BREP_kernel 0.9.0

//! The RUNOUT: a convex stripe's march continued with a CONCAVE STRIPE's own
//! surface as one of its carriers.
//!
//! # The shape
//!
//! A convex edge that DIES on the concave edges it meets — the rib spine
//! running out into the pad ring on the floor, case
//! `inbox-20260909-fillet-rib-spine-corner` — has no corner ball at that
//! vertex: the face the convex stripe shares with a concave one is approached
//! from both sides.  What the rolling ball actually does there is not a corner
//! at all, it is a CARRIER SWITCH, and the switch happens twice.
//!
//! Take the rib's tip.  Three selected edges meet: the convex spine (roof ∩
//! side), and two concave ones (floor ∩ side, floor ∩ roof).  The floor is the
//! face the two concave stripes SHARE and the convex stripe does not touch.
//!
//!  1. The convex ball rolls on the roof and the side until it becomes tangent
//!     to the FLOOR — the third face at the corner.  That is the TRI-TANGENT
//!     STATION, and it is a ball-tangency condition on an input face, not an
//!     incidence: [`plan_runout`] solves `dist(centre, floor) = r` along the
//!     edge, which is transversal.
//!
//!     At that station the ball is automatically tangent to the floor-side
//!     stripe's own surface as well, and its contact on the SIDE sits exactly
//!     on that stripe's rail there.  Both follow from one fact: a ball of
//!     radius r tangent to both faces of a concave edge from the material side
//!     is the MIRROR, across either face, of the ball that generates that
//!     edge's radius-r fillet, so their centres are 2r apart and the two balls
//!     touch — at the contact point itself.  The tangency is therefore a
//!     GRAZE: the centre's distance to the stripe's SURFACE has a MINIMUM
//!     equal to r there rather than crossing it, a double root, which is why
//!     the station is solved on the floor and only CHECKED against the stripe.
//!
//!  2. Past it the ball rolls on the roof and on the floor-side stripe's
//!     SURFACE.  That is the runout march this module builds: the same §4.9
//!     tangency system with the stripe's fitted surface as the second carrier
//!     and the section plane still riding the convex edge (extended past its
//!     own vertex).
//!
//!  3. It stops where its contact with the kept mate ENDS — where that contact
//!     reaches the OTHER concave stripe's rail on the roof.  By the same
//!     mirror identity that is exactly where the ball grazes the second
//!     stripe, so the stop is found as the MINIMUM of the distance to it
//!     (well conditioned, unlike the double root) and reported with its
//!     residual.  Past the stop the first carrier switches too, and the ball
//!     rolls on the two STRIPES.
//!
//!  4. That last stretch ends in a POLE: the two concave stripes' rails on the
//!     face they share MEET, the ball's two contacts merge there, and the
//!     blend section shrinks to a point.  The pole is a curve-curve
//!     intersection of two rails the march already fitted, not a solve.
//!
//! # What this module is for
//!
//! It measures the class: the station, the carrier pair, the marched stretch
//! with its contact residuals, the stop and the pole. The mixed-convexity
//! refusal (`fillet/edges.rs`) reports those numbers instead of only naming
//! the corner, so the refusal says WHICH construction is missing and where.

use crate::{KernelRefusal, KernelStage, OrRefuse};
use crate::topology::{BrepSolid, CoedgeRecord, EdgeRecord, FaceRecord, LoopRecord, VertexRecord};
use crate::{intersect_curves, project_point_to_curve, project_point_to_surface, NurbsCurve, NurbsSurface, Vec3, Vec4};

use super::edge::{
    extrusion_direction, fit_open_rows, fit_rows_breaking, fresh_id_source, interpolate_piece,
    march_open_stations, prune_orphan_vertices, trim_edge_at, vertex_stations,
};
use super::network::{stripe_mates, tangent_away_from, MixedCorner, OVERSHOOTS};
use super::stations::*;

/// Samples used to bracket a station along the convex edge, and bisections
/// spent on it afterwards.
const BRACKET_SAMPLES: usize = 64;
const BISECTIONS: usize = 60;

/// Stations marched over the runout stretch, and over the search window the
/// stop is minimised in.
const RUNOUT_STATIONS: usize = 32;
const STOP_SAMPLES: usize = 96;

/// One carrier switch of a runout march.
pub(crate) struct RunoutStation {
    /// Parameter on the convex edge (EXTENDED past its own vertex where the
    /// station sits beyond it).
    #[allow(dead_code)]
    pub(crate) t: f64,
    /// Arc distance from the mixed-convexity vertex, along the convex edge.
    pub(crate) s: f64,
    /// The ball centre there.
    pub(crate) center: Vec3,
    /// The contact that reached the end of its carrier.
    #[allow(dead_code)]
    pub(crate) contact: Vec3,
    /// `|dist(centre, the stripe's surface) − r|` at the station: zero says
    /// the input-face solve and the stripe agree that this is the switch.
    pub(crate) stripe_residual: f64,
}

/// What a convex stripe's march needs past the tri-tangent station, measured.
pub(crate) struct RunoutPlan {
    /// The mixed-convexity vertex.
    #[allow(dead_code)]
    pub(crate) vertex: u64,
    /// The convex edge that runs out there.
    pub(crate) convex_edge: u64,
    /// The face the two concave stripes share and the convex stripe does not
    /// touch — the one the tri-tangent station is solved against.
    pub(crate) shared_face: u64,
    /// The convex stripe's mate that is KEPT past the station, and the one the
    /// concave stripe replaces.
    pub(crate) kept_mate: u64,
    pub(crate) replaced_mate: u64,
    /// Station A: where the second carrier becomes `second_carrier`'s surface.
    pub(crate) station: RunoutStation,
    /// The concave edge whose stripe is the second carrier past station A.
    pub(crate) second_carrier: u64,
    /// Station B: where the kept mate's contact ends on `first_carrier`'s
    /// rail, so the FIRST carrier switches to that stripe too.
    pub(crate) stop: Option<RunoutStation>,
    /// The concave edge whose stripe is the first carrier past station B.
    pub(crate) first_carrier: Option<u64>,
    /// Where the two concave stripes' rails on the shared face meet: the
    /// degenerate terminus of the last stretch.
    pub(crate) pole: Option<Vec3>,
    /// Stations marched between A and B with the stripe as second carrier.
    pub(crate) marched: usize,
    /// The worst excursion of either contact OUTSIDE its carrier's own
    /// parameter domain over those stations, as a fraction of that domain's
    /// span.  `|centre − contact| − r` would be a tautology here — the centre
    /// IS the offset point the Newton converged on — where this is the
    /// question that matters: `evaluate_extended` continues an open direction
    /// along the boundary tangent plane without bound, so a converged station
    /// can rest on a fictitious surface (`check_supports_on_carriers`).
    ///
    /// It is NOT zero on this class, and that is the shape rather than a
    /// defect: the kept mate's contact walks toward the strip the concave
    /// stripe's pad ADDS to that face, which the original face's own patch
    /// does not cover, so it leaves the patch before the switch station.  The
    /// bar is `check_supports_on_carriers`'s — one full span — and 5.3e-2 of
    /// one is what the rib measures.
    pub(crate) carrier_excursion: f64,
    /// Worst distance between the fitted runout rows' rails and the marched
    /// contacts they were fitted through.  Measured against the STATIONS, not
    /// by projecting onto the carriers: projection clamps at the carrier's
    /// patch boundary, and on this class the kept mate's contact deliberately
    /// walks off that patch (see `carrier_excursion`), so a projected residual
    /// would report the excursion again instead of the fit.
    pub(crate) fit_residual: f64,
}

/// The ball of a stripe at one section, solved and evaluated.
struct Ball {
    uv: [f64; 4],
    p1: Vec3,
    p2: Vec3,
    center: Vec3,
}

#[allow(clippy::too_many_arguments)]
fn solve_ball(
    surface1: &NurbsSurface,
    surface2: &NurbsSurface,
    rho: [f64; 2],
    seed: [f64; 4],
    edge: &EdgeRecord,
    t: f64,
    scale: f64,
) -> Result<Ball, KernelRefusal> {
    let (section_point, section_tangent) = section_frame(edge, t)?;
    let uv = solve_station(
        surface1,
        surface2,
        rho,
        seed,
        section_point,
        section_tangent,
        scale,
    )?;
    let (_, p1, p2, center) = tangency_residual(
        surface1,
        surface2,
        rho,
        uv,
        section_point,
        section_tangent,
    )?;
    Ok(Ball {
        uv,
        p1,
        p2,
        center,
    })
}

/// How far `uv` lies OUTSIDE `surface`'s own domain, as a fraction of the
/// domain span, worst of the two directions.  Closed directions are exempt:
/// every periodic image is the same real surface.
fn domain_excursion(surface: &NurbsSurface, uv: [f64; 2]) -> Result<f64, KernelRefusal> {
    let [u0, u1] = surface.domain_u().or_refuse(KernelStage::Refine, "domain_u")?;
    let [v0, v1] = surface.domain_v().or_refuse(KernelStage::Refine, "domain_v")?;
    let (closed_u, closed_v) = surface.closed_directions().or_refuse(KernelStage::Refine, "closed_directions")?;
    let mut worst = 0.0f64;
    for (closed, value, low, high) in [
        (closed_u, uv[0], u0, u1),
        (closed_v, uv[1], v0, v1),
    ] {
        if closed {
            continue;
        }
        let span = (high - low).abs();
        if !(span > 0.0) {
            continue;
        }
        worst = worst.max((low - value).max(value - high).max(0.0) / span);
    }
    Ok(worst)
}

/// Distance from `point` to `surface`, and the parameters it projects to.
fn surface_gap(surface: &NurbsSurface, point: Vec3) -> Result<(f64, f64, f64), KernelRefusal> {
    let projection = project_point_to_surface(surface, point).or_refuse(KernelStage::Refine, "project_point_to_surface")?;
    Ok((projection.distance, projection.u, projection.v))
}

/// March and fit one stripe the way `blend_star_network` step 4 does, taking
/// the first rung of the overshoot ladder that fits.
///
/// `widest` takes the LAST rung instead, which the pole needs: the pole sits
/// where the two stripes' rails on their shared face cross, and that crossing
/// is PAST the end of at least one of the two edges (on the rib it is half a
/// radius beyond the tip edge's own end, against that edge's 0.08·2.0 = 0.16
/// first-rung overshoot), so the narrow fit's rails do not reach each other.
fn fit_stripe(
    edge: &EdgeRecord,
    first: &BlendMate,
    second: &BlendMate,
    radius: f64,
    widest: bool,
) -> Result<FittedRows, KernelRefusal> {
    let radius_at = |_: f64| radius;
    let mut last = None;
    let ladder: Vec<f64> = if widest {
        OVERSHOOTS.iter().rev().copied().collect()
    } else {
        OVERSHOOTS.to_vec()
    };
    for &overshoot in &ladder {
        match march_open_stations(edge, first, second, &radius_at, overshoot) {
            Ok((stations, vertex_indices)) => {
                let parameters = station_parameters(&stations);
                let extrusion = extrusion_direction(edge, first, second);
                match fit_open_rows(&stations, &parameters, false, Some(vertex_indices), extrusion) {
                    Ok(mut rows) => {
                        rows.vertex_stations = vertex_stations(&parameters, vertex_indices);
                        return Ok(rows);
                    }
                    Err(error) => last = Some(error),
                }
            }
            Err(error) => last = Some(error),
        }
    }
    Err(last.unwrap_or_else(|| KernelRefusal::internal(KernelStage::Refine, "runout_march", format!("blend runout: edge {} did not march", edge.id))))
}

/// Arc distance from `vertex_t` to `t` along `curve`, by chord sampling (the
/// convex edges this runs on are straight, and a chord sum is exact there and
/// conservative elsewhere).
fn arc_distance(curve: &NurbsCurve, vertex_t: f64, t: f64) -> Result<f64, KernelRefusal> {
    const STEPS: usize = 64;
    let mut total = 0.0;
    let mut previous = curve.evaluate_extended(vertex_t).or_refuse(KernelStage::Refine, "evaluate_extended")?;
    for step in 1..=STEPS {
        let at = vertex_t + (t - vertex_t) * step as f64 / STEPS as f64;
        let point = curve.evaluate_extended(at).or_refuse(KernelStage::Refine, "evaluate_extended")?;
        total += point.sub(previous).length();
        previous = point;
    }
    Ok(total)
}

/// Measure what the convex stripe at `corner` needs past its tri-tangent
/// station.  `Err` says the class could not be measured on this selection —
/// the caller keeps its unmeasured refusal rather than dropping it.
pub(crate) fn plan_runout(
    solid: &BrepSolid,
    edge_ids: &[u64],
    radius: f64,
    corner: &MixedCorner,
) -> Result<RunoutPlan, KernelRefusal> {
    let mates = stripe_mates(solid, edge_ids, radius)?;
    let convex_index = *corner
        .convex
        .first()
        .ok_or(KernelRefusal::internal(KernelStage::Classify, "runout_convex_edge", "blend runout: the corner names no convex edge"))?;
    let (convex_edge, convex_first, convex_second) = &mates[convex_index];
    // The corner vertex is the one the convex edge shares with the concave
    // ones; `mixed_convexity_corner` reports its point, so match on that.
    let vertex = [convex_edge.start_vertex_id, convex_edge.end_vertex_id]
        .into_iter()
        .find(|id| {
            solid
                .vertices
                .iter()
                .find(|vertex| vertex.id == *id)
                .map(|vertex| vertex.point.sub(corner.point).length() <= 1e-9 * (1.0 + radius))
                .unwrap_or(false)
        })
        .ok_or(KernelRefusal::internal(KernelStage::Classify, "runout_convex_edge", "blend runout: the convex edge does not end at the reported corner"))?;
    let concave: Vec<usize> = corner
        .concave
        .iter()
        .copied()
        .filter(|index| {
            let edge = mates[*index].0;
            edge.start_vertex_id == vertex || edge.end_vertex_id == vertex
        })
        .collect();
    if concave.len() < 2 {
        return Err(KernelRefusal::unsupported(KernelStage::Classify, "runout_concave_edges", "blend runout: fewer than two concave edges reach the corner"));
    }
    // The face the concave stripes SHARE and the convex stripe does not touch.
    let convex_faces = [convex_first.face.id, convex_second.face.id];
    let mut shared: Option<&FaceRecord> = None;
    for &index in &concave {
        for mate in [&mates[index].1, &mates[index].2] {
            if convex_faces.contains(&mate.face.id) {
                continue;
            }
            let carried_by_all = concave.iter().all(|other| {
                [&mates[*other].1, &mates[*other].2]
                    .iter()
                    .any(|candidate| candidate.face.id == mate.face.id)
            });
            if carried_by_all {
                shared = Some(mate.face);
            }
        }
    }
    let shared = shared.ok_or(
        KernelRefusal::unsupported(KernelStage::Classify, "runout_shared_face", "blend runout: the concave edges at the corner share no face the convex edge misses",)
    )?;

    // ---- Station A: the convex ball tangent to the shared face. ----
    let vertex_t = if convex_edge.start_vertex_id == vertex {
        convex_edge.t0
    } else {
        convex_edge.t1
    };
    let far_t = if convex_edge.start_vertex_id == vertex {
        convex_edge.t1
    } else {
        convex_edge.t0
    };
    let scale = march_model_scale(&convex_edge.curve, convex_edge.t0, convex_edge.t1, radius)?;
    let rho = [convex_first.rho, convex_second.rho];
    let surface1 = &convex_first.face.surface;
    let surface2 = &convex_second.face.surface;
    let seed_at = |t: f64| -> Result<[f64; 4], KernelRefusal> {
        let clamped = t.clamp(convex_edge.t0, convex_edge.t1);
        let uv1 = edge_uv_on_face(convex_first.coedge, convex_edge, clamped)?;
        let uv2 = edge_uv_on_face(convex_second.coedge, convex_edge, clamped)?;
        Ok([uv1[0], uv1[1], uv2[0], uv2[1]])
    };
    // Walk from the FAR end toward the corner so the first sign change found
    // is the first switch the ball meets.
    let mut seed = seed_at(far_t)?;
    let mut previous: Option<(f64, f64)> = None;
    let mut bracket: Option<(f64, f64)> = None;
    for sample in 0..=BRACKET_SAMPLES {
        let t = far_t + (vertex_t - far_t) * sample as f64 / BRACKET_SAMPLES as f64;
        let ball = solve_ball(surface1, surface2, rho, seed, convex_edge, t, scale)?;
        seed = ball.uv;
        let (gap, _, _) = surface_gap(&shared.surface, ball.center)?;
        let value = gap - radius;
        if let Some((previous_t, previous_value)) = previous {
            if previous_value.signum() != value.signum() {
                bracket = Some((previous_t, t));
                break;
            }
        }
        previous = Some((t, value));
    }
    let (mut low, mut high) = bracket.ok_or_else(|| {
        format!(
            "blend runout: the ball on edge {} never becomes tangent to face {} along it",
            convex_edge.id, shared.id
        )
    }).or_refuse(KernelStage::Refine, "ok_or_else")?;
    let station_ball = {
        let mut seed = seed_at(low)?;
        let mut best = solve_ball(surface1, surface2, rho, seed, convex_edge, low, scale)?;
        let mut low_value = surface_gap(&shared.surface, best.center)?.0 - radius;
        for _ in 0..BISECTIONS {
            let middle = 0.5 * (low + high);
            let ball = solve_ball(
                surface1,
                surface2,
                rho,
                seed,
                convex_edge,
                middle,
                scale,
            )?;
            seed = ball.uv;
            let value = surface_gap(&shared.surface, ball.center)?.0 - radius;
            if value.signum() == low_value.signum() {
                low = middle;
                low_value = value;
            } else {
                high = middle;
            }
            best = ball;
        }
        best
    };
    let station_t = 0.5 * (low + high);
    let station_s = arc_distance(&convex_edge.curve, vertex_t, station_t)?;

    // Which concave stripe becomes the second carrier: the one the ball is
    // tangent to at the station.  Its stripe is fitted against the ORIGINAL
    // solid, exactly as the network marches it.
    let mut stripes: Vec<(usize, FittedRows)> = Vec::new();
    for &index in &concave {
        let (edge, first, second) = &mates[index];
        stripes.push((index, fit_stripe(edge, first, second, radius, false)?));
    }
    let mut ranked: Vec<(usize, f64)> = Vec::new();
    for (index, rows) in &stripes {
        let (gap, _, _) = surface_gap(&rows.surface, station_ball.center)?;
        ranked.push((*index, (gap - radius).abs()));
    }
    ranked.sort_by(|a, b| a.1.partial_cmp(&b.1).unwrap_or(std::cmp::Ordering::Equal));
    let (second_index, stripe_residual) = ranked[0];
    let (_, second_rows) = stripes
        .iter()
        .find(|(index, _)| *index == second_index)
        .expect("ranked stripe present");
    let other_index = ranked
        .get(1)
        .map(|(index, _)| *index)
        .ok_or(KernelRefusal::unsupported(KernelStage::Classify, "runout_concave_stripes", "blend runout: only one concave stripe at the corner"))?;
    let (_, other_rows) = stripes
        .iter()
        .find(|(index, _)| *index == other_index)
        .expect("ranked stripe present");

    // Which mate the stripe replaces: the one whose contact is ON the stripe.
    let gap1 = surface_gap(&second_rows.surface, station_ball.p1)?.0;
    let gap2 = surface_gap(&second_rows.surface, station_ball.p2)?.0;
    let (kept, replaced, kept_surface, kept_rho, contact) = if gap2 <= gap1 {
        (
            convex_first.face.id,
            convex_second.face.id,
            surface1,
            convex_first.rho,
            station_ball.p2,
        )
    } else {
        (
            convex_second.face.id,
            convex_first.face.id,
            surface2,
            convex_second.rho,
            station_ball.p1,
        )
    };
    let station = RunoutStation {
        t: station_t,
        s: station_s,
        center: station_ball.center,
        contact,
        stripe_residual,
    };

    // ---- The pole: the two stripes' rails on the shared face meet. ----
    let rail_on = |index: usize, rows: &FittedRows| -> Result<NurbsCurve, KernelRefusal> {
        let (_, first, second) = &mates[index];
        if first.face.id == shared.id {
            Ok(rows.cr.clone())
        } else if second.face.id == shared.id {
            Ok(rows.cs.clone())
        } else {
            Err(KernelRefusal::internal(KernelStage::Refine, "runout_shared_face", format!(
                "blend runout: edge {} does not touch the shared face {}",
                mates[index].0.id, shared.id
            )))
        }
    };
    // The rails are FITS, so their crossing is only as sharp as the fit: the
    // band is the kernel's own committed fit accuracy at this scale, the same
    // one `march_stations` refines toward, not a hand-picked number.
    let band = crate::KernelTolerances::for_scale(scale, 1e-7).intersection_fit;
    let cross = |a: &NurbsCurve, b: &NurbsCurve| -> Option<Vec3> {
        intersect_curves(a, b, band)
            .ok()
            .and_then(|hits| {
                hits.into_iter().min_by(|x, y| {
                    x.gap.partial_cmp(&y.gap).unwrap_or(std::cmp::Ordering::Equal)
                })
            })
            .map(|hit| hit.point)
    };
    let mut pole = match (
        rail_on(second_index, second_rows),
        rail_on(other_index, other_rows),
    ) {
        (Ok(a), Ok(b)) => cross(&a, &b),
        _ => None,
    };
    if pole.is_none() {
        // The crossing is past at least one edge's own end, so refit both
        // stripes on the WIDEST rung of the ladder and look again.  Only the
        // rails are taken from this fit; the surfaces above stay the ones the
        // network itself would march.
        let mut wide: Vec<(usize, FittedRows)> = Vec::new();
        for &index in &[second_index, other_index] {
            let (edge, first, second) = &mates[index];
            if let Ok(rows) = fit_stripe(edge, first, second, radius, true) {
                wide.push((index, rows));
            }
        }
        if wide.len() == 2 {
            if let (Ok(a), Ok(b)) = (rail_on(wide[0].0, &wide[0].1), rail_on(wide[1].0, &wide[1].1))
            {
                pole = cross(&a, &b);
            }
        }
    }

    // ---- The runout march, and station B. ----
    // The stripe's signed radius: the centre sits one radius off its surface
    // along the RAW normal, and which way is read off the station itself.
    let (_, stripe_u, stripe_v) = surface_gap(&second_rows.surface, station_ball.center)?;
    let stripe_normal = raw_normal(&second_rows.surface, stripe_u, stripe_v)?;
    let stripe_point = second_rows.surface.evaluate(stripe_u, stripe_v).or_refuse(KernelStage::Refine, "evaluate")?;
    let stripe_rho = radius * station_ball.center.sub(stripe_point).dot(stripe_normal).signum();
    let runout_rho = [kept_rho, stripe_rho];
    let kept_uv = if kept == convex_first.face.id {
        [station_ball.uv[0], station_ball.uv[1]]
    } else {
        [station_ball.uv[2], station_ball.uv[3]]
    };
    let runout_seed = [kept_uv[0], kept_uv[1], stripe_u, stripe_v];
    // The window the stop is searched in: from the station to the pole's own
    // section plane (or to the corner vertex when there is no pole).
    let window_end = match pole {
        Some(point) => section_parameter(convex_edge, point, station_t, vertex_t)?,
        None => vertex_t,
    };
    let graze = |t: f64, seed: [f64; 4]| -> Result<(f64, Ball), KernelRefusal> {
        let ball = solve_ball(
            kept_surface,
            &second_rows.surface,
            runout_rho,
            seed,
            convex_edge,
            t,
            scale,
        )?;
        let (gap, _, _) = surface_gap(&other_rows.surface, ball.center)?;
        Ok(((gap - radius).abs(), ball))
    };
    let mut seed = runout_seed;
    let mut samples: Vec<(f64, f64, [f64; 4])> = Vec::new();
    for sample in 0..=STOP_SAMPLES {
        let t = station_t + (window_end - station_t) * sample as f64 / STOP_SAMPLES as f64;
        let Ok((value, ball)) = graze(t, seed) else {
            break;
        };
        seed = ball.uv;
        samples.push((t, value, ball.uv));
    }
    // Station B grazes the other stripe, so it is the MINIMUM of that distance,
    // not a crossing of it — and a minimum read off the sample grid is only
    // located to half a step (0.028 of arc here, which would report the mirror
    // identity as holding to 1e-4 when it holds far tighter).  Golden-section
    // the bracketing interval down to the sample seeds' own accuracy.
    let best = samples
        .iter()
        .enumerate()
        .min_by(|a, b| a.1 .1.partial_cmp(&b.1 .1).unwrap_or(std::cmp::Ordering::Equal))
        .map(|(index, sample)| (index, *sample))
        .filter(|(index, _)| *index > 0 && *index + 1 < samples.len());
    let (stop_station, stop_t) = match best {
        Some((index, _)) => {
            const GOLDEN: f64 = 0.618_033_988_749_894_9;
            let seed = samples[index].2;
            let mut low = samples[index - 1].0;
            let mut high = samples[index + 1].0;
            let mut x1 = high - GOLDEN * (high - low);
            let mut x2 = low + GOLDEN * (high - low);
            let mut f1 = graze(x1, seed)?.0;
            let mut f2 = graze(x2, seed)?.0;
            for _ in 0..BISECTIONS {
                if f1 < f2 {
                    high = x2;
                    x2 = x1;
                    f2 = f1;
                    x1 = high - GOLDEN * (high - low);
                    f1 = graze(x1, seed)?.0;
                } else {
                    low = x1;
                    x1 = x2;
                    f1 = f2;
                    x2 = low + GOLDEN * (high - low);
                    f2 = graze(x2, seed)?.0;
                }
            }
            let t = 0.5 * (low + high);
            let (residual, ball) = graze(t, seed)?;
            (
                Some(RunoutStation {
                    t,
                    s: arc_distance(&convex_edge.curve, vertex_t, t)?,
                    center: ball.center,
                    contact: ball.p1,
                    stripe_residual: residual,
                }),
                t,
            )
        }
        None => (None, window_end),
    };

    // ---- The stretch itself, marched and fitted. ----
    let mut stations: Vec<Station> = Vec::new();
    let mut carrier_excursion: f64 = 0.0;
    let mut seed = runout_seed;
    for sample in 0..=RUNOUT_STATIONS {
        let t = station_t + (stop_t - station_t) * sample as f64 / RUNOUT_STATIONS as f64;
        let (section_point, section_tangent) = section_frame(convex_edge, t)?;
        let uv = solve_station(
            kept_surface,
            &second_rows.surface,
            runout_rho,
            seed,
            section_point,
            section_tangent,
            scale,
        )?;
        seed = uv;
        let (_, p1, p2, center) = tangency_residual(
            kept_surface,
            &second_rows.surface,
            runout_rho,
            uv,
            section_point,
            section_tangent,
        )?;
        let n1 = raw_normal(kept_surface, uv[0], uv[1])?;
        let n2 = raw_normal(&second_rows.surface, uv[2], uv[3])?;
        let cos_alpha = runout_rho[0].signum() * runout_rho[1].signum() * n1.dot(n2);
        let weight = ((1.0 + cos_alpha) * 0.5).max(0.0).sqrt();
        if weight <= 1e-6 {
            break;
        }
        let apex = apex_point(p1, n1, p2, n2, center)?;
        carrier_excursion = carrier_excursion
            .max(domain_excursion(kept_surface, [uv[0], uv[1]])?)
            .max(domain_excursion(&second_rows.surface, [uv[2], uv[3]])?);
        stations.push(Station {
            uv1: [uv[0], uv[1]],
            uv2: [uv[2], uv[3]],
            p1,
            p2,
            center,
            weight,
            apex,
        });
    }
    let mut fit_residual = f64::NAN;
    if stations.len() >= 4 {
        let parameters = station_parameters(&stations);
        if let Ok(rows) = fit_open_rows(&stations, &parameters, false, None, None) {
            let mut worst: f64 = 0.0;
            for (station, u) in stations.iter().zip(&parameters) {
                worst = worst
                    .max(rows.cr.evaluate(*u).or_refuse(KernelStage::Refine, "evaluate")?.sub(station.p1).length())
                    .max(rows.cs.evaluate(*u).or_refuse(KernelStage::Refine, "evaluate")?.sub(station.p2).length());
            }
            fit_residual = worst;
        }
    }

    Ok(RunoutPlan {
        vertex,
        convex_edge: convex_edge.id,
        shared_face: shared.id,
        kept_mate: kept,
        replaced_mate: replaced,
        station,
        second_carrier: mates[second_index].0.id,
        stop: stop_station,
        first_carrier: Some(mates[other_index].0.id),
        pole,
        marched: stations.len(),
        carrier_excursion,
        fit_residual,
    })
}

/// The section plane of `edge` at `t`: the extended point and the edge's unit
/// tangent, read as the one-sided limit where the parameterization is
/// stationary. The runout rides the convex edge's section planes past its own
/// vertex, so the read is the extended one, and past a stationary open end it
/// advances by the chord of the open march's nominal station step
/// (`march_section`). The runout's own reads (bracketing, bisection, its
/// stretch) have no one step, so every one of them takes the stripe march's
/// `span / STATIONS`, which keeps them on one family of section planes.
fn section_frame(edge: &EdgeRecord, t: f64) -> Result<(Vec3, Vec3), KernelRefusal> {
    let station_step = (edge.t1 - edge.t0).abs() / STATIONS as f64;
    march_section(edge, t, station_step, "blend runout")
}

/// The parameter on `edge` whose normal (section) plane contains `point`,
/// bracketed by `from`..`to`.  The section planes of the convex edge are what
/// the runout march rides, so a point past the edge's own vertex still has
/// one.
fn section_parameter(
    edge: &EdgeRecord,
    point: Vec3,
    from: f64,
    to: f64,
) -> Result<f64, KernelRefusal> {
    let value = |t: f64| -> Result<f64, KernelRefusal> {
        let (section_point, section_tangent) = section_frame(edge, t)?;
        Ok(point.sub(section_point).dot(section_tangent))
    };
    // The section plane sweeps monotonically along a straight or mildly
    // curved edge, so walk the window for a sign change and bisect.
    let span = to - from;
    let mut low = from;
    let mut low_value = value(from)?;
    let mut high = to + span;
    let mut found = false;
    for sample in 1..=BRACKET_SAMPLES {
        let t = from + (to + span - from) * sample as f64 / BRACKET_SAMPLES as f64;
        let current = value(t)?;
        if current.signum() != low_value.signum() {
            high = t;
            found = true;
            break;
        }
        low = t;
        low_value = current;
    }
    if !found {
        return Err(KernelRefusal::internal(KernelStage::Refine, "runout_section_plane", "blend runout: the pole has no section plane on the convex edge"));
    }
    for _ in 0..BISECTIONS {
        let middle = 0.5 * (low + high);
        let current = value(middle)?;
        if current.signum() == low_value.signum() {
            low = middle;
            low_value = current;
        } else {
            high = middle;
        }
    }
    Ok(0.5 * (low + high))
}

// ======================================================================
// The composition: one wall per spine, pole to pole, on the ring result.
// ======================================================================

/// Stations per unit of the convex edge's parameter on the middle stretch,
/// read as the open march's `STATIONS` over the stretch between the two
/// tri-tangent stations; the shorter stretches take the same density with a
/// floor of `FIT_DEGREE + 1` stations so every row piece carries the fit.
const RUNOUT_DENSITY_STATIONS: usize = STATIONS;
const MIN_STRETCH_STATIONS: usize = FIT_DEGREE + 1;

/// One stretch of the runout wall: the carrier under each rail and the
/// stations marched on them, in increasing convex-edge parameter.  `first`
/// carries the FIRST rail (`Station::p1`), `second` the second, on every
/// stretch — the carriers change along the wall, the rails' sides do not.
struct Stretch {
    first: u64,
    second: u64,
    stations: Vec<Station>,
}

/// What the ring built at one end of a spine's G1 ridge: the two chain edges
/// past the spine, the pole they end on, the pads under them, the face the
/// pads share, and the two pad rails the switch vertices land on.
struct EndChain {
    /// The spine's own vertex at this end (v118 on the document).
    spine_vertex: u64,
    /// The kept mate ∩ pad_s edge (#120) and its far vertex (v117).
    e1: u64,
    mid_vertex: u64,
    /// The pad_r ∩ pad_s seam (#119) and the pole it ends on (v107).
    e2: u64,
    pole: u64,
    pole_point: Vec3,
    /// The concave stripe whose surface replaces the spine's `replaced` mate
    /// past station A, and the one replacing the kept mate past B.
    pad_s: u64,
    pad_r: u64,
    /// The face both pads run along and the spine misses: the one station A
    /// is solved against.
    floor: u64,
    /// pad_s's rail on the replaced mate, ending at `spine_vertex`, and
    /// pad_r's rail on the kept mate, ending at `mid_vertex`.
    side_rail: u64,
    roof_rail: u64,
}

// ======================================================================
//
// A convex spine whose fillet dies on ONE selected concave edge at a
// three-valent vertex whose third edge is unselected.  On the ring the
// spine's ridge continues smoothly into the pad's END ARC on the kept mate
// (`e1`) and then into the unselected edge itself (`e2`, kept ∩ floor): no
// seam, no pole.  Past the tri-tangent station the ball rolls on the kept
// mate and the pad, and the stripe ends
//
//  * where its kept-mate rail meets the unselected edge's line extended
//    (C5's `RAIL_X_EXTENDED_EDGEUSE`), closed by a planar CLIFF in the final
//    section's own plane, when that edge is CONVEX (the cliff keeps it
//    sharp; the Pappus body of `one_ended_runout_corner.py`);
//  * at the pad's far rail, rolling on the kept mate EXTENDED into the
//    floor and trimmed by the floor, when that edge is CONCAVE (not built
//    in this commit; named).
//
// Behind `BREP_BLEND_ONE_ENDED_RUNOUT` until its census.
// ======================================================================

/// `BREP_BLEND_ONE_ENDED_RUNOUT=1` turns the one-ended runout on, `=0` off;
/// `set_one_ended_runout_for_test` forces it on a test's own thread.  Off,
/// the public entry refuses the class exactly as before.
const ONE_ENDED_RUNOUT_DEFAULT: bool = false;

thread_local! {
    static ONE_ENDED_RUNOUT_FOR_TEST: std::cell::Cell<Option<bool>> = const { std::cell::Cell::new(None) };
}

pub(crate) fn one_ended_runout_on() -> bool {
    if let Some(forced) = ONE_ENDED_RUNOUT_FOR_TEST.with(|forced| forced.get()) {
        return forced;
    }
    match std::env::var("BREP_BLEND_ONE_ENDED_RUNOUT").as_deref() {
        Ok("0") | Ok("off") => false,
        Ok("1") | Ok("on") => true,
        _ => ONE_ENDED_RUNOUT_DEFAULT,
    }
}


/// What the ring built at a ONE-ENDED end of a spine: the pad's end arc on
/// the kept mate continuing the spine, the unselected edge continuing that
/// arc onto the floor, the pad, the floor, and the pad's rail on the
/// replaced mate that ends at the spine's vertex.
struct OneEndChain {
    spine_vertex: u64,
    /// kept ∩ pad: the pad's end arc, from `spine_vertex` to `mid_vertex`.
    e1: u64,
    mid_vertex: u64,
    /// kept ∩ floor: the UNSELECTED edge, continuing `e1` past `mid_vertex`.
    e2: u64,
    pad: u64,
    floor: u64,
    /// The pad's rail on the replaced mate, ending at `spine_vertex`.
    side_rail: u64,
    /// The unselected edge's convexity: convex takes the C5 cliff.
    third_convex: bool,
}

/// Read a one-ended end on the ring, or say why this end is not one.
fn one_ended_chain(
    ring: &BrepSolid,
    spine: &EdgeRecord,
    spine_vertex: u64,
    mates: [u64; 2],
    radius: f64,
) -> Result<(OneEndChain, u64, u64), KernelRefusal> {
    let refuse = |what: &str, message: String| KernelRefusal::unsupported(KernelStage::Classify, what, format!("blend one-ended runout: {message}"));
    let e1_id = continuation(ring, spine, spine_vertex)?.ok_or_else(|| refuse(
        "one_ended_chain",
        format!("the spine {} does not continue smoothly past vertex {spine_vertex} on the ring result", spine.id),
    ))?;
    let e1 = edge_by_id(ring, e1_id)?;
    let e1_faces = faces_of(ring, e1_id);
    let kept = *mates.iter().find(|id| e1_faces.contains(id)).ok_or_else(|| refuse(
        "one_ended_chain",
        format!("edge {e1_id} past the spine runs along neither of the spine's mates"),
    ))?;
    let replaced = if kept == mates[0] { mates[1] } else { mates[0] };
    let pad = *e1_faces.iter().find(|id| **id != kept).ok_or_else(|| KernelRefusal::internal(
        KernelStage::Classify, "one_ended_chain", format!("blend one-ended runout: edge {e1_id} has one face"),
    ))?;
    let mid_vertex = other_vertex(e1, spine_vertex);
    let e2_id = continuation(ring, e1, mid_vertex)?.ok_or_else(|| refuse(
        "one_ended_chain",
        format!("the pad's end arc {e1_id} does not continue smoothly past vertex {mid_vertex}"),
    ))?;
    let e2_faces = faces_of(ring, e2_id);
    if !e2_faces.contains(&kept) || e2_faces.contains(&pad) {
        return Err(refuse("one_ended_chain", format!(
            "the edge {e2_id} past vertex {mid_vertex} runs along faces {e2_faces:?}, not along the kept mate {kept} and a third face (a two-ended ridge runs along the pad here)"
        )));
    }
    let floor = *e2_faces.iter().find(|id| **id != kept).ok_or_else(|| KernelRefusal::internal(
        KernelStage::Classify, "one_ended_chain", format!("blend one-ended runout: edge {e2_id} has one face"),
    ))?;
    if floor == replaced {
        return Err(refuse("one_ended_chain", format!(
            "the edge {e2_id} past the pad's end arc runs back onto the spine's replaced mate {replaced}"
        )));
    }
    // The pad's rail on the replaced mate ends at the spine's vertex.
    let side_rail = edges_at(ring, spine_vertex)
        .into_iter()
        .filter(|id| *id != spine.id && *id != e1_id)
        .find(|id| {
            let faces = faces_of(ring, *id);
            faces.contains(&replaced) && faces.contains(&pad)
        })
        .ok_or_else(|| refuse("one_ended_chain", format!(
            "no rail of pad {pad} on face {replaced} ends at vertex {spine_vertex}"
        )))?;
    let third_convex = super::network::selection_convexity(ring, &[e2_id], radius)
        .and_then(|convexity| convexity.first().copied())
        .ok_or_else(|| refuse("one_ended_chain", format!("the convexity of the unselected edge {e2_id} could not be read")))?;
    Ok((
        OneEndChain { spine_vertex, e1: e1_id, mid_vertex, e2: e2_id, pad, floor, side_rail, third_convex },
        kept,
        replaced,
    ))
}

/// One end of a spine's ridge on the ring: a two-ended chain (seam and
/// pole) or a one-ended end (the pad's end arc and the unselected edge).
enum RingEnd {
    Chain(EndChain),
    One(OneEndChain),
}

/// Read either end kind at `spine_vertex`.  A two-ended chain is read first
/// (the shipped class); the one-ended end only with its switch on, and the
/// chain's own refusal is kept when neither reads.
fn read_end(
    ring: &BrepSolid,
    spine: &EdgeRecord,
    spine_vertex: u64,
    mates: [u64; 2],
    radius: f64,
) -> Result<(RingEnd, u64, u64), KernelRefusal> {
    match end_chain(ring, spine, spine_vertex, mates) {
        Ok((chain, kept, replaced)) => Ok((RingEnd::Chain(chain), kept, replaced)),
        Err(chain_error) => {
            if !one_ended_runout_on() {
                return Err(chain_error);
            }
            match one_ended_chain(ring, spine, spine_vertex, mates, radius) {
                Ok((chain, kept, replaced)) => Ok((RingEnd::One(chain), kept, replaced)),
                Err(one_error) => Err(chain_error.with_message(|message| format!("{message}; nor a one-ended end: {one_error}"))),
            }
        }
    }
}

/// Every ONE-ENDED mixed corner of a selection: a three-valent vertex where
/// exactly two selected edges meet, one convex and one concave, sharing one
/// face, the third edge unselected.  Reported as a [`MixedCorner`] with one
/// concave index, which `mixed_convexity_corners` (two or more concave) never
/// lists, so the runout composition can count the end.  Empty with the
/// switch off.
pub(crate) fn one_ended_runout_corners(solid: &BrepSolid, edge_ids: &[u64], radius: f64) -> Vec<MixedCorner> {
    let mut found = Vec::new();
    if !one_ended_runout_on() {
        return found;
    }
    let Ok(mates) = stripe_mates(solid, edge_ids, radius) else {
        return found;
    };
    let mut by_vertex: Vec<(u64, Vec<usize>)> = Vec::new();
    for (index, (edge, _, _)) in mates.iter().enumerate() {
        for vertex in [edge.start_vertex_id, edge.end_vertex_id] {
            match by_vertex.iter_mut().find(|(id, _)| *id == vertex) {
                Some((_, indices)) => indices.push(index),
                None => by_vertex.push((vertex, vec![index])),
            }
        }
    }
    for (vertex_id, indices) in by_vertex {
        if indices.len() != 2 || edges_at(solid, vertex_id).len() != 3 {
            continue;
        }
        let (a, b) = (indices[0], indices[1]);
        let (Ok(a_convex), Ok(b_convex)) = (
            super::network::edge_is_convex(mates[a].0, &mates[a].1, &mates[a].2),
            super::network::edge_is_convex(mates[b].0, &mates[b].1, &mates[b].2),
        ) else {
            continue;
        };
        if a_convex == b_convex {
            continue;
        }
        let faces_a = [mates[a].1.face.id, mates[a].2.face.id];
        let faces_b = [mates[b].1.face.id, mates[b].2.face.id];
        if faces_a.iter().filter(|id| faces_b.contains(id)).count() != 1 {
            continue;
        }
        let Some(point) = solid.vertices.iter().find(|vertex| vertex.id == vertex_id).map(|vertex| vertex.point) else {
            continue;
        };
        let (convex, concave) = if a_convex { (a, b) } else { (b, a) };
        found.push(MixedCorner { vertex: vertex_id, point, convex: vec![convex], concave: vec![concave] });
    }
    found
}

fn face_by_id(solid: &BrepSolid, id: u64) -> Result<&FaceRecord, KernelRefusal> {
    solid
        .shells
        .iter()
        .flat_map(|shell| &shell.faces)
        .find(|face| face.id == id)
        .ok_or_else(|| KernelRefusal::internal(KernelStage::Refine, "runout_face", format!("blend runout: face {id} missing")))
}

fn edge_by_id(solid: &BrepSolid, id: u64) -> Result<&EdgeRecord, KernelRefusal> {
    solid
        .edges
        .iter()
        .find(|edge| edge.id == id)
        .ok_or_else(|| KernelRefusal::internal(KernelStage::Refine, "runout_edge", format!("blend runout: edge {id} missing")))
}

fn vertex_point(solid: &BrepSolid, id: u64) -> Result<Vec3, KernelRefusal> {
    solid
        .vertices
        .iter()
        .find(|vertex| vertex.id == id)
        .map(|vertex| vertex.point)
        .ok_or_else(|| KernelRefusal::internal(KernelStage::Refine, "runout_vertex", format!("blend runout: vertex {id} missing")))
}

/// The faces whose loops use `edge_id`.
fn faces_of(solid: &BrepSolid, edge_id: u64) -> Vec<u64> {
    let mut out = Vec::new();
    for face in solid.shells.iter().flat_map(|shell| &shell.faces) {
        if face.loops.iter().any(|lp| lp.coedges.iter().any(|c| c.edge_id == edge_id)) {
            out.push(face.id);
        }
    }
    out
}

fn edges_at(solid: &BrepSolid, vertex: u64) -> Vec<u64> {
    solid
        .edges
        .iter()
        .filter(|edge| !edge.degenerate && (edge.start_vertex_id == vertex || edge.end_vertex_id == vertex))
        .map(|edge| edge.id)
        .collect()
}

fn other_vertex(edge: &EdgeRecord, vertex: u64) -> u64 {
    if edge.start_vertex_id == vertex {
        edge.end_vertex_id
    } else {
        edge.start_vertex_id
    }
}

/// The edge continuing `from` smoothly (G1) through `vertex`, if exactly one does.
fn continuation(solid: &BrepSolid, from: &EdgeRecord, vertex: u64) -> Result<Option<u64>, KernelRefusal> {
    let own = tangent_away_from(from, vertex)?;
    let mut found = None;
    for id in edges_at(solid, vertex) {
        if id == from.id {
            continue;
        }
        let edge = edge_by_id(solid, id)?;
        let other = tangent_away_from(edge, vertex)?;
        if own.dot(other) <= -(1.0 - 1e-6) {
            if found.is_some() {
                return Err(KernelRefusal::unsupported(KernelStage::Classify, "runout_chain", format!(
                    "blend runout: vertex {vertex} continues edge {} smoothly along more than one edge",
                    from.id
                )));
            }
            found = Some(id);
        }
    }
    Ok(found)
}

/// Read the ring's ridge past one end of the spine: kept ∩ pad_s, then
/// pad_r ∩ pad_s, then the pole.  `kept` and `replaced` are the spine's two
/// mates; which is which is decided HERE, by which mate the first chain edge
/// runs along.
fn end_chain(
    ring: &BrepSolid,
    spine: &EdgeRecord,
    spine_vertex: u64,
    mates: [u64; 2],
) -> Result<(EndChain, u64, u64), KernelRefusal> {
    let e1_id = continuation(ring, spine, spine_vertex)?.ok_or_else(|| KernelRefusal::unsupported(
        KernelStage::Classify, "runout_chain",
        format!("blend runout: the spine {} does not continue smoothly past vertex {spine_vertex} on the ring result", spine.id),
    ))?;
    let e1 = edge_by_id(ring, e1_id)?;
    let e1_faces = faces_of(ring, e1_id);
    let kept = *mates.iter().find(|id| e1_faces.contains(id)).ok_or_else(|| KernelRefusal::unsupported(
        KernelStage::Classify, "runout_chain",
        format!("blend runout: edge {e1_id} past the spine runs along neither of the spine's mates"),
    ))?;
    let replaced = if kept == mates[0] { mates[1] } else { mates[0] };
    let pad_s = *e1_faces.iter().find(|id| **id != kept).ok_or_else(|| KernelRefusal::internal(
        KernelStage::Classify, "runout_chain", format!("blend runout: edge {e1_id} has one face"),
    ))?;
    let mid_vertex = other_vertex(e1, spine_vertex);
    let e2_id = continuation(ring, e1, mid_vertex)?.ok_or_else(|| KernelRefusal::unsupported(
        KernelStage::Classify, "runout_chain",
        format!("blend runout: edge {e1_id} does not continue smoothly past vertex {mid_vertex}"),
    ))?;
    let e2 = edge_by_id(ring, e2_id)?;
    let e2_faces = faces_of(ring, e2_id);
    if !e2_faces.contains(&pad_s) {
        return Err(KernelRefusal::unsupported(KernelStage::Classify, "runout_chain", format!(
            "blend runout: the seam {e2_id} past vertex {mid_vertex} does not run along pad {pad_s}"
        )));
    }
    let pad_r = *e2_faces.iter().find(|id| **id != pad_s).ok_or_else(|| KernelRefusal::internal(
        KernelStage::Classify, "runout_chain", format!("blend runout: edge {e2_id} has one face"),
    ))?;
    let pole = other_vertex(e2, mid_vertex);
    if continuation(ring, e2, pole)?.is_some() {
        return Err(KernelRefusal::unsupported(KernelStage::Classify, "runout_chain", format!(
            "blend runout: the ridge continues smoothly past the pole vertex {pole}; the class is a ridge of exactly three edges per end"
        )));
    }
    // The floor: the face both pad rails at the pole lie on, other than the pads.
    let mut floor = None;
    let rails: Vec<u64> = edges_at(ring, pole).into_iter().filter(|id| *id != e2_id).collect();
    if rails.len() != 2 {
        return Err(KernelRefusal::unsupported(KernelStage::Classify, "runout_chain", format!(
            "blend runout: the pole vertex {pole} carries {} edges beside the seam (expected two pad rails)",
            rails.len()
        )));
    }
    let faces_a = faces_of(ring, rails[0]);
    let faces_b = faces_of(ring, rails[1]);
    for id in &faces_a {
        if faces_b.contains(id) && *id != pad_s && *id != pad_r {
            floor = Some(*id);
        }
    }
    let floor = floor.ok_or_else(|| KernelRefusal::unsupported(KernelStage::Classify, "runout_chain", format!(
        "blend runout: the two pad rails at the pole {pole} share no face beside the pads"
    )))?;
    // pad_s's rail on the replaced mate ends at the spine's vertex.
    let side_rail = edges_at(ring, spine_vertex)
        .into_iter()
        .filter(|id| *id != spine.id && *id != e1_id)
        .find(|id| {
            let faces = faces_of(ring, *id);
            faces.contains(&replaced) && faces.contains(&pad_s)
        })
        .ok_or_else(|| KernelRefusal::unsupported(KernelStage::Classify, "runout_chain", format!(
            "blend runout: no rail of pad {pad_s} on face {replaced} ends at vertex {spine_vertex}"
        )))?;
    // pad_r's rail on the kept mate ends at the mid vertex.
    let roof_rail = edges_at(ring, mid_vertex)
        .into_iter()
        .filter(|id| *id != e1_id && *id != e2_id)
        .find(|id| {
            let faces = faces_of(ring, *id);
            faces.contains(&kept) && faces.contains(&pad_r)
        })
        .ok_or_else(|| KernelRefusal::unsupported(KernelStage::Classify, "runout_chain", format!(
            "blend runout: no rail of pad {pad_r} on face {kept} ends at vertex {mid_vertex}"
        )))?;
    Ok((
        EndChain {
            spine_vertex,
            e1: e1_id,
            mid_vertex,
            e2: e2_id,
            pole,
            pole_point: vertex_point(ring, pole)?,
            pad_s,
            pad_r,
            floor,
            side_rail,
            roof_rail,
        },
        kept,
        replaced,
    ))
}

/// A station of the runout wall from a converged `uv` on `(surface1, surface2)`.
fn make_station(
    surface1: &NurbsSurface,
    surface2: &NurbsSurface,
    rho: [f64; 2],
    uv: [f64; 4],
    edge: &EdgeRecord,
    t: f64,
) -> Result<Station, KernelRefusal> {
    let (section_point, section_tangent) = section_frame(edge, t)?;
    let (_, p1, p2, center) = tangency_residual(surface1, surface2, rho, uv, section_point, section_tangent)?;
    let n1 = raw_normal(surface1, uv[0], uv[1])?;
    let n2 = raw_normal(surface2, uv[2], uv[3])?;
    let cos_alpha = rho[0].signum() * rho[1].signum() * n1.dot(n2);
    let weight = ((1.0 + cos_alpha) * 0.5).max(0.0).sqrt();
    if weight <= 1e-6 {
        return Err(KernelRefusal::unsupported(KernelStage::Refine, "tangent_station", "blend runout: the carriers are tangent at a station (α = π)"));
    }
    let apex = apex_point(p1, n1, p2, n2, center)?;
    Ok(Station { uv1: [uv[0], uv[1]], uv2: [uv[2], uv[3]], p1, p2, center, weight, apex })
}

/// The signed radius a convex ball marching on `surface` takes there, read
/// off a centre already seated against it: `centre = p + ρ·n_raw(p)`.
fn signed_rho(surface: &NurbsSurface, center: Vec3, radius: f64) -> Result<(f64, [f64; 2]), KernelRefusal> {
    let (_, u, v) = surface_gap(surface, center)?;
    let normal = raw_normal(surface, u, v)?;
    let point = surface.evaluate(u, v).or_refuse(KernelStage::Refine, "evaluate")?;
    Ok((radius * center.sub(point).dot(normal).signum(), [u, v]))
}

/// Measured residuals of one wall, for the record and the debug trace.
#[derive(Default, Debug, Clone)]
pub(crate) struct RunoutWallReport {
    pub(crate) spine: u64,
    /// Per end (t0 side then t1 side): the mirror-identity residuals at A and
    /// B against the ring's own pads, the pole station's two-centre miss, and
    /// how many stations the (pad, pad) stretch held before the pole.
    pub(crate) station_a_residual: [f64; 2],
    pub(crate) station_b_residual: [f64; 2],
    pub(crate) pole_center_miss: [f64; 2],
    pub(crate) pole_stretch_stations: [usize; 2],
    /// The switch vertices' distance off the pad rails they are trimmed on.
    pub(crate) switch_off_rail: [f64; 4],
    pub(crate) stations: usize,
    /// Worst `1 − t·t′` of a rail's one-sided unit tangents at a switch, and
    /// worst `1 − |n·n′|` of the wall's normal against a carrier's at a
    /// station.
    pub(crate) switch_turn: f64,
    pub(crate) plane_lean: f64,
    /// Worst `1 − n·n′` of the fitted wall's normals across a switch's break
    /// knot, along the section.
    pub(crate) switch_lean: f64,
}

thread_local! {
    static WALL_REPORTS: std::cell::RefCell<Vec<RunoutWallReport>> = const { std::cell::RefCell::new(Vec::new()) };
}

/// The reports of the walls the last [`build_runout_walls`] built.
pub(crate) fn take_runout_reports() -> Vec<RunoutWallReport> {
    WALL_REPORTS.with(|reports| std::mem::take(&mut *reports.borrow_mut()))
}

/// One thing a spine's ridge claims on the ring: an EDGE (its ridge edge
/// kept∩pad, its pad∩pad seam, or the pad's side rail it trims) or its pole
/// VERTEX.  Edges share one id namespace whatever role they play, poles
/// another, so sharing is read per namespace, never per role: spine 1's ridge
/// edge and spine 2's side rail with the same id are the same edge.
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub(crate) enum ChainRole {
    Edge(&'static str),
    Pole,
}

impl ChainRole {
    fn namespace(&self) -> &'static str {
        match self {
            ChainRole::Edge(_) => "edge",
            ChainRole::Pole => "pole vertex",
        }
    }
    fn name(&self) -> &'static str {
        match self {
            ChainRole::Edge(what) => what,
            ChainRole::Pole => "pole",
        }
    }
}

#[derive(Clone, Copy, Debug)]
pub(crate) struct ChainClaim {
    pub(crate) spine: u64,
    pub(crate) role: ChainRole,
    pub(crate) id: u64,
}

/// The first pair of claims by DIFFERENT spines on one id of one namespace,
/// whatever their roles.  Sharing a ROOF rail is not a claim at all (it is
/// the reporter's own shape, each spine trimming that rail at its own end),
/// so roof rails are never passed in.
pub(crate) fn shared_chain_claim(claims: &[ChainClaim]) -> Option<(ChainClaim, ChainClaim)> {
    for (index, a) in claims.iter().enumerate() {
        for b in &claims[index + 1..] {
            if a.spine != b.spine && a.id == b.id && a.role.namespace() == b.role.namespace() {
                return Some((*a, *b));
            }
        }
    }
    None
}

/// Build the runout wall of every spine in `spines` on `ring`, the result of
/// blending the concave edges on their own: the composition §5 of the
/// 2026-09-30 runout record designs.  Each spine's wall runs from pole to pole
/// in the spine's own section planes, switching carriers at the tri-tangent
/// station A (solved against the floor, the face the two concave stripes
/// share) and at B (where the kept mate's contact reaches the other pad's rail,
/// a graze minimum), and closes on the ring's own pole vertices across a
/// degenerate edge.  The rails are split at the switches into one edge per
/// carrier; nothing is cut and nothing is intersected.
pub(crate) fn build_runout_walls(
    ring: &BrepSolid,
    spines: &[(u64, Option<String>)],
    radius: f64,
) -> Result<BrepSolid, KernelRefusal> {
    let mut result = ring.clone();
    let mut take_id = fresh_id_source(ring);
    WALL_REPORTS.with(|reports| reports.borrow_mut().clear());
    // Every spine's chains are read from the RING while the walls are built
    // one after another in RESULT, so two spines whose ridges share a chain
    // edge, a pole or a side rail would have the second wall trim an edge the
    // first already consumed (its old vertex gone) and fail inside the
    // surgery as an internal error.  Sharing a ROOF rail is the reporter's own
    // shape (both spines run out on the tip pad and on the wall∩roof pad,
    // each trimming that rail at its own end) and is built; anything else
    // refuses by name before any wall is cut.
    {
        let mut claims: Vec<ChainClaim> = Vec::new();
        for (spine_id, _) in spines {
            let mates = stripe_mates(ring, &[*spine_id], radius)?;
            let (spine, first, second) = &mates[0];
            let mate_ids = [first.face.id, second.face.id];
            for vertex in [spine.start_vertex_id, spine.end_vertex_id] {
                let (end, _, _) = read_end(ring, spine, vertex, mate_ids, radius)?;
                match end {
                    RingEnd::Chain(chain) => {
                        for (role, id) in [(ChainRole::Edge("ridge edge"), chain.e1), (ChainRole::Edge("seam"), chain.e2), (ChainRole::Edge("side rail"), chain.side_rail), (ChainRole::Pole, chain.pole)] {
                            claims.push(ChainClaim { spine: *spine_id, role, id });
                        }
                    }
                    // A one-ended end cuts the pad's end arc and the side
                    // rail; the unselected edge is left as it is.
                    RingEnd::One(chain) => {
                        for (role, id) in [(ChainRole::Edge("ridge edge"), chain.e1), (ChainRole::Edge("side rail"), chain.side_rail)] {
                            claims.push(ChainClaim { spine: *spine_id, role, id });
                        }
                    }
                }
            }
        }
        if let Some((a, b)) = shared_chain_claim(&claims) {
            return Err(KernelRefusal::unsupported(KernelStage::Classify, "runout_shared_chain", format!(
                "blend runout: spines {} and {} share {} {} on the ring (as {} and {}); two runouts on one ridge are not built",
                a.spine, b.spine, a.role.namespace(), a.id, a.role.name(), b.role.name()
            )));
        }
    }
    for (spine_id, name) in spines {
        let report = build_runout_wall(ring, &mut result, &mut take_id, *spine_id, radius, name.as_deref())?;
        WALL_REPORTS.with(|reports| reports.borrow_mut().push(report));
    }
    prune_orphan_vertices(&mut result);
    Ok(result)
}

fn build_runout_wall(
    ring: &BrepSolid,
    result: &mut BrepSolid,
    take_id: &mut dyn FnMut() -> u64,
    spine_id: u64,
    radius: f64,
    name: Option<&str>,
) -> Result<RunoutWallReport, KernelRefusal> {
    let debug = std::env::var("BREP_DEBUG_NETWORK").is_ok();
    let mates = stripe_mates(ring, &[spine_id], radius)?;
    let (spine, first, second) = &mates[0];
    let mate_ids = [first.face.id, second.face.id];
    // The ridge past each end, and which mate is KEPT past A (the one the
    // chain's first edge runs along).  Both ends must agree.
    let (end0_ring, kept0, replaced0) = read_end(ring, spine, spine.start_vertex_id, mate_ids, radius)?;
    let (end1_ring, kept1, _) = read_end(ring, spine, spine.end_vertex_id, mate_ids, radius)?;
    if kept0 != kept1 {
        return Err(KernelRefusal::unsupported(KernelStage::Classify, "runout_chain", format!(
            "blend runout: the two ends of spine {spine_id} keep different mates ({kept0} and {kept1})"
        )));
    }
    let (kept, replaced) = (kept0, replaced0);
    let kept_is_first = kept == first.face.id;
    // Each end runs out against ITS OWN shared face: the floor at the rib's
    // tip, the back wall at the other end.
    let ring_ends = [end0_ring, end1_ring];
    let scale = march_model_scale(&spine.curve, spine.t0, spine.t1, radius)?;
    let rho = [first.rho, second.rho];
    let surface1 = &first.face.surface;
    let surface2 = &second.face.surface;
    let seed_at = |t: f64| -> Result<[f64; 4], KernelRefusal> {
        let clamped = t.clamp(spine.t0.min(spine.t1), spine.t0.max(spine.t1));
        let uv1 = edge_uv_on_face(first.coedge, spine, clamped)?;
        let uv2 = edge_uv_on_face(second.coedge, spine, clamped)?;
        Ok([uv1[0], uv1[1], uv2[0], uv2[1]])
    };
    let t_mid = 0.5 * (spine.t0 + spine.t1);
    let mut report = RunoutWallReport { spine: spine_id, ..Default::default() };

    // ---- Per end: the pole's section, station A and station B, solved
    //      once.  The stretches between them are marched below on a station
    //      ladder.
    /// How a wall end terminates: at the ring's pole after station B (a
    /// two-ended chain), or at C5's stop with a planar cliff (a one-ended
    /// end).
    enum Terminus {
        Pole {
            t_b: f64,
            t_pole: f64,
            /// (pad_r, pad_s) carriers past B.
            last: (u64, u64, [f64; 2], [f64; 4]),
            pole_station: Station,
        },
        Cliff {
            t_stop: f64,
            stop: Station,
        },
    }
    struct EndSolve {
        t_a: f64,
        pad_s_first: bool,
        /// (kept, pad_s) carriers past A: surfaces by face id, signed radii
        /// and the seed at A.
        middle: (u64, u64, [f64; 2], [f64; 4]),
        terminus: Terminus,
    }
    let mut ends: Vec<EndSolve> = Vec::with_capacity(2);
    for (slot, ring_end) in ring_ends.iter().enumerate() {
        let vertex_t = if slot == 0 { spine.t0 } else { spine.t1 };
        // Station A is bracketed from the spine's middle to the pole's
        // section on a chain, and to half a spine past the vertex on a
        // one-ended end (the ball meets the floor's plane near the vertex).
        let (t_pole, floor_id, pad_s_id) = match ring_end {
            RingEnd::Chain(chain) => (section_parameter(spine, chain.pole_point, t_mid, vertex_t)?, chain.floor, chain.pad_s),
            RingEnd::One(chain) => (vertex_t + (vertex_t - t_mid), chain.floor, chain.pad),
        };
        let floor = face_by_id(ring, floor_id)?;
        // A one-ended end reads the floor's distance on its PLANE (C5's
        // side-face extension): the convex ball's centre lies past the
        // floor's own patch there (under the slot floor on the channel), where
        // the projector clamps and the gap never closes.  A chain end reads
        // the projected distance exactly as before.
        let floor_plane: Option<(Vec3, Vec3)> = match ring_end {
            RingEnd::Chain(_) => None,
            RingEnd::One(chain) => {
                let Some(crate::AnalyticSurface::Plane { origin, u_dir, v_dir, .. }) = floor.surface.analytic() else {
                    return Err(KernelRefusal::unsupported(KernelStage::Classify, "one_ended_floor", format!(
                        "blend one-ended runout: the floor face {floor_id} at vertex {} is not planar; C5's side-face extension is only exact on a plane",
                        chain.spine_vertex
                    )));
                };
                Some((*origin, u_dir.cross(*v_dir).normalized().or_refuse(KernelStage::Refine, "normalized")?))
            }
        };
        let floor_distance = |point: Vec3| -> Result<f64, KernelRefusal> {
            match floor_plane {
                Some((origin, normal)) => Ok(point.sub(origin).dot(normal).abs()),
                None => Ok(surface_gap(&floor.surface, point)?.0),
            }
        };
        // Station A: the convex ball on (first, second) tangent to the floor.
        let gap_at = |t: f64, seed: [f64; 4]| -> Result<(f64, Ball), KernelRefusal> {
            let ball = solve_ball(surface1, surface2, rho, seed, spine, t, scale)?;
            let gap = floor_distance(ball.center)? - radius;
            Ok((gap, ball))
        };
        let mut seed = seed_at(t_mid)?;
        let mut previous: Option<(f64, f64)> = None;
        let mut bracket: Option<(f64, f64)> = None;
        // A sample landing ON the root (a one-ended end whose ring vertex is
        // the tangency itself: the pad's rail at 5 + r, a station sample)
        // is the station; bisecting from a re-evaluated end whose rounding
        // flipped its sign walked to the far end of the bracket instead.
        let exact = 1e-12 * (1.0 + scale);
        let mut hit: Option<(f64, Ball)> = None;
        for sample in 0..=BRACKET_SAMPLES {
            let t = t_mid + (t_pole - t_mid) * sample as f64 / BRACKET_SAMPLES as f64;
            let (value, ball) = gap_at(t, seed)?;
            seed = ball.uv;
            if matches!(ring_end, RingEnd::One(_)) && value.abs() <= exact {
                hit = Some((t, ball));
                break;
            }
            if let Some((previous_t, previous_value)) = previous {
                if previous_value.signum() != value.signum() {
                    bracket = Some((previous_t, t));
                    break;
                }
            }
            previous = Some((t, value));
        }
        let (t_a, ball_a) = if let Some((t, ball)) = hit {
            (t, ball)
        } else {
        let (mut low, mut high) = bracket.ok_or_else(|| KernelRefusal::unsupported(KernelStage::Refine, "runout_station_a", format!(
            "blend runout: the ball on spine {spine_id} never becomes tangent to face {floor_id} toward vertex {}",
            if slot == 0 { spine.start_vertex_id } else { spine.end_vertex_id }
        )))?;
        let mut seed = seed_at(low)?;
        let (mut low_value, mut ball_a) = gap_at(low, seed)?;
        for _ in 0..BISECTIONS {
            let middle = 0.5 * (low + high);
            let (value, ball) = gap_at(middle, seed)?;
            seed = ball.uv;
            if value.signum() == low_value.signum() {
                low = middle;
                low_value = value;
            } else {
                high = middle;
            }
            ball_a = ball;
        }
        (0.5 * (low + high), ball_a)
        };
        let pad_s = face_by_id(ring, pad_s_id)?;
        report.station_a_residual[slot] = (surface_gap(&pad_s.surface, ball_a.center)?.0 - radius).abs();
        if debug {
            eprintln!(
                "runout spine {spine_id} end {slot}: station A t={t_a:.6} (spine t0 {:.4} t1 {:.4}, vertex t {vertex_t:.4}, window to {t_pole:.4}); centre ({:.6}, {:.6}, {:.6}) p1 ({:.6}, {:.6}, {:.6}) p2 ({:.6}, {:.6}, {:.6}); floor gap {:.3e}",
                spine.t0, spine.t1, ball_a.center.x, ball_a.center.y, ball_a.center.z, ball_a.p1.x, ball_a.p1.y, ball_a.p1.z, ball_a.p2.x, ball_a.p2.y, ball_a.p2.z,
                floor_distance(ball_a.center)? - radius
            );
        }
        // Past A the replaced mate's slot carries pad_s.
        let (pad_s_rho, pad_s_uv) = signed_rho(&pad_s.surface, ball_a.center, radius)?;
        let pad_s_first = !kept_is_first;
        let (middle_s1, middle_s2, middle_rho, middle_seed): (&NurbsSurface, &NurbsSurface, [f64; 2], [f64; 4]) = if pad_s_first {
            (&pad_s.surface, surface2, [pad_s_rho, rho[1]], [pad_s_uv[0], pad_s_uv[1], ball_a.uv[2], ball_a.uv[3]])
        } else {
            (surface1, &pad_s.surface, [rho[0], pad_s_rho], [ball_a.uv[0], ball_a.uv[1], pad_s_uv[0], pad_s_uv[1]])
        };
        let middle_ids = if pad_s_first { (pad_s_id, kept) } else { (kept, pad_s_id) };
        let terminus = match ring_end {
        RingEnd::Chain(chain) => {
        let pad_r = face_by_id(ring, chain.pad_r)?;
        // Station B: the graze minimum of the centre's distance to pad_r.
        let graze = |t: f64, seed: [f64; 4]| -> Result<(f64, Ball), KernelRefusal> {
            let ball = solve_ball(middle_s1, middle_s2, middle_rho, seed, spine, t, scale)?;
            let gap = surface_gap(&pad_r.surface, ball.center)?.0;
            Ok(((gap - radius).abs(), ball))
        };
        let mut seed = middle_seed;
        let mut samples: Vec<(f64, f64, [f64; 4])> = Vec::new();
        for sample in 0..=STOP_SAMPLES {
            let t = t_a + (t_pole - t_a) * sample as f64 / STOP_SAMPLES as f64;
            let Ok((value, ball)) = graze(t, seed) else { break };
            seed = ball.uv;
            samples.push((t, value, ball.uv));
        }
        let best = samples
            .iter()
            .enumerate()
            .min_by(|a, b| a.1 .1.partial_cmp(&b.1 .1).unwrap_or(std::cmp::Ordering::Equal))
            .map(|(index, sample)| (index, *sample))
            .filter(|(index, _)| *index > 0 && *index + 1 < samples.len())
            .ok_or_else(|| KernelRefusal::unsupported(KernelStage::Refine, "runout_station_b", format!(
                "blend runout: the kept mate's contact on spine {spine_id} never reaches pad {}'s rail toward vertex {} ({} stations marched)",
                chain.pad_r, chain.spine_vertex, samples.len()
            )))?;
        const GOLDEN: f64 = 0.618_033_988_749_894_9;
        let (index, _) = best;
        let seed = samples[index].2;
        let mut low = samples[index - 1].0;
        let mut high = samples[index + 1].0;
        let mut x1 = high - GOLDEN * (high - low);
        let mut x2 = low + GOLDEN * (high - low);
        let mut f1 = graze(x1, seed)?.0;
        let mut f2 = graze(x2, seed)?.0;
        for _ in 0..BISECTIONS {
            if f1 < f2 {
                high = x2;
                x2 = x1;
                f2 = f1;
                x1 = high - GOLDEN * (high - low);
                f1 = graze(x1, seed)?.0;
            } else {
                low = x1;
                x1 = x2;
                f1 = f2;
                x2 = low + GOLDEN * (high - low);
                f2 = graze(x2, seed)?.0;
            }
        }
        let t_b = 0.5 * (low + high);
        let (residual_b, ball_b) = graze(t_b, seed)?;
        report.station_b_residual[slot] = residual_b;
        // Past B the kept mate's slot carries pad_r.
        let (pad_r_rho, pad_r_uv) = signed_rho(&pad_r.surface, ball_b.center, radius)?;
        let (last_s1, last_s2, last_rho, last_seed): (&NurbsSurface, &NurbsSurface, [f64; 2], [f64; 4]) = if pad_s_first {
            (&pad_s.surface, &pad_r.surface, [middle_rho[0], pad_r_rho], [ball_b.uv[0], ball_b.uv[1], pad_r_uv[0], pad_r_uv[1]])
        } else {
            (&pad_r.surface, &pad_s.surface, [pad_r_rho, middle_rho[1]], [pad_r_uv[0], pad_r_uv[1], ball_b.uv[2], ball_b.uv[3]])
        };
        let last_ids = if pad_s_first { (chain.pad_s, chain.pad_r) } else { (chain.pad_r, chain.pad_s) };
        // The pole station, analytically: both contacts ARE the pole, the
        // centre sits one radius off it along either pad's normal.
        let pole = chain.pole_point;
        let (_, u1, v1) = surface_gap(last_s1, pole)?;
        let (_, u2, v2) = surface_gap(last_s2, pole)?;
        let n1 = raw_normal(last_s1, u1, v1)?;
        let n2 = raw_normal(last_s2, u2, v2)?;
        let center1 = pole.add(n1.scale(last_rho[0]));
        let center2 = pole.add(n2.scale(last_rho[1]));
        report.pole_center_miss[slot] = center1.sub(center2).length();
        let cos_alpha = last_rho[0].signum() * last_rho[1].signum() * n1.dot(n2);
        let weight = ((1.0 + cos_alpha) * 0.5).max(0.0).sqrt();
        let pole_station = Station { uv1: [u1, v1], uv2: [u2, v2], p1: pole, p2: pole, center: center1, weight, apex: pole };
        // The march rides the spine's section planes, and a station's CENTRE
        // lies in its plane.  The pole ball's centre sits a radius off the
        // pole along the shared normal, which is not in the pole's own section
        // plane (0.46 out of it on the wedge's wall end), so the pole's
        // section is the one through that CENTRE: measured with the plane
        // through P instead, the marched contacts stood 0.7 from the pole at
        // every station count.
        let t_pole_center = section_parameter(spine, center1, t_mid, t_pole)?;
        if debug {
            eprintln!(
                "runout spine {spine_id} end {slot}: A t={t_a:.6} (pad residual {:.3e}), B t={t_b:.6} (residual {:.3e}), pole section t={t_pole:.6} through P, {t_pole_center:.6} through its centre; pole centre miss {:.3e}, weight {weight:.6}",
                report.station_a_residual[slot], residual_b, report.pole_center_miss[slot]
            );
        }
        Terminus::Pole {
            t_b,
            t_pole: t_pole_center,
            last: (last_ids.0, last_ids.1, last_rho, last_seed),
            pole_station,
        }
        }
        RingEnd::One(chain) => {
            // Past A the ball rolls on the pad, so A must BE the pad's graze:
            // true when the concave edge is right-angled (the mirror identity
            // of the module doc), where the ball tangent to the floor is the
            // reflection of the pad's own ball across the shared mate.  A
            // sloped pad (the wedge's wall ∩ roof) grazes elsewhere — the
            // two-ended lane's station B kind — and is named here.
            let graze_bar = 1e-9 * (1.0 + scale);
            if report.station_a_residual[slot] > graze_bar {
                return Err(KernelRefusal::unsupported(KernelStage::Classify, "one_ended_pad_graze", format!(
                    "blend one-ended runout: at vertex {} the ball tangent to the floor {floor_id} stands {:.3e} off the pad {}'s graze (bar {graze_bar:.1e}); the concave edge is not right-angled, so the one-ended end is not the floor-tangent station",
                    chain.spine_vertex, report.station_a_residual[slot], chain.pad
                )));
            }
            if !chain.third_convex {
                return Err(KernelRefusal::unsupported(KernelStage::Classify, "one_ended_concave_third", format!(
                    "blend one-ended runout: the unselected edge {} at vertex {} is concave; the roll on the kept mate extended to the pad's far rail, trimmed by the floor, is not built here (the convex case ends at C5's extended edge with a cliff)",
                    chain.e2, chain.spine_vertex
                )));
            }
            // C5's stop: the KEPT contact reaches the floor's plane extended,
            // which is the unselected edge's line extended (that edge is
            // kept ∩ floor).
            let (floor_origin, floor_normal) = floor_plane.expect("a one-ended end has a planar floor");
            let kept_contact = |ball: &Ball| if kept_is_first { ball.p1 } else { ball.p2 };
            let height = |t: f64, seed: [f64; 4]| -> Result<(f64, Ball), KernelRefusal> {
                let ball = solve_ball(middle_s1, middle_s2, middle_rho, seed, spine, t, scale)?;
                Ok((kept_contact(&ball).sub(floor_origin).dot(floor_normal), ball))
            };
            // The stop lies within a few radii of A along the spine's
            // direction; the window is read in the spine's own parameter.
            let chord = spine.curve.evaluate(spine.t1).or_refuse(KernelStage::Refine, "evaluate")?
                .sub(spine.curve.evaluate(spine.t0).or_refuse(KernelStage::Refine, "evaluate")?).length();
            let speed = (chord / (spine.t1 - spine.t0).abs()).max(1e-300);
            let direction = (vertex_t - t_mid).signum();
            let t_far = t_a + direction * 2.5 * radius / speed;
            let mut seed = middle_seed;
            let (mut low, mut low_value) = (t_a, f64::NAN);
            let mut bracket: Option<(f64, f64)> = None;
            let mut hit: Option<(f64, Ball)> = None;
            for sample in 0..=STOP_SAMPLES {
                let t = t_a + (t_far - t_a) * sample as f64 / STOP_SAMPLES as f64;
                let (value, ball) = height(t, seed)?;
                seed = ball.uv;
                if sample > 0 && value.abs() <= exact {
                    hit = Some((t, ball));
                    break;
                }
                if sample > 0 && value.signum() != low_value.signum() {
                    bracket = Some((low, t));
                    break;
                }
                low = t;
                low_value = value;
            }
            let (t_stop, ball_stop) = if let Some(hit) = hit {
                hit
            } else {
            let (mut low, mut high) = bracket.ok_or_else(|| KernelRefusal::unsupported(KernelStage::Refine, "one_ended_stop", format!(
                "blend one-ended runout: the kept mate's contact on spine {spine_id} never reaches the unselected edge {}'s line within 2.5 r past station A",
                chain.e2
            )))?;
            let (mut low_value, mut ball_stop) = height(low, seed)?;
            for _ in 0..BISECTIONS {
                let middle = 0.5 * (low + high);
                let (value, ball) = height(middle, seed)?;
                seed = ball.uv;
                if value.signum() == low_value.signum() {
                    low = middle;
                    low_value = value;
                } else {
                    high = middle;
                }
                ball_stop = ball;
            }
            (0.5 * (low + high), ball_stop)
            };
            report.station_b_residual[slot] = kept_contact(&ball_stop).sub(floor_origin).dot(floor_normal).abs();
            let stop = make_station(middle_s1, middle_s2, middle_rho, ball_stop.uv, spine, t_stop)?;
            if debug {
                eprintln!(
                    "runout spine {spine_id} end {slot} (one-ended): A t={t_a:.6} (pad residual {:.3e}), C5 stop t={t_stop:.6} (kept contact {:.3e} off the unselected edge's plane) at ({:.6}, {:.6}, {:.6})",
                    report.station_a_residual[slot], report.station_b_residual[slot], kept_contact(&ball_stop).x, kept_contact(&ball_stop).y, kept_contact(&ball_stop).z
                );
            }
            Terminus::Cliff { t_stop, stop }
        }
        };
        ends.push(EndSolve {
            t_a,
            pad_s_first,
            middle: (middle_ids.0, middle_ids.1, middle_rho, middle_seed),
            terminus,
        });
    }
    let [end0, end1] = match ends.len() {
        2 => {
            let end1 = ends.pop().expect("two ends");
            let end0 = ends.pop().expect("two ends");
            [end0, end1]
        }
        _ => return Err(KernelRefusal::internal(KernelStage::Refine, "runout_ends", "blend runout: two ends expected")),
    };
    if !((end1.t_a - end0.t_a) * (spine.t1 - spine.t0) > 0.0) {
        return Err(KernelRefusal::unsupported(KernelStage::Refine, "runout_stations_cross", format!(
            "blend runout: the two tri-tangent stations of spine {spine_id} cross each other (t {:.6} and {:.6}); the spine is shorter than its two runouts",
            end0.t_a, end1.t_a
        )));
    }

    // ---- The five stretches, marched on a station ladder: each stretch
    //      doubles its stations until the rails fitted through them stand on
    //      their carriers to the network's own rail bar (`rails_off_carriers`
    //      sampling, half `intersection_fit`), read between the stations. ----
    let rail_bar = 0.5 * crate::KernelTolerances::for_solid(ring, 1e-7).intersection_fit;
    let span = (spine.t1 - spine.t0).abs();
    let count_for = |t_from: f64, t_to: f64| -> usize {
        ((((t_to - t_from).abs() / span) * RUNOUT_DENSITY_STATIONS as f64).ceil() as usize)
            .max(MIN_STRETCH_STATIONS - 1)
    };
    // Stretch k: (first carrier, second carrier, rho, seed at t_from, t_from, t_to, pole at the far end?)
    struct StretchSpec {
        first: u64,
        second: u64,
        rho: [f64; 2],
        seed: [f64; 4],
        t_from: f64,
        t_to: f64,
        /// `Some(pole station)` when `t_to` is the pole and the march stops
        /// short of it on the singular Newton.
        pole: Option<Station>,
        /// Seed at the MIDDLE and march outward (the centre stretch).
        seed_middle: bool,
    }
    // An end's stretches, outermost first: a chain's last stretch (to the
    // pole) and its middle stretch (A to B); a one-ended end's middle
    // stretch alone (A to the C5 stop).
    let end_specs = |end: &EndSolve| -> Vec<StretchSpec> {
        match &end.terminus {
            Terminus::Pole { t_b, t_pole, last, pole_station } => vec![
                StretchSpec { first: last.0, second: last.1, rho: last.2, seed: last.3, t_from: *t_b, t_to: *t_pole, pole: Some(*pole_station), seed_middle: false },
                StretchSpec { first: end.middle.0, second: end.middle.1, rho: end.middle.2, seed: end.middle.3, t_from: end.t_a, t_to: *t_b, pole: None, seed_middle: false },
            ],
            Terminus::Cliff { t_stop, .. } => vec![
                StretchSpec { first: end.middle.0, second: end.middle.1, rho: end.middle.2, seed: end.middle.3, t_from: end.t_a, t_to: *t_stop, pole: None, seed_middle: false },
            ],
        }
    };
    let mut specs: Vec<StretchSpec> = end_specs(&end0);
    let n0 = specs.len();
    specs.push(StretchSpec { first: first.face.id, second: second.face.id, rho, seed: seed_at(t_mid)?, t_from: end0.t_a, t_to: end1.t_a, pole: None, seed_middle: true });
    let mut end1_specs = end_specs(&end1);
    end1_specs.reverse();
    specs.extend(end1_specs);
    // The end-0 stretches are marched from the middle outward (toward
    // decreasing t) and reversed into increasing t below.
    let march_stretch = |spec: &StretchSpec, count: usize| -> Result<Vec<Station>, KernelRefusal> {
        let s1 = &face_by_id(ring, spec.first)?.surface;
        let s2 = &face_by_id(ring, spec.second)?.surface;
        // Toward a pole the stations are spaced quadratically in the distance
        // to the pole's section, on the expectation that the two contacts
        // close on the pole like a square root (the ball is tangent to both
        // pads AT the pole).  Added while the pole's section was still taken
        // through P, where the last chord stood 2.6e-3 off the pads at 4, 33
        // and 1024 stations alike; the spacing did not move that stall — the
        // section through the pole ball's CENTRE did (see `t_pole_center`).
        // Kept, not separately A/B'd against uniform spacing.
        let t_of = |k: usize| {
            let fraction = k as f64 / count as f64;
            let fraction = if spec.pole.is_some() { 1.0 - (1.0 - fraction) * (1.0 - fraction) } else { fraction };
            spec.t_from + (spec.t_to - spec.t_from) * fraction
        };
        let mut out: Vec<Station> = Vec::with_capacity(count + 1);
        if spec.seed_middle {
            let mut solved: Vec<Option<[f64; 4]>> = vec![None; count + 1];
            let mid = count / 2;
            let mut seed = spec.seed;
            for k in (0..=mid).rev() {
                let (section_point, section_tangent) = section_frame(spine, t_of(k))?;
                let uv = solve_station(s1, s2, spec.rho, seed, section_point, section_tangent, scale)?;
                seed = uv;
                solved[k] = Some(uv);
            }
            let mut seed = solved[mid].expect("middle solved");
            for k in mid + 1..=count {
                let (section_point, section_tangent) = section_frame(spine, t_of(k))?;
                let uv = solve_station(s1, s2, spec.rho, seed, section_point, section_tangent, scale)?;
                seed = uv;
                solved[k] = Some(uv);
            }
            for (k, uv) in solved.into_iter().enumerate() {
                out.push(make_station(s1, s2, spec.rho, uv.expect("solved"), spine, t_of(k))?);
            }
            return Ok(out);
        }
        let mut seed = spec.seed;
        let last_regular = if spec.pole.is_some() { count - 1 } else { count };
        for k in 0..=last_regular {
            let (section_point, section_tangent) = section_frame(spine, t_of(k))?;
            let uv = match solve_station(s1, s2, spec.rho, seed, section_point, section_tangent, scale) {
                Ok(uv) => uv,
                Err(error) => {
                    if spec.pole.is_none() || k < MIN_STRETCH_STATIONS - 1 {
                        return Err(error.with_message(|error| format!(
                            "blend runout: spine {spine_id}'s stretch from t {:.6} to {:.6} holds only {k} stations: {error}",
                            spec.t_from, spec.t_to
                        )));
                    }
                    break;
                }
            };
            seed = uv;
            out.push(make_station(s1, s2, spec.rho, uv, spine, t_of(k))?);
        }
        if let Some(pole) = &spec.pole {
            out.push(*pole);
        }
        Ok(out)
    };
    let mut counts: Vec<usize> = specs.iter().map(|spec| count_for(spec.t_from, spec.t_to)).collect();
    const MAX_STRETCH_STATIONS: usize = 64;
    let (stations, ranges, parameters, rows, stretches) = loop {
        let mut stretches: Vec<Stretch> = Vec::with_capacity(5);
        for (index, spec) in specs.iter().enumerate() {
            let mut marched = march_stretch(spec, counts[index])?;
            if index < n0 {
                marched.reverse();
            }
            stretches.push(Stretch { first: spec.first, second: spec.second, stations: marched });
        }
        // Global station list (switch stations once) with each stretch's index range.
        let mut stations: Vec<Station> = Vec::new();
        let mut ranges: Vec<[usize; 2]> = Vec::new();
        for (index, stretch) in stretches.iter().enumerate() {
            let skip = usize::from(index > 0);
            let start = stations.len().saturating_sub(skip);
            stations.extend(stretch.stations.iter().skip(skip).cloned());
            ranges.push([start, stations.len() - 1]);
        }
        let breaks: Vec<usize> = ranges.iter().skip(1).map(|range| range[0]).collect();
        let parameters = station_parameters(&stations);
        let rows = fit_rows_breaking(&stations, &parameters, &breaks)?;
        // Each stretch's rails off its own carriers, between its stations.
        let mut over: Vec<usize> = Vec::new();
        let mut worst_all: f64 = 0.0;
        for (index, (stretch, range)) in stretches.iter().zip(&ranges).enumerate() {
            let s1 = &face_by_id(ring, stretch.first)?.surface;
            let s2 = &face_by_id(ring, stretch.second)?.surface;
            let mut worst: f64 = 0.0;
            for pair in parameters[range[0]..=range[1]].windows(2) {
                for fraction in [0.25, 0.5, 0.75] {
                    let u = pair[0] + (pair[1] - pair[0]) * fraction;
                    let p1 = rows.cr.evaluate(u).or_refuse(KernelStage::Refine, "evaluate")?;
                    let p2 = rows.cs.evaluate(u).or_refuse(KernelStage::Refine, "evaluate")?;
                    worst = worst.max(surface_gap(s1, p1)?.0).max(surface_gap(s2, p2)?.0);
                }
            }
            if debug {
                eprintln!(
                    "runout spine {spine_id}: stretch {index} on faces ({}, {}) at {} stations: rails {worst:.3e} off their carriers (bar {rail_bar:.3e})",
                    stretch.first, stretch.second, stretch.stations.len()
                );
                if worst > rail_bar {
                    let range_of = |pick: fn(&Station) -> [f64; 2]| {
                        stretch.stations.iter().fold([f64::INFINITY, f64::NEG_INFINITY, f64::INFINITY, f64::NEG_INFINITY], |acc, st| {
                            let uv = pick(st);
                            [acc[0].min(uv[0]), acc[1].max(uv[0]), acc[2].min(uv[1]), acc[3].max(uv[1])]
                        })
                    };
                    for (which, surface, uv_range) in [("first", s1, range_of(|st| st.uv1)), ("second", s2, range_of(|st| st.uv2))] {
                        eprintln!(
                            "    {which} carrier domain u {:?} v {:?}; stations span u [{:.4}, {:.4}] v [{:.4}, {:.4}]",
                            surface.domain_u().unwrap_or([f64::NAN; 2]), surface.domain_v().unwrap_or([f64::NAN; 2]),
                            uv_range[0], uv_range[1], uv_range[2], uv_range[3]
                        );
                    }
                    // Where the worst sample sits: which interval, and the two stations' contacts.
                    let mut worst_at = (0usize, 0.0f64);
                    for (k, pair) in parameters[range[0]..=range[1]].windows(2).enumerate() {
                        for fraction in [0.25, 0.5, 0.75] {
                            let u = pair[0] + (pair[1] - pair[0]) * fraction;
                            let p1 = rows.cr.evaluate(u).or_refuse(KernelStage::Refine, "evaluate")?;
                            let p2 = rows.cs.evaluate(u).or_refuse(KernelStage::Refine, "evaluate")?;
                            let off = surface_gap(s1, p1)?.0.max(surface_gap(s2, p2)?.0);
                            if off > worst_at.1 { worst_at = (k, off); }
                        }
                    }
                    let (k, off) = worst_at;
                    let a = &stretch.stations[k];
                    let b = &stretch.stations[k + 1];
                    eprintln!(
                        "    worst {off:.3e} in interval {k}/{}: p1 ({:.5},{:.5},{:.5})->({:.5},{:.5},{:.5}) p2 ({:.5},{:.5},{:.5})->({:.5},{:.5},{:.5}) weights {:.4}->{:.4}",
                        stretch.stations.len() - 1,
                        a.p1.x, a.p1.y, a.p1.z, b.p1.x, b.p1.y, b.p1.z, a.p2.x, a.p2.y, a.p2.z, b.p2.x, b.p2.y, b.p2.z, a.weight, b.weight
                    );
                }
            }
            worst_all = worst_all.max(worst);
            if worst > rail_bar {
                over.push(index);
            }
        }
        if over.is_empty() {
            break (stations, ranges, parameters, rows, stretches);
        }
        if over.iter().any(|index| counts[*index] >= MAX_STRETCH_STATIONS) {
            return Err(KernelRefusal::non_convergence(KernelStage::Refine, super::miter::MARCHED_FIT_OFF_CARRIERS_WHAT, format!(
                "blend runout: spine {spine_id}'s rails still stand {worst_all:.3e} off their carriers (bar {rail_bar:.3e}) with {MAX_STRETCH_STATIONS} stations on a stretch"
            )));
        }
        for index in over {
            counts[index] *= 2;
        }
    };
    report.stations = stations.len();
    let breaks: Vec<usize> = ranges.iter().skip(1).map(|range| range[0]).collect();
    // ---- G1 at every switch.  The RAIL kinks there and that is the
    //      geometry: a contact is the centre displaced by r along the
    //      carrier's normal, and that normal's rate jumps from zero on a plane
    //      to its value on the pad cylinder (measured: the unit tangents' dot
    //      reads 0.946 at B on both shapes).  What is G1 is the WALL: the
    //      centre path is G1 across carriers that are themselves tangent,
    //      so the envelope's tangent plane is continuous across the switch.
    //      Read off the FITTED surface: its normal just before and just
    //      after each break knot, along the whole section, must agree to the
    //      network's own reading of a smooth continuation (1 − 1e-6 on the
    //      unit normals' dot, `tangent_away_from`'s bar), not a bar set here.
    //      The rails' turns are reported, never judged. ----
    let mut worst_switch_turn: f64 = 0.0;
    let mut worst_switch_lean: f64 = 0.0;
    let [u_low, u_high] = rows.u_domain;
    let u_step = 1e-6 * (u_high - u_low).abs();
    for &index in &breaks {
        let u = parameters[index];
        for row in [&rows.cr, &rows.cs] {
            let (left, right) = row.split(u).or_refuse(KernelStage::Refine, "split")?;
            let [_, left_end] = left.domain().or_refuse(KernelStage::Refine, "domain")?;
            let [right_start, _] = right.domain().or_refuse(KernelStage::Refine, "domain")?;
            let before = left.derivatives(left_end, 1).or_refuse(KernelStage::Refine, "derivatives")?[1];
            let after = right.derivatives(right_start, 1).or_refuse(KernelStage::Refine, "derivatives")?[1];
            if let (Ok(before), Ok(after)) = (before.normalized(), after.normalized()) {
                worst_switch_turn = worst_switch_turn.max((1.0 - before.dot(after)).max(0.0));
            }
        }
        for k in 0..=8 {
            let v = k as f64 / 8.0;
            let before = raw_normal(&rows.surface, u - u_step, v)?;
            let after = raw_normal(&rows.surface, u + u_step, v)?;
            let lean = (1.0 - before.dot(after)).max(0.0);
            worst_switch_lean = worst_switch_lean.max(lean);
            if before.dot(after) < 1.0 - 1e-6 {
                return Err(KernelRefusal::unsupported(KernelStage::Refine, "runout_switch_not_g1", format!(
                    "blend runout: spine {spine_id}'s wall is not G1 across its carrier switch at station {index} (v = {v:.3}: unit normals' dot {:.9} across the break)",
                    before.dot(after)
                )));
            }
        }
    }
    report.switch_lean = worst_switch_lean;
    report.switch_turn = worst_switch_turn;
    // The wall's tangent plane against each carrier's at the contacts: the
    // section arc is tangent to both carriers by construction, read back off
    // the FITTED surface at every station.
    let mut worst_plane_lean: f64 = 0.0;
    for (stretch, range) in stretches.iter().zip(&ranges) {
        let s1 = &face_by_id(ring, stretch.first)?.surface;
        let s2 = &face_by_id(ring, stretch.second)?.surface;
        for (j, station) in stretch.stations.iter().enumerate() {
            let u = parameters[range[0] + j];
            if station.p1.sub(station.p2).length() < 1e-9 * (1.0 + scale) {
                continue; // the pole: the section is a point
            }
            for (v, surface, uv) in [(0.0, s1, station.uv1), (1.0, s2, station.uv2)] {
                let Ok(wall) = raw_normal(&rows.surface, u, v) else { continue };
                let carrier = raw_normal(surface, uv[0], uv[1])?;
                worst_plane_lean = worst_plane_lean.max(1.0 - wall.dot(carrier).abs());
            }
        }
    }
    report.plane_lean = worst_plane_lean;
    if debug {
        eprintln!(
            "runout spine {spine_id}: rails turn at the switches by at most 1 − t·t′ = {worst_switch_turn:.3e} (the geometry); the wall's normal across a switch moves by at most 1 − n·n′ = {worst_switch_lean:.3e}; the wall's tangent plane leans off its carriers by at most 1 − |n·n′| = {worst_plane_lean:.3e}"
        );
    }
    if debug {
        let mut worst: f64 = 0.0;
        for (station, u) in stations.iter().zip(&parameters) {
            worst = worst
                .max(rows.cr.evaluate(*u).or_refuse(KernelStage::Refine, "evaluate")?.sub(station.p1).length())
                .max(rows.cs.evaluate(*u).or_refuse(KernelStage::Refine, "evaluate")?.sub(station.p2).length());
        }
        eprintln!("runout spine {spine_id}: {} stations, breaks {breaks:?}, rows reproduce their stations to {worst:.3e}", stations.len());
    }

    // ---- The rail pieces: one edge per run of stretches sharing a carrier
    //      on that side. ----
    struct Piece {
        carrier: u64,
        /// Global station index range, inclusive.
        range: [usize; 2],
        /// uv on `carrier` per station of the range.
        uvs: Vec<[f64; 2]>,
    }
    // The uv samples come from each STRETCH's own stations: at a switch the
    // global list keeps the earlier stretch's copy of the shared station,
    // whose uv on the switching side is on the earlier carrier.
    let pieces_on = |side_first: bool| -> Vec<Piece> {
        let mut pieces: Vec<Piece> = Vec::new();
        for (stretch, range) in stretches.iter().zip(&ranges) {
            let carrier = if side_first { stretch.first } else { stretch.second };
            let uvs: Vec<[f64; 2]> = stretch.stations.iter().map(|s| if side_first { s.uv1 } else { s.uv2 }).collect();
            match pieces.last_mut() {
                Some(last) if last.carrier == carrier => {
                    last.range[1] = range[1];
                    last.uvs.extend(uvs.into_iter().skip(1));
                }
                _ => pieces.push(Piece { carrier, range: *range, uvs }),
            }
        }
        pieces
    };
    let pieces_cr = pieces_on(true);
    let pieces_cs = pieces_on(false);
    // A chain end adds a pad piece on both sides, a one-ended end on the
    // replaced side only: the replaced side always has three pieces, the
    // kept side one more than its chain ends.
    let chain_ends = ring_ends.iter().filter(|end| matches!(end, RingEnd::Chain(_))).count();
    let (kept_pieces, replaced_pieces) = if kept_is_first { (pieces_cr.len(), pieces_cs.len()) } else { (pieces_cs.len(), pieces_cr.len()) };
    if replaced_pieces != 3 || kept_pieces != 1 + chain_ends {
        return Err(KernelRefusal::internal(KernelStage::Refine, "runout_pieces", format!(
            "blend runout: spine {spine_id}'s rails split into {kept_pieces} kept-side and {replaced_pieces} replaced-side pieces ({} and three expected)",
            1 + chain_ends
        )));
    }

    // ---- Topology. ----
    // Each end's two rail vertices (first-mate side, second-mate side): the
    // ring's pole for a chain end, two fresh vertices at the stop station's
    // contacts for a one-ended end.
    let mut end_vertices = [[0u64; 2]; 2];
    for (slot, (ring_end, end)) in ring_ends.iter().zip([&end0, &end1]).enumerate() {
        end_vertices[slot] = match (ring_end, &end.terminus) {
            (RingEnd::Chain(chain), _) => [chain.pole, chain.pole],
            (RingEnd::One(_), Terminus::Cliff { stop, .. }) => {
                let mut fresh = [0u64; 2];
                for (side, point) in [stop.p1, stop.p2].into_iter().enumerate() {
                    let id = take_id();
                    result.vertices.push(VertexRecord { id, point });
                    fresh[side] = id;
                }
                fresh
            }
            (RingEnd::One(_), Terminus::Pole { .. }) => {
                return Err(KernelRefusal::internal(KernelStage::Refine, "runout_terminus", "blend runout: a one-ended end solved a pole"));
            }
        };
    }
    // The switch vertices: the contact at each interior piece boundary.
    let mut switch_off_rail = Vec::new();
    let mut commit_pieces = |pieces: &[Piece], side_first: bool, result: &mut BrepSolid, take_id: &mut dyn FnMut() -> u64| -> Result<Vec<(u64, u64, u64, NurbsCurve, u64)>, KernelRefusal> {
        // (edge id, start vertex, end vertex, mate pcurve in station order, carrier)
        let row = if side_first { &rows.cr } else { &rows.cs };
        let side = usize::from(!side_first);
        let mut out = Vec::new();
        let mut vertices: Vec<u64> = vec![end_vertices[0][side]];
        for pair in pieces.windows(2) {
            let index = pair[0].range[1];
            let point = if side_first { stations[index].p1 } else { stations[index].p2 };
            let id = take_id();
            result.vertices.push(VertexRecord { id, point });
            vertices.push(id);
        }
        vertices.push(end_vertices[1][side]);
        for (k, piece) in pieces.iter().enumerate() {
            let params = &parameters[piece.range[0]..=piece.range[1]];
            let samples: Vec<Vec4> = piece.uvs.iter().map(|uv| Vec4::from_point(Vec3::new(uv[0], uv[1], 0.0), 1.0)).collect();
            if samples.len() != params.len() {
                return Err(KernelRefusal::internal(KernelStage::Refine, "runout_piece", "blend runout: a rail piece's uv samples do not match its stations"));
            }
            // The row breaks that fall inside this piece, as local indices.
            let local_breaks: Vec<usize> = breaks
                .iter()
                .filter(|b| **b > piece.range[0] && **b < piece.range[1])
                .map(|b| b - piece.range[0])
                .collect();
            let pcurve = interpolate_piece(&samples, params, &local_breaks)?;
            if debug {
                let carrier = face_by_id(ring, piece.carrier)?;
                let (mut fit_worst, mut data_worst) = (0.0f64, 0.0f64);
                for (j, (uv, u)) in piece.uvs.iter().zip(params).enumerate() {
                    let fitted = pcurve.evaluate(*u).or_refuse(KernelStage::Refine, "evaluate")?;
                    fit_worst = fit_worst.max(((fitted.x - uv[0]).powi(2) + (fitted.y - uv[1]).powi(2)).sqrt());
                    let on_face = carrier.surface.evaluate(uv[0], uv[1]).or_refuse(KernelStage::Refine, "evaluate")?;
                    let station = &stations[piece.range[0] + j];
                    let contact = if side_first { station.p1 } else { station.p2 };
                    data_worst = data_worst.max(on_face.sub(contact).length());
                }
                eprintln!(
                    "runout spine {spine_id}: {} piece {k} on face {} ({} samples, u [{:.4}, {:.4}], breaks {local_breaks:?}): pcurve off its samples {fit_worst:.3e}, samples off the contacts {data_worst:.3e}",
                    if side_first { "cr" } else { "cs" }, piece.carrier, piece.uvs.len(), params[0], params[params.len() - 1]
                );
            }
            let id = take_id();
            result.edges.push(EdgeRecord {
                id,
                curve: row.clone(),
                t0: params[0],
                t1: params[params.len() - 1],
                start_vertex_id: vertices[k],
                end_vertex_id: vertices[k + 1],
                degenerate: false,
                name: None,
            });
            out.push((id, vertices[k], vertices[k + 1], pcurve, piece.carrier));
        }
        Ok(out)
    };
    let rails_cr = commit_pieces(&pieces_cr, true, result, take_id)?;
    let rails_cs = commit_pieces(&pieces_cs, false, result, take_id)?;
    // Which rail side runs on the kept mate (and hence on the pad_r pads)?
    let (kept_rails, replaced_rails) = if kept_is_first { (&rails_cr, &rails_cs) } else { (&rails_cs, &rails_cr) };
    // Trim the pad rails at the switch vertices: a chain's roof rail at B
    // and side rail at A; a one-ended end's side rail at A.
    let kept_last = kept_rails.len() - 1;
    let replaced_last = replaced_rails.len() - 1;
    for (slot, ring_end) in ring_ends.iter().enumerate() {
        let (b_vertex, a_vertex) = if slot == 0 {
            (kept_rails[0].2, replaced_rails[0].2)
        } else {
            (kept_rails[kept_last].1, replaced_rails[replaced_last].1)
        };
        let trims: Vec<(u64, u64, u64, usize)> = match ring_end {
            RingEnd::Chain(chain) => vec![
                (chain.roof_rail, chain.mid_vertex, b_vertex, 0usize),
                (chain.side_rail, chain.spine_vertex, a_vertex, 1usize),
            ],
            RingEnd::One(chain) => vec![(chain.side_rail, chain.spine_vertex, a_vertex, 1usize)],
        };
        for (rail_id, old_vertex, new_vertex, which) in trims {
            let rail = edge_by_id(result, rail_id)?;
            let point = vertex_point(result, new_vertex)?;
            let projection = project_point_to_curve(&rail.curve, point).or_refuse(KernelStage::Refine, "project_point_to_curve")?;
            switch_off_rail.push(projection.distance);
            if debug {
                let fraction = (projection.u - rail.t0) / (rail.t1 - rail.t0);
                eprintln!(
                    "runout spine {spine_id} end {slot}: {} rail {rail_id} trimmed at fraction {fraction:.6} (vertex {new_vertex} {:.3e} off the rail)",
                    if which == 0 { "roof" } else { "side" }, projection.distance
                );
            }
            trim_edge_at(result, rail_id, projection.u, old_vertex, new_vertex)?;
        }
    }
    for (k, value) in switch_off_rail.iter().enumerate().take(4) {
        report.switch_off_rail[k] = *value;
    }

    // ---- The one-ended ends' cliffs: at each, the pad's end arc is cut
    // where the stop's section plane crosses it (q), and three edges close
    // the stripe — the stop's section arc between the two rail ends, the
    // pad's generator from its contact to q, and the segment on the kept mate
    // from q to the kept rail's end — bounding a planar face in that
    // section's plane. (edge ids: arc, generator, segment; q; the face)
    struct CliffEdges {
        arc: u64,
        generator: u64,
        segment: u64,
        q: u64,
        plane: NurbsSurface,
    }
    let mut cliffs: Vec<Option<CliffEdges>> = vec![None, None];
    let mut cliff_segments: Vec<(usize, u64, NurbsCurve)> = Vec::new();
    let mut cliff_planes: Vec<(usize, NurbsCurve, NurbsCurve, NurbsCurve)> = Vec::new();
    let pcurve_band = crate::KernelTolerances::for_solid(ring, 1e-7).pcurve_consistency;
    for (slot, (ring_end, end)) in ring_ends.iter().zip([&end0, &end1]).enumerate() {
        let (RingEnd::One(chain), Terminus::Cliff { stop, .. }) = (ring_end, &end.terminus) else { continue };
        let centre = stop.center;
        let x_axis = stop.p1.sub(centre).normalized().or_refuse(KernelStage::Refine, "normalized")?;
        let to_p2 = stop.p2.sub(centre);
        let plane_normal = x_axis.cross(to_p2).normalized().or_refuse(KernelStage::Refine, "normalized")?;
        let y_axis = plane_normal.cross(x_axis).normalized().or_refuse(KernelStage::Refine, "normalized")?;
        let angle = to_p2.dot(y_axis).atan2(to_p2.dot(x_axis));
        // q: the pad's end arc crosses the stop's section plane once.
        let e1 = edge_by_id(result, chain.e1)?;
        let plane_height = |t: f64| -> Result<f64, KernelRefusal> {
            Ok(e1.curve.evaluate(t).or_refuse(KernelStage::Refine, "evaluate")?.sub(centre).dot(plane_normal))
        };
        let mut low = e1.t0;
        let mut low_value = plane_height(low)?;
        let mut high = None;
        for sample in 1..=BRACKET_SAMPLES {
            let t = e1.t0 + (e1.t1 - e1.t0) * sample as f64 / BRACKET_SAMPLES as f64;
            let value = plane_height(t)?;
            if value.signum() != low_value.signum() {
                high = Some(t);
                break;
            }
            low = t;
            low_value = value;
        }
        let mut high = high.ok_or_else(|| KernelRefusal::unsupported(KernelStage::Refine, "one_ended_cliff_q", format!(
            "blend one-ended runout: the pad's end arc {} does not cross the stop's section plane on spine {spine_id}",
            chain.e1
        )))?;
        for _ in 0..BISECTIONS {
            let middle = 0.5 * (low + high);
            let value = plane_height(middle)?;
            if value.signum() == low_value.signum() {
                low = middle;
                low_value = value;
            } else {
                high = middle;
            }
        }
        let t_q = 0.5 * (low + high);
        let q_point = e1.curve.evaluate(t_q).or_refuse(KernelStage::Refine, "evaluate")?;
        let (e1_t0, e1_t1) = (e1.t0, e1.t1);
        let q = take_id();
        result.vertices.push(VertexRecord { id: q, point: q_point });
        if debug {
            eprintln!(
                "runout spine {spine_id} end {slot} (one-ended): q on the pad's end arc {} at fraction {:.6}: ({:.6}, {:.6}, {:.6}); section angle {:.6}",
                chain.e1, (t_q - e1_t0) / (e1_t1 - e1_t0), q_point.x, q_point.y, q_point.z, angle
            );
        }
        trim_edge_at(result, chain.e1, t_q, chain.spine_vertex, q)?;
        // The stop's contacts by role.
        let (kept_point, pad_point) = if kept_is_first { (stop.p1, stop.p2) } else { (stop.p2, stop.p1) };
        let (kept_vertex, pad_vertex) = if kept_is_first { (end_vertices[slot][0], end_vertices[slot][1]) } else { (end_vertices[slot][1], end_vertices[slot][0]) };
        let pad = face_by_id(ring, chain.pad)?;
        let kept_face = face_by_id(ring, kept)?;
        // The pad's generator q -> pad contact must lie on the pad (a ruling
        // of a straight concave edge's pad); a curved pad is named.
        let generator_mid = q_point.add(pad_point).scale(0.5);
        let off_pad = surface_gap(&pad.surface, generator_mid)?.0;
        if off_pad > pcurve_band {
            return Err(KernelRefusal::unsupported(KernelStage::Refine, "one_ended_pad_generator", format!(
                "blend one-ended runout: the pad {}'s section in the stop's plane is not its straight generator ({off_pad:.3e} off at the middle, band {pcurve_band:.1e}); a curved concave edge's pad is not cut here",
                chain.pad
            )));
        }
        let uv_of = |surface: &NurbsSurface, point: Vec3| -> Result<[f64; 2], KernelRefusal> {
            let projection = project_point_to_surface(surface, point).or_refuse(KernelStage::Refine, "project_point_to_surface")?;
            Ok([projection.u, projection.v])
        };
        let line_pcurve = |surface: &NurbsSurface, from: Vec3, to: Vec3| -> Result<NurbsCurve, KernelRefusal> {
            let a = uv_of(surface, from)?;
            let b = uv_of(surface, to)?;
            crate::sweep_topology::parameter_line(a[0], a[1], b[0], b[1]).or_refuse(KernelStage::Refine, "parameter_line")
        };
        // The three edges.  The arc runs from the first-mate rail's end to
        // the second's (v = 0 to v = 1 on the wall), exactly on the ball.
        let arc_curve = crate::make_arc(centre, x_axis, y_axis, radius, 0.0, angle).or_refuse(KernelStage::Refine, "make_arc")?;
        let [arc_t0, arc_t1] = arc_curve.domain().or_refuse(KernelStage::Refine, "domain")?;
        let arc = take_id();
        result.edges.push(EdgeRecord { id: arc, curve: arc_curve.clone(), t0: arc_t0, t1: arc_t1, start_vertex_id: end_vertices[slot][0], end_vertex_id: end_vertices[slot][1], degenerate: false, name: None });
        let generator_curve = crate::make_line(q_point, pad_point).or_refuse(KernelStage::Refine, "make_line")?;
        let generator = take_id();
        result.edges.push(EdgeRecord { id: generator, curve: generator_curve.clone(), t0: 0.0, t1: 1.0, start_vertex_id: q, end_vertex_id: pad_vertex, degenerate: false, name: None });
        let segment_curve = crate::make_line(kept_point, q_point).or_refuse(KernelStage::Refine, "make_line")?;
        let segment = take_id();
        result.edges.push(EdgeRecord { id: segment, curve: segment_curve.clone(), t0: 0.0, t1: 1.0, start_vertex_id: kept_vertex, end_vertex_id: q, degenerate: false, name: None });
        // The cliff's plane, large enough to hold the section with margin so
        // the projector never clamps.
        let extent = 6.0 * radius;
        let plane_origin = centre.sub(x_axis.scale(0.5 * extent)).sub(y_axis.scale(0.5 * extent));
        let plane = crate::make_plane(plane_origin, x_axis, y_axis, extent, extent).or_refuse(KernelStage::Refine, "make_plane")?;
        // Pcurves on the pad and the kept mate (lines in their uv), and on
        // the plane (the arc's control points mapped through the plane's
        // affine inverse, exact for a rational arc).
        let generator_on_pad = line_pcurve(&pad.surface, q_point, pad_point)?;
        let segment_on_kept = line_pcurve(&kept_face.surface, kept_point, q_point)?;
        {
            let mid = generator_on_pad.evaluate(0.5).or_refuse(KernelStage::Refine, "evaluate")?;
            let on_pad = pad.surface.evaluate(mid.x, mid.y).or_refuse(KernelStage::Refine, "evaluate")?;
            let miss = on_pad.sub(generator_mid).length();
            if miss > pcurve_band {
                return Err(KernelRefusal::unsupported(KernelStage::Refine, "one_ended_pad_generator", format!(
                    "blend one-ended runout: the pad {}'s uv line misses its generator by {miss:.3e} at the middle (band {pcurve_band:.1e})",
                    chain.pad
                )));
            }
        }
        let arc_on_plane = {
            let mut points = Vec::with_capacity(arc_curve.control_points.len());
            for cp in &arc_curve.control_points {
                let w = cp.w;
                let uv = uv_of(&plane, Vec3::new(cp.x / w, cp.y / w, cp.z / w))?;
                points.push(Vec4 { x: uv[0] * w, y: uv[1] * w, z: 0.0, w });
            }
            NurbsCurve::new(arc_curve.degree, arc_curve.knots.clone(), points).or_refuse(KernelStage::Refine, "new")?
        };
        let generator_on_plane = line_pcurve(&plane, q_point, pad_point)?;
        let segment_on_plane = line_pcurve(&plane, kept_point, q_point)?;
        // The pad gains the generator and its rail piece between its end arc
        // (now ending at q) and its side rail (now starting at A); the kept
        // mate gains the segment with its rail pieces (spliced below with
        // the ridge run).  The cliff face: its arc coedge opposite to the
        // wall's, the other two chained by their vertices.
        let k = if slot == 0 { 0 } else { replaced_last };
        splice_loop(result, chain.pad, &[], Some((chain.e1, chain.side_rail)), &[(generator, generator_on_pad), (replaced_rails[k].0, replaced_rails[k].3.clone())], take_id)?;
        cliffs[slot] = Some(CliffEdges { arc, generator, segment, q, plane });
        // Kept-mate pcurve of the segment, and the plane pcurves, are
        // carried into the splice and the face below.
        cliff_segments.push((slot, segment, segment_on_kept));
        cliff_planes.push((slot, arc_on_plane, generator_on_plane, segment_on_plane));
    }

    // The mate loops: each loses its run of ridge coedges and gains the rail
    // piece(s) that replace it.  (face, removed edges, inserted rail pieces)
    let kept_centre = if matches!(ring_ends[0], RingEnd::Chain(_)) { 1 } else { 0 };
    let mut kept_run: Vec<u64> = Vec::new();
    if let RingEnd::Chain(chain) = &ring_ends[0] {
        kept_run.push(chain.e1);
    }
    kept_run.push(spine_id);
    if let RingEnd::Chain(chain) = &ring_ends[1] {
        kept_run.push(chain.e1);
    }
    let mut kept_inserted: Vec<(u64, NurbsCurve)> = vec![(kept_rails[kept_centre].0, kept_rails[kept_centre].3.clone())];
    for (_, segment, segment_on_kept) in &cliff_segments {
        kept_inserted.push((*segment, segment_on_kept.clone()));
    }
    let mut splices: Vec<(u64, Vec<u64>, Vec<(u64, NurbsCurve)>)> = vec![
        (kept, kept_run, kept_inserted),
        (replaced, vec![spine_id], vec![(replaced_rails[1].0, replaced_rails[1].3.clone())]),
    ];
    for (slot, ring_end) in ring_ends.iter().enumerate() {
        let RingEnd::Chain(chain) = ring_end else { continue };
        let k = if slot == 0 { 0 } else { replaced_last };
        let kk = if slot == 0 { 0 } else { kept_last };
        splices.push((chain.pad_s, vec![chain.e1, chain.e2], vec![(replaced_rails[k].0, replaced_rails[k].3.clone())]));
        splices.push((chain.pad_r, vec![chain.e2], vec![(kept_rails[kk].0, kept_rails[kk].3.clone())]));
    }
    // The wall's use of each rail edge: cr pieces walked forward exactly
    // when the blend loop's walk is station order (`build_open_surgery`).
    let blend_cr_forward = !first.coedge.forward;
    let is_rail = |edge_id: u64| rails_cr.iter().chain(rails_cs.iter()).any(|rail| rail.0 == edge_id);
    let wall_forward_of = |edge_id: u64| -> bool {
        if rails_cr.iter().any(|rail| rail.0 == edge_id) { blend_cr_forward } else { !blend_cr_forward }
    };
    for (face_id, removed, inserted) in splices {
        let senses = replace_run(result, face_id, &removed, &inserted, take_id)?;
        for (edge_id, mate_forward) in senses {
            if is_rail(edge_id) && mate_forward == wall_forward_of(edge_id) {
                return Err(KernelRefusal::internal(KernelStage::Sew, "runout_pairing", format!(
                    "blend runout: face {face_id}'s loop walks rail {edge_id} in the same sense as the wall would (manifold pairing broken)"
                )));
            }
        }
    }
    if debug {
        // Every new rail's pcurve on each mate against the rail itself.
        for rail in rails_cr.iter().chain(rails_cs.iter()) {
            let edge = edge_by_id(result, rail.0)?;
            for face in result.shells.iter().flat_map(|shell| &shell.faces) {
                for lp in &face.loops {
                    for coedge in lp.coedges.iter().filter(|c| c.edge_id == rail.0) {
                        let mut worst: f64 = 0.0;
                        for k in 0..=16 {
                            let t = edge.t0 + (edge.t1 - edge.t0) * k as f64 / 16.0;
                            let on_edge = edge.curve.evaluate(t).or_refuse(KernelStage::Refine, "evaluate")?;
                            // A reversed pcurve is mirrored over its own domain.
                            let s = if coedge.forward { t } else { edge.t0 + edge.t1 - t };
                            let uv = coedge.pcurve.evaluate(s).or_refuse(KernelStage::Refine, "evaluate")?;
                            let on_face = face.surface.evaluate(uv.x, uv.y).or_refuse(KernelStage::Refine, "evaluate")?;
                            worst = worst.max(on_face.sub(on_edge).length());
                        }
                        let [d0, d1] = coedge.pcurve.domain().unwrap_or([f64::NAN; 2]);
                        eprintln!(
                            "runout spine {spine_id}: rail {} on face {} (forward {}) edge t [{:.4}, {:.4}] pcurve domain [{d0:.4}, {d1:.4}] deviation {worst:.3e}",
                            rail.0, face.id, coedge.forward, edge.t0, edge.t1
                        );
                    }
                }
            }
        }
    }
    // Delete the ridge edges: the spine, and a chain end's two chain edges
    // (a one-ended end's arc was trimmed, its unselected edge is untouched).
    let mut ridge: Vec<u64> = vec![spine_id];
    for ring_end in &ring_ends {
        if let RingEnd::Chain(chain) = ring_end {
            ridge.push(chain.e1);
            ridge.push(chain.e2);
        }
    }
    result.edges.retain(|edge| !ridge.contains(&edge.id));

    // ---- The wall face. ----
    let u_pole = [parameters[0], parameters[parameters.len() - 1]];
    // The wall's closing edge at each end: a chain's degenerate pole edge, a
    // one-ended end's section arc (first-mate side to second-mate side).
    let mut pole_edges = [0u64; 2];
    for slot in 0..2 {
        match &ring_ends[slot] {
            RingEnd::Chain(chain) => {
                let point = chain.pole_point;
                let id = take_id();
                result.edges.push(EdgeRecord {
                    id,
                    curve: crate::make_line(point, point).or_refuse(KernelStage::Refine, "make_line")?,
                    t0: 0.0,
                    t1: 1.0,
                    start_vertex_id: chain.pole,
                    end_vertex_id: chain.pole,
                    degenerate: true,
                    name: None,
                });
                pole_edges[slot] = id;
            }
            RingEnd::One(_) => {
                pole_edges[slot] = cliffs[slot].as_ref().ok_or_else(|| KernelRefusal::internal(KernelStage::Sew, "runout_cliff", "blend runout: a one-ended end has no cliff"))?.arc;
            }
        }
    }
    let line = |u0: f64, v0: f64, u1: f64, v1: f64| crate::sweep_topology::parameter_line(u0, v0, u1, v1).or_refuse(KernelStage::Refine, "parameter_line");
    let rail_coedges = |rails: &[(u64, u64, u64, NurbsCurve, u64)], v: f64, forward: bool, take_id: &mut dyn FnMut() -> u64| -> Result<Vec<CoedgeRecord>, KernelRefusal> {
        let mut out = Vec::new();
        let ordered: Vec<&(u64, u64, u64, NurbsCurve, u64)> = if forward { rails.iter().collect() } else { rails.iter().rev().collect() };
        for rail in ordered {
            let edge = edge_by_id(result, rail.0)?;
            let (a, b) = (edge.t0, edge.t1);
            out.push(CoedgeRecord {
                id: take_id(),
                edge_id: rail.0,
                forward,
                pcurve: if forward { line(a, v, b, v)? } else { line(b, v, a, v)? },
            });
        }
        Ok(out)
    };
    // A pole edge is walked `forward` whichever way (both ends are the
    // pole); a section arc runs from v = 0 to v = 1, so a walk from v = 1 to
    // v = 0 is its reverse.
    let pole_coedge = |slot: usize, from_v: f64, to_v: f64, take_id: &mut dyn FnMut() -> u64| -> Result<CoedgeRecord, KernelRefusal> {
        let forward = matches!(ring_ends[slot], RingEnd::Chain(_)) || from_v < to_v;
        Ok(CoedgeRecord { id: take_id(), edge_id: pole_edges[slot], forward, pcurve: line(u_pole[slot], from_v, u_pole[slot], to_v)? })
    };
    let mut coedges = Vec::new();
    if blend_cr_forward {
        coedges.extend(rail_coedges(&rails_cr, 0.0, true, take_id)?);
        coedges.push(pole_coedge(1, 0.0, 1.0, take_id)?);
        coedges.extend(rail_coedges(&rails_cs, 1.0, false, take_id)?);
        coedges.push(pole_coedge(0, 1.0, 0.0, take_id)?);
    } else {
        coedges.extend(rail_coedges(&rails_cr, 0.0, false, take_id)?);
        coedges.push(pole_coedge(0, 0.0, 1.0, take_id)?);
        coedges.extend(rail_coedges(&rails_cs, 1.0, true, take_id)?);
        coedges.push(pole_coedge(1, 1.0, 0.0, take_id)?);
    }
    // Orientation: the wall's normal against the first mate's outward normal
    // at the centre stretch's middle station.
    let centre_range = ranges[n0];
    let mid_index = (centre_range[0] + centre_range[1]) / 2;
    let mid_u = parameters[mid_index];
    let blend_normal = raw_normal(&rows.surface, mid_u, 0.0)?;
    let uv1 = stations[mid_index].uv1;
    let n1 = raw_normal(&first.face.surface, uv1[0], uv1[1])?;
    let out1 = if first.face.same_sense { n1 } else { n1.scale(-1.0) };
    let same_sense = blend_normal.dot(out1) >= 0.0;
    let face_id = take_id();
    let shell_index = result
        .shells
        .iter()
        .position(|shell| shell.faces.iter().any(|face| face.id == kept))
        .ok_or(KernelRefusal::internal(KernelStage::Sew, "mate_shell", "blend runout: the kept mate's shell is missing"))?;
    let wall_arc_forward: Vec<(usize, bool)> = (0..2)
        .filter(|slot| matches!(ring_ends[*slot], RingEnd::One(_)))
        .map(|slot| (slot, coedges.iter().find(|coedge| coedge.edge_id == pole_edges[slot]).map(|coedge| coedge.forward).unwrap_or(true)))
        .collect();
    result.shells[shell_index].faces.push(FaceRecord {
        id: face_id,
        surface: rows.surface.clone(),
        same_sense,
        loops: vec![LoopRecord { id: take_id(), coedges }],
        name: name.map(str::to_string),
    });

    // ---- The cliff faces: planar, three edges, the arc walked
    // opposite to the wall, the normal into the air (back toward
    // station A, where the removed material was). ----
    for (slot, arc_on_plane, generator_on_plane, segment_on_plane) in cliff_planes {
        let cliff = cliffs[slot].take().ok_or_else(|| KernelRefusal::internal(KernelStage::Sew, "runout_cliff", "blend runout: a cliff's edges are missing"))?;
        let arc_forward = !wall_arc_forward.iter().find(|(s, _)| *s == slot).map(|(_, forward)| *forward).unwrap_or(true);
        let endpoints = |result: &BrepSolid, edge_id: u64| -> Result<(u64, u64), KernelRefusal> {
            let edge = edge_by_id(result, edge_id)?;
            Ok((edge.start_vertex_id, edge.end_vertex_id))
        };
        let mut loop_coedges = vec![CoedgeRecord {
            id: take_id(),
            edge_id: cliff.arc,
            forward: arc_forward,
            pcurve: if arc_forward { arc_on_plane.clone() } else { arc_on_plane.reversed().or_refuse(KernelStage::Refine, "reversed")? },
        }];
        let (arc_start, arc_end) = endpoints(result, cliff.arc)?;
        let (loop_start, mut cursor) = if arc_forward { (arc_start, arc_end) } else { (arc_end, arc_start) };
        let mut pending: Vec<(u64, NurbsCurve)> = vec![(cliff.generator, generator_on_plane), (cliff.segment, segment_on_plane)];
        while !pending.is_empty() {
            let position = pending
                .iter()
                .position(|(edge_id, _)| endpoints(result, *edge_id).map(|(s, e)| s == cursor || e == cursor).unwrap_or(false))
                .ok_or_else(|| KernelRefusal::internal(KernelStage::Sew, "runout_cliff_loop", format!("blend runout: the cliff's edges do not continue from vertex {cursor}")))?;
            let (edge_id, pcurve) = pending.remove(position);
            let (s, e) = endpoints(result, edge_id)?;
            let forward = s == cursor;
            cursor = if forward { e } else { s };
            loop_coedges.push(CoedgeRecord {
                id: take_id(),
                edge_id,
                forward,
                pcurve: if forward { pcurve } else { pcurve.reversed().or_refuse(KernelStage::Refine, "reversed")? },
            });
        }
        if cursor != loop_start {
            return Err(KernelRefusal::internal(KernelStage::Sew, "runout_cliff_loop", "blend runout: the cliff's loop does not close"));
        }
        // Normal into the air: toward station A's centre.  The plane's
        // normal is read at its middle.
        let end = if slot == 0 { &end0 } else { &end1 };
        let Terminus::Cliff { stop, .. } = &end.terminus else { unreachable!("a cliff end") };
        let a_ball = {
            let (section_point, section_tangent) = section_frame(spine, end.t_a)?;
            let (_, _, _, centre_a) = tangency_residual(
                &face_by_id(ring, end.middle.0)?.surface,
                &face_by_id(ring, end.middle.1)?.surface,
                end.middle.2,
                end.middle.3,
                section_point,
                section_tangent,
            )?;
            centre_a
        };
        let [u0, u1] = cliff.plane.domain_u().or_refuse(KernelStage::Refine, "domain_u")?;
        let [v0, v1] = cliff.plane.domain_v().or_refuse(KernelStage::Refine, "domain_v")?;
        let plane_normal = cliff.plane.normal(0.5 * (u0 + u1), 0.5 * (v0 + v1)).or_refuse(KernelStage::Refine, "normal")?;
        let same_sense = plane_normal.dot(a_ball.sub(stop.center)) > 0.0;
        let _ = cliff.q;
        result.shells[shell_index].faces.push(FaceRecord {
            id: take_id(),
            surface: cliff.plane,
            same_sense,
            loops: vec![LoopRecord { id: take_id(), coedges: loop_coedges }],
            name: name.map(|value| format!("{value}_cliff")),
        });
    }
    Ok(report)
}

/// In `face_id`'s loop that carries `removed[0]`, replace the contiguous run
/// of coedges on `removed` with coedges on `inserted` (edge id, pcurve in the
/// edge's own parameter direction), each oriented so the loop stays
/// connected.  Returns each inserted edge's sense in that loop.
fn replace_run(
    result: &mut BrepSolid,
    face_id: u64,
    removed: &[u64],
    inserted: &[(u64, NurbsCurve)],
    take_id: &mut dyn FnMut() -> u64,
) -> Result<Vec<(u64, bool)>, KernelRefusal> {
    splice_loop(result, face_id, removed, None, inserted, take_id)
}

/// [`replace_run`] with the run located either by `removed[0]` or, for an
/// EMPTY run, by `anchor`: two edges whose coedges are adjacent in the loop
/// (they met at a vertex a trim has since moved), between which the inserted
/// edges go.  The inserted edges are chained by their vertices from the
/// loop's cursor in whatever order they are given.
fn splice_loop(
    result: &mut BrepSolid,
    face_id: u64,
    removed: &[u64],
    anchor: Option<(u64, u64)>,
    inserted: &[(u64, NurbsCurve)],
    take_id: &mut dyn FnMut() -> u64,
) -> Result<Vec<(u64, bool)>, KernelRefusal> {
    let endpoints = |result: &BrepSolid, coedge: &CoedgeRecord| -> Result<(u64, u64), KernelRefusal> {
        let edge = edge_by_id(result, coedge.edge_id)?;
        Ok(if coedge.forward { (edge.start_vertex_id, edge.end_vertex_id) } else { (edge.end_vertex_id, edge.start_vertex_id) })
    };
    let new_edges: Vec<(u64, u64, u64)> = inserted
        .iter()
        .map(|(id, _)| edge_by_id(result, *id).map(|edge| (*id, edge.start_vertex_id, edge.end_vertex_id)))
        .collect::<Result<_, _>>()?;
    let face_index = result
        .shells
        .iter()
        .flat_map(|shell| &shell.faces)
        .position(|face| face.id == face_id)
        .ok_or_else(|| KernelRefusal::internal(KernelStage::Sew, "runout_face", format!("blend runout: face {face_id} missing")))?;
    let locator = match (removed.first(), anchor) {
        (Some(first), _) => *first,
        (None, Some((before, _))) => before,
        (None, None) => return Err(KernelRefusal::internal(KernelStage::Sew, "runout_run", "blend runout: a splice with nothing removed needs an anchor")),
    };
    let mut loop_index = None;
    {
        let face = result.shells.iter().flat_map(|shell| &shell.faces).nth(face_index).expect("face present");
        for (index, lp) in face.loops.iter().enumerate() {
            if lp.coedges.iter().any(|coedge| coedge.edge_id == locator) {
                loop_index = Some(index);
            }
        }
    }
    let loop_index = loop_index.ok_or_else(|| KernelRefusal::internal(KernelStage::Sew, "runout_loop", format!(
        "blend runout: face {face_id} has no loop on edge {locator}"
    )))?;
    // Read the loop, compute the splice, then write it back.
    let old: Vec<CoedgeRecord> = result
        .shells
        .iter()
        .flat_map(|shell| &shell.faces)
        .nth(face_index)
        .expect("face present")
        .loops[loop_index]
        .coedges
        .clone();
    let count = old.len();
    let positions: Vec<usize> = old.iter().enumerate().filter(|(_, c)| removed.contains(&c.edge_id)).map(|(i, _)| i).collect();
    if positions.len() != removed.len() {
        return Err(KernelRefusal::internal(KernelStage::Sew, "runout_run", format!(
            "blend runout: face {face_id}'s loop carries {} of the {} ridge edges {removed:?}",
            positions.len(), removed.len()
        )));
    }
    // The run starts at the removed coedge whose predecessor is kept; an
    // empty run starts at the anchor's later coedge.
    let start = if positions.is_empty() {
        let (a, b) = anchor.expect("an empty run has an anchor");
        let (pa, pb) = (old.iter().position(|c| c.edge_id == a), old.iter().position(|c| c.edge_id == b));
        let (Some(pa), Some(pb)) = (pa, pb) else {
            return Err(KernelRefusal::internal(KernelStage::Sew, "runout_run", format!(
                "blend runout: face {face_id}'s loop does not carry both anchor edges {a} and {b}"
            )));
        };
        if (pa + 1) % count == pb {
            pb
        } else if (pb + 1) % count == pa {
            pa
        } else {
            return Err(KernelRefusal::internal(KernelStage::Sew, "runout_run", format!(
                "blend runout: the anchor edges {a} and {b} are not adjacent in face {face_id}'s loop"
            )));
        }
    } else {
        *positions
            .iter()
            .find(|&&p| !positions.contains(&((p + count - 1) % count)))
            .ok_or_else(|| KernelRefusal::internal(KernelStage::Sew, "runout_run", format!("blend runout: face {face_id}'s loop is wholly the ridge")))?
    };
    for k in 0..positions.len() {
        if !positions.contains(&((start + k) % count)) {
            return Err(KernelRefusal::internal(KernelStage::Sew, "runout_run", format!(
                "blend runout: the ridge edges {removed:?} are not contiguous in face {face_id}'s loop"
            )));
        }
    }
    let before = old[(start + count - 1) % count].clone();
    let after = old[(start + positions.len()) % count].clone();
    let (_, mut cursor) = endpoints(result, &before)?;
    let (after_start, _) = endpoints(result, &after)?;
    let mut senses = Vec::new();
    let mut fresh = Vec::new();
    let mut pending: Vec<usize> = (0..inserted.len()).collect();
    while !pending.is_empty() {
        // The next inserted edge is the one touching the cursor (the given
        // order first, so a single rail reads exactly as before).
        let next = pending
            .iter()
            .position(|&index| new_edges[index].1 == cursor || new_edges[index].2 == cursor)
            .ok_or_else(|| {
                let names: Vec<u64> = pending.iter().map(|&index| new_edges[index].0).collect();
                KernelRefusal::internal(KernelStage::Sew, "runout_run", format!(
                    "blend runout: none of the rails {names:?} continues face {face_id}'s loop from vertex {cursor}"
                ))
            })?;
        let index = pending.remove(next);
        let (id, pcurve) = (&inserted[index].0, &inserted[index].1);
        let (_, s, e) = &new_edges[index];
        let forward = *s == cursor;
        cursor = if forward { *e } else { *s };
        fresh.push(CoedgeRecord {
            id: take_id(),
            edge_id: *id,
            forward,
            pcurve: if forward { pcurve.clone() } else { pcurve.reversed().or_refuse(KernelStage::Refine, "reversed")? },
        });
        senses.push((*id, forward));
    }
    if cursor != after_start {
        return Err(KernelRefusal::internal(KernelStage::Sew, "runout_run", format!(
            "blend runout: the rails spliced into face {face_id}'s loop end at vertex {cursor}, not at {after_start} where the loop continues"
        )));
    }
    let mut rebuilt: Vec<CoedgeRecord> = Vec::with_capacity(count - positions.len() + fresh.len());
    let mut k = (start + positions.len()) % count;
    for _ in 0..count - positions.len() {
        rebuilt.push(old[k].clone());
        k = (k + 1) % count;
    }
    rebuilt.extend(fresh);
    let face = result
        .shells
        .iter_mut()
        .flat_map(|shell| &mut shell.faces)
        .nth(face_index)
        .expect("face present");
    face.loops[loop_index].coedges = rebuilt;
    Ok(senses)
}