
/// point is a constrained stationary optimum iff this projected norm is ~0,
/// matching `opt`'s `GradientTolerance{ projected: true }` exit. With no bounds
/// this is just `‖g‖₂`.
///
/// This matches the KKT convention used by the P-IRLS
/// [`crate::pirls::newton_solve::projected_gradient_norm`] (which drops
/// the `g_i > 0` infeasible-multiplier at an active lower bound and keeps the
/// `g_i < 0` feasible descent). An earlier version had both branches inverted —
/// it zeroed the *feasible-descent* component and kept the infeasible pull — so a
/// coordinate with a real interior descent off an active bound was reported as
/// constrained-stationary, which let the outer cost-stall guard certify a railed
/// optimum as converged (the #1074 quakes-trend / #1082 / #1426 railing).
#[inline]
/// KKT-projected gradient VECTOR at a box-constrained point.
///
/// Zeros the infeasible (bound-multiplier) component on every axis pinned at a
/// box bound, keeping only the feasible-descent part — the exact split
/// [`projected_gradient_norm`] takes the norm of. Callers that need the
/// direction (e.g. the curvature-scaled flat-valley Newton decrement in
/// `certify_outer_optimality`) consume this; `projected_gradient_norm` is its
/// Euclidean norm.
pub(crate) fn project_gradient_vector(
    x: &Array1<f64>,
    gradient: &Array1<f64>,
    bounds: Option<&(Array1<f64>, Array1<f64>)>,
) -> Array1<f64> {
    match bounds {
        Some((lower, upper)) => Array1::from_iter((0..gradient.len()).map(|i| {
            let gi = gradient[i];
            // Active lower bound: feasible moves are upward, so a positive g_i
            // (its downward step `-g_i` exits the box) is the infeasible
            // KKT-multiplier pull → drop it, keeping the feasible-descent
            // negative part.
            let gi = if x[i] <= lower[i] { gi.min(0.0) } else { gi };
            // Active upper bound: feasible moves are downward, so a negative g_i
            // (its upward step `-g_i` exits the box) is the infeasible pull →
            // drop it, keeping the feasible-descent positive part.
            if x[i] >= upper[i] { gi.max(0.0) } else { gi }
        })),
        None => gradient.clone(),
    }
}

pub(crate) fn projected_gradient_norm(
    x: &Array1<f64>,
    gradient: &Array1<f64>,
    bounds: Option<&(Array1<f64>, Array1<f64>)>,
) -> f64 {
    project_gradient_vector(x, gradient, bounds)
        .iter()
        .map(|v| v * v)
        .sum::<f64>()
        .sqrt()
}

/// Apply the same strict-complementarity critical-cone reduction used by the
/// optimizer, then adjudicate second-order stationarity with the final outer
/// certificate's PSD rule. This keeps the infeasible-stall guard from treating
/// an interior (or weak-bound) strict saddle as a terminal neighbourhood.
pub(crate) fn reduced_hessian_psd_at_point(
    x: &Array1<f64>,
    gradient: &Array1<f64>,
    hessian: &Array2<f64>,
    bounds: Option<(&Array1<f64>, &Array1<f64>)>,
) -> Option<bool> {
    let n = x.len();
    if gradient.len() != n || hessian.nrows() != n || hessian.ncols() != n {
        return None;
    }
    if bounds.is_some_and(|(lower, upper)| lower.len() != n || upper.len() != n) {
        return None;
    }
    let free: Vec<usize> = (0..n)
        .filter(|&index| {
            let Some((lower, upper)) = bounds else {
                return true;
            };
            let fixed = lower[index] == upper[index];
            let strict_lower = x[index] <= lower[index] + 1.0e-10 && gradient[index] > 0.0;
            let strict_upper = x[index] >= upper[index] - 1.0e-10 && gradient[index] < 0.0;
            !(fixed || strict_lower || strict_upper)
        })
        .collect();
    if free.is_empty() {
        return Some(true);
    }
    let reduced = Array2::from_shape_fn((free.len(), free.len()), |(row, column)| {
        hessian[[free[row], free[column]]]
    });
    certificate_hessian_is_psd(&reduced)
}

#[cfg(test)]
#[path = "projected_gradient_tests.rs"]
mod projected_gradient_tests;

pub(crate) const LOWER_BOUND_SEPARATION_ACTIVE_MIN: usize = 2;

/// Count log-precision axes pinned at their lower box bound while the raw
/// gradient still points farther out of bounds. On near-separable softmax fits,
/// those λ→0 axes can keep lowering the raw REML score even though the move is
/// infeasible; once several such axes are active, repeated ARC trials there are
/// constrained-stationary separation probes, not useful descent.
///
/// Sign convention (mirrors the KKT split in [`projected_gradient_norm`], fixed
/// in a14b712 for the #1074/#1082 railing class): the optimizer MINIMIZES the
/// cost and `gradient` is ∂cost/∂ρ, so at an active lower bound the descent step
/// `-g_i` exits the box exactly when `g_i > 0` — a POSITIVE gradient is the
/// infeasible outward/separation pull this counts. A NEGATIVE `g_i` is feasible
/// interior descent (kept by `projected_gradient_norm`'s `g_i.min(0.0)`) and is
/// NOT outward, so it must NOT be counted; doing so (the prior `< -outward_floor`
/// inversion) certified railed/under-fit axes that still had real descent as
/// separation-stationary.
#[inline]
pub(crate) fn lower_bound_outward_active_count(
    x: &Array1<f64>,
    gradient: &Array1<f64>,
    bounds: Option<&(Array1<f64>, Array1<f64>)>,
    grad_threshold: f64,
) -> usize {
    let Some((lower, _upper)) = bounds else {
        return 0;
    };
    let tol = 1.0e-10;
    let outward_floor = grad_threshold.max(COST_STALL_PROJECTED_GRAD_FLOOR);
    (0..x.len().min(gradient.len()).min(lower.len()))
        .filter(|&i| x[i] <= lower[i] + tol && gradient[i] > outward_floor)
        .count()
}

#[cfg(test)]
#[path = "lower_bound_outward_tests.rs"]
mod lower_bound_outward_tests;

#[inline]
pub(crate) fn project_to_bounds(
    x: &Array1<f64>,
    bounds: Option<&(Array1<f64>, Array1<f64>)>,
) -> Array1<f64> {
    match bounds {
        Some((lower, upper)) => {
            let mut out = x.clone();
            for idx in 0..out.len() {
                out[idx] = out[idx].clamp(lower[idx], upper[idx]);
            }
            out
        }
        None => x.clone(),
    }
}

/// Translate an `OuterEval`'s Hessian into the `Option<Array2<f64>>`
/// shape expected by `opt::SecondOrderSample`, enforcing the contract
/// implied by the planner's `HessianSource`.
///
/// For `HessianSource::Analytic` (the exact second-order route) a missing
/// or non-materializable Hessian is FATAL: returning `None` here would
/// invite `opt::SecondOrderCache::finite_difference_hessian` to silently
/// estimate the Hessian by finite-differencing the gradient, which (a)
/// throws away the analytic structure the route was selected for, and
/// (b) costs O(K) full outer evaluations per ARC iteration — at large-scale
/// scale, hours of work per silently-mis-routed step. The right
/// behavior on a planner/runtime mismatch is to surface it loudly so
/// the seed loop can either retry, demote the plan, or fail the seed.
///
/// Operator Hessians that *are* cheaply materializable (the operator's
/// `materialization` reports `Explicit` / `BatchedHvp` and the
/// dimension is below `materialize_operator_max_dim`) are converted to
/// dense in-place so dense ARC can run an exact factorization. Operator
/// Hessians that are NOT cheaply materializable should never arrive
/// here: the seed loop routes those to `run_operator_trust_region`
/// before constructing the bridge. Reaching this branch on the analytic
/// route means the runtime contradicted the seed-time decision, which
/// is the same kind of mismatch we treat as fatal.
///
/// For `HessianSource::BfgsApprox`, `EfsFixedPoint`, and
/// `HybridEfsFixedPoint` we deliberately return `None`: those routes do
/// not consume an analytic Hessian and feed the Hessian into a
/// quasi-Newton/fixed-point update instead. (Today these `HessianSource`
/// variants don't actually drive `opt`'s second-order solvers, but the
/// match preserves the original behavior in case a future routing
/// reuses this bridge.)
pub(crate) fn build_bridge_hessian_for_source(
    source: HessianSource,
    hessian: HessianValue,
    materialize_operator_max_dim: usize,
) -> Result<Option<Array2<f64>>, ObjectiveEvalError> {
    match source {
        HessianSource::Analytic => match hessian {
            HessianValue::Dense(h) => Ok(Some(h)),
            HessianValue::Operator(op)
                if op.materialization().is_available()
                    && op.dim() <= materialize_operator_max_dim =>
            {
                op.materialize_dense()
                    .map(Some)
                    .map_err(|error| ObjectiveEvalError::Fatal {
                        message: format!("outer Hessian operator materialization failed: {error}"),
                    })
            }
            HessianValue::Operator(op) => Err(ObjectiveEvalError::Fatal {
                message: format!(
                    "outer plan declared HessianSource::Analytic but the runtime returned a \
                     non-materializable Hessian operator (dim={}, materialization={:?}); \
                     finite-difference Hessian estimation is not permitted on the analytic route",
                    op.dim(),
                    op.materialization(),
                ),
            }),
            HessianValue::Unavailable => Err(ObjectiveEvalError::Fatal {
                message: "outer plan declared HessianSource::Analytic but the runtime returned \
                          HessianValue::Unavailable; finite-difference Hessian estimation is \
                          not permitted on the analytic route"
                    .to_string(),
            }),
        },
        HessianSource::BfgsApprox
        | HessianSource::EfsFixedPoint
        | HessianSource::HybridEfsFixedPoint => Ok(None),
    }
}

pub(crate) struct OuterFixedPointBridge<'a> {
    pub(crate) obj: &'a mut dyn OuterObjective,
    pub(crate) layout: OuterThetaLayout,
    pub(crate) barrier_config: Option<BarrierConfig>,
    pub(crate) fixed_point_tolerance: f64,
    /// Consecutive HybridEFS iterations whose ψ block was zeroed after
    /// exhausting backtracking. When this reaches
    /// [`MAX_CONSECUTIVE_PSI_STAGNATION`], the bridge surfaces the
    /// [`EFS_FIRST_ORDER_FALLBACK_MARKER`] error so the runner aborts the
    /// HybridEFS attempt and the fallback ladder routes to a joint
    /// gradient-based solver where ψ stationarity ∇_ψ V = 0 can be enforced.
    pub(crate) consecutive_psi_zero_iters: usize,
    /// Restore streak reported by the previous fixed-point evaluation.  Keeping
    /// this in the bridge requires the certificate to recur across two distinct
    /// outer evaluations; multiple inner-refinement chunks at one rho cannot
    /// terminate the outer walk by themselves.
    pub(crate) last_restored_incumbent_streak: Option<usize>,
    /// Publication slot for the recurrent-restored-incumbent stop (#2235
    /// verdict 2). The bridge is moved into the `opt::FixedPoint` driver, so
    /// when the model-state fixed point fires the streak count is handed back
    /// to `run_fixed_point_outer_solver` through this shared cell and stamped
    /// onto the returned [`OuterResult`] as
    /// [`OuterConvergedVia::RecurrentIncumbent`]. `None` slot after the run
    /// means the walk stopped through the ordinary step-norm test instead.
    pub(crate) recurrent_incumbent_exit: Arc<Mutex<Option<usize>>>,
}

impl OuterFixedPointBridge<'_> {
    fn reject_nonstationary_tiny_psi_step(
        &self,
        step: &Array1<f64>,
        psi_indices: Option<&[usize]>,
        psi_gradient: Option<&Array1<f64>>,
        cost: f64,
    ) -> Result<(), ObjectiveEvalError> {
        let Some(psi_indices) = psi_indices else {
            return Ok(());
        };
        let Some(psi_gradient) = psi_gradient else {
            return Ok(());
        };
        let psi_step_inf = psi_indices
            .iter()
            .map(|&idx| step[idx].abs())
            .fold(0.0_f64, f64::max);
        let psi_grad_inf = psi_gradient.iter().map(|v| v.abs()).fold(0.0_f64, f64::max);
        if psi_step_inf <= self.fixed_point_tolerance && psi_grad_inf > self.fixed_point_tolerance {
            return Err(ObjectiveEvalError::recoverable(format!(
                "{} HybridEFS ψ nonstationary: ||Δψ||∞={:.3e} <= tol={:.3e} \
                 but raw ||gψ||∞={:.3e} (rho_dim={}, psi_dim={}, n_params={}, cost={:.6e})",
                EFS_FIRST_ORDER_FALLBACK_MARKER,
                psi_step_inf,
                self.fixed_point_tolerance,
                psi_grad_inf,
                self.layout.rho_dim(),
                self.layout.psi_dim,
                self.layout.n_params,
                cost,
            )));
        }
        Ok(())
    }
}

/// Maximum number of α halvings for the cost line search wrapping the EFS
/// step.
///
/// The Wood–Fasiolo paper proves that the EFS update direction is an *ascent
/// direction* for REML/LAML on penalty-like coordinates, but full-step
/// monotonicity is not guaranteed — both the original Fellner–Schall paper
/// and the extension recommend step-length control. We backtrack the entire
/// θ vector by halving α ∈ {1, 1/2, …, 1/2⁸ ≈ 0.004}, accepting the first
/// trial point with a strictly lower cost. With 8 halvings the smallest
/// trial step is ≈ 0.4% of the raw EFS step in every coordinate, which is
/// enough to clear pathologies near the identifiability boundary while
/// staying inside one cache-warm Hessian factorization budget.
pub(crate) const MAX_EFS_BACKTRACK: usize = 8;

/// Step components below this threshold (in θ-space) are treated as zero
/// for backtracking purposes — there is no point line-searching a step of
/// magnitude `1e-12`, and skipping the trial keeps the convergence path
/// numerically clean (no spurious cost decreases from ULP noise).
pub(crate) const EFS_NEGLIGIBLE_STEP: f64 = 1e-12;

/// Maximum infinity-norm of the EFS step (in θ-space) at which we skip the
/// cost line search and trust the multiplicative formula's quadratic
/// convergence. Above this, we always backtrack.
///
/// At small step magnitudes the canonical formula `Δρ = log((d−t)/q_eff)`
/// is itself a Newton step on the REML stationarity equation, with
/// quadratic local convergence. Under Wood–Fasiolo's Loewner-order
/// assumptions on the penalty derivative, sufficiently small steps are
/// always descent on `V`, so the line search would add an inner P-IRLS
/// solve per outer iteration with essentially zero chance of finding a
/// halving that beats the full step. The threshold is set to ~exp(0.5)
/// ≈ 1.65× change in any single λ_i (well inside the local-convergence
/// regime) and gates only the line-search call — the step itself is
/// applied unchanged, so correctness is preserved.
pub(crate) const EFS_LINESEARCH_THRESHOLD: f64 = 0.5;

/// Relative tolerance for the descent condition `c < current_cost` during
/// EFS backtracking. Without this, ULP-level cost noise near a fixed point
/// can cause spurious backtracking even when the step is mathematically
/// correct. We accept any trial whose cost is within
/// `EFS_COST_DESCENT_TOL · |current_cost|` of the current value.
pub(crate) const EFS_COST_DESCENT_TOL: f64 = 1e-12;

/// Maximum number of consecutive HybridEFS iterations whose ψ block was
/// zeroed before the bridge bails out and triggers a solver switch.
///
/// On hard problems (Matérn additive at large scale, Duchon60, anisotropic
/// joint penalties) a single zeroed-ψ iteration after exhausted backtracking
/// is already strong evidence the EFS ψ direction is not descent-correlated
/// at the current iterate; continuing on ρ alone with Δψ = 0 cannot enforce
/// ∇_ψ V = 0 and burns outer iterations on a non-stationary direction.
/// Bail out immediately so the fallback ladder routes to a joint
/// gradient-based solver (BFGS / L-BFGS) where ψ stationarity is part of
/// the optimality condition.
pub(crate) const MAX_CONSECUTIVE_PSI_STAGNATION: usize = 1;

impl FixedPointObjective for OuterFixedPointBridge<'_> {
    fn eval_step(&mut self, x: &Array1<f64>) -> Result<FixedPointSample, ObjectiveEvalError> {
        self.layout.validate_point_len(x, "outer EFS eval failed")?;
        let eval = match self.obj.eval_efs(x) {
            Ok(eval) => eval,
            Err(err @ EstimationError::GradientUnavailable { .. })
                if requests_immediate_first_order_fallback(&err.to_string()) =>
            {
                log::warn!(
                    "[STAGE] EFS -> gradient fallback: gradient unavailable at \
                     fixed-point dispatch; retrying with fixed-point disabled \
                     (rho_dim={}, psi_dim={}, n_params={})",
                    self.layout.rho_dim(),
                    self.layout.psi_dim,
                    self.layout.n_params,
                );
                return Err(ObjectiveEvalError::recoverable(format!(
                    "outer EFS eval failed: {err}"
                )));
            }
            // A REML/inner-solve failure is POINT-LOCAL evidence about this ρ —
            // another seed can solve fine — so it must surface as recoverable:
            // at seed evaluation the driver maps it to SeedRejected and the
            // seed cascade tries the next candidate without spending the
            // rejected slot's budget (#2367). Blanket-classifying it fatal
            // turned one invalid startup seed into a whole-fit
            // "Fatal outer-objective evaluation failure". Structural errors
            // (dimension mismatches, invalid layouts) remain fatal below.
            Err(err @ EstimationError::RemlOptimizationFailed(_)) => {
                return Err(ObjectiveEvalError::recoverable(format!(
                    "outer EFS eval failed: {err}"
                )));
            }
            Err(err) => return Err(into_objective_error("outer EFS eval failed", err)),
        };
        self.layout
            .validate_efs_eval(&eval, "outer EFS eval failed")?;
        if !eval.cost.is_finite() {
            return Err(ObjectiveEvalError::recoverable(
                "outer EFS eval failed: objective returned a non-finite cost".to_string(),
            ));
        }
        // Reject non-finite EFS step components at the bridge boundary with
        // full diagnostic context (which coord, its value, and whether it is
        // a ρ or ψ coord). Without this, a NaN/Inf step flows into the
        // hybrid-EFS backtrack loop, which halves it via `NaN * 0.5^k = NaN`
        // until backtracking exhausts, then silently zeros the ψ block and
        // applies only the ρ step — masking the analytic-gradient bug that
        // produced the NaN. The opt crate's FixedPoint::run also detects
        // this downstream (opt 0.2.2 lib.rs:4949) but surfaces only the bare
        // `NonFiniteStep` variant with no context, which is not actionable.
        if let Some((idx, value)) = eval.steps.iter().enumerate().find(|(_, v)| !v.is_finite()) {
            let psi_indices = eval.psi_indices.as_deref();
            let coord_kind = match psi_indices {
                Some(indices) if indices.contains(&idx) => "ψ",
                Some(_) => "ρ/τ",
                None => "ρ",
            };
            return Err(ObjectiveEvalError::recoverable(format!(
                "outer EFS eval failed: non-finite {coord_kind} step at coord {idx} \
                 (step[{idx}]={value}, rho_dim={}, psi_dim={}, n_params={}, cost={:.6e})",
                self.layout.rho_dim(),
                self.layout.psi_dim,
                self.layout.n_params,
                eval.cost,
            )));
        }
        if let Some(ref barrier_cfg) = self.barrier_config
            && let Some(ref beta) = eval.beta
        {
            // Scale-free precondition check for EFS. Wood–Fasiolo's
            // multiplicative log-λ update is derived under the
            // assumption that the inner Hessian is ≈ X'WX + S. A log
            // barrier adds τ/(β_j−l_j)² to the Hessian diagonal at the
            // constrained coords; when the tightest slack is much
            // smaller than the typical slack, that diagonal becomes
            // locally dominant and the EFS direction is no longer
            // guaranteed-ascent. Comparing slack *ratios* is
            // dimensionless — independent of τ, β scale, and the
            // inner-Hessian magnitude — which is exactly the regime
            // change EFS cannot represent. The earlier criterion
            // `barrier_curvature_is_significant(β, ref_diag=1.0, 0.01)`
            // was dimensionful and depended on three quantities the
            // bridge has no way to set correctly.
            //
            // Two principled triggers, each catching a distinct
            // failure mode of the EFS precondition:
            //  • `ratio = 0.1`        — asymmetric concentration:
            //    the worst slack is ≥10× tighter than the median.
            //    Catches the common "one coefficient hits its bound
            //    while others stay healthy" case.
            //  • `saturation = 1.0`   — absolute saturation:
            //    `max_j τ/Δ_j² ≥ 1`, i.e. at least one barrier-
            //    diagonal entry has reached the natural unit penalty
            //    scale. Catches the symmetric near-boundary regime
            //    that ratio-only checks would let through (median Δ
            //    also small, so min/median ratio stays near 1, but
            //    EFS's "ignore the barrier diagonal" assumption is
            //    still violated everywhere on the active set).
            const LOCAL_CONCENTRATION_RATIO: f64 = 0.1;
            const BARRIER_CURVATURE_SATURATION: f64 = 1.0;
            const BARRIER_CURVATURE_RELATIVE_THRESHOLD: f64 = 0.05;
            if let Some(hessian_scale) = eval.inner_hessian_scale
                && hessian_scale.is_finite()
                && hessian_scale > 0.0
                && barrier_cfg.barrier_curvature_is_significant(
                    beta,
                    hessian_scale,
                    BARRIER_CURVATURE_RELATIVE_THRESHOLD,
                )
            {
                return Err(ObjectiveEvalError::recoverable(format!(
                    "{} EFS barrier curvature significant relative to inner Hessian \
                         (rho_dim={}, psi_dim={}, n_params={}, cost={:.6e}, ref_diag={:.3e})",
                    EFS_FIRST_ORDER_FALLBACK_MARKER,
                    self.layout.rho_dim(),
                    self.layout.psi_dim,
                    self.layout.n_params,
                    eval.cost,
                    hessian_scale,
                )));
            }
            if barrier_cfg.barrier_curvature_locally_concentrated(
                beta,
                LOCAL_CONCENTRATION_RATIO,
                BARRIER_CURVATURE_SATURATION,
            ) {
                return Err(ObjectiveEvalError::recoverable(format!(
                    "{} EFS barrier curvature locally concentrated \
                         (rho_dim={}, psi_dim={}, n_params={}, cost={:.6e})",
                    EFS_FIRST_ORDER_FALLBACK_MARKER,
                    self.layout.rho_dim(),
                    self.layout.psi_dim,
                    self.layout.n_params,
                    eval.cost,
                )));
            }
        }
        let status = FixedPointStatus::Continue;

        let raw_step = Array1::from_vec(eval.steps);
        let psi_indices = eval.psi_indices.clone();
        self.reject_nonstationary_tiny_psi_step(
            &raw_step,
            psi_indices.as_deref(),
            eval.psi_gradient.as_ref(),
            eval.cost,
        )?;
        let max_step_abs = raw_step.iter().map(|s| s.abs()).fold(0.0_f64, f64::max);
        let current_cost = eval.cost;
        // #2241 — an objective may certify that consecutive inner solves
        // returned to the same banked incumbent after non-monotone boundary
        // mutations. One restore is merely a repair; a second consecutive
        // restore is the first possible recurrence and proves that another outer
        // update did not change the fitted state. Terminate on that model-state
        // certificate instead of guessing flatness from a relative-cost scalar
        // or exhausting an iteration budget. The current rho/cost is retained;
        // `FixedPointStatus::Stop` prevents the non-stationary proposal from
        // being applied.
        let restored_incumbent_recurred = match (
            self.last_restored_incumbent_streak,
            eval.consecutive_restored_incumbents,
        ) {
            (Some(previous), Some(current)) => current > previous && current > 1,
            _ => false,
        };
        self.last_restored_incumbent_streak = eval.consecutive_restored_incumbents;
        if restored_incumbent_recurred {
            let restores = eval.consecutive_restored_incumbents.unwrap_or_default();
            if psi_indices.is_some() {
                self.consecutive_psi_zero_iters = 0;
            }
            if let Ok(mut slot) = self.recurrent_incumbent_exit.lock() {
                *slot = Some(restores);
            }
            log::info!(
                "[OUTER] fixed-point convergence by recurrent restored incumbent: \
                 consecutive_restores={} cost={current_cost:.6e} rho_dim={} psi_dim={}",
                restores,
                self.layout.rho_dim(),
                self.layout.psi_dim,
            );
            return Ok(FixedPointSample {
                value: current_cost,
                step: raw_step,
                status: FixedPointStatus::Stop,
            });
        }
        if self.fixed_point_step_converged(x, &raw_step, psi_indices.as_deref()) {
            if psi_indices.is_some() {
                self.consecutive_psi_zero_iters = 0;
            }
            return Ok(FixedPointSample {
                value: current_cost,
                step: raw_step,
                status: FixedPointStatus::Stop,
            });
        }

        // Negligible raw step — the iteration is at (or numerically
        // indistinguishable from) a fixed point. Pass it through so the
        // outer step-norm convergence check fires; no point evaluating the
        // cost at x + 1e-30·s to chase ULP-level "improvements".
        if max_step_abs < EFS_NEGLIGIBLE_STEP {
            if psi_indices.is_some() {
                self.consecutive_psi_zero_iters = 0;
            }
            return Ok(FixedPointSample {
                value: current_cost,
                step: raw_step,
                status,
            });
        }

        // Small-step fast path. The canonical Wood–Fasiolo formula is
        // locally quadratically convergent, so once we are inside the
        // multiplicative-Newton basin (`||Δθ||∞ < EFS_LINESEARCH_THRESHOLD`)
        // a halving is essentially never accepted over the full step. Skip
        // the inner P-IRLS solve we'd otherwise burn on backtracking. When a
        // barrier is configured, every accepted rho-step must still pass
        // through the barrier-aware cost because feasibility can change even
        // under a small smoothing-parameter move. For hybrid runs we still
        // need to reset the ψ-stagnation counter.
        if self.barrier_config.is_none() && max_step_abs < EFS_LINESEARCH_THRESHOLD {
            if psi_indices.is_some() {
                self.consecutive_psi_zero_iters = 0;
            }
            return Ok(FixedPointSample {
                value: current_cost,
                step: raw_step,
                status,
            });
        }

        // ── Stage 1: full-vector cost backtracking ──
        //
        // Wood–Fasiolo gives ascent in the EFS direction but not full-step
        // monotonicity, so backtrack α ∈ {1, 1/2, …} on the *whole* step
        // vector (not just ψ). This is a uniform requirement: even on the
        // pure-ρ path, the additive log-λ formula is exact only at the
        // fixed point and is otherwise just a Newton-flavoured Wood–Fasiolo
        // surrogate that benefits from line search at large iterations.
        if let Some(scaled) = self.efs_backtrack(x, &raw_step, current_cost, MAX_EFS_BACKTRACK)? {
            if psi_indices.is_some() {
                self.consecutive_psi_zero_iters = 0;
            }
            return Ok(FixedPointSample {
                value: current_cost,
                step: scaled,
                status,
            });
        }

        // ── Stage 2 (hybrid only): ψ-zeroed retry ──
        //
        // Full-vector backtracking exhausted means *every* α we tried gave
        // a worse cost. On the hybrid path, the most common cause is a
        // bad ψ direction polluting an otherwise-good ρ step (preconditioned
        // gradient step on a near-singular ψ-ψ Gram matrix overshoots).
        // Try the ρ/τ block alone with the same backtracking schedule. If
        // that succeeds, we make progress on ρ this iteration; the ψ
        // stagnation counter advances and triggers the joint-solver
        // fallback once it crosses MAX_CONSECUTIVE_PSI_STAGNATION.
        if let Some(psi_idx) = psi_indices.as_ref() {
            let mut rho_only = raw_step.clone();
            for &i in psi_idx {
                rho_only[i] = 0.0;
            }
            let max_rho_abs = rho_only.iter().map(|s| s.abs()).fold(0.0_f64, f64::max);
            if max_rho_abs >= EFS_NEGLIGIBLE_STEP
                && let Some(scaled) =
                    self.efs_backtrack(x, &rho_only, current_cost, MAX_EFS_BACKTRACK)?
            {
                self.consecutive_psi_zero_iters = self.consecutive_psi_zero_iters.saturating_add(1);
                log::info!(
                    "[HYBRID-EFS] full-vector backtrack exhausted; ρ/τ-only step \
                         accepted. Consecutive ψ-zero iters = {}",
                    self.consecutive_psi_zero_iters,
                );
                if self.consecutive_psi_zero_iters >= MAX_CONSECUTIVE_PSI_STAGNATION {
                    log::info!(
                        "[STAGE] HybridEFS -> joint gradient (BFGS/L-BFGS) fallback: \
                             {} consecutive ψ-zero iterations after exhausted backtracking \
                             (rho_dim={}, psi_dim={}, n_params={}, cost={:.6e})",
                        self.consecutive_psi_zero_iters,
                        self.layout.rho_dim(),
                        self.layout.psi_dim,
                        self.layout.n_params,
                        current_cost,
                    );
                    return Err(ObjectiveEvalError::recoverable(format!(
                        "{} HybridEFS ψ stagnation: {} consecutive iterations \
                             exhausted backtracking and zeroed ψ step \
                             (rho_dim={}, psi_dim={}, n_params={}, cost={:.6e})",
                        EFS_FIRST_ORDER_FALLBACK_MARKER,
                        self.consecutive_psi_zero_iters,
                        self.layout.rho_dim(),
                        self.layout.psi_dim,
                        self.layout.n_params,
                        current_cost,
                    )));
                }
                return Ok(FixedPointSample {
                    value: current_cost,
                    step: scaled,
                    status,
                });
            }
            // ρ/τ-only backtracking also failed — surface the joint-solver
            // fallback marker so the runner abandons EFS for this attempt.
            log::info!(
                "[STAGE] HybridEFS -> joint gradient fallback: ρ/τ-only step also \
                 failed all {} halvings (rho_dim={}, psi_dim={}, n_params={}, \
                 cost={:.6e})",
                MAX_EFS_BACKTRACK,
                self.layout.rho_dim(),
                self.layout.psi_dim,
                self.layout.n_params,
                current_cost,
            );
            return Err(ObjectiveEvalError::recoverable(format!(
                "{} HybridEFS step rejected after {} halvings on full vector \
                 and {} halvings on ρ/τ-only fallback \
                 (rho_dim={}, psi_dim={}, n_params={}, cost={:.6e})",
                EFS_FIRST_ORDER_FALLBACK_MARKER,
                MAX_EFS_BACKTRACK,
                MAX_EFS_BACKTRACK,
                self.layout.rho_dim(),
                self.layout.psi_dim,
                self.layout.n_params,
                current_cost,
            )));
        }

        // Pure-EFS path with full backtracking exhausted: there is no ψ
        // block to escape to. Surface the same fallback marker so the
        // runner switches to a gradient-based solver instead of looping.
        log::info!(
            "[STAGE] EFS -> gradient fallback: no α ∈ {{1, …, 2^-{}}} decreased the \
             cost (rho_dim={}, n_params={}, cost={:.6e})",
            MAX_EFS_BACKTRACK,
            self.layout.rho_dim(),
            self.layout.n_params,
            current_cost,
        );
        Err(ObjectiveEvalError::recoverable(format!(
            "{} EFS step rejected after {} halvings on pure-ρ vector \
             (rho_dim={}, n_params={}, cost={:.6e})",
            EFS_FIRST_ORDER_FALLBACK_MARKER,
            MAX_EFS_BACKTRACK,
            self.layout.rho_dim(),
            self.layout.n_params,
            current_cost,
        )))
    }
}

impl OuterFixedPointBridge<'_> {
    /// Backtrack the cost along `raw_step` by halving α ∈ {1, 1/2, …, 2^-k}
    /// up to `max_halvings` times. Returns `Some(α·raw_step)` for the first
    /// α that yields a strictly lower finite cost, `None` if every trial
    /// failed or evaluation errored. Eval errors at trial points are
    /// treated as step rejection (a common pathology in inner solves at
    /// over-aggressive λ jumps), not propagated.
    fn efs_backtrack(
        &mut self,
        x: &Array1<f64>,
        raw_step: &Array1<f64>,
        current_cost: f64,
        max_halvings: usize,
    ) -> Result<Option<Array1<f64>>, ObjectiveEvalError> {
        // Relaxed Armijo: accept any trial within ULP noise of the current
        // cost. Pure `<` rejects ULP-noise dithering on flat regions of V
        // and forces unnecessary halvings.
        let cost_floor = current_cost + EFS_COST_DESCENT_TOL * current_cost.abs().max(1.0);
        // `bt` counts trials so the accepted step can report its halving count
        // (trial `bt` runs at α = 2^-bt). An eval error is an INVALID trial
        // (`Ok(None)`): the search halves without consulting the acceptance
        // test, exactly as the pre-migration loop swallowed `Err(_)`.
        let mut bt = 0usize;
        let accepted = backtracking_line_search::<_, ObjectiveEvalError>(
            BacktrackConfig {
                max_steps: max_halvings + 1,
                ..BacktrackConfig::default()
            },
            |alpha| {
                bt += 1;
                let trial_step = raw_step * alpha;
                let trial = x + &trial_step;
                match self.obj.eval_cost(&trial) {
                    Ok(c) => {
                        if !(c.is_finite() && c <= cost_floor) {
                            log::trace!(
                                "[EFS] backtrack α=2^-{bt}={alpha:.4e}: trial cost {c:.6e} \
                                 not below current {current_cost:.6e}, halving",
                                bt = bt - 1,
                            );
                        }
                        Ok(Some((c, trial_step)))
                    }
                    Err(err) => {
                        log::trace!(
                            "[EFS] backtrack α=2^-{bt}={alpha:.4e}: trial eval failed \
                             ({err}), halving",
                            bt = bt - 1,
                        );
                        Ok(None)
                    }
                }
            },
            |_alpha, c| c.is_finite() && c <= cost_floor,
        )?;
        Ok(accepted.map(|step| {
            let halvings = bt - 1;
            if halvings > 0 {
                log::debug!(
                    "[EFS] backtrack accepted at α=2^-{halvings}={alpha:.4e} \
                     after {halvings} halvings (cost: {current_cost:.6e} → {c:.6e})",
                    alpha = step.step,
                    c = step.value,
                );
            }
            step.payload
        }))
    }

    fn fixed_point_step_converged(
        &self,
        x: &Array1<f64>,
        step: &Array1<f64>,
        psi_indices: Option<&[usize]>,
    ) -> bool {
        if x.len() != step.len() {
            return false;
        }
        for idx in 0..step.len() {
            let scale = match psi_indices {
                Some(indices) if indices.contains(&idx) => x[idx].abs().max(1.0),
                _ => 1.0,
            };
            let normalized = step[idx].abs() / scale;
            if !normalized.is_finite() || normalized > self.fixed_point_tolerance {
                return false;
            }
        }
        true
    }
}

pub(crate) fn solution_into_outer_result(
    solution: Solution,
    converged: bool,
    plan_used: OuterPlan,
) -> OuterResult {
    let mut result = OuterResult::new(
        solution.final_point,
        solution.final_value,
        solution.iterations,
        converged,
        plan_used,
    );
    result.final_grad_norm = solution.final_gradient_norm;
    result.final_gradient = solution.final_gradient;
    result.final_hessian = solution.final_hessian;
    result
}

pub(crate) fn outer_result_with_gradient_norm(
    rho: Array1<f64>,
    final_value: f64,
    iterations: usize,
    final_grad_norm: Option<f64>,
    converged: bool,
    plan_used: OuterPlan,
) -> OuterResult {
    let mut result = OuterResult::new(rho, final_value, iterations, converged, plan_used);
    result.final_grad_norm = final_grad_norm;
    result
}

pub(crate) fn outer_result_with_gradient(
    rho: Array1<f64>,
    final_value: f64,
    iterations: usize,
    final_grad_norm: Option<f64>,
    final_gradient: Option<Array1<f64>>,
    converged: bool,
    plan_used: OuterPlan,
) -> OuterResult {
    let mut result = outer_result_with_gradient_norm(
        rho,
        final_value,
        iterations,
        final_grad_norm,
        converged,
        plan_used,
    );
    result.final_gradient = final_gradient;
    result
}

use gam_problem::diagnostics::format_top_abs as format_top_abs_components;

pub(crate) fn bfgs_line_search_failure_message(
    context: &str,
    solution: &Solution,
    max_attempts: usize,
    failure_reason: impl std::fmt::Debug,
) -> String {
    let grad_norm = solution
        .final_gradient_norm
        .or_else(|| {
            solution
                .final_gradient
                .as_ref()
                .map(|gradient| gradient.iter().map(|v| v * v).sum::<f64>().sqrt())
        })
        .unwrap_or(f64::NAN);
    let gradient_detail = solution
        .final_gradient
        .as_ref()
        .map(|gradient| format_top_abs_components(gradient, "top_abs_gradient", 6))
        .unwrap_or_else(|| "top_abs_gradient=<unavailable>".to_string());
    format!(
        "{context}: BFGS line search failed; reason={failure_reason:?} \
         max_attempts={max_attempts} iterations={} final_value={:.6e} \
         |g|={:.3e} func_evals={} grad_evals={} {} {}",
        solution.iterations,
        solution.final_value,
        grad_norm,
        solution.func_evals,
        solution.grad_evals,
        format_top_abs_components(&solution.final_point, "top_abs_rho", 6),
        gradient_detail,
    )
}