brep_kernel/geometry/

pcurve.rs

use crate::{
    interpolate_curve, project_point_to_surface, project_point_to_surface_seeded, AnalyticSurface,
    DiagnosticSeverity, KernelDiagnostics, KernelStage, KnotVector, NurbsCurve, NurbsSurface, Vec3,
    Vec4, KNOT_IDENTITY_TOL,
};

const EPSILON: f64 = 1e-12;
const LINEAR_TOLERANCE: f64 = 1e-7;

fn surface_domains(surface: &NurbsSurface) -> Result<([f64; 2], [f64; 2]), String> {
    Ok((
        KnotVector::new(surface.knots_u.clone(), surface.degree_u)?.domain(),
        KnotVector::new(surface.knots_v.clone(), surface.degree_v)?.domain(),
    ))
}

/// The carrier's closed directions, as [`NurbsSurface::closed_directions`]
/// reads them (cached on the surface, seam tolerance 1e-6, and a sliver patch
/// whose rows coincide without the surface travelling between them is OPEN).
/// Until 2026-09-26 this was a second reading with its own tolerance; the
/// unwrap in [`build_interpolant`] and the wrap in `evaluate_extended` must
/// agree on what a seam is, or a trim across a sliver reads a period wide.
fn surface_closedness(surface: &NurbsSurface) -> Result<(bool, bool), String> {
    surface.closed_directions()
}

fn invert_checked(surface: &NurbsSurface, point: Vec3) -> Result<([f64; 2], f64), String> {
    if surface.is_affine()? {
        let ([u0, _], [v0, _]) = surface_domains(surface)?;
        let (origin, du, dv) = surface.deriv1(u0, v0)?;
        let delta = point.sub(origin);
        let uu = du.dot(du);
        let uv = du.dot(dv);
        let vv = dv.dot(dv);
        let along_u = delta.dot(du);
        let along_v = delta.dot(dv);
        let determinant = uu * vv - uv * uv;
        if determinant.abs() > EPSILON {
            let parameter = [
                u0 + (along_u * vv - along_v * uv) / determinant,
                v0 + (along_v * uu - along_u * uv) / determinant,
            ];
            // The standoff is a MEASUREMENT here as on every other carrier:
            // until 2026-09-26 this returned 0.0 for "affine, not measured",
            // and a curve 1e-2 off its plane read as lying on it.
            let foot = origin.add(du.scale(parameter[0] - u0)).add(dv.scale(parameter[1] - v0));
            return Ok((parameter, point.sub(foot).length()));
        }
    }
    let projection = project_point_to_surface(surface, point)?;
    Ok(([projection.u, projection.v], projection.distance))
}

/// Whether an end station at a cone's apex takes its nearest regular
/// station's u (`true`), as in `build_interpolant`, or the projector's
/// arbitrary u there as before 2026-09-27. `BREP_PCURVE_CONE_APEX_U=0`
/// restores the projector's u, for a same-binary A/B.
fn cone_apex_u() -> bool {
    static ON: std::sync::OnceLock<bool> = std::sync::OnceLock::new();
    *ON.get_or_init(|| std::env::var_os("BREP_PCURVE_CONE_APEX_U").is_none_or(|value| value != "0"))
}

/// Whether a round with more candidates than the cap has room for keeps its
/// WORST probes (`true`), or the first in span order as before 2026-09-27.
/// `BREP_PCURVE_WORST_FIRST=0` restores span order, for a same-binary A/B.
fn worst_first() -> bool {
    static ON: std::sync::OnceLock<bool> = std::sync::OnceLock::new();
    *ON.get_or_init(|| std::env::var_os("BREP_PCURVE_WORST_FIRST").is_none_or(|value| value != "0"))
}

fn seeded_inserts() -> bool {
    static ON: std::sync::OnceLock<bool> = std::sync::OnceLock::new();
    *ON.get_or_init(|| std::env::var_os("BREP_PCURVE_SEEDED_INSERTS").is_none_or(|value| value != "0"))
}

/// [`invert_checked`] for a station inserted between two whose feet are known:
/// Newton from the pcurve's own `(u, v)` at that fraction, which already lies
/// between the neighbours' feet on their branch, and the global search only
/// when that local foot stands off further than the neighbours do (plus the
/// refinement floor) — a local minimum the curve does not sit at. The global
/// search was 30 s of a 54 s `abc_00000011` import (732k calls, 41 µs each,
/// measured 2026-09-26). Planes and analytic carriers keep their closed forms.
/// `BREP_PCURVE_SEEDED_INSERTS=0` restores the global search at every insert,
/// for a same-binary A/B.
fn invert_near(
    surface: &NurbsSurface,
    point: Vec3,
    seed: [f64; 2],
    neighbour_gap: f64,
) -> Result<([f64; 2], f64), String> {
    if !seeded_inserts() || surface.is_affine()? || surface.analytic().is_some() {
        return invert_checked(surface, point);
    }
    let seeded = project_point_to_surface_seeded(surface, point, seed[0], seed[1])?;
    if seeded.distance <= neighbour_gap + PCURVE_REFINEMENT_TOLERANCE {
        return Ok(([seeded.u, seeded.v], seeded.distance));
    }
    invert_checked(surface, point)
}

fn unwrap_periodic(values: &mut [f64], minimum: f64, maximum: f64) {
    let period = maximum - minimum;
    for index in 1..values.len() {
        while values[index] - values[index - 1] > period / 2.0 {
            values[index] -= period;
        }
        while values[index] - values[index - 1] < -period / 2.0 {
            values[index] += period;
        }
    }
    if values.len() > 1 {
        while values[0] - values[1] > period / 2.0 {
            values[0] -= period;
        }
        while values[0] - values[1] < -period / 2.0 {
            values[0] += period;
        }
    }
    // Recenter the whole (now continuous) sequence into the domain by the
    // whole-period shift that leaves the LEAST parameter outside [minimum,
    // maximum]. A single middle/mean sample is NOT representative of a curve
    // that bulges out to touch — or straddle — the periodic seam: when that
    // sample is a boundary-kissing point (the at-seam projection unwraps a hair
    // past the boundary), a single-sample test misfires and shoves the entire
    // curve a full period out of range (ABC helmet 00000011/12 side channels:
    // a touching edge's parameter run snapped a whole period off its true
    // interior, tearing the loop open in parameter space).
    if values.len() > 1 {
        let excursion = |shift: f64| -> f64 {
            values
                .iter()
                .map(|value| {
                    let v = value + shift;
                    (minimum - v).max(0.0) + (v - maximum).max(0.0)
                })
                .sum::<f64>()
        };
        let mean = values.iter().copied().sum::<f64>() / values.len() as f64;
        let center = 0.5 * (minimum + maximum);
        let base_k = ((center - mean) / period).round() as i64;
        let mut best_shift = 0.0;
        let mut best_excursion = f64::INFINITY;
        for k in (base_k - 1)..=(base_k + 1) {
            let shift = k as f64 * period;
            let value = excursion(shift);
            if value < best_excursion {
                best_excursion = value;
                best_shift = shift;
            }
        }
        if best_shift != 0.0 {
            for value in values.iter_mut() {
                *value += best_shift;
            }
        }
    }
}

fn build_interpolant(
    surface: &NurbsSurface,
    raw_parameters: &[[f64; 2]],
    curve_parameters: &[f64],
) -> Result<NurbsCurve, String> {
    let ([u0, u1], [v0, v1]) = surface_domains(surface)?;
    let (closed_u, closed_v) = surface_closedness(surface)?;
    let mut parameters = raw_parameters.to_vec();
    // A cone's apex has no longitude either: every u there is the same point,
    // and the projector answers one arbitrarily (0, which unwraps to 1). A trim
    // that ENDS at the apex then turns sideways in its last span —
    // `abc 00008080` face #163 edge 476, a generatrix at u = 0.80255 whose
    // apex station read u = 1.0 (2026-09-27). An apex station at either end of
    // the run takes the u of its nearest regular station, so a generatrix's
    // pcurve is the straight u = const it is. Interior apex crossings are left
    // to the branch logic below. `BREP_PCURVE_CONE_APEX_U=0` switches this off,
    // for a same-binary A/B.
    if cone_apex_u() && matches!(surface.analytic(), Some(AnalyticSurface::RuledRevolution { .. })) && parameters.len() >= 2 {
        let speeds = parameters
            .iter()
            .map(|parameter| surface.deriv1(parameter[0], parameter[1]).map(|(_, su, _)| su.length()))
            .collect::<Result<Vec<_>, _>>()?;
        let reference = speeds.iter().copied().fold(0.0_f64, f64::max);
        let singular = |index: usize| speeds[index] <= reference * 1e-2;
        if let (Some(first), Some(last)) = (
            (0..parameters.len()).find(|&index| !singular(index)),
            (0..parameters.len()).rfind(|&index| !singular(index)),
        ) {
            for index in 0..first {
                parameters[index][0] = parameters[first][0];
            }
            for index in last + 1..parameters.len() {
                parameters[index][0] = parameters[last][0];
            }
        }
    }
    // Longitude is undefined at a sphere pole: the surface's u tangent
    // vanishes there, and closest-point inversion is free to return any u.
    // Letting that arbitrary value participate in periodic unwrapping can move
    // the first real meridian by a whole period (e.g. 0.75 -> -0.25). Analytic
    // carriers are subsequently clamped, collapsing that meridian onto u=0.
    //
    // Keep this deliberately sphere- and singularity-gated. Split the u values
    // into maximal non-pole runs, unwrap each run independently, and leave the
    // pole samples untouched. The existing interpolant degree stays unchanged;
    // lowering it here would also change endpoint-tangent decisions made later
    // while stitching the face loop.
    let sphere_u_singular = if matches!(surface.analytic(), Some(AnalyticSurface::Sphere { .. })) {
        let speeds = parameters
            .iter()
            .map(|parameter| {
                surface
                    .deriv1(parameter[0], parameter[1])
                    .map(|(_, su, _)| su.length())
            })
            .collect::<Result<Vec<_>, _>>()?;
        let reference = speeds.iter().copied().fold(0.0_f64, f64::max);
        speeds
            .into_iter()
            // Uniform edge stations need not hit the pole exactly (ABC 5360
            // bottoms out at v=5.5e-4). In this small conditioning band,
            // longitude is already numerically arbitrary and must not connect
            // the two meridian runs across the pole.
            .map(|speed| speed <= reference * 1e-2)
            .collect::<Vec<_>>()
    } else {
        vec![false; parameters.len()]
    };
    // Do not perturb ordinary pole-touching trims. The special path is needed
    // only when a pole band is bracketed by nonsingular samples and its raw
    // longitude changes by at least half a period: a true through-pole branch
    // reset whose two meridian runs must be unwrapped independently.
    // Prefix/suffix pole samples on normal sphere caps retain the legacy
    // interpolation byte-for-byte. Include the exactly-half-period case so
    // floating-point noise cannot decide which entire run is lifted (ABC 5603).
    let has_interior_pole_crossing = sphere_u_singular
        .iter()
        .position(|singular| !singular)
        .zip(sphere_u_singular.iter().rposition(|singular| !singular))
        .is_some_and(|(first, last)| sphere_u_singular[first..=last].iter().any(|value| *value));
    let period = (u1 - u0).abs();
    let interior_pole_branch_reset = closed_u
        && has_interior_pole_crossing
        && parameters.windows(2).zip(sphere_u_singular.windows(2)).any(
            |(parameter_pair, singular_pair)| {
                (singular_pair[0] || singular_pair[1])
                    && (parameter_pair[1][0] - parameter_pair[0][0]).abs()
                        >= 0.5 * period - 1e-12 * period.max(1.0)
            },
        );
    if closed_u {
        let mut values = parameters.iter().map(|value| value[0]).collect::<Vec<_>>();
        if interior_pole_branch_reset {
            let mut start = 0;
            while start < values.len() {
                while start < values.len() && sphere_u_singular[start] {
                    start += 1;
                }
                let mut end = start;
                while end < values.len() && !sphere_u_singular[end] {
                    end += 1;
                }
                unwrap_periodic(&mut values[start..end], u0, u1);
                start = end;
            }
        } else {
            unwrap_periodic(&mut values, u0, u1);
        }
        for (parameter, value) in parameters.iter_mut().zip(values) {
            parameter[0] = value;
        }
    }
    if closed_v {
        let mut values = parameters.iter().map(|value| value[1]).collect::<Vec<_>>();
        unwrap_periodic(&mut values, v0, v1);
        for (parameter, value) in parameters.iter_mut().zip(values) {
            parameter[1] = value;
        }
    }
    // A closed direction WRAPS: an edge that straddles the seam has no single
    // in-domain parameter run, so keep the unwrapped (possibly slightly
    // out-of-domain) values and let the periodic evaluator wrap them — clamping
    // them onto the seam boundary would collapse the straddling geometry (and,
    // via the interpolant, spike neighbouring stations). This applies to
    // analytic carriers too: a refitted cylinder can have stepped rims that
    // cross its seam. evaluate_extended wraps every closed direction.
    // Open directions must stay in-domain; their extension is a ruling.
    let points = parameters
        .into_iter()
        .map(|parameter| {
            let u = if closed_u {
                parameter[0]
            } else {
                parameter[0].clamp(u0, u1)
            };
            let v = if closed_v {
                parameter[1]
            } else {
                parameter[1].clamp(v0, v1)
            };
            Vec3::new(u, v, 0.0)
        })
        .collect::<Vec<_>>();
    interpolate_curve(&points, 3usize.min(points.len() - 1), curve_parameters)
}

/// The 3D residual a pcurve's image must reach before `build_pcurve_on_surface`
/// accepts it, and the sample ceiling it refines up to.
///
/// A trimmed face's boundary is only ever as accurate as these two numbers, and
/// every mass property is an integral over that boundary — so this is the floor
/// under area, volume and centroid for every curved trim in the kernel.
///
/// These were `min(1e-3, 1e-4 * scale)` and 160, where `scale` is
/// `1 + ||curve midpoint||` — the DISTANCE FROM THE WORLD ORIGIN of the curve's
/// midpoint, not the part's size. That expression pinned to an absolute 1e-3
/// once the midpoint sat more than about nine units from the origin, so on an
/// ordinary part the bar was 1e-3 and tracked nothing.
///
/// Measured on `offset-shell-box-bore-dished-face`, whose dished wall is a
/// closed-form identity at 2468.431782123 (three independent routes agree). The
/// floor is one of TWO errors there; the other is the refitted sphere carrier
/// that `sphere_offset_surface` now builds in closed form, and before both moved
/// they partly CANCELLED — which is why either alone reads as a regression:
///
/// ```text
/// floor     Box_O.S2            abs err     relative   gate total  worst case
/// 1e-3   2468.442270355      +1.049e-02     4.25e-06        (old)
/// 1e-6   2468.431785429      +3.306e-06     1.34e-09       181.4 s     31.2 s
/// 1e-7   2468.431782170      +4.746e-08     1.92e-11       218.5 s     32.1 s
/// 1e-9   2468.431782103      -2.037e-08    -8.25e-12       305.5 s     46.8 s
/// ```
///
/// 1e-7 is chosen: 70x the accuracy of 1e-6 for +20% gate time, where 1e-9 costs
/// a further +40% for a 2.3x residual no oracle in this corpus could resolve.
/// All four cells are one session on one binary, so the times are comparable to
/// each other. Every case carrying a closed-form oracle moves TOWARD it and none
/// away (`PushFaceTest` 339x, `PushFaceTest2` 275x); no case changes class at
/// any floor.
///
/// A flat ABSOLUTE is what was measured, not what was designed. This codebase
/// otherwise keys identity bands to part size (`KernelTolerances::heal_band`,
/// the `1e-11*(1+scale)` blend family), and on a 3000 mm part 1e-7 is 3e-11
/// relative — refinement will exhaust `MAX_PCURVE_SAMPLES` and return whatever
/// it reached. Until 2026-09-12 it did so SILENTLY; now the fit's
/// [`PcurveFitReport`] carries the residual actually achieved and a
/// [`PcurveFitExit::SampleCeiling`] exit, and the owning operation's
/// diagnostics count it (`pcurve.unmet_floor`). That is far better than the
/// 1e-3 it replaces but it is not a size-aware bar; deriving one from the
/// surface's own extent is separate, unmeasured work. Being absolute does remove this gate's dependence on
/// distance from the origin as a side effect — a real translation-VARIANCE
/// defect, the same anti-pattern `tolerance::merge_scale` records being removed
/// elsewhere. `drop_tolerance` and `endpoint_tolerance` below still carry it.
pub const PCURVE_REFINEMENT_TOLERANCE: f64 = 1e-7;

/// Sample ceiling for a fit's FIRST refinement pass. 160 could not reach
/// [`PCURVE_REFINEMENT_TOLERANCE`] on a curved trim of any length; 600 is the
/// value every cell of the table above was measured at.
///
/// It BINDS. On `boolean_fuzz_corpus/25_cube_pierce_solid` the fitter stopped
/// here 1.65e-3 of volume short of the converged answer (2000 and 4000 samples
/// agree exactly). Raising it globally to 2000 was costed and rejected: +23%
/// case-gate time, concentrated in three offset/fillet cases, and one case
/// (`inbox-20260910-offset-shell-collapsed-fillet`) driven 26x outside its
/// volume band. So it is not raised here: a fit that stops at this ceiling
/// with a residual still worse than its edge curve's own distance from the
/// surface re-enters refinement with [`PCURVE_RAISED_SAMPLES`]
/// ([`refine_pcurve`]); one whose residual is already that gap stops here and
/// says so ([`PcurveFitReport::off_surface`]), because no sample count can
/// close it.
const MAX_PCURVE_SAMPLES: usize = 600;

/// Sample ceiling for the ONE raised pass a fit may take past
/// [`MAX_PCURVE_SAMPLES`]. 2000 is the ceiling the convergence sweep measured
/// on `25_cube_pierce_solid`: its 2000 and 4000 cells agree to every printed
/// digit because refinement there stops itself near 1540 samples, so the cap
/// is not what binds.
const PCURVE_RAISED_SAMPLES: usize = 2000;

/// What a pcurve fit ACHIEVED, returned beside the curve so a ceiling exit can
/// never be mistaken for a met floor.
///
/// [`fit_pcurve_on_surface`] asks for [`PCURVE_REFINEMENT_TOLERANCE`] and stops
/// at [`MAX_PCURVE_SAMPLES`] — or, where that ceiling bound and the floor is
/// reachable, at [`PCURVE_RAISED_SAMPLES`]. Until 2026-09-12 a trim that ran
/// out of samples came back exactly like one that met the floor; on
/// `boolean_fuzz_corpus/25_cube_pierce_solid` that silence hid a 1.65e-3 volume
/// error. Now every fit says which it was: `residual` is measured on the curve
/// that is actually returned and `exit` names why refinement stopped.
///
/// This is a MEASUREMENT taken at a construction, not a band.
#[derive(Clone, Copy, Debug, PartialEq)]
pub struct PcurveFitReport {
    /// Largest 3D distance between the returned pcurve's image on the surface
    /// and the edge curve. On a [`PcurveFitExit::Converged`] exit whose final
    /// round probed every span this is that round's maximum — the refinement's
    /// own acceptance test. On every other exit, including a converged round
    /// that skipped spans narrower than 1e-3 of the parameter range, it is a
    /// sweep over EVERY span of the returned curve.
    pub residual: f64,
    /// The residual the refinement was asked to reach.
    pub floor: f64,
    /// Interpolation samples in the returned curve.
    pub samples: usize,
    /// Why refinement stopped.
    pub exit: PcurveFitExit,
    /// Largest distance from the edge curve to the surface at every station
    /// the fit inverted — the part of `residual` no pcurve can remove, since a
    /// pcurve's image lies ON the surface. Zero for an affine carrier. A value
    /// above `floor` says the floor is unreachable for this input; a ceiling
    /// exit is retried only while `residual` exceeds it by more than the floor,
    /// i.e. while the returned curve is worse than its input.
    pub off_surface: f64,
    /// Whether the fit re-entered refinement with [`PCURVE_RAISED_SAMPLES`]
    /// after [`MAX_PCURVE_SAMPLES`] bound with the floor unmet.
    pub raised: bool,
}

impl PcurveFitReport {
    /// Whether the returned curve reaches the floor it was asked for.
    pub fn met_floor(&self) -> bool {
        self.residual <= self.floor
    }

    /// Whether the returned curve is on the ACCEPTANCE BAR: the floor it was
    /// asked for, plus the stations' own standoff from the carrier.
    ///
    /// A pcurve's image lies on the carrier, so at a station standing `g` off
    /// it no image can come closer than `g`; a fit whose residual is within
    /// the floor of `off_surface` is as good as the input allows, and one over
    /// it is worse than its input — the same condition [`refine_pcurve`] uses
    /// to decide that a raise can still buy accuracy. This is the bar every
    /// station builder refuses over ([`build_pcurve_on_surface_stations`]).
    /// It is blind to sideways error quadratically (a station tracked `d`
    /// aside of the projection of a point `g` off the surface reads only
    /// `d²/(2g)` further away), which is stated rather than hidden: on the
    /// import corpus the stations that stand off their carriers are the
    /// FILE's curves, and the bar there is the file's own number.
    pub fn on_bar(&self) -> bool {
        self.residual <= self.floor + self.off_surface
    }
}

/// Why a pcurve refinement stopped. Only [`PcurveFitExit::Converged`] means the
/// floor was met; every other variant is a BUDGET exit, and the report's
/// residual was measured on the returned curve rather than assumed.
#[derive(Clone, Copy, Debug, PartialEq, Eq, Hash)]
pub enum PcurveFitExit {
    /// A probe round found every station within the floor. An affine carrier
    /// maps exactly and reports this with a zero residual.
    Converged,
    /// The sample ceiling bound before the floor was met: [`MAX_PCURVE_SAMPLES`]
    /// when the raise was declined (the residual is already explained by
    /// [`PcurveFitReport::off_surface`]), [`PCURVE_RAISED_SAMPLES`] when it was
    /// taken and bound too.
    SampleCeiling,
    /// Every refinement round was spent and the last one still inserted
    /// samples, so the curve it built was never re-checked by the loop.
    RoundBudget,
    /// A round found stations over the floor but could improve none of them:
    /// the edge point could not be inverted inside the drop band, or (in the
    /// raised pass) the pcurve already tracks the surface's nearest point to
    /// within the floor. Either way the edge curve itself sits off the surface
    /// there and no pcurve can close that gap; the residual reports how far.
    Stalled,
    /// The last probe round found every span it PROBED within the floor, but
    /// skipped spans narrower than the loop's own resolution floor — 1e-3 of
    /// the parameter range in [`refine_pcurve`], [`SPAN_FLOOR`] (1e-6) in
    /// [`fit_pcurve_on_surface_stations`] — and a sweep over those found the
    /// floor unmet.
    /// Until 2026-09-12 this exit was reported as `Converged` with the probed
    /// spans' maximum: on `25_cube_pierce_solid` a 1539-sample fit with 1537
    /// narrow spans read 7e-8 that way while every span read 1.9e-4.
    SpanFloor,
}

/// A fitted pcurve with what the fit achieved.
#[derive(Clone, Debug)]
pub struct PcurveFit {
    pub curve: NurbsCurve,
    pub report: PcurveFitReport,
}

/// Per-operation tally of pcurve fits and how many missed their floor.
///
/// [`build_pcurve_on_surface`] and its marched twin are called from some forty
/// sites, most of which have no diagnostics record to write into. Rather than
/// thread one through every layer, an operation that returns
/// [`KernelDiagnostics`] opens a [`PcurveFitScope`]; every fit made on that
/// thread while the scope is open is tallied here, and closing the scope hands
/// the tally back for [`PcurveFitLedger::report_into`]. This is the shape the
/// boolean's `CONFORMANCE_REPAIRS` counter already uses. With no scope open,
/// recording is a no-op.
///
/// Scopes nest: CLOSING a child folds it into its parent, so an offset shell
/// sees the fits of the booleans it ran. A scope that is DROPPED without being
/// closed is discarded, not folded — a boolean attempt that refused (and may be
/// retried under perturbation) must not leave its fits on the caller's tally.
///
/// Thread-local by construction: a fit made on a rayon worker (the `parallel`
/// feature's STEP body import) is not seen by a scope on the calling thread. No
/// diagnostics-returning operation builds pcurves off-thread today.
#[derive(Clone, Copy, Debug, Default, PartialEq)]
pub struct PcurveFitLedger {
    /// Fits recorded, exact affine ones included.
    pub fits: u64,
    /// Fits whose returned curve did not reach its floor.
    pub unmet: u64,
    /// Unmet fits that stopped at a sample ceiling.
    pub sample_ceiling: u64,
    /// Unmet fits that spent every refinement round.
    pub round_budget: u64,
    /// Unmet fits that could insert nothing.
    pub stalled: u64,
    /// Unmet fits whose remaining over-floor spans were all below the loop's
    /// 1e-3 span floor.
    pub span_floor: u64,
    /// Largest residual among the unmet fits; zero when there are none.
    pub worst_unmet_residual: f64,
    /// Fits that re-entered refinement with [`PCURVE_RAISED_SAMPLES`], met or
    /// not.
    pub raised: u64,
    /// Unmet ceiling exits that were NOT raised because the residual is
    /// already within the floor of the edge curve's own gap from the surface.
    pub raise_declined: u64,
    /// Largest [`PcurveFitReport::off_surface`] over every fit.
    pub worst_off_surface: f64,
}

impl PcurveFitLedger {
    fn record(&mut self, report: &PcurveFitReport) {
        self.fits += 1;
        if report.raised {
            self.raised += 1;
        }
        self.worst_off_surface = self.worst_off_surface.max(report.off_surface);
        if report.met_floor() {
            return;
        }
        self.unmet += 1;
        match report.exit {
            PcurveFitExit::Converged => {}
            PcurveFitExit::SampleCeiling => {
                self.sample_ceiling += 1;
                if !report.raised {
                    self.raise_declined += 1;
                }
            }
            PcurveFitExit::RoundBudget => self.round_budget += 1,
            PcurveFitExit::Stalled => self.stalled += 1,
            PcurveFitExit::SpanFloor => self.span_floor += 1,
        }
        self.worst_unmet_residual = self.worst_unmet_residual.max(report.residual);
    }

    fn fold(&mut self, child: &PcurveFitLedger) {
        self.fits += child.fits;
        self.unmet += child.unmet;
        self.sample_ceiling += child.sample_ceiling;
        self.round_budget += child.round_budget;
        self.stalled += child.stalled;
        self.span_floor += child.span_floor;
        self.worst_unmet_residual = self.worst_unmet_residual.max(child.worst_unmet_residual);
        self.raised += child.raised;
        self.raise_declined += child.raise_declined;
        self.worst_off_surface = self.worst_off_surface.max(child.worst_off_surface);
    }

    /// Write the tally into an operation's diagnostics: the `pcurve.*`
    /// counters and measurements always, plus one `pcurve.unmet_floor` event at
    /// [`DiagnosticSeverity::Degraded`] when any fit missed its floor. Degraded,
    /// not Error: the result is still the kernel's best construction and stays
    /// shippable, but it no longer claims the accuracy its floor states.
    pub fn report_into(&self, diagnostics: &mut KernelDiagnostics) {
        diagnostics.count_n("pcurve.fits", self.fits);
        diagnostics.count_n("pcurve.unmet_floor", self.unmet);
        diagnostics.count_n("pcurve.exit.sample_ceiling", self.sample_ceiling);
        diagnostics.count_n("pcurve.exit.round_budget", self.round_budget);
        diagnostics.count_n("pcurve.exit.stalled", self.stalled);
        diagnostics.count_n("pcurve.exit.span_floor", self.span_floor);
        diagnostics.count_n("pcurve.budget_raised", self.raised);
        diagnostics.count_n("pcurve.budget_raise_declined", self.raise_declined);
        diagnostics.measure_max("pcurve.worst_unmet_residual", self.worst_unmet_residual);
        diagnostics.measure_max("pcurve.worst_off_surface", self.worst_off_surface);
        if self.unmet > 0 {
            diagnostics.event(
                DiagnosticSeverity::Degraded,
                KernelStage::Refine,
                "pcurve.unmet_floor",
                format!(
                    "{} of {} pcurve fits did not reach the {:.0e} floor \
                     (sample ceiling {}, round budget {}, stalled {}, span floor {}); \
                     worst residual {:.3e}; budget raised on {}, declined on {}; \
                     edge curves sit off their surfaces by up to {:.3e}",
                    self.unmet,
                    self.fits,
                    PCURVE_REFINEMENT_TOLERANCE,
                    self.sample_ceiling,
                    self.round_budget,
                    self.stalled,
                    self.span_floor,
                    self.worst_unmet_residual,
                    self.raised,
                    self.raise_declined,
                    self.worst_off_surface,
                ),
            );
        }
    }
}

thread_local! {
    /// The open [`PcurveFitScope`]s on this thread, innermost last.
    static PCURVE_FIT_SCOPES: std::cell::RefCell<Vec<PcurveFitLedger>> =
        const { std::cell::RefCell::new(Vec::new()) };
}

/// An open tally of the pcurve fits made on this thread — see
/// [`PcurveFitLedger`]. Open it where an operation starts, [`close`] it where
/// the operation returns its diagnostics; a `?` that unwinds past it drops the
/// tally.
///
/// [`close`]: PcurveFitScope::close
pub struct PcurveFitScope {
    /// Not `Send`: the scope must close on the thread that opened it.
    _thread_bound: std::marker::PhantomData<*const ()>,
}

impl PcurveFitScope {
    pub fn open() -> Self {
        PCURVE_FIT_SCOPES.with(|scopes| scopes.borrow_mut().push(PcurveFitLedger::default()));
        Self {
            _thread_bound: std::marker::PhantomData,
        }
    }

    /// Take the tally, folding it into the enclosing scope if there is one.
    pub fn close(self) -> PcurveFitLedger {
        let ledger = PCURVE_FIT_SCOPES.with(|scopes| {
            let mut scopes = scopes.borrow_mut();
            let ledger = scopes.pop().unwrap_or_default();
            if let Some(parent) = scopes.last_mut() {
                parent.fold(&ledger);
            }
            ledger
        });
        std::mem::forget(self);
        ledger
    }
}

impl Drop for PcurveFitScope {
    fn drop(&mut self) {
        PCURVE_FIT_SCOPES.with(|scopes| {
            scopes.borrow_mut().pop();
        });
    }
}

fn record_fit(report: &PcurveFitReport) {
    // Per-fit trace for the convergence probes (`trim_floor_sweep_probe` and
    // its offset-shell sibling): one line per fit with the whole report, the
    // way `BREP_DEBUG_PCURVE` traces an endpoint failure.
    if std::env::var_os("BREP_DEBUG_PCURVE_FIT").is_some() {
        eprintln!("PCURVE-FIT {report:?}");
    }
    PCURVE_FIT_SCOPES.with(|scopes| {
        if let Some(open) = scopes.borrow_mut().last_mut() {
            open.record(report);
        }
    });
}

/// Largest 3D distance between `pcurve`'s image and `curve` at the
/// quarter-points of every span of `parameters` — the stations refinement
/// probes, over EVERY span. Evaluations only; nothing is inverted. The sweep
/// measures, it does not decide.
fn probe_residual(
    surface: &NurbsSurface,
    curve: &NurbsCurve,
    pcurve: &NurbsCurve,
    parameters: &[f64],
    [t0, t1]: [f64; 2],
    [u0, u1]: [f64; 2],
    [v0, v1]: [f64; 2],
) -> Result<f64, String> {
    let mut worst = 0.0_f64;
    for span in parameters.windows(2) {
        for local in [0.25, 0.5, 0.75] {
            let fraction = span[0] + (span[1] - span[0]) * local;
            let parameter = pcurve.evaluate(fraction)?;
            let on_surface =
                surface.evaluate(parameter.x.clamp(u0, u1), parameter.y.clamp(v0, v1))?;
            let on_curve = curve.evaluate(t0 + (t1 - t0) * fraction)?;
            worst = worst.max(on_surface.sub(on_curve).length());
        }
    }
    Ok(worst)
}

/// What one call to [`refine_pcurve`] achieved, beside the curve it left in
/// place.
struct Refinement {
    exit: PcurveFitExit,
    residual: f64,
    off_surface: f64,
    raised: bool,
}

/// Adaptive 3D-residual refinement of an interpolant, with the PER-TRIM sample
/// budget. Shared by [`fit_pcurve_on_surface`] (global inversion) and
/// [`fit_pcurve_on_surface_marched`] (seeded projection); `insert` is the one
/// step that differs — given a probe station's fraction, the current pcurve's
/// parameter there and the edge point, it returns the surface parameter to
/// interpolate through (or `None` to decline the station) and the distance
/// from the edge point to the surface at that station, its GAP.
///
/// **First pass.** Four rounds at [`MAX_PCURVE_SAMPLES`], each probing the
/// quarter-points of every span wider than 1e-3 of the parameter range and
/// inserting the stations over `floor`. Byte-for-byte the pre-2026-09-12
/// loop; every exit is classified ([`PcurveFitExit`]) and the residual is
/// measured on the RETURNED curve — over every span, unless the final round
/// probed every span, in which case that round's maximum already is.
///
/// **The raise.** A [`PcurveFitExit::SampleCeiling`] exit re-enters the four
/// rounds ONCE with [`PCURVE_RAISED_SAMPLES`], continuing from the samples it
/// has, when the residual exceeds `off_surface` by more than the floor — the
/// returned curve is worse than the edge curve's own distance from the
/// surface, so more samples CAN improve it. Measured on `25_cube_pierce_solid`
/// (2026-09-12): the edge curves there sit off their surfaces by up to 1.9e-4,
/// four of the eight ceiling exits already read their gap to four digits, and
/// raising those spent 900 inversions each for no change in the swept
/// residual. The raised pass therefore inserts only where the pcurve's surface
/// point is further than the floor from the PROJECTION of the edge point — its
/// tracking error, which the inversion just computed: it stops when the pcurve
/// tracks the surface's nearest point to the floor, which is the best any
/// pcurve can do, and reports [`PcurveFitExit::Stalled`] with the deviation it
/// actually reached when that is still over the floor. The limit of that pass
/// as the floor tightens is the projected edge curve itself, so a volume built
/// on it can be shown to converge; the deviation test alone cannot say that on
/// off-surface input. Grow-once, capped, deterministic.
#[allow(clippy::too_many_arguments)]
fn refine_pcurve(
    surface: &NurbsSurface,
    curve: &NurbsCurve,
    domain: [f64; 2],
    u_domain: [f64; 2],
    v_domain: [f64; 2],
    floor: f64,
    parameters: &mut Vec<f64>,
    raw: &mut Vec<[f64; 2]>,
    pcurve: &mut NurbsCurve,
    off_surface: f64,
    mut insert: impl FnMut(f64, Vec3, Vec3) -> Result<(Option<[f64; 2]>, f64), String>,
) -> Result<Refinement, String> {
    let mut best = refine_pcurve_pass(surface, curve, domain, u_domain, v_domain,
        floor, parameters, raw, pcurve, off_surface, false, 1.0, &mut insert)?;
    if best.off_surface > floor { return Ok(best); }
    // Each target starts from the same baseline station set: retaining the
    // coarser target's inserts can exhaust the cap before the finer target
    // places its own error-directed stations. Keep the best converged rung.
    let baseline_parameters = parameters.clone();
    let baseline_raw = raw.clone();
    let baseline_curve = pcurve.clone();
    let baseline_off_surface = best.off_surface;
    for fraction in [0.1, 0.01] {
        if best.residual <= floor * fraction { continue; }
        let mut refined_parameters = baseline_parameters.clone();
        let mut refined_raw = baseline_raw.clone();
        let mut refined_curve = baseline_curve.clone();
        let candidate = refine_pcurve_pass(surface, curve, domain, u_domain, v_domain,
            floor, &mut refined_parameters, &mut refined_raw, &mut refined_curve,
            baseline_off_surface, true, fraction, &mut insert);
        match candidate {
            Ok(candidate) if candidate.exit == PcurveFitExit::Converged
                && candidate.residual <= floor && candidate.residual < best.residual => {
                *parameters = refined_parameters;
                *raw = refined_raw;
                *pcurve = refined_curve;
                best = candidate;
            }
            _ => break,
        }
    }
    Ok(best)
}

fn refine_pcurve_pass(
    surface: &NurbsSurface,
    curve: &NurbsCurve,
    [t0, t1]: [f64; 2],
    [u0, u1]: [f64; 2],
    [v0, v1]: [f64; 2],
    floor: f64,
    parameters: &mut Vec<f64>,
    raw: &mut Vec<[f64; 2]>,
    pcurve: &mut NurbsCurve,
    mut off_surface: f64,
    track_projection: bool,
    tracking_fraction: f64,
    mut insert: impl FnMut(f64, Vec3, Vec3) -> Result<(Option<[f64; 2]>, f64), String>,
) -> Result<Refinement, String> {
    let tracking_floor = if track_projection { floor * tracking_fraction } else { floor };
    let mut max_samples = MAX_PCURVE_SAMPLES;
    let mut raised = false;
    loop {
        // Every way out of this loop is CLASSIFIED (`PcurveFitExit`); a budget
        // exit returns the curve it built, but says so. Falling off the end
        // after a fourth round of inserts is the round-budget exit.
        let mut exit = PcurveFitExit::RoundBudget;
        let mut residual = None;
        for _ in 0..4 {
            if parameters.len() >= max_samples {
                exit = PcurveFitExit::SampleCeiling;
                break;
            }
            let mut inserts = Vec::new();
            let mut round_max = 0.0_f64;
            let mut truncated = false;
            let mut skipped_narrow = false;
            for index in 0..parameters.len() - 1 {
                if parameters.len() + inserts.len() >= max_samples {
                    truncated = true;
                    break;
                }
                let start = parameters[index];
                let end = parameters[index + 1];
                if end - start < if track_projection { SPAN_FLOOR } else { 1e-3 } {
                    skipped_narrow = true;
                    continue;
                }
                for local_fraction in [0.25, 0.5, 0.75] {
                    if parameters.len() + inserts.len() >= max_samples {
                        truncated = true;
                        break;
                    }
                    let fraction = start + (end - start) * local_fraction;
                    let parameter = pcurve.evaluate(fraction)?;
                    let on_surface =
                        surface.evaluate(parameter.x.clamp(u0, u1), parameter.y.clamp(v0, v1))?;
                    let on_curve = curve.evaluate(t0 + (t1 - t0) * fraction)?;
                    let deviation = on_surface.sub(on_curve).length();
                    round_max = round_max.max(deviation);
                    if deviation <= tracking_floor {
                        continue;
                    }
                    let (surface_parameter, gap) = insert(fraction, parameter, on_curve)?;
                    off_surface = off_surface.max(gap);
                    if let Some(surface_parameter) = surface_parameter {
                        // The raised pass stops where the pcurve TRACKS the
                        // edge curve's projection to the floor — the best any
                        // pcurve can do at this station. Not `deviation - gap`:
                        // a surface point at tangential distance d from the
                        // projection of a point g off the surface is only
                        // d^2/(2g) further from it, so the deviation test is
                        // quadratically blind to sideways error on off-surface
                        // input (6e-6 of slop at g = 1.9e-4, floor 1e-7).
                        if raised || track_projection {
                            let projected =
                                surface.evaluate(surface_parameter[0], surface_parameter[1])?;
                            if on_surface.sub(projected).length() <= tracking_floor {
                                continue;
                            }
                        }
                        inserts.push((index + 1, fraction, surface_parameter));
                    }
                }
            }
            if inserts.is_empty() {
                // Nothing to insert: every probed station is within the floor
                // (converged — unless a narrow span was skipped, in which case
                // the sweep below decides), or the ones over it could not be
                // improved (stalled — the edge curve is off the surface there).
                exit = if truncated {
                    PcurveFitExit::SampleCeiling
                } else if round_max > floor {
                    PcurveFitExit::Stalled
                } else if skipped_narrow {
                    PcurveFitExit::SpanFloor
                } else {
                    residual = Some(round_max);
                    PcurveFitExit::Converged
                };
                break;
            }
            for (at, fraction, surface_parameter) in inserts.into_iter().rev() {
                parameters.insert(at, fraction);
                raw.insert(at, surface_parameter);
            }
            *pcurve = build_interpolant(surface, raw, parameters)?;
            if truncated {
                exit = PcurveFitExit::SampleCeiling;
                break;
            }
        }
        let residual = match residual {
            Some(measured) => measured,
            // A budget exit never re-checked the curve it is returning, and a
            // converged round that skipped narrow spans never checked those:
            // measure every span.
            None => probe_residual(
                surface,
                curve,
                pcurve,
                parameters,
                [t0, t1],
                [u0, u1],
                [v0, v1],
            )?,
        };
        if exit == PcurveFitExit::SpanFloor && residual <= floor {
            exit = PcurveFitExit::Converged;
        }
        // The per-trim budget decision. Deterministic, grow-once, capped (I3):
        // the ceiling bound and the returned curve is worse than the edge
        // curve's own gap from the surface by more than the floor, so samples
        // can still buy accuracy. A curve already at its gap stops here —
        // reported, not retried.
        if exit == PcurveFitExit::SampleCeiling && !raised && residual > off_surface + floor {
            raised = true;
            max_samples = PCURVE_RAISED_SAMPLES;
            continue;
        }
        return Ok(Refinement {
            exit,
            residual,
            off_surface,
            raised,
        });
    }
}

/// The exact inverse of an affine carrier's parameterization.
///
/// An affine patch (2x2 net, degree 1 both ways, equal weights) maps `uv` to 3D
/// by an affine map, so the inverse — a 3D point to the `uv` of its orthogonal
/// projection — is affine too. An affine map commutes with the rational basis:
/// sending a curve's control points through it and keeping the weights
/// produces the EXACT image of that curve in parameter space, not a fit of it.
struct AffineInversion {
    origin: Vec3,
    du: Vec3,
    dv: Vec3,
    u0: f64,
    v0: f64,
    uu: f64,
    uv: f64,
    vv: f64,
    determinant: f64,
}

impl AffineInversion {
    /// `None` when `surface` is not affine, or when its two directions are
    /// parallel enough that the inverse is not determined. The caller decides
    /// what that means: a refusal on the whole-curve lane, the sampled lane on
    /// the range lane.
    fn of(surface: &NurbsSurface) -> Result<Option<Self>, String> {
        if !surface.is_affine()? {
            return Ok(None);
        }
        let ([u0, _], [v0, _]) = surface_domains(surface)?;
        let (origin, du, dv) = surface.deriv1(u0, v0)?;
        let uu = du.dot(du);
        let uv = du.dot(dv);
        let vv = dv.dot(dv);
        let determinant = uu * vv - uv * uv;
        if determinant.abs() <= 1e-18 {
            return Ok(None);
        }
        Ok(Some(Self {
            origin,
            du,
            dv,
            u0,
            v0,
            uu,
            uv,
            vv,
            determinant,
        }))
    }

    fn uv_of(&self, point: Vec3) -> [f64; 2] {
        let delta = point.sub(self.origin);
        let along_u = delta.dot(self.du);
        let along_v = delta.dot(self.dv);
        [
            self.u0 + (along_u * self.vv - along_v * self.uv) / self.determinant,
            self.v0 + (along_v * self.uu - along_u * self.uv) / self.determinant,
        ]
    }

    /// The exact image of `curve` in this carrier's parameter space, with the
    /// curve's own degree, knots and weights.
    fn image_of(&self, curve: &NurbsCurve) -> Result<NurbsCurve, String> {
        let control_points = curve
            .control_points
            .iter()
            .map(|control| {
                let [u, v] = self.uv_of(control.point()?);
                Ok(Vec4::from_point(Vec3::new(u, v, 0.0), control.w))
            })
            .collect::<Result<Vec<_>, String>>()?;
        NurbsCurve::new(curve.degree, curve.knots.clone(), control_points)
    }
}

/// Build a parameter-space curve for a 3D curve lying on a surface, and say
/// what the fit achieved.
///
/// This mirrors the reference imprint implementation: affine patches map
/// homogeneous control points exactly; general patches use sampled inversion,
/// periodic seam unwrapping, and adaptive 3D residual refinement. The report
/// beside the curve carries the residual the RETURNED curve reaches and why
/// refinement stopped ([`PcurveFitReport`]); the curve itself is what
/// [`build_pcurve_on_surface`] has always returned.
pub fn fit_pcurve_on_surface(
    surface: &NurbsSurface,
    curve: &NurbsCurve,
) -> Result<PcurveFit, String> {
    let [t0, t1] = curve.domain()?;
    if surface.is_affine()? {
        let Some(inversion) = AffineInversion::of(surface)? else {
            return Err("build_pcurve_on_surface: singular affine parameterization".into());
        };
        let pcurve = inversion.image_of(curve)?;
        return Ok(PcurveFit {
            report: PcurveFitReport {
                residual: 0.0,
                floor: PCURVE_REFINEMENT_TOLERANCE,
                samples: pcurve.control_points.len(),
                exit: PcurveFitExit::Converged,
                off_surface: 0.0,
                raised: false,
            },
            curve: pcurve,
        });
    }

    let scale = 1.0 + curve.evaluate((t0 + t1) / 2.0)?.length();
    let drop_tolerance = 1e-4 * scale;
    let endpoint_tolerance = 1e-3 * scale;
    let refinement_tolerance = PCURVE_REFINEMENT_TOLERANCE;
    let ([u0, u1], [v0, v1]) = surface_domains(surface)?;
    let invert =
        |fraction: f64| invert_checked(surface, curve.evaluate(t0 + (t1 - t0) * fraction)?);

    let mut parameters = Vec::with_capacity(25);
    let mut raw_surface_parameters = Vec::with_capacity(25);
    // The edge curve's own distance from the surface at every station inverted,
    // accepted or not: the floor below which no pcurve can bring the residual.
    let mut off_surface = 0.0_f64;
    for index in 0..=24 {
        let fraction = index as f64 / 24.0;
        let (parameter, distance) = invert(fraction)?;
        off_surface = off_surface.max(distance);
        if distance > drop_tolerance && index != 0 && index != 24 {
            continue;
        }
        if distance > endpoint_tolerance {
            if std::env::var("BREP_DEBUG_PCURVE").is_ok() {
                let p3 = curve.evaluate(t0 + (t1 - t0) * fraction).ok();
                let p0 = curve.evaluate(t0).ok();
                let p1 = curve.evaluate(t1).ok();
                eprintln!(
                    "PCURVE-FAIL idx={index} frac={fraction} dist={distance} endpt_tol={endpoint_tolerance} scale={scale}\n  curve3D@frac={p3:?}\n  curve3D@t0={p0:?} curve3D@t1={p1:?}\n  surf_domain=u[{u0},{u1}] v[{v0},{v1}] invpar={parameter:?}\n  surf@invpar={:?}",
                    surface.evaluate(parameter[0], parameter[1]).ok()
                );
            }
            return Err(format!(
                "build_pcurve_on_surface: endpoint projection failed (distance={distance})"
            ));
        }
        parameters.push(fraction);
        raw_surface_parameters.push(parameter);
    }

    let mut pcurve = build_interpolant(surface, &raw_surface_parameters, &parameters)?;
    let refinement = refine_pcurve(
        surface,
        curve,
        [t0, t1],
        [u0, u1],
        [v0, v1],
        refinement_tolerance,
        &mut parameters,
        &mut raw_surface_parameters,
        &mut pcurve,
        off_surface,
        |fraction, _parameter, _on_curve| {
            let (surface_parameter, distance) = invert(fraction)?;
            Ok((
                (distance <= drop_tolerance).then_some(surface_parameter),
                distance,
            ))
        },
    )?;
    Ok(PcurveFit {
        curve: pcurve,
        report: PcurveFitReport {
            residual: refinement.residual,
            floor: refinement_tolerance,
            samples: parameters.len(),
            exit: refinement.exit,
            off_surface: refinement.off_surface,
            raised: refinement.raised,
        },
    })
}

/// [`fit_pcurve_on_surface`] for the callers that only need the curve. The
/// report is not dropped: it is tallied on the open [`PcurveFitScope`], if any,
/// so the operation that owns this fit can say in its diagnostics whether every
/// trim reached the floor.
pub fn build_pcurve_on_surface(
    surface: &NurbsSurface,
    curve: &NurbsCurve,
) -> Result<NurbsCurve, String> {
    let fit = fit_pcurve_on_surface(surface, curve)?;
    record_fit(&fit.report);
    Ok(fit.curve)
}

/// SECTION-PCURVE SEEDED MARCH (t222: cone × ABC 00000327, non-integral genus).
///
/// `build_pcurve_on_surface` inverts every sample by GLOBAL closest point. Where
/// the carrier surface is COMPRESSED / self-overlapping along the row the
/// section rides, the global search aliases interior samples onto a DISTANT
/// preimage sheet, folding the pcurve out past the trim boundary and re-crossing
/// it. The arrangement then splits the shared section at that phantom crossing,
/// while the mate face (whose carrier is not folded there) keeps the section
/// whole — so the two operands' fragments disagree and the section strands
/// one-use (non-integral genus). Ground truth for t222: 00000327 face 496's
/// v≈0.7875 row maps u∈[0.55,0.81] into a ~0.18mm neighbourhood of the section
/// endpoint A; the global pcurve for piece 9 folded out to u=0.806 and back,
/// re-crossing trim edge 349 at v≈0.784, 0.003 below the corner vertex A.
///
/// This marches the interior inversions with each seeded from the PREVIOUS
/// accepted parameters, so a marched sample
/// adopts the nearby branch instead of the momentarily-closest distant sheet.
/// The two ENDPOINT samples keep the deterministic global inversion (they are
/// the piece's shared junction vertices — same reasoning as
/// `repair_branch_jumps`). Additive + fail-soft: the marched chain is used ONLY
/// when it CLOSES onto the global far endpoint (its last interior sample is
/// parameter-adjacent to it, measured against the chain's own median step);
/// otherwise the plain global build is returned byte-for-byte. A single-preimage
/// carrier (the cone side of the same section) marches to the same samples the
/// global search already had. Use the analytic-aware seeded projector for both
/// the march and inserted stations: raw Newton stops at 1e-7 even on an exact
/// cylinder, introducing avoidable trim error. General folded carriers retain
/// their local seeded Newton path; analytic carriers take their closed form.
///
/// Escape hatch: `BREP_SECTION_PCURVE_MARCH=0`.
pub fn fit_pcurve_on_surface_marched(
    surface: &NurbsSurface,
    curve: &NurbsCurve,
) -> Result<PcurveFit, String> {
    let global = fit_pcurve_on_surface(surface, curve)?;
    if std::env::var("BREP_SECTION_PCURVE_MARCH").as_deref() == Ok("0") || surface.is_affine()? {
        return Ok(global);
    }
    let [t0, t1] = curve.domain()?;
    // A closed (ring) section has no distinct endpoints to anchor the march;
    // leave it to the existing path (closed rings are handled elsewhere).
    if curve.evaluate(t0)?.sub(curve.evaluate(t1)?).length() <= 1e-6 {
        return Ok(global);
    }
    const SAMPLES: usize = 24;
    let fractions: Vec<f64> = (0..=SAMPLES).map(|i| i as f64 / SAMPLES as f64).collect();
    let edge: Vec<Vec3> = fractions
        .iter()
        .map(|fraction| curve.evaluate(t0 + (t1 - t0) * fraction))
        .collect::<Result<Vec<_>, _>>()?;
    let scale = 1.0 + curve.evaluate((t0 + t1) / 2.0)?.length();
    let endpoint_tolerance = 1e-3 * scale;
    let (first, first_distance) = invert_checked(surface, edge[0])?;
    let (last, last_distance) = invert_checked(surface, edge[SAMPLES])?;
    if first_distance > endpoint_tolerance || last_distance > endpoint_tolerance {
        return Ok(global);
    }
    // The edge curve's distance from the surface at every station this chain
    // inverts — on the MARCHED branch, which is the one being fitted.
    let mut off_surface = first_distance.max(last_distance);
    let ([u0, u1], [v0, v1]) = surface_domains(surface)?;
    let u_span = (u1 - u0).abs().max(EPSILON);
    let v_span = (v1 - v0).abs().max(EPSILON);
    let (closed_u, closed_v) = surface_closedness(surface)?;
    let norm_step = |a: [f64; 2], b: [f64; 2]| -> f64 {
        let mut du = a[0] - b[0];
        if closed_u {
            while du > 0.5 * u_span {
                du -= u_span;
            }
            while du < -0.5 * u_span {
                du += u_span;
            }
        }
        let mut dv = a[1] - b[1];
        if closed_v {
            while dv > 0.5 * v_span {
                dv -= v_span;
            }
            while dv < -0.5 * v_span {
                dv += v_span;
            }
        }
        ((du / u_span).powi(2) + (dv / v_span).powi(2)).sqrt()
    };
    // Forward seeded march over the interior samples.
    let mut raw = vec![first];
    let mut previous = first;
    for index in 1..SAMPLES {
        let seeded =
            crate::projection::project_point_to_surface_from_seed(surface, edge[index], previous)?;
        let (_, global_distance) = invert_checked(surface, edge[index])?;
        // The seeded footpoint must still sit on the surface — as tight as the
        // fit band or the global answer already had it. If it fell off (the
        // seed led Newton into a valley), abandon the march and keep global.
        if seeded.distance > 4.0 * global_distance + endpoint_tolerance {
            return Ok(global);
        }
        off_surface = off_surface.max(seeded.distance);
        previous = [seeded.u, seeded.v];
        raw.push(previous);
    }
    raw.push(last);
    // CLOSURE GATE: the last marched interior sample must be parameter-adjacent
    // to the global far endpoint — the march stayed on ONE branch the whole way.
    // Measure the endpoint gap against the chain's OWN median step so a genuine
    // long edge is not rejected while a sheet-hop (which leaves a big gap to the
    // endpoint) is.
    let mut steps: Vec<f64> = (1..raw.len())
        .map(|i| norm_step(raw[i], raw[i - 1]))
        .collect();
    let endpoint_gap = steps.pop().unwrap_or(0.0);
    steps.sort_by(f64::total_cmp);
    let median = steps
        .get(steps.len() / 2)
        .copied()
        .unwrap_or(0.0)
        .max(EPSILON);
    if endpoint_gap > (4.0 * median).max(0.05) {
        return Ok(global);
    }
    // Build the interpolant from the marched samples, then refine — re-projecting
    // each insert SEEDED from the interpolant (already on the marched branch), so
    // a mid-refinement global inversion cannot re-alias onto the far sheet.
    let mut parameters = fractions;
    let mut pcurve = build_interpolant(surface, &raw, &parameters)?;
    let refinement_tolerance = PCURVE_REFINEMENT_TOLERANCE;
    // Same classified exits and budget ladder as `fit_pcurve_on_surface`. A
    // seeded insert always lands, so this loop cannot stall.
    let refinement = refine_pcurve(
        surface,
        curve,
        [t0, t1],
        [u0, u1],
        [v0, v1],
        refinement_tolerance,
        &mut parameters,
        &mut raw,
        &mut pcurve,
        off_surface,
        |_fraction, seed, on_curve| {
            let seeded = crate::projection::project_point_to_surface_from_seed(surface, on_curve, [seed.x, seed.y])?;
            Ok((Some([seeded.u, seeded.v]), seeded.distance))
        },
    )?;
    Ok(PcurveFit {
        curve: pcurve,
        report: PcurveFitReport {
            residual: refinement.residual,
            floor: refinement_tolerance,
            samples: parameters.len(),
            exit: refinement.exit,
            off_surface: refinement.off_surface,
            raised: refinement.raised,
        },
    })
}

/// [`fit_pcurve_on_surface_marched`] for the callers that only need the curve;
/// the report is tallied on the open [`PcurveFitScope`] exactly as
/// [`build_pcurve_on_surface`] does.
pub fn build_pcurve_on_surface_marched(
    surface: &NurbsSurface,
    curve: &NurbsCurve,
) -> Result<NurbsCurve, String> {
    let fit = fit_pcurve_on_surface_marched(surface, curve)?;
    record_fit(&fit.report);
    Ok(fit.curve)
}

/// Build a pcurve for a represented subrange of a larger edge curve.
///
/// On an affine carrier the answer is exact and is taken as such
/// ([`affine_pcurve_on_range`]). Otherwise this samples only the represented
/// interval, which is essential when off-interval control points do not lie on
/// the target carrier — the exact lane restricts the net by knot insertion
/// first, so those control points are removed rather than ignored.
pub fn fit_pcurve_on_surface_range(
    surface: &NurbsSurface,
    curve: &NurbsCurve,
    edge_start: f64,
    edge_end: f64,
    forward: bool,
    tolerance: f64,
) -> Result<PcurveFit, String> {
    if let Some(exact) =
        affine_pcurve_on_range(surface, curve, edge_start, edge_end, forward, tolerance)?
    {
        return Ok(exact);
    }
    fit_pcurve_on_surface_range_dense(
        surface, curve, edge_start, edge_end, forward, tolerance, 64, 3, 513,
    )
}

/// [`fit_pcurve_on_surface_range`] for the callers that only need the curve:
/// the fit is ACCEPTED or REFUSED on its bar ([`accept_fit`]), never returned
/// off it.
pub fn build_pcurve_on_surface_range(
    surface: &NurbsSurface,
    curve: &NurbsCurve,
    edge_start: f64,
    edge_end: f64,
    forward: bool,
    tolerance: f64,
) -> Result<NurbsCurve, String> {
    accept_fit(fit_pcurve_on_surface_range(
        surface, curve, edge_start, edge_end, forward, tolerance,
    )?)
}

/// `BREP_DEBUG_STATION_FIT`: trace every station fit round by round, and the
/// stations around the worst span of a fit off its bar. Read once per process.
fn station_fit_tracing() -> bool {
    static ON: std::sync::OnceLock<bool> = std::sync::OnceLock::new();
    *ON.get_or_init(|| std::env::var_os("BREP_DEBUG_STATION_FIT").is_some())
}

/// Narrowest span, as a fraction of the parameter range, that
/// [`fit_pcurve_on_surface_stations`] subdivides. `refine_pcurve` floors at
/// 1e-3 from 25 base samples; this fitter starts from 64 and quarters, so a
/// floor of 1e-3 bound on its first pass and refused 67 of the 57-document
/// corpus's bodies (measured 2026-09-26). 1e-6 admits every halving a real
/// trim needs and stops the runaway that fits stations 4e-7 apart.
const SPAN_FLOOR: f64 = 1e-6;

/// Rounds the raised pass of [`fit_pcurve_on_surface_stations`] may spend:
/// enough halvings for any span the sample cap can resolve.
const RAISED_ROUNDS: usize = 40;

/// The refusal every station builder returns for a fit off its bar.
pub const PCURVE_OFF_BAR: &str = "pcurve fit refused: the trim misses its stations by";

/// The one acceptance decision behind [`build_pcurve_on_surface_range`],
/// [`build_pcurve_on_surface_range_dense`] and
/// [`build_pcurve_on_surface_stations`]: a fit on its bar
/// ([`PcurveFitReport::on_bar`]) is the curve, a fit off it is a refusal that
/// names the residual, the bar, what was asked, how far the stations stand off
/// the carrier and why refinement stopped.
///
/// Until 2026-09-26 these builders returned whatever their last round built
/// without measuring it (the 2026-09-18 census: 4564 of 54556 imported pcurves
/// over the ask they were built to, 2072 of them at a station the refinement
/// itself had tested and given up on). Nothing here widens a band: the bar is
/// the caller's own ask plus what the input makes unreachable.
fn accept_fit(fit: PcurveFit) -> Result<NurbsCurve, String> {
    let report = fit.report;
    if report.on_bar() {
        return Ok(fit.curve);
    }
    Err(off_bar_message(&report))
}

fn off_bar_message(report: &PcurveFitReport) -> String {
    format!(
        "{PCURVE_OFF_BAR} {:.3e} out of sample against a bar of {:.3e} (asked {:.1e}, the \
         stations stand {:.3e} off the carrier; {:?} at {} samples{})",
        report.residual,
        report.floor + report.off_surface,
        report.floor,
        report.off_surface,
        report.exit,
        report.samples,
        if report.raised { ", after the raise" } else { "" },
    )
}

/// The class slug of a pcurve fit refused off its bar
/// ([`crate::RefusalClass::NonConvergence`]): minted by
/// [`build_pcurve_on_surface_range_classed`] and read by
/// [`is_pcurve_off_bar`], so a dispatcher decides on the class, never on the
/// message text.
pub(crate) const PCURVE_OFF_BAR_WHAT: &str = "pcurve_fit_off_bar";

/// Whether `refusal` is a pcurve fit that could not reach its bar.
pub(crate) fn is_pcurve_off_bar(refusal: &crate::KernelRefusal) -> bool {
    matches!(&refusal.class, crate::RefusalClass::NonConvergence { what } if what == PCURVE_OFF_BAR_WHAT)
}

/// [`build_pcurve_on_surface_range`] with a typed refusal: a fit off its bar
/// is `NonConvergence { what: PCURVE_OFF_BAR_WHAT }` at `stage`, with the same
/// message; any other failure of the fit is `Internal` under `what`.
pub(crate) fn build_pcurve_on_surface_range_classed(
    surface: &NurbsSurface,
    curve: &NurbsCurve,
    edge_start: f64,
    edge_end: f64,
    forward: bool,
    tolerance: f64,
    stage: KernelStage,
    what: &'static str,
) -> Result<NurbsCurve, crate::KernelRefusal> {
    use crate::OrRefuse;
    let fit = fit_pcurve_on_surface_range(surface, curve, edge_start, edge_end, forward, tolerance)
        .or_refuse(stage, what)?;
    accept_fit_classed(fit, stage)
}

/// [`accept_fit`] with the refusal typed: `NonConvergence { what:
/// PCURVE_OFF_BAR_WHAT }` at `stage`, the message unchanged.
pub(crate) fn accept_fit_classed(fit: PcurveFit, stage: KernelStage) -> Result<NurbsCurve, crate::KernelRefusal> {
    if fit.report.on_bar() {
        return Ok(fit.curve);
    }
    Err(crate::KernelRefusal {
        class: crate::RefusalClass::NonConvergence { what: PCURVE_OFF_BAR_WHAT.into() },
        stage,
        message: off_bar_message(&fit.report),
    })
}

/// The EXACT pcurve for a coedge's subrange of `curve` on an affine carrier,
/// or `None` when there is no such answer and the sampled lane must run.
///
/// [`fit_pcurve_on_surface`] has always mapped a WHOLE curve's control net
/// exactly on an affine carrier. A coedge's trim is that same curve restricted
/// to `[edge_start, edge_end]` and oriented by `forward`, and both operations
/// are exact on a NURBS: [`NurbsCurve::split`] subdivides by knot insertion,
/// [`NurbsCurve::reversed`] mirrors the net, and rescaling the knots to
/// `[0, 1]` is affine in the edge's own parameter — which is precisely the
/// fraction convention the sampled lane uses, so the pcurve's correspondence
/// with the edge is the same one either lane produces.
///
/// Sampling where an exact answer exists leaves a real error: measured
/// 2026-09-17, a booleaned band rim's two planar trims sat 2.379e-4 and
/// 2.395e-4 off their edges, where this lane lands 1.4e-13 off.
///
/// The image is verified against the caller's own `tolerance` before it is
/// returned. The exact image of a curve that does NOT lie on this carrier is
/// its orthogonal projection, which is not that coedge's trim, so every such
/// case falls through to the sampled lane and keeps the refusals that lane
/// makes today.
fn affine_pcurve_on_range(
    surface: &NurbsSurface,
    curve: &NurbsCurve,
    edge_start: f64,
    edge_end: f64,
    forward: bool,
    tolerance: f64,
) -> Result<Option<PcurveFit>, String> {
    // Opt-out for A/B bisection of this lane against the sampled one, in the
    // shape of the two hatches beside it (`BREP_NO_PCURVE_REPAIR`,
    // `BREP_SECTION_PCURVE_MARCH`). It is what showed that the boolean's
    // planar residuals do NOT come through here: the same binary with this set
    // and unset read identically to every digit on 21_band_rim.
    if std::env::var("BREP_AFFINE_RANGE_PCURVE").as_deref() == Ok("0") {
        return Ok(None);
    }
    let Some(inversion) = AffineInversion::of(surface)? else {
        return Ok(None);
    };
    if !(edge_start.is_finite() && edge_end.is_finite() && edge_start < edge_end) {
        // The sampled lane owns this refusal; it is not this lane's to reword.
        return Ok(None);
    }
    let [t0, t1] = curve.domain()?;
    let start = edge_start.max(t0);
    let end = edge_end.min(t1);
    // A range narrower than the knot identity tolerance cannot be split onto
    // its own endpoints at all.
    if end - start <= 2.0 * KNOT_IDENTITY_TOL {
        return Ok(None);
    }
    let mut restricted = curve.clone();
    if end < t1 - KNOT_IDENTITY_TOL {
        restricted = restricted.split(end)?.0;
    }
    if start > t0 + KNOT_IDENTITY_TOL {
        restricted = restricted.split(start)?.1;
    }
    let mut pcurve = inversion.image_of(&restricted)?;
    if !forward {
        pcurve = pcurve.reversed()?;
    }
    let first = pcurve.knots[0];
    let last = pcurve.knots[pcurve.knots.len() - 1];
    if !(last - first > 0.0) {
        return Ok(None);
    }
    let knots = pcurve
        .knots
        .iter()
        .map(|knot| (knot - first) / (last - first))
        .collect::<Vec<_>>();
    let pcurve = NurbsCurve::new(pcurve.degree, knots, pcurve.control_points.clone())?;

    // Verify against the range the CALLER asked for, not the one the
    // restriction produced: `split` snaps onto a nearby knot, so a snap that
    // moved the correspondence reads here as a deviation. Sampling at
    // fractions that are not the stations any lane interpolates through keeps
    // this a check on the whole trim rather than on its endpoints.
    let scale = 1.0 + curve.evaluate((start + end) / 2.0)?.length();
    let bar = tolerance.max(1e-11 * scale);
    let mut residual = 0.0_f64;
    for index in 0..=32 {
        let fraction = index as f64 / 32.0;
        let uv = pcurve.evaluate(fraction)?;
        // `evaluate` CLAMPS out-of-domain uv. A planar carrier's domain is a
        // bounding rectangle that a booleaned trim can leave, and an affine
        // patch's extension is the same plane, so read it unclamped.
        let on_carrier = surface.evaluate_extended(uv.x, uv.y)?;
        let edge_fraction = if forward { fraction } else { 1.0 - fraction };
        let on_edge = curve.evaluate(start + (end - start) * edge_fraction)?;
        residual = residual.max(on_carrier.sub(on_edge).length());
        if residual > bar {
            return Ok(None);
        }
    }
    let samples = pcurve.control_points.len();
    Ok(Some(PcurveFit {
        curve: pcurve,
        report: PcurveFitReport {
            residual,
            // The floor this lane VERIFIED against: the caller's ask, or the
            // floating-point floor of an exact map on a part this size. An
            // exact image is on its bar by construction; a caller asking for
            // less than the map's own precision is not refused for it.
            floor: bar,
            samples,
            exit: PcurveFitExit::Converged,
            off_surface: 0.0,
            raised: false,
        },
    }))
}

fn align_collapsed_endpoints(surface: &NurbsSurface, raw: &mut [[f64; 2]]) -> Result<(), String> {
    let anchor = surface.control_points[0][0].point()?;
    let mut extent = 0.0_f64;
    for control in surface.control_points.iter().flatten() {
        extent = extent.max(control.point()?.sub(anchor).length());
    }
    // This is a geometric identity check, independent of the fitting allowance
    // and of world position. Positive rational weights keep the whole iso-curve
    // inside the convex hull of its Euclidean controls. Cap the size coupling
    // at the linear tolerance so a large patch cannot erase a thin finite row.
    let pole_band = (1e-10 * (1.0 + extent)).min(LINEAR_TOLERANCE);
    for (endpoint, neighbor) in [(0, 1), (raw.len() - 1, raw.len() - 2)] {
        let point = surface.evaluate(raw[endpoint][0], raw[endpoint][1])?;
        for axis in 0..2 {
            let mut candidate = raw[endpoint];
            candidate[axis] = raw[neighbor][axis];
            if candidate == raw[endpoint]
                || surface
                    .evaluate(candidate[0], candidate[1])?
                    .sub(point)
                    .length()
                    > pole_band
            {
                continue;
            }
            let row = if axis == 0 {
                surface.iso_curve_v(raw[endpoint][1])?
            } else {
                surface.iso_curve_u(raw[endpoint][0])?
            };
            let mut collapsed = true;
            for control in &row.control_points {
                if control.point()?.sub(point).length() > pole_band {
                    collapsed = false;
                    break;
                }
            }
            if collapsed {
                raw[endpoint] = candidate;
            }
        }
    }
    Ok(())
}

/// Re-seat any raw inversion sample that BRANCH-JUMPED — snapped to a distant
/// fold of a self-overlapping general carrier — back onto the branch traced by
/// its neighbours.
///
/// Each entry in `raw` is an INDEPENDENT global closest-point inversion of the
/// corresponding 3D edge sample. Every one is geometrically valid (on the
/// surface, at the edge), but they need not be CONTIGUOUS: where a rational
/// B-spline surface folds back over the small trimmed patch it carries, the
/// momentarily-closest fold flips from one sample to the next, so the raw
/// polygon zig-zags across the whole domain and self-crosses — and the region
/// its interpolant bounds no longer covers the true face (its surface flux can
/// exceed its own area). We anchor on the LONGEST run of mutually-continuous
/// samples — the fold the trimmed patch actually lies on, since a small face
/// lives on ONE fold and the jumped samples are the minority the global search
/// snapped elsewhere — then walk outward in both directions, re-seeding Newton
/// from the last good parameters and adopting the continuous footpoint whenever
/// it is geometrically as valid as the global one. Anchoring on consensus (not
/// blindly on sample 0, which can itself be an outlier that would drag the
/// whole edge onto the wrong branch and tear the loop open at its endpoint)
/// leaves continuous nonsingular fits unchanged. Certified collapsed endpoint
/// rows first adopt their neighbour's branch; a bad seed in the subsequent
/// jump repair cannot make a sample worse than the global search already had it.
fn repair_branch_jumps(
    surface: &NurbsSurface,
    edge_points: &[Vec3],
    raw: &mut [[f64; 2]],
    tolerance: f64,
) -> Result<(), String> {
    if raw.len() < 3 {
        return Ok(());
    }
    // Affine carriers invert linearly and exactly — no folds, nothing to chase.
    // Checked first so the common planar/affine face never touches the env.
    if surface.is_affine()? {
        return Ok(());
    }
    // Opt-out for A/B bisection of a STEP-import regression against this repair.
    if std::env::var("BREP_NO_PCURVE_REPAIR").is_ok() {
        return Ok(());
    }
    // A pole has no unique coordinate along its collapsed row. Continuing the
    // adjacent sample's branch is safe only when the ENTIRE row is collapsed;
    // coincident points across an ordinary seam or fold are not enough.
    align_collapsed_endpoints(surface, raw)?;
    let ([u0, u1], [v0, v1]) = surface_domains(surface)?;
    let u_span = (u1 - u0).abs().max(EPSILON);
    let v_span = (v1 - v0).abs().max(EPSILON);
    let (closed_u, closed_v) = surface_closedness(surface)?;
    // Normalised, seam-aware step between two parameter samples. Wrapping the
    // closed directions keeps a legitimate seam crossing SMALL, so the seam /
    // periodic-unwrap machinery elsewhere is never disturbed by this repair.
    let norm_step = |a: [f64; 2], b: [f64; 2]| -> f64 {
        let mut du = a[0] - b[0];
        if closed_u {
            while du > 0.5 * u_span {
                du -= u_span;
            }
            while du < -0.5 * u_span {
                du += u_span;
            }
        }
        let mut dv = a[1] - b[1];
        if closed_v {
            while dv > 0.5 * v_span {
                dv -= v_span;
            }
            while dv < -0.5 * v_span {
                dv += v_span;
            }
        }
        ((du / u_span).powi(2) + (dv / v_span).powi(2)).sqrt()
    };
    // A jump crosses a large fraction of the WHOLE domain — orders above the
    // per-sample motion of any real (even domain-spanning) edge, which advances
    // ~1/N of its traversal between consecutive stations.
    const JUMP_THRESHOLD: f64 = 0.2;
    let count = raw.len();

    // Locate the longest maximal run of consecutive continuous samples — the
    // branch to anchor on. If nothing jumps, this spans [0, count-1] and both
    // re-seat passes below are empty, leaving `raw` untouched.
    let (mut best_start, mut best_len, mut run_start) = (0usize, 1usize, 0usize);
    for index in 1..count {
        if norm_step(raw[index], raw[index - 1]) > JUMP_THRESHOLD {
            if index - run_start > best_len {
                best_len = index - run_start;
                best_start = run_start;
            }
            run_start = index;
        }
    }
    if count - run_start > best_len {
        best_len = count - run_start;
        best_start = run_start;
    }
    let (spine_lo, spine_hi) = (best_start, best_start + best_len - 1);

    // Adopt the continuous footpoint only when it (a) still sits on the surface
    // — as tight as the fit band or the global answer — and (b) genuinely
    // closes the jump rather than trading it for another. Returns the parameter
    // to carry forward as the next seed (the repaired one, or the untouched
    // global when no repair applies).
    let reseat = |edge_point: Vec3,
                  previous: [f64; 2],
                  global: [f64; 2]|
     -> Result<[f64; 2], String> {
        let global_step = norm_step(global, previous);
        if global_step <= JUMP_THRESHOLD {
            return Ok(global);
        }
        let seeded =
            project_point_to_surface_seeded(surface, edge_point, previous[0], previous[1])?;
        let candidate = [seeded.u, seeded.v];
        let global_residual = surface
            .evaluate(global[0], global[1])?
            .sub(edge_point)
            .length();
        let on_surface = seeded.distance <= 4.0 * global_residual + tolerance.max(1e-12);
        if on_surface && norm_step(candidate, previous) < 0.5 * global_step {
            if std::env::var("BREP_DEBUG_PCURVE").is_ok() {
                eprintln!(
                    "pcurve repair: jump {global_step:.4} ({global:?}) -> {:.4} ({candidate:?}) res {:.2e}->{:.2e}",
                    norm_step(candidate, previous),
                    global_residual,
                    seeded.distance
                );
            }
            Ok(candidate)
        } else {
            Ok(global)
        }
    };

    // Walk forward off the spine's high end, then backward off its low end,
    // chaining each repaired sample as the next seed so continuity propagates.
    //
    // Apart from the certified collapsed rows above, the two ENDPOINT samples
    // (fraction 0 and 1) are NEVER moved: they are the
    // edge's shared loop vertices. The global inversion is deterministic, so the
    // two coedges meeting at a vertex land it at the SAME parameters even when
    // that vertex sits on a fold reachable from two branches — moving one side
    // to a different branch tears the loop open there (`synthesize_pole_edge`
    // then rejects the non-collapsed gap). A vertex's genuine fold transition
    // (an edge whose interior rides u≈0.08 but whose endpoint must meet its
    // neighbour at u≈0.95) is exactly this case and must be preserved, not
    // "continuity-repaired" back onto the interior branch.
    let mut previous = raw[spine_hi];
    for index in (spine_hi + 1)..count.saturating_sub(1) {
        raw[index] = reseat(edge_points[index], previous, raw[index])?;
        previous = raw[index];
    }
    let mut previous = raw[spine_lo];
    for index in (1..spine_lo).rev() {
        raw[index] = reseat(edge_points[index], previous, raw[index])?;
        previous = raw[index];
    }
    Ok(())
}

/// Range fitter with explicit sampling knobs, reporting what the fit achieved.
/// [`fit_pcurve_on_surface_range`] keeps the long-standing (base 64, 3
/// refinement rounds, 513 cap) budget; a fit that binds at that cap with the
/// bar unmet grows once to [`PCURVE_RAISED_SAMPLES`]
/// ([`fit_pcurve_on_surface_stations`]).
///
/// NOTE (2026-09-03): this comment used to say "STEP import retries failed fits
/// with a denser budget". No caller passes these knobs any more — that retry
/// was removed and the sentence outlived it. The knobs are kept because they
/// are the honest way to ASK whether a fit had headroom left, which is what
/// `examples/pcurve_residual_probe.rs` uses them for: on every ABC corpus
/// coedge that failed `validate`'s pcurve check, budgets up to (512, 8, 8193)
/// with a 1000x tighter target reproduced the coarse fit's deviation to six
/// decimals, because that deviation is the edge-vs-surface closest-point
/// residual and a curve ON the surface cannot beat it.
#[allow(clippy::too_many_arguments)]
pub fn fit_pcurve_on_surface_range_dense(
    surface: &NurbsSurface,
    curve: &NurbsCurve,
    edge_start: f64,
    edge_end: f64,
    forward: bool,
    tolerance: f64,
    base_samples: usize,
    refinement_rounds: usize,
    parameter_cap: usize,
) -> Result<PcurveFit, String> {
    if !(edge_start.is_finite() && edge_end.is_finite() && edge_start < edge_end) {
        return Err("build_pcurve_on_surface_range: invalid edge interval".into());
    }
    let evaluate_edge = |fraction: f64| {
        let edge_fraction = if forward { fraction } else { 1.0 - fraction };
        curve.evaluate(edge_start + (edge_end - edge_start) * edge_fraction)
    };
    let breaks = range_breaks(curve, edge_start, edge_end, forward);
    fit_pcurve_on_surface_stations(
        surface,
        &evaluate_edge,
        &breaks,
        tolerance,
        base_samples,
        refinement_rounds,
        parameter_cap,
    )
}

/// [`fit_pcurve_on_surface_range_dense`] for the callers that only need the
/// curve, accepted or refused on its bar ([`accept_fit`]).
#[allow(clippy::too_many_arguments)]
pub fn build_pcurve_on_surface_range_dense(
    surface: &NurbsSurface,
    curve: &NurbsCurve,
    edge_start: f64,
    edge_end: f64,
    forward: bool,
    tolerance: f64,
    base_samples: usize,
    refinement_rounds: usize,
    parameter_cap: usize,
) -> Result<NurbsCurve, String> {
    accept_fit(fit_pcurve_on_surface_range_dense(
        surface,
        curve,
        edge_start,
        edge_end,
        forward,
        tolerance,
        base_samples,
        refinement_rounds,
        parameter_cap,
    )?)
}

/// The fractions of `[edge_start, edge_end]`, walked in the coedge's sense, at
/// which `curve` is not C2: interior knots of multiplicity at least its
/// degree. A rational-quadratic arc past a quarter turn is two segments joined
/// at a double knot, and an interpolant through uniform samples across that
/// joint cannot represent the second-derivative jump: on the 2026-09-18 import
/// census 1511 of the 2979 arc trims blamed on the fitter had their worst
/// miss at fraction 0.49–0.51, the mid double knot, exactly as the fillet's
/// corner patch had measured (`blend/track_fit.rs::curve_breaks`, which this
/// mirrors for a range).
pub(crate) fn range_breaks(
    curve: &NurbsCurve,
    edge_start: f64,
    edge_end: f64,
    forward: bool,
) -> Vec<f64> {
    let mut fractions = Vec::new();
    let span = edge_end - edge_start;
    if !(span.abs() > 0.0) {
        return fractions;
    }
    let (low, high) = (edge_start.min(edge_end), edge_start.max(edge_end));
    let mut index = 0;
    while index < curve.knots.len() {
        let value = curve.knots[index];
        let run = curve.knots[index..]
            .iter()
            .take_while(|knot| **knot == value)
            .count();
        if value > low && value < high && run >= curve.degree.max(1) {
            let along = (value - edge_start) / span;
            fractions.push(if forward { along } else { 1.0 - along });
        }
        index += run;
    }
    fractions.sort_by(f64::total_cmp);
    fractions
}

/// The body of [`fit_pcurve_on_surface_range_dense`], parameterized by the
/// 3D station the pcurve must pass through at each coedge fraction instead of
/// by an edge curve and a range.
///
/// The station supplier is the whole difference between deriving a pcurve and
/// reading one. `fit_pcurve_on_surface_range_dense` supplies points off the
/// edge's own 3D curve, so the answer is "the nearest point of the carrier",
/// which is ambiguous exactly where a trim is interesting — a seam branch, a
/// near-tangential approach, a pole. The STEP importer's supplied-pcurve lane
/// supplies points off the VENDOR's stated trim instead, and the ambiguity is
/// gone because the vendor resolved it.
///
/// `stations(fraction)` must be continuous in `fraction` over `[0, 1]` and land
/// on (or very near) `surface`; `breaks` are the fractions at which it is not
/// C2 ([`range_breaks`]), each of which becomes a station so no kink sits
/// between two nodes. Everything else — inversion, branch repair, refinement
/// and the measurement — is shared with the derived lane by construction.
///
/// The SURFACE's knot-line crossings are deliberately not stations here,
/// although `blend/track_fit.rs` makes them so. Measured 2026-09-26 over the
/// 57-document import corpus: with each crossing bisected on the fraction and
/// projected seeded from the left station, 34 bodies refused (19 with a foot
/// that misses its global distance rejected) against 15 without the pass —
/// every extra refusal a fit stalled around a crossing station, on
/// `AssemblyExample-Assembly` and `abc_00000014` among others. The raised
/// pass resolves the O(h²) tail across a crossing on its own.
///
/// # What the fit is measured against, and when it stops
///
/// The stations' own standoff from the carrier is read FIRST, from the base
/// inversions, and the refinement target is the ask plus that standoff
/// ([`PcurveFitReport::on_bar`]): a station standing `g` off the surface
/// cannot be imaged closer than `g`, so chasing it to the ask alone spends the
/// whole budget for nothing (the 2026-09-18 census: 2052 imported pcurves at
/// the 513 cap, 1958 of them on a curve that is not on its carrier).
///
/// Every probe and the final measurement read the carrier through
/// `evaluate_extended`: a pcurve riding a periodic seam legitimately carries
/// parameters past the domain, and the clamped `evaluate` this loop used until
/// 2026-09-26 read every such station as a gross miss — an insert per round
/// until the cap, on a trim that was right.
///
/// The RETURNED curve is measured out of sample — at the quarter points of
/// every span between its nodes, where an interpolant is exact at the nodes
/// and extremal between — unless its last round probed every span and found
/// nothing to insert, in which case that round's maximum already is that
/// measurement. A fit that binds at `parameter_cap` or spends its rounds with
/// the bar still unmet grows ONCE to [`PCURVE_RAISED_SAMPLES`]
/// (deterministic). The report says which exit it took and what it reached;
/// the `build_*` wrappers refuse over the bar, this function never does.
#[allow(clippy::too_many_arguments)]
pub fn fit_pcurve_on_surface_stations(
    surface: &NurbsSurface,
    stations: &dyn Fn(f64) -> Result<Vec3, String>,
    breaks: &[f64],
    tolerance: f64,
    base_samples: usize,
    refinement_rounds: usize,
    parameter_cap: usize,
) -> Result<PcurveFit, String> {
    let evaluate_edge = stations;
    let base_samples = base_samples.max(1);
    // No fit reaches below the refinement floor: the projector places a foot
    // to about `LINEAR_TOLERANCE` (1e-7), so a caller asking for 1e-9 asks for
    // more than any station can deliver. Chased to such an ask the fit spends
    // its cap and reads its own noise — the direct-edit heals asked 1e-9 on
    // stations 9.9e-8 off their carrier and were refused by 8.5e-9
    // (2026-09-26). The floor is the ask's floor; the report says what was
    // reached against it.
    let tolerance = tolerance.max(PCURVE_REFINEMENT_TOLERANCE);
    let mut parameters = (0..=base_samples)
        .map(|index| index as f64 / base_samples as f64)
        .collect::<Vec<_>>();
    // A break within the span floor of a station IS that station: two nodes
    // closer than that make the interpolant swing between them, and over a
    // stretch that short the kink moves the fit by nothing a bar can read.
    // `abc_00000011` face #549's curve carries a triple knot 4e-7 before its
    // end; as a station it put the cubic 1.594e-2 off between it and the last
    // one (measured 2026-09-26), where the file's curve sits 1e-7 off.
    for &fraction in breaks {
        if !(fraction > 0.0 && fraction < 1.0) {
            continue;
        }
        let at = parameters.partition_point(|parameter| *parameter < fraction);
        let near_left = at > 0 && fraction - parameters[at - 1] <= SPAN_FLOOR;
        let near_right = at < parameters.len() && parameters[at] - fraction <= SPAN_FLOOR;
        if near_left || near_right {
            continue;
        }
        parameters.insert(at, fraction);
    }
    // The 3D edge point behind each raw inversion sample, kept parallel so the
    // continuity repair can re-seed a jumped station from its own footpoint.
    let mut edge_points = parameters
        .iter()
        .map(|fraction| evaluate_edge(*fraction))
        .collect::<Result<Vec<_>, _>>()?;
    // The stations' standoff from the carrier: the part of every residual no
    // pcurve can remove, known before a single probe is spent.
    let mut off_surface = 0.0_f64;
    let mut raw = Vec::with_capacity(edge_points.len());
    let mut gaps = Vec::with_capacity(edge_points.len());
    for point in &edge_points {
        let (parameter, gap) = invert_checked(surface, *point)?;
        off_surface = off_surface.max(gap);
        raw.push(parameter);
        gaps.push(gap);
    }
    repair_branch_jumps(surface, &edge_points, &mut raw, tolerance)?;
    let mut pcurve = build_interpolant(surface, &raw, &parameters)?;

    let mut cap = parameter_cap.max(parameters.len());
    let mut rounds = refinement_rounds;
    let mut raised = false;
    loop {
        let mut exit = PcurveFitExit::RoundBudget;
        let mut converged_max = None;
        for _ in 0..rounds {
            if parameters.len() >= cap {
                exit = PcurveFitExit::SampleCeiling;
                break;
            }
            // The target is the ASK, bare: a station standing off the carrier
            // is chased to the foot it projects to, as the fitter has always
            // done. A target of ask + standoff (one landing, 2026-09-26) let
            // such a fit stop at its base stations with the interpolant
            // wandering between them, folded a molex wall's mesh and put a
            // trim of abc_00000023 2.2e-5 off its own parameter; the bar
            // below still reads what it reached.
            let target = tolerance;
            let mut candidates = Vec::new();
            let mut round_max = 0.0_f64;
            let mut truncated = false;
            let mut skipped_narrow = false;
            let worst_first = worst_first();
            'spans: for index in 0..parameters.len() - 1 {
                // The loop's own resolution floor: a span narrower than this
                // is not subdivided. Without it the
                // raised pass inserted into a vanishing span at a trim's end
                // until the cap — `abc_00000011` face #549's last stations sat
                // 4e-7 apart with their feet clamped at the patch edge, and the
                // cubic through them swung 1.594e-2 off between them (measured
                // 2026-09-26). A miss that survives only inside such a span is
                // reported as [`PcurveFitExit::SpanFloor`].
                if parameters[index + 1] - parameters[index] < SPAN_FLOOR {
                    skipped_narrow = true;
                    continue;
                }
                for local in [0.25, 0.5, 0.75] {
                    if !worst_first && parameters.len() + candidates.len() >= cap {
                        truncated = true;
                        break 'spans;
                    }
                    let fraction =
                        parameters[index] + (parameters[index + 1] - parameters[index]) * local;
                    let uv = pcurve.evaluate(fraction)?;
                    let represented = surface.evaluate_extended(uv.x, uv.y)?;
                    let edge_point = evaluate_edge(fraction)?;
                    let deviation = represented.sub(edge_point).length();
                    round_max = round_max.max(deviation);
                    if deviation <= target {
                        continue;
                    }
                    candidates.push((index, fraction, edge_point, uv, deviation));
                }
            }
            // A round with more candidates than the cap has room for spends
            // that room on its WORST probes, kept in curve order, rather than
            // on the first spans in order: truncating in span order left the
            // late spans of an off-carrier ring at their base spacing and its
            // image 1.523e-6 off its feet; worst first reaches 5.84e-8 on the
            // same 513 samples. An even stride along the curve reached 2.87e-7
            // there but diluted the inserts a local miss needs and lost fits
            // over the corpus (a 1.3e-5 trim of `abc_00000011` fell back to its
            // 4.4e-4 size-ask fit), measured 2026-09-27.
            let room = cap.saturating_sub(parameters.len());
            if candidates.len() > room {
                truncated = true;
                let mut order: Vec<usize> = (0..candidates.len()).collect();
                order.sort_by(|&a, &b| candidates[b].4.total_cmp(&candidates[a].4).then(a.cmp(&b)));
                order.truncate(room);
                order.sort_unstable();
                candidates = order.into_iter().map(|index| candidates[index]).collect();
            }
            let mut inserts = Vec::with_capacity(candidates.len());
            for (index, fraction, edge_point, uv, _) in candidates {
                let (parameter, gap) = invert_near(
                    surface,
                    edge_point,
                    [uv.x, uv.y],
                    gaps[index].max(gaps[index + 1]),
                )?;
                inserts.push((index + 1, fraction, edge_point, parameter, gap));
            }
            if station_fit_tracing() {
                eprintln!(
                    "STATION-FIT round: samples={} target={target:.3e} round_max={round_max:.3e} inserts={} truncated={truncated}",
                    parameters.len(),
                    inserts.len()
                );
            }
            if inserts.is_empty() {
                exit = if truncated {
                    PcurveFitExit::SampleCeiling
                } else if skipped_narrow {
                    // Every probed span is within the target; the narrow ones
                    // were not probed, and the sweep below measures them.
                    PcurveFitExit::SpanFloor
                } else {
                    converged_max = Some(round_max);
                    PcurveFitExit::Converged
                };
                break;
            }
            for (index, fraction, edge_point, parameter, gap) in inserts.into_iter().rev() {
                parameters.insert(index, fraction);
                edge_points.insert(index, edge_point);
                raw.insert(index, parameter);
                gaps.insert(index, gap);
                off_surface = off_surface.max(gap);
            }
            // Inserts are independent global inversions too — re-run the repair
            // so a fold-flip introduced mid-refinement cannot poison the next
            // round's deviation interpolant.
            repair_branch_jumps(surface, &edge_points, &mut raw, tolerance)?;
            pcurve = build_interpolant(surface, &raw, &parameters)?;
            if truncated {
                exit = PcurveFitExit::SampleCeiling;
                break;
            }
        }
        let residual = match converged_max {
            Some(measured) => measured,
            None => {
                station_residual(surface, evaluate_edge, &pcurve, &parameters, tolerance, &mut off_surface)?
            }
        };
        if station_fit_tracing() {
            eprintln!(
                "STATION-FIT pass raised={raised} exit={exit:?} samples={} residual={residual:.3e} off_surface={off_surface:.3e}",
                parameters.len()
            );
            if residual > tolerance + off_surface && (exit == PcurveFitExit::Converged || raised) {
                // Where the miss sits, and the uv track around it: the trace a
                // refusal needs to be attributed to a fold, a seam or the file.
                let (mut worst, mut at) = (0.0f64, 0usize);
                for (index, span) in parameters.windows(2).enumerate() {
                    let fraction = 0.5 * (span[0] + span[1]);
                    let uv = pcurve.evaluate(fraction)?;
                    let deviation = surface.evaluate_extended(uv.x, uv.y)?.sub(evaluate_edge(fraction)?).length();
                    if deviation > worst {
                        worst = deviation;
                        at = index;
                    }
                }
                let ([u0, u1], [v0, v1]) = surface_domains(surface)?;
                let (closed_u, closed_v) = surface_closedness(surface)?;
                eprintln!(
                    "STATION-FIT OFF BAR: worst {worst:.3e} between stations {at} and {}; surface degree {}x{} net {}x{} domain u[{u0},{u1}] v[{v0},{v1}] closed ({closed_u},{closed_v}) analytic {:?}",
                    at + 1,
                    surface.degree_u,
                    surface.degree_v,
                    surface.control_points.len(),
                    surface.control_points[0].len(),
                    surface.analytic().map(|_| "yes").unwrap_or("no")
                );
                {
                    let fraction = 0.5 * (parameters[at] + parameters[at + 1]);
                    let uv = pcurve.evaluate(fraction)?;
                    let image = surface.evaluate_extended(uv.x, uv.y)?;
                    let station = evaluate_edge(fraction)?;
                    let foot = project_point_to_surface(surface, station)?;
                    let seeded = project_point_to_surface_seeded(surface, station, raw[at][0], raw[at][1])?;
                    eprintln!(
                        "    at the worst midpoint: deviation {:.4e}, station gap {:.4e} (seeded {:.4e}), image-to-foot {:.4e} (to seeded foot {:.4e}), foot uv ({:.7}, {:.7}) seeded ({:.7}, {:.7}) image uv ({:.7}, {:.7})",
                        image.sub(station).length(), foot.distance, seeded.distance, image.sub(foot.point).length(), image.sub(seeded.point).length(), foot.u, foot.v, seeded.u, seeded.v, uv.x, uv.y
                    );
                }
                let lo = at.saturating_sub(3);
                let hi = (at + 4).min(parameters.len() - 1);
                for index in lo..=hi {
                    eprintln!(
                        "    station {index}: fraction {:.6} uv ({:.6}, {:.6}) point ({:.4}, {:.4}, {:.4})",
                        parameters[index], raw[index][0], raw[index][1], edge_points[index].x, edge_points[index].y, edge_points[index].z
                    );
                }
            }
        }
        if exit == PcurveFitExit::SpanFloor && residual <= tolerance + off_surface {
            exit = PcurveFitExit::Converged;
        }
        if exit != PcurveFitExit::Converged
            && !raised
            && residual > tolerance + off_surface
            && cap < PCURVE_RAISED_SAMPLES
        {
            raised = true;
            cap = PCURVE_RAISED_SAMPLES;
            // The raised pass is bounded by SAMPLES, not rounds: a kink
            // converges O(h²), a round per halving, and the three rounds of
            // the first pass are what left the 2026-09-18 census's fits over
            // their ask with budget to spare (RoundBudget at 158–400 of 2000
            // samples, measured 2026-09-26). Forty rounds cover every halving
            // the cap admits; a fit that is not converging stalls out first.
            rounds = RAISED_ROUNDS;
            continue;
        }
        return Ok(PcurveFit {
            curve: pcurve,
            report: PcurveFitReport {
                residual,
                floor: tolerance,
                samples: parameters.len(),
                exit,
                off_surface,
                raised,
            },
        });
    }
}

/// [`fit_pcurve_on_surface_stations`] for the callers that only need the
/// curve, accepted or refused on its bar ([`accept_fit`]).
pub fn build_pcurve_on_surface_stations(
    surface: &NurbsSurface,
    stations: &dyn Fn(f64) -> Result<Vec3, String>,
    breaks: &[f64],
    tolerance: f64,
    base_samples: usize,
    refinement_rounds: usize,
    parameter_cap: usize,
) -> Result<NurbsCurve, String> {
    accept_fit(fit_pcurve_on_surface_stations(
        surface,
        stations,
        breaks,
        tolerance,
        base_samples,
        refinement_rounds,
        parameter_cap,
    )?)
}

/// The largest 3D distance between `pcurve`'s image and its stations at the
/// quarter points of every span of `parameters` — out of sample by
/// construction, since the interpolant is exact at its nodes. Unclamped
/// evaluations; a station read over the running bar is inverted so that
/// `off_surface` covers the standoff AT the stations the residual is taken at,
/// not only at the nodes (a curve lifted off its plane by a bump stands
/// further off between two nodes than at either).
fn station_residual(
    surface: &NurbsSurface,
    stations: &dyn Fn(f64) -> Result<Vec3, String>,
    pcurve: &NurbsCurve,
    parameters: &[f64],
    tolerance: f64,
    off_surface: &mut f64,
) -> Result<f64, String> {
    let mut worst = 0.0_f64;
    for span in parameters.windows(2) {
        for local in [0.25, 0.5, 0.75] {
            let fraction = span[0] + (span[1] - span[0]) * local;
            let uv = pcurve.evaluate(fraction)?;
            let represented = surface.evaluate_extended(uv.x, uv.y)?;
            let station = stations(fraction)?;
            let deviation = represented.sub(station).length();
            worst = worst.max(deviation);
            if deviation > tolerance + *off_surface {
                *off_surface = off_surface.max(invert_checked(surface, station)?.1);
            }
        }
    }
    Ok(worst)
}