pounce_sensitivity/

solver.rs

//! `Solver` — value-typed session API that holds an `IpoptApplication`,
//! its TNLP, and the converged KKT factor between calls.
//!
//! This is Phase 3a of the factor-reuse work tracked in
//! [pounce#16](https://github.com/jkitchin/pounce/issues/16). It is
//! the public surface for callers who want to:
//!
//! 1. Run a normal IPM solve, then
//! 2. Issue many cheap operations against the converged factor
//!    (`kkt_solve`, `parametric_step`) without going through the
//!    [`set_on_converged`] callback shape that [`crate::SensSolve`]
//!    requires.
//!
//! [`set_on_converged`]: pounce_algorithm::IpoptApplication::set_on_converged
//!
//! # Usage
//!
//! ```ignore
//! use pounce_sensitivity::Solver;
//! use std::cell::RefCell;
//! use std::rc::Rc;
//!
//! let app = make_configured_app();
//! let tnlp: Rc<RefCell<dyn TNLP>> = Rc::new(RefCell::new(MyTnlp));
//! let mut solver = Solver::new(app, tnlp);
//!
//! let status = solver.solve();
//! assert!(solver.converged().is_some());
//!
//! // Issue any number of back-solves against the same factor:
//! let dim = solver.kkt_dim().unwrap();
//! let mut lhs = vec![0.0; dim];
//! let rhs = vec![1.0; dim];
//! solver.kkt_solve(&rhs, &mut lhs).unwrap();
//!
//! // Parametric step with respect to a set of pinned equality
//! // constraints (same interpretation as [`crate::SensSolve`]):
//! let dx = solver.parametric_step(&[2, 3], &[-0.5, 0.0]).unwrap();
//! ```
//!
//! # Scope of Phase 3a
//!
//! - **In**: `solve()`, `converged()`, `kkt_solve()`, `parametric_step()`,
//!   `block_dims()` / `kkt_dim()`.
//! - **Deferred to Phase 3b**: `resolve()` (warm-start that reuses the
//!   linear backend pool) and the `parametric_mpc` /
//!   `sensitivity_session` example binaries.
//!   ([`Solver::compute_reduced_hessian`] has since landed on the Solver
//!   and is no longer `SensSolve`-only.)

use std::cell::{Ref, RefCell};
use std::rc::Rc;

use pounce_algorithm::application::IpoptApplication;
use pounce_common::types::{Index, Number};
use pounce_nlp::TNLP;
use pounce_nlp::return_codes::ApplicationReturnStatus;

use crate::PdSensBacksolver;
use crate::activity::{ActivityReport, ReducedActivityReport, ReducedRowActivityReport};
use crate::backsolver::SensBacksolver;
use crate::boundcheck::PathOperator;
use crate::index::{FullXSlice, VarToFull, VarX};
use crate::schur_data::IndexSchurData;
use crate::sens_app::{SensApplication, SensOptions};
use crate::vec_util::dense_to_vec;

/// Sign of the barrier correction term, set from a comparison
/// against sIPOPT rather than derived.
pub const BARRIER_SIGN: Number = -1.0;

/// The bound geometry the bound-aware parametric steps share. See
/// [`Solver::bound_context`].
struct BoundContext {
    /// Length of the `x` block: how much of the box below is
    /// variables, and how much of a step is the caller's answer.
    n_x: usize,
    /// Lower bounds over the `(x, s)` prefix of the compound KKT
    /// vector, in the model's own units.
    ///
    /// The `s` half is the limits of the inequality rows, `d_l`. A
    /// limit written as a constraint — `g(x) <= cap` — is a bound like
    /// any other, on the row's slack instead of on a variable, and
    /// leaving it out of the box is what let a step walk straight
    /// through a cap with no breakpoint and no warning (gh#928). The
    /// two blocks are adjacent in the compound vector, so the box is
    /// one contiguous slice and a consumer needs no second index
    /// space; `n_x` is where it changes meaning.
    lo: Vec<Number>,
    /// Upper bounds, likewise: variable `x_u` then row `d_u`.
    hi: Vec<Number>,
    /// The converged point over that same `(x, s)` prefix.
    x_curr: Vec<Number>,
    /// How far outside a bound still counts as on it.
    eps: Number,
    /// How far negative a bound multiplier has to go before its bound
    /// is released. Always the solve's own margin, whatever `eps` is.
    release_eps: Number,
    /// Bound multipliers at the base point, in the solve's own
    /// coordinates, with the compound row each occupies: `z_l`, `z_u`,
    /// then `v_l`, `v_u`. The `v` half is what lets an active
    /// constraint limit be *released* when a step drives its
    /// multiplier through zero, the mirror of the `s` half of the box.
    mults: Vec<crate::boundcheck::BoundMultiplier>,
}

impl BoundContext {
    /// Distance from the base point to each bound, for one var-x row.
    ///
    /// Typed because the callers read a full-x `ActivityReport` in the
    /// same scope: indexing `lo` / `hi` / `x_curr` with a full-x value
    /// is the swap `crate::index` exists to prevent. Those three now
    /// run past the `x` block into `s`, so the type is doing more work
    /// than it was: a var-x row is in range for the whole box and a
    /// full-x row that overshoots `n_x` no longer runs off the end,
    /// it silently reads a slack.
    fn slacks_at(&self, row: VarX) -> (Number, Number) {
        let i = row.get();
        (self.x_curr[i] - self.lo[i], self.hi[i] - self.x_curr[i])
    }
}

/// Errors returned by post-convergence operations on [`Solver`].
#[derive(Debug, Clone)]
#[non_exhaustive]
pub enum SolverError {
    /// The solver has not yet converged, or the last solve failed
    /// before producing a usable KKT factor.
    NotConverged,
    /// An input slice's length did not match the KKT dimension or the
    /// parameter count.
    BadShape {
        /// Human description of the mismatched buffer.
        what: &'static str,
        /// Length the caller passed.
        got: usize,
        /// Length expected.
        expected: usize,
    },
    /// The underlying back-solve failed (singular factor, numerical
    /// breakdown).
    BacksolveFailed,
    /// The underlying [`SensApplication`] step failed (e.g. row mapping
    /// invalid for the current problem).
    SensComputationFailed(String),
    /// An option the requested computation depends on holds an
    /// incompatible value; the message names the option and the value
    /// required.
    BadOptions(String),
}

/// State captured at convergence: the user-visible iterate plus the
/// `PdSensBacksolver` that wraps the converged KKT factor.
///
/// Read this via [`Solver::converged`].
pub struct ConvergedState {
    /// IPM return status of the most recent solve.
    pub status: ApplicationReturnStatus,
    /// Final primal iterate `x*` (length `n_x`), in the user's own
    /// units: a `user-scaling` change of variables is undone here, so
    /// this is `x`, never the algorithm's `x̃ = d ⊙ x` (gh#486).
    pub x: Vec<Number>,
    /// Final objective value `f(x*)`.
    pub obj_val: Number,
    /// `bound_relax_factor` **as the solve that produced this state
    /// ran with it**, not as the application's options read today.
    /// The bounds were relaxed (or not) once, during this solve; a
    /// later `set_numeric_value` cannot change what the held slacks
    /// were measured against, so post-solve calls whose validity
    /// depends on unrelaxed bounds must guard on this value. See
    /// [`Solver::classify_activity`].
    pub bound_relax_factor: Number,
    /// Whether the solve computed exact Hessians, **as it ran**. A
    /// `limited-memory` solve's `IpoptData::w` is the quasi-Newton
    /// matrix, and there is no exact Hessian to evaluate at another
    /// point, so the corrector keeps that matrix instead of
    /// refreshing it at the predicted iterate.
    pub exact_hessian: bool,
    /// Converged KKT-factor wrapper. Owns `Rc` handles to the
    /// `PdFullSpaceSolver`, the IpoptData / Cq, and the NLP, so it
    /// outlives the IPM call frame.
    backsolver: PdSensBacksolver,
}

impl ConvergedState {
    /// Block dimensions of the compound KKT vector in
    /// `(x, s, y_c, y_d, z_l, z_u, v_l, v_u)` order.
    pub fn block_dims(&self) -> [usize; 8] {
        self.backsolver.block_dims()
    }

    /// Total dimension of the compound KKT vector (sum of `block_dims`).
    pub fn kkt_dim(&self) -> usize {
        self.backsolver.dim()
    }
}

/// Session-style solver: holds an [`IpoptApplication`], its TNLP, and
/// the converged factor between calls.
pub struct Solver {
    app: IpoptApplication,
    tnlp: Rc<RefCell<dyn TNLP>>,
    /// Side channel populated by the `on_converged` callback installed
    /// in [`Self::solve`]. The `RefCell<Option<…>>` shape mirrors the
    /// pattern in [`crate::convenience`] (the callback closure needs
    /// shared mutable access; the `Option` is `None` before the first
    /// solve and gets overwritten on each call).
    state: Rc<RefCell<Option<ConvergedState>>>,
}

impl Solver {
    /// Build a new session. The `app` should already have its options
    /// configured and `initialize()` called.
    pub fn new(app: IpoptApplication, tnlp: Rc<RefCell<dyn TNLP>>) -> Self {
        Self {
            app,
            tnlp,
            state: Rc::new(RefCell::new(None)),
        }
    }

    /// Borrow the underlying `IpoptApplication` (e.g. to read its
    /// options table after a solve). Mutation between `solve` calls is
    /// supported via [`Self::app_mut`].
    pub fn app(&self) -> &IpoptApplication {
        &self.app
    }

    /// Mutable borrow of the underlying `IpoptApplication`. Useful for
    /// reconfiguring options before a follow-up `solve()`. Note that
    /// changing options that affect the KKT linear system between
    /// calls will invalidate the cached factor; the next `solve()`
    /// rebuilds it.
    pub fn app_mut(&mut self) -> &mut IpoptApplication {
        &mut self.app
    }

    /// Run the IPM to convergence. On a successful solve the
    /// [`ConvergedState`] (including the KKT backsolver) is stashed
    /// inside the `Solver` and accessible via [`Self::converged`].
    ///
    /// Each call to `solve()` overwrites the previous converged
    /// state; the previously held factor is dropped.
    pub fn solve(&mut self) -> ApplicationReturnStatus {
        // Clear any previous state so a failed re-solve doesn't leave
        // a stale factor visible.
        self.state.borrow_mut().take();

        // Snapshot the options this solve will run under, before it
        // runs. `bound_relax_factor` is consumed once, when the NLP
        // relaxes its bounds; reading it back at query time would
        // describe the application's options rather than the state
        // being queried. The registry supplies its own default when
        // the option is unset, so no second copy of the default lives
        // here.
        let brf = self
            .app
            .options()
            .get_numeric_value("bound_relax_factor", "")
            .map(|(v, _)| v)
            .expect("bound_relax_factor is a registered core option");
        let exact_hessian = self
            .app
            .options()
            .get_string_value("hessian_approximation", "")
            .map(|(v, _)| v == "exact")
            .expect("hessian_approximation is a registered core option");

        let state_cb = Rc::clone(&self.state);
        // NOTE (gh#884 follow-up): `set_on_converged` fires once per
        // *attempt*. When a later attempt loses and an earlier one's answer
        // is replayed through the three-sink floor -- the mu fallback
        // (pounce#870, on by DEFAULT) or the gh#884 dual-divergence retry --
        // the converged KKT state this closure reads belongs to the
        // DISCARDED attempt, while the status, objective and statistics
        // reported alongside it are the winner's.
        // `IpoptApplication::answer_restored_from_floor()` reports that this
        // happened; the CLI's main path consults it and re-reads the point
        // from the `finalize_solution` payload. This site does not, because
        // what it needs is the factorization and the KKT state, which the
        // payload does not carry and which cannot be rewound. Pre-existing
        // and unfixed: a sensitivity result taken across a floored solve
        // describes the attempt that lost.
        self.app
            .set_on_converged(Box::new(move |data, cq, nlp, pd| {
                let curr = match data.borrow().curr.clone() {
                    Some(c) => c,
                    None => return,
                };
                let backsolver = match PdSensBacksolver::new(data, cq, nlp, Rc::clone(&pd)) {
                    Ok(b) => b,
                    Err(e) => {
                        // No session state is stored, so post-solve
                        // calls will report NotConverged; at least say
                        // why on stderr rather than failing silently.
                        eprintln!("pounce: Solver could not capture the KKT factor: {e}");
                        return;
                    }
                };
                // The algorithm's iterate is `x̃ = d ⊙ x` when the
                // solve ran under a change of variables (gh#486): this
                // capture reads the iterate, not the
                // `finalize_solution` payload, so it undoes the
                // substitution itself. The backsolver already read the
                // factors off the NLP, in this same var-x space.
                let mut x = dense_to_vec(&*curr.x);
                if let Some(d) = backsolver.variable_scaling() {
                    debug_assert_eq!(x.len(), d.len());
                    for (xi, &di) in x.iter_mut().zip(d.iter()) {
                        *xi /= di;
                    }
                }
                let obj_val = cq.borrow_mut().curr_f();
                // Status is overwritten with the real value after
                // optimize_tnlp returns.
                *state_cb.borrow_mut() = Some(ConvergedState {
                    status: ApplicationReturnStatus::InternalError,
                    x,
                    obj_val,
                    bound_relax_factor: brf,
                    exact_hessian,
                    backsolver,
                });
            }));

        let status = crate::optimize_tnlp_for_sensitivity(&mut self.app, Rc::clone(&self.tnlp));
        if let Some(s) = self.state.borrow_mut().as_mut() {
            s.status = status;
        }
        status
    }

    /// Borrow the converged state, if a successful solve has been
    /// run. Returns `None` if no solve has run or if the most recent
    /// solve failed before reaching convergence.
    pub fn converged(&self) -> Option<Ref<'_, ConvergedState>> {
        let r = self.state.borrow();
        r.as_ref()?;
        Some(Ref::map(r, |o| {
            o.as_ref()
                .unwrap_or_else(|| unreachable!("checked is_some above"))
        }))
    }

    /// Total dimension of the compound KKT vector (sum of
    /// `block_dims`). Returns `None` if no converged factor is held.
    pub fn kkt_dim(&self) -> Option<usize> {
        self.converged().map(|c| c.kkt_dim())
    }

    /// Block dimensions of the compound KKT vector in
    /// `(x, s, y_c, y_d, z_l, z_u, v_l, v_u)` order. Returns `None` if
    /// no converged factor is held.
    pub fn block_dims(&self) -> Option<[usize; 8]> {
        self.converged().map(|c| c.block_dims())
    }

    /// Classify every bounded variable and every finite-bounded
    /// inequality row of the converged solve by activity: see
    /// [`crate::activity`] and
    /// `dev-notes/covariance-information-roadmap.md` item 0 (gh #362).
    ///
    /// Requires the held solve to have run with `bound_relax_factor=0`
    /// (the Ipopt default is `1e-8`): with relaxed bounds the solver's
    /// slacks are measured against perturbed bounds, and the
    /// complementarity products the classifier reads no longer track
    /// `μ`.
    ///
    /// The guard reads
    /// [`ConvergedState::bound_relax_factor`] — the value that solve
    /// ran under — not the application's current options. Setting the
    /// option after the fact neither unlocks a state whose bounds were
    /// relaxed nor invalidates one whose bounds were not; re-solve to
    /// change the answer.
    ///
    /// # Neither classes' `q` is a reduced curvature
    ///
    /// A variable's ratio is `Σ_i/|H_ii|`, and at a kink the
    /// multiplier is generated by the curvature **reduced** along that
    /// coordinate, not by the diagonal. The two agree only where the
    /// coordinate is decoupled, so a genuine kink coupled to a
    /// neighbour reads [`AMBIGUOUS`](crate::activity::AMBIGUOUS) here
    /// at any tolerance (gh#763). Do not read that class as "probably
    /// not a kink": use [`Self::reduced_activity`], which normalizes
    /// by the reduced curvature at one back-solve per coordinate.
    ///
    /// A row's ratio divides by the curvature along the row's own
    /// gradient instead, which is a genuine directional curvature but
    /// still not a reduced one, so the same warning and the same
    /// remedy apply there: its ratio is `reduced/directional` and
    /// [`Self::reduced_row_activity`] answers the kink question
    /// (gh#804).
    pub fn classify_activity(&self) -> Result<ActivityReport, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let brf = state.bound_relax_factor;
        if brf != 0.0 {
            return Err(SolverError::BadOptions(format!(
                "classify_activity requires bound_relax_factor=0, but the \
                 held solve ran with {brf:e}: relaxed bounds shift the \
                 slacks the classifier reads. Set the option and solve() \
                 again — changing it now does not re-measure the slacks."
            )));
        }
        Ok(crate::activity::compute(&state.backsolver))
    }

    /// [`Self::classify_activity`]'s per-variable verdict for
    /// `user_vars`, re-measured against the **reduced** curvature
    /// along each coordinate instead of the Hessian diagonal — one
    /// back-solve against the held factor per variable (gh#763).
    ///
    /// `classify_activity` normalizes a variable's `Σ` by `H_ii`, but
    /// the multiplier at a kink is generated by the curvature left
    /// after the other free variables re-optimize. The two agree only
    /// where the coordinate is decoupled, so a genuine kink coupled to
    /// a neighbour reads [`AMBIGUOUS`](crate::activity::AMBIGUOUS)
    /// there at any tolerance — the ratio is `μ`-independent. Ask here
    /// and the same kink reads
    /// [`WEAKLY_ACTIVE`](crate::activity::WEAKLY_ACTIVE).
    ///
    /// Indices are **user space** (full-x), as the report's are. The
    /// intended call is over a report's ambiguous entries:
    ///
    /// ```ignore
    /// let report = solver.classify_activity()?;
    /// let ask: Vec<usize> = (0..report.var_status.len())
    ///     .filter(|&i| report.var_status[i] == AMBIGUOUS)
    ///     .collect();
    /// let refined = solver.reduced_activity(&ask)?;
    /// ```
    ///
    /// The cost is one back-solve per index, so it is a refinement to
    /// call over the entries in question, not over every bounded
    /// variable of a large model. See
    /// [`crate::activity::reduced_activity`] for the algebra and the
    /// edge cases.
    ///
    /// Requires the held solve to have run with `bound_relax_factor=0`
    /// for the same reason [`Self::classify_activity`] does: both read
    /// the slacks relaxed bounds shift.
    pub fn reduced_activity(
        &self,
        user_vars: &[usize],
    ) -> Result<ReducedActivityReport, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let brf = state.bound_relax_factor;
        if brf != 0.0 {
            return Err(SolverError::BadOptions(format!(
                "reduced_activity requires bound_relax_factor=0, but the \
                 held solve ran with {brf:e}: relaxed bounds shift the \
                 slacks the classifier reads. Set the option and solve() \
                 again — changing it now does not re-measure the slacks."
            )));
        }
        crate::activity::reduced_activity(&state.backsolver, user_vars).map_err(|e| match e {
            crate::activity::ReducedActivityError::OutOfRange { got, n_full_x } => {
                SolverError::BadShape {
                    what: "reduced_activity variable index",
                    got,
                    expected: n_full_x,
                }
            }
            crate::activity::ReducedActivityError::Backsolve => SolverError::BacksolveFailed,
        })
    }

    /// [`Self::classify_activity`]'s per-ROW verdict for `user_rows`,
    /// re-measured against the **reduced** curvature along each row's
    /// gradient instead of the directional curvature `∇dᵀH∇d/‖∇d‖²` —
    /// one back-solve against the held factor per row (gh#804).
    ///
    /// The row counterpart of [`Self::reduced_activity`], and the same
    /// defect one block over. A row's directional denominator is a
    /// genuine curvature along the row's own gradient — strictly
    /// better than the variable path's bare `H_ii`, which is why
    /// gh#763 fixed the variables first — but it is still not
    /// *reduced*: it does not account for the other free coordinates
    /// re-optimizing, and the multiplier is generated by what is left
    /// after they do. So a row's ratio there is
    /// `reduced/directional`, equal to `1` only where the row's
    /// direction is decoupled from the remaining free space, and a
    /// genuine row kink that is coupled reads
    /// [`AMBIGUOUS`](crate::activity::AMBIGUOUS) at any tolerance —
    /// the ratio is `μ`-independent. Ask here and the same kink reads
    /// [`WEAKLY_ACTIVE`](crate::activity::WEAKLY_ACTIVE).
    ///
    /// Indices are **user space** (full-g), as the report's are —
    /// equality rows included, which report
    /// [`EQUALITY`](crate::activity::EQUALITY) rather than being an
    /// error. The intended call is over a report's ambiguous rows:
    ///
    /// ```ignore
    /// let report = solver.classify_activity()?;
    /// let ask: Vec<usize> = (0..report.row_status.len())
    ///     .filter(|&j| report.row_status[j] == AMBIGUOUS)
    ///     .collect();
    /// let refined = solver.reduced_row_activity(&ask)?;
    /// ```
    ///
    /// The cost is one back-solve per index, so it is a refinement to
    /// call over the rows in question, not over every bounded row of a
    /// large model. See [`crate::activity::reduced_row_activity`] for
    /// the algebra and the edge cases.
    ///
    /// Requires the held solve to have run with `bound_relax_factor=0`
    /// for the same reason [`Self::classify_activity`] does: both read
    /// the slacks relaxed bounds shift.
    pub fn reduced_row_activity(
        &self,
        user_rows: &[usize],
    ) -> Result<ReducedRowActivityReport, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let brf = state.bound_relax_factor;
        if brf != 0.0 {
            return Err(SolverError::BadOptions(format!(
                "reduced_row_activity requires bound_relax_factor=0, but the \
                 held solve ran with {brf:e}: relaxed bounds shift the \
                 slacks the classifier reads. Set the option and solve() \
                 again — changing it now does not re-measure the slacks."
            )));
        }
        crate::activity::reduced_row_activity(&state.backsolver, user_rows).map_err(|e| match e {
            crate::activity::ReducedRowActivityError::OutOfRange { got, n_full_g } => {
                SolverError::BadShape {
                    what: "reduced_row_activity constraint index",
                    got,
                    expected: n_full_g,
                }
            }
            crate::activity::ReducedRowActivityError::Backsolve => SolverError::BacksolveFailed,
        })
    }

    /// The gradient of user constraint row `user_row` at the converged
    /// iterate, in user variable order (length `n_full_x`) and in
    /// **natural (unscaled) units**: the internal Jacobian row carries
    /// the solver's per-row `c_scale`/`d_scale`, which is divided out
    /// here, so this is the gradient of the row as the user wrote it.
    /// Equality and inequality rows alike; entries for fixed
    /// (`make_parameter`-removed) variables are 0 because the solve
    /// dropped their columns. Errors on an out-of-range row.
    ///
    /// Serves the covariance roadmap's item 1: a binding row's normal
    /// restricted to the fitted block is the projection direction.
    pub fn row_normal(&self, user_row: usize) -> Result<Vec<Number>, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        crate::activity::row_normal(&state.backsolver, user_row).map_err(|m| {
            SolverError::BadShape {
                what: "row_normal constraint index",
                got: user_row,
                expected: m,
            }
        })
    }

    /// The exact Lagrangian Hessian times a user-space vector, in
    /// user variable order and natural units (see
    /// [`crate::activity::hessian_vec`]). Errors on a length mismatch.
    pub fn hessian_vec(&self, v: &[Number]) -> Result<Vec<Number>, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        crate::activity::hessian_vec(&state.backsolver, v).map_err(|n| SolverError::BadShape {
            what: "hessian_vec vector length",
            got: v.len(),
            expected: n,
        })
    }

    /// Solve `K · lhs = rhs` against the converged KKT factor. Both
    /// slices must have length `kkt_dim()`; the layout is the flat
    /// `x || s || y_c || y_d || z_l || z_u || v_l || v_u` packing.
    ///
    /// `K` here is the **natural-units** (unscaled) KKT matrix: when
    /// the IPM solved with active NLP scaling, the backsolver scales
    /// the RHS/solution (all eight blocks, including the z/v
    /// bound-multiplier rows) so callers pass and receive data in the
    /// user's own units (pounce#128) — see
    /// [`crate::PdSensBacksolver::solve`]. For the raw scaled-space
    /// back-solve use [`Self::kkt_solve_scaled`].
    pub fn kkt_solve(&self, rhs: &[Number], lhs: &mut [Number]) -> Result<(), SolverError> {
        self.kkt_solve_impl(rhs, lhs, false)
    }

    /// [`Self::kkt_solve`] without the natural-units conjugation: the
    /// back-solve runs against the factor exactly as the IPM holds it
    /// (the solver's internal scaled space). Identical to `kkt_solve`
    /// when no NLP scaling is active. "Scaled space" includes a
    /// `user-scaling` change of variables (gh#486), so on such a solve
    /// the `x` and `z` blocks here are in the substituted coordinates
    /// `x̃ = d ⊙ x`, not the model's.
    pub fn kkt_solve_scaled(&self, rhs: &[Number], lhs: &mut [Number]) -> Result<(), SolverError> {
        self.kkt_solve_impl(rhs, lhs, true)
    }

    fn kkt_solve_impl(
        &self,
        rhs: &[Number],
        lhs: &mut [Number],
        scaled: bool,
    ) -> Result<(), SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let total = state.backsolver.dim();
        if rhs.len() != total {
            return Err(SolverError::BadShape {
                what: "rhs",
                got: rhs.len(),
                expected: total,
            });
        }
        if lhs.len() != total {
            return Err(SolverError::BadShape {
                what: "lhs",
                got: lhs.len(),
                expected: total,
            });
        }
        let ok = if scaled {
            state.backsolver.solve_scaled_space(rhs, lhs)
        } else {
            state.backsolver.solve(rhs, lhs)
        };
        if ok {
            Ok(())
        } else {
            Err(SolverError::BacksolveFailed)
        }
    }

    /// Batched-RHS back-solve. `rhs_flat` and `lhs_flat` are row-major
    /// `(n_rhs, kkt_dim)` buffers; each row is solved against the
    /// same converged factor. Equivalent in result to looping
    /// [`Self::kkt_solve`] but reuses one `IteratesVector` for the
    /// RHS and one for the result across all `n_rhs` calls — see
    /// [`crate::algorithm_backsolver::PdSensBacksolver::solve_many`].
    pub fn kkt_solve_many(
        &self,
        rhs_flat: &[Number],
        lhs_flat: &mut [Number],
        n_rhs: usize,
    ) -> Result<(), SolverError> {
        self.kkt_solve_many_impl(rhs_flat, lhs_flat, n_rhs, false)
    }

    /// [`Self::kkt_solve_many`] without the natural-units
    /// conjugation (the batched sibling of [`Self::kkt_solve_scaled`]).
    pub fn kkt_solve_many_scaled(
        &self,
        rhs_flat: &[Number],
        lhs_flat: &mut [Number],
        n_rhs: usize,
    ) -> Result<(), SolverError> {
        self.kkt_solve_many_impl(rhs_flat, lhs_flat, n_rhs, true)
    }

    fn kkt_solve_many_impl(
        &self,
        rhs_flat: &[Number],
        lhs_flat: &mut [Number],
        n_rhs: usize,
        scaled: bool,
    ) -> Result<(), SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let total = state.backsolver.dim();
        let expected = n_rhs * total;
        if rhs_flat.len() != expected {
            return Err(SolverError::BadShape {
                what: "rhs",
                got: rhs_flat.len(),
                expected,
            });
        }
        if lhs_flat.len() != expected {
            return Err(SolverError::BadShape {
                what: "lhs",
                got: lhs_flat.len(),
                expected,
            });
        }
        let ok = if scaled {
            state
                .backsolver
                .solve_many_scaled_space(rhs_flat, lhs_flat, n_rhs)
        } else {
            state.backsolver.solve_many(rhs_flat, lhs_flat, n_rhs)
        };
        if ok {
            Ok(())
        } else {
            Err(SolverError::BacksolveFailed)
        }
    }

    /// First-order parametric step `Δx ≈ ∂x*/∂p · Δp` for a set of
    /// pinned equality constraints. `pin_constraint_indices` are
    /// 0-based indices into the user's `g(x)`; `deltas` is the
    /// perturbation `Δp` (same length).
    ///
    /// Returns the `n_x`-long primal step. For the full KKT-space
    /// step, use [`Self::kkt_solve`] directly.
    pub fn parametric_step(
        &self,
        pin_constraint_indices: &[Index],
        deltas: &[Number],
    ) -> Result<Vec<Number>, SolverError> {
        if pin_constraint_indices.len() != deltas.len() {
            return Err(SolverError::BadShape {
                what: "deltas",
                got: deltas.len(),
                expected: pin_constraint_indices.len(),
            });
        }
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;

        // Map user g-indices to y_c rows through the NLP's c/d-split
        // permutation (pounce#128; matches `convenience.rs`).
        let dims = state.backsolver.block_dims();
        let n_x = dims[0];
        let param_rows = state
            .backsolver
            .map_pin_g_to_kkt_rows(pin_constraint_indices)
            .map_err(SolverError::SensComputationFailed)?;
        let signs = vec![1; pin_constraint_indices.len()];
        let a_data = IndexSchurData::from_parts(param_rows, signs)
            .map_err(|e| SolverError::SensComputationFailed(format!("{e:?}")))?;

        let opts = SensOptions {
            run_sens: true,
            ..SensOptions::default()
        };
        let sens_app = SensApplication::new(a_data, state.backsolver.clone(), opts);
        let n_full = state.backsolver.dim();
        let mut dx_full = vec![0.0; n_full];
        if !sens_app.parametric_step(deltas, &mut dx_full) {
            return Err(SolverError::SensComputationFailed(
                "SensApplication::parametric_step failed".into(),
            ));
        }
        // carry the step from the barrier problem's solution toward the
        // original problem's (the paper's equation 11)
        let corr = self.barrier_correction(state)?;
        for (d, c) in dx_full.iter_mut().zip(corr.iter()) {
            *d += *c * BARRIER_SIGN;
        }
        dx_full.truncate(n_x);
        Ok(dx_full)
        // NOTE: parametric_step_full below applies the same correction,
        // so the two agree on their shared block.
    }

    /// The right-hand side [`Self::parametric_step_full`] answers,
    /// barrier term included. That method adds the term as a correction
    /// to the solution rather than to the right-hand side, which is the
    /// same thing by linearity.
    ///
    /// The parameter rows go through `map_pin_g_to_kkt_rows` exactly as
    /// they do there. Passing the constraint indices raw instead puts
    /// the perturbation on the x rows, where it contributes nothing --
    /// a release then sees only its own multiplier shift and lands on
    /// the wrong answer without failing.
    fn parametric_rhs_full(
        &self,
        pin_constraint_indices: &[Index],
        deltas: &[Number],
    ) -> Result<Vec<Number>, SolverError> {
        let state = self.converged().ok_or(SolverError::NotConverged)?;
        let state = &*state;
        let dims = state.backsolver.block_dims();
        let n_full = state.backsolver.dim();
        let param_rows = state
            .backsolver
            .map_pin_g_to_kkt_rows(pin_constraint_indices)
            .map_err(SolverError::SensComputationFailed)?;
        let signs = vec![1; pin_constraint_indices.len()];
        let a_data = IndexSchurData::from_parts(param_rows, signs)
            .map_err(|e| SolverError::SensComputationFailed(format!("{e:?}")))?;
        let opts = SensOptions {
            run_sens: true,
            ..SensOptions::default()
        };
        let sens_app = SensApplication::new(a_data, state.backsolver.clone(), opts);
        let mut rhs = vec![0.0; n_full];
        if !sens_app.parametric_rhs(deltas, &mut rhs) {
            return Err(SolverError::SensComputationFailed(
                "SensApplication::parametric_rhs failed".into(),
            ));
        }
        let mu = state.backsolver.barrier_mu();
        let start = dims[0] + dims[1] + dims[2] + dims[3];
        let end = start + dims[4] + dims[5] + dims[6] + dims[7];
        for r in rhs.iter_mut().take(end).skip(start) {
            *r += mu * BARRIER_SIGN;
        }
        Ok(rhs)
    }

    /// The barrier correction of the parametric step: the paper's
    /// equation 11 term, which carries the step from the solution of
    /// the barrier problem at `mu > 0` toward the one at `mu = 0`.
    ///
    /// [`Self::parametric_step`] is taken against a factorization held
    /// at the final `mu`, so it estimates where the BARRIER problem's
    /// solution moves, not where the original problem's does. The two
    /// differ by `O(mu)`, which is negligible at a tight tolerance and
    /// is not at a loose one. Measured against sIPOPT on a nonlinear
    /// model, the uncorrected step agrees to 2e-9 at `tol = 1e-8` and
    /// differs by 9e-6 at `tol = 1e-3`.
    ///
    /// The term is one more backsolve against the same factor, with
    /// `mu` in the complementarity rows, which are the bound multiplier
    /// blocks of the compound vector.
    ///
    /// Returns the correction over the whole compound vector, to be
    /// added to the step.
    fn barrier_correction(&self, state: &ConvergedState) -> Result<Vec<Number>, SolverError> {
        let dims = state.backsolver.block_dims();
        let n_full = state.backsolver.dim();
        let mu = state.backsolver.barrier_mu();
        // z_l, z_u, v_l, v_u: the rows carrying the complementarity
        // conditions, which are the ones the barrier perturbs
        let start = dims[0] + dims[1] + dims[2] + dims[3];
        let end = start + dims[4] + dims[5] + dims[6] + dims[7];
        let mut rhs = vec![0.0; n_full];
        for r in rhs.iter_mut().take(end).skip(start) {
            *r = mu;
        }
        let mut corr = vec![0.0; n_full];
        if !state.backsolver.solve(&rhs, &mut corr) {
            return Err(SolverError::BacksolveFailed);
        }
        Ok(corr)
    }

    /// Parametric step with the bounds respected by pinning, not by
    /// clamping. Returns the `n_x`-long primal step, the rows it
    /// constrained to reach it, and why the refinement stopped.
    ///
    /// [`Self::parametric_step`] answers where the linear predictor
    /// points, which can be outside the box. Clamping a coordinate
    /// back to its bound leaves every other coordinate at its
    /// predictor value, so the answer is feasible but no longer
    /// consistent with the KKT relations. This instead adds a row
    /// pinning each offending coordinate at its bound and re-solves, so
    /// the others move to stay consistent under the pins, which is the
    /// refinement upstream runs under `sens_boundcheck`.
    ///
    /// A pass takes every crossing it can see, pins them together, and
    /// re-solves, so the loop ends when nothing is left outside rather
    /// than when the passes run out. Each pass rebuilds the Schur
    /// complement over the pins so far, so a pass carrying `k` of them
    /// costs one dense `k × k` solve and `k + 1` back-solves; the
    /// factorization itself is never rebuilt for a pin.
    ///
    /// What counts as outside a bound is the `eps` argument when the
    /// caller passes one, and the solve's own margin when it passes
    /// `None`: the solve was willing to leave a converged point
    /// `bound_relax_factor` outside its bound, so anything within that
    /// is on the bound. An unrelaxed solve gets a roundoff floor.
    ///
    /// Passes stop when nothing is outside its bound by that much, when
    /// a pin cannot be achieved because the pins have exhausted the
    /// problem's degrees of freedom, or at `max_iter`, which is a
    /// safety limit rather than a budget: it took one pin per pass
    /// until gh#732, where a model with more crossings than passes had
    /// its answer picked by the limit. None of those is an error, and
    /// the returned [`crate::boundcheck::RefineStop`] says which
    /// happened.
    pub fn parametric_step_bounded(
        &self,
        pin_constraint_indices: &[Index],
        deltas: &[Number],
        max_iter: usize,
        bound_eps: Option<Number>,
    ) -> Result<(Vec<Number>, Vec<Index>, crate::boundcheck::RefineStop), SolverError> {
        let dx_full = self.parametric_step_full(pin_constraint_indices, deltas)?;
        let rhs_plain = self.parametric_rhs_full(pin_constraint_indices, deltas)?;
        let ctx = self.bound_context(bound_eps)?;
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let (dx, pinned, stop) = crate::boundcheck::refine_step_onto_bounds(
            &state.backsolver,
            &dx_full,
            &ctx.x_curr,
            &ctx.lo,
            &ctx.hi,
            &ctx.mults,
            &rhs_plain,
            ctx.eps,
            ctx.release_eps,
            max_iter,
        )
        .map_err(SolverError::SensComputationFailed)?;
        Ok((
            dx[..ctx.n_x].to_vec(),
            pinned.into_iter().map(|p| p as Index).collect(),
            stop,
        ))
    }

    /// Parametric step applied a little at a time instead of taken
    /// whole, stopping wherever the active set changes and continuing
    /// from there under the new one. Returns the primal step and the
    /// breakpoints crossed.
    ///
    /// [`Self::parametric_step_bounded`] decides every condition at the
    /// base point, which is upstream's fix-relax. This is past it: the
    /// result is piecewise linear in the parameter, exact for a QP
    /// because a QP's solution is piecewise affine in the parameter,
    /// and still a predictor for an NLP because nothing is
    /// re-linearized between breakpoints.
    ///
    /// `max_iter` caps the breakpoints crossed. It is in practice a
    /// budget on factorizations, since a pin is a back-solve against
    /// the held factor while a release re-factors.
    pub fn parametric_step_path(
        &self,
        pin_constraint_indices: &[Index],
        deltas: &[Number],
        max_iter: usize,
    ) -> Result<(Vec<Number>, Vec<crate::boundcheck::PathSegment>), SolverError> {
        let rhs_plain = self.parametric_rhs_full(pin_constraint_indices, deltas)?;
        // Nothing here decides the weak rows -- that is what the
        // "decided" variant below is for -- but the walk still has to
        // be told which they are, or it reads their order-one sigma as
        // a bound the factorization enforces and lets the variable
        // walk out of its box (gh#852). A relaxed solve shifts the
        // slacks the classifier reads, so this comes back empty there
        // and the walk behaves as it did before -- the same silence
        // `weakly_active_bounds` hands every other caller.
        let weak_rows: Vec<usize> = self.weakly_active_bounds()?.iter().map(|w| w.row).collect();
        let ctx = self.bound_context(None)?;
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let (dx, segments) = crate::boundcheck::step_along_path(
            &state.backsolver,
            &rhs_plain,
            &ctx.x_curr,
            &ctx.lo,
            &ctx.hi,
            &ctx.mults,
            max_iter,
            &[],
            &[],
            &weak_rows,
            ctx.eps,
        )
        .map_err(SolverError::SensComputationFailed)?;
        Ok((dx[..ctx.n_x].to_vec(), segments))
    }

    /// [`Self::parametric_step_path`] with the weak-row
    /// decision supplied by the caller instead of searched for.
    /// `held_var_rows` names the var-x rows of the weakly active
    /// bounds the direction holds; every other weakly active bound is
    /// forced into the walk's base-activity table as a leaving row.
    /// A row left there is still reachable, so a caller that hands in
    /// an empty held list — every weak row declared a leaver, which is
    /// what an undecided study of the all-released step does — gets
    /// the bound back at the fraction the walk finds the direction
    /// pressing into it, rather than an answer outside the box
    /// (gh#852). Study surface for an externally solved eq. 14 QP.
    pub fn parametric_step_path_decided(
        &self,
        pin_constraint_indices: &[Index],
        deltas: &[Number],
        max_iter: usize,
        held_var_rows: &[Index],
    ) -> Result<(Vec<Number>, Vec<crate::boundcheck::PathSegment>), SolverError> {
        let weak = self.weakly_active_bounds()?;
        if weak.is_empty() {
            return self.parametric_step_path(pin_constraint_indices, deltas, max_iter);
        }
        let mut rhs_plain = self.parametric_rhs_full(pin_constraint_indices, deltas)?;
        for w in &weak {
            rhs_plain[w.row] = 0.0;
        }
        let held: std::collections::HashSet<usize> =
            held_var_rows.iter().map(|&r| r as usize).collect();
        let ctx = self.bound_context(None)?;
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let holds: Vec<(usize, bool)> = weak
            .iter()
            .filter(|w| held.contains(&w.var_row))
            .map(|w| (w.var_row, w.lower))
            .collect();
        let forced_active: Vec<usize> = weak
            .iter()
            .filter(|w| !held.contains(&w.var_row))
            .map(|w| w.row)
            .collect();
        // Every weak row, held or leaving. For a held one the flag is
        // inert -- it arrives released and pinned already -- and for a
        // leaving one it is what lets the walk take the bound back
        // when the direction turns out to press into it, which is the
        // whole of an empty held list on a holding perturbation
        // (gh#852).
        let weak_rows: Vec<usize> = weak.iter().map(|w| w.row).collect();
        let (dx, segments) = crate::boundcheck::step_along_path(
            &state.backsolver,
            &rhs_plain,
            &ctx.x_curr,
            &ctx.lo,
            &ctx.hi,
            &ctx.mults,
            max_iter,
            &forced_active,
            &holds,
            &weak_rows,
            ctx.eps,
        )
        .map_err(SolverError::SensComputationFailed)?;
        Ok((dx[..ctx.n_x].to_vec(), segments))
    }

    /// Newton iterations on the barrier system, refining a step that
    /// some mode already produced.
    ///
    /// `step` is a full compound step, the shape
    /// [`Self::parametric_step_full`] returns, so any mode's result
    /// can be handed in. Every correction pays one derivative
    /// evaluation and one factorization at the predicted point, and
    /// each iteration after that costs one back-solve. Returns the
    /// refined step and a [`CorrectorReport`] saying what the
    /// iterations bought.
    ///
    /// The corrector aims at the barrier solution at the μ the solve
    /// finished on, not at a re-solve, so the accuracy it can reach is
    /// bounded by that offset. Its operator is assembled at the
    /// PREDICTED point, every block: the Hessian, the constraint
    /// Jacobians, and the barrier diagonal all evaluated at the
    /// stepped iterate with the step's own multipliers, and the
    /// predictor's active set applied to the diagonal in that frame.
    /// A base solve the sigma ceiling (gh#737) touched, or one that
    /// crossed over into the declared frame (gh#654), is no
    /// exception: both rules are re-derived at the predicted point
    /// rather than read from the base-point diagonals stored for the
    /// held factor's own back-solves.
    /// A chord iteration contracts at the rate the distance between
    /// its operator and the true Jacobian sets, and the predicted
    /// point is where the truth is. Under a `limited-memory` solve
    /// the quasi-Newton matrix is kept as is, since no exact Hessian
    /// exists to evaluate elsewhere. Where the perturbation needs a
    /// bound to leave the active set that the step's endpoint does
    /// not show, no released row is applied: the step's clamped
    /// multiplier leaves a weak diagonal entry there, the iterations
    /// can move the coordinate partway off the bound, and the answer
    /// is not the re-solve. The release-deciding modes are the ones
    /// that cross exactly. `CorrectorReport::improved` reports
    /// whether the residual fell; when it did not, the step handed
    /// back is the caller's own.
    ///
    /// The returned point always satisfies the variable bounds, since
    /// the barrier residual is undefined outside them and the
    /// fraction-to-boundary rule keeps every iterate inside. A step
    /// that arrives pointing out of the box is therefore put back in
    /// before the first iteration, which means `max_iter = 0` is not a
    /// no-op: it costs the derivative evaluation and the residual
    /// evaluation, no back-solve, and reports the residual the
    /// caller's step leaves.
    ///
    /// Errors with [`SolverError::SensComputationFailed`] when the
    /// barrier residual at that starting point is not finite, which is
    /// what a predicted point outside the domain of one of the model's
    /// functions gives (gh#845). A *declared bound* is protection here,
    /// since the clamp above puts the coordinate back inside it; a
    /// variable held in a function's domain by a **constraint** has no
    /// bound to be put back inside, and an ordinary `log`, `sqrt` or
    /// reciprocal is then reachable by a large enough perturbation.
    /// There is no correction to make from such a point, so it is an
    /// error rather than a report -- and never a step full of NaN
    /// carrying `residual = 0.0` and `converged = true`.
    pub fn correct_step(
        &self,
        pin_constraint_indices: &[Index],
        deltas: &[Number],
        step: &[Number],
        max_iter: usize,
    ) -> Result<(Vec<Number>, crate::corrector::CorrectorReport), SolverError> {
        let ctx = self.bound_context(None)?;
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let bs = &state.backsolver;
        let dim = bs.dim();
        if step.len() != dim {
            return Err(SolverError::BadShape {
                what: "step",
                got: step.len(),
                expected: dim,
            });
        }
        // `>= 0`, not `> 0`: `barrier_mu` reports exactly zero for a
        // point whose bound multipliers were zeroed on the way out (see
        // its doc comment), and that is a barrier level, not a missing
        // one. The complementarity rows are then already satisfied
        // where they stand, which is what the corrector should measure.
        let mu = {
            let m = bs.barrier_mu();
            if m >= 0.0 && m.is_finite() {
                m
            } else {
                return Err(SolverError::SensComputationFailed(
                    "corrector: the solve reported no barrier parameter".into(),
                ));
            }
        };
        let base = {
            let mut flat = vec![0.0; dim];
            bs.curr_flat(&mut flat).map_err(|_| {
                SolverError::SensComputationFailed(
                    "corrector: converged iterate unavailable".into(),
                )
            })?;
            flat
        };
        // The pinned equalities' KKT rows and the row scales the
        // algorithm applied to them. A user `g` index is not the KKT
        // row: the two differ once an inequality precedes the pin in
        // `g(x)` (pounce#128), and the residual the corrector measures
        // sits in the algorithm's scaled equality block, so the deltas
        // have to carry the same factors.
        let (pin_rows, pin_scales) = bs
            .pin_rows_and_c_scales(pin_constraint_indices)
            .map_err(SolverError::SensComputationFailed)?;
        let pin_rows: Vec<usize> = pin_rows.iter().map(|&r| r as usize).collect();
        let scaled_deltas: Vec<Number> = deltas
            .iter()
            .zip(&pin_scales)
            .map(|(&d, &c)| d * c)
            .collect();
        crate::corrector::run(
            bs,
            &base,
            step,
            &pin_rows,
            &scaled_deltas,
            &ctx.lo,
            &ctx.hi,
            mu,
            max_iter,
            state.exact_hessian,
        )
    }

    /// [`Self::parametric_step_bounded`] with the weak-row
    /// decision supplied by the caller instead of searched for. The
    /// direction is computed for the given working set (all weak rows
    /// released, the held variables pinned through Schur rows), then
    /// refined onto the bounds exactly as the searched variant does.
    /// Study surface for an externally solved eq. 14 QP.
    pub fn parametric_step_bounded_decided(
        &self,
        pin_constraint_indices: &[Index],
        deltas: &[Number],
        max_iter: usize,
        held_var_rows: &[Index],
        bound_eps: Option<Number>,
    ) -> Result<(Vec<Number>, Vec<Index>, crate::boundcheck::RefineStop), SolverError> {
        let weak = self.weakly_active_bounds()?;
        if weak.is_empty() {
            return self.parametric_step_bounded(
                pin_constraint_indices,
                deltas,
                max_iter,
                bound_eps,
            );
        }
        let rhs_plain = self.parametric_rhs_full(pin_constraint_indices, deltas)?;
        let ctx = self.bound_context(bound_eps)?;
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let released: Vec<usize> = weak.iter().map(|w| w.row).collect();
        let pinned_rows: Vec<usize> = held_var_rows.iter().map(|&r| r as usize).collect();
        let (d, _) = crate::boundcheck::path_direction(
            &state.backsolver,
            &rhs_plain,
            &released,
            &pinned_rows,
        )
        .map_err(SolverError::SensComputationFailed)?;
        let (dx, pinned, stop) = crate::boundcheck::refine_step_onto_bounds(
            &state.backsolver,
            &d,
            &ctx.x_curr,
            &ctx.lo,
            &ctx.hi,
            &ctx.mults,
            &rhs_plain,
            ctx.eps,
            ctx.release_eps,
            max_iter,
        )
        .map_err(SolverError::SensComputationFailed)?;
        Ok((
            dx[..ctx.n_x].to_vec(),
            pinned.into_iter().map(|p| p as Index).collect(),
            stop,
        ))
    }

    /// [`crate::boundcheck::path_direction`] for a working set the
    /// caller names, and the force each held row carries under it.
    /// Study surface, and the seam the two pins are measured against
    /// each other through.
    ///
    /// `released_bound_rows` are compound bound-multiplier rows, as
    /// [`Self::weakly_active_bounds`] reports them;
    /// `held_primal_rows` are primal rows -- `x` block for a
    /// variable, `s` block for a constraint's own limit (gh#928).
    ///
    /// `operator` picks which *operator* the walk's (single, exact)
    /// Schur pin is applied to.
    /// [`Plain`](PathOperator::Plain) is the released system as the
    /// factorization already holds it -- one factorization for the
    /// whole segment, and no inverse at all when releasing two
    /// curvature-free variables that share a row leaves the
    /// stationarity rows dependent (gh#930).
    /// [`Regularized`](PathOperator::Regularized) raises the pinned
    /// diagonals until it is invertible, at the cost of rebuilding
    /// the diagonal per solve, so the factorization cache misses.
    /// [`Preferred`](PathOperator::Preferred) is what the walk itself
    /// runs: the plain one, falling back when it fails *or when its
    /// pins do not take*, which is not the same test -- see
    /// [`crate::boundcheck::path_direction`].
    ///
    /// Both operators give the same answer: the Schur row enforces
    /// `Eᵀ w = 0`, which annihilates the added diagonal, so the
    /// system solved is the released one either way and both return
    /// values agree in value, frame and units.
    /// `issue_930_two_curvature_free_releases.rs` measures that.
    pub fn path_direction_decided(
        &self,
        pin_constraint_indices: &[Index],
        deltas: &[Number],
        released_bound_rows: &[Index],
        held_primal_rows: &[Index],
        operator: PathOperator,
    ) -> Result<(Vec<Number>, Vec<Number>), SolverError> {
        let rhs_plain = self.parametric_rhs_full(pin_constraint_indices, deltas)?;
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let released: Vec<usize> = released_bound_rows.iter().map(|&r| r as usize).collect();
        let held: Vec<usize> = held_primal_rows.iter().map(|&r| r as usize).collect();
        crate::boundcheck::path_direction_with(
            &state.backsolver,
            &rhs_plain,
            &released,
            &held,
            operator,
        )
        .map_err(SolverError::SensComputationFailed)
    }

    /// The all-released step: the plain parametric step solved with
    /// every weakly active bound's row released, and nothing decided.
    ///
    /// This is [`Self::parametric_step_directional`]'s first
    /// back-solve returned as the answer instead of refined. The
    /// caller trades the engagement's budget for whatever violations
    /// the released direction carries at weak bounds the perturbation
    /// actually holds, which come back as crossings for the mode's
    /// clamp, pins, or path segments, or for a correction, to handle.
    /// A clean base point takes the plain step. Returns the direction
    /// over the model's variables and the number of rows released.
    pub fn parametric_step_release_all(
        &self,
        pin_constraint_indices: &[Index],
        deltas: &[Number],
    ) -> Result<(Vec<Number>, usize), SolverError> {
        let rhs_plain = self.parametric_rhs_full(pin_constraint_indices, deltas)?;
        let weak = self.weakly_active_bounds()?;
        let ctx = self.bound_context(None)?;
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let released: Vec<usize> = weak.iter().map(|w| w.row).collect();
        let mut d = vec![0.0; state.backsolver.dim()];
        // solve_released is the whole mechanism: an empty released set
        // is the plain solve, and shift = false matches the
        // directional path's all-released solve, whose rationale lives
        // on `solve_released_inner`.
        if !state
            .backsolver
            .solve_released(&released, &rhs_plain, &mut d)
        {
            return Err(SolverError::BacksolveFailed);
        }
        Ok((d[..ctx.n_x].to_vec(), released.len()))
    }

    /// The eq. 14 directional derivative, decided by pounce-qp over
    /// the weak rows the direction engages.
    ///
    /// One released factorization serves the whole decision: the
    /// released `Σ` is built once and every solve passes the same
    /// object, so the factorization cache reuses the factor across the
    /// all-released direction and the basis columns. The decision
    /// itself is the dual of eq. 14 restricted to the weak rows the
    /// direction engages: with `a_k` the signed unit vector of weak
    /// row `k` (positive for a lower bound), `X_k = K_rel^{-1} a_k`,
    /// `S = aᵀX` and `m = aᵀd0`, the pin forces `λ` solve
    ///
    /// ```text
    ///     min  ½ λᵀ S λ + mᵀ λ    s.t.  λ ≥ 0
    /// ```
    ///
    /// whose KKT conditions are eq. 14's complementarity: a released
    /// row moves to its feasible side (the QP gradient `Sλ + m ≥ 0`)
    /// and a held row's pin force is nonnegative. Rows outside the
    /// engaged set are verified against the decided direction and the
    /// set expands until no new row violates. Nothing reads the
    /// perturbation's size, so the decision is linear in the step.
    ///
    /// An engaged row is decided only when its bound is at a kink,
    /// read off `kappa = sigma * S_kk`, the barrier weight times the
    /// row's own diagonal of the reduced matrix. `sigma` equals the
    /// curvature reduced along the coordinate at an exact kink, and
    /// `S_kk` is that reduced curvature's inverse, so `kappa` is 1
    /// there at any curvature, coupling, or scaling, and it falls as
    /// the squared ratio of kink width to slack away from one. A row
    /// below `KAPPA_MIN` is dropped from the engaged set and its
    /// plain movement stands: its bound is too far from a kink for a
    /// pin force to decide, and the error of leaving it undecided is
    /// bounded by its own slack, order `sqrt(mu)` at the threshold.
    /// A coordinate an equality pins is the limiting case, `S_kk`
    /// exactly zero, dropped by the same test.
    ///
    /// `max_iter` is the total back-solve budget: the all-released
    /// solve, every basis column, and the combined solve that recovers
    /// the direction all count against it. A budget of zero errs
    /// before any work. Any budget above that pays the all-released
    /// factorization first, because which rows engage is only known
    /// once that solve has run, and the shortfall is reported when the
    /// basis columns cannot fit. Either way the caller falls back to
    /// the one-sided step. Returns the direction, the var-x rows held,
    /// and the back-solves spent.
    pub fn parametric_step_directional(
        &self,
        pin_constraint_indices: &[Index],
        deltas: &[Number],
        max_iter: usize,
    ) -> Result<(Vec<Number>, Vec<usize>, usize), SolverError> {
        use pounce_common::types::NLP_UPPER_BOUND_INF;
        use pounce_linalg::triplet::{GenTMatrix, GenTMatrixSpace, SymTMatrix, SymTMatrixSpace};
        use pounce_qp::QpStatus;
        use pounce_qp::options::QpOptions;
        use pounce_qp::problem::{HessianInertia, QpProblem};
        use pounce_qp::solver::{ParametricActiveSetSolver, QpSolver};

        const EPS_REL: Number = 1e-9;
        /// A row whose `kappa = sigma * S_kk` is below this is not at
        /// a kink and is dropped from the QP. `kappa` is 1 at an
        /// exact kink and equals the squared ratio of kink width to
        /// slack, so a row at the threshold sits about 30 widths from
        /// its bound and the cost of deciding it either way is
        /// bounded by that slack. Measured populations: exact fixture
        /// kinks 1.0, held solves near a release 4e-2 and 6e-3,
        /// genuinely interior rows 3e-8 and below, pin-owned rows at
        /// or below zero.
        const KAPPA_MIN: Number = 1e-3;

        let rhs_plain = self.parametric_rhs_full(pin_constraint_indices, deltas)?;
        let weak = self.weakly_active_bounds()?;
        let ctx = self.bound_context(None)?;
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let bs = &state.backsolver;
        let dim = bs.dim();
        let n_x = ctx.n_x;
        let nw = weak.len();
        let mut work = 0usize;
        // A weak bound's slack and multiplier are both of order
        // sqrt(mu) and their uncertainty equals their magnitude, so a
        // movement below sqrt(mu) of the direction's scale cannot be
        // resolved against the bound and does not warrant an exact
        // complementarity decision. The engagement and expansion
        // tests use this band; acceptance-level roundoff tests keep
        // EPS_REL.
        let band = bs.barrier_mu().max(0.0).sqrt().max(EPS_REL);

        if weak.is_empty() {
            // a clean base point takes the plain step and no decision
            // happens, so the reported decision work is zero
            let mut d = vec![0.0; dim];
            if !bs.solve(&rhs_plain, &mut d) {
                return Err(SolverError::BacksolveFailed);
            }
            return Ok((d[..n_x].to_vec(), Vec::new(), 0));
        }

        // What the caller needs is the number to raise
        // `degeneracy_iter` to, so the message reports the engaged
        // count rather than the weak-set size: engagement is the retry
        // price, and on a model with hundreds of weak bounds the two
        // differ by enough that raising one at a time is dozens of
        // retries. The engaged set can still grow on a later pass, so
        // the figure is a floor and says so.
        //
        // `engaged_now + 2` prices a decision that finishes on one
        // pass. Each expansion round pays another combined solve, so
        // on a multi-pass decision that total is short, and once the
        // engaged set stops growing it stops moving at all: the
        // combined solve of the last round would otherwise be told to
        // raise the budget to the number already spent, which is a
        // retry that buys nothing and reads as self-contradictory.
        // Flooring at `spent + 1` keeps the advice strictly larger
        // than what is gone, so every retry makes progress.
        let budget = |engaged_now: usize, spent: usize| {
            let need = (engaged_now + 2).max(spent + 1);
            SolverError::SensComputationFailed(format!(
                "directional derivative: {spent} of {max_iter} back-solve(s) \
                 spent, and {engaged_now} of {nw} weakly active bound(s) are \
                 engaged so far. Raise degeneracy_iter to at least {need}; \
                 the engaged set can still grow, so that is a floor."
            ))
        };
        let fail = |what: &str| {
            SolverError::SensComputationFailed(format!("directional derivative: {what}"))
        };

        let released: Vec<usize> = weak.iter().map(|w| w.row).collect();
        // `weak` comes from `weakly_active_bounds`, which is keyed by
        // a var-x row and so can only ever name a bound in the `x`
        // block. Constraint-row limits (gh#928) live in the `s` block
        // and would need the second diagonal below; releasing one here
        // with the `s` diagonal left at its base value would solve a
        // system that still enforces the bound it claims to have
        // released, silently. Asked for rather than assumed: the
        // second slot is `None` exactly when nothing in `released`
        // touched the `s` block, so a non-`None` here means the
        // premise this arm rests on has stopped holding.
        let (sigma, sigma_s) = bs
            .released_sigmas(&released)
            .ok_or_else(|| fail("released sigma unavailable"))?;
        if sigma_s.is_some() {
            return Err(fail(
                "a released bound reached the `s` block. The directional \
                 decision is built on `weakly_active_bounds`, whose rows are \
                 var-x, so this cannot happen without that contract having \
                 changed -- and releasing an `s`-block bound needs the slack \
                 diagonal this arm does not carry.",
            ));
        }
        if work + 1 > max_iter {
            // Nothing is engaged before the all-released solve, so
            // this fires only at a budget of zero, and the floor is
            // the one solve the decision cannot start without.
            return Err(SolverError::SensComputationFailed(format!(
                "directional derivative: degeneracy_iter is {max_iter}, and the \
                 decision cannot start without one back-solve over the {nw} \
                 weakly active bound(s). Raise degeneracy_iter to at least 2."
            )));
        }
        let mut d0 = vec![0.0; dim];
        // shift = false, matching `path_direction`'s all-released
        // solve: a weak bound's multiplier is order sqrt(mu) and the
        // released convention holds it at exactly zero, so the step
        // shift's multiplier injection is deliberately omitted.
        if !bs.solve_released_prebuilt(
            &released,
            Rc::clone(&sigma),
            None,
            None,
            &rhs_plain,
            &mut d0,
            false,
        ) {
            return Err(SolverError::BacksolveFailed);
        }
        work += 1;

        // movement of weak row k under a direction: positive is the
        // feasible side for that row's bound
        let sign = |k: usize| if weak[k].lower { 1.0 } else { -1.0 };
        let movement = |k: usize, d: &[Number]| -> Number { sign(k) * d[weak[k].var_row] };
        // The barrier weight of each weak row's variable, in natural
        // units to match the natural-units response the back-solves
        // return, so `kappa` below is frame-invariant. The classifier
        // succeeded inside `weakly_active_bounds`, so a nonempty weak
        // set implies this call succeeds too.
        // `var_sigma` is a FULL-x array read from a VAR-x row, and the
        // `unwrap_or(0.0)` below turns a miss into a zero that silently
        // drops the row from the engaged set rather than raising. That
        // is the shape gh#672 finding 1 shipped, so the index is typed:
        // `sigma.at` takes a `FullX` and a bare `w.var_row` will not
        // compile. See `crate::index`.
        let nat_sigma: Vec<Number> = {
            let report = self.classify_activity()?;
            let (_, _, nlp) = state.backsolver.activity_handles();
            let nl = nlp.borrow();
            let sigma = FullXSlice::new(&report.var_sigma);
            let map = VarToFull::build(ctx.n_x, |r| nl.var_x_to_full_x(r.as_index()) as usize);
            weak.iter()
                .map(|w| {
                    map.full_of(VarX::new(w.var_row))
                        .and_then(|full| sigma.at(full))
                        .unwrap_or(0.0)
                })
                .collect()
        };
        let scale_of = |d: &[Number]| -> Number {
            d[..n_x]
                .iter()
                .fold(0.0_f64, |a, &b| a.max(b.abs()))
                .max(1e-300)
        };

        let tol0 = band * scale_of(&d0);
        let mut engaged: Vec<usize> = (0..nw).filter(|&k| movement(k, &d0) < -tol0).collect();
        if engaged.is_empty() {
            return Ok((d0[..n_x].to_vec(), Vec::new(), work));
        }

        // Each basis column is only ever read at the weak rows' own
        // variables, once to build `S` and never again: the direction
        // it contributes is recovered below in a single solve. So the
        // column is projected onto those `nw` entries and the
        // full-length vector dropped, which bounds this by the weak
        // set rather than by `dim` times the budget. Holding the full
        // columns costs about 114 MB on a 62k model at 230 engaged
        // rows, and grows with `degeneracy_iter`.
        let mut proj: Vec<Option<Vec<Number>>> = vec![None; nw];
        let mut d = d0.clone();
        let held: Vec<usize>;
        // A weak row is decided only when its bound is at a kink,
        // and `kappa = sigma * S_kk` measures exactly that: 1 at an
        // exact kink, falling as the squared ratio of kink width to
        // slack away from one. A row below the threshold is dropped
        // and its plain movement stands, since a pin force there
        // holds the coordinate a full slack from where the bound
        // actually is, and the error of not deciding is bounded by
        // that same slack. The limiting cases fall out of the one
        // test: a coordinate an equality owns has `S_kk` exactly
        // zero (its pin absorbs any bound force, and admitting it
        // puts a zero diagonal beside a nonzero gradient, an
        // unbounded QP), and a negative diagonal, which the QP could
        // not bound either, is likewise below the threshold.
        let mut inert: Vec<usize> = Vec::new();
        loop {
            for &k in &engaged {
                if proj[k].is_some() {
                    continue;
                }
                if work + 1 > max_iter {
                    return Err(budget(engaged.len(), work));
                }
                let mut unit = vec![0.0; dim];
                unit[weak[k].var_row] = sign(k);
                let mut xk = vec![0.0; dim];
                if !bs.solve_released_prebuilt(
                    &released,
                    Rc::clone(&sigma),
                    None,
                    None,
                    &unit,
                    &mut xk,
                    false,
                ) {
                    return Err(SolverError::BacksolveFailed);
                }
                work += 1;
                let col: Vec<Number> = weak.iter().map(|w| xk[w.var_row]).collect();
                let own = sign(k) * col[k];
                if nat_sigma[k] * own < KAPPA_MIN {
                    inert.push(k);
                }
                proj[k] = Some(col);
            }
            engaged.retain(|k| !inert.contains(k));
            if engaged.is_empty() {
                return Ok((d[..n_x].to_vec(), Vec::new(), work));
            }

            // dense reduced data over the engaged rows, upper triangle
            let ke = engaged.len();
            let mut irows = Vec::new();
            let mut jcols = Vec::new();
            let mut vals = Vec::new();
            for i in 0..ke {
                for j in i..ke {
                    let col_j = proj[engaged[j]].as_ref().expect("column built");
                    let col_i = proj[engaged[i]].as_ref().expect("column built");
                    // S_ij = a_i^T X_j; symmetrize, since S is
                    // symmetric in exact arithmetic. The projection
                    // holds one entry per weak row, so a weak row's
                    // own index is where its `a` picks the column out.
                    let s_ij = 0.5
                        * (sign(engaged[i]) * col_j[engaged[i]]
                            + sign(engaged[j]) * col_i[engaged[j]]);
                    // pounce-linalg triplets are one-based
                    irows.push((i + 1) as Index);
                    jcols.push((j + 1) as Index);
                    vals.push(s_ij);
                }
            }
            // The engine's feasibility and optimality tolerances are
            // absolute and act on the QP's variables, which are the
            // pin forces, so both sides of the problem are scaled to
            // order one: the gradient against the direction's scale
            // (a 1e-10 perturbation must decide the same way a 1e-2
            // one does) and S against its largest entry, which is a
            // compliance in the model's units. The joint scaling maps
            // the solution by g_scale / s_scale exactly, so the
            // scaled solve loses nothing.
            let g_raw: Vec<Number> = engaged.iter().map(|&k| movement(k, &d0)).collect();
            let g_scale = g_raw
                .iter()
                .fold(0.0_f64, |a, &b| a.max(b.abs()))
                .max(1e-300);
            let g: Vec<Number> = g_raw.iter().map(|&v| v / g_scale).collect();
            let s_scale = vals
                .iter()
                .fold(0.0_f64, |a, &b| a.max(b.abs()))
                .max(1e-300);
            let vals_scaled: Vec<Number> = vals.iter().map(|&v| v / s_scale).collect();
            let space = SymTMatrixSpace::new(ke as Index, irows, jcols);
            let mut h = SymTMatrix::new(space);
            h.set_values(&vals_scaled);
            let a_space = GenTMatrixSpace::new(0, ke as Index, Vec::new(), Vec::new());
            let a = GenTMatrix::new(a_space);
            let xl = vec![0.0; ke];
            let xu = vec![NLP_UPPER_BOUND_INF; ke];
            let qp = QpProblem {
                n: ke,
                m: 0,
                h: &h,
                g: &g,
                a: &a,
                bl: &[],
                bu: &[],
                xl: &xl,
                xu: &xu,
                hessian_inertia: HessianInertia::Unknown,
            };
            let opts = QpOptions {
                max_iter: (10 * ke as u32).max(200),
                // the engine's Schur-update path (use_schur_updates)
                // hits MaxIter on a dense reduced problem of hundreds
                // of rows where the refactorizing path terminates
                // Optimal, so the default stays; the heavy-direction
                // exact decision pays engine refactorizations and is
                // priced accordingly in the docs
                ..QpOptions::default()
            };
            let mut engine =
                ParametricActiveSetSolver::new(Box::new(pounce_feral::FeralSolverInterface::new()));
            let sol = engine
                .solve(&qp, None, &opts)
                .map_err(|e| fail(&format!("reduced QP failed: {e:?}")))?;
            if sol.status != QpStatus::Optimal {
                return Err(fail(&format!(
                    "reduced QP terminated {:?} over {ke} engaged row(s)",
                    sol.status
                )));
            }
            let lambda: Vec<Number> = sol.x.iter().map(|&v| v * (g_scale / s_scale)).collect();

            // plus, not minus: the QP's optimality gradient is
            // S lambda + m, so the direction's movement must be
            // m + lambda S, which is d0 + Σ λ_k X_k here.
            //
            // Each `X_k` is `K_rel⁻¹ a_k`, so that sum is
            // `K_rel⁻¹ (Σ λ_k a_k)` and one solve on the combined
            // right-hand side gives it. That is why the columns above
            // need not be kept: the only thing they were held for is
            // recovered here, in a single back-solve, at the price of
            // one more against the budget per expansion round.
            d.copy_from_slice(&d0);
            if lambda.iter().any(|&l| l != 0.0) {
                if work + 1 > max_iter {
                    return Err(budget(engaged.len(), work));
                }
                let mut comb = vec![0.0; dim];
                for (i, &k) in engaged.iter().enumerate() {
                    comb[weak[k].var_row] += lambda[i] * sign(k);
                }
                let mut corr = vec![0.0; dim];
                if !bs.solve_released_prebuilt(
                    &released,
                    Rc::clone(&sigma),
                    None,
                    None,
                    &comb,
                    &mut corr,
                    false,
                ) {
                    return Err(SolverError::BacksolveFailed);
                }
                work += 1;
                for (dv, &cv) in d.iter_mut().zip(corr.iter()) {
                    *dv += cv;
                }
            }

            let tol = band * scale_of(&d);
            let mut grew = false;
            for k in 0..nw {
                if engaged.contains(&k) || inert.contains(&k) {
                    continue;
                }
                if movement(k, &d) < -tol {
                    engaged.push(k);
                    grew = true;
                }
            }
            if !grew {
                // relative to the largest pin force, with no absolute
                // floor: a 1e-10-scale perturbation's pins are as real
                // as a 1e-2 one's, and a floor here silently unlabels
                // them while the direction still carries the pin
                let lam_scale = lambda
                    .iter()
                    .fold(0.0_f64, |a, &b| a.max(b.abs()))
                    .max(1e-300);
                held = engaged
                    .iter()
                    .enumerate()
                    .filter(|(i, _)| lambda[*i] > EPS_REL * lam_scale)
                    .map(|(_, &k)| weak[k].var_row)
                    .collect();
                break;
            }
        }

        Ok((d[..n_x].to_vec(), held, work))
    }

    /// The bounds the activity classifier could not call at the base
    /// point: on the bound with a multiplier of the same order as the
    /// slack. Each entry is a bound row present in the held
    /// factorization, with the side taken from the smaller slack,
    /// which is the only side an ambiguous label can come from.
    ///
    /// Both [`WEAKLY_ACTIVE`](crate::activity::WEAKLY_ACTIVE) and
    /// [`AMBIGUOUS`](crate::activity::AMBIGUOUS) count as weak here,
    /// deliberately: the ambiguous class contains genuine kinks whose
    /// coordinate is coupled to a neighbour (gh#763), so treating it
    /// as "not a kink" would drop real weak rows. That is why the
    /// mislabeling is not a wrong answer in the step path — see
    /// [`Self::reduced_activity`] for the class itself.
    ///
    /// The classifier reports per user variable, in full-x, while the
    /// bound context and the factor's rows are var-x, and the two
    /// index spaces diverge from the first fixed variable on. Each
    /// var-x row's status is read through the same map the classifier
    /// scattered through, so a fixed variable shifts nothing. Using
    /// the full-x index as a factor row instead returns a NEIGHBORING
    /// variable's answer, plausible and wrong, which is the gh#450
    /// hazard the `primal_row` discipline exists to prevent.
    pub fn weakly_active_bounds(&self) -> Result<Vec<crate::boundcheck::WeakBound>, SolverError> {
        use crate::activity::{AMBIGUOUS, WEAKLY_ACTIVE};

        // A relaxed solve shifts the slacks the classifier reads, so
        // degeneracy is undetectable there: the callers take the plain
        // step, the same choice `estimate_report` makes when it fills
        // `bounds_relaxed` instead of raising.
        let report = match self.classify_activity() {
            Ok(r) => r,
            Err(SolverError::BadOptions(_)) => return Ok(Vec::new()),
            Err(e) => return Err(e),
        };
        let ctx = self.bound_context(None)?;
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let Some(rows) = state.backsolver.bound_rows() else {
            return Ok(Vec::new());
        };
        // Three index spaces are live in the loop below: `report` is
        // FULL-x, `ctx` is VAR-x, and `br.row` is a bound row. The
        // first two coincide until the first `make_parameter`-removed
        // variable and diverge after it, so a swap reads a NEIGHBOURING
        // variable's status -- in range, plausible, wrong (gh#450, then
        // gh#672 finding 1). The typed indices make that a compile
        // error; `sens_invariance_legs.rs` leg 3 is what covers the
        // site already written. See `crate::index`.
        let map = {
            let (_, _, nlp) = state.backsolver.activity_handles();
            let nl = nlp.borrow();
            VarToFull::build(ctx.n_x, |r| nl.var_x_to_full_x(r.as_index()) as usize)
        };
        let status = FullXSlice::new(&report.var_status);
        let mut out = Vec::new();
        for row in map.rows() {
            let Some(full) = map.full_of(row) else {
                continue;
            };
            let Some(st) = status.at(full) else {
                continue;
            };
            if st != WEAKLY_ACTIVE && st != AMBIGUOUS {
                continue;
            }
            let (s_lo, s_hi) = ctx.slacks_at(row);
            let lower = s_lo <= s_hi;
            // a table lookup, so a space-swap here would fail loudly
            let var_row = row.get();
            // `rows` now carries constraint-row limits too, whose
            // `var_row` is an `s`-block row (gh#928). They can never
            // match here -- `map.rows()` runs over the `x` block, so
            // `var_row < ctx.n_x`, and every `s` row is at or above it
            // -- but that is a property of two index ranges not
            // overlapping, which is exactly the kind of thing that
            // silently stops being true. Said out loud so it is a
            // filter rather than a coincidence.
            debug_assert!(var_row < ctx.n_x);
            if let Some(br) = rows
                .iter()
                .filter(|b| b.var_row < ctx.n_x)
                .find(|b| b.var_row == var_row && b.lower == lower)
            {
                out.push(crate::boundcheck::WeakBound {
                    row: br.row,
                    var_row,
                    lower,
                });
            }
        }
        Ok(out)
    }

    /// The bound geometry both bound-aware steps read: the primal
    /// block's size and base point, its bounds in the model's own
    /// units, the tolerance that decides what counts as on a bound, and
    /// the bound multipliers at the base point.
    ///
    /// Shared rather than assembled twice. The two callers have to
    /// agree on all of it, and the unit and index-space conversions
    /// below are exactly what went wrong when a second caller wrote its
    /// own.
    ///
    /// `bound_eps` overrides the margin. `None` keeps how far outside
    /// the solve itself was willing to settle, floored so an unrelaxed
    /// solve does not pin on roundoff.
    fn bound_context(&self, bound_eps: Option<Number>) -> Result<BoundContext, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let dims = state.backsolver.block_dims();
        let n_x = dims[0];
        let n_s = dims[1];

        // Expanded once, before any re-solve: reading the compressed
        // form means borrowing the NLP, and the solves below re-borrow
        // it.
        //
        // Both primal blocks, in one pass over the same borrow: `x`
        // against `x_l` / `x_u` through `px_l` / `px_u`, and `s`
        // against `d_l` / `d_u` through `pd_l` / `pd_u`. The second
        // half is gh#928's subject — an inequality row's limit is a
        // bound on its slack, and a box that stopped at `n_x` left
        // every such limit unwatched.
        let (mut lo, mut hi, s_lo, s_hi) = {
            let (_, _, nlp) = state.backsolver.activity_handles();
            let nl = nlp.borrow();
            let (lo, hi) =
                crate::boundcheck::expand_bounds(n_x, &nl.px_l(), &nl.px_u(), nl.x_l(), nl.x_u());
            let (s_lo, s_hi) =
                crate::boundcheck::expand_bounds(n_s, &nl.pd_l(), &nl.pd_u(), nl.d_l(), nl.d_u());
            (lo, hi, s_lo, s_hi)
        };
        // Those bounds bound the algorithm's `x̃ = d ⊙ x`, while
        // `state.x` and the step are both in the model's own units
        // (gh#486 stage 3). Undo the change of variables on the bounds
        // so all three agree, rather than projecting onto the wrong box.
        // A negative factor reflects the interval, so the sides swap.
        // `variable_scaling`, not `variable_scaling_full`: `lo` / `hi`
        // are var-x length, and the two index spaces diverge from the
        // first fixed variable on.
        if let Some(d) = state.backsolver.variable_scaling() {
            for i in 0..n_x {
                let di = d[i];
                if di == 0.0 || di == 1.0 {
                    continue;
                }
                let (a, b) = (lo[i] / di, hi[i] / di);
                lo[i] = a.min(b);
                hi[i] = a.max(b);
            }
        }

        // The `s` block gets the same treatment for the row scaling it
        // was solved under. `F` — the vector every back-solve
        // post-multiplies its answer by — is `1/dd_r` on the `s` rows,
        // so a scaled quantity times `F` is the natural one, which is
        // exactly the conversion the `x` block just made by dividing
        // by `d`. Using `F` rather than re-reading `d_scale` keeps
        // this pinned to the same numbers the step arrives in: the
        // point of the conversion is that `x_curr`, the box and the
        // step agree, not that the arithmetic matches a formula.
        //
        // The base slack `s* - d_l` is scale-invariant in sign but not
        // in size, and it is compared against a multiplier that gets
        // the same treatment inside the walk, so both sides move
        // together. `variable_scaling_sensitivity.rs` is the general
        // statement of why that has to be checked rather than assumed.
        let mut s_curr: Vec<Number> = {
            let (data, _, _) = state.backsolver.activity_handles();
            let d = data.borrow();
            let curr = d.curr.as_ref().ok_or(SolverError::NotConverged)?;
            crate::vec_util::dense_to_vec(&*curr.s)
        };
        let mut s_lo = s_lo;
        let mut s_hi = s_hi;
        if s_curr.len() != n_s {
            return Err(SolverError::BadShape {
                what: "slack block of the converged iterate",
                got: s_curr.len(),
                expected: n_s,
            });
        }
        if let Some(f) = state.backsolver.natural_units_factor() {
            for i in 0..n_s {
                let fi = f[n_x + i];
                if fi == 1.0 {
                    continue;
                }
                s_curr[i] *= fi;
                let (a, b) = (s_lo[i] * fi, s_hi[i] * fi);
                s_lo[i] = a.min(b);
                s_hi[i] = a.max(b);
            }
        }

        // What counts as outside a bound is the solve's own answer: it
        // was willing to leave a converged point `bound_relax_factor`
        // outside, so anything within that is on the bound, not past
        // it. A floor keeps an unrelaxed solve from pinning on
        // roundoff.
        let floor = crate::boundcheck::release_floor(state.bound_relax_factor);
        // Rejected here rather than at each entry point, so the pyo3
        // binding and every Rust caller get the check the CLI's
        // `sens_bound_eps` gets from its strict lower bound. Zero
        // reinstates the roundoff pinning the floor prevents, and NaN
        // makes `over > eps` false everywhere, so the refinement pins
        // nothing and still reports settled — both return a plausible
        // vector rather than failing, which is the worse outcome.
        // `> 0.0` is false for NaN, as `pyomo_pounce`'s own check is.
        let eps = match bound_eps {
            None => floor,
            Some(e) if e > 0.0 => e,
            Some(e) => {
                return Err(SolverError::BadOptions(format!(
                    "bound_eps must be a positive number, got {e}"
                )));
            }
        };
        // A caller's `bound_eps` is a primal margin and says nothing
        // about when a multiplier has changed sign, so the release test
        // keeps the solve's own margin.
        let release_eps = floor;
        // The bound multipliers at the base point, with the compound
        // row each one occupies, so a step that drives one negative can
        // release that bound.
        let mults = {
            let (z_l_off, z_u_off) = (
                dims[0] + dims[1] + dims[2] + dims[3],
                dims[0] + dims[1] + dims[2] + dims[3] + dims[4],
            );
            let (data, _, _) = state.backsolver.activity_handles();
            let d = data.borrow();
            let curr = d.curr.as_ref().ok_or(SolverError::NotConverged)?;
            let (v_l_off, v_u_off) = (z_u_off + dims[5], z_u_off + dims[5] + dims[6]);
            let mut out = Vec::new();
            for (off, v) in [
                (z_l_off, &curr.z_l),
                (z_u_off, &curr.z_u),
                (v_l_off, &curr.v_l),
                (v_u_off, &curr.v_u),
            ] {
                for (k, &base) in crate::vec_util::dense_to_vec(&**v).iter().enumerate() {
                    out.push(crate::boundcheck::BoundMultiplier { row: off + k, base });
                }
            }
            out
        };
        lo.extend_from_slice(&s_lo);
        hi.extend_from_slice(&s_hi);
        let mut x_curr = state.x[..n_x].to_vec();
        x_curr.extend_from_slice(&s_curr);
        Ok(BoundContext {
            n_x,
            lo,
            hi,
            x_curr,
            eps,
            release_eps,
            mults,
        })
    }

    /// Full KKT-space parametric step for a set of pinned equality
    /// constraints: the same computation as [`Self::parametric_step`],
    /// returned WITHOUT truncating to the primal block. The layout is
    /// the compound KKT vector `(x, s, y_c, y_d, z_l, z_u, v_l, v_u)`;
    /// use [`Self::block_dims`] for the block sizes and
    /// [`Self::g_multiplier_rows`] to locate a constraint's multiplier
    /// row. This exposes the multiplier sensitivities `∂λ*/∂p`
    /// alongside the primal step.
    pub fn parametric_step_full(
        &self,
        pin_constraint_indices: &[Index],
        deltas: &[Number],
    ) -> Result<Vec<Number>, SolverError> {
        if pin_constraint_indices.len() != deltas.len() {
            return Err(SolverError::BadShape {
                what: "deltas",
                got: deltas.len(),
                expected: pin_constraint_indices.len(),
            });
        }
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;

        let param_rows = state
            .backsolver
            .map_pin_g_to_kkt_rows(pin_constraint_indices)
            .map_err(SolverError::SensComputationFailed)?;
        let signs = vec![1; pin_constraint_indices.len()];
        let a_data = IndexSchurData::from_parts(param_rows, signs)
            .map_err(|e| SolverError::SensComputationFailed(format!("{e:?}")))?;

        let opts = SensOptions {
            run_sens: true,
            ..SensOptions::default()
        };
        let sens_app = SensApplication::new(a_data, state.backsolver.clone(), opts);
        let n_full = state.backsolver.dim();
        let mut dx_full = vec![0.0; n_full];
        if !sens_app.parametric_step(deltas, &mut dx_full) {
            return Err(SolverError::SensComputationFailed(
                "SensApplication::parametric_step failed".into(),
            ));
        }
        let corr = self.barrier_correction(state)?;
        for (d, c) in dx_full.iter_mut().zip(corr.iter()) {
            *d += *c * BARRIER_SIGN;
        }
        Ok(dx_full)
    }

    /// Flat rows of the compound KKT vector holding the equality
    /// multipliers `y_c` for the given 0-based **full-g** constraint
    /// indices. `None` for inequalities — their multipliers live in
    /// the `y_d` block, which [`Self::d_multiplier_rows`] addresses.
    /// Row `r` of a [`Self::parametric_step_full`] result is then
    /// `∂λ_g/∂p · Δp`.
    pub fn g_multiplier_rows(
        &self,
        g_indices: &[Index],
    ) -> Result<Vec<Option<Index>>, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let dims = state.backsolver.block_dims();
        let y_c_offset = (dims[0] + dims[1]) as Index;
        Ok(g_indices
            .iter()
            .map(|&g| {
                state
                    .backsolver
                    .full_g_to_c_block(g)
                    .map(|pos| y_c_offset + pos)
            })
            .collect())
    }

    /// Flat rows of the compound KKT vector holding the **inequality**
    /// multipliers `y_d` for the given 0-based **full-g** constraint
    /// indices. `None` for equalities (those are
    /// [`Self::g_multiplier_rows`]'s). The `y_d` counterpart of that
    /// accessor, added by gh#910: `parametric_step_full` already
    /// returned the `y_d` block, and this is the map that says which
    /// row of it belongs to which user constraint.
    ///
    /// **Reading the row is not the same as the row being a
    /// derivative.** `y_d` holds a number for every inequality, in all
    /// three activity regimes, and only one of them has a two-sided
    /// `∂λ/∂p` at all:
    ///
    /// * **strictly active** (`s ≈ 0`, `λ > 0`): the row behaves as an
    ///   equality over a neighbourhood of the solved point, and this
    ///   row is the same back-solve an equality gets. Well defined.
    /// * **inactive** (`s > 0`, `λ ≈ 0`): the derivative is a
    ///   structural zero over a neighbourhood; the KKT row carries the
    ///   barrier's residue rather than a derivative of anything.
    /// * **weakly active** (a kink: `s ≈ 0` *and* `λ ≈ 0`): the two
    ///   one-sided derivatives differ and no two-sided value exists.
    ///   The entry holds whichever side the factorization landed on —
    ///   the silently-wrong-while-reporting-success class.
    ///   [`Self::parametric_step_directional`] is what answers a kink,
    ///   and it needs a direction.
    ///
    /// So a caller reading these rows as `∂λ/∂p` must gate on the
    /// regime first, and the classifier that answers it is
    /// [`Self::reduced_row_activity`], **not**
    /// [`Self::classify_activity`]: a genuine kink whose row couples
    /// to the remaining free space reports `INACTIVE` on the
    /// directional normalizer at strong enough coupling (gh#804), and
    /// `INACTIVE` is the one class whose derivative a caller may
    /// legitimately read as a structural zero. Gating on the cheap
    /// classifier would therefore answer "it does not move" about a
    /// kink — a wrong answer wearing a refusal's clothes. That
    /// inference, reading an activity class as a proxy for kink-ness,
    /// is what shipped gh#756.
    pub fn d_multiplier_rows(
        &self,
        g_indices: &[Index],
    ) -> Result<Vec<Option<Index>>, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let dims = state.backsolver.block_dims();
        let y_d_offset = (dims[0] + dims[1] + dims[2]) as Index;
        Ok(g_indices
            .iter()
            .map(|&g| {
                state
                    .backsolver
                    .full_g_to_d_block(g)
                    .map(|pos| y_d_offset + pos)
            })
            .collect())
    }

    /// Flat rows of the compound KKT vector holding an inequality's
    /// **slack** `s`, for the given 0-based full-g row indices; `None`
    /// for a row that is not an inequality.
    ///
    /// The primal counterpart of [`Self::d_multiplier_rows`], and the
    /// discriminator a consumer of [`Self::parametric_step_path`]
    /// needs. A limit written as `g(x) <= cap` bounds this slack
    /// rather than any variable, so a breakpoint on it carries a
    /// primal KKT row in the `s` block (gh#928). Reading such a row as
    /// a var-x index returns a neighbouring variable's answer, the
    /// gh#450 hazard, so a caller that maps rows back to model objects
    /// resolves them here.
    ///
    /// The `s` block sits immediately after `x`, and is indexed by
    /// d-block position exactly as `y_d` is, so this row and
    /// [`Self::d_multiplier_rows`]'s row name the same inequality from
    /// the two sides of its complementarity pair.
    pub fn d_slack_rows(&self, g_indices: &[Index]) -> Result<Vec<Option<Index>>, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let s_offset = state.backsolver.block_dims()[0] as Index;
        Ok(g_indices
            .iter()
            .map(|&g| {
                state
                    .backsolver
                    .full_g_to_d_block(g)
                    .map(|pos| s_offset + pos)
            })
            .collect())
    }

    /// Flat rows of the compound KKT vector holding the primal values
    /// `x` for the given 0-based **full-x** variable indices. `None`
    /// where the solve removed the column (`x_l == x_u` under
    /// `fixed_variable_treatment = make_parameter`), which has no row
    /// in the factor at all.
    ///
    /// The `x` counterpart of [`Self::g_multiplier_rows`], and needed
    /// for the same reason: a caller holding user-space indices — from
    /// the `.col` file, from [`Self::classify_activity`], from
    /// [`Self::row_normal`] — cannot index the factor with them
    /// directly. Row `r` of a [`Self::parametric_step_full`] result is
    /// then `∂x/∂p · Δp` for that variable, and `e_r` is the unit
    /// vector selecting its column in a [`Self::kkt_solve`].
    pub fn x_primal_rows(&self, x_indices: &[Index]) -> Result<Vec<Option<Index>>, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let n_full = state.backsolver.n_full_x();
        // out of range must not masquerade as "removed as fixed": the
        // NLP map returns None for both, and the caller's whole reason
        // for asking is that it cannot tell the spaces apart itself
        if let Some(&bad) = x_indices.iter().find(|&&i| i < 0 || i >= n_full) {
            return Err(SolverError::BadShape {
                what: "x_primal_rows variable index",
                got: bad as usize,
                expected: n_full as usize,
            });
        }
        // the x block starts at flat index 0, so the var-x position IS
        // the KKT row; the offset stays explicit for the day it is not
        Ok(x_indices
            .iter()
            .map(|&i| state.backsolver.full_x_to_var_x(i))
            .collect())
    }

    /// The user TNLP's variable count: the length of a full-x report
    /// and the domain of [`Self::x_primal_rows`].
    pub fn n_full_x(&self) -> Result<usize, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        Ok(state.backsolver.n_full_x() as usize)
    }

    /// The user TNLP's constraint count: the length of a full-g
    /// report and the domain of [`Self::reduced_row_activity`].
    pub fn n_full_g(&self) -> Result<usize, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        Ok(state.backsolver.n_full_g() as usize)
    }

    /// Reduced Hessian over the pinned equality-constraint rows:
    /// `obj_scal · B K⁻¹ Bᵀ`, where `B` selects the
    /// `pin_constraint_indices` rows of the y_c block and `K` is the
    /// **natural-units** (unscaled) KKT matrix — active NLP scaling
    /// is undone by the backsolver, so `−inv` of the returned matrix
    /// is directly the parameter covariance regardless of
    /// `nlp_scaling_method` (pounce#128). `obj_scal` survives as a
    /// plain extra multiplier (default 1.0); it is no longer needed to
    /// recover natural units. Returns the `n²`-long column-major dense
    /// matrix (`n = pin_constraint_indices.len()`).
    ///
    /// # Sign convention: this returns `−H_R`, not `H_R` (gh#937)
    ///
    /// The matrix is the **negated** reduced Hessian. On a model whose
    /// objective Hessian is `[[2, 1], [1, 2]]` with both variables
    /// pinned, this returns `[[−2, −1], [−1, −2]]`. So a well-posed
    /// minimum reports an all-*negative* spectrum; that is the
    /// convention, not an indefiniteness or convergence bug.
    ///
    /// The minus is the augmented system's, and it is why the
    /// covariance recipe above negates: pin indices map to the `y_c`
    /// multiplier block, and for `K = [[H, Aᵀ], [A, 0]]` the
    /// `(y_c, y_c)` block of `K⁻¹` is `−(A H⁻¹ Aᵀ)⁻¹` — so over pin
    /// rows `B K⁻¹ Bᵀ` is the multiplier sensitivity
    /// `∂λ/∂p = −∂²f*/∂p²`, i.e. `±H_R` itself and not a submatrix of
    /// an inverse. (The `x` block of `K⁻¹` *is* an inverse. The two
    /// blocks sit on opposite sides of one inversion, which is what
    /// makes the CLI's `red_hessian` suffix path — upstream sIPOPT's,
    /// selecting x rows — a different quantity rather than the same
    /// one with a different sign.)
    ///
    /// Negate to read curvature: `−hr` is `H_R`, and `−inv(hr)` is the
    /// covariance. Pinned by
    /// `tests/issue_937_reduced_hessian_sign.rs`; demonstrated by
    /// `examples/rh_orientation_check.rs`.
    ///
    /// Equivalent to [`crate::SensSolve::with_reduced_hessian`] but
    /// usable post-hoc on a held `Solver`. For the solver-space
    /// (pre-#128) value use [`Self::compute_reduced_hessian_scaled`];
    /// the factors themselves are exposed via [`Self::nlp_scaling`] /
    /// [`Self::pin_g_scaling`].
    pub fn compute_reduced_hessian(
        &self,
        pin_constraint_indices: &[Index],
        obj_scal: Number,
    ) -> Result<Vec<Number>, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let n = pin_constraint_indices.len();
        let param_rows = state
            .backsolver
            .map_pin_g_to_kkt_rows(pin_constraint_indices)
            .map_err(SolverError::SensComputationFailed)?;
        let signs = vec![1; n];
        let a_data = IndexSchurData::from_parts(param_rows, signs)
            .map_err(|e| SolverError::SensComputationFailed(format!("{e:?}")))?;
        let opts = SensOptions {
            compute_red_hessian: true,
            obj_scal,
            ..SensOptions::default()
        };
        let mut sens_app = SensApplication::new(a_data, state.backsolver.clone(), opts);
        let mut hr = vec![0.0; n * n];
        if !sens_app.compute_reduced_hessian(&mut hr) {
            return Err(SolverError::SensComputationFailed(
                "SensApplication::compute_reduced_hessian failed".into(),
            ));
        }
        Ok(hr)
    }

    /// [`Self::compute_reduced_hessian`] plus its eigendecomposition —
    /// `(H_R, eigenvalues, eigenvectors)`.
    ///
    /// The curvature on the null space of the active constraints is the
    /// question; its **spectrum** is what answers "is this parameter
    /// identifiable, and along which direction". `SensSolve` has offered that
    /// since gh#561 ([`crate::SensSolve::with_reduced_hessian_eigen`]), and
    /// the session API did not — so a caller holding a `Solver` had to
    /// re-solve the whole NLP through the one-shot builder to get a
    /// decomposition of a matrix it already had. That is the gap this closes;
    /// the numbers are the one-shot path's, from the same
    /// [`pounce_linalg::symmetric_eigen`].
    ///
    /// Eigenvectors are column-major, length `n²`, column `j` belonging to
    /// eigenvalue `j`, and sign-pinned by `symmetric_eigen` so a column read
    /// as a direction reproduces across builds.
    ///
    /// # This is the spectrum of `−H_R`, so ascending runs stiffest first
    ///
    /// [`Self::compute_reduced_hessian`] returns the **negated** reduced
    /// Hessian (gh#937, and see its docs for why). The eigenvalues here are
    /// that matrix's, in ascending order — which on `−H_R` runs from most
    /// negative to least, i.e. **stiffest mode first and softest last**, the
    /// reverse of what the identifiability reading wants. On `H = [[2, 1],
    /// [1, 2]]` fully pinned they come back `[−3, −1]`: the leading column is
    /// the curvature-3 stiff direction, the trailing one the curvature-1 soft
    /// direction.
    ///
    /// So a caller taking the leading columns as the least-identifiable
    /// directions gets the best-identified ones, and nothing looks wrong —
    /// the vectors are unit-norm, sign-pinned and entirely plausible. Either
    /// negate the eigenvalues and reverse the order, or read the *trailing*
    /// columns as the soft modes. Pinned by
    /// `tests/issue_937_reduced_hessian_sign.rs`.
    pub fn compute_reduced_hessian_eigen(
        &self,
        pin_constraint_indices: &[Index],
        obj_scal: Number,
    ) -> Result<(Vec<Number>, Vec<Number>, Vec<Number>), SolverError> {
        let hr = self.compute_reduced_hessian(pin_constraint_indices, obj_scal)?;
        let n = pin_constraint_indices.len();
        let mut vals = vec![0.0; n];
        let mut vecs = vec![0.0; n * n];
        if !pounce_linalg::symmetric_eigen(&hr, n, &mut vals, &mut vecs) {
            return Err(SolverError::SensComputationFailed(
                "the reduced Hessian's eigendecomposition did not converge".into(),
            ));
        }
        Ok((hr, vals, vecs))
    }

    /// The reduced Hessian as the solver's internal **scaled** space
    /// sees it — the value [`Self::compute_reduced_hessian`] returned
    /// before pounce#128: `H̃_ij = (df / (dc_i·dc_j)) · H_ij`.
    /// Identical to `compute_reduced_hessian` when no NLP scaling is
    /// active.
    ///
    /// Sign: this is [`Self::compute_reduced_hessian`]'s `−H_R`
    /// multiplied through by `df / (dc_i·dc_j)` (gh#937), so unlike the
    /// natural-units value its orientation is **not** fixed. Measured on
    /// a fully pinned `[[2, 1], [1, 2]]`: `[[−2, −1], [−1, −2]]` by
    /// default, but `[[2, 1], [1, 2]]` under `obj_scaling_factor = −1`,
    /// where `df` carries the minus that makes a maximization a
    /// minimization. Read the sign off the reported factors
    /// ([`Self::nlp_scaling`], [`Self::pin_g_scaling`]) rather than
    /// assuming it, or use the natural-units value, whose `−H_R` holds
    /// whatever the scaling.
    pub fn compute_reduced_hessian_scaled(
        &self,
        pin_constraint_indices: &[Index],
        obj_scal: Number,
    ) -> Result<Vec<Number>, SolverError> {
        let mut hr = self.compute_reduced_hessian(pin_constraint_indices, obj_scal)?;
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        let df = state.backsolver.obj_scaling_factor();
        let dc = state
            .backsolver
            .pin_c_scales(pin_constraint_indices)
            .map_err(SolverError::SensComputationFailed)?;
        crate::reduced_hessian::scale_to_solver_space(&mut hr, df, &dc);
        Ok(hr)
    }

    /// Effective NLP scaling the IPM applied on the most recent
    /// converged solve: `(obj_scaling_factor, c_scale, d_scale)`.
    /// `(1.0, None, None)` ⇔ no scaling was active. The vectors are
    /// per-row factors over the algorithm's equality (`c`) and
    /// inequality (`d`) blocks.
    pub fn nlp_scaling(
        &self,
    ) -> Result<(Number, Option<Vec<Number>>, Option<Vec<Number>>), SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        Ok(state.backsolver.nlp_scaling())
    }

    /// The per-variable `user-scaling` factors `d` the held solve ran
    /// under (gh#486), in the user TNLP's **full-x** space, or `None`
    /// when the solve applied no change of variables.
    ///
    /// Every accessor on this type already reports natural units, so
    /// this is diagnostic rather than a correction a caller has to
    /// apply — it answers "was this solve conditioned, and by how
    /// much", the x-axis counterpart of [`Self::nlp_scaling`].
    pub fn variable_scaling(&self) -> Result<Option<Vec<Number>>, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        Ok(state.backsolver.variable_scaling_full().map(|d| d.to_vec()))
    }

    /// Inertia-correction perturbations `(δ_x, δ_s, δ_c, δ_d)` baked
    /// into the held KKT factor. All zero ⇔ the final factorization
    /// was unregularized and the natural-units back-solves invert the
    /// exact KKT matrix — see
    /// [`crate::PdSensBacksolver::kkt_perturbations`].
    pub fn kkt_perturbations(&self) -> Result<[Number; 4], SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        Ok(state.backsolver.kkt_perturbations())
    }

    /// Per-pin equality-row scaling factors `dc_i` (1.0 entries when
    /// no constraint scaling is active), ordered like
    /// `pin_constraint_indices`.
    pub fn pin_g_scaling(
        &self,
        pin_constraint_indices: &[Index],
    ) -> Result<Vec<Number>, SolverError> {
        let state = self.state.borrow();
        let state = state.as_ref().ok_or(SolverError::NotConverged)?;
        state
            .backsolver
            .pin_c_scales(pin_constraint_indices)
            .map_err(SolverError::SensComputationFailed)
    }
}