brep_kernel/blending/fillet/

edges.rs

use super::*;

/// Constant-radius rolling-ball fillet of one edge (Golovanov §4.9 march,
/// §6.9 surgery; the cutter only where the march refuses).
pub fn fillet_edge(
    solid: &BrepSolid,
    edge_id: u64,
    radius: f64,
    name: Option<&str>,
) -> Result<BrepSolid, KernelRefusal> {
    // One blend operation: the reports of an earlier one, never consumed, are
    // dropped as it starts (nested calls keep this operation's).
    let _operation = crate::blend::BlendOperation::enter();
    check_mixed_concavity(solid, edge_id, "fillet_edge")?;
    check_support_extent_clipping(solid, edge_id, radius, "fillet_edge")?;
    fillet_or_chamfer(solid, edge_id, radius, false, name, ToolEnds::default(), Lane::GeneralFirst)
}

/// `fillet_edge` with the cutter first: the input the corner-closure lanes
/// (`round_convex_corner`, the mixed-convexity closures) were written
/// against.  Their tests build it directly; production reaches those lanes
/// only through the group's sequential composition, which is cutter-first
/// for the same reason.
pub(crate) fn fillet_edge_cutter(
    solid: &BrepSolid,
    edge_id: u64,
    radius: f64,
    name: Option<&str>,
) -> Result<BrepSolid, KernelRefusal> {
    check_mixed_concavity(solid, edge_id, "fillet_edge")?;
    check_support_extent_clipping(solid, edge_id, radius, "fillet_edge")?;
    fillet_or_chamfer(solid, edge_id, radius, false, name, ToolEnds::default(), Lane::CutterFirst)
        .map(settle_cutter_scaffold)
}

/// Equal-leg chamfer of one edge (Golovanov §6.11: the same construction
/// with the arc's chord).
pub fn chamfer_edge(
    solid: &BrepSolid,
    edge_id: u64,
    distance: f64,
    name: Option<&str>,
) -> Result<BrepSolid, KernelRefusal> {
    // One blend operation: the reports of an earlier one, never consumed, are
    // dropped as it starts (nested calls keep this operation's).
    let _operation = crate::blend::BlendOperation::enter();
    check_mixed_concavity(solid, edge_id, "chamfer_edge")?;
    check_support_extent(solid, edge_id, distance, "chamfer_edge")?;
    fillet_or_chamfer(solid, edge_id, distance, true, name, ToolEnds::default(), Lane::GeneralFirst)
}

/// Build and apply the chamfer tool for one straight edge from an already-built
/// cross-section profile (curve index 1 is the chamfer chord / blend wall).
/// Shared by the two-distance and distance-angle asymmetric entries.
fn apply_chamfer_offsets_profile(
    solid: &BrepSolid,
    cross: &EdgeCross,
    profile: &[NurbsCurve],
    name: Option<&str>,
    edge_id: u64,
) -> Result<BrepSolid, KernelRefusal> {
    // Run through to the faces the edge's ends meet where the rails go on past
    // them (`BREP_CHAMFER_THROUGH_END_FACES`, default on; `=0` restores); the flush cutter
    // below answers everything else.
    if let Some(result) = super::tool::chamfer_through_end_faces(solid, edge_id, cross, profile, name) {
        return Ok(settle_cutter_scaffold(result));
    }
    let mut tool = match &cross.path {
        EdgePath::Straight { direction, length } => {
            extrude_profile_brep(profile, *direction, *length)?
        }
        EdgePath::Circular { .. } => {
            return Err(KernelRefusal::unsupported(
                KernelStage::Classify,
                "asymmetric_curved_edge",
                "chamfer_edge_asymmetric: only straight edges on planar faces are supported \
                 in this slice (asymmetric chamfer on general/curved edges is out of scope)",
            ));
        }
    };
    // Side faces are emitted in input-curve order; the chamfer wall is the
    // second profile curve (index 1). Every other face is the cutter's
    // scaffold, tagged and settled exactly as the symmetric cutter's is.
    let mut side_index = 0usize;
    for shell in &mut tool.shells {
        for face in &mut shell.faces {
            if side_index == 1 {
                if face.name.is_none() {
                    face.name = name.map(str::to_string);
                }
            } else {
                face.name = Some(scaffold_tag(name));
            }
            side_index += 1;
        }
    }
    apply_tool(solid, &tool, cross.convex).map(settle_cutter_scaffold)
}

/// Asymmetric (two-distance) chamfer of one STRAIGHT edge between two planar
/// faces (Golovanov §6.11): setback `d1` along face 1 and `d2` along face 2 —
/// the standard CAD "d1 × d2" bevel.  General/curved edges are out of scope for
/// this slice and return a clear error.
pub fn chamfer_edge_asymmetric(
    solid: &BrepSolid,
    edge_id: u64,
    d1: f64,
    d2: f64,
    name: Option<&str>,
) -> Result<BrepSolid, KernelRefusal> {
    // One blend operation: the reports of an earlier one, never consumed, are
    // dropped as it starts (nested calls keep this operation's).
    let _operation = crate::blend::BlendOperation::enter();
    if !(d1 > 0.0) || !(d2 > 0.0) || !d1.is_finite() || !d2.is_finite() {
        return Err(KernelRefusal::input(
            KernelStage::Collect,
            "setback_distances",
            "chamfer_edge_asymmetric: both setback distances must be positive",
        ));
    }
    // `analyze_edge` only uses the radius to size the orientation probe step;
    // the smaller setback keeps that probe inside both faces.
    let cross = analyze_edge(solid, edge_id, d1.min(d2))?;
    let profile = chamfer_cross_section_offsets(&cross, d1, d2)?;
    apply_chamfer_offsets_profile(solid, &cross, &profile, name, edge_id)
}

/// Distance-angle chamfer of one STRAIGHT edge between two planar faces
/// (Golovanov §6.11): setback `d1` along face 1 and angle `angle_rad` between
/// the chamfer face and face 1.  `d2` is constructed geometrically in the
/// cross-section plane (see `chamfer_angle_second_distance`), then the
/// two-distance builder is applied.
pub fn chamfer_edge_angle(
    solid: &BrepSolid,
    edge_id: u64,
    d1: f64,
    angle_rad: f64,
    name: Option<&str>,
) -> Result<BrepSolid, KernelRefusal> {
    // One blend operation: the reports of an earlier one, never consumed, are
    // dropped as it starts (nested calls keep this operation's).
    let _operation = crate::blend::BlendOperation::enter();
    if !(d1 > 0.0) || !d1.is_finite() {
        return Err(KernelRefusal::input(
            KernelStage::Collect,
            "setback_d1",
            "chamfer_edge_angle: setback distance d1 must be positive",
        ));
    }
    let cross = analyze_edge(solid, edge_id, d1)?;
    let d2 = chamfer_angle_second_distance(&cross, d1, angle_rad)?;
    let profile = chamfer_cross_section_offsets(&cross, d1, d2)?;
    apply_chamfer_offsets_profile(solid, &cross, &profile, name, edge_id)
}

/// Snap small trim/intersection gaps between blend edges and their vertices.
/// The boolean endpoint welder preserves gaps already inside the validation
/// band and limits repairs to `max(|radius| * 1e-3, 1e-4)`.
pub(super) fn heal_edge_vertex_gaps(
    solid: &mut BrepSolid,
    radius: f64,
) -> Result<(), KernelRefusal> {
    let search = (radius.abs() * 1e-3).max(1e-7);
    crate::boolean::commit_nearby_edge_endpoints(solid, search)
}

/// The fillet pipeline's exit acceptance: a solid with a validation ERROR is
/// refused by name, never returned.
///
/// Warnings do not refuse; only the issues `validate()` itself calls errors.
fn refuse_invalid_result(result: &BrepSolid, entry: &str) -> Result<(), KernelRefusal> {
    let errors: Vec<_> = result.validate_memoised().into_iter().filter(|issue| issue.severity == "error").collect();
    let Some(first) = errors.first() else {
        return Ok(());
    };
    Err(KernelRefusal::new(
        RefusalClass::InvalidResultTopology { issues: errors.len() as u32 },
        KernelStage::Validate,
        format!(
            "{entry}: the blend built a solid that fails validation ({} errors, first: {}); \
             refused rather than returned",
            errors.len(),
            first.message
        ),
    ))
}

/// Resolve a point within 1e-3 of a trimmed edge.
pub(super) fn resolve_edge_by_point(solid: &BrepSolid, point: Vec3) -> Result<u64, KernelRefusal> {
    match crate::topology::nearest_edge(solid, point) {
        Some((edge_id, distance)) if distance <= 1e-3 => Ok(edge_id),
        Some((_, distance)) => Err(KernelRefusal::input(
            KernelStage::Collect,
            "edge_point",
            format!(
                "fillet_edges: no edge within tolerance of the point (nearest {distance:.6})"
            ),
        )),
        None => Err(KernelRefusal::input(
            KernelStage::Collect,
            "no_edges",
            "fillet_edges: solid has no edges",
        )),
    }
}

/// Lane R1 row 3b: a fillet operation opens the validation coedge memo
/// (`CoedgeMemoScope`). On by default (promoted 2026-10-18 on the 98 + 144
/// census); `BREP_FILLET_COEDGE_MEMO=0` (or `off`) opens none.
pub(crate) fn fillet_coedge_memo_on() -> bool {
    switch!("BREP_FILLET_COEDGE_MEMO", true)
}


/// Fillet (or chamfer) a GROUP of edges as ONE operation, and — for fillets —
/// round the convex "star" vertices where three or more of the selected edges
/// meet (Golovanov §6.9.7).  This is the whole multi-edge fillet in a single
/// kernel call: the caller passes the object plus one 3D point on each edge,
/// and the kernel orchestrates the filleting and corner blending against the
/// full topology (so acute corners resolve coherently instead of being
/// stitched edge-by-edge by the app).  A corner the kernel cannot round (e.g.
/// non-orthogonal beyond support, or a general no-common-ball star) is left as
/// the edge fillets rather than failing the whole group.
///
/// When the WHOLE selection cannot be blended as one unit — a shared convex
/// corner where a revolve axis/pole edge meets the adjacent cap edges can
/// defeat the sequential corner surgery even though each edge and every proper
/// SUBSET of the selection blends cleanly (the three fillets converge on the
/// pole with no single end face across the corner) — a maximal-valid-SUBSET
/// search blends the largest subset that works, dropping only the edges that
/// cannot co-blend.  That is a real outcome and [`fillet_edges_reported`]
/// returns it, WITH the list of edges it could not blend.
///
/// This entry point does not.  A solid holding eight of the ten walls that
/// were asked for is not the answer to the question that was asked, and
/// returning it as a plain `Ok` reports success for a different result; so an
/// incomplete selection is an error here, naming every edge that was dropped
/// and the group refusal behind them.  A caller that genuinely wants the
/// partial answer asks for it by name.
///
/// `edge_names` (when `Some`) is the per-edge blend-FACE name parallel to
/// `edge_points` — each grown wall is named after ITS originating edge; `None`
/// names every wall with the single base `name` (the legacy/test behavior,
/// byte-identical to before). The whole `*_edges` family takes the same
/// `edge_names` slot in the same position.
pub fn fillet_edges(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    chamfer: bool,
    name: Option<&str>,
) -> Result<BrepSolid, KernelRefusal> {
    // A selected closed rim with a corner at its one vertex, on a mate creased
    // along its own seam, is blended on the body canonicalised for it — the
    // mate split along a smooth meridian so the network's miter closes the
    // crease — and the walls are merged back afterwards
    // (`blend/creased_rim.rs`; `BREP_CREASED_RIM_SPLIT=0` turns it off). Every
    // other selection takes the body as it came.
    let prepared = if crate::blend::creased_rim::creased_rim_split_on() {
        crate::blend::creased_rim::split_creased_rims(solid, edge_points, edge_names, &resolve_edge_by_point)?
    } else {
        None
    };
    match prepared {
        Some(prepared) => {
            let names = prepared.names.as_deref();
            let built = fillet_edges_prepared(&prepared.solid, &prepared.points, names, radius, chamfer, name)?;
            Ok(prepared.rejoin(built))
        }
        None => {
            // A chamfer selection holding a MIXED STAR (one concave and two
            // convex selected edges at a vertex, the convex two on one cap) is
            // built in two phases (`fillet/chamfer_star.rs`;
            // `BREP_CHAMFER_MIXED_STAR=0` turns it off); a phase that refuses
            // hands the selection to the ordinary path below.
            if chamfer && super::chamfer_star::chamfer_mixed_star_on() {
                let build = |body: &BrepSolid, points: &[Vec3], names: Option<&[String]>| {
                    fillet_edges_prepared(body, points, names, radius, true, name)
                };
                if let Some(built) = super::chamfer_star::chamfer_mixed_stars(solid, edge_points, edge_names, &build) {
                    return Ok(built);
                }
            }
            let ordinary = fillet_edges_prepared(solid, edge_points, edge_names, radius, chamfer, name);
            // A closed rim written as a chain of edges along the halves of a
            // split carrier, refused because only the closed-rim rollover
            // follows its ball and that lane takes one closed edge, is blended
            // on the body with those halves merged and the chain joined into one
            // closed edge (`blend/split_rim.rs`; default on,
            // `BREP_SPLIT_RIM_MERGE=0` off). Only that refusal asks: a selection
            // the ordinary path builds keeps its body and names.
            let answered = match ordinary {
                Err(refusal)
                    if crate::blend::split_rim::split_rim_merge_on()
                        && matches!(&refusal.class, crate::RefusalClass::UnsupportedGeometry { what } if what == super::analyze::TANGENT_ROLLOVER_ONLY_WHAT) =>
                {
                    match crate::blend::split_rim::merge_split_rims(solid, edge_points, edge_names, &resolve_edge_by_point) {
                        // The merged attempt's own reason is appended: the
                        // first refusal names the lane that is missing, the
                        // second why the canonical body did not answer either
                        // (lane F's open item).
                        Some(merged) => fillet_edges_prepared(&merged.solid, &merged.points, merged.names.as_deref(), radius, chamfer, name)
                            .map_err(|merged_refusal| {
                                refusal.with_message(|message| {
                                    format!("{message} The rim merged into one closed edge refused too: {}", merged_refusal.message)
                                })
                            }),
                        None => Err(refusal),
                    }
                }
                other => other,
            };
            // A sheet's edge that runs out at the free boundary has no face
            // across its end: each such end is closed by a temporary planar
            // face, the body blended, and the faces removed again
            // (`blend/sheet_ends.rs`; default on, `BREP_SHEET_FREE_END=0` off).
            // Only a refused sheet selection asks; the refusal stands unless
            // the reopened sheet validates.
            match answered {
                Err(refusal) if solid.kind == crate::BodyKind::Sheet && crate::blend::sheet_ends::sheet_free_end_on() => {
                    let edge_ids = edge_points.iter().map(|point| resolve_edge_by_point(solid, *point)).collect::<Result<Vec<_>, _>>();
                    let Ok(edge_ids) = edge_ids else { return Err(refusal) };
                    let Some(closed) = crate::blend::sheet_ends::close_free_ends(solid, &edge_ids) else { return Err(refusal) };
                    let count = crate::blend::sheet_ends::temporary_faces(&closed);
                    match fillet_edges_prepared(&closed, edge_points, edge_names, radius, chamfer, name) {
                        Ok(built) => crate::blend::sheet_ends::open_free_ends(built, count).ok_or(refusal),
                        Err(closed_refusal) => Err(refusal.with_message(|message| {
                            format!("{message} The sheet with its free ends closed refused too: {}", closed_refusal.message)
                        })),
                    }
                }
                other => other,
            }
        }
    }
}

fn fillet_edges_prepared(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    chamfer: bool,
    name: Option<&str>,
) -> Result<BrepSolid, KernelRefusal> {
    // One blend operation: the reports of an earlier one, never consumed, are
    // dropped as it starts (nested calls keep this operation's).
    let _operation = crate::blend::BlendOperation::enter();
    // Each operation pays for the first read of every body it validates
    // (`BrepSolid::validate_memoised`, lane R1 row 3).
    BrepSolid::clear_validate_memo();
    // One coedge-check memo per operation, as a boolean opens one (lane R1
    // row 3b, `BREP_FILLET_COEDGE_MEMO`): the surgery's, the group lane's
    // and the exit's validations re-read the coedges the rims left unchanged.
    let _coedge_memo = crate::topology::CoedgeMemoScope::open(fillet_coedge_memo_on() && crate::topology::coedge_memo_enabled());
    let entry = if chamfer { "chamfer_edges" } else { "fillet_edges" };
    // The search is capped at one dropped edge here (lane F row 3);
    // `BREP_SUBSET_SEARCH_FULL=1` restores the full search.
    let cap = std::env::var_os("BREP_SUBSET_SEARCH_FULL").is_none().then_some(1);
    let _cap = SubsetDropCap::set(cap);
    let (result, selection) = match fillet_edges_reported(solid, edge_points, edge_names, radius, chamfer, name) {
        // A wall that folds is relieved where it folds, not refused
        // (`blend/relief.rs`).
        Err(fold) if !chamfer && crate::blend::is_wall_fold(&fold) && crate::blend::relief::radius_relief_on() => {
            let misfit = crate::blend::relief::is_misfit_fold(&fold);
            let relieved = relieve_fold(solid, edge_points, edge_names, radius, name, fold);
            // A misfit (a lane's refusal read as the ball not fitting) took
            // the group off the ladder that follows the network; when no
            // relief builds, that ladder answers, as it did before the relief.
            if relieved.is_err() && misfit {
                let _off = crate::blend::relief::MisfitOff::enter();
                return fillet_edges(solid, edge_points, edge_names, radius, chamfer, name);
            }
            return relieved;
        }
        reported => reported?,
    };
    if selection.is_complete() {
        return Ok(result);
    }
    // Every dropped edge BY NAME: the caller's own per-edge name when it gave
    // one, and always its index and the point it was selected by.
    let dropped: Vec<String> = selection
        .rejected
        .iter()
        .map(|index| {
            let point = edge_points[*index];
            let named = edge_names
                .and_then(|names| names.get(*index))
                .map(|name| format!("`{name}` "))
                .unwrap_or_default();
            format!(
                "{named}#{index} at ({:.6}, {:.6}, {:.6})",
                point.x, point.y, point.z
            )
        })
        .collect();
    // The selection, as specified, has no single answer the kernel will
    // stand behind: the group refusal behind the drop travels as TEXT in
    // `BlendSelection::reason`, so this refusal carries its own class.
    Err(KernelRefusal::ill_posed(
        KernelStage::Select,
        "incomplete_selection",
        format!(
            "{entry}: {} of the {} selected edges were blended — {} could not co-blend with the \
             rest of the selection and carry no wall in the result ({}). The group refusal was: \
             {}. Blend them separately, or call the reported entry point to accept the partial \
             selection deliberately.",
            selection.applied.len(),
            selection.requested,
            selection.rejected.len(),
            dropped.join(", "),
            selection
                .reason
                .as_deref()
                .unwrap_or("the whole group built but not every edge grew a wall"),
        ),
    ))
}

/// The fillet a constant `radius` would fold: rebuilt with the radius reduced
/// only where it does not fit (`blend/relief.rs`), and a note saying where
/// and to what. Selected edges joined (through shared vertices) to no relief
/// zone are built at the constant radius first; each relieved group is then
/// built by the taper network on that body. A relief that still folds is
/// planned again with a wider margin; when none builds, the fold stands with
/// the last relief's reason appended.
fn relieve_fold(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    name: Option<&str>,
    fold: KernelRefusal,
) -> Result<BrepSolid, KernelRefusal> {
    let _relieving = crate::blend::relief::Relieving::enter();
    let debug = std::env::var("BREP_DEBUG_NETWORK").is_ok();
    let selected: Vec<u64> = edge_points.iter().map(|point| resolve_edge_by_point(solid, *point)).collect::<Result<_, _>>()?;
    // One edge per carrier continued (`relief::join_continuations`): the
    // selection is read again on the joined body, two points on one joined
    // edge selecting it once.
    let joined = crate::blend::relief::join_continuations(solid, &selected);
    let solid = joined.as_ref().unwrap_or(solid);
    let mut ids: Vec<u64> = Vec::with_capacity(edge_points.len());
    let mut points: Vec<Vec3> = Vec::with_capacity(edge_points.len());
    let mut names: Vec<String> = Vec::with_capacity(edge_points.len());
    for (index, point) in edge_points.iter().enumerate() {
        let id = resolve_edge_by_point(solid, *point)?;
        if !ids.contains(&id) {
            ids.push(id);
            points.push(*point);
            if let Some(name) = edge_names.and_then(|names| names.get(index)) {
                names.push(name.clone());
            }
        }
    }
    let (edge_points, edge_names) = (&points[..], edge_names.map(|_| &names[..]));
    if debug && joined.is_some() {
        eprintln!("radius relief: joined continuation edges, {} selected edges become {}", selected.len(), ids.len());
    }
    let started = web_time::Instant::now();
    let measured = match crate::blend::relief::measure_ceilings(solid, &ids, radius) {
        Ok(measured) => measured,
        Err(refusal) => return Err(fold.with_message(|message| format!("{message}; the relief could not measure the edges: {refusal}"))),
    };
    if debug {
        eprintln!("radius relief: ceilings measured in {:.1} ms", started.elapsed().as_secs_f64() * 1e3);
    }
    // A relief is LOCAL: the requested radius stands where it fits. Where it
    // fits nowhere on the selection (a ball wider than a rod's rim, the whole
    // rim short alike) a radius lowered everywhere is a different request,
    // and the fold stands, naming the largest ball that does fit.
    let largest = measured.iter().flatten().map(|(_, ceiling)| *ceiling).fold(0.0_f64, f64::max);
    if largest < radius {
        return Err(fold.with_message(|message| {
            format!("{message}; r {radius} fits nowhere along the selection, so it is not reduced locally: the largest ball that fits anywhere on it is {largest:.6}")
        }));
    }
    let mut last: Option<KernelRefusal> = None;
    for attempt in 0..crate::blend::relief::ATTEMPTS {
        let plan = match crate::blend::relief::plan_relief(solid, &ids, radius, &measured, attempt) {
            Ok(Some(plan)) => plan,
            // The ball fits everywhere the probe reads: the fold is not one
            // a smaller radius answers.
            Ok(None) => break,
            Err(refusal) => {
                last = Some(refusal);
                break;
            }
        };
        let mark = crate::blend::blend_notes_mark();
        match build_relief_plan(solid, edge_points, edge_names, &ids, radius, name, &plan) {
            Ok(body) => {
                for zone in &plan.zones {
                    let near: Vec<String> = zone
                        .edges
                        .iter()
                        .filter_map(|index| solid.edges.iter().find(|edge| edge.id == ids[*index]))
                        .map(|edge| edge.name.clone().unwrap_or_else(|| format!("edge #{}", edge.id)))
                        .map(|label| format!("`{label}`"))
                        .collect();
                    // The note states what was built: where the law never
                    // climbs back to the requested radius on the zone's edges
                    // (report 5e49bb42's rim, 1.08 to 1.78 against r 2.2), it
                    // says so instead of "locally".
                    let (low, high) = zone
                        .edges
                        .iter()
                        .flat_map(|index| plan.stops[*index].iter().map(|(_, r)| *r))
                        .fold((f64::INFINITY, 0.0_f64), |(low, high), r| (low.min(r), high.max(r)));
                    let note = if high < radius * (1.0 - 1e-9) {
                        format!(
                            "radius reduced over the whole of {}: {low:.2} to {high:.2}, so r {radius} is reached nowhere \
                             on them (where they turn tightest the largest ball that fits is {:.2})",
                            near.join(", "),
                            zone.ceiling,
                        )
                    } else {
                        format!(
                            "radius reduced to {:.2} locally near {}: r {radius} does not fit there (the largest ball that does is {:.2})",
                            zone.radius,
                            near.join(", "),
                            zone.ceiling,
                        )
                    };
                    crate::blend::record_blend_note(note);
                }
                return Ok(body);
            }
            Err(refusal) => {
                crate::blend::blend_notes_truncate(mark);
                if debug {
                    eprintln!("radius relief attempt {attempt} refused at {:.1} ms: {refusal}", started.elapsed().as_secs_f64() * 1e3);
                }
                last = Some(refusal);
            }
        }
    }
    Err(match last {
        Some(last) => fold.with_message(|message| format!("{message}; a locally reduced radius did not build either: {last}")),
        None => fold,
    })
}

/// Build one relief plan (see [`relieve_fold`]).
fn build_relief_plan(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    ids: &[u64],
    radius: f64,
    name: Option<&str>,
    plan: &crate::blend::relief::ReliefPlan,
) -> Result<BrepSolid, KernelRefusal> {
    // Groups of selected edges joined through shared vertices.
    let count = ids.len();
    let mut group: Vec<usize> = (0..count).collect();
    let find = |group: &mut Vec<usize>, mut index: usize| {
        while group[index] != index {
            group[index] = group[group[index]];
            index = group[index];
        }
        index
    };
    let vertices: Vec<[u64; 2]> = ids
        .iter()
        .map(|id| {
            solid.edges.iter().find(|edge| edge.id == *id).map(|edge| [edge.start_vertex_id, edge.end_vertex_id]).unwrap_or([u64::MAX; 2])
        })
        .collect();
    for a in 0..count {
        for b in a + 1..count {
            if vertices[a].iter().any(|vertex| *vertex != u64::MAX && vertices[b].contains(vertex)) {
                let (root_a, root_b) = (find(&mut group, a), find(&mut group, b));
                group[root_a] = root_b;
            }
        }
    }
    let reduced: Vec<bool> = plan.stops.iter().map(|stops| stops.iter().any(|(_, r)| *r < radius)).collect();
    let relieved_group: Vec<bool> = (0..count)
        .map(|index| {
            let root = find(&mut group, index);
            (0..count).any(|other| reduced[other] && find(&mut group, other) == root)
        })
        .collect();
    let pick = |want: bool| -> (Vec<Vec3>, Option<Vec<String>>, Vec<Vec<(f64, f64)>>) {
        let indices: Vec<usize> = (0..count).filter(|index| relieved_group[*index] == want).collect();
        (
            indices.iter().map(|index| edge_points[*index]).collect(),
            edge_names.map(|names| indices.iter().filter_map(|index| names.get(*index).cloned()).collect()),
            indices.iter().map(|index| plan.stops[*index].clone()).collect(),
        )
    };
    let (constant_points, constant_names, _) = pick(false);
    let (relieved_points, relieved_names, relieved_stops) = pick(true);
    let mut body = solid.clone();
    if !constant_points.is_empty() {
        body = fillet_edges(&body, &constant_points, constant_names.as_deref(), radius, false, name)?;
    }
    let built = fillet_edges_variable_impl(
        &body,
        &relieved_points,
        relieved_names.as_deref(),
        &relieved_stops,
        false,
        name,
        "fillet_edges",
        false,
    )?;
    let issues = built.validate();
    if !issues.is_empty() {
        return Err(KernelRefusal::internal(
            KernelStage::Validate,
            "relief_validate",
            format!("radius relief: the relieved fillet does not validate: {issues:?}"),
        ));
    }
    Ok(built)
}

/// What a blend selection actually achieved.
///
/// The kernel can answer a group blend with fewer walls than were asked for —
/// the maximal-valid-subset search drops the edges that cannot co-blend at a
/// shared corner rather than failing outright. This says WHICH, so partial
/// fulfilment is a statement instead of a silence.
#[derive(Clone, Debug)]
pub struct BlendSelection {
    /// How many edges the caller selected.
    pub requested: usize,
    /// Indices into that selection that carry a blend wall in the result.
    pub applied: Vec<usize>,
    /// Indices that do not.
    pub rejected: Vec<usize>,
    /// Why the whole selection could not be built, when something was dropped.
    pub reason: Option<String>,
}

impl BlendSelection {
    /// Every requested edge carries a wall.
    pub fn is_complete(&self) -> bool {
        self.rejected.is_empty()
    }
}

/// [`fillet_edges`] with the selection outcome reported rather than refused.
///
/// The result is the same solid `fillet_edges` would build; the difference is
/// that an incomplete selection comes back as `Ok` with the misses named,
/// which is the explicit partial-result contract. Every named refusal that
/// applies to the WHOLE selection — mixed concavity, a radius the local wall
/// cannot hold, a mixed-convexity corner — is still an `Err`: dropping edges
/// cannot rescue those, so a partial answer would be a different defect.
/// [`fillet_edges_reported`] with the solid and the edge points translated by
/// `offset` for the operation and the result translated back. Refusal
/// messages name the translated coordinates.
pub fn fillet_edges_reported_at_offset(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    chamfer: bool,
    name: Option<&str>,
    offset: Vec3,
) -> Result<(BrepSolid, BlendSelection), KernelRefusal> {
    fillet_translated(solid, edge_points, edge_names, radius, chamfer, name, offset, crate::recentre::EntryKind::Control)
}

/// The translated run behind the control and the guard.
#[allow(clippy::too_many_arguments)]
fn fillet_translated(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    chamfer: bool,
    name: Option<&str>,
    offset: Vec3,
    kind: crate::recentre::EntryKind,
) -> Result<(BrepSolid, BlendSelection), KernelRefusal> {
    let _entry = crate::recentre::TranslatedEntry::open(kind);
    let translate = |solid: &BrepSolid, offset: Vec3| {
        crate::recentre::translate_solid(solid, offset).map_err(|message| {
            KernelRefusal::internal(
                KernelStage::Collect,
                "translation_control",
                format!("fillet_edges: the translation control could not move the solid: {message}"),
            )
        })
    };
    let moved = translate(solid, offset)?;
    let points: Vec<Vec3> = edge_points.iter().map(|point| point.add(offset)).collect();
    let (result, selection) = fillet_edges_reported(&moved, &points, edge_names, radius, chamfer, name)?;
    Ok((translate(&result, offset.scale(-1.0))?, selection))
}

pub fn fillet_edges_reported(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    chamfer: bool,
    name: Option<&str>,
) -> Result<(BrepSolid, BlendSelection), KernelRefusal> {
    // The translation control (`BREP_TRANSLATION_CONTROL`, `brep/recentre.rs`):
    // the outermost entry runs the fillet on a translated solid.
    if let Some(offset) = crate::recentre::control_offset_at_entry() {
        return fillet_edges_reported_at_offset(solid, edge_points, edge_names, radius, chamfer, name, offset);
    }
    // The recentre guard: a solid far from the origin in its own units
    // is filleted at the origin.
    let bounds = crate::recentre::solid_bounds(solid);
    if let Some(offset) = crate::recentre::guard_offset(bounds) {
        crate::recentre::profile_fired("fillet", offset, bounds);
        return fillet_translated(solid, edge_points, edge_names, radius, chamfer, name, offset, crate::recentre::EntryKind::Guard);
    }
    // One blend operation: the reports of an earlier one, never consumed, are
    // dropped as it starts (nested calls keep this operation's).
    let _operation = crate::blend::BlendOperation::enter();
    // Dropped before `_operation`, so the total is on the ledger when the
    // outermost operation prints it.
    let _total = crate::blend::BlendTimer::start("timing.total_ms");
    if !(radius > 0.0) || !radius.is_finite() {
        return Err(KernelRefusal::input(
            KernelStage::Collect,
            "radius",
            "fillet_edges: radius must be positive",
        ));
    }
    if edge_points.is_empty() {
        return Err(KernelRefusal::input(
            KernelStage::Collect,
            "selection",
            "fillet_edges: no edges selected",
        ));
    }
    // The rolling ball must reach both supports of every SELECTED edge — see
    // `check_support_extent`.  Checked ONCE, here, against the solid the
    // selection was made on: inside the group build the faces have already
    // been eaten into by earlier blends of the same selection, and a fillet
    // legitimately runs off that remainder (two rounds pinching on a shared
    // face).  Dropping edges cannot rescue an oversized radius either, so
    // this runs before the subset search rather than inside it.
    let entry = if chamfer { "chamfer_edges" } else { "fillet_edges" };
    let mut selected_ids = Vec::with_capacity(edge_points.len());
    if crate::blend::blend_operation_outermost() {
        super::analyze::clear_rollover_only();
    }
    let selection_checks = crate::blend::BlendTimer::start("timing.selection_checks_ms");
    let _boundaries = super::analyze::SupportBoundaryScope::open();
    for point in edge_points {
        let resolve = crate::blend::BlendTimer::start("timing.selection_checks.resolve_ms");
        let edge_id = resolve_edge_by_point(solid, *point)?;
        resolve.stop();
        let concavity = crate::blend::BlendTimer::start("timing.selection_checks.concavity_ms");
        check_mixed_concavity(solid, edge_id, entry)?;
        concavity.stop();
        let _extent = crate::blend::BlendTimer::start("timing.selection_checks.extent_ms");
        // A lone fillet edge may take the clipped concave fill (lane F row 5).
        if edge_points.len() == 1 && !chamfer {
            check_support_extent_clipping(solid, edge_id, radius, entry)?;
        } else {
            check_support_extent(solid, edge_id, radius, entry)?;
        }
        selected_ids.push(edge_id);
    }
    drop(_boundaries);
    selection_checks.stop();
    // A corner where the selection mixes convexity — a convex edge dying into
    // the concave edges it meets — has no rolling-ball closure at all: the
    // blends run out against each other partway along, which is the vertex
    // blend the stripe network does not construct yet.  It is TERMINAL, and
    // for the same reason as the two checks above it runs once, here: neither
    // the cutter composition nor the subset search below can rescue it.  The
    // cutter "succeeds" on this shape by shredding the solid (a sliver end cap
    // per corner, the carrier face split and renamed), which is worse than
    // the named refusal — see the 2026-09-09 rib-spine report.
    //
    // Nor can PARTITIONING the selection at the corner rescue it, which is why
    // this is still a refusal and not a split (measured 2026-09-13, the
    // partition record): the two convexity-pure halves each build on the
    // network, and composing them walls every selected edge and leaves every
    // other face's area identical to twelve digits — but the convex wall comes
    // out as the full untrimmed quarter-round prism (2·(1 − π/4)·r²·L to
    // 2.2e-10, with NO rolled-corner term) because its contact rail runs off
    // its own face at the TRI-TANGENT STATION and the surgery splices the
    // overhang in rather than clipping it. The faces it runs along come back
    // with self-crossing loops that `validate()`, `solid_self_intersections`,
    // the connectivity report and the Euler check all pass — and that
    // `check_loop_self_crossings` REFUSES since 9e50f6d55, so the split is not
    // a route to ten walls either: blend the concave half first and the
    // acceptance refuses every convex edge whose wall runs out into it — on
    // the rib both spines, at that same station, so that feature fails
    // outright; blend the convex half first and the maximal-valid-subset
    // search drops one ring edge and returns NINE walls of ten. The
    // construction that is missing is the runout itself: past that station the
    // ball rolls on one input face and on the concave stripe's own surface, so
    // the march needs a STRIPE as its second carrier.
    //
    // And more than that, measured 2026-09-13 (`blend/runout.rs`, which this
    // refusal now reports): the KEPT mate runs out too, at the mirror-image
    // station, so the FIRST carrier becomes the other concave stripe and the
    // last stretch rolls on two stripes; that stretch ends in a POLE where the
    // two stripes' rails on the face they share meet and the section shrinks
    // to a point. Two carrier switches and a degenerate terminus, not a march
    // with a stop.
    if !chamfer {
        // The class is BUILT where the runout composition applies — the ring
        // first, then one wall per spine from pole to pole (`blend/runout.rs`,
        // the 2026-09-30 record §5) — and refused by name, with the
        // composition's own reason appended, where it does not.
        let corner_analysis = crate::blend::BlendTimer::start("timing.corner_analysis_ms");
        let mut corners = crate::blend::mixed_convexity_corners(solid, &selected_ids, radius);
        corners.extend(crate::blend::runout::one_ended_runout_corners(solid, &selected_ids, radius));
        corner_analysis.stop();
        if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
            for corner in &corners {
                eprintln!(
                    "mixed corner at vertex {} ({:.4}, {:.4}, {:.4}): convex {:?} concave {:?}",
                    corner.vertex, corner.point.x, corner.point.y, corner.point.z,
                    corner.convex.iter().map(|index| selected_ids[*index]).collect::<Vec<_>>(),
                    corner.concave.iter().map(|index| selected_ids[*index]).collect::<Vec<_>>()
                );
            }
        }
        if !corners.is_empty() {
            // BREP lane L: begin walls_missing_tie_all_stations_20261015
            // Cleared on entry: only THIS composition's runout may set it.
            RUNOUT_RING_IS_NETWORK.with(|flag| flag.set(false));
            // BREP lane L: end walls_missing_tie_all_stations_20261015
            let composed = rounded_boss_composition(
                solid, &selected_ids, edge_points, edge_names, radius, name, &corners,
            ).or_else(|error| {
                if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                    eprintln!("rounded boss composition refused: {error}");
                }
                runout_composition(
                    solid, &selected_ids, edge_points, edge_names, radius, name, &corners, entry,
                )
            }).or_else(|error| {
                if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                    eprintln!("runout composition refused: {error}");
                }
                // The open-pad composition (2026-10-17 face-ring report): one
                // convex and one concave edge at each mixed corner over a
                // planar face, the concave pads continued onto the convex
                // walls. Declines by name on any other corner shape, so the
                // runout's refusal stands where it stood.
                open_pad_composition(
                    solid, &selected_ids, edge_points, edge_names, radius, name, &corners, entry,
                ).map_err(|pad_error| {
                    if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                        eprintln!("open-pad composition refused: {pad_error}");
                    }
                    // The decline is carried in the refusal's CLASS (`what`),
                    // never in its text: 2026-10-18 the text test let the
                    // decline replace the runout's named refusal on the
                    // nine-pick wedge (three lib tests).
                    let declined = matches!(&pad_error.class, crate::RefusalClass::UnsupportedGeometry { what } if what == "open_pad_shape");
                    if declined { error } else { pad_error }
                })
            }).or_else(|error| {
                // The cusp composition (reports 80c5d9d1 and 6dfd5a22): every
                // mixed corner a CUSP, the convex and concave edges leaving it
                // tangent. Declines by class on any other corner, and every
                // other refusal keeps the one before it: it is the last try.
                cusp_composition(solid, &selected_ids, edge_points, edge_names, radius, name, &corners).map_err(|cusp_error| {
                    if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                        eprintln!("cusp composition refused: {cusp_error}");
                    }
                    error
                })
            });
            match composed {
                Ok(result) => {
                    // BREP lane L: begin walls_missing_tie_all_stations_20261015
                    // The runout composition's walls are built and checked by
                    // construction; a station its OWN neighbouring wall consumes
                    // (a runout trimmed at an edge's end) is not a missing wall.
                    // Only on the NETWORK's ring: there every concave edge has its
                    // own wall by construction and a runout trims at most its
                    // ends; a ring the cutter fallback built keeps the strict
                    // first-station rule (the rounded boss path too).
                    let network_ring = RUNOUT_RING_IS_NETWORK.with(|flag| flag.replace(false));
                    // Integrator merge 2026-10-16 (lanes F + L): the flag above is consumed
                    // BEFORE lane F's exit acceptance can return early, so a refused result
                    // never leaves it set for the next call on this thread.
                    refuse_invalid_result(&result, entry)?;
                    let missing = walls_missing_ties(
                        solid, &result, edge_points, &(0..edge_points.len()).collect::<Vec<_>>(), radius, chamfer, entry, network_ring,
                    )?;
                    // BREP lane L: end walls_missing_tie_all_stations_20261015
                    if !missing.is_empty() {
                        return Err(KernelRefusal::internal(
                            KernelStage::Validate,
                            "runout_walls_missing",
                            format!(
                                "{entry}: the composition built a solid with no wall on {}",
                                missing.iter().map(|index| selection_label(*index, edge_points, edge_names, name)).collect::<Vec<_>>().join(", ")
                            ),
                        ));
                    }
                    let n = edge_points.len();
                    return Ok((
                        result,
                        BlendSelection { requested: n, applied: (0..n).collect(), rejected: Vec::new(), reason: None },
                    ));
                }
                Err(error) => {
                    if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                        eprintln!("runout composition refused: {error}");
                    }
                    check_mixed_corner_convexity(
                        solid, &selected_ids, edge_points, edge_names, radius, name, Some(&error),
                    )?;
                }
            }
        }
        // A selection whose corner setbacks consume a whole edge — the exact
        // degenerate radius, every face shrunk to a point — has no strip for
        // the network to build and nothing the cutter or the subset search
        // can rescue (the search answered the r = 10 cube with nine walls and
        // a 5174 volume against the sphere's 4189).  Terminal, by name, here.
        let corner_analysis = crate::blend::BlendTimer::start("timing.corner_analysis_ms");
        check_degenerate_corner_setbacks(
            solid,
            &selected_ids,
            edge_points,
            edge_names,
            radius,
            name,
        )?;
        corner_analysis.stop();
    }

    let n = edge_points.len();
    let complete = |applied: Vec<usize>| BlendSelection {
        requested: n,
        applied,
        rejected: Vec::new(),
        reason: None,
    };
    // Every subset below is built against the SAME solid, and a selection of
    // several components is composed one component at a time in input order,
    // so two subsets that keep the same leading components rebuild the same
    // intermediate solids. The memo keeps each component's result under the
    // chain of components composed before it, which is everything that result
    // depends on here (the solid, radius, chamfer and name are fixed for the
    // call), so the search returns what it did without the memo, only without
    // the rebuilding. On the 2026-09-26 cross-drilled sphere report four bored
    // rims were re-blended ahead of the same failing component in each of the
    // (up to 372) subsets tried: 13 min 25 s for one refusal, 34 s with it.
    let mut memo = ComponentMemo::default();
    let profile = crate::profile::profile_enabled();
    let group_started = web_time::Instant::now();
    let group = fillet_edges_group_in(
        solid,
        edge_points,
        edge_names,
        radius,
        chamfer,
        name,
        Some(&mut memo),
    );
    if profile {
        profile_line!(
            "blend.selection group edges={n} ok={} ms={:.3}",
            group.is_ok(),
            group_started.elapsed().as_secs_f64() * 1e3
        );
    }
    let group_err = match group {
        // A group solid that fails `validate()` is never returned: it becomes
        // the group's refusal, so the subset search below (which validates
        // every subset it builds) still gets its turn, and a selection with
        // nothing to drop refuses by that name.
        Ok(result) if refuse_invalid_result(&result, entry).is_err() => {
            refuse_invalid_result(&result, entry).unwrap_err()
        }
        Ok(result) => {
            // The group returned a solid, which is NOT the same as the group
            // having blended every edge.  `validate()` cannot tell the
            // difference — it checks incidence, and a solid missing a wall is
            // perfectly incident — so the walls are COUNTED.
            let missing =
                walls_missing(solid, &result, edge_points, &(0..n).collect::<Vec<_>>(), radius, chamfer, entry)?;
            if missing.is_empty() {
                return Ok((result, complete((0..n).collect())));
            }
            let applied: Vec<usize> = (0..n).filter(|index| !missing.contains(index)).collect();
            return Ok((
                result,
                BlendSelection {
                    requested: n,
                    applied,
                    rejected: missing,
                    reason: None,
                },
            ));
        }
        Err(error) => error,
    };
    // A wall that FOLDS THROUGH ITSELF is terminal for the whole selection, and
    // for the reason the two checks above are: dropping edges cannot rescue it.
    // The subset search would happily answer this document's two collars with
    // the ONE that does not fold, and return a solid whose volume is a blend
    // short — the same silent partial `both_collars_add_independently` exists
    // to catch.  Reported by name instead (`blend/fold.rs`).
    // A miter past the sharp edge's far vertex needs support-face switching,
    // not a smaller selection. Do not let the subset search mask the limit.
    // Nor can dropping another edge close a tangential end whose taper failed.
    if crate::blend::is_wall_fold(&group_err) || crate::blend::is_miter_support_overrun(&group_err)
        || crate::blend::is_tangential_transverse_end(&group_err)
    {
        return Err(group_err);
    }
    // Fewer than two edges: nothing to drop, so the group error is final.
    // Cap the combinatorial search so a large malformed selection cannot
    // explode (the full group carries the common case; the search is a
    // rare fallback).
    if n < 2 || n > 12 {
        return Err(group_err);
    }
    // Drop the fewest edges first (largest surviving subset), trying the
    // drop-sets in lexicographic order so the result is deterministic.
    // Take the first subset that blends to a valid solid IN WHICH EVERY KEPT
    // EDGE GREW A WALL: a subset that itself silently drops one of its own
    // edges is not the maximal subset, it is a smaller one wearing a larger
    // one's label.
    let max_drop = SUBSET_DROP_CAP.with(|cap| cap.get()).map_or(n - 1, |cap| cap.min(n - 1));
    for drop in 1..=max_drop {
        for dropped in index_combinations(n, drop) {
            let keep: Vec<usize> = (0..n).filter(|i| !dropped.contains(i)).collect();
            let points: Vec<Vec3> = keep.iter().map(|i| edge_points[*i]).collect();
            // Subset the per-edge blend-face names with the IDENTICAL
            // drop-set so `kept_names[k]` still names `kept[k]`.
            let kept_names: Option<Vec<String>> =
                edge_names.map(|names| keep.iter().map(|i| names[*i].clone()).collect());
            crate::blend::record_count("count.subset.builds", 1);
            let subset_started = web_time::Instant::now();
            let subset = fillet_edges_group_in(
                solid,
                &points,
                kept_names.as_deref(),
                radius,
                chamfer,
                name,
                Some(&mut memo),
            );
            if profile {
                profile_line!(
                    "blend.selection subset dropped={dropped:?} ok={} ms={:.3}",
                    subset.is_ok(),
                    subset_started.elapsed().as_secs_f64() * 1e3
                );
            }
            let result = match subset {
                Ok(result) => result,
                Err(error) => {
                    if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                        eprintln!("subset dropped={dropped:?} refused: {error}");
                    }
                    continue;
                }
            };
            if !result.validate().is_empty() {
                continue;
            }
            if !walls_missing(solid, &result, edge_points, &keep, radius, chamfer, entry)?.is_empty() {
                continue;
            }
            return Ok((
                result,
                BlendSelection {
                    requested: n,
                    applied: keep,
                    rejected: dropped,
                    reason: Some(group_err.message),
                },
            ));
        }
    }
    Err(group_err)
}

/// Lane R1: [`walls_missing`] reads a station's side off its nearest face
/// before casting a ray. On by default (promoted 2026-10-19 on the fixed
/// reader: 98 + 144 census 0 differ); `BREP_WALLS_MISSING_NEAREST=0` (or
/// `off`) casts the ray at every station.
pub(crate) fn walls_missing_nearest_on() -> bool {
    switch!("BREP_WALLS_MISSING_NEAREST", true)
}


/// Which of `selected` (indices into the caller's own selection) carry NO blend
/// wall in `result` — decided by the MATERIAL, not by a name.
///
/// A blended edge is an edge whose material changed: a convex blend removes
/// the sliver under the edge, so a point of the original edge ends up OUTSIDE
/// the answer; a concave one pads the crease, so it ends up INSIDE. An edge
/// the selection dropped is still the sharp edge it was, and its points still
/// lie ON the skin. That statement holds for every caller and every lane,
/// which the previous attribution did not: it looked for one face per edge
/// carrying the caller's per-edge name, so a caller passing `None` (the wasm
/// ABI, `fillet_edges_json`) got NO count at all, and a caller passing names
/// was told a wall was missing where the network had merged two coaxial walls
/// into one face (the U-channel at half its width keeps one name for two
/// edges) or collapsed the walls of a full-width star into its corner patch
/// (`{base}:CORNER:…`, and no wall name at all). Both were measured
/// 2026-09-26; the named path REFUSED those complete blends.
///
/// The classifier answers `On` inside a band, so the check states how far
/// the blend of THIS size moves the edge at each probed station — `r·(1/cos(α/2)
/// − 1)` for a fillet and `d·sin(α/2)` for a chamfer, `α` the angle between the
/// two faces' outward normals there — and probes the station the blend moves
/// MOST. An edge that is tangent at one end (the miter seam two rounds leave
/// runs to 180° at its pole) is still attributed where it is sharp. Only when
/// no station clears the band is the selection refused by name, rather than
/// an unseeable move read as a missing wall.
pub(super) fn walls_missing(
    original: &BrepSolid,
    result: &BrepSolid,
    edge_points: &[Vec3],
    selected: &[usize],
    size: f64,
    chamfer: bool,
    entry: &str,
) -> Result<Vec<usize>, KernelRefusal> {
    walls_missing_ties(original, result, edge_points, selected, size, chamfer, entry, false)
}

// BREP lane L: begin walls_missing_tie_all_stations_20261015
/// [`walls_missing`], with `tie_majority`: on a clearance TIE (a uniform
/// dihedral, every station moving the same) the first station stands for
/// all of them unless this is set, when a wall is missing only if MORE THAN
/// HALF of the tied stations still lie on the skin (an unwalled edge whose
/// two ends neighbouring walls consume — the wedge nine-pick's unselected
/// side A ∩ wall — still reads missing; "every tied station" did not).  Only the runout composition sets it: its
/// own neighbouring wall may consume an edge's end (the rib nine-edge
/// selection's trimmed runout over the wall ∩ roof pad's first tenth read
/// as that pad having no wall), while elsewhere the first station is what
/// catches a cutter composition's wrong corners (the one-ended channel with
/// its runout switched off: stations ON at both ends of an edge whose middle
/// is walled — a measured 1253.219 against its 1253.373 body) and a station
/// reading alone cannot tell the two apart.
#[allow(clippy::too_many_arguments)]
pub(super) fn walls_missing_ties(
    original: &BrepSolid,
    result: &BrepSolid,
    edge_points: &[Vec3],
    selected: &[usize],
    size: f64,
    chamfer: bool,
    entry: &str,
    tie_majority: bool,
) -> Result<Vec<usize>, KernelRefusal> {
// BREP lane L: end walls_missing_tie_all_stations_20261015
    use crate::{PointClass, SolidClassifier};
    if selected.is_empty() {
        return Ok(Vec::new());
    }
    let _timing = crate::blend::BlendTimer::start("timing.walls_missing_ms");
    crate::classification::classify_profile_begin();
    crate::classification::containment_profile_begin();
    let band = crate::KernelTolerances::for_solid(result, 1e-7).model;
    // `classify` reads `On` within ten times its band; attribution needs the
    // move to clear that by the same margin again.
    let visible = band * 100.0;
    let build_timing = crate::blend::BlendTimer::start("timing.walls_missing.classifier_ms");
    let classifier = SolidClassifier::new(result, band)
        .or_refuse(KernelStage::Validate, "wall_classifier")?;
    // Lane R1 (`BREP_WALLS_MISSING_NEAREST`): a station's On-or-not is read
    // off its nearest face first (`NearestFaceSide`, the classifier's own On
    // band of ten `band`s, answering only where the read is rigorous); the
    // ray answers whatever it declines.
    let nearest = if walls_missing_nearest_on() {
        Some(super::analyze::NearestFaceSide::new(result, band).or_refuse(KernelStage::Validate, "wall_nearest_face")?)
    } else {
        None
    };
    let reads_on = |point: Vec3| -> Result<bool, KernelRefusal> {
        if let Some(class) = nearest.as_ref().and_then(|nearest| nearest.side(point)) {
            crate::blend::record_count("count.walls_missing.nearest", 1);
            return Ok(class == PointClass::On);
        }
        Ok(classifier.classify(point).or_refuse(KernelStage::Validate, "wall_classify")?.class == PointClass::On)
    };
    build_timing.stop();
    const STATIONS: usize = 9;
    let mut missing = Vec::new();
    for &index in selected {
        let point = edge_points[index];
        let edge_id = resolve_edge_by_point(original, point)?;
        let (edge, faces) = super::analyze::edge_and_faces(original, edge_id)?;
        // The station the blend moves most, and how far.
        let mut best: Option<(Vec3, f64)> = None;
        // BREP lane L: begin walls_missing_tie_all_stations_20261015
        let mut read: Vec<(Vec3, f64)> = Vec::with_capacity(STATIONS);
        // BREP lane L: end walls_missing_tie_all_stations_20261015
        for step in 1..=STATIONS {
            let t = edge.t0 + (edge.t1 - edge.t0) * step as f64 / (STATIONS + 1) as f64;
            let Ok(station) = edge.curve.evaluate(t) else {
                continue;
            };
            let clearance = match faces[..] {
                [first, second] => {
                    let (Ok((_, a)), Ok((_, b))) = (
                        super::analyze::face_plane_normal(first, station),
                        super::analyze::face_plane_normal(second, station),
                    ) else {
                        continue;
                    };
                    let half: f64 = 0.5 * a.dot(b).clamp(-1.0, 1.0).acos();
                    if chamfer {
                        size * half.sin()
                    } else {
                        size * (1.0 / half.cos().max(1e-12) - 1.0)
                    }
                }
                // No dihedral to read: probe anyway, unguarded.
                _ => f64::INFINITY,
            };
            if best.map_or(true, |(_, known)| clearance > known) {
                best = Some((station, clearance));
            }
            // BREP lane L: begin walls_missing_tie_all_stations_20261015
            read.push((station, clearance));
            // BREP lane L: end walls_missing_tie_all_stations_20261015
        }
        let Some((station, clearance)) = best else {
            return Err(KernelRefusal::internal(
                KernelStage::Validate,
                "wall_attribution_stations",
                format!(
                    "{entry}: cannot attribute a wall to edge #{index} at ({:.6}, {:.6}, {:.6}) — no \
                     station of it could be evaluated",
                    point.x, point.y, point.z
                ),
            ));
        };
        if !(clearance > visible) {
            // The faces themselves put the move under the band: the input
            // leaves the attribution undecidable, and it is refused rather
            // than guessed.
            return Err(KernelRefusal::ill_posed(
                KernelStage::Validate,
                "wall_attribution_band",
                format!(
                    "{entry}: cannot attribute a wall to edge #{index} at ({:.6}, {:.6}, {:.6}) — \
                     its faces are so nearly tangent that a size-{size} blend moves the edge by at \
                     most {clearance:.3e}, inside the {visible:.3e} the classifier can see; the \
                     selection is refused rather than reported on a guess",
                    point.x, point.y, point.z
                ),
            ));
        }
        let classify_timing = crate::blend::BlendTimer::start("timing.walls_missing.classify_ms");
        // The question is On or not On, never In against Out: a station the
        // certificate below puts farther than `visible` from every face of
        // the result cannot read On by the classifier's band pass (a foot on
        // a carrier within ten bands) nor by its ray pass (a hit within ten
        // bands), so it is not asked. Anything the certificate declines is
        // classified as before.
        let certified = station_clear_of_skin(result, station, band * 10.0, visible)
            || station_clear_of_trimmed_skin(result, station, band * 10.0, visible);
        let on = if certified {
            crate::blend::record_count("count.walls_missing.certified", 1);
            false
        } else {
            crate::blend::record_count("count.walls_missing.classified", 1);
            reads_on(station)?
        };
        classify_timing.stop();
        // BREP lane L: begin walls_missing_tie_all_stations_20261015
        let on = on && (!tie_majority || {
            // The most-moved station read On; on a tie, so must more than
            // half of the tied stations (itself included).
            let tolerance = 1e-12 * clearance.abs().max(1.0);
            let (mut tied, mut on_skin) = (1usize, 1usize);
            for (other, other_clearance) in read.iter().copied() {
                if other.sub(station).length() == 0.0 || clearance - other_clearance > tolerance {
                    continue;
                }
                tied += 1;
                if !station_clear_of_skin(result, other, band * 10.0, visible) && reads_on(other)? {
                    on_skin += 1;
                }
            }
            2 * on_skin > tied
        });
        // BREP lane L: end walls_missing_tie_all_stations_20261015
        if on {
            missing.push(index);
        }
    }
    crate::classification::classify_profile_report("blend.walls_missing");
    crate::classification::containment_profile_report("blend.walls_missing");
    Ok(missing)
}



/// Is `station` certainly farther than `margin` from every face of `result`,
/// trims included, read without a classifier query? `true` only when every
/// face is certified by one of three readings, each a lower bound on the
/// distance from the station to that face's trimmed carrier:
///
/// - its control-point box, widened by `on_band`, does not hold the station
///   (the carrier lies in the hull of positively weighted control points);
/// - its carrier is a recognised revolution (`NurbsSurface::analytic`,
///   matched to 1e-9 relative): a sphere or torus farther than `margin` in
///   closed form; a cone, cylinder or straight-generatrix partial sweep
///   whose PATCH is farther than `margin` by the larger of three lower
///   bounds, the meridian distance to the generatrix line, the axial
///   clearance from the generatrix's slab and the clearance from the
///   sweep's wedge (a station on one box edge lies ON the infinite cylinder
///   of each neighbouring wall, a radius from its centre line, and is a
///   radius short of its slab);
/// - its carrier is a recognised plane, the station's foot on it is
///   `Outside` the trims (`parameter_point_in_face`, the kernel's own lane
///   at the band the classifier would use), every loop pcurve is a straight
///   two-point unit-weight chord (so its image under the affine chart is a
///   straight 3-D segment, and the trim polygon is exact), and the station
///   is farther than `margin` from every chord's segment: a foot outside the
///   region is nearer to the region's boundary than to any of its points.
///
/// Any face none of the three reads (an unrecognised carrier, a plane the
/// station lies on whose trims are curved, a foot inside or on the trims)
/// declines the whole station to the classifier. The recogniser's 1e-9
/// relative bar sits four orders below the `margin` the caller passes
/// (a hundred model bands), which is what absorbs it.
fn station_clear_of_skin(result: &BrepSolid, station: Vec3, on_band: f64, margin: f64) -> bool {
    use crate::AnalyticSurface;
    // The station in a revolution frame: (ρ, h) in the meridian plane and
    // its azimuth θ in [0, 2π) from the frame's x axis.
    let meridian = |frame: &crate::RevolutionFrame| -> (f64, f64, f64) {
        let offset = station.sub(frame.origin);
        let axial = offset.dot(frame.axis);
        let radial = offset.sub(frame.axis.scale(axial));
        let mut azimuth = radial.dot(frame.y_axis).atan2(radial.dot(frame.x_axis));
        if azimuth < 0.0 {
            azimuth += std::f64::consts::TAU;
        }
        (radial.length(), axial, azimuth)
    };
    // A patch of a revolution lies inside the axial slab of its generatrix
    // and inside the wedge of its sweep, so the station's clearance from
    // either is a lower bound on its distance to the patch.
    let axial_clearance = |axial: f64, h0: f64, h1: f64| (h0.min(h1) - axial).max(axial - h0.max(h1)).max(0.0);
    let wedge_clearance = |radial: f64, azimuth: f64, sweep: f64| -> f64 {
        if sweep >= std::f64::consts::TAU || azimuth <= sweep {
            return 0.0;
        }
        // Outside the sweep: the angle to the nearer bounding meridian.
        let turn = (azimuth - sweep).min(std::f64::consts::TAU - azimuth);
        if turn >= std::f64::consts::FRAC_PI_2 { radial } else { radial * turn.sin() }
    };
    // Distance in the meridian plane from (ρ, h) to the LINE through two
    // generatrix points: a lower bound for a cone or cylinder's carrier.
    let line_distance = |radial: f64, axial: f64, (rho0, h0): (f64, f64), (rho1, h1): (f64, f64)| -> Option<f64> {
        let (du, dv) = (rho1 - rho0, h1 - h0);
        let length = (du * du + dv * dv).sqrt();
        (length > 0.0).then(|| ((radial - rho0) * dv - (axial - h0) * du).abs() / length)
    };
    let clear = |distance: f64| distance.is_finite() && distance > margin;
    for face in result.shells.iter().flat_map(|shell| &shell.faces) {
        let Ok(controls) = crate::Aabb::from_surface_controls(&face.surface) else { return false };
        if !controls.expanded(on_band).contains(station) {
            continue;
        }
        let certified = match face.surface.analytic() {
            Some(AnalyticSurface::RuledRevolution { frame, rho0, rho1, height }) => {
                let (radial, axial, _) = meridian(frame);
                let Some(line) = line_distance(radial, axial, (*rho0, 0.0), (*rho1, *height)) else { return false };
                clear(line.max(axial_clearance(axial, 0.0, *height)))
            }
            Some(AnalyticSurface::Sphere { frame, radius }) => clear((station.sub(frame.origin).length() - radius).abs()),
            Some(AnalyticSurface::Torus { frame, major_radius, minor_radius }) => {
                let (radial, axial, _) = meridian(frame);
                let tube = radial - major_radius;
                clear(((tube * tube + axial * axial).sqrt() - minor_radius).abs())
            }
            Some(AnalyticSurface::Revolution { frame, sweep, generatrix, .. }) => {
                // A STRAIGHT generatrix in the frame's zero meridian (the
                // extruded-arc walls: a line parallel to the axis at the
                // radius): its two control points read as (ρ, h), and the
                // patch's distance is bounded below by the meridian line
                // distance, the axial clearance and the wedge clearance
                // together. A curved or off-meridian generatrix declines.
                let [a, b] = generatrix.control_points.as_slice() else { return false };
                if generatrix.degree != 1 || a.w != 1.0 || b.w != 1.0 {
                    return false;
                }
                let meridian_of = |point: Vec3| -> Option<(f64, f64)> {
                    let offset = point.sub(frame.origin);
                    let axial = offset.dot(frame.axis);
                    let radial = offset.sub(frame.axis.scale(axial));
                    let rho = radial.length();
                    let on_meridian = radial.dot(frame.y_axis).abs() <= 1e-9 * rho.max(1.0) && radial.dot(frame.x_axis) >= -1e-9 * rho.max(1.0);
                    on_meridian.then_some((rho, axial))
                };
                let (Some(p0), Some(p1)) = (meridian_of(Vec3::new(a.x, a.y, a.z)), meridian_of(Vec3::new(b.x, b.y, b.z))) else {
                    return false;
                };
                let (radial, axial, azimuth) = meridian(frame);
                let Some(line) = line_distance(radial, axial, p0, p1) else { return false };
                // The TRIMMED face's slab is narrower than the surface's:
                // a station on one box edge lies on the untrimmed surface
                // of each perpendicular wall (that wall's rail line runs
                // through the corner), a radius short of its trimmed face.
                let slab = trimmed_axial_slab(face, frame).unwrap_or((p0.1, p1.1));
                clear(line.max(axial_clearance(axial, slab.0, slab.1)).max(wedge_clearance(radial, azimuth, *sweep)))
            }
            Some(plane @ AnalyticSurface::Plane { origin, u_dir, v_dir, .. }) => {
                let Ok(normal) = u_dir.cross(*v_dir).normalized() else { return false };
                if clear(station.sub(*origin).dot(normal).abs()) {
                    true
                } else {
                    planar_trims_clear_of(face, plane, station, on_band, margin)
                }
            }
            None => false,
        };
        if !certified {
            return false;
        }
    }
    true
}

/// A tighter sufficient Not-On certificate. General carriers use subdivided
/// positive-weight hulls. Analytic carriers retain the projector and trim
/// band used by the nearest-face reader; an exterior foot also needs every
/// exact boundary edge farther than the existing attribution margin.
/// Any inconclusive read leaves the original nearest/ray query in charge.
fn station_clear_of_trimmed_skin(result: &BrepSolid, point: Vec3, on_band: f64, margin: f64) -> bool {
    let _timer = crate::blend::BlendTimer::start("timing.walls_missing.trimmed_bound_ms");
    if !point.x.is_finite() || !point.y.is_finite() || !point.z.is_finite()
        || !margin.is_finite() || !(margin > on_band && on_band > 0.0)
    { return false; }
    for face in result.shells.iter().flat_map(|shell| &shell.faces) {
        let Ok(bounds) = crate::Aabb::from_surface_controls(&face.surface) else { return false; };
        if !bounds.expanded(margin).contains(point) { continue; }
        if face.surface.analytic().is_none() {
            if general_carrier_clear(&face.surface, point, margin) { continue; }
            return false;
        }
        // Within `margin`, these regular carriers have a unique nearest
        // foot; every other stationary foot is farther than the margin.
        // The sphere/torus medial sets and spindle/horn tori decline. Thus
        // an exterior nearest foot can meet a trimmed region within the
        // margin only by crossing a boundary edge inside that same ball.
        match face.surface.analytic() {
            Some(crate::AnalyticSurface::Plane { .. }) => {},
            Some(crate::AnalyticSurface::Sphere { radius, .. }) if *radius > 4.0 * margin => {},
            Some(crate::AnalyticSurface::Torus { major_radius, minor_radius, .. })
                if *minor_radius > 4.0 * margin && *major_radius > *minor_radius + 4.0 * margin => {},
            _ => return false,
        }
        let Ok(projection) = crate::project_point_to_surface(&face.surface, point) else { return false; };
        if !projection.distance.is_finite() { return false; }
        if projection.distance > margin { continue; }
        let band = crate::classification::surface_uv_band(&face.surface, projection.u, projection.v, on_band);
        let Ok(inside) = crate::parameter_point_in_face(face, crate::Vec2 { x: projection.u, y: projection.v }, band) else { return false; };
        if inside != crate::PolygonClass::Outside { return false; }
        let mut edges = 0;
        for coedge in face.loops.iter().flat_map(|lp| &lp.coedges) {
            let Some(edge) = result.edges.iter().find(|edge| edge.id == coedge.edge_id) else { return false; };
            let Some(piece) = super::analyze::edge_piece(&edge.curve, edge.t0.min(edge.t1), edge.t0.max(edge.t1)) else { return false; };
            let Ok(foot) = crate::project_point_to_curve(&piece, point) else { return false; };
            if !foot.distance.is_finite() || foot.distance <= margin { return false; }
            edges += 1;
        }
        if edges == 0 { return false; }
    }
    true
}

/// A sufficient clearance test for a general carrier, without finding its
/// nearest point. Positive-weight Bezier patches lie inside their control
/// hulls. Decompose both clamped knot vectors, then bisect only boxes still
/// within `margin` of the query. Any unsupported net or exhausted work budget
/// declines to the existing classifier; no approximate distance is accepted.
fn general_carrier_clear(surface: &crate::NurbsSurface, point: Vec3, margin: f64) -> bool {
    use crate::Vec4;
    let rows = surface.control_points.len();
    let columns = surface.control_points.first().map_or(0, Vec::len);
    if rows == 0 || columns == 0 || rows.saturating_mul(columns) > 4096
        || surface.degree_u > 16 || surface.degree_v > 16 || surface.control_points.iter().any(|row| row.len() != columns)
        || !margin.is_finite() || margin <= 0.0
        || !point.x.is_finite() || !point.y.is_finite() || !point.z.is_finite()
    { return false; }
    let mut scale = 1.0_f64;
    for control in surface.control_points.iter().flatten() {
        if !(control.w > 0.0) || !control.w.is_finite() { return false; }
        let Ok(p) = control.point() else { return false; };
        if !p.x.is_finite() || !p.y.is_finite() || !p.z.is_finite() { return false; }
        scale = scale.max(p.x.abs()).max(p.y.abs()).max(p.z.abs());
    }
    // Leave a rounding cushion far below the caller's existing model margin.
    // Unsupported scales decline rather than consume the classification band.
    let roundoff = 256.0 * f64::EPSILON * scale * (surface.degree_u + 1) as f64 * (surface.degree_v + 1) as f64;
    if roundoff >= margin * 0.01 { return false; }
    let zero = Vec4 { x: 0.0, y: 0.0, z: 0.0, w: 0.0 };
    let blend = |a: Vec4, b: Vec4, alpha: f64| a.scale(alpha).add(b.scale(1.0 - alpha));
    let mut fibres = Vec::with_capacity(columns);
    for column in 0..columns {
        let controls: Vec<_> = surface.control_points.iter().map(|row| row[column]).collect();
        let Some(spans) = crate::curve::bezier_decompose_raw(&surface.knots_u, &controls, surface.degree_u, zero, blend) else { return false; };
        fibres.push(spans);
    }
    let mut budget = 256usize;
    for u_span in 0..fibres[0].len() {
        let mut across = Vec::with_capacity(surface.degree_u + 1);
        for row in 0..=surface.degree_u {
            let controls: Vec<_> = fibres.iter().map(|fibre| fibre[u_span][row]).collect();
            let Some(spans) = crate::curve::bezier_decompose_raw(&surface.knots_v, &controls, surface.degree_v, zero, blend) else { return false; };
            across.push(spans);
        }
        for v_span in 0..across[0].len() {
            let net = across.iter().map(|row| row[v_span].clone()).collect();
            if !bezier_box_clear(net, point, margin + roundoff, 0, &mut budget) { return false; }
        }
    }
    true
}

fn bezier_box_clear(net: Vec<Vec<crate::Vec4>>, point: Vec3, margin: f64, depth: usize, budget: &mut usize) -> bool {
    if *budget == 0 { return false; }
    *budget -= 1;
    let mut low = Vec3::new(f64::INFINITY, f64::INFINITY, f64::INFINITY);
    let mut high = Vec3::new(f64::NEG_INFINITY, f64::NEG_INFINITY, f64::NEG_INFINITY);
    for control in net.iter().flatten() {
        if !(control.w > 0.0) || !control.w.is_finite() { return false; }
        let Ok(p) = control.point() else { return false; };
        if !p.x.is_finite() || !p.y.is_finite() || !p.z.is_finite() { return false; }
        low.x = low.x.min(p.x); low.y = low.y.min(p.y); low.z = low.z.min(p.z);
        high.x = high.x.max(p.x); high.y = high.y.max(p.y); high.z = high.z.max(p.z);
    }
    // One separating coordinate is enough, and avoids a rounded square root.
    if point.x < low.x - margin || point.x > high.x + margin
        || point.y < low.y - margin || point.y > high.y + margin
        || point.z < low.z - margin || point.z > high.z + margin
    { return true; }
    if depth == 10 { return false; }
    let rows = net.len();
    let columns = net[0].len();
    let mut left = net.clone();
    let mut right = net.clone();
    // Alternate directions: every child covers exactly half the parent's
    // parameter rectangle; both must prove clearance.
    if depth % 2 == 0 {
        for column in 0..columns {
            let mut work: Vec<_> = net.iter().map(|row| row[column]).collect();
            for level in 1..rows {
                for i in 0..rows - level { work[i] = work[i].scale(0.5).add(work[i + 1].scale(0.5)); }
                left[level][column] = work[0];
                right[rows - level - 1][column] = work[rows - level - 1];
            }
        }
    } else {
        for row in 0..rows {
            let mut work = net[row].clone();
            for level in 1..columns {
                for i in 0..columns - level { work[i] = work[i].scale(0.5).add(work[i + 1].scale(0.5)); }
                left[row][level] = work[0];
                right[row][columns - level - 1] = work[columns - level - 1];
            }
        }
    }
    bezier_box_clear(left, point, margin, depth + 1, budget)
        && bezier_box_clear(right, point, margin, depth + 1, budget)
}

/// The axial range `(h_low, h_high)` of a face's TRIMMED region on a
/// revolution whose generatrix is straight: the region lies in the uv box
/// of its straight-chord loops (every pcurve a two-point unit-weight
/// chord), the surface is a tensor product in which the axial coordinate
/// `h = (S − origin)·axis` depends on one parameter only and affinely
/// (degree 1, two control points along it), so `h` over the box is the
/// range between its values at the box's corners. `None` when any of that
/// is not read: a curved chord, an `h` that moves along both parameters,
/// a degree or control count that is not affine, an evaluation that fails.
fn trimmed_axial_slab(face: &FaceRecord, frame: &crate::RevolutionFrame) -> Option<(f64, f64)> {
    let (mut u_low, mut u_high, mut v_low, mut v_high) = (f64::INFINITY, f64::NEG_INFINITY, f64::INFINITY, f64::NEG_INFINITY);
    for coedge in face.loops.iter().flat_map(|loop_record| &loop_record.coedges) {
        let pcurve = &coedge.pcurve;
        let [a, b] = pcurve.control_points.as_slice() else { return None };
        if pcurve.degree != 1 || a.w != 1.0 || b.w != 1.0 {
            return None;
        }
        for control in [a, b] {
            u_low = u_low.min(control.x);
            u_high = u_high.max(control.x);
            v_low = v_low.min(control.y);
            v_high = v_high.max(control.y);
        }
    }
    if !(u_low <= u_high && v_low <= v_high) {
        return None;
    }
    let axial_at = |u: f64, v: f64| face.surface.evaluate(u, v).ok().map(|point| point.sub(frame.origin).dot(frame.axis));
    let corners = [axial_at(u_low, v_low)?, axial_at(u_high, v_low)?, axial_at(u_low, v_high)?, axial_at(u_high, v_high)?];
    let scale = corners.iter().fold(1.0_f64, |scale, h| scale.max(h.abs()));
    let same = |a: f64, b: f64| (a - b).abs() <= 1e-9 * scale;
    let surface = &face.surface;
    // h constant along u (the generatrix runs along v), or along v.
    let affine_in_v = same(corners[0], corners[1]) && same(corners[2], corners[3])
        && surface.degree_v == 1 && surface.control_points.first().is_some_and(|row| row.len() == 2);
    let affine_in_u = same(corners[0], corners[2]) && same(corners[1], corners[3])
        && surface.degree_u == 1 && surface.control_points.len() == 2;
    if !(affine_in_v || affine_in_u) {
        return None;
    }
    let low = corners.iter().copied().fold(f64::INFINITY, f64::min);
    let high = corners.iter().copied().fold(f64::NEG_INFINITY, f64::max);
    (low.is_finite() && high.is_finite()).then_some((low, high))
}

/// The planar reading of [`station_clear_of_skin`]: the station's foot on
/// the plane lies `Outside` the face's trims and every straight loop chord's
/// 3-D segment is farther than `margin` from the station.
fn planar_trims_clear_of(
    face: &FaceRecord,
    plane: &crate::AnalyticSurface,
    station: Vec3,
    on_band: f64,
    margin: f64,
) -> bool {
    use crate::PolygonClass;
    let Some(foot) = plane.project(&face.surface, station) else { return false };
    let uv_band = crate::classification::surface_uv_band(&face.surface, foot.u, foot.v, on_band);
    match crate::parameter_point_in_face(face, crate::Vec2 { x: foot.u, y: foot.v }, uv_band) {
        Ok(PolygonClass::Outside) => {}
        _ => return false,
    }
    for coedge in face.loops.iter().flat_map(|loop_record| &loop_record.coedges) {
        let pcurve = &coedge.pcurve;
        let [a, b] = pcurve.control_points.as_slice() else { return false };
        if pcurve.degree != 1 || a.w != 1.0 || b.w != 1.0 {
            return false;
        }
        let (Ok(start), Ok(end)) = (face.surface.evaluate(a.x, a.y), face.surface.evaluate(b.x, b.y)) else { return false };
        let chord = end.sub(start);
        let length_squared = chord.dot(chord);
        let offset = station.sub(start);
        let along = if length_squared > 0.0 { (offset.dot(chord) / length_squared).clamp(0.0, 1.0) } else { 0.0 };
        let distance = offset.sub(chord.scale(along)).length();
        if !(distance.is_finite() && distance > margin) {
            return false;
        }
    }
    true
}

/// The blend-FACE name for the `i`-th selected edge: its per-edge name when the
/// caller supplied the parallel `edge_names` (feature path — each wall named
/// after its originating edge, `{fid}:BLEND:{edge}`), else the single base
/// `name` for every wall (the legacy/test path, byte-identical to before).
fn per_edge_name<'a>(
    edge_names: Option<&'a [String]>,
    base: Option<&'a str>,
    i: usize,
) -> Option<&'a str> {
    match edge_names {
        Some(names) => names.get(i).map(|value| value.as_str()),
        None => base,
    }
}

/// The star-corner patch name: `{base}:CORNER:{sorted+join of adjacent edge
/// names}` when per-edge names were supplied (feature path), else the single
/// `base` (legacy/test path — the corner keeps the wall name, pre-change
/// behavior).  `base` is `{fid}:BLEND` and each `edge_names[i]` is the composed
/// `{fid}:BLEND:{edge}`, so stripping the `{base}:` prefix recovers the bare
/// originating-edge name for the join.  Unique per corner: two distinct corners
/// never share the same set of >=3 selected edges.
fn corner_face_name(
    edge_names: Option<&[String]>,
    base: Option<&str>,
    adjacent: &[usize],
) -> Option<String> {
    match (edge_names, base) {
        (Some(names), Some(base)) => {
            let prefix = format!("{base}:");
            let mut raws: Vec<&str> = adjacent
                .iter()
                .filter_map(|&i| names.get(i))
                .map(|composed| composed.strip_prefix(&prefix).unwrap_or(composed.as_str()))
                .collect();
            raws.sort_unstable();
            raws.dedup();
            Some(format!("{base}:CORNER:{}", raws.join("+")))
        }
        _ => base.map(|value| value.to_string()),
    }
}

/// Round a connected group of convex edges first, including its ordinary
/// convex corner patches, then blend the concave rim across a curved bridge
/// at each transverse mixed corner. The spine's carriers and the cap may be
/// planes or curved analytic faces alike: transversality is read from the
/// three normals at the corner, and the rim network marches the bridge on the
/// actual carriers. Without these bridges the rim stripes would remain
/// disconnected. Redispatch the remaining selection: other mixed corners may
/// need another convex group or a pole-to-pole runout.
/// Each step removes the rounded group from the pending selection.
#[allow(clippy::too_many_arguments)]
fn rounded_boss_composition(
    solid: &BrepSolid,
    selected_ids: &[u64],
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    name: Option<&str>,
    corners: &[crate::blend::MixedCorner],
) -> Result<BrepSolid, KernelRefusal> {
    // The legacy API gives every wall the same base name (or none). Use
    // distinct temporary identities through the recursive composition so a
    // bridge can be attributed to its own convex wall. Restore that API's
    // naming contract after the whole selection has been built.
    if edge_names.is_none() {
        let prefix = "FILLET:COMPOSITION";
        let names: Vec<_> = selected_ids
            .iter()
            .map(|id| format!("{prefix}:{id}"))
            .collect();
        let mut result = rounded_boss_composition(
            solid,
            selected_ids,
            edge_points,
            Some(&names),
            radius,
            Some(prefix),
            corners,
        )?;
        for face in result.shells.iter_mut().flat_map(|shell| &mut shell.faces) {
            if face
                .name
                .as_deref()
                .is_some_and(|value| value.starts_with("FILLET:COMPOSITION:"))
            {
                face.name = name.map(str::to_owned);
            }
        }
        return Ok(result);
    }
    let decline = || {
        KernelRefusal::unsupported(KernelStage::Classify, "rounded_boss_selection",
        "the rounded boss composition requires a convex group meeting transverse mixed corners at one end of each spine")
    };
    if std::env::var("BREP_NO_NETWORK").is_ok() {
        return Err(decline());
    }
    // Select by topology id so reversing the requested edges preserves the
    // construction order. Two-ended spines are left for runout composition.
    let corner = corners
        .iter()
        .filter(|corner| {
            corner.convex.len() == 1
                && corner.concave.len() == 2
                && corners
                    .iter()
                    .filter(|other| other.convex.contains(&corner.convex[0]))
                    .count()
                    == 1
        })
        .min_by_key(|corner| selected_ids[corner.convex[0]])
        .ok_or_else(decline)?;
    let convexity =
        crate::blend::selection_convexity(solid, selected_ids, radius).ok_or_else(decline)?;
    if !convexity[corner.convex[0]] {
        return Err(decline());
    }
    let spine_index = corner.convex[0];
    // Resolve once so a missing selected edge refuses before connectivity
    // traversal rather than panicking inside the adjacency predicate.
    let selected_edges: Vec<&EdgeRecord> = selected_ids
        .iter()
        .map(|id| {
            solid
                .edges
                .iter()
                .find(|edge| edge.id == *id)
                .ok_or_else(|| {
                    KernelRefusal::internal(
                        KernelStage::Collect,
                        "rounded_boss_selected_edge",
                        format!("rounded boss composition: selected edge {id} is missing"),
                    )
                })
        })
        .collect::<Result<_, _>>()?;
    // Round the connected convex selection together. Its far corners need
    // their usual vertex patches before a concave rim can cross these walls.
    let mut rounded_indices = vec![spine_index];
    loop {
        let next = (0..selected_ids.len()).find(|index| {
            convexity[*index]
                && !rounded_indices.contains(index)
                && rounded_indices.iter().any(|other| {
                    let edge = selected_edges[*index];
                    let mate = selected_edges[*other];
                    [edge.start_vertex_id, edge.end_vertex_id]
                        .iter()
                        .any(|v| *v == mate.start_vertex_id || *v == mate.end_vertex_id)
                })
        });
        match next {
            Some(index) => rounded_indices.push(index),
            None => break,
        }
    }
    // Keep complete stars together. A selected concave edge between two
    // convex rims is part of their vertex blend, even though it is not part
    // of the convex flood above. Rounding the rims without it builds a horn
    // torus for an *unselected* third edge; adding that edge later leaves a
    // flat cap against the torus instead of the selected star's tangent sector.
    // Only absorb an edge whose selected neighbours are already in this
    // group at BOTH ends. In particular, never pull in a concave rim leading
    // to another mixed corner: that rim still needs the bridge below.
    loop {
        let next = (0..selected_ids.len()).find(|&index| {
            if convexity[index] || rounded_indices.contains(&index) {
                return false;
            }
            let edge = selected_edges[index];
            let mut has_star = false;
            for vertex in [edge.start_vertex_id, edge.end_vertex_id] {
                let neighbours: Vec<_> = selected_edges.iter().enumerate()
                    .filter(|(other, mate)| *other != index
                        && (mate.start_vertex_id == vertex || mate.end_vertex_id == vertex))
                    .map(|(other, _)| other)
                    .collect();
                if neighbours.iter().any(|other| !rounded_indices.contains(other)) {
                    return false;
                }
                has_star |= neighbours.len() >= 2;
            }
            has_star
        });
        match next {
            Some(index) => rounded_indices.push(index),
            None => break,
        }
    }
    rounded_indices.sort_by_key(|index| selected_ids[*index]);
    let mixed: Vec<_> = corners
        .iter()
        .filter(|corner| {
            corner
                .convex
                .iter()
                .any(|index| rounded_indices.contains(index))
        })
        .collect();
    if mixed.iter().any(|corner| {
        corner.convex.len() != 1
            || corner.concave.len() != 2
            || corners
                .iter()
                .filter(|other| other.convex.contains(&corner.convex[0]))
                .count()
                != 1
    }) {
        return Err(decline());
    }
    let uses = |face: &FaceRecord, edge: u64| {
        face.loops
            .iter()
            .flat_map(|rim| &rim.coedges)
            .any(|coedge| coedge.edge_id == edge)
    };
    let mut caps = Vec::new();
    for corner in &mixed {
        let spine_id = selected_ids[corner.convex[0]];
        let cap = solid
            .shells
            .iter()
            .flat_map(|shell| &shell.faces)
            .find(|face| {
                corner
                    .concave
                    .iter()
                    .all(|index| uses(face, selected_ids[*index]))
            })
            .ok_or_else(decline)?;
        let faces: Vec<_> = solid
            .shells
            .iter()
            .flat_map(|shell| &shell.faces)
            .filter(|face| face.id == cap.id || uses(face, spine_id))
            .collect();
        // The normals at the corner: a plane's from its frame, any other
        // carrier's at the corner's own foot on it.  The 2026-10-12 sprocket
        // report (`inbox_20261012_sprocket_fillet`) was the first curved CAP;
        // the 2026-10-13 reports on the same part put the SPINE on a cylinder
        // (the rim top's arc, rim wall ∩ rim top, dying into the spoke wall)
        // and were declined here before anything was tried, which handed the
        // whole selection to the runout lane and its refusal
        // (`inbox_20261013_sprocket_spoke_fillet`).
        let normals: Option<Vec<Vec3>> = faces
            .iter()
            .map(|face| match face.surface.analytic() {
                Some(crate::AnalyticSurface::Plane { u_dir, v_dir, .. }) => {
                    u_dir.cross(*v_dir).normalized().ok()
                }
                _ => {
                    let uv = crate::project_point_to_surface(&face.surface, corner.point).ok()?;
                    face.surface.normal(uv.u, uv.v).ok()
                }
            })
            .collect();
        let normals = normals.ok_or_else(decline)?;
        // The cap need not be perpendicular to the spine's two carriers.
        // An oblique or cylindrical cap cuts the rounded wall in a curved
        // bridge. The rim network marches that bridge on the actual carriers.
        // Test transversality at the corner; a cylindrical cap need not be
        // replaced by its tangent plane to build this composition.
        if normals.len() != 3
            || normals[0].cross(normals[1]).dot(normals[2]).abs() < 1e-6
        {
            return Err(decline());
        }
        caps.push(cap.id);
    }
    // Give unnamed walls distinct identities too: bridges are located by
    // their two carriers, never proximity to the old corner.
    let rounded_names: Vec<String> = rounded_indices
        .iter()
        .map(|index| {
            per_edge_name(edge_names, name, *index)
                .map(str::to_owned)
                .unwrap_or_else(|| format!("FILLET:ROUNDED_BOSS_SPINE:{index}"))
        })
        .collect();
    let rounded_points: Vec<Vec3> = rounded_indices
        .iter()
        .map(|index| edge_points[*index])
        .collect();
    let rounded = fillet_edges_group_in(
        solid,
        &rounded_points,
        Some(&rounded_names),
        radius,
        false,
        name,
        None,
    )?;
    let mut bridges = Vec::new();
    for (corner, cap) in mixed.iter().zip(&caps) {
        let wall_index = rounded_indices
            .iter()
            .position(|index| *index == corner.convex[0])
            .ok_or_else(|| {
                KernelRefusal::internal(
                    KernelStage::Classify,
                    "rounded_boss_bridge_wall",
                    format!(
                        "rounded boss composition: no convex wall for corner {}",
                        corner.vertex
                    ),
                )
            })?;
        let candidates: Vec<_> = rounded
            .edges
            .iter()
            .filter(|edge| {
                let faces: Vec<_> = rounded
                    .shells
                    .iter()
                    .flat_map(|shell| &shell.faces)
                    .filter(|face| uses(face, edge.id))
                    .collect();
                faces.iter().any(|face| face.id == *cap)
                    && faces
                        .iter()
                        .any(|face| face.name.as_deref() == Some(&rounded_names[wall_index]))
            })
            .collect();
        if candidates.len() != 1 {
            return Err(decline());
        }
        bridges.push(candidates[0]);
    }
    let rim_ids: Vec<u64> = selected_ids
        .iter()
        .enumerate()
        .filter(|(index, _)| !rounded_indices.contains(index))
        .map(|(_, id)| *id)
        .collect();
    // Both ends must join requested rim edges: never extend a selection
    // into an unselected perimeter.
    if !bridges.iter().all(|bridge| {
        [bridge.start_vertex_id, bridge.end_vertex_id]
            .iter()
            .all(|vertex| {
                rounded.edges.iter().any(|edge| {
                    rim_ids.contains(&edge.id)
                        && (edge.start_vertex_id == *vertex || edge.end_vertex_id == *vertex)
                })
            })
    }) {
        return Err(decline());
    }
    let mut points = Vec::new();
    let mut names = Vec::new();
    for (index, id) in selected_ids.iter().enumerate() {
        if rounded_indices.contains(&index) {
            continue;
        }
        let edge = rounded
            .edges
            .iter()
            .find(|edge| edge.id == *id)
            .ok_or_else(decline)?;
        points.push(
            edge.curve
                .evaluate(0.5 * (edge.t0 + edge.t1))
                .or_refuse(KernelStage::Collect, "rim_midpoint")?,
        );
        names.push(
            per_edge_name(edge_names, name, index)
                .unwrap_or("FILLET:RIM")
                .to_owned(),
        );
    }
    for (corner, bridge) in mixed.iter().zip(&bridges) {
        points.push(
            bridge
                .curve
                .evaluate(0.5 * (bridge.t0 + bridge.t1))
                .or_refuse(KernelStage::Collect, "bridge_midpoint")?,
        );
        let mut adjacent = corner.concave.clone();
        adjacent.push(corner.convex[0]);
        names.push(
            corner_face_name(edge_names, name, &adjacent)
                .unwrap_or_else(|| format!("FILLET:BOSS_CORNER:{}", corner.vertex)),
        );
    }
    // A partial rim would leave an open mixed-corner construction. Require
    // every pending wall, including each bridge, before accepting composition.
    fillet_edges_reported(&rounded, &points, Some(&names), radius, false, name).and_then(
        |(result, selection)| {
            if selection.applied.len() != points.len() {
                Err(decline())
            } else {
                Ok(result)
            }
        },
    )
}

/// A corner with two concave edges and a convex one requires composition.
///
/// Where selected edges meet, the rolling ball has to touch every face around
/// the vertex from ONE side. A convex edge running into the concave edges it
/// meets (a rib spine dying into the fillets at its own base) asks the ball to
/// sit under the shared face for one blend and over it for the other, so no
/// ball seats there and the corner has no closure: physically the convex blend
/// runs out against the concave beads partway along the edge, which is a vertex
/// blend the stripe network does not construct.
///
/// The re-entrant vertex of a notch — ONE concave edge, the rest convex — is
/// NOT this: the ball rolls round that single concave edge from one convex
/// blend to the next, and `round_concave_chain_corner` / `round_convex_corner`
/// close it with the horn-torus sector.  Only its mirror is refused.
///
/// This is TERMINAL — deliberately not a fall-through to the cutter
/// composition. The cutter answers this shape with a watertight but shredded
/// solid: a sliver end cap at each unclosed corner and the carrier face split
/// so the original name lands on a fragment, which breaks every downstream
/// reference to it. A named refusal that says which edges to separate is
/// worth more than that solid. The runout composition (the 2026-09-30 record
/// §5): a selection whose convex edges die into concave ones at
/// mixed-convexity corners is built in two stages — the CONCAVE edges first,
/// through the ordinary group path (exactly the ring the user gets selecting
/// them alone), then one runout wall per convex spine on that ring result,
/// marched from pole to pole with its carriers switching onto the ring's own
/// pads. Accepted under the network's own checks.
///
/// The class built here: every convex edge of the selection ends on a mixed
/// corner at BOTH ends and shares no vertex with another convex edge, and
/// every other selected edge is concave.  Anything else keeps the named
/// refusal.
#[allow(clippy::too_many_arguments)]
/// `BREP_BLEND_OPEN_PAD=0`/`off` is the escape hatch for [`open_pad_composition`].
fn open_pad_on() -> bool {
    switch!(live "BREP_BLEND_OPEN_PAD", true)
}

/// The OPEN-PAD composition (the 2026-10-17 "fillet takes way too long"
/// face-ring report): a selection whose every mixed corner is ONE convex edge
/// meeting ONE concave edge over the planar face they share, the third edge
/// at the corner either a tangent rail (the convex edge is a blend's own end
/// cut by the plane, the concave one the boss it was blended to: a zero-angle
/// sliver on the plane) or a sharp concave edge (the boss meeting the convex
/// edge's other mate). Neither the rounded-boss nor the runout composition
/// admits it (flush-joined convex chains, concave pads ending at the corners,
/// no concave ring).
///
/// Built in two stages on the stripe network, each an ordinary group build:
///
/// 1. Every CONVEX edge first. A convex chain's free end at a mixed corner
///    is the §6.9 transverse construction against the boss as the end face:
///    the wall is trimmed where its envelope enters the boss, the sliver tip
///    of the planar face goes, and the boss skin that lay under it is exposed
///    along the trim curve (no cliff: the trim runs from the wall's floor
///    rail to its rail on the other mate, which at a tangent-rail corner
///    lands on that rail where wall and boss are tangent).
/// 2. Every CONCAVE edge on the stage-1 body, EXTENDED by the trim curves it
///    now continues into G1 at each trimmed end (floor|boss into wall|boss
///    across the wall's floor rail, a tangent seam): a flush join with a seam
///    at the trim's start, and at the trim's far end a pole where the wall
///    is tangent to the boss, or a cap where the trim meets an unselected
///    sharp edge. The pad rides the wall past the floor rails' crossing.
///
/// The walls are named after their edges; a trim continuation is named
/// `{base}:RUNOUT:{pad}+{wall}` from the two bare edge names.
#[allow(clippy::too_many_arguments)]
fn open_pad_composition(
    solid: &BrepSolid,
    selected_ids: &[u64],
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    name: Option<&str>,
    corners: &[crate::blend::MixedCorner],
    entry: &str,
) -> Result<BrepSolid, KernelRefusal> {
    let shape = |message: String| KernelRefusal::unsupported(KernelStage::Classify, "open_pad_shape", message);
    if !open_pad_on() {
        return Err(shape("the open-pad composition is switched off (BREP_BLEND_OPEN_PAD)".into()));
    }
    if std::env::var("BREP_NO_NETWORK").is_ok() {
        return Err(shape("the open-pad composition is a network construction and BREP_NO_NETWORK is set".into()));
    }
    if corners.is_empty() || corners.iter().any(|corner| corner.convex.len() != 1 || corner.concave.len() != 1) {
        return Err(shape("the open-pad composition is built for corners of one convex and one concave edge".into()));
    }
    let convexity = crate::blend::selection_convexity(solid, selected_ids, radius)
        .ok_or_else(|| shape("the selection's convexity could not be read edge by edge".into()))?;
    let concave: Vec<usize> = (0..selected_ids.len()).filter(|index| !convexity[*index]).collect();
    let convex: Vec<usize> = (0..selected_ids.len()).filter(|index| convexity[*index]).collect();
    if concave.is_empty() || convex.is_empty() {
        return Err(shape("the open-pad composition needs convex and concave edges".into()));
    }
    // Every concave edge ends at mixed corners (an open pad), and the face
    // the two edges share at each corner is planar.
    let mates = crate::blend::selection_mates(solid, selected_ids, radius)
        .map_err(|error| shape(format!("the selection's mates could not be read: {error}")))?;
    for corner in corners {
        let (_, convex_faces) = &mates[corner.convex[0]];
        let (_, concave_faces) = &mates[corner.concave[0]];
        let Some(shared) = convex_faces.iter().find(|face| concave_faces.iter().any(|other| other.id == face.id)) else {
            return Err(shape(format!("the two edges at vertex {} share no face", corner.vertex)));
        };
        if !matches!(shared.surface.analytic(), Some(crate::AnalyticSurface::Plane { .. })) {
            return Err(shape(format!("the face {} the edges share at vertex {} is not planar", shared.id, corner.vertex)));
        }
    }
    for &index in &concave {
        let edge = mates[index].0;
        for vertex in [edge.start_vertex_id, edge.end_vertex_id] {
            if !corners.iter().any(|corner| corner.vertex == vertex && corner.concave[0] == index) {
                return Err(shape(format!(
                    "concave {} does not end at a mixed corner at vertex {vertex}; the open-pad composition is built for pads that do",
                    selection_label(index, edge_points, edge_names, name)
                )));
            }
        }
    }
    let base = name.unwrap_or("FILLET:OPEN_PAD");
    let bare = |index: usize| -> String {
        per_edge_name(edge_names, name, index)
            .map(|composed| composed.strip_prefix(&format!("{base}:")).unwrap_or(composed).to_string())
            .unwrap_or_else(|| selected_ids[index].to_string())
    };
    let debug = std::env::var("BREP_DEBUG_NETWORK").is_ok();

    // ---- Stage 1: the convex edges, as the group they are. ----
    let convex_points: Vec<Vec3> = convex.iter().map(|index| edge_points[*index]).collect();
    let convex_names: Vec<String> = convex
        .iter()
        .map(|index| per_edge_name(edge_names, name, *index).map(str::to_owned).unwrap_or_else(|| format!("{base}:WALL:{}", selected_ids[*index])))
        .collect();
    // Stage 1's body is an intermediate: the bowtie and soundness scans run
    // once, on stage 2's body (and the caller's acceptance after it), not on
    // a body the pads are about to change (1.2 s of tessellation on the
    // reported ring).
    // Both stages run as the open-pad composition's network
    // (`with_open_pad_network`: carrier-boundary stations, the march
    // extended to a section end outside the overshoot); no other network
    // takes those, so every body the plain network built stays as it was.
    let stage1 = crate::blend::with_open_pad_network(|| {
        fillet_edges_group_with_corner_policy(solid, &convex_points, Some(&convex_names), radius, false, name, None, true, false)
    })
    .map_err(|error| error.with_message(|error| format!("open-pad composition, the convex walls: {error}")))?;
    if debug {
        eprintln!("open-pad composition: stage 1 built the {} convex walls", convex.len());
    }

    // ---- Stage 2: each pad with its trim continuations, on the stage-1 body. ----
    let uses = |face: &FaceRecord, edge_id: u64| face.loops.iter().flat_map(|l| &l.coedges).any(|c| c.edge_id == edge_id);
    let mut points: Vec<Vec3> = Vec::new();
    let mut names: Vec<String> = Vec::new();
    for &index in &concave {
        let (edge, faces) = &mates[index];
        let boss_id = if matches!(faces[0].surface.analytic(), Some(crate::AnalyticSurface::Plane { .. })) { faces[1].id } else { faces[0].id };
        let pad = stage1
            .edges
            .iter()
            .find(|candidate| candidate.id == edge.id)
            .ok_or_else(|| KernelRefusal::internal(KernelStage::Classify, "open_pad_edge", format!(
                "open-pad composition: concave edge {} vanished from the stage-1 body", edge.id
            )))?;
        points.push(pad.curve.evaluate(0.5 * (pad.t0 + pad.t1)).or_refuse(KernelStage::Collect, "pad_midpoint")?);
        names.push(per_edge_name(edge_names, name, index).map(str::to_owned).unwrap_or_else(|| format!("{base}:PAD:{}", edge.id)));
        // The trim continuations: at each end of the (trimmed) pad, the edge
        // that borders the boss and a stage-1 wall.
        for vertex in [pad.start_vertex_id, pad.end_vertex_id] {
            let trims: Vec<&EdgeRecord> = stage1
                .edges
                .iter()
                .filter(|candidate| {
                    candidate.id != pad.id
                        && !candidate.degenerate
                        && (candidate.start_vertex_id == vertex || candidate.end_vertex_id == vertex)
                })
                .filter(|candidate| {
                    let faces: Vec<&FaceRecord> = stage1.shells.iter().flat_map(|shell| &shell.faces).filter(|face| uses(face, candidate.id)).collect();
                    faces.iter().any(|face| face.id == boss_id)
                        && faces.iter().any(|face| face.name.as_deref().is_some_and(|value| convex_names.iter().any(|wall| wall == value)))
                })
                .collect();
            match trims.as_slice() {
                [trim] => {
                    let wall_name = stage1
                        .shells
                        .iter()
                        .flat_map(|shell| &shell.faces)
                        .filter(|face| uses(face, trim.id))
                        .find_map(|face| face.name.clone().filter(|value| convex_names.contains(value)))
                        .unwrap_or_default();
                    let wall_bare = wall_name.strip_prefix(&format!("{base}:")).unwrap_or(&wall_name).to_string();
                    points.push(trim.curve.evaluate(0.5 * (trim.t0 + trim.t1)).or_refuse(KernelStage::Collect, "trim_midpoint")?);
                    names.push(format!("{base}:RUNOUT:{}+{wall_bare}", bare(index)));
                    if debug {
                        eprintln!("open-pad composition: pad {} continues into trim edge {} (wall {wall_name}) at vertex {vertex}", edge.id, trim.id);
                    }
                }
                [] => {
                    return Err(KernelRefusal::unsupported(KernelStage::Classify, "open_pad_trim", format!(
                        "open-pad composition: at vertex {vertex} the concave {} meets no trim curve of a stage-1 wall on the boss",
                        selection_label(index, edge_points, edge_names, name)
                    )));
                }
                _ => {
                    return Err(KernelRefusal::unsupported(KernelStage::Classify, "open_pad_trim", format!(
                        "open-pad composition: at vertex {vertex} the concave {} meets {} trim curves",
                        selection_label(index, edge_points, edge_names, name), trims.len()
                    )));
                }
            }
        }
    }
    let result = crate::blend::with_open_pad_network(|| fillet_edges_group_in(&stage1, &points, Some(&names), radius, false, name, None))
        .map_err(|error| error.with_message(|error| format!("open-pad composition, the pads on the walls: {error}")))?;
    let _ = entry;
    Ok(result)
}

/// The CUSP composition (2026-10-24/25 reports 80c5d9d1 and 6dfd5a22): a
/// selection whose every mixed corner is a CUSP — a convex edge and a concave
/// edge leaving the vertex tangent to each other, the face they share
/// narrowing to zero width between them. Report 6dfd5a22 selects the whole
/// of E6's start-face sliver outside F2's r 5 round: its two straight convex
/// edges run on G1 into the concave arc at both tips.
///
/// No other composition admits it: the pads of the open-pad composition
/// trim into the boss, and here both blends must TAPER to the tip, each
/// against the face the other leaves. Built in two stages, each an ordinary
/// group build, the order report 80c5d9d1 took by hand (F9, then F10):
///
/// 1. Every CONCAVE edge first: its round tapers to zero at each cusp
///    (`blend/taper_runout.rs`), leaving a hairline of the shared face
///    beside it.
/// 2. Every CONVEX edge on the stage-1 body: its round tapers to zero at the
///    same tips, with the width read against the stage-1 round's rail (the
///    face's boundary past the cusp line, `taper_runout::boundary_chain`).
///    The two rounds are G1 to the hairline between them.
fn cusp_composition(
    solid: &BrepSolid,
    selected_ids: &[u64],
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    name: Option<&str>,
    corners: &[crate::blend::MixedCorner],
) -> Result<BrepSolid, KernelRefusal> {
    let shape = |message: String| KernelRefusal::unsupported(KernelStage::Classify, "cusp_composition_shape", message);
    if std::env::var("BREP_NO_NETWORK").is_ok() {
        return Err(shape("the cusp composition is a network construction and BREP_NO_NETWORK is set".into()));
    }
    let edge_of = |index: usize| solid.edges.iter().find(|edge| edge.id == selected_ids[index]);
    // The unit tangent leaving `vertex` into the edge.
    let away = |index: usize, vertex: u64| -> Option<Vec3> {
        let edge = edge_of(index)?;
        let at_start = edge.start_vertex_id == vertex;
        let tangent = edge.curve.unit_tangent(if at_start { edge.t0 } else { edge.t1 }, edge.t0, edge.t1).ok()?;
        Some(if at_start { tangent } else { tangent.scale(-1.0) })
    };
    if corners.is_empty() {
        return Err(shape("no mixed corners".into()));
    }
    for corner in corners {
        let cusped = corner.convex.iter().all(|&convex| {
            corner.concave.iter().any(|&concave| {
                matches!((away(convex, corner.vertex), away(concave, corner.vertex)), (Some(a), Some(b)) if a.dot(b) >= 1.0 - 1e-6)
            })
        });
        if !cusped || corner.convex.is_empty() || corner.concave.is_empty() {
            return Err(shape(format!("the mixed corner at vertex {} is not a cusp", corner.vertex)));
        }
    }
    let convexity = crate::blend::selection_convexity(solid, selected_ids, radius)
        .ok_or_else(|| shape("the selection's convexity could not be read edge by edge".into()))?;
    let concave: Vec<usize> = (0..selected_ids.len()).filter(|index| !convexity[*index]).collect();
    let convex: Vec<usize> = (0..selected_ids.len()).filter(|index| convexity[*index]).collect();
    if concave.is_empty() || convex.is_empty() {
        return Err(shape("the cusp composition needs convex and concave edges".into()));
    }
    let names_of = |indices: &[usize]| -> Option<Vec<String>> { edge_names.map(|names| indices.iter().map(|index| names[*index].clone()).collect()) };
    let points_of = |indices: &[usize]| -> Vec<Vec3> { indices.iter().map(|index| edge_points[*index]).collect() };
    let debug = std::env::var("BREP_DEBUG_NETWORK").is_ok();
    let stage1 = fillet_edges_group_in(solid, &points_of(&concave), names_of(&concave).as_deref(), radius, false, name, None)
        .map_err(|error| error.with_message(|error| format!("cusp composition, the concave rounds: {error}")))?;
    if debug {
        eprintln!("cusp composition: stage 1 built the {} concave rounds", concave.len());
    }
    // Each convex edge is read again on the stage-1 body by its point: the
    // taper split it only at its hairline's end, beside the tip.
    for &index in &convex {
        resolve_edge_by_point(&stage1, edge_points[index])
            .map_err(|error| error.with_message(|error| format!("cusp composition: a convex edge is not on the stage-1 body: {error}")))?;
    }
    let result = fillet_edges_group_in(&stage1, &points_of(&convex), names_of(&convex).as_deref(), radius, false, name, None)
        .map_err(|error| error.with_message(|error| format!("cusp composition, the convex rounds: {error}")))?;
    // Every round here tapers to zero at a tip, and the two rounds meet on
    // the hairline of the face between them: an edge's end stations stay ON
    // the skin (the concave arc's first tenth, where the convex round's
    // contact lands back on the arc), and the wall check reads the majority
    // of a tie, as it does for the runout composition's network ring.
    RUNOUT_RING_IS_NETWORK.with(|flag| flag.set(true));
    Ok(result)
}

fn runout_composition(
    solid: &BrepSolid,
    selected_ids: &[u64],
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    name: Option<&str>,
    corners: &[crate::blend::MixedCorner],
    entry: &str,
) -> Result<BrepSolid, KernelRefusal> {
    if std::env::var("BREP_NO_NETWORK").is_ok() {
        return Err(KernelRefusal::unsupported(KernelStage::Classify, "runout_no_network", "the runout composition is a network construction and BREP_NO_NETWORK is set"));
    }
    let convexity = crate::blend::selection_convexity(solid, selected_ids, radius)
        .ok_or(KernelRefusal::unsupported(KernelStage::Classify, "runout_convexity", "the selection's convexity could not be read edge by edge"))?;
    let mut spine_ends = vec![0usize; selected_ids.len()];
    for corner in corners {
        if corner.convex.len() != 1 {
            return Err(KernelRefusal::unsupported(KernelStage::Classify, "runout_corner_shape", format!(
                "the corner at ({:.3}, {:.3}, {:.3}) has {} convex edges; the runout is built for one convex edge dying into concave ones",
                corner.point.x, corner.point.y, corner.point.z, corner.convex.len()
            )));
        }
        spine_ends[corner.convex[0]] += 1;
    }
    let mut spines: Vec<usize> = Vec::new();
    let mut concave: Vec<usize> = Vec::new();
    for (index, ends) in spine_ends.iter().enumerate() {
        match (*ends, convexity[index]) {
            (2, true) => spines.push(index),
            (1, true) if crate::blend::runout::one_ended_runout_on() => spines.push(index),
            (0, false) => concave.push(index),
            (ends, convex) => {
                return Err(KernelRefusal::unsupported(KernelStage::Classify, "runout_selection_shape", format!(
                    "{} is {} and meets a mixed corner at {ends} of its ends; the runout is built for a convex edge running out at both ends among concave edges",
                    selection_label(index, edge_points, edge_names, name),
                    if convex { "convex" } else { "concave" },
                )));
            }
        }
    }
    if concave.is_empty() {
        return Err(KernelRefusal::unsupported(KernelStage::Classify, "runout_selection_shape", "no concave edges to build the ring from"));
    }
    // Stage 1: the ring, exactly as the concave edges build on their own.
    let ring_points: Vec<Vec3> = concave.iter().map(|index| edge_points[*index]).collect();
    let ring_names: Option<Vec<String>> = edge_names.map(|names| concave.iter().map(|index| names[*index].clone()).collect());
    LAST_NETWORK_REFUSAL.with(|last| *last.borrow_mut() = None);
    let ring = fillet_edges_group_with_corner_policy(solid, &ring_points, ring_names.as_deref(), radius, false, name, None, false, true)
        .map_err(|error| {
            // The ring's network classification is the first-order reason
            // when the group fell through to the cutter and that refused as
            // well (the wedge nine-pick: the cutter says "dihedral angle too
            // degenerate" where the network named the inside corner at
            // (8, 1, 2)); an inside corner is reported by its own name.
            let network = LAST_NETWORK_REFUSAL.with(|last| last.borrow_mut().take());
            match network {
                Some(network) if crate::blend::is_reentrant_concave(&network) => network.with_message(|network| format!(
                    "the concave ring did not build: {network} (the cutter fallback then refused: {error})"
                )),
                _ => error.with_message(|error| format!("the concave ring did not build: {error}")),
            }
        })?;
    // The ring is the NETWORK's ring.  When the network declined it at an
    // inside corner and the cutter fallback built a body for the same edges
    // instead (the wedge nine-pick: the cutter leaves the concave star at
    // (10, 1, 2) as bare edge fillets, and the walls marched on that body
    // read "dihedral angle too degenerate"), the composition declines here,
    // by the network's own classification — the corner named, not the
    // march's reading of a body that is not the ring.  Gated to that class:
    // any other fallback ring is composed exactly as before.
    let network_declined = LAST_NETWORK_REFUSAL.with(|last| last.borrow_mut().take());
    // BREP lane L: begin walls_missing_tie_all_stations_20261015
    let ring_is_network = network_declined.is_none();
    // BREP lane L: end walls_missing_tie_all_stations_20261015
    if let Some(network) = network_declined {
        if crate::blend::is_reentrant_concave(&network) {
            return Err(network.with_message(|network| format!(
                "the concave ring did not build as the network's ring: {network}; the cutter fallback built a body for the concave edges without that closure, and the runout walls are marched only on the network's ring"
            )));
        }
    }
    // Stage 2: the runout walls on it.
    let spine_walls: Vec<(u64, Option<String>)> = spines
        .iter()
        .map(|index| (selected_ids[*index], per_edge_name(edge_names, name, *index).map(str::to_string)))
        .collect();
    let spine_ids: Vec<u64> = spine_walls.iter().map(|(id, _)| *id).collect();
    let mut result = crate::blend::build_runout_walls(&ring, &spine_walls, radius)?;
    heal_edge_vertex_gaps(&mut result, radius)?;
    crate::blend::fit_planar_charts_to_trims(&ring, &mut result)?;
    let issues = result.validate();
    if !issues.is_empty() {
        return Err(KernelRefusal::new(
            RefusalClass::InvalidResultTopology { issues: issues.len() as u32 },
            KernelStage::Validate,
            format!("the runout walls do not validate: {issues:?}"),
        ));
    }
    check_blend_interference(&ring, &result, &spine_ids, entry)?;
    check_loop_self_crossings(&result, entry)?;
    check_blend_wall_loops(&result, entry)?;
    crate::soundness::census::profile_operation(&result, &[solid], None, entry);
    let accepted = crate::accept_sound(result, entry)?;
    // BREP lane L: begin walls_missing_tie_all_stations_20261015
    RUNOUT_RING_IS_NETWORK.with(|flag| flag.set(ring_is_network));
    // BREP lane L: end walls_missing_tie_all_stations_20261015
    Ok(accepted)
}

// BREP lane L: begin walls_missing_tie_all_stations_20261015
thread_local! {
    /// Set by a runout composition that SUCCEEDED on the network's ring (no
    /// network refusal recorded while its ring was built), and by a cusp
    /// composition that built (`cusp_composition`), read and cleared by the
    /// acceptance: only there does a tie take the majority.
    static RUNOUT_RING_IS_NETWORK: std::cell::Cell<bool> = const { std::cell::Cell::new(false) };
}
// BREP lane L: end walls_missing_tie_all_stations_20261015

fn check_mixed_corner_convexity(
    solid: &BrepSolid,
    edge_ids: &[u64],
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    name: Option<&str>,
    composition_error: Option<&KernelRefusal>,
) -> Result<(), KernelRefusal> {
    match crate::blend::mixed_convexity_corner(solid, edge_ids, radius) {
        Some(corner) => {
            // The refusal carries the RUNOUT's own numbers when they can be
            // measured — the station, the carrier pair, the stop and the pole
            // — so it says which construction is missing and where, not only
            // that the corner has none.  A measurement that fails leaves the
            // refusal exactly as it was: the corner is refused either way.
            let measured = match crate::blend::plan_runout(solid, edge_ids, radius, &corner) {
                Ok(plan) => Some(plan),
                Err(error) => {
                    if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                        eprintln!("runout not measured: {error}");
                    }
                    None
                }
            };
            // "this kernel does not build that runout here": a named deferral,
            // carrying why the composition declined this selection.
            let mut message = mixed_corner_message(
                &corner,
                edge_points,
                edge_names,
                name,
                measured.as_ref(),
            );
            if let Some(error) = composition_error {
                message.push_str(&format!(" The runout composition was tried on this selection and refused: {error}"));
            }
            Err(KernelRefusal::unsupported(
                KernelStage::Classify,
                "mixed_corner_convexity",
                message,
            ))
        }
        None => Ok(()),
    }
}

/// Refuse a selection one of whose edges keeps NO blend strip between the
/// corner balls at its two ends (see `degenerate_corner_setbacks`).
fn check_degenerate_corner_setbacks(
    solid: &BrepSolid,
    edge_ids: &[u64],
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    name: Option<&str>,
) -> Result<(), KernelRefusal> {
    let Some(setback) = crate::blend::degenerate_corner_setbacks(solid, edge_ids, radius) else {
        return Ok(());
    };
    let [from, to] = setback.stops;
    let how = if setback.crossed {
        format!(
            "run {:.3e} PAST each other along it (the radius is beyond the limit)",
            setback.strip
        )
    } else {
        format!(
            "meet there ({:.3e} apart against a bar of {:.3e})",
            setback.strip, setback.bar
        )
    };
    // "The radius is at or past the degenerate limit … reduce it": a
    // parameter the geometry refuses, read at `Classify`.
    Err(KernelRefusal::input(
        KernelStage::Classify,
        "degenerate_corner_setbacks",
        format!(
            "fillet_edges: at r = {radius} the corner setbacks consume the whole of {} — the rolling \
             balls seated at its two corners touch face {} at ({:.4}, {:.4}, {:.4}) and ({:.4}, \
             {:.4}, {:.4}) and {how}, so no blend strip is left between them and the faces it lies \
             on are consumed whole. The radius is at or past the degenerate limit for this \
             selection; reduce it.",
            selection_label(setback.edge, edge_points, edge_names, name),
            setback.face_id,
            from.x,
            from.y,
            from.z,
            to.x,
            to.y,
            to.z,
        ),
    ))
}

/// How a refusal names the `index`-th selected edge: its originating edge
/// name when the feature layer supplied one (with the feature's base prefix
/// stripped), else the picked point.
fn selection_label(
    index: usize,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    name: Option<&str>,
) -> String {
    let named = edge_names.and_then(|names| names.get(index)).map(|composed| match name {
        Some(base) => composed
            .strip_prefix(&format!("{base}:"))
            .unwrap_or(composed)
            .to_string(),
        None => composed.clone(),
    });
    named.unwrap_or_else(|| match edge_points.get(index) {
        Some(point) => format!("the edge at ({:.3}, {:.3}, {:.3})", point.x, point.y, point.z),
        None => format!("selection #{}", index + 1),
    })
}

/// The refusal text for [`check_mixed_corner_convexity`], naming the edges the
/// way the user selected them (their originating edge names when the feature
/// layer supplied them, else the picked point).
fn mixed_corner_message(
    corner: &crate::blend::MixedCorner,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    name: Option<&str>,
    measured: Option<&crate::blend::RunoutPlan>,
) -> String {
    let label = |index: usize| selection_label(index, edge_points, edge_names, name);
    let list = |indices: &[usize]| -> String {
        let parts: Vec<String> = indices.iter().map(|index| label(*index)).collect();
        match parts.split_last() {
            None => "none".to_string(),
            Some((last, [])) => last.clone(),
            Some((last, rest)) => format!("{} and {last}", rest.join(", ")),
        }
    };
    let mut message = format!(
        "fillet_edges: the selection mixes convexity at the corner ({:.3}, {:.3}, {:.3}) — convex \
         {} meets concave {} there. One rolling ball cannot touch the face they share from both \
         sides at once, so that corner has no closure: the convex blend runs out against the \
         concave ones partway along the edge, and this kernel does not build that runout yet. \
         Blending the concave edges in one fillet and the convex ones in another does not reach \
         a wall on every one of them either: the convex wall comes out at FULL length in either \
         order, and its contact leaves the faces it runs along where their own loops would then \
         cross. Blend the concave edges first and the blend acceptance refuses each convex edge \
         whose wall runs out into them; blend the convex ones first and the subset search drops \
         the concave edges it cannot compose instead, leaving those unfilleted.",
        corner.point.x,
        corner.point.y,
        corner.point.z,
        list(&corner.convex),
        list(&corner.concave),
    );
    // The runout, measured: where the ball's carriers switch, what they switch
    // to, and where the last stretch degenerates.  These are the numbers the
    // construction owes, so the refusal states them rather than leaving the
    // reader to derive them (`blend/runout.rs`).
    if let Some(plan) = measured {
        message.push_str(&format!(
            " Measured on this selection: the convex edge {}'s ball becomes tangent to face {} — \
             the face its two concave neighbours share — at s = {:.6} from the corner, centred \
             ({:.6}, {:.6}, {:.6}), where its contact on face {} reaches edge {}'s own rail \
             (ball-to-stripe residual {:.3e}). Past that station the second carrier is edge {}'s \
             STRIPE SURFACE, not a face of the input, and the march holds on it for {} \
             stations that fit rows reproducing their own contacts to {:.3e}. On the way the \
             kept mate's contact walks {:.3e} of a span off that face's own patch — into the \
             strip the concave stripe's pad ADDS to it — which is the shape, not an escape: \
             the bar a march is refused at is a full span.",
            plan.convex_edge,
            plan.shared_face,
            plan.station.s,
            plan.station.center.x,
            plan.station.center.y,
            plan.station.center.z,
            plan.replaced_mate,
            plan.second_carrier,
            plan.station.stripe_residual,
            plan.second_carrier,
            plan.marched,
            plan.fit_residual,
            plan.carrier_excursion,
        ));
        match (&plan.stop, plan.first_carrier) {
            (Some(stop), Some(first)) => message.push_str(&format!(
                " It does not simply stop there: at s = {:.6} the KEPT mate (face {}) runs out too \
                 — the ball's contact on it reaches edge {}'s rail, residual {:.3e} — so the FIRST \
                 carrier becomes that stripe as well and the last stretch rolls on two stripes.",
                stop.s, plan.kept_mate, first, stop.stripe_residual,
            )),
            _ => message.push_str(&format!(
                " Where the kept mate (face {}) runs out was not measurable on this selection.",
                plan.kept_mate
            )),
        }
        if let Some(pole) = plan.pole {
            message.push_str(&format!(
                " That stretch ends at a POLE, ({:.6}, {:.6}, {:.6}), where the two concave \
                 stripes' rails on face {} meet and the ball's two contacts merge, so the blend \
                 section shrinks to a point. Two carrier switches and a degenerate terminus is \
                 more than a stop station on a stripe, and none of it is composed yet.",
                pole.x, pole.y, pole.z, plan.shared_face,
            ));
        }
    }
    message
}

/// All ways to choose `k` distinct indices from `0..n`, in lexicographic order.
fn index_combinations(n: usize, k: usize) -> Vec<Vec<usize>> {
    let mut out = Vec::new();
    if k == 0 || k > n {
        return out;
    }
    let mut idx: Vec<usize> = (0..k).collect();
    loop {
        out.push(idx.clone());
        // Advance to the next combination (like counting with carry).
        let mut i = k;
        loop {
            if i == 0 {
                return out;
            }
            i -= 1;
            if idx[i] != i + n - k {
                break;
            }
        }
        idx[i] += 1;
        for j in (i + 1)..k {
            idx[j] = idx[j - 1] + 1;
        }
    }
}

thread_local! {
    /// Running count (per thread) of group builds that reached the CUTTER
    /// COMPOSITION — every network lane above it refused or was rejected.  A
    /// zero delta across a call is the witness that the network answered it;
    /// a face count or a volume cannot say which lane built a solid, since the
    /// cutter can build the same one.  Test-observable; not part of any result.
    pub(crate) static CUTTER_COMPOSITIONS: std::cell::Cell<u64> =
        const { std::cell::Cell::new(0) };
}

/// Blend the WHOLE selection as one group (the single-shot multi-edge fillet).
/// Errors if any selected edge or the shared-corner surgery cannot compose;
/// `fillet_edges` wraps this with a maximal-valid-subset fallback.
/// [`fillet_edges_group_unchecked`] held to the ONE soundness question every
/// other acceptance in this file passes: no face of the result may have a trim
/// loop that crosses itself ([`check_loop_self_crossings`]).
///
/// It sits here, at the group's single exit, rather than beside each `Ok`
/// inside it, because the group has four of them — the closed rim, the stripe
/// network, the per-component composition and the cutter composition — and the
/// cutter's is the one no other check guards at all. The network and the rim
/// ALSO ask it inside, where a bowtie falls through to the next lane instead of
/// failing the group; here there is no next lane, so it refuses.
///
/// The maximal-valid-subset search in [`fillet_edges_reported`] calls this
/// entry, so a subset that blends to a bowtie is not a valid subset either.
fn fillet_edges_group(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    chamfer: bool,
    name: Option<&str>,
) -> Result<BrepSolid, KernelRefusal> {
    fillet_edges_group_in(solid, edge_points, edge_names, radius, chamfer, name, None)
}

thread_local! {
    /// The most edges the maximal-valid-subset search may drop on this
    /// thread; `None` searches every subset. Set by [`fillet_edges`] for its
    /// own call (lane F row 3, 2026-10-15): that entry refuses any partial
    /// selection, so the search there only words the refusal, and a search
    /// past one dropped edge cost lCount_125 117 s of booleans (112 subsets)
    /// to refuse what the drop-one level refuses in 11 s.
    static SUBSET_DROP_CAP: std::cell::Cell<Option<usize>> = const { std::cell::Cell::new(None) };
}

/// Restores the previous [`SUBSET_DROP_CAP`] when the strict entry returns.
struct SubsetDropCap(Option<usize>);

impl SubsetDropCap {
    fn set(cap: Option<usize>) -> Self {
        Self(SUBSET_DROP_CAP.with(|current| current.replace(cap)))
    }
}

impl Drop for SubsetDropCap {
    fn drop(&mut self) {
        SUBSET_DROP_CAP.with(|current| current.set(self.0));
    }
}

thread_local! {
    /// Running count (per thread) of [`fillet_edges_group`] builds, nested
    /// component builds included. Test-observable; not part of any result.
    pub(crate) static GROUP_BUILDS: std::cell::Cell<u64> = const { std::cell::Cell::new(0) };
    /// The network's last refusal in a group build on this thread, read
    /// back by the runout composition when its ring stage fails.
    static LAST_NETWORK_REFUSAL: std::cell::RefCell<Option<KernelRefusal>> = const { std::cell::RefCell::new(None) };
    /// The wall fold the network refused THIS group build's selection by,
    /// scoped to one [`fillet_edges_group_unchecked`] call (nested component
    /// builds save and restore it).
    static GROUP_NETWORK_FOLD: std::cell::RefCell<Option<KernelRefusal>> = const { std::cell::RefCell::new(None) };
}

/// One component of a selection: its edge points (as bits, so the key is
/// exact) and its per-edge wall names.
type ComponentKey = (Vec<[u64; 3]>, Option<Vec<String>>);

/// One step of the cutter composition's sequential build: the edge's
/// selection point (as bits), its wall name, the cutter's end pads (as bits)
/// and the composition lane (0 = general first, 1 = cutter first). The pads
/// and the lane are part of the key because both are read off the WHOLE
/// selection (its corners), so the same edge can take a different cutter in
/// a subset that drops a corner's third edge.
type SequentialKey = ([u64; 3], Option<String>, [u64; 2], u8);

/// Component results of one [`fillet_edges_reported`] call, keyed by the chain
/// of components composed up to and including the one built. Valid only for
/// the solid, radius, chamfer flag and name of that call.
///
/// `sequential` is the same memo one level down, for a single connected
/// component: the evolving body of the cutter composition after each edge
/// blended, keyed by the chain of steps composed up to and including it. The
/// subset search's one-drop subsets of a connected selection mostly open
/// with the same edges in the same order, so without it every such subset
/// rebuilt the same cutter booleans before reaching the edge that refuses.
/// On the 2026-10-17 "fillet takes way too long" report (a 7-edge face ring
/// whose second edge refuses after the first edge's 6-8 s cutter boolean)
/// that boolean ran six times for one refusal: for the group and for the
/// five subsets that keep the first edge.
///
/// A one-drop subset of a connected ring can split it into two components
/// (the ring minus an edge that is not an end of a path), and the component
/// chains then differ from subset to subset while the cutter composition's
/// first steps do not; so the sequential memo is keyed by the chain of
/// components composed BEFORE the body being built (`prefix`, empty for a
/// single-component selection) together with the steps, and the component
/// builds carry the memo down instead of `None`. A FIRST component's steps
/// key under the empty prefix like a single-component selection's, which
/// is sound because both start from the same body: the component build
/// re-heals the selection's already healed input, and `heal_operands`
/// leaves a clean input byte-identical (its contract, stated at the heal
/// above; the cross-partition hits on the reported ring are its evidence).
#[derive(Default)]
struct ComponentMemo {
    built: std::collections::HashMap<Vec<ComponentKey>, Result<BrepSolid, KernelRefusal>>,
    /// The components composed before the build in progress, from the
    /// outermost selection down (set around each component build).
    prefix: Vec<ComponentKey>,
    sequential: std::collections::HashMap<
        (Vec<ComponentKey>, Vec<SequentialKey>),
        Result<BrepSolid, KernelRefusal>,
    >,
}

/// [`fillet_edges_group`], with the subset search's component memo when
/// `memo` is `Some`. The search passes it on its own solid, and the component
/// builds pass it on down with `ComponentMemo::prefix` naming the components
/// composed before them (see `ComponentMemo::sequential`).
fn fillet_edges_group_in(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    chamfer: bool,
    name: Option<&str>,
    memo: Option<&mut ComponentMemo>,
) -> Result<BrepSolid, KernelRefusal> {
    fillet_edges_group_with_corner_policy(solid, edge_points, edge_names, radius, chamfer, name, memo, true, true)
}

fn fillet_edges_group_with_corner_policy(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    chamfer: bool,
    name: Option<&str>,
    memo: Option<&mut ComponentMemo>,
    wrap_concave: bool,
    accept: bool,
) -> Result<BrepSolid, KernelRefusal> {
    GROUP_BUILDS.with(|count| count.set(count.get() + 1));
    let entry = if chamfer { "chamfer_edges" } else { "fillet_edges" };
    let GroupBody { body: result, bowtie_clean } =
        fillet_edges_group_unchecked(solid, edge_points, edge_names, radius, chamfer, name, memo, wrap_concave)?;
    if !accept {
        return Ok(result);
    }
    // A body the network or the closed-rim lane already scanned clean and
    // handed up untouched is not scanned again (the scan is a pure function
    // of the body, 2026-10-12). The composite road and the cutter road hand
    // up bodies no lane scanned, and are scanned here.
    if bowtie_clean {
        crate::blend::record_count("count.bowtie.lane_scan_reused", 1);
    } else {
        check_loop_self_crossings(&result, entry)?;
        check_blend_wall_loops(&result, entry)?;
    }
    let soundness = crate::blend::BlendTimer::start("timing.soundness_ms");
    // The BODY's own soundness, beside the loop's. A blend wall is a FITTED
    // surface and a fit can carry two parameters of one wall to one point in
    // space — measured on `inbox-20260902-fillet-two-torus-box-edges`, whose
    // wall `F8:BLEND:Box_NZ|P.T5_Side[0]` does exactly that while `validate()`,
    // connectivity, Euler parity, the loop scan and the face-versus-face scan
    // are all silent. Two walls crossing each other is the same class and is
    // repaired where it can be. Neither is visible to a volume oracle, so a
    // result carrying one reports success for a different shape.
    crate::soundness::census::profile_operation(&result, &[solid], None, entry);
    let accepted = crate::accept_sound(result, entry);
    soundness.stop();
    accepted
}

/// What the unchecked group lane hands up: the body, and whether a lane's
/// own acceptance already ran the bowtie scan on exactly this body and
/// found it clean (the network and closed-rim exits return the scanned
/// body unchanged; the composite and cutter exits do not scan).
struct GroupBody {
    body: BrepSolid,
    bowtie_clean: bool,
}

/// [`fillet_edges_group_lanes`], answering a selection the network refused
/// by a WALL FOLD and the cutter composition then failed too by that fold,
/// not by the composition's own text: the fold is the reason, and the
/// composition's last cutter ("mates share their end boundary edge" on the
/// 2026-10-21 sculpted cap, report 5e49bb42) names nothing the user can act
/// on. The fold keeps its class, so the subset search above stops on it.
fn fillet_edges_group_unchecked(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    chamfer: bool,
    name: Option<&str>,
    memo: Option<&mut ComponentMemo>,
    wrap_concave: bool,
) -> Result<GroupBody, KernelRefusal> {
    let outer = GROUP_NETWORK_FOLD.with(|fold| fold.borrow_mut().take());
    let built = fillet_edges_group_lanes(solid, edge_points, edge_names, radius, chamfer, name, memo, wrap_concave);
    let fold = GROUP_NETWORK_FOLD.with(|fold| std::mem::replace(&mut *fold.borrow_mut(), outer));
    match (built, fold) {
        (Err(error), Some(fold)) if !crate::blend::is_wall_fold(&error) => Err(fold),
        (built, _) => built,
    }
}

fn fillet_edges_group_lanes(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radius: f64,
    chamfer: bool,
    name: Option<&str>,
    mut memo: Option<&mut ComponentMemo>,
    wrap_concave: bool,
) -> Result<GroupBody, KernelRefusal> {
    use rustc_hash::FxHashSet as HashSet;

    // Fuse-first operand heal (Lever A) before the multi-edge surgery: snap the
    // input's near-coincident / off-plane vertices (e.g. a revolve pole apex
    // sitting a few microns off the axis) to exact and re-anchor incident
    // edges.  The selected edges are resolved geometrically below, so a
    // sub-heal_tol vertex move never changes which edges are picked; a clean
    // input is left byte-identical.
    let heal = crate::blend::BlendTimer::start("timing.heal_operands_ms");
    let mut healed_input = solid.clone();
    let heal_policy = crate::KernelTolerances::for_solid(&healed_input, 1e-7);
    crate::heal::heal_operands(&mut healed_input, &heal_policy)?;
    let solid = &healed_input;
    heal.stop();
    let group_analysis = crate::blend::BlendTimer::start("timing.group_analysis_ms");

    // 1. Detect convex corners from the ORIGINAL solid: a point that is an
    //    endpoint of >=3 of the selected edges (a cube/prism-style vertex).
    let mut endpoints: Vec<(Vec3, usize)> = Vec::with_capacity(edge_points.len() * 2);
    // The extent each selected edge has BEFORE any blend trims it, so the
    // sequential build can run every cutter through the shared corners.
    let mut original_extents: Vec<(Vec3, Vec3)> = Vec::with_capacity(edge_points.len());
    let mut selected_edge_ids: Vec<u64> = Vec::with_capacity(edge_points.len());
    for (i, point) in edge_points.iter().enumerate() {
        let edge_id = resolve_edge_by_point(solid, *point)?;
        let edge = solid
            .edges
            .iter()
            .find(|e| e.id == edge_id)
            .ok_or_else(|| {
                KernelRefusal::internal(
                    KernelStage::Collect,
                    "resolved_edge",
                    "fillet_edges: resolved edge vanished",
                )
            })?;
        let (start, end) = (
            edge.curve
                .evaluate(edge.t0)
                .or_refuse(KernelStage::Collect, "edge_start")?,
            edge.curve
                .evaluate(edge.t1)
                .or_refuse(KernelStage::Collect, "edge_end")?,
        );
        endpoints.push((start, i));
        endpoints.push((end, i));
        original_extents.push((start, end));
        selected_edge_ids.push(edge_id);
    }
    let mut corners: Vec<Vec3> = Vec::new();
    // The selected-edge INPUT INDICES meeting at each star corner (parallel to
    // `corners`), sorted — used to name the corner patch after its adjacent
    // edges (`{fid}:BLEND:CORNER:{e_a}+{e_b}+…`), UNIQUE per corner because no
    // two distinct corners share the same set of >=3 selected edges.
    let mut corner_edges: Vec<Vec<usize>> = Vec::new();
    let mut chain_corner_count = 0usize;
    let mut chain_corners: Vec<(Vec3, [usize; 2])> = Vec::new();
    let mut used = vec![false; endpoints.len()];
    for i in 0..endpoints.len() {
        if used[i] {
            continue;
        }
        used[i] = true;
        let mut edges_here: HashSet<usize> = HashSet::default();
        edges_here.insert(endpoints[i].1);
        for j in (i + 1)..endpoints.len() {
            if used[j] {
                continue;
            }
            if endpoints[i].0.sub(endpoints[j].0).length() < 1e-6 {
                used[j] = true;
                edges_here.insert(endpoints[j].1);
            }
        }
        if edges_here.len() >= 3 {
            corners.push(endpoints[i].0);
            let mut adjacent: Vec<usize> = edges_here.into_iter().collect();
            adjacent.sort_unstable();
            corner_edges.push(adjacent);
        } else if edges_here.len() == 2 {
            chain_corner_count += 1;
            let mut adjacent = edges_here.into_iter().collect::<Vec<_>>();
            adjacent.sort_unstable();
            chain_corners.push((endpoints[i].0, [adjacent[0], adjacent[1]]));
        }
    }

    // A FACE selection may contain several disconnected boundary components
    // (the common example is an outer perimeter plus a circular hole rim).
    // Miter composition is only meaningful inside one connected edge graph.
    // Feeding every component into the same INTERSECT/UNION combines multiple
    // independently filleted copies of otherwise untouched support faces; the
    // boolean then imprints those coincident copies and can leave redundant
    // seams (the reported through-hole cylinder split into three faces).
    //
    // Partition by shared original endpoints and process components in input
    // order.  Each recursive call sees one connected component, so it follows
    // the existing miter/sequential path without recursion cycling.  Distinct
    // components are then composed sequentially on the evolving solid.
    let mut component_of = vec![usize::MAX; edge_points.len()];
    let mut components: Vec<Vec<usize>> = Vec::new();
    for seed in 0..edge_points.len() {
        if component_of[seed] != usize::MAX {
            continue;
        }
        let component_index = components.len();
        component_of[seed] = component_index;
        let mut component = vec![seed];
        let mut cursor = 0;
        while cursor < component.len() {
            let current = component[cursor];
            cursor += 1;
            for candidate in 0..edge_points.len() {
                if component_of[candidate] != usize::MAX {
                    continue;
                }
                let connected = [original_extents[current].0, original_extents[current].1]
                    .into_iter()
                    .any(|a| {
                        [original_extents[candidate].0, original_extents[candidate].1]
                            .into_iter()
                            .any(|b| a.sub(b).length() < 1e-6)
                    });
                if connected {
                    component_of[candidate] = component_index;
                    component.push(candidate);
                }
            }
        }
        component.sort_unstable();
        components.push(component);
    }
    group_analysis.stop();
    if components.len() > 1 {
        let mut separated = solid.clone();
        // The chain continues the memo's prefix, so a component composed
        // inside another's build keys under the whole path to it.
        let mut chain: Vec<ComponentKey> = memo.as_deref().map(|memo| memo.prefix.clone()).unwrap_or_default();
        chain.reserve(components.len());
        for component in components {
            let points = component
                .iter()
                .map(|index| edge_points[*index])
                .collect::<Vec<_>>();
            let names = edge_names.map(|all| {
                component
                    .iter()
                    .map(|index| all[*index].clone())
                    .collect::<Vec<_>>()
            });
            let accept_component = !crate::blend::accept_once_on();
            let Some(memo) = memo.as_deref_mut() else {
                separated = fillet_edges_group_with_corner_policy(
                    &separated,
                    &points,
                    names.as_deref(),
                    radius,
                    chamfer,
                    name,
                    None,
                    wrap_concave,
                    accept_component,
                )?;
                continue;
            };
            let bits = points.iter().map(|p| [p.x.to_bits(), p.y.to_bits(), p.z.to_bits()]);
            chain.push((bits.collect(), names.clone()));
            let built = match memo.built.get(&chain) {
                Some(built) => built.clone(),
                None => {
                    // The component builds on the body the chain before it
                    // composed: its own cutter steps key under that chain.
                    let outer_prefix = std::mem::replace(&mut memo.prefix, chain[..chain.len() - 1].to_vec());
                    let built = fillet_edges_group_with_corner_policy(
                        &separated,
                        &points,
                        names.as_deref(),
                        radius,
                        chamfer,
                        name,
                        Some(&mut *memo),
                        wrap_concave,
                        accept_component,
                    );
                    memo.prefix = outer_prefix;
                    memo.built.insert(chain.clone(), built.clone());
                    built
                }
            };
            separated = built?;
        }
        return Ok(GroupBody { body: separated, bowtie_clean: false });
    }

    // 2a. The stripe network (blend/network.rs) is the lane for EVERY
    //     selection: each stripe is marched against THIS solid, each shared
    //     vertex is solved before anything is cut — a star closed by a patch
    //     of the corner ball, a two-edge corner by the seam between the two
    //     blends, a re-entrant corner by a horn torus, a tangent pair by a
    //     flush join, an unselected tangent continuation by a cap — and
    //     nothing is subtracted, so no cutter can overshoot into a neighbour
    //     and no leftover cap has to be identified afterwards.  It refuses
    //     BY NAME on what it does not yet construct (mixed-convexity corners,
    //     no-common-ball stars, pinched edges), and those fall through to the
    //     cutter composition below.
    //
    //     A CLOSED edge is one of those refusals and it has an answer the
    //     network is simply not the lane for: a closed rim has no corner to
    //     share, so `blend_star_network` says "edge N is closed and has no
    //     corner to share" and the composition took it.  `blend_closed_edge`
    //     builds exactly that case and the single-edge ladder in `tool.rs`
    //     has always tried it first — but this function reached only the
    //     network, so a closed rim inside a GROUP went to a cutter that the
    //     same rim selected on its own never saw.  A bored hole's rim splits
    //     out as its own one-edge component (see the component partition
    //     above), so that was every hole rim chamfered as part of a face
    //     selection.  Try it here too, under the same acceptance the network
    //     result gets, and under the same `BREP_NO_NETWORK` A/B.
    if std::env::var("BREP_NO_NETWORK").is_err() {
        if selected_edge_ids.len() == 1 {
            let edge_id = selected_edge_ids[0];
            let closed = solid
                .edges
                .iter()
                .find(|edge| edge.id == edge_id)
                .is_some_and(|edge| edge.start_vertex_id == edge.end_vertex_id);
            if closed {
                let rim_name = per_edge_name(edge_names, name, 0);
                match crate::blend::blend_closed_edge(solid, edge_id, radius, chamfer, rim_name) {
                    Ok(mut rim) => {
                        let entry = if chamfer { "chamfer_edges" } else { "fillet_edges" };
                        let heal = crate::blend::BlendTimer::start("timing.heal_vertex_gaps_ms");
                        let healed = heal_edge_vertex_gaps(&mut rim, radius);
                        heal.stop();
                        // A trim the surgery ran outside its planar carrier's
                        // chart is a wrong body the three checks below all pass
                        // (`crate::blend::fit_planar_charts_to_trims`), so the
                        // carrier is widened here, after the heal has settled
                        // the edges the trims are rebuilt from and before the
                        // bowtie scan reads them.  Its refusals are TERMINAL:
                        // the composition is not a second opinion on a body
                        // whose trims left their carrier.
                        let charts = crate::blend::BlendTimer::start("timing.planar_charts_ms");
                        crate::blend::fit_planar_charts_to_trims(solid, &mut rim)?;
                        charts.stop();
                        let validate = crate::blend::BlendTimer::start("timing.validate_ms");
                        let issues = rim.validate_memoised();
                        validate.stop();
                        let interference =
                            check_blend_interference(solid, &rim, &selected_edge_ids, entry);
                        let bowtie = check_loop_self_crossings(&rim, entry).and_then(|()| check_blend_wall_loops(&rim, entry));
                        if healed.is_ok()
                            && issues.is_empty()
                            && interference.is_ok()
                            && bowtie.is_ok()
                        {
                            return Ok(GroupBody { body: rim, bowtie_clean: true });
                        }
                        if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                            eprintln!(
                                "closed-rim result rejected: heal={healed:?} issues={issues:?} \
                                 interference={interference:?} bowtie={bowtie:?}"
                            );
                        }
                    }
                    Err(refusal) => {
                        if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                            eprintln!("closed-rim refused: {refusal}");
                        }
                    }
                }
            }
        }
        let network_names: Vec<Option<String>> = (0..edge_points.len())
            .map(|index| per_edge_name(edge_names, name, index).map(str::to_string))
            .collect();
        let network_corner_name =
            |adjacent: &[usize]| corner_face_name(edge_names, name, adjacent);
        // BREP lane L: begin lcount229_knot_aware_group_network_20261016 A
        // support pcurve off the floor ends the group's network here, and
        // the ladder would go on to the cutter composition: re-run the
        // network ONCE with the knot-aware support fit; a re-run that also
        // fails keeps the first refusal.
        let group_network = || {
            crate::blend::blend_star_network_with_corner_policy(
                solid,
                &selected_edge_ids,
                radius,
                chamfer,
                &network_names,
                &network_corner_name,
                wrap_concave,
            )
        };
        let network_result = group_network().or_else(|first| {
            if crate::blend::is_pcurve_floor_refusal(&first) {
                crate::blend::with_pcurve_knot_retry(group_network).map_err(|_| first)
            } else {
                Err(first)
            }
        });
        // BREP lane L: end lcount229_knot_aware_group_network_20261016
        // The network's result under its acceptance: `Ok(Some)` accepted,
        // `Ok(None)` rejected (the ladder goes on), `Err` terminal.
        let accept_network = |mut network: BrepSolid, built_radius: f64| -> Result<Option<GroupBody>, KernelRefusal> {
                // Fail-safe like the rest of the ladder: a heal or validation
                // problem in the network result falls through to the cutter,
                // it does not fail the group.  So does an UNTRIMMED result: the
                // network re-trims only each stripe's two mates, and
                // `check_blend_interference` is what notices a third face
                // crossing the swept volume -- `validate` cannot, because a
                // self-intersecting solid is still a watertight one.
                let entry = if chamfer { "chamfer_edges" } else { "fillet_edges" };
                let heal = crate::blend::BlendTimer::start("timing.heal_vertex_gaps_ms");
                let healed = heal_edge_vertex_gaps(&mut network, built_radius);
                heal.stop();
                // See the closed-rim lane above: widen a planar carrier the
                // surgery overran, or refuse — terminally — rather than fall
                // through to a composition that cannot answer that question.
                let charts = crate::blend::BlendTimer::start("timing.planar_charts_ms");
                crate::blend::fit_planar_charts_to_trims(solid, &mut network)?;
                charts.stop();
                let validate = crate::blend::BlendTimer::start("timing.validate_ms");
                let issues = network.validate_memoised();
                validate.stop();
                let interference =
                    check_blend_interference(solid, &network, &selected_edge_ids, entry);
                // And the fourth question, which the other three all pass: does
                // any face of the result have a loop that crosses ITSELF? A
                // wall whose rail ran off its own face comes back as a bowtie,
                // and a bowtie is watertight, incident and connected.
                let bowtie = check_loop_self_crossings(&network, entry).and_then(|()| check_blend_wall_loops(&network, entry));
                if healed.is_ok() && issues.is_empty() && interference.is_ok() && bowtie.is_ok() {
                    return Ok(Some(GroupBody { body: network, bowtie_clean: true }));
                }
                if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                    eprintln!(
                        "network result rejected: heal={healed:?} issues={issues:?} \
                         interference={interference:?} bowtie={bowtie:?}"
                    );
                    dump_loops_debug("INPUT", solid);
                    dump_loops_debug("RESULT", &network);
                }
                Ok(None)
        };
        match network_result {
            Ok(network) => {
                if let Some(body) = accept_network(network, radius)? {
                    return Ok(body);
                }
            }
            Err(refusal) => {
                if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                    eprintln!("network refused: {refusal}");
                }
                // A stripe whose TAPER to a cusp reaches its full radius only
                // inside the corner setback at its other end has no constant
                // run at all (report 80c5d9d1's F10, 2026-10-24: r 1 on the two
                // straight edges of a sliver at most 2 wide within 0.9 of its
                // square corner). The round is built at the largest smaller
                // radius whose taper clears the setback, and says so, as the
                // relief says it over a whole rim. Before 2026-10-26 the group
                // fell through to the cutter composition, which closed each
                // round on a web and a cap under the neighbouring blend (an
                // undercut) in 9 minutes.
                let pinched_taper = matches!(&refusal.class, crate::RefusalClass::UnsupportedGeometry { what }
                    if what == crate::blend::STRIPE_PINCH_TAPER_WHAT);
                if !chamfer && pinched_taper && crate::blend::relief::radius_relief_on() {
                    let labels: Vec<String> =
                        (0..edge_points.len()).map(|index| selection_label(index, edge_points, edge_names, name)).collect();
                    for step in 1..=10u32 {
                        let reduced = radius * (1.0 - 0.05 * f64::from(step));
                        let built = crate::blend::blend_star_network_with_corner_policy(
                            solid, &selected_edge_ids, reduced, chamfer, &network_names, &network_corner_name, wrap_concave,
                        );
                        if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                            eprintln!(
                                "taper pinch ladder: r {reduced:.4} -> {}",
                                match &built {
                                    Ok(_) => "built".to_string(),
                                    Err(error) => format!("refused: {error}"),
                                }
                            );
                        }
                        match built {
                            Ok(network) => {
                                let Some(body) = accept_network(network, reduced)? else { break };
                                crate::blend::record_blend_note(format!(
                                    "radius reduced over the whole of {} to {reduced:.4}: at r {radius} the round tapering to \
                                     its tip reaches the full radius only inside the corner setback at its other end, so \
                                     r {radius} is reached nowhere",
                                    labels.join(", ")
                                ));
                                return Ok(body);
                            }
                            Err(again) if matches!(&again.class, crate::RefusalClass::UnsupportedGeometry { what }
                                if what == crate::blend::STRIPE_PINCH_TAPER_WHAT) => continue,
                            Err(_) => break,
                        }
                    }
                }
                // The runout composition reads the ring's own classification
                // back when the ring stage falls through to the cutter and
                // that refuses too (`LAST_NETWORK_REFUSAL`).
                LAST_NETWORK_REFUSAL.with(|last| *last.borrow_mut() = Some(refusal.clone()));
                if crate::blend::is_wall_fold(&refusal) {
                    GROUP_NETWORK_FOLD.with(|fold| *fold.borrow_mut() = Some(refusal.clone()));
                }
                // A refusal that is not a fold where the ball does not fit
                // (report 37e16630 at r 1.0, a ball the size of the round it
                // rides: its centre curve collapses to a point and the station
                // Newton is singular) is the fold the relief answers, not a
                // question for the composition, which took 143 s over it.
                if !chamfer && !crate::blend::is_wall_fold(&refusal) && crate::blend::relief::radius_relief_on() {
                    let started = web_time::Instant::now();
                    let misfit = crate::blend::relief::first_misfit(solid, &selected_edge_ids, radius);
                    if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                        eprintln!("radius relief: misfit probe {misfit:?} in {:.1} ms", started.elapsed().as_secs_f64() * 1e3);
                    }
                    if let Some((edge_id, least)) = misfit {
                        return Err(crate::blend::relief::misfit_fold(edge_id, least, radius, &refusal));
                    }
                }
                // The fail-safe above is for a refusal that says "not this
                // lane".  A marched fit that left its carriers does not say
                // that, and the cutter's answer to the same selection is not a
                // second opinion on it: on the notched cap's stuck connector
                // the composition reads 1724.506048919, +1.81e-2 against the
                // closed form, and it VALIDATES.  A refusal that falls through
                // to a validating wrong solid is not a refusal.
                //
                // Not every WALL FOLD, though the single-edge tool reads them all
                // as terminal (`is_terminal_blend_refusal`): a fold of the lens
                // between the rails is one the composition's cutter carves
                // (`inbox_20260902_corner_torus_collar_fillet::each_collar_has_its_own_fold_threshold`,
                // the r 2.1 collar at fold factor 1.036). One that reaches a
                // RAIL is not: the ball pokes out through the face it rides,
                // there is no lens to carve, and every cutter sweeps the same
                // envelope. Until 2026-10-21 the group fell through on that too:
                // report 37e16630's end face at r 1.7 around an r 1 round on its
                // outline ran the composition and its subset search for over 20
                // minutes. A lens fold the composition cannot rescue either is
                // answered by the fold (`fillet_edges_group_unchecked`).
                //
                // A blend stopping within a sliver of its face's width that
                // cannot be snapped onto the far edge without opening the shell
                // is terminal for the same reason: the SIZE is the question
                // (`blend::CONSUMED_SNAP_UNSOUND`), and the composition's tool
                // cut a sliver short of an edge leaves the sliver no lane keeps.
                //
                // And a FULL-WIDTH corner the network cannot collapse soundly
                // (`blend::RAIL_COLLAPSE_UNSUPPORTED`,
                // `blend::FULL_WIDTH_COLLAPSE_UNSOUND`): the composition's answer
                // to a full-width corner was measured wrong — a 5F/8E/5V miter
                // 8.0e-4 off its closed form at 1e-5 short of the width, and the
                // picture-frame corner at a full-width chamfer star.
                //
                // And a trim that left its planar carrier's chart
                // (`blend::PLANAR_CHART_EDGE_OFF_PLANE`,
                // `blend::PLANAR_CHART_WIDEN_UNSOUND`): the clamped body the
                // composition would answer with is the defect being refused.
                // A tangential end whose taper could not be planned also has
                // no supported closure for cutter composition to recover.
                if crate::blend::is_rail_fold(&refusal)
                    || crate::blend::is_marched_fit_off_carriers(&refusal)
                    || crate::blend::is_consumed_snap(&refusal)
                    || crate::blend::is_rail_collapse(&refusal)
                    || crate::blend::is_full_width_collapse(&refusal)
                    || crate::blend::is_planar_chart_refusal(&refusal)
                    || crate::blend::is_miter_support_overrun(&refusal)
                    || crate::blend::is_tangential_transverse_end(&refusal)
                {
                    return Err(refusal);
                }
            }
        }
    }

    // An edge whose ball reaches its support only by rolling over a tangent
    // neighbour (`analyze::rollover_only_refusal`): the composition's tool
    // rides the mates' own carriers past the tangent edge and fills over air
    // (lCount_409 at r 2: +1.096 against the quadrature's 1103.315046), so
    // the reach refusal the unsigned check gave stands.
    if let Some(refusal) = super::analyze::rollover_only_refusal(&selected_edge_ids) {
        return Err(refusal);
    }
    CUTTER_COMPOSITIONS.with(|count| count.set(count.get() + 1));
    crate::blend::record_count("count.cutter_fallback", 1);
    let _cutter = crate::blend::BlendTimer::start("timing.cutter_ms");
    // 2. Build the blends.
    //
    //    CHAIN corners (§6.9.6 — exactly TWO selected edges share a vertex):
    //    the sequential build truncates the later blend where it runs into the
    //    earlier one and closes it with a flat bulkhead across the fillet
    //    channel — a hard step, not a transition.  For selections containing
    //    chain corners, blend each edge FULL-LENGTH on the ORIGINAL solid and
    //    INTERSECT the per-edge results instead: the removal volumes union, so
    //    adjacent blends run through the shared corner and trim each other
    //    along their intersection curve — the standard MITER corner, tangent
    //    to the shared face at the seam's tangency end.  Non-adjacent edges
    //    are unaffected (their removals are disjoint, intersection ≡
    //    sequential).
    //
    //    Selections without chain corners keep the sequential build unchanged
    //    (star corners are rounded in step 3 against exactly the sequential
    //    geometry round_convex_corner was built for).
    //    CONVEXITY GUARD: a convex blend REMOVES material (fillet = orig −
    //    cut), a concave blend ADDS it (orig + pad).  Full-length blends
    //    combine as orig − ∪cuts + ∪pads, so the per-edge results compose by
    //    INTERSECTION when every edge is convex and by UNION when every edge
    //    is concave; a mixed selection has no single composition and falls
    //    back to the sequential build.
    // Fillet stars keep the sequential topology required by their sphere
    // patches. Chamfer stars have no subsequent corner patch: compose their
    // full-length removals too, so all bevel planes meet at the corner.
    let miter_operation = if (chain_corner_count > 0 || !corners.is_empty())
        && (chamfer || corners.is_empty())
    {
        let mut any_convex = false;
        let mut any_concave = false;
        for point in edge_points {
            // Convexity is a DIHEDRAL property, so it is read from the general
            // scan (`scan_dihedral`) — the same one `check_mixed_concavity`
            // has already run over every selected edge.  `analyze_edge` used
            // to answer here and it ALSO refuses an edge the EXACT CUTTER
            // cannot build: a straight edge whose mates are not both planes,
            // a circular one whose mates do not share its axis.  Reading that
            // refusal as "unknown convexity" dropped the §6.9.6 miter for
            // every selection touching a curved carrier — the groove rim of a
            // bored box, say — and the sequential build then left a bulkhead
            // face standing in the shared corner (the 2026-09-10 "chamfer
            // produced a spurious face" report).
            match resolve_edge_by_point(solid, *point)
                .and_then(|edge_id| super::analyze::scan_dihedral(solid, edge_id))
            {
                // A mixed edge has no single composition either; it takes the
                // `_ => None` arm below through both flags.
                Ok(profile) if profile.samples > 0 && !profile.is_mixed() => {
                    any_convex |= profile.any_convex;
                    any_concave |= profile.any_concave;
                }
                // Unknown edge class: let the sequential path produce its own
                // (more specific) error or result.
                _ => {
                    any_convex = true;
                    any_concave = true;
                    break;
                }
            }
        }
        match (any_convex, any_concave) {
            (true, false) => Some(crate::BooleanOperation::Intersect),
            (false, true) => Some(crate::BooleanOperation::Union),
            _ => None,
        }
    } else {
        None
    };

    // Fillet/chamfer each edge in turn, resolving its point on the evolving
    // solid (ids shift as earlier fillets rewrite topology; the midpoint of an
    // edge is untouched by the corner surgery of the others).  This is the
    // baseline build used directly for non-miter selections AND as the
    // fallback when the miter composition below cannot reassemble.
    // The composition's per-edge lane: cutter-first ONLY when a corner
    // closure will run afterwards and read cutter-shaped topology (a star or
    // a chain corner).  A selection with no shared vertex — a lone closed
    // rim, a lone chamfer, disconnected edges — has no such closure, so each
    // edge gets the march first there too.
    let composition_lane = if corners.is_empty() && chain_corner_count == 0 {
        Lane::GeneralFirst
    } else {
        Lane::CutterFirst
    };
    // The order the sequential build blends the selection in: the input
    // order, except that a CONCAVE edge ending at a star corner goes before
    // every convex edge (`sequential_build_order`). The §6.9.7 torus-sector
    // closure of a re-entrant vertex whose concave edge is selected too
    // (`round_mixed_concave_corner`) reads the concave blend's tangency
    // vertices off the convex cap cutters, so it exists only when the concave
    // edge was cut FIRST; a user who picks the keyway's rim and then its
    // inner corners (the 2026-10-13 sprocket report) got flat caps at both
    // corners from the input order, and the closure's own closed form from
    // this one. Every other selection keeps its input order, byte for byte.
    let concave_at_star = concave_star_edges(solid, &selected_edge_ids, &corner_edges, radius, chamfer);
    // And at a star corner whose edges the ladder answers with DIFFERENT
    // cutters, the marched edges go before the exact ones and the revolved
    // cutters last (`mixed_star_class_ranks`): the 2026-10-13 sprocket
    // report whose slit walls were selected as faces.
    let class_ranks = mixed_star_class_ranks(solid, &selected_edge_ids, &corner_edges, radius, chamfer);
    let sequential_order = sequential_build_order(edge_points.len(), &concave_at_star, &class_ranks);
    // And the corners close in the same precedence: a star with a concave
    // edge (the torus sector) before the convex stars, which on the report's
    // keyway share a 3.8-long rim edge with it — the convex star's surgery
    // read an open edge when it ran first (measured 2026-10-13), and closes
    // cleanly after the sector has trimmed that rim.
    let (corners, corner_edges) = corners_concave_first(corners, corner_edges, &concave_at_star);
    let mut build_sequential = || -> Result<BrepSolid, KernelRefusal> {
        let mut sequential = solid.clone();
        // The prefix chain of this build, for the subset search's memo (see
        // `ComponentMemo::sequential`); unused without one.
        let mut chain: Vec<SequentialKey> = Vec::with_capacity(sequential_order.len());
        for &index in &sequential_order {
            let point = &edge_points[index];
            let edge_id = resolve_edge_by_point(&sequential, *point)?;
            // Earlier cutters in this loop TRIM the edges that share a corner
            // with them; extend this cutter back over what they took so the
            // two removal volumes union through the corner (§6.9.6 miter)
            // instead of leaving a wedge of material standing behind a flush
            // end cap.  Untouched edges get a zero pad and the historical
            // flush cutter.
            let ends = tool_ends_to_original_extent(
                &sequential,
                edge_id,
                radius,
                original_extents[index],
                if chamfer { &[] } else { &corners },
            );
            // This edge's blend wall carries the name of THIS input edge
            // (`edge_names[index]`); a smooth chain that engulfs several edges
            // is named after the FIRST such input edge processed here (input
            // order), the chain's representative.
            let edge_name = per_edge_name(edge_names, name, index);
            if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                eprintln!(
                    "sequential build: selection #{index} {} is edge {edge_id} on the evolving body (ends {ends:?}, {composition_lane:?})",
                    selection_label(index, edge_points, edge_names, name)
                );
            }
            // Every step is a deterministic function of the body it starts
            // from and these four inputs, and the body is itself the chain of
            // the steps before it: a chain the search already composed on
            // this solid is answered from the memo, refusals included.
            chain.push((
                [point.x.to_bits(), point.y.to_bits(), point.z.to_bits()],
                edge_name.map(str::to_string),
                ends.key_bits(),
                u8::from(composition_lane == Lane::CutterFirst),
            ));
            if let Some(memo) = memo.as_deref_mut() {
                let key = (memo.prefix.clone(), chain.clone());
                let hit = memo.sequential.get(&key);
                if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                    eprintln!(
                        "sequential memo: {} components before, step {} {} ({} chains stored)",
                        memo.prefix.len(),
                        chain.len(),
                        if hit.is_some() { "HIT" } else { "miss" },
                        memo.sequential.len(),
                    );
                }
                if let Some(built) = hit {
                    crate::blend::record_count("count.sequential.memo_hits", 1);
                    sequential = built.clone()?;
                    continue;
                }
            }
            crate::blend::record_count("count.sequential.steps_built", 1);
            let built = fillet_or_chamfer(
                &sequential,
                edge_id,
                radius,
                chamfer,
                edge_name,
                ends,
                composition_lane,
            );
            if let Some(memo) = memo.as_deref_mut() {
                memo.sequential.insert((memo.prefix.clone(), chain.clone()), built.clone());
            }
            sequential = built?;
        }
        Ok(sequential)
    };

    let mut result = if let Some(operation) = miter_operation {
        // The per-edge results still carry their cutters' standing scaffold,
        // which the group settles once its corners are closed. A scaffold face
        // merged here would hand its tag to whatever it merged with.
        let options = crate::BooleanOptions {
            keep_unmerged_name_substrs: vec![CUTTER_SCAFFOLD_NAME.to_string()],
            ..crate::BooleanOptions::default()
        };
        let mut combined: Result<Option<BrepSolid>, KernelRefusal> = Ok(None);
        // The units the miter composes: one per selected edge, except that
        // edges joined at a TANGENT chain corner blend as one unit through
        // this same group build (see `miter_tangent_units`).
        let units = miter_tangent_units(solid, &selected_edge_ids, &chain_corners, edge_points.len());
        for unit in &units {
            let index = unit[0];
            let point = &edge_points[index];
            let blended = if unit.len() > 1 {
                let points = unit.iter().map(|&member| edge_points[member]).collect::<Vec<_>>();
                let names = edge_names.map(|all| unit.iter().map(|&member| all[member].clone()).collect::<Vec<_>>());
                fillet_edges_group_with_corner_policy(
                    solid,
                    &points,
                    names.as_deref(),
                    radius,
                    chamfer,
                    name,
                    None,
                    wrap_concave,
                    !crate::blend::accept_once_on(),
                )?
            } else {
                let edge_id = resolve_edge_by_point(solid, *point)?;
                let edge_name = per_edge_name(edge_names, name, index);
                let entry = if chamfer { "chamfer_edges" } else { "fillet_edges" };
                check_mixed_concavity(solid, edge_id, entry)?;
                check_support_extent(solid, edge_id, radius, entry)?;
                fillet_or_chamfer(
                    solid,
                    edge_id,
                    radius,
                    chamfer,
                    edge_name,
                    ToolEnds::default(),
                    Lane::CutterFirst,
                )?
            };
            combined = match combined {
                Err(e) => Err(e),
                Ok(None) => Ok(Some(blended)),
                Ok(Some(previous)) => {
                    crate::boolean_operation(&previous, &blended, operation, &options)
                        .map(Some)
                        .map_err(|error| {
                            // The boolean's own class survives the wrap.
                            error.with_message(|error| {
                                format!("fillet_edges: chain-corner miter composition failed: {error}")
                            })
                        })
                }
            };
            if combined.is_err() {
                break;
            }
        }
        // A §6.9.6 miter (per-edge blends intersected/unioned through the
        // shared chain corners) can fail to reassemble on faces whose
        // fragmented boundary does not close — e.g. a planar cap whose ENTIRE
        // perimeter is selected, where fragment_face reports an "incomplete
        // run".  Rather than let `fillet_edges` silently DROP a selected edge
        // to recover a valid subset (the reported defect: not every edge of
        // the face gets a fillet), fall back to the sequential build, which
        // blends EVERY selected edge (chain corners get a flat bulkhead
        // instead of a miter).  Only if that also fails to produce a valid
        // solid do we surface the miter error so the caller's
        // maximal-valid-subset search can still run.
        match combined {
            Ok(Some(mut mitered)) => {
                drop_slit_shells(&mut mitered);
                mitered
            }
            Ok(None) => {
                return Err(KernelRefusal::input(
                    KernelStage::Collect,
                    "miter_selection",
                    "fillet_edges: no edges selected",
                ))
            }
            Err(miter_err) => match build_sequential() {
                Ok(seq) if seq.validate().is_empty() => seq,
                _ => return Err(miter_err),
            },
        }
    } else {
        build_sequential()?
    };

    // A re-entrant vertex of a selected face perimeter has two selected,
    // convex cap-wall edges and one UNSELECTED concave wall-wall edge.  The
    // two cutter volumes only touch there, so the boolean miter leaves a
    // triangular planar end-cap instead of carrying the rolling ball around
    // the corner.  Close that exact orthogonal class with its horn-torus
    // sector (major radius = minor radius = fillet radius).
    // BREP lane L: begin keyway_floor_open_mixed_corner_20261014
    let mut open_mixed_corners: Vec<(Vec3, [u64; 2], Option<String>)> = Vec::new();
    // BREP lane L: end keyway_floor_open_mixed_corner_20261014
    if !chamfer {
        for (corner, adjacent) in &chain_corners {
            let has_concave_unselected_edge = solid.edges.iter().any(|edge| {
                if selected_edge_ids.contains(&edge.id) {
                    return false;
                }
                let Ok(a) = edge.curve.evaluate(edge.t0) else {
                    return false;
                };
                let Ok(b) = edge.curve.evaluate(edge.t1) else {
                    return false;
                };
                (a.sub(*corner).length() < 1e-6 || b.sub(*corner).length() < 1e-6)
                    && analyze_edge(solid, edge.id, radius)
                        .map(|cross| !cross.convex)
                        .unwrap_or(false)
            });
            // BREP lane L: begin keyway_floor_open_mixed_corner_20261014
            // The OPEN mixed corner (the 2026-10-13 sprocket's keyway floor):
            // one selected convex rim, one selected concave vertical and an
            // UNSELECTED convex third edge. The rim's cutter ran through the
            // concave pad and left a scaffold square and a misplaced
            // bulkhead there; the closure is the one-ended runout's body —
            // the §6.9.7 torus sector to C5's stop, ended by its cliff
            // (`blend/corner/mixed_open.rs`). It reads the SETTLED body (the
            // pad's flush end merged into the cap), so the corner is noted
            // here and closed after `settle_cutter_scaffold` below.
            if !has_concave_unselected_edge {
                let convexity = |edge_id: u64| analyze_edge(solid, edge_id, radius).map(|cross| cross.convex).ok();
                let selected_pair = [convexity(selected_edge_ids[adjacent[0]]), convexity(selected_edge_ids[adjacent[1]])];
                let one_concave = matches!(selected_pair, [Some(true), Some(false)] | [Some(false), Some(true)]);
                let has_convex_unselected_edge = solid.edges.iter().any(|edge| {
                    if selected_edge_ids.contains(&edge.id) {
                        return false;
                    }
                    let (Ok(a), Ok(b)) = (edge.curve.evaluate(edge.t0), edge.curve.evaluate(edge.t1)) else {
                        return false;
                    };
                    (a.sub(*corner).length() < 1e-6 || b.sub(*corner).length() < 1e-6)
                        && analyze_edge(solid, edge.id, radius).map(|cross| cross.convex).unwrap_or(false)
                });
                if one_concave && has_convex_unselected_edge && crate::blend::open_mixed_corner_on() {
                    open_mixed_corners.push((
                        *corner,
                        [selected_edge_ids[adjacent[0]], selected_edge_ids[adjacent[1]]],
                        corner_face_name(edge_names, name, adjacent),
                    ));
                }
                continue;
            }
            // BREP lane L: end keyway_floor_open_mixed_corner_20261014
            let corner_name = corner_face_name(edge_names, name, adjacent);
            if let Ok(rounded) = crate::blend::round_concave_chain_corner(
                &result,
                solid,
                *corner,
                [
                    selected_edge_ids[adjacent[0]],
                    selected_edge_ids[adjacent[1]],
                ],
                radius,
                corner_name.as_deref(),
            ) {
                result = rounded;
            }
        }
    }

    // 3. Round the convex corners (fillets only — chamfers keep sharp
    //    vertices).  A corner that cannot be rounded is left as the edge
    //    fillets so the group still succeeds.  Each corner patch is named after
    //    the selected edges meeting there (`{fid}:BLEND:CORNER:{e_a}+…`) so no
    //    two corners collide and the patch stays under the `{fid}:BLEND` prefix.
    if !chamfer {
        for (ci, corner) in corners.iter().enumerate() {
            let corner_name = corner_face_name(edge_names, name, &corner_edges[ci]);
            match crate::blend::round_convex_corner(&result, *corner, radius, corner_name.as_deref()) {
                Ok(rounded) => result = rounded,
                Err(error) => {
                    if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                        eprintln!("corner at {corner:?} left as the edge fillets: {error}");
                    }
                }
            }
        }
    }

    // Heal any residual vertex/edge gaps introduced by the corner-rounding
    // surgery (the per-edge results are already healed inside fillet_or_chamfer).
    heal_edge_vertex_gaps(&mut result, radius)?;
    crate::blend::fit_planar_charts_to_trims(solid, &mut result)?;
    // BREP lane L: begin keyway_floor_open_mixed_corner_20261014
    let mut body = settle_cutter_scaffold(result);
    for (corner, pair, corner_name) in &open_mixed_corners {
        match crate::blend::round_mixed_concave_open_corner(&body, solid, *corner, *pair, radius, corner_name.as_deref()) {
            Ok(rounded) => body = rounded,
            Err(error) => {
                if std::env::var("BREP_DEBUG_NETWORK").is_ok() {
                    eprintln!("open mixed corner at {corner:?} left as the composition built it: {error}");
                }
            }
        }
    }
    Ok(GroupBody { body, bowtie_clean: false })
    // BREP lane L: end keyway_floor_open_mixed_corner_20261014
}

/// The selected edges (indices into the selection) that are CONCAVE and end
/// at a star corner (`corner_edges`: three or more selected edges at one
/// vertex), in input order. Empty for a chamfer, a selection with no star
/// corner, one whose star corners carry no concave edge, and one whose
/// convexity cannot be read — the cases the composition builds in input
/// order, exactly as before.
/// Whether the miter composition drops a stray SLIT shell: on unless
/// `BREP_FILLET_MITER_DROP_SLIT_SHELLS=0`/`off`.
fn drop_slit_shells_on() -> bool {
    switch!(live "BREP_FILLET_MITER_DROP_SLIT_SHELLS", true)
}

/// Remove from a composed miter body every shell beyond the first whose faces
/// are all SLITS — each loop runs along each of its edges once forward and
/// once back — over edges no other shell uses, then the edges and vertices
/// nothing references any more. Returns how many shells went.
///
/// Such a shell bounds no area and no volume. The 2026-11-07 "fillet fails"
/// report's five-edge selection composed to the right body (the volume of
/// the same composition on the serialised input, which has one shell) plus
/// one face of the hole's floor plane, `Box_NZ_Offset_1_1`, whose single loop
/// ran 1.25 out along one edge and straight back; `validate()` accepts it.
fn drop_slit_shells(body: &mut BrepSolid) -> usize {
    if !drop_slit_shells_on() || body.shells.len() < 2 {
        return 0;
    }
    let mut users: std::collections::HashMap<u64, std::collections::HashSet<usize>> = std::collections::HashMap::new();
    for (index, shell) in body.shells.iter().enumerate() {
        for coedge in shell.faces.iter().flat_map(|face| &face.loops).flat_map(|lp| &lp.coedges) {
            users.entry(coedge.edge_id).or_default().insert(index);
        }
    }
    let slit_face = |face: &crate::topology::FaceRecord| {
        face.loops.iter().all(|lp| {
            let mut runs: std::collections::HashMap<u64, (usize, usize)> = std::collections::HashMap::new();
            for coedge in &lp.coedges {
                let entry = runs.entry(coedge.edge_id).or_default();
                if coedge.forward {
                    entry.0 += 1;
                } else {
                    entry.1 += 1;
                }
            }
            !runs.is_empty() && runs.values().all(|&(forward, back)| forward == 1 && back == 1)
        })
    };
    let mut index = 0usize;
    let mut dropped = 0usize;
    body.shells.retain(|shell| {
        let this = index;
        index += 1;
        let slit = this > 0
            && !shell.faces.is_empty()
            && shell.faces.iter().all(|face| slit_face(face))
            && shell
                .faces
                .iter()
                .flat_map(|face| &face.loops)
                .flat_map(|lp| &lp.coedges)
                .all(|coedge| users.get(&coedge.edge_id).is_some_and(|set| set.len() == 1));
        dropped += usize::from(slit);
        !slit
    });
    if dropped == 0 {
        return 0;
    }
    let used_edges: std::collections::HashSet<u64> = body
        .shells
        .iter()
        .flat_map(|shell| &shell.faces)
        .flat_map(|face| &face.loops)
        .flat_map(|lp| &lp.coedges)
        .map(|coedge| coedge.edge_id)
        .collect();
    body.edges.retain(|edge| used_edges.contains(&edge.id));
    let used_vertices: std::collections::HashSet<u64> =
        body.edges.iter().flat_map(|edge| [edge.start_vertex_id, edge.end_vertex_id]).collect();
    body.vertices.retain(|vertex| used_vertices.contains(&vertex.id));
    dropped
}

/// Whether the miter composition blends a TANGENT run of the selection as
/// one unit: on unless `BREP_FILLET_MITER_TANGENT_UNITS=0`/`off`.
fn miter_tangent_units_on() -> bool {
    switch!(live "BREP_FILLET_MITER_TANGENT_UNITS", true)
}

/// The units the §6.9.6 miter composition blends and then intersects (or
/// unions), in input order of each unit's first edge: every selected edge on
/// its own, except that edges meeting at a chain corner where their tangents
/// CONTINUE each other (a G1 joint, the 1e-6 bar of the tangent-chain walk)
/// form one unit.
///
/// The miter exists for SHARP chain corners: two full-length blends run
/// through the corner and trim each other. At a tangent joint there is
/// nothing to trim — the two walls meet flush — and two full-length
/// per-edge blends of the edges either side of it, each capped where its
/// tangent neighbour was left sharp, overlap along that joint in a sliver
/// the boolean cannot conform (2026-11-07 "fillet fails" report: a revolve
/// end cap's hole loop, one cylinder edge meeting a ceiling plane edge that
/// runs on tangentially through an offset blend into a wall; "one-use edge
/// conformance did not converge after 8 passes" on the plane-edge result
/// intersected with the blend-edge result). The network lane builds such a
/// tangent run whole with flush joins; it refused the selection only for
/// its sharp re-entrant corners, so a unit that is a STRICT part of the
/// selection goes back through the group build, which reaches it there.
/// A selection that is one tangent run keeps per-edge units, so the
/// recursion always shrinks the selection.
fn miter_tangent_units(
    solid: &BrepSolid,
    selected_edge_ids: &[u64],
    chain_corners: &[(Vec3, [usize; 2])],
    count: usize,
) -> Vec<Vec<usize>> {
    let singles = || (0..count).map(|index| vec![index]).collect::<Vec<_>>();
    if !miter_tangent_units_on() || selected_edge_ids.len() != count {
        return singles();
    }
    // The tangent along `edge` pointing AWAY from its end at `corner`.
    let leaving = |edge_id: u64, corner: Vec3| -> Option<Vec3> {
        let edge = solid.edges.iter().find(|edge| edge.id == edge_id)?;
        if edge.degenerate {
            return None;
        }
        let start = edge.curve.evaluate(edge.t0).ok()?;
        let end = edge.curve.evaluate(edge.t1).ok()?;
        let at_start = start.sub(corner).length() <= end.sub(corner).length();
        let t = if at_start { edge.t0 } else { edge.t1 };
        let tangent = edge.curve.unit_tangent(t, edge.t0, edge.t1).ok()?;
        Some(if at_start { tangent } else { tangent.scale(-1.0) })
    };
    let mut parent: Vec<usize> = (0..count).collect();
    fn root(parent: &mut [usize], mut index: usize) -> usize {
        while parent[index] != index {
            parent[index] = parent[parent[index]];
            index = parent[index];
        }
        index
    }
    for (corner, [a, b]) in chain_corners {
        let (Some(ta), Some(tb)) = (leaving(selected_edge_ids[*a], *corner), leaving(selected_edge_ids[*b], *corner)) else {
            continue;
        };
        // Leaving the corner, a tangent continuation points the opposite way.
        if ta.dot(tb) <= -(1.0 - 1e-6) {
            let (ra, rb) = (root(&mut parent, *a), root(&mut parent, *b));
            if ra != rb {
                parent[ra.max(rb)] = ra.min(rb);
            }
        }
    }
    let mut units: Vec<Vec<usize>> = Vec::new();
    let mut unit_of_root: std::collections::HashMap<usize, usize> = std::collections::HashMap::new();
    for index in 0..count {
        let r = root(&mut parent, index);
        let slot = *unit_of_root.entry(r).or_insert_with(|| {
            units.push(Vec::new());
            units.len() - 1
        });
        units[slot].push(index);
    }
    if units.len() <= 1 || units.iter().all(|unit| unit.len() == 1) {
        return singles();
    }
    units
}

fn concave_star_edges(
    solid: &BrepSolid,
    selected_edge_ids: &[u64],
    corner_edges: &[Vec<usize>],
    radius: f64,
    chamfer: bool,
) -> Vec<usize> {
    if chamfer || corner_edges.is_empty() || !concave_first_on() {
        return Vec::new();
    }
    let Some(convexity) = crate::blend::selection_convexity(solid, selected_edge_ids, radius) else {
        return Vec::new();
    };
    (0..selected_edge_ids.len())
        .filter(|&index| !convexity[index] && corner_edges.iter().any(|adjacent| adjacent.contains(&index)))
        .collect()
}

/// The cutter the sequential build's single-edge ladder answers a selected
/// edge with, read on the ORIGINAL solid: an exact extrusion (a straight
/// edge between planes), an exact revolution (a circular edge on
/// straight-meridian carriers about its own axis) or, for every other edge,
/// the general march (`fillet_or_chamfer_exact` refuses it, so the march
/// builds it with direct surgery on the evolving body).
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
enum CutterClass {
    Marched,
    Straight,
    Circular,
}

fn cutter_class(solid: &BrepSolid, edge_id: u64, radius: f64) -> CutterClass {
    match super::analyze::analyze_edge(solid, edge_id, radius) {
        Ok(cross) => match cross.path {
            super::analyze::EdgePath::Straight { .. } => CutterClass::Straight,
            super::analyze::EdgePath::Circular { .. } => CutterClass::Circular,
        },
        Err(_) => CutterClass::Marched,
    }
}

/// The sequential build's rank of every selected edge, from the cutter
/// classes meeting at the selection's star corners: `1` for a marched edge
/// and `3` for a revolved cutter at a star whose edges MIX classes, `2` for
/// everything else (a star of one class, a chain corner, a lone edge).
/// Within a rank the input order holds ([`sequential_build_order`]).
///
/// At such a corner the input order fails in two ways, both measured on the
/// 2026-10-13 sprocket report whose slit walls were selected as faces
/// (`inbox_20261013_sprocket_keyway_slit_fillet`): a revolved cutter leaves
/// a TORUS tangent to the hub's top plane along its rail, and the next
/// straight cutter's scaffold face lies in that very plane and ends on that
/// rail, so the boolean meets a tangent plane/torus contact along a 0.148
/// arc that the imprint traces but the assembly cannot close (`unsupported
/// tangent-curve surface intersection`, open edges, a 0.067 curve/vertex
/// gap); and the slit's vertical, a straight edge on the hub CYLINDER that
/// only the march builds, finds "no single end face across the corner" once
/// its star mates have been cut. The marched edge cut first sees the pristine
/// corner, and a straight cutter cut before the revolved one meets the
/// scaffold plane of a CYLINDER tangent along a line — the ordinary planar
/// star's own geometry, which the arrangement has always closed. Every
/// selection without a mixed-class star keeps its order, byte for byte.
fn mixed_star_class_ranks(
    solid: &BrepSolid,
    selected_edge_ids: &[u64],
    corner_edges: &[Vec<usize>],
    radius: f64,
    chamfer: bool,
) -> Vec<u8> {
    let mut ranks = vec![2u8; selected_edge_ids.len()];
    if chamfer || corner_edges.is_empty() || !class_order_on() {
        return ranks;
    }
    let classes: Vec<CutterClass> = selected_edge_ids
        .iter()
        .map(|&edge_id| cutter_class(solid, edge_id, radius))
        .collect();
    for adjacent in corner_edges {
        let Some(&head) = adjacent.first() else {
            continue;
        };
        if adjacent.iter().all(|&index| classes[index] == classes[head]) {
            continue;
        }
        for &index in adjacent {
            match classes[index] {
                CutterClass::Marched => ranks[index] = 1,
                CutterClass::Circular => ranks[index] = 3,
                CutterClass::Straight => {}
            }
        }
    }
    ranks
}

/// The order the cutter composition's sequential build blends a selection
/// in: the input order, except that the edges in `first` (a star corner's
/// concave edges, [`concave_star_edges`]) are moved ahead of every other
/// edge and the rest follow by `ranks` ([`mixed_star_class_ranks`], lower
/// first), keeping the input order among the moved edges and within each
/// rank. With `first` empty and every rank equal this is the input order.
pub(super) fn sequential_build_order(count: usize, first: &[usize], ranks: &[u8]) -> Vec<usize> {
    let mut order: Vec<usize> = (0..count).collect();
    // A stable sort: ties keep the input order.
    order.sort_by_key(|&index| if first.contains(&index) { 0 } else { ranks.get(index).copied().unwrap_or(2).max(1) });
    order
}

/// Whether the sequential build ranks a mixed-class star's edges by cutter
/// class ([`mixed_star_class_ranks`]); always, outside a test that forces it
/// off.
fn class_order_on() -> bool {
    true
}


/// `corners` and their `corner_edges` (parallel), with every corner that one
/// of the `first` edges ends at moved ahead of the rest, both runs in their
/// detection order. With `first` empty both come back as they were.
fn corners_concave_first(
    corners: Vec<Vec3>,
    corner_edges: Vec<Vec<usize>>,
    first: &[usize],
) -> (Vec<Vec3>, Vec<Vec<usize>>) {
    if first.is_empty() {
        return (corners, corner_edges);
    }
    let mut ordered: Vec<(Vec3, Vec<usize>)> = Vec::with_capacity(corners.len());
    let pairs: Vec<(Vec3, Vec<usize>)> = corners.into_iter().zip(corner_edges).collect();
    let with_concave = |adjacent: &Vec<usize>| adjacent.iter().any(|index| first.contains(index));
    ordered.extend(pairs.iter().filter(|(_, adjacent)| with_concave(adjacent)).cloned());
    ordered.extend(pairs.iter().filter(|(_, adjacent)| !with_concave(adjacent)).cloned());
    ordered.into_iter().unzip()
}

/// Whether the sequential build moves a star corner's concave edge first
/// ([`sequential_build_order`]); always, outside a test that forces it off.
fn concave_first_on() -> bool {
    true
}


/// Variable-radius fillet/chamfer of a GROUP of edges (§4.9.5), the app entry
/// for tapered blends: each selected edge (resolved by a point on it) is
/// blended with the SAME radius profile `radii` — a list of (edge-fraction,
/// radius) stops in `[0,1]` — applied along that edge's own parameterization.
/// Edges are blended independently (no shared-vertex corner rounding; a
/// variable-radius star has no single tangent ball), so this is the tapered
/// counterpart of `fillet_edges` for the constant case.
pub fn fillet_edges_variable(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    radii: &[(f64, f64)],
    chamfer: bool,
    name: Option<&str>,
) -> Result<BrepSolid, KernelRefusal> {
    // One blend operation: the reports of an earlier one, never consumed, are
    // dropped as it starts (nested calls keep this operation's).
    let _operation = crate::blend::BlendOperation::enter();
    if edge_points.is_empty() {
        return Err(KernelRefusal::input(
            KernelStage::Collect,
            "variable_selection",
            "fillet_edges_variable: no edges selected",
        ));
    }
    let per_edge_stops: Vec<Vec<(f64, f64)>> = vec![radii.to_vec(); edge_points.len()];
    fillet_edges_variable_impl(
        solid,
        edge_points,
        edge_names,
        &per_edge_stops,
        chamfer,
        name,
        "fillet_edges_variable",
        true,
    )
}

/// The shared variable-radius group core: each selected edge `i` is blended
/// with ITS OWN stop list `per_edge_stops[i]` (the legacy entry replicates one
/// list; the law entries sample a chain-abscissa [`crate::law::RadiusLaw`] per
/// edge).  `allow_tapered_sequential` keeps the legacy entry's sequential
/// fallback for tapered chains (whose end state is the honest
/// "mismatched radii" validation gate); the law entries pass `false` because
/// their stop fractions are computed against the ORIGINAL edges — the
/// sequential build re-resolves edges on the evolving solid whose shared
/// corners are already TRIMMED by earlier blends, which would silently distort
/// the law's abscissa mapping (constant stops are immune, so they may still
/// fall back).
#[allow(clippy::too_many_arguments)]
fn fillet_edges_variable_impl(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    per_edge_stops: &[Vec<(f64, f64)>],
    chamfer: bool,
    name: Option<&str>,
    entry: &str,
    allow_tapered_sequential: bool,
) -> Result<BrepSolid, KernelRefusal> {
    // The translation control (`BREP_TRANSLATION_CONTROL`, `brep/recentre.rs`)
    // and the recentre guard, as `fillet_edges_reported` applies them.
    let bounds = crate::recentre::solid_bounds(solid);
    let translated = match crate::recentre::control_offset_at_entry() {
        Some(offset) => Some((offset, crate::recentre::EntryKind::Control)),
        None => crate::recentre::guard_offset(bounds).map(|offset| {
            crate::recentre::profile_fired(entry, offset, bounds);
            (offset, crate::recentre::EntryKind::Guard)
        }),
    };
    if let Some((offset, kind)) = translated {
        let _entry = crate::recentre::TranslatedEntry::open(kind);
        let translate = |solid: &BrepSolid, offset: Vec3| {
            crate::recentre::translate_solid(solid, offset).map_err(|message| {
                KernelRefusal::internal(
                    KernelStage::Collect,
                    "translation_control",
                    format!("{entry}: the translation control could not move the solid: {message}"),
                )
            })
        };
        let moved = translate(solid, offset)?;
        let points: Vec<Vec3> = edge_points.iter().map(|point| point.add(offset)).collect();
        let result = fillet_edges_variable_impl(
            &moved,
            &points,
            edge_names,
            per_edge_stops,
            chamfer,
            name,
            entry,
            allow_tapered_sequential,
        )?;
        return translate(&result, offset.scale(-1.0));
    }
    debug_assert_eq!(per_edge_stops.len(), edge_points.len());
    // A law that is one constant everywhere IS the constant-radius fillet:
    // hand it to the constant group, whose corners are constructed (the
    // stripe network) rather than composed by boolean.
    if let Some(constant) = per_edge_stops
        .first()
        .and_then(|stops| stops.first())
        .map(|(_, radius)| *radius)
    {
        let uniform = per_edge_stops.iter().all(|stops| {
            stops
                .iter()
                .all(|(_, radius)| (radius - constant).abs() <= 1e-12 * (1.0 + constant.abs()))
        });
        if uniform && constant > 0.0 {
            return fillet_edges(solid, edge_points, edge_names, constant, chamfer, name);
        }
    }
    let max_radius = per_edge_stops
        .iter()
        .flat_map(|stops| stops.iter())
        .map(|(_, r)| r.abs())
        .fold(0.0_f64, f64::max);

    // Tried first behind its switch; whatever the network does not build (an
    // unequal-radius corner, a flush join or re-entrant vertex at a varying
    // radius, a result its acceptance rejects) falls through to the
    // composition below exactly as before.
    if crate::blend::taper_network_on() && edge_points.len() >= 2 {
        match fillet_variable_network(
            solid, edge_points, edge_names, per_edge_stops, chamfer, name, entry, max_radius,
        )? {
            Ok(built) => return Ok(built),
            // A radius relief's law is the network's to build: the
            // composition below took 75 s a plan on report 37e16630 at r 1.0
            // and built none, so the relief's next plan is asked instead.
            Err(refusal) if crate::blend::relief::relieving() => return Err(refusal),
            Err(_) => {}
        }
    }

    // Chain corners miter exactly like the constant-radius group (§6.9.6):
    // per-edge blends on the ORIGINAL solid composed by boolean — Intersect
    // when every edge is convex, Union when every edge is concave.  Mixed or
    // unclassifiable selections keep the sequential build.  Variable blends
    // never round star vertices, so unlike the constant group there is no
    // sequential-only star path to protect.
    let mut chain_corner = false;
    {
        use rustc_hash::FxHashSet as HashSet;
        let mut endpoints: Vec<(Vec3, usize)> = Vec::with_capacity(edge_points.len() * 2);
        for (i, point) in edge_points.iter().enumerate() {
            if let Ok(edge_id) = resolve_edge_by_point(solid, *point) {
                if let Some(edge) = solid.edges.iter().find(|e| e.id == edge_id) {
                    if let (Ok(a), Ok(b)) =
                        (edge.curve.evaluate(edge.t0), edge.curve.evaluate(edge.t1))
                    {
                        endpoints.push((a, i));
                        endpoints.push((b, i));
                    }
                }
            }
        }
        let mut used = vec![false; endpoints.len()];
        for i in 0..endpoints.len() {
            if used[i] {
                continue;
            }
            used[i] = true;
            let mut edges_here: HashSet<usize> = HashSet::default();
            edges_here.insert(endpoints[i].1);
            for j in (i + 1)..endpoints.len() {
                if used[j] {
                    continue;
                }
                if endpoints[i].0.sub(endpoints[j].0).length() < 1e-6 {
                    used[j] = true;
                    edges_here.insert(endpoints[j].1);
                }
            }
            if edges_here.len() == 2 {
                chain_corner = true;
            }
        }
    }
    let miter_operation = if chain_corner {
        let probe_radius = if max_radius > 0.0 { max_radius } else { 1.0 };
        let mut any_convex = false;
        let mut any_concave = false;
        for point in edge_points {
            match resolve_edge_by_point(solid, *point)
                .and_then(|edge_id| analyze_edge(solid, edge_id, probe_radius))
            {
                Ok(cross) if cross.convex => any_convex = true,
                Ok(_) => any_concave = true,
                Err(_) => {
                    any_convex = true;
                    any_concave = true;
                    break;
                }
            }
        }
        match (any_convex, any_concave) {
            (true, false) => Some(crate::BooleanOperation::Intersect),
            (false, true) => Some(crate::BooleanOperation::Union),
            _ => None,
        }
    } else {
        None
    };

    // Try the miter first; the variable blend's FITTED boundary curves are
    // only ~1e-3 accurate at blend-blend tangencies (unlike the exact
    // constant-radius cylinders), so the composition can fail — fall back to
    // the sequential build then, which is never worse than the pre-miter
    // behavior.  Tightening the taper surface's endpoint fitting is the
    // documented follow-up that would make the miter stick.
    let miter_attempt: Option<BrepSolid> = if let Some(operation) = miter_operation {
        let options = crate::BooleanOptions::default();
        let mut combined: Option<BrepSolid> = None;
        let mut failed = false;
        for (index, point) in edge_points.iter().enumerate() {
            let Ok(edge_id) = resolve_edge_by_point(solid, *point) else {
                failed = true;
                break;
            };
            let edge_name = per_edge_name(edge_names, name, index);
            let Ok(blended) = crate::blend::blend_edge_variable(
                solid,
                edge_id,
                &per_edge_stops[index],
                chamfer,
                edge_name,
            ) else {
                failed = true;
                break;
            };
            let next = match combined.take() {
                None => blended,
                Some(previous) => {
                    match crate::boolean_operation(&previous, &blended, operation, &options) {
                        Ok(next) => next,
                        Err(_) => {
                            failed = true;
                            break;
                        }
                    }
                }
            };
            combined = Some(next);
        }
        if failed {
            None
        } else {
            combined.filter(|s| s.validate().is_empty())
        }
    } else {
        None
    };
    let mut result = match miter_attempt {
        Some(mitered) => mitered,
        None => {
            // Any stop list that is NOT radius-uniform (the same 1e-12
            // relative criterion `blend_edge_variable` uses for its exact
            // constant-radius degeneration) makes the sequential build
            // abscissa-distorting on trimmed chain edges; law entries refuse
            // instead of silently shifting the law.
            let tapered = per_edge_stops.iter().any(|stops| {
                stops.first().is_some_and(|(_, first)| {
                    stops
                        .iter()
                        .any(|(_, r)| (r - first).abs() > 1e-12 * (1.0 + first.abs()))
                })
            });
            if chain_corner && tapered && !allow_tapered_sequential {
                // The transition patch that would carry a taper through the
                // corner is not built: a named deferral.
                return Err(KernelRefusal::unsupported(
                    KernelStage::Sew,
                    "tapered_chain_miter",
                    format!(
                        "{entry}: the tapered blends across the selected chain's shared corners \
                         did not compose to a valid solid (the fitted blend boundaries could not \
                         be mitered); fillet fewer edges per operation or reduce the taper"
                    ),
                ));
            }
            let mut sequential = solid.clone();
            for (index, point) in edge_points.iter().enumerate() {
                let edge_id = resolve_edge_by_point(&sequential, *point)?;
                let edge_name = per_edge_name(edge_names, name, index);
                sequential = crate::blend::blend_edge_variable(
                    &sequential,
                    edge_id,
                    &per_edge_stops[index],
                    chamfer,
                    edge_name,
                )?;
            }
            sequential
        }
    };
    // Heal §6.9 surgery so re-trimmed edges meet their vertices exactly; scale
    // the heal bound by the largest radius stop in the taper profile.
    heal_edge_vertex_gaps(&mut result, max_radius)?;
    crate::blend::fit_planar_charts_to_trims(solid, &mut result)?;
    // Final honesty gate: a genuinely TAPERED chain (different radii at the
    // shared corner) has mismatched trim stations there — the blends cannot
    // meet without a transition patch (not implemented), and the sequential
    // surgery silently left broken topology before this gate existed.
    let issues = result.validate();
    if !issues.is_empty() {
        // Two verdicts share the text's shape: the legacy entry names the
        // transition patch it does not build (a deferral); the law entries,
        // whose taper is continuous through the corner, report the
        // composition failing its own validation (a check that tripped).
        let (detail, class) = if allow_tapered_sequential {
            (
                "tapered blends meet at a shared chain vertex with \
                 mismatched radii — the radius-transition corner patch is not implemented; \
                 fillet the edges in separate operations or use matching stop radii",
                RefusalClass::UnsupportedGeometry {
                    what: "taper_transition_patch".into(),
                },
            )
        } else {
            (
                "the composed radius-law blend produced invalid topology — the blends \
                 across a shared chain corner failed to reassemble",
                RefusalClass::Internal {
                    what: "law_chain_topology".into(),
                },
            )
        };
        return Err(KernelRefusal::new(
            class,
            KernelStage::Validate,
            format!(
                "{entry}: {detail} ({} validation issues, first: {})",
                issues.len(),
                issues
                    .first()
                    .map(|issue| issue.message.clone())
                    .unwrap_or_default()
            ),
        ));
    }
    // The variable-radius lane's own soundness acceptance, the same one the
    // constant-radius group exit takes. `validate()` above is an INCIDENCE
    // test: a tapered wall that passes through a neighbour, or through itself,
    // satisfies every check it makes and still measures a volume.
    crate::soundness::census::profile_operation(&result, &[solid], None, entry);
    crate::accept_sound(result, entry)
}

/// `Ok(None)` is "not this lane" — the network refused, or its result
/// failed the same acceptance the constant group holds the network to —
/// and the caller composes as before; `Err` is a refusal nothing below
/// can answer.
#[allow(clippy::too_many_arguments)]
fn fillet_variable_network(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    per_edge_stops: &[Vec<(f64, f64)>],
    chamfer: bool,
    name: Option<&str>,
    entry: &str,
    max_radius: f64,
) -> Result<Result<BrepSolid, KernelRefusal>, KernelRefusal> {
    let mut edge_ids = Vec::with_capacity(edge_points.len());
    for point in edge_points {
        edge_ids.push(resolve_edge_by_point(solid, *point)?);
    }
    let laws: Vec<crate::blend::StripeRadius> =
        per_edge_stops.iter().map(|stops| crate::blend::StripeRadius::from_stops(stops)).collect();
    let network_names: Vec<Option<String>> = (0..edge_points.len())
        .map(|index| per_edge_name(edge_names, name, index).map(str::to_string))
        .collect();
    let network_corner_name = |adjacent: &[usize]| corner_face_name(edge_names, name, adjacent);
    let debug = std::env::var("BREP_DEBUG_NETWORK").is_ok();
    // A support pcurve off the floor is re-run ONCE with the knot-aware
    // support fit, as the constant group's network is; a re-run that also
    // fails keeps the first refusal. A radius relief's law crowds the
    // march's stations into its ramp, and the rail's knot spans there are
    // finer than the uniform track can follow (a reported model: r 1.4 on
    // the outline of a cut through a sub-D body, its relieved rail spans
    // 7.7e-5 against a track spacing of 2.4e-4, 1.5e-6 off at 4097
    // samples).
    let law_network = || {
        crate::blend::blend_star_network_laws(solid, &edge_ids, &laws, chamfer, &network_names, &network_corner_name, true)
    };
    let network_result = law_network().or_else(|first| {
        if crate::blend::is_pcurve_floor_refusal(&first) {
            crate::blend::with_pcurve_knot_retry(law_network).map_err(|_| first)
        } else {
            Err(first)
        }
    });
    let mut network = match network_result {
        Ok(network) => network,
        Err(refusal) => {
            if debug {
                eprintln!("law network refused: {refusal}");
            }
            return Ok(Err(refusal));
        }
    };
    // The constant group's acceptance of a network result, in its order:
    // the vertex-gap heal, the planar charts, validation, the interference
    // classifier and the bowtie scan; then the soundness acceptance the
    // variable lane's exit takes.
    let healed = heal_edge_vertex_gaps(&mut network, max_radius);
    crate::blend::fit_planar_charts_to_trims(solid, &mut network)?;
    let issues = network.validate();
    let interference = check_blend_interference(solid, &network, &edge_ids, entry);
    let bowtie = check_loop_self_crossings(&network, entry).and_then(|()| check_blend_wall_loops(&network, entry));
    if healed.is_ok() && issues.is_empty() && interference.is_ok() && bowtie.is_ok() {
        crate::soundness::census::profile_operation(&network, &[solid], None, entry);
        match crate::accept_sound(network, entry) {
            Ok(sound) => return Ok(Ok(sound)),
            Err(refusal) => {
                if debug {
                    eprintln!("law network result unsound: {refusal}");
                }
                return Ok(Err(refusal));
            }
        }
    }
    if debug {
        eprintln!(
            "law network result rejected: heal={healed:?} issues={issues:?} \
             interference={interference:?} bowtie={bowtie:?}"
        );
    }
    Ok(Err(KernelRefusal::unsupported(
        KernelStage::Validate,
        "law_network_rejected",
        format!(
            "{entry}: the tapered network's result was rejected (heal {}, {} validation issues, interference {}, loops {})",
            if healed.is_ok() { "ok" } else { "failed" },
            issues.len(),
            if interference.is_ok() { "ok" } else { "found" },
            if bowtie.is_ok() { "ok" } else { "found" },
        ),
    )))
}

pub(crate) fn fillet_edges_variable_stops(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    per_edge_stops: &[Vec<(f64, f64)>],
    chamfer: bool,
    name: Option<&str>,
) -> Result<BrepSolid, KernelRefusal> {
    let _operation = crate::blend::BlendOperation::enter();
    if edge_points.is_empty() || per_edge_stops.len() != edge_points.len() {
        return Err(KernelRefusal::input(
            KernelStage::Collect,
            "variable_selection",
            "fillet_edges_variable_stops: one stop list per selected edge is required",
        ));
    }
    fillet_edges_variable_impl(
        solid,
        edge_points,
        edge_names,
        per_edge_stops,
        chamfer,
        name,
        "fillet_edges_variable_stops",
        false,
    )
}

/// One selected edge placed on the ordered chain, with its arc-length
/// parameterization (the caller-side mapping from chain abscissa to the
/// `blend_edge_variable` per-edge parameter-fraction stop seam).
struct ChainLink {
    /// Position of this edge in the caller's `edge_points` selection.
    input_index: usize,
    /// True when the edge's own t0→t1 parameter direction runs WITH the chain.
    forward: bool,
    /// Cumulative chord-length table from the edge's t0 end:
    /// `(parameter fraction, arc length)`, uniformly spaced in fraction.
    arc: Vec<(f64, f64)>,
    /// Total arc length of the edge.
    length: f64,
    /// Chain abscissa at the link's ENTRY vertex (the end reached first when
    /// walking the chain from its start).
    abscissa: f64,
}

/// Cumulative chord-length table of one edge, `(parameter fraction, arc
/// length)` at uniform fractions.  The sample count doubles until two
/// successive total-length estimates agree within `tol` (chord length
/// converges O(N⁻²) for smooth curves, so the agreement of the N and 2N
/// estimates bounds the remaining error at the same order); straight edges
/// converge on the first doubling.  The 16-sample start resolves any
/// single-span arc of up to half a turn to sub-percent before refinement; the
/// 4096 cap (8 doublings) guards adversarial curves — beyond it the table is
/// two orders denser than the blend march's station grid, so finer chords
/// cannot move any station's sampled radius meaningfully.
fn edge_arc_table(
    curve: &NurbsCurve,
    t0: f64,
    t1: f64,
    tol: f64,
) -> Result<(Vec<(f64, f64)>, f64), KernelRefusal> {
    let evaluate = |t: f64| curve.evaluate(t).or_refuse(KernelStage::Collect, "arc_table");
    let build = |n: usize| -> Result<(Vec<(f64, f64)>, f64), KernelRefusal> {
        let mut table = Vec::with_capacity(n + 1);
        let mut cumulative = 0.0;
        let mut previous = evaluate(t0)?;
        table.push((0.0, 0.0));
        for j in 1..=n {
            let fraction = j as f64 / n as f64;
            let point = evaluate(t0 + (t1 - t0) * fraction)?;
            cumulative += point.sub(previous).length();
            previous = point;
            table.push((fraction, cumulative));
        }
        Ok((table, cumulative))
    };
    let mut n = 16usize;
    let (mut table, mut length) = build(n)?;
    while n < 4096 {
        n *= 2;
        let (next_table, next_length) = build(n)?;
        let converged = (next_length - length).abs() <= tol;
        table = next_table;
        length = next_length;
        if converged {
            break;
        }
    }
    Ok((table, length))
}

/// Arc length from the edge's t0 end at `fraction` of its parameter span,
/// linearly interpolated in the uniform chord table.
fn arc_length_at_fraction(table: &[(f64, f64)], fraction: f64) -> f64 {
    let fraction = fraction.clamp(0.0, 1.0);
    let intervals = table.len() - 1;
    let scaled = fraction * intervals as f64;
    let index = (scaled.floor() as usize).min(intervals - 1);
    let local = scaled - index as f64;
    let (_, a) = table[index];
    let (_, b) = table[index + 1];
    a + (b - a) * local
}

/// Resolve the selected edges and order them into ONE OPEN CHAIN with
/// cumulative arc-length abscissas.  The chain starts at the free endpoint
/// belonging to the EARLIEST-selected end edge (so users get the natural
/// "first pick carries the law start" orientation); a single selected edge is
/// its own chain oriented t0→t1 (which also admits a closed edge — the law's
/// end radii must then match, enforced downstream by `blend_edge_variable`).
/// Branching (a vertex shared by 3+ selected edges), closed rings of several
/// edges, and disconnected selections refuse with named errors.
fn resolve_selected_chain(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    entry: &str,
    tol: f64,
) -> Result<Vec<ChainLink>, KernelRefusal> {
    // Endpoint identity uses the kernel-wide COINCIDENCE_DISTANCE_FLOOR — the
    // same band the constant-radius group's corner detector applies.
    let band = crate::tolerance::COINCIDENCE_DISTANCE_FLOOR;

    struct Resolved {
        edge_id: u64,
        start: Vec3,
        end: Vec3,
        arc: Vec<(f64, f64)>,
        length: f64,
    }
    let mut resolved: Vec<Resolved> = Vec::with_capacity(edge_points.len());
    for point in edge_points {
        let edge_id = resolve_edge_by_point(solid, *point)?;
        if resolved.iter().any(|r| r.edge_id == edge_id) {
            return Err(KernelRefusal::input(
                KernelStage::Collect,
                "duplicate_edge",
                format!("{entry}: the same edge was selected more than once"),
            ));
        }
        let edge = solid
            .edges
            .iter()
            .find(|e| e.id == edge_id)
            .ok_or_else(|| {
                KernelRefusal::internal(
                    KernelStage::Collect,
                    "chain_resolved_edge",
                    format!("{entry}: resolved edge vanished"),
                )
            })?;
        let (arc, length) = edge_arc_table(&edge.curve, edge.t0, edge.t1, tol)?;
        if !(length > 0.0) {
            return Err(KernelRefusal::input(
                KernelStage::Collect,
                "zero_length_edge",
                format!("{entry}: selected edge has zero length"),
            ));
        }
        resolved.push(Resolved {
            edge_id,
            start: edge
                .curve
                .evaluate(edge.t0)
                .or_refuse(KernelStage::Collect, "chain_edge_start")?,
            end: edge
                .curve
                .evaluate(edge.t1)
                .or_refuse(KernelStage::Collect, "chain_edge_end")?,
            arc,
            length,
        });
    }
    let n = resolved.len();
    if n == 1 {
        let only = resolved.remove(0);
        return Ok(vec![ChainLink {
            input_index: 0,
            forward: true,
            arc: only.arc,
            length: only.length,
            abscissa: 0.0,
        }]);
    }

    // Cluster the 2n endpoints; each entry is (edge index, is_start_end).
    let mut clusters: Vec<(Vec3, Vec<(usize, bool)>)> = Vec::new();
    for (i, r) in resolved.iter().enumerate() {
        for (point, is_start) in [(r.start, true), (r.end, false)] {
            match clusters
                .iter_mut()
                .find(|(anchor, _)| anchor.sub(point).length() < band)
            {
                Some((_, members)) => members.push((i, is_start)),
                None => clusters.push((point, vec![(i, is_start)])),
            }
        }
    }
    if clusters.iter().any(|(_, members)| members.len() > 2) {
        return Err(KernelRefusal::input(
            KernelStage::Collect,
            "chain_branching",
            format!(
                "{entry}: selected edges must form one open chain \
                 (a vertex is shared by three or more selected edges)"
            ),
        ));
    }
    let free: Vec<usize> = clusters
        .iter()
        .enumerate()
        .filter(|(_, (_, members))| members.len() == 1)
        .map(|(c, _)| c)
        .collect();
    if free.len() != 2 {
        return Err(KernelRefusal::input(
            KernelStage::Collect,
            "chain_open",
            format!(
                "{entry}: selected edges must form one OPEN chain \
                 (closed rings and disconnected selections are not supported)"
            ),
        ));
    }
    // Start at the free end whose edge appears EARLIEST in the selection.
    let start_cluster = *free
        .iter()
        .min_by_key(|&&c| clusters[c].1[0].0)
        .expect("two free ends");

    // Walk the chain.
    let mut links: Vec<ChainLink> = Vec::with_capacity(n);
    let mut visited = vec![false; n];
    let mut abscissa = 0.0_f64;
    let mut cluster = start_cluster;
    for _ in 0..n {
        let Some(&(edge_index, entered_at_start)) = clusters[cluster]
            .1
            .iter()
            .find(|(edge_index, _)| !visited[*edge_index])
        else {
            return Err(KernelRefusal::input(
                KernelStage::Collect,
                "chain_walk",
                format!(
                    "{entry}: selected edges are not connected into one chain"
                ),
            ));
        };
        visited[edge_index] = true;
        let r = &resolved[edge_index];
        links.push(ChainLink {
            input_index: edge_index,
            forward: entered_at_start,
            arc: r.arc.clone(),
            length: r.length,
            abscissa,
        });
        abscissa += r.length;
        let exit_point = if entered_at_start { r.end } else { r.start };
        cluster = clusters
            .iter()
            .position(|(anchor, _)| anchor.sub(exit_point).length() < band)
            .ok_or_else(|| {
                KernelRefusal::internal(
                    KernelStage::Collect,
                    "chain_cluster",
                    format!("{entry}: chain walk lost an endpoint cluster"),
                )
            })?;
    }
    if visited.iter().any(|v| !v) {
        return Err(KernelRefusal::input(
            KernelStage::Collect,
            "chain_connected",
            format!(
                "{entry}: selected edges are not connected into one chain"
            ),
        ));
    }
    Ok(links)
}

/// Sample the law into one edge's `(parameter fraction, radius)` stop list —
/// the caller-side bridge from chain abscissa into the `radius_at` closure
/// seam that `blend_edge_variable` builds over its stops.
///
/// `scale` maps chain abscissa into the law's own abscissa units
/// (`law.total_length() / chain_length` — proportional, so a law built from
/// the measured chain lengths maps 1:1).  The base stop count derives from
/// the law's curvature: piecewise-linear sampling of a C1, piecewise-C2
/// function over step `h` errs at most `h²·max|r''|/8`, so
/// `h = sqrt(8·tol/max|r''|)` holds the sampling error under the SSI fit
/// tolerance the variable lane is built to.  That bound is exact for
/// arc-length-linear (straight) edges; curved edges bend the
/// parameter→abscissa map, so each interval is additionally midpoint-checked
/// against the law and bisected on violation (up to 8 halvings — a 4⁸ ≈ 6·10⁴
/// error reduction, decisive for any C1 law).  The base count is capped at
/// 256 intervals: 4× the blend march's 64-station density
/// (blend/stations.rs), beyond which denser stops cannot move any station's
/// sampled radius by more than the fit tolerance.
fn law_stops_for_link(
    link: &ChainLink,
    law: &crate::law::RadiusLaw,
    scale: f64,
    tol: f64,
) -> Vec<(f64, f64)> {
    let radius_at_fraction = |fraction: f64| -> f64 {
        let arc = arc_length_at_fraction(&link.arc, fraction);
        let chain_s = if link.forward {
            link.abscissa + arc
        } else {
            link.abscissa + (link.length - arc)
        };
        law.radius_at(chain_s * scale)
    };
    // Curvature of the law in edge-fraction units: d²r/df² ≤
    // max|r''|·(scale·length)² for the arc-length-linear map.
    let curvature = law
        .max_second_derivative(link.abscissa * scale, (link.abscissa + link.length) * scale)
        * (scale * link.length).powi(2);
    let base = if curvature * 0.125 <= tol {
        1usize
    } else {
        ((curvature / (8.0 * tol)).sqrt().ceil() as usize).clamp(1, 256)
    };
    let mut stops: Vec<(f64, f64)> = (0..=base)
        .map(|j| {
            let fraction = j as f64 / base as f64;
            (fraction, radius_at_fraction(fraction))
        })
        .collect();
    // Midpoint refinement for curved parameter→abscissa maps.
    let mut depth = 0usize;
    while depth < 8 {
        let mut refined: Vec<(f64, f64)> = Vec::with_capacity(stops.len());
        let mut inserted = false;
        for pair in stops.windows(2) {
            refined.push(pair[0]);
            let mid_fraction = 0.5 * (pair[0].0 + pair[1].0);
            let law_mid = radius_at_fraction(mid_fraction);
            let linear_mid = 0.5 * (pair[0].1 + pair[1].1);
            if (law_mid - linear_mid).abs() > tol {
                refined.push((mid_fraction, law_mid));
                inserted = true;
            }
        }
        refined.push(*stops.last().expect("at least two stops"));
        stops = refined;
        if !inserted {
            break;
        }
        depth += 1;
    }
    stops
}

/// Fillet (or chamfer) a chain of edges under a composable radius law
/// evaluated on the chain's cumulative arc-length abscissa (the OCCT
/// `Law_Composite` model; see [`crate::law::RadiusLaw`]).  The selected edges
/// must form ONE OPEN CHAIN (or be a single edge); the chain starts at the
/// free end of the earliest-selected end edge, and the law's abscissa maps
/// proportionally onto the chain's measured arc length (a law built with the
/// chain's own lengths — e.g. [`crate::law::RadiusLaw::from_vertex_radii`] —
/// maps 1:1).  Endpoint radii are met exactly; radii at shared chain vertices
/// match by the law's continuity, so the per-edge blends miter through the
/// corners; a chain whose blends cannot be mitered REFUSES rather than
/// distorting the law through the sequential rebuild.
pub fn fillet_edges_variable_law(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    law: &crate::law::RadiusLaw,
    chamfer: bool,
    name: Option<&str>,
) -> Result<BrepSolid, KernelRefusal> {
    // One blend operation: the reports of an earlier one, never consumed, are
    // dropped as it starts (nested calls keep this operation's).
    let _operation = crate::blend::BlendOperation::enter();
    const ENTRY: &str = "fillet_edges_variable_law";
    if edge_points.is_empty() {
        return Err(KernelRefusal::input(
            KernelStage::Collect,
            "law_selection",
            format!("{ENTRY}: no edges selected"),
        ));
    }
    let tolerances = crate::KernelTolerances::for_solid(solid, 1e-7);
    let chain = resolve_selected_chain(solid, edge_points, ENTRY, tolerances.intersection_fit)?;
    fillet_variable_law_on_chain(
        solid,
        edge_points,
        edge_names,
        &chain,
        law,
        chamfer,
        name,
        ENTRY,
        tolerances.intersection_fit,
    )
}

/// The natural per-vertex user model: radius `vertex_radii[i]` at chain
/// vertex `i` (in CHAIN order, starting at the free end of the
/// earliest-selected edge), smoothly interpolated along the chain
/// (monotone C1 — every vertex radius met exactly, no overshoot).  Requires
/// exactly one radius per chain vertex (`edges + 1`).
pub fn fillet_edges_variable_vertex_radii(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    vertex_radii: &[f64],
    chamfer: bool,
    name: Option<&str>,
) -> Result<BrepSolid, KernelRefusal> {
    // One blend operation: the reports of an earlier one, never consumed, are
    // dropped as it starts (nested calls keep this operation's).
    let _operation = crate::blend::BlendOperation::enter();
    const ENTRY: &str = "fillet_edges_variable_vertex_radii";
    if edge_points.is_empty() {
        return Err(KernelRefusal::input(
            KernelStage::Collect,
            "vertex_radii_selection",
            format!("{ENTRY}: no edges selected"),
        ));
    }
    let tolerances = crate::KernelTolerances::for_solid(solid, 1e-7);
    let chain = resolve_selected_chain(solid, edge_points, ENTRY, tolerances.intersection_fit)?;
    let lengths: Vec<f64> = chain.iter().map(|link| link.length).collect();
    let law = crate::law::RadiusLaw::from_vertex_radii(&lengths, vertex_radii)
        .map_err(|error| error.with_message(|error| format!("{ENTRY}: {error}")))?;
    fillet_variable_law_on_chain(
        solid,
        edge_points,
        edge_names,
        &chain,
        &law,
        chamfer,
        name,
        ENTRY,
        tolerances.intersection_fit,
    )
}

/// Shared law-entry tail: sample per-edge stop lists from the law over the
/// resolved chain and run the variable group core (miter-or-refuse: no
/// sequential fallback for tapered chains — see `fillet_edges_variable_impl`).
#[allow(clippy::too_many_arguments)]
fn fillet_variable_law_on_chain(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    chain: &[ChainLink],
    law: &crate::law::RadiusLaw,
    chamfer: bool,
    name: Option<&str>,
    entry: &str,
    tol: f64,
) -> Result<BrepSolid, KernelRefusal> {
    let chain_length: f64 = chain.iter().map(|link| link.length).sum();
    let scale = law.total_length() / chain_length;
    let mut per_edge_stops: Vec<Vec<(f64, f64)>> = vec![Vec::new(); edge_points.len()];
    for link in chain {
        per_edge_stops[link.input_index] = law_stops_for_link(link, law, scale, tol);
    }
    fillet_edges_variable_impl(
        solid,
        edge_points,
        edge_names,
        &per_edge_stops,
        chamfer,
        name,
        entry,
        false,
    )
}

/// Asymmetric (two-distance) chamfer of a GROUP of edges, the app entry: each
/// selected edge (resolved by a point on it) gets a `d1 × d2` bevel (§6.11).
/// Edges are chamfered independently — asymmetric chamfers keep sharp vertices,
/// so there is no shared-corner blending.
pub fn chamfer_edges_asymmetric(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    d1: f64,
    d2: f64,
    name: Option<&str>,
) -> Result<BrepSolid, KernelRefusal> {
    // One blend operation: the reports of an earlier one, never consumed, are
    // dropped as it starts (nested calls keep this operation's).
    let _operation = crate::blend::BlendOperation::enter();
    if edge_points.is_empty() {
        return Err(KernelRefusal::input(
            KernelStage::Collect,
            "asymmetric_selection",
            "chamfer_edges_asymmetric: no edges selected",
        ));
    }
    let mut result = solid.clone();
    // `edge_names[index]` is keyed by INPUT position; the loop resolves each
    // point on the EVOLVING `result`, but enumerates the input points in order,
    // so the index alignment holds.
    for (index, point) in edge_points.iter().enumerate() {
        let edge_id = resolve_edge_by_point(&result, *point)?;
        let edge_name = per_edge_name(edge_names, name, index);
        result = chamfer_edge_asymmetric(&result, edge_id, d1, d2, edge_name)?;
    }
    refuse_invalid_result(&result, "chamfer_edges_asymmetric")?;
    Ok(result)
}

/// Distance-angle chamfer of a GROUP of edges, the app entry: each selected
/// edge (resolved by a point on it) gets a setback `d1` on face 1 and a chamfer
/// face at `angle_rad` from face 1 (§6.11); `d2` is constructed per edge.
pub fn chamfer_edges_angle(
    solid: &BrepSolid,
    edge_points: &[Vec3],
    edge_names: Option<&[String]>,
    d1: f64,
    angle_rad: f64,
    name: Option<&str>,
) -> Result<BrepSolid, KernelRefusal> {
    // One blend operation: the reports of an earlier one, never consumed, are
    // dropped as it starts (nested calls keep this operation's).
    let _operation = crate::blend::BlendOperation::enter();
    if edge_points.is_empty() {
        return Err(KernelRefusal::input(
            KernelStage::Collect,
            "angle_selection",
            "chamfer_edges_angle: no edges selected",
        ));
    }
    let mut result = solid.clone();
    for (index, point) in edge_points.iter().enumerate() {
        let edge_id = resolve_edge_by_point(&result, *point)?;
        let edge_name = per_edge_name(edge_names, name, index);
        result = chamfer_edge_angle(&result, edge_id, d1, angle_rad, edge_name)?;
    }
    refuse_invalid_result(&result, "chamfer_edges_angle")?;
    Ok(result)
}

/// Debug aid (`BREP_DEBUG_NETWORK`): every face loop as its coedge walk, with
/// each edge's stored vertices, so a loop the surgery left open can be read
/// against the input it was built from.
fn dump_loops_debug(label: &str, solid: &BrepSolid) {
    for shell in &solid.shells {
        for face in &shell.faces {
            for (index, loop_record) in face.loops.iter().enumerate() {
                let walk: Vec<String> = loop_record
                    .coedges
                    .iter()
                    .map(|coedge| {
                        // The pcurve's walk endpoints in the face's (u, v), so a
                        // loop that closes in space but not in parameter space
                        // (a periodic carrier's seam) is visible in the dump.
                        let uv = coedge
                            .pcurve
                            .domain()
                            .ok()
                            .and_then(|[q0, q1]| {
                                let a = coedge.pcurve.evaluate(q0).ok()?;
                                let b = coedge.pcurve.evaluate(q1).ok()?;
                                Some(format!("uv[({:.4},{:.4})->({:.4},{:.4})]", a.x, a.y, b.x, b.y))
                            })
                            .unwrap_or_default();
                        match solid.edges.iter().find(|e| e.id == coedge.edge_id) {
                            Some(edge) => format!(
                                "c{}:e{}{}({}->{}){uv}",
                                coedge.id,
                                edge.id,
                                if coedge.forward { "+" } else { "-" },
                                edge.start_vertex_id,
                                edge.end_vertex_id
                            ),
                            None => format!("c{}:e{}?MISSING", coedge.id, coedge.edge_id),
                        }
                    })
                    .collect();
                eprintln!(
                    "{label} face {} ({:?}) loop {index}: {}",
                    face.id,
                    face.name,
                    walk.join(" ")
                );
            }
        }
    }
}