brep_kernel/offset/offset_shell/

pipeline.rs

use crate::{KernelRefusal, KernelStage, OrRefuse};
use super::*;

/// Put every face on one normal convention (surface normal on the CCW
/// parameter-winding side) and then restore outward normals. The offset shell's
/// source and offset skins are assembled as independent components whose
/// per-face `same_sense` can follow opposite conventions; `validate()` only
/// checks coedge-direction coherence (the `forward` flags), so a joined,
/// watertight, coedge-coherent manifold can still be normal-incoherent and
/// report the wrong volume. Re-deriving `same_sense` from the loop winding makes
/// the coedge-coherent manifold normal-coherent; the signed-volume flip then
/// orients it outward. Only invoked after a non-coplanar rim weld, so the
/// coplanar cylinder/prism shells keep their exact assembled orientation.
/// `BREP_OFFSET_ONE_CALL_IMPRINT` (lane P3b row 6, 2026-10-26): one
/// `build_imprints` over a composite of every carrier instead of one call per
/// pair. On by default (promoted 2026-10-27 after the 98 + 144 case oracle
/// census); `0`/`off` retains the per-pair calls. Re-pinned by independent
/// oracles, not byte-identical (piece and vertex numbering is the single
/// call's).
fn one_call_imprint_enabled() -> bool {
    switch!("BREP_OFFSET_ONE_CALL_IMPRINT", true)
}

thread_local! {
    static ONE_CALL_RUNS: std::cell::Cell<u64> = const { std::cell::Cell::new(0) };
}
/// Shell or offset operations whose imprint the one call produced on this thread since the last call.
pub(super) fn take_one_call_runs() -> u64 {
    ONE_CALL_RUNS.with(|count| count.replace(0))
}


fn coheres_face_normals(solid: &mut BrepSolid) -> Result<(), KernelRefusal> {
    for face in solid.shells.iter_mut().flat_map(|shell| &mut shell.faces) {
        face.same_sense = parameter_space_area(face).or_refuse(KernelStage::Sew, "parameter_space_area")? > 0.0;
    }
    // Outward is a PER-SHELL property: a through-hole shelled at both ends
    // legitimately produces two disjoint closed components (the outer wall
    // tube and the hole wall tube), and a single global signed-volume flip
    // cannot orient both.
    for shell_index in 0..solid.shells.len() {
        let single = BrepSolid {
            mass_properties_cache: Default::default(),
            cells: Default::default(),
            id: solid.id,
            vertices: solid.vertices.clone(),
            edges: solid.edges.clone(),
            shells: vec![solid.shells[shell_index].clone()],
            genus: 0,
            kind: crate::BodyKind::Solid,
        };
        // Read for its SIGN only, through the per-face term memo
        // (`solid_signed_volume_for_sign`: the verdict against the bar is the
        // exact integral's, and a face read about the same reference before —
        // the next feature's push, the boolean assembler's own check — is
        // served its term). The exact read here cost 0.26 s on the shelled-box
        // report's 34-face shell, 42 of its faces fitted.
        if crate::signed_volume_against(&single, 0.0).or_refuse(KernelStage::Sew, "solid_signed_volume")? < 0.0 {
            flip_shell_faces(&mut solid.shells[shell_index])?;
        }
    }
    Ok(())
}

/// Mint a vertex at every SELF-TOUCH of the source's faces before the shell
/// reads their trims.
///
/// A face whose loops touch at an EDGE INTERIOR is a pinched region: the
/// reported 2026-09-09 document's r = 6 through-hole kisses both r = 4 fillet
/// setbacks on the top face, at (16, 20, 10) and (10, 20, 16), with no vertex
/// there.  The fragment stage's 2D arrangement resolves such a pinch by
/// carving the touched-off corner into its own region, whose loop walks the
/// closed hole chain in a PARTIAL run — which `build_loop` can only assemble
/// when the touch is a vertex of both touching edges, and otherwise refuses
/// with "chain N fragmented into an incomplete run".
///
/// The boolean lane mints that vertex inside `build_imprints`
/// (`imprint/self_touch.rs`).  Nothing minted it here: the shell fragments
/// its SOURCE against an EMPTY imprint, and its offset carriers — whose trims
/// are copied from the source faces — deliberately keep their boundaries
/// un-split, so both arrived at the arrangement pinched.  Splitting the
/// source once, before the carriers are built from it, fixes both: every
/// carrier inherits the split trim by construction.
///
/// Returns `None` when no face self-touches, so every part that was never
/// pinched keeps its exact previous path.  Escape hatch:
/// `BREP_SELF_TOUCH_SPLIT=0`, the same one the imprint lane honours.
fn split_self_touching_source(source: &BrepSolid) -> Result<Option<BrepSolid>, KernelRefusal> {
    if std::env::var("BREP_SELF_TOUCH_SPLIT").as_deref() == Ok("0") {
        return Ok(None);
    }
    // The same derivation `offset_shell_impl` uses for every other band, so
    // the source scan and the carrier-pair scans agree on what "touching"
    // means.
    let scale = crate::solid_scale(source);
    let tolerance = 1e-7f64.max(scale * 1e-9);
    let edge_splits = crate::imprint::self_touch_edge_splits(source, SOURCE_OPERAND, tolerance, scale)?;
    if edge_splits.is_empty() {
        return Ok(None);
    }
    os_debug!(
        "self-touch: source split on {} edge(s): {:?}",
        edge_splits.len(),
        edge_splits
            .iter()
            .map(|split| split.edge_id)
            .collect::<Vec<_>>(),
    );
    let imprint = ImprintResultRecord {
        edge_splits,
        ..empty_imprint()
    };
    Ok(Some(apply_edge_splits(source, SOURCE_OPERAND, &imprint)?))
}

/// How far past the grown outline a planar opening wall that meets a RULED
/// opening is padded, in units of the offset-skin band the wall selection reads
/// (see the wall construction).
///
/// Measured on the torus wedge through the pipeline, grown by 0.05, 0.1, 0.5
/// and 0.999 (band 2e-3): a clearance of 2 or 2.5 bands still shreds the cap
/// into 22 to 36 faces with the area 5% to 11% high; from 4 bands on every
/// member builds its 6 faces, and from 4 to 32 bands each member's volume moves
/// by at most 1.2e-7. 16 sits four times above the first clean reading.
const RULED_NEIGHBOUR_CLEARANCE_BANDS: f64 = 16.0;

/// How far past its opening plane an inward carrier is grown, as a fraction of
/// the shell distance (see the fall-short reach in the carrier loop).
const REACH_CLEARANCE: f64 = 0.25;

/// What the pipeline assembles.
#[derive(Clone, Copy, Debug, Eq, PartialEq)]
pub(super) enum Body {
    /// A shell: the retained source faces, their offset skins and the walls
    /// at the openings, bounding the material between source and offset.
    Shell,
    /// The offset body itself: the offset skins alone, closed by their
    /// mutual sections; `opening_face_ids` are HELD faces, copied at
    /// distance zero as walls, with no source face assembled and no skin
    /// flipped. The whole-solid offset and the boolean fallback's offset
    /// body are built this way.
    Whole,
}


#[cfg(not(test))]
fn forced_direct_refusal() -> Option<KernelRefusal> {
    None
}

/// `BREP_OFFSET_EVERY_SKIN_PIECE` (default on; `=0` keeps the seed's one
/// piece): see EVERY SKIN PIECE in the offset selection.
fn every_skin_piece_enabled() -> bool {
    crate::switch!("BREP_OFFSET_EVERY_SKIN_PIECE", true)
}

/// `BREP_OFFSET_EVERY_SEEDED_SKIN_PIECE` (default on; `=0` restores): when an
/// inward shell's assembly fails, EVERY SKIN PIECE is retried on the seeded
/// path too. Past 2.0 a reported box and pin model's bore skin comes apart into one piece
/// per corner channel and the seed answered for one of them. Only as a retry:
/// a shell the seed's choice closes keeps that answer.
fn every_seeded_skin_piece_enabled() -> bool {
    crate::switch!("BREP_OFFSET_EVERY_SEEDED_SKIN_PIECE", true)
}

pub fn offset_shell(
    source: &BrepSolid,
    opening_face_ids: &[u64],
    distance: f64,
) -> Result<OffsetShellResultRecord, KernelRefusal> {
    let direct = match forced_direct_refusal() {
        Some(forced) => Err(forced),
        None => offset_body(source, opening_face_ids, distance, Body::Shell, "offset_shell"),
    };
    let mut shelled = offset_shell_lanes(source, opening_face_ids, distance, direct)?;
    if membrane_shells_on() {
        drop_folded_shells(&mut shelled.solid, distance);
        let membranes = membrane_shell_indices(&shelled.solid, distance);
        let paired = paired_face_ids(&shelled.solid);
        if !membranes.is_empty() || !paired.is_empty() {
            for (first, second) in &paired {
                let shell = shelled.solid.shells.iter().position(|shell| shell.faces.iter().any(|face| face.id == *first));
                os_debug!(
                    "membrane pair: faces {first} and {second} in shell {shell:?} of {:?} faces",
                    shell.map(|index| shelled.solid.shells[index].faces.len())
                );
            }
            return Err(KernelRefusal::unsupported(KernelStage::Validate, "offset_shell_membrane", format!(
                "offset_shell: at this distance the result has {} shell(s) enclosing no volume and \
                 {} pair(s) of faces bounded by the same edges{} (zero-thickness membranes \
                 beside the cavity), which no solid carries",
                membranes.len(),
                paired.len(),
                paired.first().map(|(a, b)| format!(", first faces {a} and {b}")).unwrap_or_default()
            )));
        }
    }
    Ok(shelled)
}

/// `BREP_OFFSET_MEMBRANE_SHELLS` (default on; `=0` restores): a shell of the
/// result that encloses nothing (made only of FOLDED faces, each loop running
/// along its edges once in each direction, closed by point edges; or of zero
/// volume over its area; or a bag, its faces pairing up on the same edges) is
/// removed with the edges and vertices only it used, when the
/// rest still validates. Otherwise, and for two faces of one real shell bounded
/// by the same edges (a zero-thickness fin), the shell is refused as a
/// membrane.
///
/// Cause (a reported box and pin model): at 2.38 and 2.40 the result reported
/// three shells, two of them single folded faces of about 6.5e-6 mm^2 at the
/// floor/wall corner; at 2.46 (with the corner-debris switch) six more, folded
/// faces and two-face bags of volume 1e-8 over area 4e-3. None shares an edge
/// with the cavity's shell. `validate()` accepts them all and the volume is
/// right, but the shell count is not, nor is anything that walks the shells.
///
/// Every test is scaled by the offset distance, since a real piece of the shell
/// is about |distance| thick: a folded or bag shell is removed only when its
/// volume is at most 1% of |distance| × its area (a sphere face, whose seam
/// also runs both ways, encloses the sealed cavity of a sphere-cut shell), and
/// the zero-volume bar never exceeds 1e-3 × |distance| (a shell 1e-7 thick is
/// a real sliver, not a membrane).
fn membrane_shells_on() -> bool {
    switch!("BREP_OFFSET_MEMBRANE_SHELLS", true)
}

/// Whether every non-degenerate edge of `face`'s loops is used once forward
/// and once reversed by the face itself.
fn face_is_folded(face: &FaceRecord, edges: &HashMap<u64, &EdgeRecord>) -> bool {
    let mut balance: HashMap<u64, (usize, usize)> = HashMap::default();
    for coedge in face.loops.iter().flat_map(|l| &l.coedges) {
        if edges.get(&coedge.edge_id).is_some_and(|edge| edge.degenerate) {
            continue;
        }
        let entry = balance.entry(coedge.edge_id).or_default();
        if coedge.forward {
            entry.0 += 1;
        } else {
            entry.1 += 1;
        }
    }
    !balance.is_empty() && balance.values().all(|&(forward, reversed)| forward == 1 && reversed == 1)
}

/// Whether every face of `shell` shares its whole set of non-degenerate edges
/// with exactly one other face of it: two-sided sheets with nothing between.
fn shell_is_bag(shell: &ShellRecord, edges: &HashMap<u64, &EdgeRecord>) -> bool {
    let mut sets: HashMap<Vec<u64>, usize> = HashMap::default();
    for face in &shell.faces {
        let mut set: Vec<u64> = face
            .loops
            .iter()
            .flat_map(|l| &l.coedges)
            .map(|coedge| coedge.edge_id)
            .filter(|id| !edges.get(id).is_some_and(|edge| edge.degenerate))
            .collect();
        set.sort_unstable();
        set.dedup();
        if set.is_empty() {
            return false;
        }
        *sets.entry(set).or_default() += 1;
    }
    !sets.is_empty() && sets.values().all(|&count| count == 2)
}

/// Removes the shells made only of folded faces, then the edges and vertices
/// no remaining face uses. Keeps `solid` unchanged when that would leave no
/// shell or a result that does not validate. A folded or bag shell qualifies
/// only when its volume is at most 1% of |distance| × its area.
pub(crate) fn drop_folded_shells(solid: &mut BrepSolid, distance: f64) {
    let edges: HashMap<u64, &EdgeRecord> = solid.edges.iter().map(|edge| (edge.id, edge)).collect();
    let empty = membrane_shell_indices(solid, distance);
    let folded: Vec<usize> = (0..solid.shells.len())
        .filter(|&index| {
            let shell = &solid.shells[index];
            empty.contains(&index)
                || ((shell.faces.iter().all(|face| face_is_folded(face, &edges)) || shell_is_bag(shell, &edges))
                    && shell_volume_and_area(solid, index)
                        .is_some_and(|(volume, area)| volume.abs() <= 1e-2 * distance.abs() * area))
        })
        .collect();
    if folded.is_empty() || folded.len() == solid.shells.len() {
        return;
    }
    let mut trimmed = solid.clone();
    let mut index = 0;
    trimmed.shells.retain(|_| {
        let keep = !folded.contains(&index);
        index += 1;
        keep
    });
    let used_edges: HashSet<u64> = trimmed
        .shells
        .iter()
        .flat_map(|shell| &shell.faces)
        .flat_map(|face| &face.loops)
        .flat_map(|l| &l.coedges)
        .map(|coedge| coedge.edge_id)
        .collect();
    trimmed.edges.retain(|edge| used_edges.contains(&edge.id));
    let used_vertices: HashSet<u64> = trimmed
        .edges
        .iter()
        .flat_map(|edge| [edge.start_vertex_id, edge.end_vertex_id])
        .collect();
    trimmed.vertices.retain(|vertex| used_vertices.contains(&vertex.id));
    crate::boolean::refresh_derived_genus(&mut trimmed);
    let issues = trimmed.validate();
    if !issues.is_empty() {
        os_debug!("folded shells {folded:?} kept: without them the result does not validate: {issues:?}");
        return;
    }
    os_debug!("dropped folded shells {folded:?}");
    *solid = trimmed;
}

/// Pairs of faces of one shell bounded by exactly the same non-degenerate
/// edges: two sides of a zero-thickness fin. By face id.
pub(crate) fn paired_face_ids(solid: &BrepSolid) -> Vec<(u64, u64)> {
    let mut pairs = Vec::new();
    let degenerate: HashSet<u64> = solid.edges.iter().filter(|edge| edge.degenerate).map(|edge| edge.id).collect();
    for shell in &solid.shells {
        let mut seen: HashMap<Vec<u64>, u64> = HashMap::default();
        for face in &shell.faces {
            let mut edges: Vec<u64> = face
                .loops
                .iter()
                .flat_map(|l| &l.coedges)
                .map(|c| c.edge_id)
                .filter(|id| !degenerate.contains(id))
                .collect();
            edges.sort_unstable();
            if edges.len() < 2 {
                continue;
            }
            match seen.get(&edges) {
                Some(&other) => pairs.push((other, face.id)),
                None => {
                    seen.insert(edges, face.id);
                }
            }
        }
    }
    pairs
}

/// The volume and area of shell `index` of `solid` on its own.
fn shell_volume_and_area(solid: &BrepSolid, index: usize) -> Option<(f64, f64)> {
    let mut alone = solid.clone();
    alone.shells = vec![solid.shells[index].clone()];
    crate::solid_mass_properties(&alone).ok().map(|mass| (mass.volume, mass.surface_area))
}

/// The shells of `solid` that enclose no volume, by index: |V| at most
/// min(1e-6 × the bounding-box diagonal, 1e-3 × |distance|) × the shell's
/// area. A shell whose mass properties cannot be measured is never reported.
pub(crate) fn membrane_shell_indices(solid: &BrepSolid, distance: f64) -> Vec<usize> {
    if solid.vertices.is_empty() {
        return Vec::new();
    }
    let mut lo = Vec3::new(f64::INFINITY, f64::INFINITY, f64::INFINITY);
    let mut hi = Vec3::new(f64::NEG_INFINITY, f64::NEG_INFINITY, f64::NEG_INFINITY);
    for vertex in &solid.vertices {
        lo = Vec3::new(lo.x.min(vertex.point.x), lo.y.min(vertex.point.y), lo.z.min(vertex.point.z));
        hi = Vec3::new(hi.x.max(vertex.point.x), hi.y.max(vertex.point.y), hi.z.max(vertex.point.z));
    }
    let thickness = (1e-6 * hi.sub(lo).length()).min(1e-3 * distance.abs());
    (0..solid.shells.len())
        .filter(|&index| {
            shell_volume_and_area(solid, index).is_some_and(|(volume, area)| volume.abs() <= thickness * area)
        })
        .collect()
}

fn offset_shell_lanes(
    source: &BrepSolid,
    opening_face_ids: &[u64],
    distance: f64,
    direct: Result<OffsetShellResultRecord, KernelRefusal>,
) -> Result<OffsetShellResultRecord, KernelRefusal> {
    match direct {
        Ok(result) => Ok(result),
        // The boolean lane answers the direct construction's own named
        // refusals; when it refuses too, the direct refusal is the one
        // reported, because it names the cause.
        Err(direct) if super::solid::direct_refusal_takes_the_boolean_lane(&direct) => {
            os_debug!("direct pipeline refused ({direct}); taking the boolean lane");
            match super::solid::offset_shell_by_boolean(source, opening_face_ids, distance) {
                Ok(result) => Ok(result),
                Err(boolean) => {
                    os_debug!("boolean lane refused too ({boolean})");
                    Err(direct)
                }
            }
        }
        Err(direct) => Err(direct),
    }
}

/// The pipeline's one timed entry: `offset_shell` and `offset_solid` both run
/// through it, so the `timing.*` split and the acceptance step are the same.
/// Lane O row 38's switch, default on since batch 17 (`0`/`off` restores the fit): it asks the offset pipeline's
/// imprints for the planar-iso lane's exact pcurves (`ImprintOptions::
/// exact_iso_pcurves`, parity slice 1): a section the lane answers is an
/// iso-curve of the ruled carrier and an affine image on the plane, so its
/// pcurves are written down instead of fitted. Booleans keep their own
/// default; `BREP_EXACT_ISO_PCURVES` still overrides every imprint.
fn offset_exact_iso_pcurves_on() -> bool {
    switch!("BREP_OFFSET_EXACT_ISO_PCURVES", true)
}


pub(super) fn offset_body(
    source: &BrepSolid,
    opening_face_ids: &[u64],
    distance: f64,
    body: Body,
    entry: &str,
) -> Result<OffsetShellResultRecord, KernelRefusal> {
    profile::reset();
    // Lane P3b row 3: one stderr write per operation for every profile line
    // emitted below (`crate::imprint::ProfileSink`).
    let _profile_sink = crate::imprint::ProfileSink::open();
    let started = web_time::Instant::now();
    let result = offset_shell_any_lane(source, opening_face_ids, distance, body).and_then(|result| {
        let mut clock = web_time::Instant::now();
        let connected = outward_shell_is_connected(source, distance, &result);
        profile::lap(&mut clock, profile::ACCEPTANCE);
        connected.map(|()| result)
    });
    profile::add(profile::TOTAL, started.elapsed());
    profile::report(entry);
    result
}

/// Every shell of an OUTWARD result must bound volume. The shell is the grown
/// body less the source with the openings held, so it can be disconnected
/// (opening a cylinder's lateral face leaves its two grown caps as two
/// slabs), but each of its shells encloses material. A shell that encloses
/// none, a lone face that happened to close, is a fragment of an assembly that
/// went wrong and still validated. Measured on 2026-09-26: a free-form D whose
/// fitted opening wall sat within the fit tolerance of its carriers' sections
/// returned its body plus 62 single-face shells of volume under 3e-24, each
/// valid, with the right total volume. So this refuses by name for EVERY
/// outward shell, not only for the lane that produced those.
pub(super) fn outward_shell_is_connected(
    source: &BrepSolid,
    distance: f64,
    result: &OffsetShellResultRecord,
) -> Result<(), KernelRefusal> {
    if distance >= 0.0 || result.solid.shells.len() <= 1 {
        return Ok(());
    }
    let scale = crate::solid_scale(source).max(1.0);
    let volumes = result
        .solid
        .shells
        .iter()
        .map(|shell| {
            let mut one = result.solid.clone();
            one.shells = vec![shell.clone()];
            crate::solid_mass_properties(&one)
                .map(|properties| properties.volume)
                .or_refuse(KernelStage::Validate, "solid_mass_properties")
        })
        .collect::<Result<Vec<f64>, KernelRefusal>>()?;
    let total = volumes.iter().map(|volume| volume.abs()).sum::<f64>();
    let floor = (1e-9 * total).max(1e-12 * scale * scale * scale);
    let empty = volumes.iter().filter(|volume| volume.abs() <= floor).count();
    if empty == 0 {
        return Ok(());
    }
    Err(refusals::empty_shell(result.solid.shells.len(), empty, floor).into())
}

fn offset_shell_any_lane(
    source: &BrepSolid,
    opening_face_ids: &[u64],
    distance: f64,
    body: Body,
) -> Result<OffsetShellResultRecord, KernelRefusal> {
    // Pinched source faces are un-modellable by the arrangement until the
    // touch is a vertex; do it once here so BOTH `offset_shell_impl` attempts
    // and every carrier built from the source see the same topology.
    let prepared = split_self_touching_source(source)?;
    let source = prepared.as_ref().unwrap_or(source);
    // The revolved-wedge lane answers a SHELL question (the material between
    // a revolve and its grown image); the offset body takes the pipeline.
    if body == Body::Whole {
        // Row 36 (`BREP_OFFSET_SPHERE_BODY`, default off): the offset body of a
        // ball is the ball scaled about its centre (`sphere_body.rs`); it
        // declines to the pipeline on anything else.
        if opening_face_ids.is_empty() {
            if let Some(result) = sphere_body::sphere_body(source, distance) {
                return Ok(result);
            }
            // `BREP_OFFSET_TORUS_APPLE` (default on; `=0` restores): a whole torus grown
            // past its hole is the revolved apple (`torus_body.rs`).
            if let Some(result) = super::torus_body::torus_apple(source, distance) {
                return Ok(result);
            }
        }
        return imprint_attempts(source, opening_face_ids, distance, body);
    }
    // An outward shell of the one circular face of a partial revolve is planar
    // in the meridian half-plane, fold band and all: build it there and revolve
    // it once (`revolved_wedge.rs`). Anything else declines and takes the
    // imprint pipeline, unchanged.
    match revolved_wedge::recognize(source, opening_face_ids, distance) {
        Ok(revolved_wedge::Recognition::Wedge(wedge)) => {
            os_debug!("revolved wedge lane: {}", wedge.report());
            return revolved_wedge::build(&wedge, source);
        }
        Ok(revolved_wedge::Recognition::Declined(reason)) => {
            os_debug!("revolved wedge lane declined: {reason}");
        }
        // A source the lane cannot even read is not one it takes; the pipeline
        // owns every refusal.
        Err(error) => {
            os_debug!("revolved wedge lane declined: could not read the source ({error})");
        }
    }
    // `BREP_OFFSET_SPHERE_CUT_SHELL` (default on; `=0` restores): an inward shell of a ball cut by
    // planes is the source less the moved ball cut by the moved planes
    // (`sphere_cut_shell.rs`); it declines to the pipeline on anything else.
    if let Some(result) = sphere_cut_shell::sphere_cut_shell(source, opening_face_ids, distance) {
        return Ok(result);
    }
    // Row 32 (`BREP_OFFSET_PLANAR_SHELL`, default on; `=0` restores): an outward shell of a
    // convex polyhedron in closed form (`planar_shell.rs`); it declines to the
    // pipeline on anything else.
    if let Some(result) = planar_shell::planar_shell(source, opening_face_ids, distance) {
        return Ok(result);
    }
    imprint_attempts(source, opening_face_ids, distance, body)
}


fn imprint_attempts(
    source: &BrepSolid,
    opening_face_ids: &[u64],
    distance: f64,
    body: Body,
) -> Result<OffsetShellResultRecord, KernelRefusal> {
    // First try with oblique hole-wall carriers extended past their opening
    // planes (the lane that closes tilted/conical through-holes). If that
    // extension fired but the arrangement could not assemble the harder
    // configuration it produced (e.g. the extended bore genuinely intersects
    // other offset walls), fall back to the un-extended pipeline so every
    // previously-supported shape keeps its exact path and every refusal keeps
    // the canonical honesty-gate message.
    let mut extension_fired = false;
    let mut reflex_rebuilds = 0usize;
    let mut shadowed_points = Vec::new();
    let mut omitted = HashSet::default();
    let mut vacated_notes = Vec::new();
    let result = match offset_shell_impl(
        source,
        opening_face_ids,
        distance,
        body,
        true,
        &mut extension_fired,
        &mut reflex_rebuilds,
        &HashSet::default(),
        &HashSet::default(),
        false,
        &mut shadowed_points,
        &mut omitted,
        &mut vacated_notes,
    ) {
        Ok(result) => {
            // Attribution marker: which pipeline produced this result. A
            // retry-produced Ok can silently mask a broken extension lane —
            // always say which attempt won. The reflex-rebuild count rides
            // along because that lane fires on BOTH attempts: "extension
            // idle" alone would misread as "the plain pipeline did this".
            os_debug!(
                "offset_shell result: FIRST attempt (extension {}, reflex-rebuilt {})",
                if extension_fired { "fired" } else { "idle" },
                reflex_rebuilds,
            );
            Ok(result)
        }
        Err(error) if extension_fired => {
            os_debug!("extended pipeline failed ({error}); retrying without carrier extension");
            let mut unused = false;
            let mut retry_reflex = 0usize;
            shadowed_points.clear();
            omitted.clear();
            let retried = offset_shell_impl(
                source,
                opening_face_ids,
                distance,
                body,
                false,
                &mut unused,
                &mut retry_reflex,
                &HashSet::default(),
                &HashSet::default(),
                false,
                &mut shadowed_points,
                &mut omitted,
                &mut vacated_notes,
            );
            os_debug!(
                "offset_shell result: RETRY without extension ({}, reflex-rebuilt {retry_reflex})",
                if retried.is_ok() { "ok" } else { "err" },
            );
            retried
        }
        Err(error) => Err(error),
    };
    // A refusal from a shell whose selection saw two offsets CROSS inside
    // material thinner than twice the distance says so first: a rib thinner
    // than 2d shelled inward is that configuration, and the assembly's own
    // message names only the symptom. A refusal that already names its cause
    // keeps it. The crossing is looked for here, on the failure path only.
    match result {
        // A refusal that already names its CAUSE keeps its own sentence; one
        // that names a symptom ("connector edge left its plane", an assembly
        // message) is worth a cause in front of it. Reading the whole prefix as
        // "anything this module wrote" was too wide: the twin bosses came out
        // led by a notch sentence when their cause was the shadow cascade, and
        // the partial cone lost its junction cause behind a connector's.
        // Each cause is put IN FRONT of the symptom with `with_message`, so the
        // symptom's class (an assembly's `NonIntegralGenus`, …) survives.
        Err(error) if !refusal_names_its_cause(&error.message) => {
            let crossing = name_offset_crossing(source, opening_face_ids, distance, &shadowed_points);
            let junction = crossing
                .is_none()
                .then(|| name_retained_opening_junction(source, opening_face_ids, distance))
                .flatten();
            Err(cause_in_front(error, crossing, junction, vacated_notes.first()))
        }
        result => result,
    }
}

/// The symptom `error` with the first named cause IN FRONT of it: an offset
/// crossing, else a retained/opening junction, else a vacated rim's note.
/// Each prefix is a `with_message`, so the symptom's CLASS (an assembly's
/// `NonIntegralGenus`, …) survives the sentence put before it.
pub(super) fn cause_in_front(
    error: KernelRefusal,
    crossing: Option<String>,
    junction: Option<String>,
    note: Option<&String>,
) -> KernelRefusal {
    if let Some(crossing) = crossing {
        return error.with_message(|error| {
            format!(
                "offset_shell: {crossing}; a skin partly shadowed that way is not yet \
                 assembled ({error})"
            )
        });
    }
    if let Some(junction) = junction {
        return error.with_message(|error| format!("offset_shell: {junction} ({error})"));
    }
    // A rim the closure could not bridge, with what it measured, in front of
    // whatever the assembly then said about the hole it left.
    match note {
        Some(note) => error.with_message(|error| {
            format!(
                "offset_shell: a rim vacated by an omitted support was left open — {note} \
                 ({error})"
            )
        }),
        None => error,
    }
}

/// The refusals this module's own cause-naming lanes produce. Anything else is
/// a symptom another stage reported, and takes a cause in front of it.
fn refusal_names_its_cause(error: &str) -> bool {
    [
        " coincide near ",
        " touch tangentially near ",
        "is sealed off from the opening",
        "cannot tell whether",
        "omitting the shadowed offsets of source faces",
        "no retained face keeps any offset skin",
        "is shadowed at every one of its",
        "a rim vacated by an omitted support was left open",
        "junction, so its offset ends inside the cavity",
        "cross near",
        // The opening-wall lane's own named refusals, so a cause of mine never
        // arrives in front of one of theirs.
        "folds where the carve",
        "is curved and not a ruled surface",
        "cannot be built",
    ]
    .iter()
    .any(|name| error.contains(name))
}

fn offset_shell_impl(
    source: &BrepSolid,
    opening_face_ids: &[u64],
    distance: f64,
    body: Body,
    extend_oblique_carriers: bool,
    extension_fired: &mut bool,
    reflex_rebuilds: &mut usize,
    shadowed_supports: &HashSet<u64>,
    cavity_bounds: &HashSet<u64>,
    every_seeded_piece: bool,
    shadowed_points: &mut Vec<(Vec3, u64)>,
    omitted_supports: &mut HashSet<u64>,
    vacated_notes: &mut Vec<String>,
) -> Result<OffsetShellResultRecord, KernelRefusal> {
    profile::attempt();
    let mut clock = web_time::Instant::now();
    omitted_supports.clone_from(shadowed_supports);
    vacated_notes.clear();
    if !distance.is_finite() || distance.abs() <= MINIMUM_DISTANCE {
        return Err(KernelRefusal::input(KernelStage::Collect, "offset_shell_distance", "offset_shell: distance must be finite and non-zero"));
    }
    if opening_face_ids.is_empty() && body == Body::Shell {
        return Err(KernelRefusal::input(KernelStage::Collect, "offset_shell_no_opening", "offset_shell: at least one opening face is required"));
    }
    let source_faces = source
        .shells
        .iter()
        .flat_map(|shell| &shell.faces)
        .collect::<Vec<_>>();
    let opening_set = opening_face_ids.iter().copied().collect::<HashSet<_>>();
    if opening_set
        .iter()
        .any(|id| !source_faces.iter().any(|face| face.id == *id))
    {
        return Err(KernelRefusal::input(KernelStage::Collect, "offset_shell_foreign_opening", "offset_shell: opening face does not belong to source"));
    }
    let retained = source_faces
        .iter()
        .filter(|face| !opening_set.contains(&face.id))
        .map(|face| face.id)
        .collect::<Vec<_>>();
    if retained.is_empty() {
        return Err(refusals::no_retained_face().into());
    }
    let mut carriers = Vec::new();
    let mut open_section_reach = false;
    if distance < 0.0 && std::env::var("BREP_OFFSET_OPEN_SECTION_REACH").as_deref() != Ok("0") {
        for face in &source_faces {
            if retained.contains(&face.id) && open_profile_has_construction_headroom(face)? {
                open_section_reach = true;
                break;
            }
        }
    }
    // Faces whose folded part was carved away, by source face id. Later stages
    // that read a retained face's TRIM — the apex-cap scan below is the one —
    // must read the carved trim, not the one the source still carries.
    let mut carved_supports = HashMap::<u64, FaceRecord>::default();
    // Lane O item 705: (neighbour face, opening face, rim edge) of every
    // tangent junction with a curved opening the neighbour cannot reach.
    let mut tangent_rim_rounds: Vec<(u64, u64, u64)> = Vec::new();
    let mut shielded_cusps = HashSet::<u64>::default();

    for face_id in &retained {
        let face = source_faces.iter().find(|face| face.id == *face_id).unwrap();
        profile::lap(&mut clock, profile::CARRIER_TRIM);
        let collapsed = profile::detail("offset_support_collapsed", || offset_support_collapsed(face, distance))?;
        profile::lap(&mut clock, profile::CARRIER_REGULARITY);
        if collapsed {
            os_debug!("omitting collapsed offset support for source face {face_id}");
            continue;
        }
        // A support a previous pass found SHADOWED contributes no skin, and its
        // carrier must not imprint its neighbours either — see the selection.
        if shadowed_supports.contains(face_id) {
            os_debug!("omitting shadowed offset support for source face {face_id}");
            continue;
        }
        // A planar carrier whose source face meets a neighbour at a REFLEX
        // (concave) edge extends its trim by |distance|: at such a junction
        // the offset skin must GROW past the source footprint to meet the
        // neighbour's offset (a pocket wall's source [5,15] → skin
        // [3.5,16.5]); a source-sized trim leaves a gap no miter can bridge.
        // The surplus at the face's convex ends is clipped by the imprints +
        // on-skin filter. Convex-only faces keep their exact source-sized
        // trim — extending them perturbs delicately-selected fragment
        // landscapes for no benefit.
        let miter = source_faces
            .iter()
            .find(|face| face.id == *face_id)
            .map(|face| face_reflex_miter_tan(source, face))
            .transpose()?
            .flatten();
        // The offsets of two faces joined at a reflex edge meet d·tan(θn/2)
        // past the source rim — d only for a perpendicular join. Scale the
        // extension to the worst sampled miter (×1.5 headroom so the true
        // intersection lands strictly INSIDE the extended trim, never on its
        // boundary where the arrangement drops boundary-coincident cuts) and
        // cap it: a near-tangential join would ask for an unbounded trim.
        let reflex_extension = miter.map(|worst_tan| {
            (distance.abs() * worst_tan * 1.5)
                .max(distance.abs())
                .min(distance.abs() * 12.0)
        });
        if let Some(extension) = reflex_extension {
            os_debug!("carrier for src face {face_id}: reflex-extended by {extension:.4}");
        }
        // OUTWARD: every retained carrier grows by |distance| across its
        // sharp and opening-facing sides (the offsets of a convex junction
        // meet past both source rims; a wall plane cuts the surplus), and
        // by nothing across a smooth junction into a retained neighbour —
        // see `outward_carrier_extension`. A reflex junction needs no
        // outward growth at all (those offsets cross inside the trims).
        let mut curved_reach = None;
        let planar_reach = planar_opening_reach(face, &source_faces, &opening_set, distance)?;
        let extension = if distance < 0.0 {
            outward_carrier_extension(source, face, &source_faces, &opening_set, distance.abs())?
        } else {
            let mut inward = crate::CarrierExtension::uniform(reflex_extension.unwrap_or(0.0));
            // `BREP_OFFSET_THIN_SKIN_PAD` (default on; `=0` restores): a skin thinner than
            // the fragmenter's graze band is padded past its source rims
            // (`thin_skin_pad`), so its sections end far from its corners.
            if let Some(pad) = thin_skin_pad(source, face, distance)? {
                inward = crate::CarrierExtension::uniform(inward.u_min.max(pad));
                os_debug!("carrier for src face {face_id}: thin skin padded by {pad:.6}");
            }
            if let Some(reach) = planar_reach {
                let grown = reach + REACH_CLEARANCE * distance;
                inward = crate::CarrierExtension::uniform(inward.u_min.max(grown));
                os_debug!("carrier for src face {face_id}: planar reach to its opening {reach:.6}");
            }
            if reflex_extension.is_some() {
                let [u_min, u_max, v_min, v_max] =
                    apex_corner_reach(source, face, &source_faces, &opening_set, distance)?;
                if u_min.max(u_max).max(v_min).max(v_max) > 0.0 {
                    inward.u_min = inward.u_min.max(u_min);
                    inward.u_max = inward.u_max.max(u_max);
                    inward.v_min = inward.v_min.max(v_min);
                    inward.v_max = inward.v_max.max(v_max);
                    os_debug!(
                        "carrier for src face {face_id}: apex corner reach u=({:.4},{:.4}) v=({:.4},{:.4})",
                        inward.u_min, inward.u_max, inward.v_min, inward.v_max,
                    );
                }
            }
            // An INWARD carrier that falls short of the planar opening it meets
            // is grown until it crosses that plane, so the imprint cuts the
            // skin there instead of leaving a rim for the ruled weld to cork
            // below the opening (`fallshort_opening_reach`).
            if let Some((at_v_max, reach)) =
                fallshort_opening_reach(source, face, &source_faces, &opening_set, distance)?
            {
                // Past the plane by a quarter of the shell's own thickness, so
                // the section is strictly inside the carrier's trim rather than
                // on its boundary. Measured 2026-09-17 over the offset-shell
                // suite: every fraction from 0.01 to 2.0 of |d| gives the same
                // 73 passing members, while a margin sized in COINCIDENCE BANDS
                // (1 to 32 of them, 2e-5 each here) breaks the sphere-under-
                // frustum member by 9.4e-2 — the fragile margins are the ones
                // far below the construction's own length.
                let grown = reach + REACH_CLEARANCE * distance.abs();
                if at_v_max {
                    inward.v_max = inward.v_max.max(grown);
                } else {
                    inward.v_min = inward.v_min.max(grown);
                }
                os_debug!(
                    "carrier for src face {face_id}: inward reach to its opening plane {reach:.6}                      at v_{} ",
                    if at_v_max { "max" } else { "min" },
                );
            }
            // A CURVED opening has no such construction, and the pipeline
            // closes the cavity with whatever completion reaches for: measured
            // 7.6% short on the hemisphere cup shelled through its dome.
            if let Some((opening, gap)) =
                offset_falls_short_of_curved_opening(source, face, &source_faces, &opening_set, distance)?
            {
                if std::env::var("BREP_OFFSET_CURVED_REACH").as_deref() != Ok("0") {
                    curved_reach = curved_opening_reach_source(source, face, distance)?;
                }
                // `BREP_OFFSET_CURVED_RULED_REACH` (default on; `=0` restores): where the
                // rectangle continuation declines (a u-closed band, a tangent
                // junction), a v-ruled carrier is grown along its rulings
                // until it crosses the curved opening, as the planar reach
                // does for a plane (`curved_fallshort_reach`).
                let mut ruled_reach = false;
                if curved_reach.is_none() {
                    if let Some((at_v_max, reach)) =
                        curved_fallshort_reach(source, face, &source_faces, &opening_set, distance)?
                    {
                        let grown = reach + REACH_CLEARANCE * distance.abs();
                        if at_v_max {
                            inward.v_max = inward.v_max.max(grown);
                        } else {
                            inward.v_min = inward.v_min.max(grown);
                        }
                        ruled_reach = true;
                        os_debug!(
                            "carrier for src face {face_id}: ruled reach to its curved opening {opening} {reach:.6} at v_{}",
                            if at_v_max { "max" } else { "min" },
                        );
                    }
                }
                // Lane O item 705 (`BREP_OFFSET_TANGENT_RIM_ROUND`, default off):
                // a face that meets its curved opening TANGENTIALLY never
                // reaches it — its skin runs parallel to the opening at
                // exactly d — and the wall there is a round of radius d about
                // the rim (`rim_round.rs`). Record the rim; the round carrier
                // is emitted beside the apex caps, and this face's carrier is
                // built as it is.
                if curved_reach.is_none() && !ruled_reach && super::rim_round::tangent_rim_round_enabled() {
                    for coedge in face.loops.iter().flat_map(|loop_record| &loop_record.coedges) {
                        let Some((mate, mate_use)) = junction_mate(&source_faces, face, coedge.edge_id) else {
                            continue;
                        };
                        if mate.id != opening || !uses_are_tangent(face, coedge, mate, mate_use)? {
                            continue;
                        }
                        os_debug!("tangent rim round: source face {face_id} meets opening {opening} tangentially along edge {}", coedge.edge_id);
                        tangent_rim_rounds.push((*face_id, opening, coedge.edge_id));
                    }
                }
                if curved_reach.is_none() && !ruled_reach && !tangent_rim_rounds.iter().any(|(id, _, _)| id == face_id) {
                    return Err(KernelRefusal::unsupported(KernelStage::Refine, "offset_shell_curved_opening_short", format!(
                        "offset_shell: the offset skin of source face {face_id} stops {gap:.3e} short of \
                         curved opening face {opening}; only a planar opening's skin is grown to meet it"
                    )));
                }
            }
            inward
        };
        if distance < 0.0 {
            os_debug!(
                "carrier for src face {face_id}: outward extension u=({:.3},{:.3}) v=({:.3},{:.3})",
                extension.u_min,
                extension.u_max,
                extension.v_min,
                extension.v_max,
            );
        }
        // CARVE: a support whose offset folds over PART of its trim keeps the
        // regular part and drops the rest, instead of being built whole with
        // the fold in it. The carrier is built over the CARVED source, so the
        // trim it clones stops at the fold locus — and, because that trim no
        // longer reaches past the offset's pinch, `offset_surface`'s apex-cone
        // pinch retrim does not fire and the cloned pcurves still name the
        // pointwise offset.
        //
        // A carve that cannot be made is not a licence to build the folded
        // support anyway: the refusal is the answer, because a carrier with an
        // inverted patch in it welds into a shell that validates and bounds the
        // wrong solid.
        profile::lap(&mut clock, profile::CARRIER_TRIM);
        let carved = profile::detail("carve_folded_support", || carve_shell_support(source, face, distance, opening_face_ids))?;
        profile::lap(&mut clock, profile::CARRIER_REGULARITY);
        // A CUT RIM — the face's rim with a planar opening running across
        // its surface rather than along a v end — has the trim rebuilt over
        // the surface's full rectangle so the opening plane truncates the
        // offset skin (`cut_rim_full_rectangle_source`); like the carve, a
        // substitute source the carrier is built over.
        let cut_rim = if carved.is_none() && curved_reach.is_none() {
            cut_rim_full_rectangle_source(source, face, &source_faces, &opening_set, distance)?
        } else {
            None
        };
        // A curved face's SHARP outward u-side: the carrier is built over the
        // surface continued past it, so it reaches the mitre with its sharp
        // neighbour's offset (`sharp_side_continued_source`). Its INWARD
        // reflex u-side widens the trim inside the surface's own domain
        // instead (`reflex_side_widened_source`).
        let sharp_side = if carved.is_none() && cut_rim.is_none() && curved_reach.is_none() {
            match sharp_side_continued_source(source, face, &source_faces, &opening_set, distance, &extension)? {
                Some(continued) => Some(continued),
                None => match band_reach_continued_source(source, face, distance, &extension)? {
                    Some(widened) => Some(widened),
                    None => reflex_side_widened_source(source, face, &source_faces, &opening_set, distance, &extension)?,
                },
            }
        } else {
            None
        };
        let carrier_source = match (&carved, &cut_rim) {
            (None, None) if curved_reach.is_some() => curved_reach.as_ref().unwrap(),
            (None, None) if sharp_side.is_some() => sharp_side.as_ref().unwrap(),
            (None, Some(rebuilt)) => rebuilt,
            (Some(carved), _) => {
                os_debug!(
                    "carving the fold off source face {face_id}: {:.1}% of its trim kept, \
                     {} new boundary pcurve(s)",
                    carved.kept_fraction * 100.0,
                    carved.fold_boundary
                );
                carved_supports.insert(*face_id, carved.face.clone());
                if carved.shielded_cusp {
                    shielded_cusps.insert(*face_id);
                }
                &carved.solid
            }
            (None, None) => source,
        };
        profile::lap(&mut clock, profile::CARRIER_TRIM);
        let (mut carrier, lane) = carrier_solid_sided_lane(carrier_source, *face_id, distance, &extension)?;
        profile::carrier_built(lane, clock.elapsed());
        clock = web_time::Instant::now();
        if (distance < 0.0 || reflex_extension.is_some()) && face.surface.is_affine().or_refuse(KernelStage::Sew, "is_affine")? {
            // PADDED planes: the pad slides the plane's net while the trim's
            // loops are cloned in uv, so an INTERIOR loop (a bore or boss rim
            // in the face) is carried outward with the grown outline instead
            // of following the neighbour's offset.
            //   OUTWARD: a bore's rim grew (r=4.68 → 5.15) while the shrunk
            //   offset bore (r=3.68) that must hole this plane fell inside the
            //   enlarged void, so neither the imprint nor the orphan-rim weld
            //   could cut it.
            //   INWARD, reflex-extended: the same slide moves a REFLEX hole
            //   rim the wrong way — a sphere sitting under a frustum's base
            //   annulus (the user's 2026-09-07 document) had its r=8.3 rim
            //   scaled to 9.22 while the shrunk offset sphere (r=7.3) crosses
            //   the offset plane at r=7.23, inside the enlarged hole: the
            //   pair imprint found nothing and the rim stayed one-use.
            // Drop the interior loops (as the wall carriers do): the
            // neighbour's carrier imprints — or its rim, welded in as a
            // hole — cut the true opening, and the seeded selection drops
            // the disk over it. A CONVEX interior rim on the same face (a
            // through-bore in a pocket floor) is safe to drop too: its
            // offset bore imprints at r+d and the ring inside is rejected by
            // class (Out) or by the on-skin filter, no seed involved.
            drop_wall_interior_loops(&mut carrier)?;
            // A free-form outline grown uniformly outward is trimmed by its
            // outline OFFSET, not the slid net's scaled trim
            // (`outline_offset_carrier`).
            let pads = [extension.u_min, extension.u_max, extension.v_min, extension.v_max];
            let widest = pads.iter().copied().fold(f64::NEG_INFINITY, f64::max);
            let narrowest = pads.iter().copied().fold(f64::INFINITY, f64::min);
            // `BREP_OFFSET_CONVEX_POLYGON_OUTLINE`: pads equal but for their last
            // bits (a tetrahedron's turned sides read 2.121 four ways, not
            // bit-equal) are the uniform pad too.
            let uniform = (extension.u_min == extension.u_max
                && extension.u_min == extension.v_min
                && extension.u_min == extension.v_max)
                || (convex_polygon_outline_enabled() && widest - narrowest <= 1e-9 * widest.abs());
            if distance < 0.0 && uniform {
                // Past the walls' sections by the same clearance a planar
                // neighbour of a ruled wall gets, so each section lies
                // strictly INSIDE the fitted trim rather than on it.
                let clearance =
                    RULED_NEIGHBOUR_CLEARANCE_BANDS * offset_skin_band(crate::solid_scale(source), distance);
                if let Some(rebuilt) = outline_offset_carrier(source, face, widest, clearance, &carrier)? {
                    os_debug!(
                        "carrier for src face {face_id}: outline trimmed by its offset by {:.4}",
                        widest + clearance
                    );
                    carrier = rebuilt;
                }
            }
        }
        // An inward reflex junction needs a construction plane extending PAST
        // the source rim. Sliding the net under a free-form trim scales that
        // trim instead: a notched revolve cap's notch grows, and clips away the
        // sections with the box's offsets. Even its curved convex boundary can
        // fall inside the neighbouring offset section after that scaling.
        // Use the extended plane's full rectangle and let the neighbouring
        // carriers cut the skin. The source face and the offset surface stay
        // unchanged. Circular outlines need the same construction: scaling a
        // reflex arc can also clip away its offset cylinder's section. Unlike
        // an outline offset, this needs no approximation of
        // the planar rim's parallel curve (which may collapse at a tight turn).
        if distance > 0.0
            && reflex_extension.is_some()
            && face.surface.is_affine().or_refuse(KernelStage::Refine, "is_affine")?
            && outline_has_curved_element(source, face)?
        {
            carrier = full_rectangle_face(&carrier, &carrier.shells[0].faces[0])?;
        }
        if planar_reach.is_some() {
            // A chamfer's trimmed polygon can cut diagonally across its
            // patch. Scaling that polygon with the net can still clip the
            // opening section. Let the opening and neighbouring offsets
            // trim the grown construction rectangle instead.
            carrier = full_rectangle_face(&carrier, &carrier.shells[0].faces[0])?;
        }
        if open_section_reach {
            if let Some(continued) = open_section_construction(source, face, &carrier, &extension, distance)? {
                carrier = continued;
                os_debug!("carrier for src face {face_id}: open section uses its natural construction rectangle");
            }
        }
        // The rim a shadowed support's removal VACATED in this carrier's trim:
        // a slot narrower than 2|d| that runs out through this face leaves a
        // notch its own skin has to close, because the dilation fills the slot.
        // The source face keeps its notch; only the carrier is bridged.
        if distance < 0.0 && !shadowed_supports.is_empty() {
            let report = close_vacated_rims(
                &mut carrier,
                source_faces
                    .iter()
                    .find(|face| face.id == *face_id)
                    .copied()
                    .unwrap_or(face),
                &source_faces,
                shadowed_supports,
            )?;
            if report.closed > 0 {
                os_debug!(
                    "carrier for src face {face_id}: {} vacated rim run(s) bridged",
                    report.closed
                );
            }
            for declined in &report.declined {
                os_debug!("carrier for src face {face_id}: vacated rim left alone — {declined}");
                vacated_notes.push(format!("source face {face_id}: {declined}"));
            }
        }
        carriers.push(Carrier {
            solid: carrier,
            source_face_id: *face_id,
            kind: OffsetFaceRole::Offset,
        });
    }
    profile::lap(&mut clock, profile::CARRIER_TRIM);
    // APEX-CAP carriers: a retained face whose DEGENERATE edge (an apex
    // point, e.g. a conic crater's tip) offsets into a RING is an EXTERIOR
    // cone offset — the parallel surface truncates at the ring and the exact
    // rolling-ball offset closes it with a SPHERE of radius |distance| about
    // the apex. Emit that sphere as an extra carrier: the ring imprint
    // (sphere × offset frustum) splits it, the on-skin filter keeps exactly
    // the cap sector (its points are |d| from the apex and ≥|d| from the
    // rest of the face), and the ring becomes two-use.
    let mut apex_cap_sources = HashSet::default();
    let mut apex_cap_pairs: Vec<(usize, usize)> = Vec::new();
    {
        let retained_count = carriers.len();
        for index in 0..retained_count {
            // A CARVED support's trim is the carved one. Its coedges name edges
            // that live only in the carved clone, so the `source.edges` lookup
            // below finds nothing and the apex-cap lane skips the face — which
            // is right: the carve already ended that carrier at the offset's own
            // pinch, and there is no apex ring left to cap.
            let carved_face = carved_supports.get(&carriers[index].source_face_id);
            let source_face = carved_face
                .or_else(|| source_by_id_lookup(&source_faces, carriers[index].source_face_id));
            let Some(source_face) = source_face else {
                continue;
            };
            let carrier_surface = carriers[index].solid.shells[0].faces[0].surface.clone();
            for coedge in source_face
                .loops
                .iter()
                .flat_map(|loop_record| &loop_record.coedges)
            {
                let Some(edge) = source
                    .edges
                    .iter()
                    .find(|edge| edge.id == coedge.edge_id)
                else {
                    continue;
                };
                if !edge.degenerate {
                    continue;
                }
                // Image of the degenerate edge on the offset surface.
                let [p0, p1] = coedge.pcurve.domain().or_refuse(KernelStage::Sew, "domain")?;
                let mut ring = Vec::new();
                for sample in 0..=8 {
                    let uv = coedge
                        .pcurve
                        .evaluate(p0 + (p1 - p0) * sample as f64 / 8.0).or_refuse(KernelStage::Sew, "evaluate")?;
                    ring.push(carrier_surface.evaluate(uv.x, uv.y).or_refuse(KernelStage::Sew, "evaluate")?);
                }
                let spread = ring
                    .iter()
                    .map(|point| point.sub(ring[0]).length())
                    .fold(0.0f64, f64::max);
                if spread <= distance.abs() * 1e-2 {
                    continue;
                }
                let apex = source
                    .vertices
                    .iter()
                    .find(|vertex| vertex.id == edge.start_vertex_id)
                    .map(|vertex| vertex.point)
                    .ok_or_else(|| "offset_shell: apex vertex missing".to_string()).or_refuse(KernelStage::Sew, "offset_shell_apex_vertex_missing")?;
                let centroid = ring
                    .iter()
                    .fold(Vec3::default(), |sum, point| sum.add(*point))
                    .scale(1.0 / ring.len() as f64);
                let axis = apex.sub(centroid);
                let axis = if axis.length() > 1e-9 {
                    axis.normalized().or_refuse(KernelStage::Sew, "normalized")?
                } else {
                    Vec3::new(0.0, 0.0, 1.0)
                };
                os_debug!(
                    "apex-cap carrier for src face {} at ({:.3},{:.3},{:.3}) r={:.3}",
                    carriers[index].source_face_id,
                    apex.x,
                    apex.y,
                    apex.z,
                    distance.abs(),
                );
                apex_cap_sources.insert(index);
                apex_cap_sources.insert(carriers.len());
                apex_cap_pairs.push((index, carriers.len()));
                carriers.push(Carrier {
                    solid: crate::make_sphere_brep(apex, distance.abs(), axis).or_refuse(KernelStage::Sew, "make_sphere_brep")?,
                    source_face_id: carriers[index].source_face_id,
                    kind: OffsetFaceRole::Offset,
                });
            }
        }
    }
    // Lane O item 705: the ROUND carriers about tangent opening rims, emitted
    // as the apex caps are (an extra Offset carrier of the neighbour face) and
    // handled by the same exemptions and coincident-edge splits for now.
    if !tangent_rim_rounds.is_empty() {
        let tolerance = 1e-7f64.max(crate::solid_scale(source) * 1e-9);
        for (neighbour_id, opening_id, edge_id) in &tangent_rim_rounds {
            let Some(neighbour_index) = carriers.iter().position(|carrier| carrier.source_face_id == *neighbour_id && matches!(carrier.kind, OffsetFaceRole::Offset)) else {
                continue;
            };
            let Some(edge) = source.edges.iter().find(|edge| edge.id == *edge_id) else { continue };
            let Some(opening_face) = source_by_id_lookup(&source_faces, *opening_id) else { continue };
            match super::rim_round::circular_rim_round(edge, opening_face, distance, tolerance)? {
                Some(round) => {
                    os_debug!("tangent rim round: torus about edge {edge_id} (neighbour {neighbour_id}, opening {opening_id}) as carrier {}", carriers.len());
                    apex_cap_sources.insert(neighbour_index);
                    apex_cap_sources.insert(carriers.len());
                    apex_cap_pairs.push((neighbour_index, carriers.len()));
                    carriers.push(Carrier { solid: round, source_face_id: *neighbour_id, kind: OffsetFaceRole::Offset });
                }
                None => {
                    return Err(KernelRefusal::unsupported(KernelStage::Refine, "offset_shell_tangent_rim_round", format!(
                        "offset_shell: the round about the tangent rim (edge {edge_id}) between source face {neighbour_id} and \
                         curved opening face {opening_id} is built only for a circular rim inside its own circle"
                    )));
                }
            }
        }
    }
    profile::lap(&mut clock, profile::APEX_CAPS);
    // A fold the carve's probe did not see (`outward_fold_the_carve_missed`):
    // a band too thin for its grid is built whole, and the torus wedge grown by
    // 1.001 then held the reflected lemon beyond its axis. Asked of every
    // retained face the carve neither carved nor omitted, on every outward
    // shell: a ruled wall is what first let the pipeline CLOSE such a shape,
    // but the geometry is the trigger. Refused by name, the tangent onset
    // included; a partial revolve of that kind is the revolved-wedge lane's
    // question, and that lane runs in front of this pipeline.
    if distance < 0.0 {
        for face_id in &retained {
            if carved_supports.contains_key(face_id)
                || !carriers.iter().any(|carrier| carrier.source_face_id == *face_id)
            {
                continue;
            }
            let face = source_by_id_lookup(&source_faces, *face_id).unwrap();
            if let Some(fold) = profile::detail("outward_fold_the_carve_missed", || outward_fold_the_carve_missed(face, distance))? {
                return Err(KernelRefusal::unsupported(KernelStage::Refine, "offset_shell_outward_fold_band", format!(
                    "offset_shell: the outward offset of source face {face_id} folds where the carve \
                     saw no fold (area factor {:.3e} at u={:.6}, v={:.6}); a fold band that thin is \
                     not built",
                    fold.factor, fold.u, fold.v
                )));
            }
        }
    }
    profile::lap(&mut clock, profile::CARRIER_REGULARITY);
    // Wall carriers built as a RULED opening face extended along itself, by
    // carrier index: the in-front guard reads their sides off the surface
    // normal where it would read a plane's.
    let mut ruled_walls = HashSet::default();
    // Inward walls at reflex rims need construction area beyond the source
    // opening too. Track them for source imprints and material-side selection.
    let mut inward_extended_walls = HashSet::default();
    let mut reflex_wall_neighbors = HashMap::default();
    // OUTWARD through a CURVED opening. A ruled one — the cone a revolved
    // straight profile edge sweeps, a cylinder — has its wall built as the
    // opening face held in place and EXTENDED along itself, exactly as a plane
    // is padded (`extended_ruled_wall`); the pad is the plane's, read off the
    // same convex junctions. Built first, so the planar walls know which
    // neighbours are ruled walls.
    //
    // Any other curved opening is refused by name. Its wall used to be the
    // unpadded source face, which never reaches the grown offset: on a torus
    // wedge whose chord revolves to a cone, the pair offset x cone cut 0
    // pieces, the cone's one fragment was dropped as a void cross-section, and
    // the result read "completion produced non-integral genus" with 12 open
    // edges. No outward shell in the kernel suites or the case corpus closed
    // through that wall (census, 2026-09-17).
    let mut ruled_built = HashMap::<u64, BrepSolid>::default();
    if distance < 0.0 {
        for face_id in opening_face_ids {
            let source_face = source_by_id_lookup(&source_faces, *face_id).unwrap();
            if source_face.surface.is_affine().or_refuse(KernelStage::Sew, "is_affine")? {
                continue;
            }
            if !ruled_opening(source_face)? {
                return Err(KernelRefusal::unsupported(KernelStage::Refine, "offset_shell_curved_opening", format!(
                    "offset_shell: opening face {face_id} is curved and not a ruled surface; an \
                     outward wall is built only on a planar or ruled opening"
                )));
            }
            let pad = outward_wall_pad(
                source,
                source_face,
                &source_faces,
                &opening_set,
                distance.abs(),
            )?;
            let tolerance = 1e-7f64.max(crate::solid_scale(source) * 1e-9);
            match extended_ruled_wall(source, source_face, pad, tolerance)? {
                Ok(wall) => {
                    os_debug!(
                        "wall for opening face {face_id}: ruled, pad {pad:.4}, sides reach {:.4?}",
                        wall.reached
                    );
                    ruled_built.insert(*face_id, wall.solid);
                }
                Err(reason) => {
                    return Err(KernelRefusal::unsupported(KernelStage::Refine, "offset_shell_ruled_wall", format!(
                        "offset_shell: the outward wall of ruled opening face {face_id} cannot be \
                         built: {reason}"
                    )));
                }
            }
        }
    }
    let ruled_openings = ruled_built
        .keys()
        .filter_map(|id| source_by_id_lookup(&source_faces, *id))
        .collect::<Vec<_>>();
    for face_id in opening_face_ids {
        let source_face = source_faces
            .iter()
            .find(|face| face.id == *face_id)
            .unwrap();
        if let Some(wall) = ruled_built.remove(face_id) {
            ruled_walls.insert(carriers.len());
            carriers.push(Carrier {
                solid: wall,
                source_face_id: *face_id,
                kind: OffsetFaceRole::Wall,
            });
            continue;
        }
        if distance > 0.0
            && source_face.surface.is_affine().or_refuse(KernelStage::Refine, "is_affine")?
        {
            if let Some((pad, neighbors)) =
                inward_wall_pad(source, source_face, &source_faces, &opening_set, distance)?
            {
                // Scaling a notched/free-form outline can still clip the new
                // rim. Use the whole plane rectangle; source and offset
                // imprints below recover both exact boundaries of the wall.
                let padded = carrier_solid(source, *face_id, 0.0, pad)?;
                let wall = full_rectangle_face(&padded, &padded.shells[0].faces[0])?;
                inward_extended_walls.insert(carriers.len());
                reflex_wall_neighbors.insert(carriers.len(), neighbors);
                carriers.push(Carrier {
                    solid: wall,
                    source_face_id: *face_id,
                    kind: OffsetFaceRole::Wall,
                });
                continue;
            }
        }
        carriers.push(Carrier {
            solid: if distance < 0.0 && source_face.surface.is_affine().or_refuse(KernelStage::Sew, "is_affine")? {
                let pad = outward_wall_pad(
                    source,
                    source_face,
                    &source_faces,
                    &opening_set,
                    distance.abs(),
                )?;
                // A plane meeting a RULED opening is padded past the grown
                // outline, not onto it. Padded by exactly the pad, its trim's
                // arc lands ON the grown offset's section wherever the join is
                // perpendicular, and the ruled wall's section then ENDS on that
                // coincident pair: on the torus wedge the cap shredded into 60
                // slivers along the grown circle. The clearance is counted in
                // the band that decides whether the strip beyond the grown
                // outline is wall or void, so the strip always resolves as its
                // own fragment and is dropped as void.
                let pad = if ruled_openings
                    .iter()
                    .any(|ruled| source_faces_adjacent(source_face, ruled))
                {
                    pad + RULED_NEIGHBOUR_CLEARANCE_BANDS
                        * offset_skin_band(crate::solid_scale(source), distance)
                } else if outline_has_free_element(source, source_face)? {
                    // A free-form outline's wall is a FIT: without the same
                    // clearance its outline sits within the fit tolerance of
                    // the carriers' own sections on the plane, and the
                    // near-coincident curves shred the wall's imprint (26 to
                    // 121 fragments on the free-form D band, 2026-09-26). With
                    // it, the surplus ring is cut away cleanly.
                    pad + RULED_NEIGHBOUR_CLEARANCE_BANDS * offset_skin_band(crate::solid_scale(source), distance)
                } else {
                    pad
                };
                os_debug!("wall for opening face {face_id}: outward pad {pad:.4}");
                // The wall is the opening's outline OFFSET by the pad in its
                // own plane (`outline_offset.rs`). The pad used to SCALE that
                // outline about the patch centre, which is the offset only for
                // a rectangle filling its patch or a circle centred on it: an
                // L-shaped opening's notch edges pass through the centre and
                // did not move at all, so the notch faces' grown offsets
                // crossed the plane outside the wall ("non-integral genus",
                // `l-step-open-bottom` at d = −1.5 in the membrane census).
                //
                // An outline this module does not offset keeps the scaled wall,
                // but only when that wall still COVERS the offset outline along
                // every edge a retained face meets — which is what the sections
                // have to land inside. Otherwise the shell refuses by name
                // rather than assembling an open rim.
                let mut scaled = carrier_solid(source, *face_id, 0.0, pad)?;
                drop_wall_interior_loops(&mut scaled)?;
                let mut wall = match profile::detail("offset_outline_wall", || offset_outline_wall(source, source_face, pad))? {
                    Ok(outline) => {
                        // Where the scaled wall already trims the offset
                        // outline's own region — a rectangle filling its patch,
                        // a circle centred on it — it is kept, so no shell that
                        // was right before is rebuilt on a different patch.
                        let same = profile::detail("walls_trim_the_same_region", || walls_trim_the_same_region(
                            &outline,
                            &scaled,
                            1e-9f64.max(crate::solid_scale(source) * 1e-11),
                        ))?;
                        if same {
                            os_debug!("wall for opening face {face_id}: scaled, which IS its outline offset");
                            scaled
                        } else {
                            os_debug!("wall for opening face {face_id}: outline offset by {pad:.4}");
                            outline
                        }
                    }
                    Err(reason) => {
                        let carrier_surfaces = carriers
                            .iter()
                            .filter(|carrier| matches!(carrier.kind, OffsetFaceRole::Offset))
                            .map(|carrier| (carrier.source_face_id, &carrier.solid.shells[0].faces[0].surface))
                            .collect::<Vec<_>>();
                        if let Err(uncovered) = scaled_wall_covers_offset(
                            source,
                            source_face,
                            &scaled,
                            pad,
                            &source_faces,
                            &opening_set,
                            &carrier_surfaces,
                        )? {
                            return Err(KernelRefusal::unsupported(KernelStage::Refine, "offset_shell_opening_wall_coverage", format!(
                                "offset_shell: opening face {face_id}'s outline is not offset \
                                 ({reason}), and its scaled outward wall does not contain every \
                                 retained neighbour's grown offset section (a point of one lies \
                                 {uncovered:.3e} outside)"
                            )));
                        }
                        os_debug!(
                            "wall for opening face {face_id}: scaled, which covers its offset ({reason})"
                        );
                        scaled
                    }
                };
                // The pad extends the wall by STRETCHING the plane's 2x2 net
                // while the trim pcurves are cloned in UV, so every loop
                // scales about the patch centre by (w+2|d|)/w. For the OUTER
                // loop of a full-domain rectangle that lands exactly on the
                // grown outline — the intended wall extent. But an INTERIOR
                // loop (a carve outline: a bore, gouge or crater rim in the
                // opening face) is scaled OUTWARD too, and the enlarged hole
                // then swallows both circles the wall must be cut by (the
                // source carve rim from the wall x source imprint and the
                // shrunk rim from the wall x offset-carrier imprint) — the
                // imprints clip to nothing and the rims dangle one-use. Drop
                // interior loops instead: the wall covers the carve mouth,
                // the imprints cut the true rims, and selection (void-wall
                // guard + on-skin filter) drops the fragments over the void.
                drop_wall_interior_loops(&mut wall)?;
                wall
            } else {
                standalone_face(source, *face_id)?
            },
            source_face_id: *face_id,
            kind: OffsetFaceRole::Wall,
        });
    }
    if carriers.len() >= SOURCE_OPERAND as usize {
        return Err(KernelRefusal::unsupported(KernelStage::Collect, "offset_shell_carrier_count", "offset_shell: too many carrier faces for current ABI"));
    }
    profile::lap(&mut clock, profile::WALLS);
    // The characteristic length every tolerance below is derived from.  This
    // was `max ‖vertex‖` — the distance from the WORLD ORIGIN — until audit
    // slice 0: that form made every band widen as the part was moved away from
    // the origin while ignoring the part's actual size, so the same shell
    // solved to different tolerances depending only on where it was modelled.
    // `solid_scale` is the bbox diagonal, floored at 1.0 exactly as the old
    // fold was, so this changes ONE thing: placement-dependence.
    let scale = crate::solid_scale(source);
    crate::report_scale_migration("offset_shell", scale, || {
        source
            .vertices
            .iter()
            .map(|vertex| vertex.point.length())
            .fold(1.0, f64::max)
    });
    let smooth = synchronize_smooth_offset_boundaries(&mut carriers, &source_faces, scale)?;
    // Reflex-rim rebuilds run on BOTH attempts: the fallback for a broken
    // extension is an honest refusal, but the fallback for a skipped reflex
    // rebuild is a silently WRONG (membrane-capped) shell.
    let reflex_rebuilt = rebuild_reflex_rim_carriers(
        &mut carriers,
        source,
        &source_faces,
        &smooth.pairs,
        scale,
        distance,
    )?;
    *reflex_rebuilds = reflex_rebuilt.len();
    if extend_oblique_carriers {
        let oblique_extended = extend_offset_carriers_past_open_hole_rims(
            &mut carriers,
            source,
            &source_faces,
            &opening_set,
            &smooth.pairs,
            scale,
        )?;
        let mut already_extended = oblique_extended.clone();
        already_extended.extend(reflex_rebuilt.iter().copied());
        let fallshort_extended = extend_fallshort_curved_carriers(
            &mut carriers,
            source,
            &source_faces,
            &opening_set,
            &smooth.pairs,
            &already_extended,
            scale,
            distance,
        )?;
        *extension_fired = !oblique_extended.is_empty() || fallshort_extended > 0;
        os_debug!(
            "extended {} oblique + {fallshort_extended} fall-short carriers past their opening planes",
            oblique_extended.len(),
        );
    }
    profile::lap(&mut clock, profile::EXTENSION);
    let tolerance = 1e-7f64.max(scale * 1e-9);
    let pair_tolerance = (scale * 1e-8).max(2e-6);
    let reach = distance.abs() * 4.0 + pair_tolerance;
    let carrier_samples = carriers
        .iter()
        .map(|carrier| carrier_extent_points(&carrier.solid))
        .collect::<Result<Vec<_>, _>>()?;
    let carrier_bounds = carrier_samples
        .iter()
        .map(|samples| Bounds::from_nonempty_points(samples))
        .collect::<Option<Vec<_>>>()
        .ok_or_else(|| "offset_shell: carrier has no boundary samples".to_string()).or_refuse(KernelStage::Sew, "offset_shell_carrier_has_no_boundary_samples")?;
    let source_by_id = source_faces
        .iter()
        .map(|face| (face.id, *face))
        .collect::<HashMap<_, _>>();
    let source_samples = source_faces
        .iter()
        .map(|face| {
            let standalone = standalone_face(source, face.id)?;
            Ok((face.id, edge_sample_points(&standalone)?))
        })
        .collect::<Result<HashMap<_, _>, KernelRefusal>>()?;
    if debug_enabled() {
        for (index, carrier) in carriers.iter().enumerate() {
            let face = &carrier.solid.shells[0].faces[0];
            let samples = &carrier_samples[index];
            let bounds =
                samples
                    .iter()
                    .fold((Vec3::default(), Vec3::default()), |(low, high), point| {
                        (
                            Vec3 {
                                x: low.x.min(point.x),
                                y: low.y.min(point.y),
                                z: low.z.min(point.z),
                            },
                            Vec3 {
                                x: high.x.max(point.x),
                                y: high.y.max(point.y),
                                z: high.z.max(point.z),
                            },
                        )
                    });
            let surface = &face.surface;
            let rational = surface
                .control_points
                .iter()
                .flatten()
                .any(|point| (point.w - 1.0).abs() > 1e-12);
            os_debug!(
                "carrier[{index}] kind={:?} source_face={} loops={} deg=({},{}) net={}x{} rational={rational} bounds=({:.3},{:.3},{:.3})..({:.3},{:.3},{:.3})",
                carrier.kind, carrier.source_face_id, face.loops.len(),
                surface.degree_u, surface.degree_v,
                surface.control_points.len(),
                surface.control_points.first().map(|row| row.len()).unwrap_or(0),
                bounds.0.x, bounds.0.y, bounds.0.z, bounds.1.x, bounds.1.y, bounds.1.z,
            );
        }
        for (index, carrier) in carriers.iter().enumerate() {
            if !matches!(carrier.kind, OffsetFaceRole::Offset) {
                continue;
            }
            let source_face = source_by_id[&carrier.source_face_id];
            let carrier_surface = &carrier.solid.shells[0].faces[0].surface;
            let (ku, kv) = (
                crate::KnotVector::new(
                    source_face.surface.knots_u.clone(),
                    source_face.surface.degree_u,
                ),
                crate::KnotVector::new(
                    source_face.surface.knots_v.clone(),
                    source_face.surface.degree_v,
                ),
            );
            if let (Ok(ku), Ok(kv)) = (ku, kv) {
                let [u0, u1] = ku.domain();
                let [v0, v1] = kv.domain();
                let mut worst = 0.0f64;
                let mut worst_at = (0.0f64, 0.0f64);
                for iu in 0..=24 {
                    for iv in 0..=24 {
                        let u = u0 + (u1 - u0) * iu as f64 / 24.0;
                        let v = v0 + (v1 - v0) * iv as f64 / 24.0;
                        let expected = face_offsets(source_face)
                            .at(u, v, -distance)
                            .map(|sample| sample.point);
                        let actual = carrier_surface.evaluate(u, v);
                        if let (Ok(expected), Ok(actual)) = (expected, actual) {
                            let error = expected.sub(actual).length();
                            if error > worst {
                                worst = error;
                                worst_at = (u, v);
                            }
                        }
                    }
                }
                let mut normal_swing = 0.0f64;
                let mut previous: Option<Vec3> = None;
                for iu in 0..=48 {
                    let u = u0 + (u1 - u0) * iu as f64 / 48.0;
                    let v = (v0 + v1) / 2.0;
                    if let Ok(normal) = face_normal(source_face, u, v) {
                        if let Some(previous) = previous {
                            normal_swing = normal_swing.max(previous.sub(normal).length());
                        }
                        previous = Some(normal);
                    }
                }
                let weights = source_face
                    .surface
                    .control_points
                    .iter()
                    .flatten()
                    .map(|point| point.w)
                    .fold((f64::MAX, f64::MIN), |(low, high), w| {
                        (low.min(w), high.max(w))
                    });
                os_debug!(
                    "carrier[{index}] offset fit error max={worst:.6} at=({:.4},{:.4}) domain_u=({u0:.4},{u1:.4}) normal_step_max={normal_swing:.4} src_net={}x{} src_deg=({},{}) src_w=({:.4},{:.4}) knots_u={:?}",
                    worst_at.0, worst_at.1,
                    source_face.surface.control_points.len(),
                    source_face.surface.control_points.first().map(|row| row.len()).unwrap_or(0),
                    source_face.surface.degree_u, source_face.surface.degree_v,
                    weights.0, weights.1,
                    &source_face.surface.knots_u,
                );
            }
        }
        os_debug!("smooth pairs: {:?}", smooth.pairs);
    }
    let mut imprint = empty_imprint();
    let mut next_piece_id = 1;
    let mut next_vertex_id = 1;
    // Lane P3b row 2 (2026-10-26): where the IMPRINT phase spends its time
    // outside `build_imprints`, one `offset_shell.imprint_split` line under
    // BREP_PROFILE: the carrier-pair builds and merges, the wall x source
    // builds, their standalone face copies and merges, the apex ring splits,
    // the smooth-split sync and the boundary-coincidence filter.
    let split_profile = crate::imprint::profile_enabled();
    let mut split_ms = [0.0f64; 9];
    let mut split_clock = split_profile.then(web_time::Instant::now);
    let mut split_lap = |slot: usize, clock: &mut Option<web_time::Instant>| {
        if let Some(started) = clock {
            let now = web_time::Instant::now();
            split_ms[slot] += (now - *started).as_secs_f64() * 1e3;
            *started = now;
        }
    };
    // The pair skip rules, one closure for both imprint paths below (the
    // per-pair calls and row 6's one-call composite).
    let pair_admitted = |first: usize, second: usize| -> bool {
            let first_source = source_by_id[&carriers[first].source_face_id];
            let second_source = source_by_id[&carriers[second].source_face_id];
            if matches!(carriers[first].kind, OffsetFaceRole::Wall)
                && matches!(carriers[second].kind, OffsetFaceRole::Wall)
            {
                // Unextended inward walls are the opening faces themselves:
                // they never overlap; adjacent ones only touch along their
                // shared edge, so there is nothing to imprint. OUTWARD walls
                // are the opening planes EXTENDED by |distance|: the walls of
                // two ADJACENT openings cross along the line of their shared
                // source edge, and each wall ring must stop there (the sheet
                // past that line lies in front of the other opening). Without
                // the cut the wall × source chain below stays open on the
                // shared-edge side (its neighbour there is an opening, not a
                // retained face), the wall never fragments into its ring, and
                // every source rim along both openings dangles one-use.
                // Extended inward walls need those mutual cuts as well.
                if (distance > 0.0
                    && !inward_extended_walls.contains(&first)
                    && !inward_extended_walls.contains(&second))
                    || !source_faces_adjacent(first_source, second_source)
                {
                    return false;
                }
            }
            if smooth.pairs.contains(&(first, second)) {
                os_debug!("pair ({first},{second}) skipped: smooth");
                return false;
            }
            if !carrier_bounds[first].intersects_upper(carrier_bounds[second], pair_tolerance) {
                os_debug!("pair ({first},{second}) skipped: bounds");
                return false;
            }
            if !source_faces_adjacent(first_source, second_source)
                && sample_separation(
                    &source_samples[&first_source.id],
                    &source_samples[&second_source.id],
                ) > reach
            {
                os_debug!("pair ({first},{second}) skipped: separation");
                return false;
            }
            true
    };
    // The wall x source pairs: an extended opening wall (outward, or inward
    // at a reflex rim) against each retained source face adjacent to its
    // opening (see the note at the per-pair loop below).
    let mut wall_source_pairs: Vec<(usize, u64)> = Vec::new();
    if body == Body::Shell && (distance < 0.0 || !inward_extended_walls.is_empty()) {
        for index in 0..carriers.len() {
            if !matches!(carriers[index].kind, OffsetFaceRole::Wall) {
                continue;
            }
            if distance > 0.0 && !inward_extended_walls.contains(&index) {
                continue;
            }
            let opening = source_by_id[&carriers[index].source_face_id];
            for retained_id in &retained {
                if source_faces_adjacent(opening, source_by_id[retained_id]) {
                    wall_source_pairs.push((index, *retained_id));
                }
            }
        }
    }
    let one_call = one_call_imprint_enabled();
    let mut one_call_done = false;
    if one_call {
        // ONE-CALL IMPRINT (`BREP_OFFSET_ONE_CALL_IMPRINT`, lane P3b row 6):
        // every carrier and every retained source face a wall is cut against
        // go into one open sheet solid (ids renumbered per member), one
        // `build_imprints` processes exactly the admitted pairs, and its
        // post-passes run once over the whole piece set. The record comes
        // back in the shell's operand space. A failure of the single call
        // falls back to the per-pair path below, which tolerates a failed
        // pair on its own.
        let mut members: Vec<(u8, &BrepSolid)> = carriers.iter().enumerate().map(|(index, carrier)| (index as u8, &carrier.solid)).collect();
        let mut source_members: Vec<(u64, BrepSolid)> = Vec::new();
        for (_, retained_id) in &wall_source_pairs {
            if !source_members.iter().any(|(id, _)| id == retained_id) {
                source_members.push((*retained_id, standalone_face(source, *retained_id)?));
            }
        }
        for (_, solid) in &source_members {
            members.push((SOURCE_OPERAND, solid));
        }
        let composite = compose_imprint_operands(&members);
        let mut admitted: HashSet<(usize, usize)> = HashSet::default();
        for first in 0..carriers.len() {
            for second in first + 1..carriers.len() {
                if pair_admitted(first, second) {
                    admitted.insert((composite.face_index[first], composite.face_index[second]));
                }
            }
        }
        for (index, retained_id) in &wall_source_pairs {
            let member = carriers.len() + source_members.iter().position(|(id, _)| id == retained_id).unwrap();
            let (a, b) = (composite.face_index[*index], composite.face_index[member]);
            admitted.insert((a.min(b), a.max(b)));
        }
        split_lap(0, &mut split_clock);
        match crate::imprint::build_carrier_imprints_within(
            &composite.solid,
            &|first, second| admitted.contains(&(first, second)),
            &ImprintOptions {
                tolerance,
                maximum_fit_points: 96,
                local_fit: false,
                fit_chunk_points: None,
                maximum_ssi_step: Some((scale * 0.0005).max(tolerance * 100.0)),
                span_budget: None,
                hermite_sections: false,
                tangent_curve_lane: false,
                exact_iso_pcurves: offset_exact_iso_pcurves_on(),
                march_starts: false,
                two_radius_junctions: false,
                tangent_curve_imprint: false,
            },
            !shielded_cusps.is_empty(),
        ) {
            Ok(record) => {
                split_lap(1, &mut split_clock);
                imprint = translate_composite_imprint(record, &composite, tolerance, scale)?;
                next_piece_id = imprint.pieces.iter().map(|piece| piece.id).max().unwrap_or(0) + 1;
                next_vertex_id = imprint.vertices.iter().map(|vertex| vertex.id).max().unwrap_or(0) + 1;
                one_call_done = true;
                ONE_CALL_RUNS.with(|count| count.set(count.get() + 1));
                split_lap(2, &mut split_clock);
                os_debug!(
                    "one-call imprint: {} members, {} admitted pairs, {} pieces",
                    members.len(),
                    admitted.len(),
                    imprint.pieces.len(),
                );
            }
            Err(error) => {
                os_debug!("one-call imprint failed, per-pair path: {error}");
            }
        }
    }
    for first in 0..carriers.len() {
        for second in first + 1..carriers.len() {
            if one_call_done {
                break;
            }
            if !pair_admitted(first, second) {
                continue;
            }
            split_lap(0, &mut split_clock);
            let pair = match crate::imprint::build_carrier_imprints(
                &carriers[first].solid,
                &carriers[second].solid,
                &ImprintOptions {
                    tolerance,
                    maximum_fit_points: 96,
                    // The GLOBAL interpolation lane. `local_fit: true` until
                    // 2026-09-13, which cost three orders on every smooth
                    // section: measured on the apex cone's base circle, the
                    // local lane needs 3,070 control points to reach 2.0e-5
                    // where the global one reaches 1.4e-8 with 384. What the
                    // local lane is for — a rim through a corner, where its
                    // monotonicity limiter holds a kink the global cubic rings
                    // across, 4.9x measured — is now the fit's own fallback:
                    // `fit_polyline` re-runs the other lane whenever the asked
                    // one cannot meet the tolerance and returns the better
                    // MEASURED curve, so this flag chooses which ladder is
                    // paid for first, never which curve is delivered.
                    local_fit: false,
                    fit_chunk_points: None,
                    // Offset carriers can meet in short, tightly curved
                    // branches. The general SSI default (diagonal / 15) is
                    // too coarse here and lets independently fitted branches
                    // miss a shared junction when thickness crosses one of
                    // those branches.
                    maximum_ssi_step: Some((scale * 0.0005).max(tolerance * 100.0)),
                    span_budget: None,
                    hermite_sections: false,
                    tangent_curve_lane: false,
                    exact_iso_pcurves: offset_exact_iso_pcurves_on(),
                    march_starts: false,
                    two_radius_junctions: false,
                    tangent_curve_imprint: false,
                },
                !shielded_cusps.is_empty(),
            ) {
                Ok(pair) => pair,
                // A failed pair imprint (e.g. the marching SSI exhausting its
                // step budget on a long closed sphere/plane intersection) is
                // not fatal to the shell: the affected rim simply stays
                // unsplit and the rim-weld passes below get to close it. The
                // final watertightness gate still refuses anything the welds
                // cannot reach, so tolerating the miss only enlarges the
                // honest success set.
                Err(error) => {
                    os_debug!("pair ({first},{second}) imprint failed: {error}");
                    continue;
                }
            };
            split_lap(1, &mut split_clock);
            let pieces_before = imprint.pieces.len();
            merge_pair_imprint(
                &mut imprint,
                pair,
                first as u8,
                second as u8,
                &mut next_piece_id,
                &mut next_vertex_id,
                tolerance,
                scale,
            );
            split_lap(2, &mut split_clock);
            os_debug!(
                "pair ({first},{second}) src=({},{}) pieces+={}",
                carriers[first].source_face_id,
                carriers[second].source_face_id,
                imprint.pieces.len() - pieces_before,
            );
        }
    }
    if body == Body::Shell && (distance < 0.0 || !inward_extended_walls.is_empty()) {
        // An extended opening-wall carrier (outward, or inward at a reflex
        // rim) no longer carries the source outline. OUTWARD it has the GROWN
        // outline. Its inner boundary — where the wall ring stops at the
        // source solid — is the source outline, and the retained SOURCE faces
        // are not carriers, so no carrier×carrier pair ever imprints it.
        // Un-cut, the wall stays one whole-plane fragment covering the
        // opening void, and the source faces' rim edges dangle one-use.
        // Imprint each wall carrier against the retained source faces that
        // share an edge with its opening: the wall then fragments into the
        // ring (kept) and the opening hole (dropped by the void guard).
        for index in 0..carriers.len() {
            if one_call_done {
                break;
            }
            if !matches!(carriers[index].kind, OffsetFaceRole::Wall) {
                continue;
            }
            if distance > 0.0 && !inward_extended_walls.contains(&index) {
                continue;
            }
            let opening = source_by_id[&carriers[index].source_face_id];
            for retained_id in &retained {
                let retained_face = source_by_id[retained_id];
                if !source_faces_adjacent(opening, retained_face) {
                    continue;
                }
                split_lap(0, &mut split_clock);
                let neighbor = standalone_face(source, *retained_id)?;
                split_lap(3, &mut split_clock);
                let pair = match build_imprints(
                    &carriers[index].solid,
                    &neighbor,
                    &ImprintOptions {
                        tolerance,
                        maximum_fit_points: 96,
                        // The global lane, as above.
                        local_fit: false,
                        fit_chunk_points: None,
                        maximum_ssi_step: Some((scale * 0.0005).max(tolerance * 100.0)),
                        span_budget: None,
                        hermite_sections: false,
                        tangent_curve_lane: false,
                        exact_iso_pcurves: offset_exact_iso_pcurves_on(),
                        march_starts: false,
                        two_radius_junctions: false,
                        tangent_curve_imprint: false,
                    },
                ) {
                    Ok(pair) => pair,
                    Err(error) => {
                        os_debug!(
                            "wall×source pair ({index},{retained_id}) imprint failed: {error}"
                        );
                        continue;
                    }
                };
                split_lap(4, &mut split_clock);
                let pieces_before = imprint.pieces.len();
                merge_pair_imprint(
                    &mut imprint,
                    pair,
                    index as u8,
                    SOURCE_OPERAND,
                    &mut next_piece_id,
                    &mut next_vertex_id,
                    tolerance,
                    scale,
                );
                split_lap(5, &mut split_clock);
                os_debug!(
                    "wall×source pair ({index},{retained_id}) pieces+={}",
                    imprint.pieces.len() - pieces_before,
                );
            }
        }
    }
    // APEX-CAP ring splits: the cap × cone section lies ON the cone
    // carrier's promoted ring edge (an overlap, not a crossing), so
    // `build_imprints` mints no edge splits for the cone side. Without them
    // the cone keeps ONE closed ring edge while the cap carries the section
    // as arcs — mismatched segmentation, both sides one-use. Record the arc
    // junctions as splits on the cone's coincident edge.
    for (cone_index, cap_index) in &apex_cap_pairs {
        let cap_face_id = carriers[*cap_index].solid.shells[0].faces[0].id;
        let piece_ids = imprint
            .by_face
            .iter()
            .find(|entry| entry.operand == *cap_index as u8 && entry.face_id == cap_face_id)
            .map(|entry| entry.piece_ids.clone())
            .unwrap_or_default();
        let coincidence_band = 2e-5f64.max(scale * 1e-7);
        for piece_id in piece_ids {
            let Some((curve, t0, t1)) = imprint
                .pieces
                .iter()
                .find(|piece| piece.id == piece_id)
                .map(|piece| (piece.curve.clone(), piece.t0, piece.t1))
            else {
                continue;
            };
            for endpoint in [curve.evaluate(t0).or_refuse(KernelStage::Sew, "evaluate")?, curve.evaluate(t1).or_refuse(KernelStage::Sew, "evaluate")?] {
                for edge in &carriers[*cone_index].solid.edges {
                    if edge.degenerate {
                        continue;
                    }
                    let projection = crate::project_point_to_curve(&edge.curve, endpoint).or_refuse(KernelStage::Sew, "project_point_to_curve")?;
                    if projection.distance > coincidence_band
                        || projection.u < edge.t0 + 1e-9
                        || projection.u > edge.t1 - 1e-9
                    {
                        continue;
                    }
                    let parameter = projection.u;
                    if let Some(existing) = imprint.edge_splits.iter_mut().find(|split| {
                        split.operand == *cone_index as u8 && split.edge_id == edge.id
                    }) {
                        if !existing
                            .parameters
                            .iter()
                            .any(|value| (*value - parameter).abs() <= 1e-8)
                        {
                            existing.parameters.push(parameter);
                        }
                    } else {
                        imprint.edge_splits.push(EdgeSplitRecord {
                            operand: *cone_index as u8,
                            edge_id: edge.id,
                            parameters: vec![parameter],
                        });
                    }
                    os_debug!(
                        "apex-cap ring split: cone carrier {cone_index} edge {} at t={parameter:.6}",
                        edge.id,
                    );
                }
            }
        }
    }
    if debug_enabled() {
        for piece in &imprint.pieces {
            let start = piece.curve.evaluate(piece.t0);
            let mid = piece.curve.evaluate((piece.t0 + piece.t1) * 0.5);
            let end = piece.curve.evaluate(piece.t1);
            if let (Ok(start), Ok(mid), Ok(end)) = (start, mid, end) {
                os_debug!(
                    "piece[{}] ({:.4},{:.4},{:.4})..({:.4},{:.4},{:.4})..({:.4},{:.4},{:.4}) on {:?}",
                    piece.id,
                    start.x,
                    start.y,
                    start.z,
                    mid.x,
                    mid.y,
                    mid.z,
                    end.x,
                    end.y,
                    end.z,
                    piece.support_faces,
                );
                for pcurve in &piece.pcurves {
                    let uv = |t: f64| pcurve.pcurve.evaluate(t).map(|p| (p.x, p.y)).unwrap_or((f64::NAN, f64::NAN));
                    let [a, b] = pcurve.pcurve.domain().unwrap_or([0.0, 1.0]);
                    os_debug!(
                        "  pcurve operand={} face={} uv {:?}..{:?}..{:?}",
                        pcurve.operand, pcurve.face_id, uv(a), uv(0.5 * (a + b)), uv(b),
                    );
                }
            }
        }
    }
    split_lap(6, &mut split_clock);
    synchronize_smooth_edge_splits(&mut imprint, &smooth.edge_pairs);
    split_lap(7, &mut split_clock);
    // This decides whether an imprint is genuinely the carrier's existing
    // trim, not whether independently fitted curves can later be sewn. The
    // looser assembly tolerance can erase a short separating branch while it
    // is still departing a boundary.
    let boundary_coincidence_tolerance = 2e-5f64.max(scale * 1e-7);
    let piece_geometry = imprint
        .pieces
        .iter()
        .map(|piece| {
            (
                piece.id,
                FragmentEdgeGeometry {
                    curve: piece.curve.clone(),
                    t0: piece.t0,
                    t1: piece.t1,
                },
            )
        })
        .collect::<HashMap<_, _>>();
    for (index, carrier) in carriers.iter().enumerate() {
        if !matches!(carrier.kind, OffsetFaceRole::Offset) {
            continue;
        }
        let boundaries = carrier
            .solid
            .edges
            .iter()
            .map(|edge| FragmentEdgeGeometry {
                curve: edge.curve.clone(),
                t0: edge.t0,
                t1: edge.t1,
            })
            .collect::<Vec<_>>();
        if let Some(by_face) = imprint
            .by_face
            .iter_mut()
            .find(|entry| entry.operand == index as u8)
        {
            let before = by_face.piece_ids.clone();
            by_face.piece_ids.retain(|piece_id| {
                let piece = &piece_geometry[piece_id];
                !boundaries.iter().any(|boundary| {
                    fragment_edge_lies_on(piece, boundary, boundary_coincidence_tolerance)
                        .unwrap_or(false)
                })
            });
            // Separate retained planes can imprint overlapping portions of the
            // same line on this carrier. Arrange that cut once: duplicate
            // segments otherwise erase the region between a blend section
            // and the line. Preserve each piece on its other carrier.
            let candidates = by_face.piece_ids.clone();
            by_face.piece_ids.retain(|id| {
                let edge = &piece_geometry[id];
                if edge.curve.degree != 1 || edge.curve.control_points.len() != 2 {
                    return true;
                }
                !candidates.iter().any(|other_id| {
                    if id == other_id { return false; }
                    let other = &piece_geometry[other_id];
                    if other.curve.degree != 1 || other.curve.control_points.len() != 2 {
                        return false;
                    }
                    fragment_edge_lies_on(edge, other, tolerance).unwrap_or(false)
                        && (other_id < id
                            || !fragment_edge_lies_on(other, edge, tolerance).unwrap_or(false))
                })
            });
            if debug_enabled() && before.len() != by_face.piece_ids.len() {
                os_debug!(
                    "carrier[{index}] dropped boundary-coincident pieces: {:?} -> {:?}",
                    before,
                    by_face.piece_ids,
                );
            }
        }
    }

    if debug_enabled() {
        for entry in &imprint.by_face {
            os_debug!(
                "by_face operand={} face={} pieces={:?}",
                entry.operand,
                entry.face_id,
                entry.piece_ids,
            );
        }
    }
    split_lap(8, &mut split_clock);
    if split_profile {
        profile_line!(
            "offset_shell.imprint_split skips={:.3} pair_build={:.3} pair_merge={:.3} standalone={:.3} wall_build={:.3} wall_merge={:.3} ring={:.3} smooth={:.3} filter={:.3}",
            split_ms[0], split_ms[1], split_ms[2], split_ms[3], split_ms[4], split_ms[5], split_ms[6], split_ms[7], split_ms[8],
        );
    }
    profile::lap(&mut clock, profile::IMPRINT);
    let mut split_carriers = Vec::new();
    let mut fragments_by_carrier = Vec::new();
    for (index, carrier) in carriers.iter().enumerate() {
        // Offset carriers keep their boundary un-split (their trims never
        // coincide with imprints — the boundary-coincidence filter drops such
        // pieces instead). An apex-cap ring breaks that assumption: the
        // frustum's ring is its TRIM boundary while the cap crosses it as TWO
        // imprint arcs — without splitting, the two sides segment differently
        // and both stay one-use. Split exactly the cap-affected carriers.
        let split = if matches!(carrier.kind, OffsetFaceRole::Offset)
            && !apex_cap_sources.contains(&index)
        {
            carrier.solid.clone()
        } else {
            apply_edge_splits(&carrier.solid, index as u8, &imprint)?
        };
        let fragment_clock = web_time::Instant::now();
        let fragments = fragment_solid(&split, index as u8, &imprint)?;
        os_debug!(
            "carrier[{index}] kind={:?} src={} fragments={} ({:.3} ms)",
            carrier.kind,
            carrier.source_face_id,
            fragments.len(),
            fragment_clock.elapsed().as_secs_f64() * 1e3,
        );
        if debug_enabled() {
            for (fragment_index, fragment) in fragments.iter().enumerate() {
                let mut boundary_count = 0usize;
                let mut imprint_pieces = Vec::new();
                let mut derived_count = 0usize;
                for coedge in fragment
                    .loops
                    .iter()
                    .flat_map(|loop_record| &loop_record.coedges)
                {
                    match &coedge.source {
                        FragmentEdgeSource::Boundary { .. }
                        | FragmentEdgeSource::SharedBoundary { .. } => boundary_count += 1,
                        FragmentEdgeSource::Imprint { piece_id } => imprint_pieces.push(*piece_id),
                        FragmentEdgeSource::Derived { .. } => derived_count += 1,
                    }
                }
                os_debug!(
                    "  frag[{index}.{fragment_index}] test=({:.3},{:.3},{:.3}) uv=({:.3},{:.3}) boundary={boundary_count} derived={derived_count} pieces={imprint_pieces:?}",
                    fragment.test_point.x, fragment.test_point.y, fragment.test_point.z,
                    fragment.test_uv.x, fragment.test_uv.y,
                );
            }
        }
        split_carriers.push(split);
        fragments_by_carrier.push(fragments);
    }
    profile::lap(&mut clock, profile::FRAGMENT);
    let mut assembly_sources = [(SOURCE_OPERAND, source)]
        .into_iter()
        .collect::<HashMap<_, _>>();
    for (index, carrier) in split_carriers.iter().enumerate() {
        assembly_sources.insert(index as u8, carrier);
    }

    let source_edge_by_id = source
        .edges
        .iter()
        .map(|edge| (edge.id, edge))
        .collect::<HashMap<_, _>>();
    let retained_faces_with_edges = retained
        .iter()
        .map(|face_id| {
            let face = source_by_id[face_id];
            let edges = face
                .loops
                .iter()
                .flat_map(|loop_record| &loop_record.coedges)
                .filter_map(|coedge| source_edge_by_id.get(&coedge.edge_id).copied())
                .collect::<Vec<_>>();
            (face, edges)
        })
        .collect::<Vec<_>>();
    let mut offset_skin_tolerance = offset_skin_band(scale, distance);
    // `BREP_OFFSET_THIN_SKIN_PAD`: the band's 2e-3 floor is a fixed length;
    // on a shell thinner than it (box-paper-thin, d = 1e-3) no point was ever
    // shadowed (`separation < d - 2e-3` cannot hold), so the opening's 1e-3
    // corner squares, holding no wall seed, read as the void and were left
    // out of the rim. A thin shell's band is held to a quarter of it.
    if thin_skin_pad_enabled() && distance > 0.0 {
        offset_skin_tolerance = offset_skin_tolerance.min(0.25 * distance.abs());
    }
    // OUTWARD shells: a wall fragment lying IN FRONT of another opening —
    // past that opening's plane on its void side, within the extent of its
    // (extended) wall carrier — is not material: an opening removes
    // everything in front of it. Such a fragment is the sheet of this wall
    // between the source outline and the grown outline on the far side of an
    // ADJACENT opening's shared edge (the x∈[−1,0] strip of an opened −Z
    // face's wall, in front of the opened −X face). It sits within |distance|
    // of a retained face's corner edge, so the void-cross-section guard reads
    // it as wall thickness; only the other opening's plane tells it apart.
    //
    // A RULED opening's wall is curved, so it has no one plane: its front is
    // read at the point's foot on the wall carrier, along the carrier's normal
    // turned to the opening face's outward side. On the torus wedge that is
    // what drops the start and end caps' strips beyond the extended chord,
    // which lie within |distance| of the torus and in front of the cone.
    let mut opening_planes: Vec<(usize, Vec3, Vec3)> = Vec::new();
    let mut opening_ruled: Vec<(usize, f64)> = Vec::new();
    if distance < 0.0 {
        for (index, carrier) in carriers.iter().enumerate() {
            if !matches!(carrier.kind, OffsetFaceRole::Wall) {
                continue;
            }
            let opening = source_by_id[&carrier.source_face_id];
            let [u0, u1] = opening.surface.domain_u().or_refuse(KernelStage::Sew, "domain_u")?;
            let [v0, v1] = opening.surface.domain_v().or_refuse(KernelStage::Sew, "domain_v")?;
            let (u, v) = ((u0 + u1) * 0.5, (v0 + v1) * 0.5);
            if ruled_walls.contains(&index) {
                let wall = &split_carriers[index].shells[0].faces[0];
                let probe = project_point_to_surface(&wall.surface, opening.surface.evaluate(u, v).or_refuse(KernelStage::Sew, "evaluate")?).or_refuse(KernelStage::Sew, "project_point_to_surface")?;
                let side = face_normal(wall, probe.u, probe.v)?.dot(face_normal(opening, u, v)?);
                opening_ruled.push((index, if side < 0.0 { -1.0 } else { 1.0 }));
                continue;
            }
            if !opening.surface.is_affine().or_refuse(KernelStage::Sew, "is_affine")? {
                continue;
            }
            opening_planes.push((
                index,
                opening.surface.evaluate(u, v).or_refuse(KernelStage::Sew, "evaluate")?,
                face_normal(opening, u, v)?,
            ));
        }
    }
    let in_front_of_other_opening = |wall_index: usize, point: Vec3| -> Result<bool, KernelRefusal> {
        let band = 2e-3f64.max(scale * 5e-5);
        for (index, plane_point, outward) in &opening_planes {
            if *index == wall_index || point.sub(*plane_point).dot(*outward) <= band {
                continue;
            }
            let face = &split_carriers[*index].shells[0].faces[0];
            let projection = project_point_to_surface(&face.surface, point).or_refuse(KernelStage::Sew, "project_point_to_surface")?;
            let uv = Vec2 {
                x: projection.u,
                y: projection.v,
            };
            if parameter_point_in_face(face, uv, 1e-8).or_refuse(KernelStage::Sew, "parameter_point_in_face")? != PolygonClass::Outside {
                return Ok(true);
            }
        }
        for (index, side) in &opening_ruled {
            if *index == wall_index {
                continue;
            }
            let face = &split_carriers[*index].shells[0].faces[0];
            let projection = project_point_to_surface(&face.surface, point).or_refuse(KernelStage::Sew, "project_point_to_surface")?;
            let outward = face_normal(face, projection.u, projection.v)?.scale(*side);
            if point.sub(projection.point).dot(outward) <= band {
                continue;
            }
            let uv = Vec2 {
                x: projection.u,
                y: projection.v,
            };
            if parameter_point_in_face(face, uv, 1e-8).or_refuse(KernelStage::Sew, "parameter_point_in_face")? != PolygonClass::Outside {
                return Ok(true);
            }
        }
        Ok(false)
    };

    let mut chosen_offsets = Vec::new();
    let mut chosen_walls = Vec::new();
    let mut newly_shadowed = HashSet::<u64>::default();
    for (index, carrier) in carriers.iter().enumerate() {
        let raw_seeds = source_seeds(source, carrier.source_face_id)?;
        let mut seeds = Vec::with_capacity(raw_seeds.len());
        for raw_seed in raw_seeds {
            let seed = if matches!(carrier.kind, OffsetFaceRole::Offset) {
                offset_seed(
                    source_by_id[&carrier.source_face_id],
                    raw_seed,
                    &split_carriers[index].shells[0].faces[0].surface,
                    distance,
                    2e-3f64.max(scale * 5e-5),
                )?
                .unwrap_or(raw_seed)
            } else if inward_extended_walls.contains(&index) {
                offset_seed(
                    source_by_id[&carrier.source_face_id],
                    raw_seed,
                    &split_carriers[index].shells[0].faces[0].surface,
                    0.0,
                    2e-3f64.max(scale * 5e-5),
                )?
                .unwrap_or(raw_seed)
            } else {
                raw_seed
            };
            if seed.sub(raw_seed).length() > 1e-9 {
                os_debug!(
                    "  seed[{index}] moved from ({:.4},{:.4}) to the offset image ({:.4},{:.4})",
                    raw_seed.x,
                    raw_seed.y,
                    seed.x,
                    seed.y
                );
            }
            seeds.push(seed);
        }
        let seed = seeds[0];
        if matches!(carrier.kind, OffsetFaceRole::Offset) {
            // Fragments whose test point sits at the full offset distance
            // from every retained source face are the true offset-skin
            // regions. Fragments closer to some other face are shadowed
            // pockets/strips that another carrier owns.
            // A fragment whose offset lies ON another retained face's offset
            // (material exactly twice the shell distance thick: a slot, fin or
            // floor at t = 2d) reads as SHADOWED by that face. One hair
            // thinner, the same fragment reads shadowed on its own and is
            // dropped by the on-skin filter, or omits its carrier by the
            // skinless rule below, so the answer is the limit of that
            // construction: the census's L∞ forms are continuous across the
            // onset. Without this the fragment read On, was selected, and the
            // shell refused as "a cavity pinched to zero thickness". The
            // coincidence test is `coincident_offset_face`'s (the other face's
            // distance within `offset_skin_tolerance` of |d|, its foot on the
            // opposite side of the point from this face's own), taken only on
            // the THIN side of the onset.
            let mut readings = Vec::with_capacity(fragments_by_carrier[index].len());
            // The construction band allows a carrier's own fit residual. It
            // must not also preserve pockets buried by neighboring offsets:
            // that coverage is what justified admitting a finite cusp. Keep
            // the own-face reading unchanged, and measure other faces at the
            // fit tolerance for this construction.
            let cusp_neighbors = if shielded_cusps.is_empty() {
                Vec::new()
            } else {
                retained_faces_with_edges.iter()
                    .filter(|(face, _)| face.id != carrier.source_face_id)
                    .cloned().collect::<Vec<_>>()
            };
            for fragment in &fragments_by_carrier[index] {
                let reading = offset_skin_reading(
                    fragment.test_point,
                    &retained_faces_with_edges,
                    distance,
                    offset_skin_tolerance,
                );
                let reading = if matches!(&reading, SkinReading::On) && !cusp_neighbors.is_empty() {
                    offset_skin_reading(
                        fragment.test_point,
                        &cusp_neighbors,
                        distance,
                        4.0 * KernelTolerances::for_solid(source, 1e-7).intersection_fit,
                    )
                } else {
                    reading
                };
                let reading = match reading {
                    SkinReading::On => match coincident_offset_face(
                        fragment.test_point,
                        carrier.source_face_id,
                        &retained_faces_with_edges,
                        distance,
                        offset_skin_tolerance,
                    )? {
                        // ONE-SIDED and tight: only material no thicker than
                        // 2|d| (to 10× the model tolerance) takes the thin
                        // construction, which is exact there. A wall inside
                        // the skin band but genuinely thicker than 2|d| keeps
                        // its sliver cavity's named refusal below; taking the
                        // thin construction there would fill a real cavity.
                        Some((face, separation)) if separation <= distance.abs() + tolerance * 10.0 => {
                            // A coincidence the fragment's other interior
                            // points do not share is a tangent TOUCH: the
                            // cavity pinches to a line there, and dropping the
                            // fragment as a sheet would discard real skin.
                            if tangent_touch_on() {
                                if let Some((farther, far)) = tangent_touch(
                                    fragment,
                                    face,
                                    &retained_faces_with_edges,
                                    distance,
                                    offset_skin_tolerance,
                                )? {
                                    return Err(KernelRefusal::unsupported(KernelStage::Select, refusals::TANGENT_OFFSETS, format!(
                                        "offset_shell: the offsets of source faces {} and {face} touch tangentially near \
                                         ({:.4}, {:.4}, {:.4}) — the material between those faces is exactly twice the \
                                         shell distance thick along that touch (separation {separation:.6} against {:.6}) \
                                         and thicker beside it (the offset reads {far:.4} from face {face} at ({:.4}, {:.4}, {:.4})), so the cavity would \
                                         pinch to a line there; a cavity pinched to a line is not supported — use a \
                                         shell distance a little below {:.4}",
                                        carrier.source_face_id,
                                        fragment.test_point.x,
                                        fragment.test_point.y,
                                        fragment.test_point.z,
                                        distance.abs(),
                                        farther.x,
                                        farther.y,
                                        farther.z,
                                        distance.abs(),
                                    )));
                                }
                            }
                            SkinReading::Shadowed { face, separation }
                        }
                        _ => SkinReading::On,
                    },
                    other => other,
                };
                // One test point cannot speak for a skin that bounds a cavity
                // the solid demonstrably has: see `cavity_witness`.
                let reading = match reading {
                    SkinReading::Shadowed { .. } if cavity_bounds.contains(&carrier.source_face_id) => SkinReading::On,
                    other => other,
                };
                readings.push(reading);
            }
            for (fragment, reading) in fragments_by_carrier[index].iter().zip(&readings) {
                if let SkinReading::Unmeasured { face, reason } = reading {
                    return Err(unmeasured_skin_refusal(fragment.test_point, *face, reason));
                }
            }
            // EVERY fragment measured nearer than the distance to some other
            // retained face: this support's offset is SHADOWED over its whole
            // extended carrier — the side walls of a slot narrower than 2|d|
            // grown outward, a pocket floor within 2d of the face under it.
            // It owns no skin. It used to switch the filter off and keep its
            // seeded fragment anyway, which built a closed, validating shell
            // with an inverted box inside it (+90 on a width-2 slot grown by
            // 1.5, with the 90 of material at its centre missing). Dropping
            // only its fragments is not enough: its carrier has already cut
            // its neighbours, and the top skin over that slot keeps a hole
            // nothing fills. So the pass is re-run with the support omitted
            // before the imprint, like a collapsed one. An apex-cap sphere
            // shares its cone's source id and is not judged here.
            //
            // Only the FIRST pass omits. A test point speaks for its fragment
            // because every other carrier's imprint bounds it; omitting a
            // carrier removes cuts, merges fragments and moves test points,
            // but changes no distance, so a support that reads shadowed only
            // AFTER an omission is reading a test point that no longer stands
            // for its fragment. Measured on two bosses 1 apart grown outward
            // by 1.5: once the facing boss walls were omitted, the plate's
            // grown top — one uncut fragment, mostly skin — read shadowed at
            // its single test point.
            //
            // "Every fragment" counts a fragment the selection would drop by
            // CLASS as not skin, as the selection does: the extended carrier's
            // surplus past a slot's end reads at least the distance from every
            // face, lies inside the source, and on a slot 1 wide grown outward
            // left the slot floor unjudged until the side walls were gone.
            // ADRIFT: a fragment whose test point is farther from its OWN face
            // than the distance, in the mitered metric, is not that face's skin
            // — it is inside the cavity, past where its neighbours' offsets cut
            // it. A rib, a pin or a divider thinner than 2d shelled inward
            // keeps such fragments, and the assembly then fails on them.
            let adrift = fragments_by_carrier[index]
                .iter()
                .map(|fragment| {
                    point_adrift_of_its_own_offset(
                        fragment.test_point,
                        carrier.source_face_id,
                        &retained_faces_with_edges,
                        distance,
                        offset_skin_tolerance,
                    )
                })
                .collect::<Vec<_>>();
            let mut skinless = readings
                .iter()
                .any(|reading| matches!(reading, SkinReading::Shadowed { .. }))
                || adrift.iter().any(|flag| *flag);
            for (fragment, reading) in fragments_by_carrier[index].iter().zip(&readings) {
                if !skinless {
                    break;
                }
                if matches!(reading, SkinReading::On)
                    && !adrift[fragments_by_carrier[index]
                        .iter()
                        .position(|other| std::ptr::eq(other, fragment))
                        .unwrap_or(0)]
                {
                    let class = classify_point(fragment.test_point, source, tolerance * 10.0).or_refuse(KernelStage::Sew, "classify_point")?.class;
                    skinless = (distance > 0.0 && class == PointClass::Out)
                        || (distance < 0.0 && class == PointClass::In);
                }
            }
            if skinless && !apex_cap_sources.contains(&index) {
                if body == Body::Shell && distance > 0.0 && cascade_omission_enabled() {
                    let held = fragments_holding_true_skin(
                        source_by_id[&carrier.source_face_id],
                        source,
                        &fragments_by_carrier[index],
                        &retained_faces_with_edges,
                        distance,
                        offset_skin_tolerance,
                    )?;
                    if !held.is_empty() {
                        os_debug!("  select[{index}] src={}: {} fragment(s) hold true skin", carrier.source_face_id, held.len());
                        chosen_offsets.extend(held);
                        continue;
                    }
                }
                if !shadowed_supports.is_empty() && !(body == Body::Shell && distance > 0.0 && cascade_omission_enabled()) {
                    let mut omitted = shadowed_supports.iter().copied().collect::<Vec<_>>();
                    omitted.sort_unstable();
                    return Err(KernelRefusal::unsupported(KernelStage::Select, "offset_shell_shadow_cascade", format!(
                        "offset_shell: omitting the shadowed offsets of source faces {omitted:?} \
                         leaves the offset of source face {} shadowed at every one of its {} \
                         fragment test points, which no longer stand for their fragments once \
                         those carriers stop cutting it",
                        carrier.source_face_id,
                        readings.len()
                    )));
                }
                os_debug!(
                    "  select[{index}] src={} SHADOWED over all {} fragment(s): {:?}",
                    carrier.source_face_id,
                    readings.len(),
                    readings
                );
                newly_shadowed.insert(carrier.source_face_id);
                continue;
            }
            // Partly shadowed: keep a bounded sample of the shadowed test
            // points, so a refusal can say where two offsets cross.
            for (fragment, reading) in fragments_by_carrier[index].iter().zip(&readings) {
                if shadowed_points.len() >= MAXIMUM_CROSSING_READS {
                    break;
                }
                if matches!(reading, SkinReading::Shadowed { .. }) {
                    shadowed_points.push((fragment.test_point, carrier.source_face_id));
                }
            }
            let mut on_skin = readings
                .iter()
                .zip(&adrift)
                .map(|(reading, adrift)| matches!(reading, SkinReading::On) && !adrift)
                .collect::<Vec<_>>();
            if open_section_reach {
                if let Some(ends) = open_profile_end_neighbours(index, &carriers, &source_faces)? {
                    // A continued variable profile can approach its original
                    // end cap inside the distance band. Its sharp boundary is
                    // the section with the OFFSET end plane, not that old cap.
                    for (keep, fragment) in on_skin.iter_mut().zip(&fragments_by_carrier[index]) {
                        *keep = !outside_continued_neighbours(fragment.test_point, &ends, &carriers, tolerance)?;
                    }
                    os_debug!("  select[{index}] open profile kept {} central regions", on_skin.iter().filter(|keep| **keep).count());
                }
            }
            // A synthetic apex sphere has no source-face seed of its own to
            // fall back to. When its whole cap is hidden by retained offsets,
            // keep its cuts but contribute no skin. Otherwise the fallback
            // selects the uncut sphere as a stray patch inside the material.
            // The parent cone remains subject to the ordinary selection.
            let skin_filter_active = on_skin.iter().any(|flag| *flag)
                || apex_cap_pairs.iter().any(|(_, cap)| *cap == index);
            let opening_limits = if inward_extended_walls.is_empty() {
                Vec::new()
            } else {
                let openings = inward_extended_walls
                    .iter()
                    .map(|index| source_by_id[&carriers[*index].source_face_id])
                    .collect::<Vec<_>>();
                opening_trim_halfspaces(source_by_id[&carrier.source_face_id], &openings, tolerance)?
            };
            let mut classified = Vec::new();
            let mut viable = Vec::new();
            for (fragment_index, fragment) in fragments_by_carrier[index].iter().enumerate() {
                let class = classify_point(fragment.test_point, source, tolerance * 10.0).or_refuse(KernelStage::Sew, "classify_point")?.class;
                let excluded_by_class = (distance > 0.0 && class == PointClass::Out)
                    || (distance < 0.0 && class == PointClass::In);
                let contact = if excluded_by_class {
                    false
                } else {
                    distance > 0.0
                        && fragment_has_sustained_source_contact(
                            fragment,
                            &source_faces,
                            &assembly_sources,
                            &imprint,
                            2e-3f64.max(scale * 5e-5),
                        )?
                };
                os_debug!(
                    "  select[{index}.{fragment_index}] class={class:?} contact={contact} on_skin={} seed_in={:?} uv=({:.3},{:.3})",
                    on_skin[fragment_index],
                    parameter_point_in_face(&fragment_as_trim(fragment), seed, 1e-8),
                    fragment.test_uv.x, fragment.test_uv.y,
                );
                let past_opening = opening_limits.iter().any(|(origin, normal)| {
                    fragment.test_point.sub(*origin).dot(*normal) < -tolerance * 10.0
                });
                if excluded_by_class
                    || past_opening
                    || (skin_filter_active && !on_skin[fragment_index])
                {
                    continue;
                }
                classified.push(fragment.clone());
                if contact {
                    continue;
                }
                viable.push(fragment.clone());
            }
            // One skin per REGION of the source face (see `source_seeds`):
            // every region of a pinched face is offset like a face of its
            // own, and the selection below answers each region's seed. An
            // ordinary face has the one seed and the one answer it always had.
            let mut region_skins = Vec::new();
            let continued_neighbours = if distance > 0.0 {
                continued_sphere_neighbours(index, &carriers, &source_faces, scale)?
            } else { None };
            if let Some(neighbours) = &continued_neighbours {
                for fragment in &classified {
                    if outside_continued_neighbours(fragment.test_point, neighbours, &carriers, tolerance)? {
                        region_skins.push(fragment.clone());
                    }
                }
                os_debug!("  select[{index}] continued sphere kept {} regions", region_skins.len());
            }
            for &seed in seeds.iter().filter(|_| continued_neighbours.is_none()) {
                os_debug!("  select[{index}] seed=({:.4},{:.4})", seed.x, seed.y);
                let seeded = viable
                    .iter()
                    .filter(|fragment| {
                        parameter_point_in_face(&fragment_as_trim(fragment), seed, 1e-8)
                            .is_ok_and(|class| class != PolygonClass::Outside)
                    })
                    .cloned()
                    .collect::<Vec<_>>();
                let seeded_classified = classified
                    .iter()
                    .filter(|fragment| {
                        parameter_point_in_face(&fragment_as_trim(fragment), seed, 1e-8)
                            .is_ok_and(|class| class != PolygonClass::Outside)
                    })
                    .cloned()
                    .collect::<Vec<_>>();
                let selected = if viable.is_empty() {
                    if !skin_filter_active {
                        // NOTHING on this carrier reads on-skin: every classified
                        // fragment's test point is measurably CLOSER than the
                        // offset distance to some retained face — a buried strip
                        // another carrier owns. Forcing the largest one in anyway
                        // plants a wrong skin patch that collides with the
                        // completion phase's reconstruction (3-use edges).
                        // Contribute nothing: `complete_opening_boundary_cycles`
                        // rebuilds the true skin from the carrier surface, and the
                        // watertight gate still refuses if it cannot.
                        None
                    } else {
                        let mut by_area = classified
                            .iter()
                            .cloned()
                            .map(|fragment| {
                                let area = parameter_space_area(&fragment_as_trim(&fragment)).or_refuse(KernelStage::Sew, "parameter_space_area")?.abs();
                                Ok((fragment, area))
                            })
                            .collect::<Result<Vec<_>, KernelRefusal>>()?;
                        by_area.sort_by(|a, b| b.1.total_cmp(&a.1));
                        by_area.into_iter().next().map(|entry| entry.0)
                    }
                } else if !seeded_classified.is_empty() {
                    let mut candidates = seeded_classified;
                    candidates.sort_by(|a, b| {
                        a.test_uv
                            .sub(seed)
                            .length()
                            .total_cmp(&b.test_uv.sub(seed).length())
                    });
                    candidates.into_iter().next()
                } else if viable.len() > 2 {
                    let mut by_area = viable
                        .iter()
                        .cloned()
                        .map(|fragment| {
                            let area = parameter_space_area(&fragment_as_trim(&fragment)).or_refuse(KernelStage::Sew, "parameter_space_area")?.abs();
                            Ok((fragment, area))
                        })
                        .collect::<Result<Vec<_>, KernelRefusal>>()?;
                    by_area.sort_by(|a, b| b.1.total_cmp(&a.1));
                    by_area.into_iter().next().map(|entry| entry.0)
                } else {
                    let mut candidates = if seeded.is_empty() { viable.clone() } else { seeded };
                    candidates.sort_by(|a, b| {
                        a.test_uv
                            .sub(seed)
                            .length()
                            .total_cmp(&b.test_uv.sub(seed).length())
                    });
                    candidates.into_iter().next()
                };
                let mut drop_debris = false;
                if let Some(fragment) = &selected {
                    os_debug!(
                        "  select[{index}] src={} chosen test point departs its own face's offset \
                         distance by {:?}",
                        carrier.source_face_id,
                        own_offset_departure(
                            fragment.test_point,
                            carrier.source_face_id,
                            &retained_faces_with_edges,
                            distance
                        )
                    );
                    os_debug!(
                        "  select[{index}] src={} chosen test point departs in the MITER metric by {:?}",
                        carrier.source_face_id,
                        own_miter_departure(
                            fragment.test_point,
                            carrier.source_face_id,
                            &retained_faces_with_edges,
                            distance
                        )
                    );
                    let coincidence = coincident_offset_face(
                        fragment.test_point,
                        carrier.source_face_id,
                        &retained_faces_with_edges,
                        distance,
                        offset_skin_tolerance,
                    )?;
                    let coincidence = match coincidence {
                        Some((other, separation))
                            if coincide_debris_on()
                                && fragment_is_band_debris(fragment, offset_skin_tolerance)? =>
                        {
                            os_debug!(
                                "  select[{index}] src={} drops corner debris coincident with {other} \
                                 (separation {separation}) at {:?}",
                                carrier.source_face_id,
                                fragment.test_point
                            );
                            drop_debris = true;
                            None
                        }
                        reading => reading,
                    };
                    if let Some((other, separation)) = coincidence {
                        return Err(KernelRefusal::unsupported(KernelStage::Select, "offset_shell_coincident_offsets", format!(
                            "offset_shell: the offsets of source faces {} and {other} coincide near \
                             ({:.4}, {:.4}, {:.4}) — the material between those faces is twice the \
                             shell distance thick (separation {separation:.6} against {:.6}, band \
                             {offset_skin_tolerance:.1e}), so the cavity between them has no \
                             thickness there; a cavity pinched to zero thickness is not supported",
                            carrier.source_face_id,
                            fragment.test_point.x,
                            fragment.test_point.y,
                            fragment.test_point.z,
                            distance.abs(),
                        )));
                    }
                }
                let selected = if drop_debris { None } else { selected };
                if let Some(fragment) = selected {
                    // A pinched face's seeds each find their own region; a
                    // fallback that ignores the seed can answer the same
                    // fragment twice, and one skin must enter once.
                    if !region_skins
                        .iter()
                        .any(|other: &FaceFragmentRecord| {
                            other.test_point.sub(fragment.test_point).length() <= 1e-12
                        })
                    {
                        region_skins.push(fragment);
                    }
                }
            }
            // EVERY SKIN PIECE (`BREP_OFFSET_EVERY_SKIN_PIECE`, default on):
            // one source region's offset skin can come out in several pieces
            // when another offset cuts it (the drilled block's bore skin,
            // radius 4, runs into the wall skin x = 19 and parts it at
            // y = 10 ± √15/2) or the carrier's seam parts it (the bore skin
            // itself, at its seam). The seed answers one piece; every
            // classified fragment already reads on the skin, inside the source
            // and short of the openings, so all of them are the skin, as for
            // the extended walls below. Left out, the pieces the seed missed
            // left their outlines one-use (shell/hole-skin-merging-wall).
            // Only where the seed could not choose: an inward shell whose every
            // on-skin fragment touches the source (`viable` empty), so the
            // largest stood in for the region; an outward shell's fragments and
            // a seeded choice keep the seed's answer. And only a piece whose
            // test point lies the distance from its OWN trimmed face: "on the
            // skin" says no retained face is nearer, which a point inside the
            // offset body also reads (the cylinder's offset past a reflex
            // junction, inside the box's offset wall: 0.139 from the trimmed
            // cylinder against 0.1, reflex_junction_under_a_filleted_top).
            if every_skin_piece_enabled()
                && distance > 0.0
                && (viable.is_empty() || every_seeded_piece)
                && skin_filter_active
                && inward_extended_walls.is_empty()
                && classified.len() > region_skins.len()
            {
                for fragment in &classified {
                    if region_skins.iter().any(|other: &FaceFragmentRecord| other.test_point.sub(fragment.test_point).length() <= 1e-12) {
                        continue;
                    }
                    let departure = own_offset_departure(fragment.test_point, carrier.source_face_id, &retained_faces_with_edges, distance);
                    os_debug!("  select[{index}] every skin piece: test point ({:.4},{:.4},{:.4}) departs its own face's offset by {departure:?}", fragment.test_point.x, fragment.test_point.y, fragment.test_point.z);
                    if departure.is_some_and(|departure| departure.abs() <= offset_skin_tolerance) {
                        region_skins.push(fragment.clone());
                    }
                }
            }
            if !inward_extended_walls.is_empty() && skin_filter_active {
                // The extended opening cuts beyond the source footprint.
                // Other finite carriers can subdivide a connected offset
                // region there (the box/revolve report splits both box caps
                // at x=1). A single source seed keeps only one of those cells.
                // Keep every cell already proved inside the source and on
                // the offset skin, then merge their shared boundaries.
                region_skins = classified;
            }
            chosen_offsets.extend(region_skins);
        } else if body == Body::Whole {
            // A HELD face of the offset body: its carrier is the face at
            // distance zero, cut by the neighbouring offset skins; the body
            // keeps the region the face's own seed lies in, whole. There is
            // no source rim to stop at and no void in front of it.
            for (fragment_index, fragment) in fragments_by_carrier[index].iter().enumerate() {
                if seeds.iter().any(|seed| {
                    parameter_point_in_face(&fragment_as_trim(fragment), *seed, 1e-8)
                        .is_ok_and(|class| class != PolygonClass::Outside)
                }) {
                    os_debug!("  held[{index}.{fragment_index}] kept: contains the face's seed");
                    chosen_walls.push(fragment.clone());
                }
            }
        } else {
            // Keep arrangement order stable. Besides making the result
            // reproducible, this matches the reference kernel's Set insertion
            // order when the final manifold-wall guard admits fragments.
            let opening = source_by_id[&carrier.source_face_id];
            let wall_seeds = opening_wall_seeds(
                opening,
                &split_carriers[index].shells[0].faces[0],
                source,
                &opening_set,
                distance,
            )?;
            // The wall-seed exemptions below rest on "the coplanar-rim weld
            // cuts the void hole later" — which only ever happens on PLANAR
            // opening carriers. A curved carrier (e.g. the sphere face used
            // directly as the opening) keeps its un-split fragment forever:
            // exempting it would seal the opening with the source face itself
            // and fabricate a closed hollow solid. Curved carriers therefore
            // get the strict void guard.
            let carrier_planar =
                surface_is_planar(&split_carriers[index].shells[0].faces[0].surface, tolerance)
                    .unwrap_or(false);
            let extended_inward = inward_extended_walls.contains(&index);
            let outside_source = |point| -> Result<bool, KernelRefusal> {
                Ok(extended_inward
                    && classify_point(point, source, tolerance * 10.0)
                        .or_refuse(KernelStage::Select, "classify_point")?.class == PointClass::Out)
            };
            let wall_on_skin = |point| {
                if extended_inward {
                    let inside_source = classify_point(point, source, tolerance * 10.0)
                        .or_refuse(KernelStage::Select, "classify_point")?.class == PointClass::In;
                    point_on_inward_opening_skin(
                        point,
                        &retained_faces_with_edges,
                        &reflex_wall_neighbors[&index],
                        inside_source,
                        distance,
                        offset_skin_tolerance,
                    )
                } else {
                    point_on_offset_skin(
                        point,
                        &retained_faces_with_edges,
                        distance,
                        offset_skin_tolerance,
                    )
                }
            };
            let mut selected_indices = Vec::new();
            for (fragment_index, fragment) in fragments_by_carrier[index].iter().enumerate() {
                if outside_source(fragment.test_point)? {
                    continue;
                }
                if parameter_point_in_face(&fragment_as_trim(fragment), seed, 1e-8).or_refuse(KernelStage::Sew, "parameter_point_in_face")?
                    == PolygonClass::Outside
                {
                    if in_front_of_other_opening(index, fragment.test_point)? {
                        os_debug!(
                            "  wall[{index}.{fragment_index}] skipped: in front of another \
                             opening at ({:.3},{:.3},{:.3})",
                            fragment.test_point.x,
                            fragment.test_point.y,
                            fragment.test_point.z,
                        );
                        continue;
                    }
                    // Wall = the material cross-section exposed at the
                    // opening: every point of a genuine wall fragment lies
                    // WITHIN the offset distance of some retained face (it is
                    // the thickness between that face and its offset skin). A
                    // fragment at ≥ distance from EVERY retained face — and
                    // containing no expected wall seed (an un-split cap
                    // fragment can span both the void AND the wall annulus;
                    // the coplanar-rim weld cuts its hole later) — is the
                    // core's own cross-section: a HOLE in the opening (e.g.
                    // the corner void where a carve's offset circle passes
                    // just inside the opening's corner). Welding it shut
                    // would fabricate a wall over the void.
                    if wall_on_skin(fragment.test_point)? && !(carrier_planar
                        && !extended_inward
                        && wall_seeds.iter().any(|wall_seed| {
                            parameter_point_in_face(&fragment_as_trim(fragment), *wall_seed, 1e-8)
                                .is_ok_and(|class| class != PolygonClass::Outside)
                        }))
                    {
                        os_debug!(
                            "  wall[{index}.{fragment_index}] skipped: void cross-section at \
                             ({:.3},{:.3},{:.3})",
                            fragment.test_point.x,
                            fragment.test_point.y,
                            fragment.test_point.z,
                        );
                        continue;
                    }
                    // A SURPLUS fragment beyond a sharp convex corner: it
                    // touches the wall's own outline, no edge of it was cut by
                    // the SOURCE (the rim every genuine wall region borders),
                    // and it holds no wall seed. The void test above reads it
                    // as material, because it measures a planar neighbour in the
                    // MITER metric (its extended plane, 0.277 from this region
                    // on the free-form D turning 83.7° at d = 0.3). The
                    // opening wall's clearance cuts such a region off between
                    // the carriers' tails past the mitre, and kept, it
                    // assembled as "non-integral genus" (8 of the D band's 42
                    // cells, 2026-09-26).
                    if distance < 0.0
                        && wall_surplus_beyond_the_sections(fragment, index as u8, |piece_id| {
                            imprint.pieces.iter().find(|piece| piece.id == piece_id).is_some_and(|piece| {
                                piece.support_faces.iter().any(|key| key.operand == SOURCE_OPERAND)
                            })
                        })
                        && !wall_seeds.iter().any(|wall_seed| {
                            parameter_point_in_face(&fragment_as_trim(fragment), *wall_seed, 1e-8)
                                .is_ok_and(|class| class != PolygonClass::Outside)
                        })
                    {
                        os_debug!(
                            "  wall[{index}.{fragment_index}] skipped: surplus beyond the sections at \
                             ({:.3},{:.3},{:.3}), touching the wall outline and no source rim",
                            fragment.test_point.x,
                            fragment.test_point.y,
                            fragment.test_point.z,
                        );
                        continue;
                    }
                    selected_indices.push(fragment_index);
                }
            }
            for wall_seed in wall_seeds {
                let fragments = &fragments_by_carrier[index];
                let nearest_containing = fragments
                    .iter()
                    .enumerate()
                    .filter(|(_, fragment)| {
                        parameter_point_in_face(&fragment_as_trim(fragment), wall_seed, 1e-8)
                            .is_ok_and(|class| class != PolygonClass::Outside)
                    })
                    .min_by(|(_, first), (_, second)| {
                        first
                            .test_uv
                            .sub(wall_seed)
                            .length()
                            .total_cmp(&second.test_uv.sub(wall_seed).length())
                    })
                    .map(|(fragment_index, _)| fragment_index);
                // Arrangement fitting can leave the seed just outside every
                // fragment by a small amount. Match the reference kernel's
                // geometric fallback instead of silently omitting the wall
                // fragment and leaving the assembled shell open.
                let nearest = if nearest_containing.is_some() {
                    nearest_containing
                } else {
                    let seed_point = split_carriers[index].shells[0].faces[0]
                        .surface
                        .evaluate(wall_seed.x, wall_seed.y).or_refuse(KernelStage::Sew, "evaluate")?;
                    fragments
                        .iter()
                        .enumerate()
                        .min_by(|(_, first), (_, second)| {
                            first
                                .test_point
                                .sub(seed_point)
                                .length()
                                .total_cmp(&second.test_point.sub(seed_point).length())
                        })
                        .map(|(fragment_index, _)| fragment_index)
                };
                if let Some(fragment_index) = nearest {
                    let candidate = &fragments_by_carrier[index][fragment_index];
                    if outside_source(candidate.test_point)?
                        || in_front_of_other_opening(index, candidate.test_point)?
                    {
                        continue;
                    }
                    // Void-cross-section guard. On a PLANAR carrier it applies
                    // to the GEOMETRIC FALLBACK only: a fragment that
                    // genuinely contains this wall seed is wall content by
                    // construction (even when un-split it also spans void —
                    // the rim weld cuts the hole later), but a merely-nearest
                    // fragment must not resurrect a hole in the opening. On a
                    // CURVED carrier no later weld cuts the hole, so even a
                    // seed-containing fragment gets the strict guard.
                    // Extended inward walls are cut by the source as well:
                    // a seed must not resurrect the opposite side of a
                    // reflex rim merely because its unsigned distance is small.
                    if (nearest_containing.is_none() || !carrier_planar || extended_inward)
                        && wall_on_skin(candidate.test_point)?
                    {
                        os_debug!(
                            "  wall[{index}.{fragment_index}] seed-hit skipped: void \
                             cross-section (planar={carrier_planar}) at ({:.3},{:.3},{:.3})",
                            candidate.test_point.x,
                            candidate.test_point.y,
                            candidate.test_point.z,
                        );
                        continue;
                    }
                    let existing = chosen_offsets.iter().cloned().chain(
                        selected_indices
                            .iter()
                            .map(|selected| fragments_by_carrier[index][*selected].clone()),
                    );
                    if !selected_indices.contains(&fragment_index)
                        && !would_overuse_existing_boundary(
                            candidate,
                            existing,
                            &assembly_sources,
                            &imprint,
                            2e-3,
                        )?
                    {
                        selected_indices.push(fragment_index);
                    }
                }
            }
            for fragment_index in selected_indices.iter().copied() {
                let mut fragment = fragments_by_carrier[index][fragment_index].clone();
                // Past a reflex rim the wall closes the cavity on the other
                // side of the opening plane, so its normal reverses too.
                if extended_inward
                    && classify_point(fragment.test_point, source, tolerance * 10.0)
                        .or_refuse(KernelStage::Select, "classify_point")?.class == PointClass::In
                {
                    flip_fragment(&mut fragment)?;
                }
                chosen_walls.push(fragment);
            }
        }
    }
    if debug_enabled() {
        for fragment in &chosen_offsets {
            os_debug!(
                "chosen offset: operand={} src={} test=({:.3},{:.3},{:.3})",
                fragment.operand,
                carriers[fragment.operand as usize].source_face_id,
                fragment.test_point.x,
                fragment.test_point.y,
                fragment.test_point.z,
            );
            for (loop_index, loop_record) in fragment.loops.iter().enumerate() {
                for coedge in &loop_record.coedges {
                    let surface = &fragment.surface;
                    let [c0, c1] = coedge.pcurve.domain().unwrap_or([0.0, 0.0]);
                    let uv0 = coedge.pcurve.evaluate(c0);
                    let uv1 = coedge.pcurve.evaluate(c1);
                    if let (Ok(uv0), Ok(uv1)) = (uv0, uv1) {
                        let p0 = surface.evaluate(uv0.x, uv0.y);
                        let p1 = surface.evaluate(uv1.x, uv1.y);
                        if let (Ok(p0), Ok(p1)) = (p0, p1) {
                            let kind = match &coedge.source {
                                FragmentEdgeSource::Boundary { edge_id, .. } => {
                                    format!("bnd:{edge_id}")
                                }
                                FragmentEdgeSource::SharedBoundary { edge_id, .. } => {
                                    format!("sbnd:{edge_id}")
                                }
                                FragmentEdgeSource::Imprint { piece_id } => {
                                    format!("imp:{piece_id}")
                                }
                                FragmentEdgeSource::Derived { .. } => "derived".to_string(),
                            };
                            os_debug!(
                                "    loop{loop_index} {kind} ({:.4},{:.4},{:.4})..({:.4},{:.4},{:.4})",
                                p0.x, p0.y, p0.z, p1.x, p1.y, p1.z,
                            );
                        }
                    }
                }
            }
        }
        for fragment in &chosen_walls {
            os_debug!(
                "chosen wall: operand={} src={} test=({:.3},{:.3},{:.3})",
                fragment.operand,
                carriers[fragment.operand as usize].source_face_id,
                fragment.test_point.x,
                fragment.test_point.y,
                fragment.test_point.z,
            );
        }
    }
    if !newly_shadowed.is_empty() {
        let mut omitted = shadowed_supports.clone();
        omitted.extend(newly_shadowed.iter().copied());
        os_debug!(
            "re-running without shadowed offset supports {:?} (omitted before: {:?})",
            newly_shadowed,
            shadowed_supports
        );
        // What this pass sampled belongs to carriers the next pass may not
        // build.
        shadowed_points.clear();
        profile::lap(&mut clock, profile::SELECT);
        let cascaded = !shadowed_supports.is_empty();
        let rebuilt = offset_shell_impl(
            source,
            opening_face_ids,
            distance,
            body,
            extend_oblique_carriers,
            extension_fired,
            reflex_rebuilds,
            &omitted,
            cavity_bounds,
            every_seeded_piece,
            shadowed_points,
            omitted_supports,
            vacated_notes,
        );
        let rebuilt = rebuilt?;
        if cascaded {
            refuse_a_cascade_that_kept_the_cavity(source, opening_face_ids, distance, &rebuilt.solid)?;
        }
        return Ok(rebuilt);
    }
    if chosen_offsets.is_empty() && !shadowed_supports.is_empty() {
        if matches!(body, Body::Shell) {
            if let Some(witness) = cavity_witness(source, opening_face_ids, distance)? {
                if !witness.bounds.is_subset(cavity_bounds) {
                    let mut bounds = cavity_bounds.clone();
                    bounds.extend(witness.bounds.iter().copied());
                    os_debug!(
                        "every skin read shadowed, but ({:.4}, {:.4}, {:.4}) lies {:.6} from every retained face: re-running with faces {:?} kept",
                        witness.point.x, witness.point.y, witness.point.z, witness.clearance, bounds
                    );
                    shadowed_points.clear();
                    profile::lap(&mut clock, profile::SELECT);
                    return offset_shell_impl(
                        source,
                        opening_face_ids,
                        distance,
                        body,
                        extend_oblique_carriers,
                        extension_fired,
                        reflex_rebuilds,
                        &HashSet::default(),
                        &bounds,
                        every_seeded_piece,
                        shadowed_points,
                        omitted_supports,
                        vacated_notes,
                    );
                }
                return Err(KernelRefusal::unsupported(KernelStage::Select, "offset_shell_cavity_unbuilt", format!(
                    "offset_shell: a cavity remains at this distance (the point ({:.4}, {:.4}, {:.4}) is {:.4} from \
                     every retained face, past the shell distance {distance}), but every offset skin around it \
                     reads as covered by another face, so it could not be built; returning the solid unchanged \
                     would be wrong — use a smaller shell distance",
                    witness.point.x, witness.point.y, witness.point.z, witness.clearance
                )));
            }
            if let Some(unchanged) = empty_cavity_shell(source, opening_face_ids, distance)? {
                return Ok(unchanged);
            }
        }
        let mut omitted = shadowed_supports.iter().copied().collect::<Vec<_>>();
        omitted.sort_unstable();
        return Err(KernelRefusal::unsupported(KernelStage::Select, "offset_shell_all_shadowed", format!(
            "offset_shell: no retained face keeps any offset skin once the shadowed offsets of \
             source faces {omitted:?} are omitted — the solid is nowhere thicker than twice the \
             shell distance, so there is no cavity to shell"
        )));
    }
    if chosen_offsets.is_empty() {
        return Err(KernelRefusal::internal(KernelStage::Select, "offset_shell_no_boundary", "offset_shell: carrier intersections produced no offset boundary"));
    }
    // Opening-wall fragments can overlap at dense carrier junctions. The
    // seed-time guard above handles geometric containment; this final pass
    // mirrors the topology-level rule and counts exact fragment edge sources.
    let mut boundary_use_count = HashMap::<[i64; 9], usize>::default();
    for fragment in &chosen_offsets {
        for coedge in fragment
            .loops
            .iter()
            .flat_map(|loop_record| &loop_record.coedges)
        {
            let key = fragment_edge_segment_key(&coedge.source, &assembly_sources, &imprint)?;
            *boundary_use_count.entry(key).or_default() += 1;
        }
    }
    let mut manifold_walls = Vec::new();
    for fragment in chosen_walls {
        let keys = fragment
            .loops
            .iter()
            .flat_map(|loop_record| &loop_record.coedges)
            .map(|coedge| fragment_edge_segment_key(&coedge.source, &assembly_sources, &imprint))
            .collect::<Result<Vec<_>, _>>()?;
        if !keys
            .iter()
            .any(|key| boundary_use_count.get(key).copied().unwrap_or(0) >= 2)
        {
            for key in keys {
                *boundary_use_count.entry(key).or_default() += 1;
            }
            manifold_walls.push(fragment);
        }
    }
    os_debug!(
        "manifold walls kept: {} operands={:?}",
        manifold_walls.len(),
        manifold_walls
            .iter()
            .map(|fragment| fragment.operand)
            .collect::<Vec<_>>(),
    );
    let chosen_walls = manifold_walls;
    // A part of the selected cavity skin with no path to an opening wall is a
    // SEALED hollow; named only if the assembly then refuses.
    let sealed = if sealed_hollow_on() && body == Body::Shell {
        sealed_skins(&chosen_offsets, &chosen_walls, &assembly_sources, &imprint, |fragment| {
            carriers[fragment.operand as usize].source_face_id
        })?
    } else {
        Vec::new()
    };
    if !sealed.is_empty() {
        os_debug!(
            "sealed skins: {:?}",
            sealed.iter().map(|skin| (&skin.sources, skin.centre, skin.edge_middles.len())).collect::<Vec<_>>()
        );
    }

    profile::lap(&mut clock, profile::SELECT);
    let stage_clock = web_time::Instant::now();
    // The offset BODY has no source face in it: it is the skins (and the held
    // faces) alone, oriented as the source is, so nothing is flipped either.
    let source_fragments = if body == Body::Whole {
        Vec::new()
    } else {
        fragment_solid(source, SOURCE_OPERAND, &empty_imprint())?
    };
    os_debug!("stage: source fragments {:.3}s", stage_clock.elapsed().as_secs_f64());
    let mut selected = source_fragments
        .into_iter()
        .filter(|fragment| retained.contains(&fragment.source_face_id))
        .collect::<Vec<_>>();
    let mut face_images = selected
        .iter()
        .map(|fragment| OffsetShellFaceImageRecord {
            role: OffsetFaceRole::Source,
            source_face_id: fragment.source_face_id,
        })
        .collect::<Vec<_>>();
    if distance < 0.0 {
        for fragment in &mut selected {
            flip_fragment(fragment)?;
        }
    }
    if distance > 0.0 && body == Body::Shell {
        for fragment in &mut chosen_offsets {
            flip_fragment(fragment)?;
        }
    }
    face_images.extend(
        chosen_offsets
            .iter()
            .map(|fragment| OffsetShellFaceImageRecord {
                role: OffsetFaceRole::Offset,
                source_face_id: carriers[fragment.operand as usize].source_face_id,
            }),
    );
    face_images.extend(
        chosen_walls
            .iter()
            .map(|fragment| OffsetShellFaceImageRecord {
                role: OffsetFaceRole::Wall,
                source_face_id: carriers[fragment.operand as usize].source_face_id,
            }),
    );
    selected.extend(chosen_offsets);
    selected.extend(chosen_walls);
    // `BREP_OFFSET_THIN_SKIN_PAD`: the same fixed 2e-3 floor as the skin band
    // welded the 1e-3 rim of box-paper-thin into points (every rim edge
    // degenerate, the rim eight one-face shells, the volume short by the
    // rim's 0.072); a thin inward shell's weld is held to a quarter of it.
    let mut assembly_weld = 2e-3f64.max(scale * 5e-5);
    if thin_skin_pad_enabled() && distance > 0.0 {
        assembly_weld = assembly_weld.min(0.25 * distance.abs());
    }
    let mut solid = profile::detail("assemble_open_fragments", || assemble_open_fragments(
        selected,
        &assembly_sources,
        &imprint,
        assembly_weld,
    ))?;
    os_debug!("stage: assembled open fragments {:.3}s", stage_clock.elapsed().as_secs_f64());
    profile::detail("orient_open_solid_faces", || orient_open_solid_faces(&mut solid))?;
    os_debug!("stage: oriented {:.3}s", stage_clock.elapsed().as_secs_f64());
    let connector_images = profile::detail("complete_sharp_offset_connectors", || complete_sharp_offset_connectors(
        &mut solid,
        source,
        &face_images,
        distance,
        2e-3f64.max(scale * 5e-5),
    ))?;
    os_debug!(
        "sharp connector completion added {} faces ({:.3}s)",
        connector_images.len(),
        stage_clock.elapsed().as_secs_f64()
    );
    if !connector_images.is_empty() {
        // Wall fragments were arranged before the sharp seam was closed, so
        // their prospective inner boundary was an open chain. Rebuild the
        // wall from the now-closed source/offset cycles instead of retaining
        // that pre-connector fragment.
        for shell in &mut solid.shells {
            shell.faces.retain(|face| {
                !matches!(
                    face_images
                        .get(face.id.saturating_sub(1) as usize)
                        .map(|image| image.role),
                    Some(OffsetFaceRole::Wall)
                )
            });
        }
        solid.shells.retain(|shell| !shell.faces.is_empty());
    }
    face_images.extend(connector_images);
    profile::detail("orient_open_solid_faces", || orient_open_solid_faces(&mut solid))?;
    // Weld curved-face offset rims that the wall fragmentation left orphaned
    // into their coplanar opening cap (cylinder / straight-hole shells). This
    // is the seam-rim consistency step the offset pipeline previously lacked.
    let welded_rims = profile::detail("weld_coplanar_orphan_rims", || weld_coplanar_orphan_rims(&mut solid, 2e-3f64.max(scale * 5e-5)))?;
    os_debug!("welded {welded_rims} coplanar orphan rims into opening caps");
    // A rim that is not a circle: a free-form or non-iso rim whose offset image
    // falls short of the opening. It runs after every earlier weld because it
    // only ever sees rims none of them closed.
    let rim_band = profile::detail("weld_free_form_rim_bands", || weld_free_form_rim_bands(
        &mut solid,
        &mut face_images,
        &carriers,
        source,
        2e-3f64.max(scale * 5e-5),
    ))?;
    let rim_bands = rim_band.welded;
    os_debug!("welded {rim_bands} free-form rim bands");
    // Weld leftover coaxial coplanar rim PAIRS (a through-hole's exposed wall
    // thickness at each opening) into new planar annulus faces.
    let pair_rims =
        profile::detail("weld_coplanar_rim_pair_annuli", || weld_coplanar_rim_pair_annuli(&mut solid, &mut face_images, 2e-3f64.max(scale * 5e-5)))?;
    os_debug!("welded {pair_rims} coplanar rim pairs into annulus walls");
    if debug_enabled() {
        for (shell_index, shell) in solid.shells.iter().enumerate() {
            for face in &shell.faces {
                os_debug!(
                    "POSTWELD shell {} face {} same_sense={} loops={}",
                    shell_index,
                    face.id,
                    face.same_sense,
                    face.loops.len()
                );
                for (loop_index, loop_record) in face.loops.iter().enumerate() {
                    let walk = loop_record
                        .coedges
                        .iter()
                        .map(|coedge| {
                            let spin = (|| -> Result<f64, KernelRefusal> {
                                let [d0, d1] = coedge.pcurve.domain().or_refuse(KernelStage::Sew, "domain")?;
                                let a = coedge.pcurve.evaluate(d0 + (d1 - d0) * 0.45).or_refuse(KernelStage::Sew, "evaluate")?;
                                let b = coedge.pcurve.evaluate(d0 + (d1 - d0) * 0.55).or_refuse(KernelStage::Sew, "evaluate")?;
                                let pa = face.surface.evaluate(a.x, a.y).or_refuse(KernelStage::Sew, "evaluate")?;
                                let pb = face.surface.evaluate(b.x, b.y).or_refuse(KernelStage::Sew, "evaluate")?;
                                Ok(pa.cross(pb).z)
                            })()
                            .unwrap_or(f64::NAN);
                            format!(
                                "{}{}(spin{:+.0})",
                                if coedge.forward { "+" } else { "-" },
                                coedge.edge_id,
                                spin.signum()
                            )
                        })
                        .collect::<Vec<_>>();
                    os_debug!("    loop {} {:?}", loop_index, walk);
                }
            }
        }
    }
    let completion_carriers = carriers.iter().collect::<Vec<_>>();
    let completion_tolerance = 2e-3f64.max(scale * 5e-5);
    // Unify geometric duplicate arc records BEFORE completion: a ring that is
    // already covered by both its faces (frustum + apex cap) must not read as
    // an open boundary, or completion papers over it with duplicate faces.
    let duplicate_welds = profile::detail("weld_duplicate_one_use_arcs", || weld_duplicate_one_use_arcs(&mut solid, completion_tolerance))?;
    os_debug!("welded {duplicate_welds} duplicate one-use boundary arcs");
    if debug_enabled() {
        let mut use_counts = HashMap::<u64, usize>::default();
        for coedge in solid
            .shells
            .iter()
            .flat_map(|shell| &shell.faces)
            .flat_map(|face| &face.loops)
            .flat_map(|loop_record| &loop_record.coedges)
        {
            *use_counts.entry(coedge.edge_id).or_default() += 1;
        }
        let open = use_counts.values().filter(|count| **count == 1).count();
        os_debug!(
            "assembled: faces={} open_edges={open}",
            solid
                .shells
                .iter()
                .map(|shell| shell.faces.len())
                .sum::<usize>()
        );
    }
    let completed_images = profile::detail("complete_opening_boundary_cycles", || complete_opening_boundary_cycles(
        &mut solid,
        &completion_carriers,
        &face_images,
        completion_tolerance,
    ))?;
    os_debug!("completion added {} faces", completed_images.len());
    face_images.extend(completed_images);
    let solid = profile::detail("merge_same_surface_faces_open", || merge_same_surface_faces_open(&solid, (1e-7f64).max(scale * 1e-9)))?;
    let assembly_tolerance = 2e-3f64.max(scale * 5e-5);
    // No faceted fallback here: an offset shell must be made of real analytic
    // surfaces. Failing loudly beats silently returning a tessellated BREP
    // with destroyed face provenance.
    // A shell the rim band lane recognized but could not close fails here as it
    // always did; the refusal now carries the lane's reason.
    let finalized = profile::detail("finalize_assembled_solid", || finalize_assembled_solid(solid, assembly_tolerance)).map_err(|error| {
        match &rim_band.decline {
            Some(reason) => error.with_message(|message| format!("{message}; offset_shell: {reason}")),
            None => error,
        }
    });
    let mut solid = match finalized {
        Ok(solid) => solid,
        Err(error) => {
            os_debug!("finalize_assembled_solid: {}", error.message);
            if !every_seeded_piece && body == Body::Shell && distance > 0.0 && every_seeded_skin_piece_enabled() {
                let retried = offset_shell_impl(
                    source,
                    opening_face_ids,
                    distance,
                    body,
                    extend_oblique_carriers,
                    extension_fired,
                    reflex_rebuilds,
                    shadowed_supports,
                    cavity_bounds,
                    true,
                    shadowed_points,
                    omitted_supports,
                    vacated_notes,
                );
                if retried.is_ok() {
                    return retried;
                }
            }
            if let Some((index, own, other, thickness)) = sealing_distance(
                &sealed,
                source,
                &retained_faces_with_edges,
                distance,
                offset_skin_tolerance,
            )? {
                let skin = &sealed[index];
                let bounding = skin.sources.iter().map(u64::to_string).collect::<Vec<_>>().join(", ");
                return Err(KernelRefusal::unsupported(KernelStage::Sew, refusals::SEALED_HOLLOW, format!(
                    "offset_shell: at this distance the cavity splits — a hollow bounded by the offsets of \
                     source faces {bounding} near ({:.4}, {:.4}, {:.4}) is sealed off from the opening, and a \
                     sealed hollow could not be assembled here. It is closed off where the offsets of source \
                     faces {own} and {other} cross, which they do from {:.4} up ({thickness:.4} of material \
                     between them); below {:.4} it opens into the rest of the cavity, so use a shell distance \
                     below that",
                    skin.centre.x,
                    skin.centre.y,
                    skin.centre.z,
                    0.5 * thickness,
                    0.5 * thickness,
                )));
            }
            return Err(error);
        }
    };
    profile::detail("reconstruct_analytic_shell", || crate::offset_retrim::reconstruct_analytic_shell(&mut solid))?;
    if rim_bands + pair_rims > 0 || !inward_extended_walls.is_empty() {
        // Rim welds and extended reflex walls join independently-oriented skins;
        // put the whole finalized manifold on one normal convention so the
        // shell reports its exact wall volume (coplanar-only shells keep
        // their assembled sense).
        profile::detail("coheres_face_normals", || coheres_face_normals(&mut solid))?;
    }
    let solid = solid;
    profile::lap(&mut clock, profile::ASSEMBLY);
    face_images = solid
        .shells
        .iter()
        .flat_map(|shell| &shell.faces)
        .map(|face| {
            face_images
                .get(face.id.saturating_sub(1) as usize)
                .cloned()
                .ok_or_else(|| "offset_shell: merged face lost provenance".to_string())
        })
        .collect::<Result<Vec<_>, _>>().or_refuse(KernelStage::Sew, "offset_shell_merged_face_lost_provenance")?;
    // A bore's end ring carries the bore wall's provenance whichever
    // construction built it (the opening's fragmentation or the rim-pair
    // weld), so its name does not depend on the path the rims took.
    let bore_rings = claim_bore_end_rings(&solid, &mut face_images)?;
    os_debug!("{bore_rings} bore end rings took the bore wall's provenance");
    // Honesty gate: a real rim left one-use is a hole in the shell that the
    // degenerate marking hides from `validate()` (degenerate edges are allowed
    // to be face-local). Refuse such a result instead of shipping a
    // non-watertight solid as valid. A rim is a CLOSED edge whose interior
    // sweeps measurably away from its seam vertex — a genuine circle, not a
    // legitimate pole point placeholder (which stays one-use by design).
    let mut rim_use_counts = HashMap::<u64, usize>::default();
    for coedge in solid
        .shells
        .iter()
        .flat_map(|shell| &shell.faces)
        .flat_map(|face| &face.loops)
        .flat_map(|loop_record| &loop_record.coedges)
    {
        *rim_use_counts.entry(coedge.edge_id).or_default() += 1;
    }
    if debug_enabled() {
        for (shell_index, shell) in solid.shells.iter().enumerate() {
            for face in &shell.faces {
                let role = face_images
                    .get(face.id.saturating_sub(1) as usize)
                    .map(|image| image.role);
                let c = face.surface.evaluate(0.5, 0.5).unwrap_or_default();
                os_debug!(
                    "DUMPFACE shell {} face {} affine={} role={:?} loops={} center=({:.3},{:.3},{:.3})",
                    shell_index, face.id, face.surface.is_affine().unwrap_or(false), role,
                    face.loops.len(), c.x, c.y, c.z
                );
                for (li, lp) in face.loops.iter().enumerate() {
                    let ids = lp.coedges.iter().map(|c| c.edge_id).collect::<Vec<_>>();
                    os_debug!("    loop {} edges {:?}", li, ids);
                }
            }
        }
        for edge in &solid.edges {
            let uses = rim_use_counts.get(&edge.id).copied().unwrap_or(0);
            let s = edge.curve.evaluate(edge.t0).or_refuse(KernelStage::Sew, "evaluate")?;
            let m = edge.curve.evaluate((edge.t0 + edge.t1) * 0.5).or_refuse(KernelStage::Sew, "evaluate")?;
            let q = edge.curve.evaluate(edge.t0 + (edge.t1 - edge.t0) * 0.25).or_refuse(KernelStage::Sew, "evaluate")?;
            os_debug!(
                "DUMPEDGE edge {} uses={} closed={} deg={} cpts={} cdeg={} knots={:?} start=({:.3},{:.3},{:.3}) q=({:.3},{:.3},{:.3}) mid=({:.3},{:.3},{:.3})",
                edge.id, uses, edge.start_vertex_id == edge.end_vertex_id, edge.degenerate,
                edge.curve.control_points.len(), edge.curve.degree, edge.curve.knots,
                s.x, s.y, s.z, q.x, q.y, q.z, m.x, m.y, m.z
            );
        }
    }
    for edge in &solid.edges {
        if rim_use_counts.get(&edge.id).copied().unwrap_or(0) != 1
            || edge.start_vertex_id != edge.end_vertex_id
        {
            continue;
        }
        let anchor = edge.curve.evaluate(edge.t0).or_refuse(KernelStage::Sew, "evaluate")?;
        let sweep_threshold = 1e-4f64.max(scale * 1e-6);
        let sweeps = [0.25, 0.5, 0.75].into_iter().any(|fraction| {
            edge.curve
                .evaluate(edge.t0 + (edge.t1 - edge.t0) * fraction)
                .map(|point| point.sub(anchor).length() > sweep_threshold)
                .unwrap_or(false)
        });
        if sweeps {
            if debug_enabled() {
                let rim_mid = edge.curve.evaluate((edge.t0 + edge.t1) * 0.5).or_refuse(KernelStage::Sew, "evaluate")?;
                os_debug!(
                    "PROBE orphan rim edge {} anchor=({:.4},{:.4},{:.4}) mid=({:.4},{:.4},{:.4})",
                    edge.id,
                    anchor.x,
                    anchor.y,
                    anchor.z,
                    rim_mid.x,
                    rim_mid.y,
                    rim_mid.z
                );
                for (shell_index, shell) in solid.shells.iter().enumerate() {
                    for face in &shell.faces {
                        let uses_rim = face
                            .loops
                            .iter()
                            .flat_map(|l| &l.coedges)
                            .any(|c| c.edge_id == edge.id);
                        let affine = face.surface.is_affine().unwrap_or(false);
                        let on = edge_on_surface(edge, &face.surface, 2e-3f64.max(scale * 5e-5))
                            .unwrap_or(false);
                        let proj = project_point_to_surface(&face.surface, rim_mid).or_refuse(KernelStage::Sew, "project_point_to_surface")?;
                        let contains = if on {
                            parameter_point_in_face(
                                face,
                                Vec2 {
                                    x: proj.u,
                                    y: proj.v,
                                },
                                2e-3f64.max(scale * 5e-5),
                            ).or_refuse(KernelStage::Sew, "parameter_point_in_face")? != PolygonClass::Outside
                        } else {
                            false
                        };
                        if on || uses_rim {
                            os_debug!(
                                "  face {} shell {} affine={} uses_rim={} on_surf={} proj_dist={:.5} contains={}",
                                face.id, shell_index, affine, uses_rim, on, proj.distance, contains
                            );
                        }
                    }
                }
            }
            return Err(refusals::unwelded_curved_rim(edge.id, anchor).into());
        }
    }
    // The shell's single exit. An offset skin that crosses another wall, or
    // folds through itself where the source turned tighter than the distance,
    // leaves a body that validates and measures — so it is asked here, and
    // repaired or refused by name rather than returned.
    let mut solid = solid;
    // Face-local rim construction and planar merging can reuse wire IDs.
    // Reserve every existing ID, then replace only duplicates so the completed
    // body can cross the arena/buffer ingestion boundary. These IDs have no
    // external references; face/edge identities and provenance stay intact.
    let mut loop_ids = HashSet::default();
    let mut coedge_ids = HashSet::default();
    for face in solid.shells.iter().flat_map(|shell| &shell.faces) {
        for wire in &face.loops {
            loop_ids.insert(wire.id);
            coedge_ids.extend(wire.coedges.iter().map(|coedge| coedge.id));
        }
    }
    let mut seen_loops = HashSet::default();
    let mut seen_coedges = HashSet::default();
    let mut next_loop = 1;
    let mut next_coedge = 1;
    for face in solid.shells.iter_mut().flat_map(|shell| &mut shell.faces) {
        for wire in &mut face.loops {
            if !seen_loops.insert(wire.id) {
                while !loop_ids.insert(next_loop) { next_loop += 1; }
                wire.id = next_loop;
            }
            for coedge in &mut wire.coedges {
                if !seen_coedges.insert(coedge.id) {
                    while !coedge_ids.insert(next_coedge) { next_coedge += 1; }
                    coedge.id = next_coedge;
                }
            }
        }
    }
    let solid = profile::detail("accept_offset_result", || super::acceptance::accept_offset_result(solid))?;
    profile::lap(&mut clock, profile::ACCEPTANCE);
    Ok(OffsetShellResultRecord { solid, face_images })
}

/// Run offset shell with stable counters and validation diagnostics suitable
/// for automated regressions and UI presentation.
pub fn offset_shell_with_diagnostics(
    source: &BrepSolid,
    opening_face_ids: &[u64],
    distance: f64,
    tolerances: Option<KernelTolerances>,
) -> Result<KernelOutcome<OffsetShellResultRecord>, KernelRefusal> {
    let policy = tolerances.unwrap_or_else(|| KernelTolerances::for_solid(source, 1e-7));
    policy.check().or_refuse(KernelStage::Sew, "check")?;
    let mut diagnostics = KernelDiagnostics::default();
    diagnostics.count_n(
        "collect.source_faces",
        source
            .shells
            .iter()
            .map(|shell| shell.faces.len() as u64)
            .sum(),
    );
    diagnostics.count_n("collect.opening_faces", opening_face_ids.len() as u64);
    diagnostics.measure_max("offset.distance", distance.abs());
    diagnostics.measure_max("tolerance.model", policy.model);
    // Tally every pcurve fit the shell makes (its own re-trims and any boolean
    // it runs) so `pcurve.unmet_floor` reports a trim that stopped at the
    // sample ceiling instead of passing it off as one that met the floor.
    let fit_scope = crate::PcurveFitScope::open();
    // Same for the polyline fits of its carrier×carrier sections, which is
    // where a marched rim's accuracy is decided (`geometry/fit.rs`).
    let polyline_scope = crate::PolylineFitScope::open();
    let result = offset_shell(source, opening_face_ids, distance)?;
    fit_scope.close().report_into(&mut diagnostics);
    polyline_scope.close().report_into(&mut diagnostics);
    // The phase split of the run that just finished (`profile.rs`), in the
    // boolean's `timing.<phase>_ms` shape so one reader serves both.
    let timings = profile::snapshot();
    diagnostics.count_n("shell.attempts", timings.attempts);
    for (phase, count) in &timings.carriers {
        diagnostics.count_n(format!("shell.{phase}"), *count);
    }
    for (phase, ms) in &timings.ms {
        diagnostics.measure_max(format!("timing.{phase}_ms"), *ms);
    }
    diagnostics.count_n(
        "sew.output_faces",
        result
            .solid
            .shells
            .iter()
            .map(|shell| shell.faces.len() as u64)
            .sum(),
    );
    diagnostics.count_n("sew.face_images", result.face_images.len() as u64);
    let validation = result.solid.validate_detailed(&policy);
    diagnostics.measure_max("validate.max_pcurve_error", validation.max_pcurve_error);
    diagnostics.count_n("validate.issues", validation.issues.len() as u64);
    diagnostics.count_n(
        "validate.wire_warnings",
        validation.wire_warnings.len() as u64,
    );
    for warning in validation.wire_warnings {
        diagnostics.event(
            DiagnosticSeverity::Warning,
            KernelStage::Validate,
            "validate.uv_wire",
            warning.message,
        );
    }
    if !validation.issues.is_empty() {
        for issue in &validation.issues {
            diagnostics.event(
                DiagnosticSeverity::Error,
                KernelStage::Validate,
                "validate.brep",
                issue.message.clone(),
            );
        }
        return Err(KernelRefusal::internal(KernelStage::Validate, "offset_shell_diagnostic_result", format!(
            "offset_shell: invalid diagnostic result: {:?}",
            validation.issues
        )));
    }
    Ok(KernelOutcome {
        value: result,
        diagnostics,
    })
}

/// Whether a wall fragment lies wholly in the wall's surplus: it has an edge on
/// the wall carrier's own outline (`Boundary` of operand `wall_operand`) and no
/// edge imprinted by the SOURCE solid, whose rim every genuine opening-wall
/// region borders. `cut_by_source` answers whether an imprint piece has the
/// source among its two support faces.
pub(super) fn wall_surplus_beyond_the_sections(
    fragment: &FaceFragmentRecord,
    wall_operand: u8,
    cut_by_source: impl Fn(u64) -> bool,
) -> bool {
    let mut on_outline = false;
    for coedge in fragment.loops.iter().flat_map(|record| &record.coedges) {
        match &coedge.source {
            FragmentEdgeSource::Boundary { operand, .. } if *operand == wall_operand => on_outline = true,
            FragmentEdgeSource::Boundary { operand, .. } | FragmentEdgeSource::SharedBoundary { operand, .. }
                if *operand == SOURCE_OPERAND =>
            {
                return false;
            }
            FragmentEdgeSource::Imprint { piece_id } => {
                if cut_by_source(*piece_id) {
                    return false;
                }
            }
            _ => {}
        }
    }
    on_outline
}


/// `BREP_OFFSET_SHELL_CASCADE_OMISSION` (default on; `=0` restores): an inward
/// shell judges a carrier that reads shadowed at every fragment's test point
/// by its true skin instead ([`fragments_holding_true_skin`]). One test point
/// cannot speak for a fragment that is partly skin: a reported cylinder boss model's
/// root fillets bound the cavity's bulge toward the boss mouth at every
/// distance past the boss limit (checked with
/// `scripts/offset_cavity_volume.py`), and were dropped as shadowed. A carrier
/// holding no skin at all may then be omitted even after an earlier omission.
fn cascade_omission_enabled() -> bool {
    switch!("BREP_OFFSET_SHELL_CASCADE_OMISSION", true)
}

/// A cascade of omissions must leave the deepest point of the true cavity
/// hollow: a carrier omitted on a test point that did not stand for its
/// fragment would show as material there.
fn refuse_a_cascade_that_kept_the_cavity(
    source: &BrepSolid,
    opening_face_ids: &[u64],
    distance: f64,
    result: &BrepSolid,
) -> Result<(), KernelRefusal> {
    let Some(witness) = cavity_witness(source, opening_face_ids, distance)? else {
        return Ok(());
    };
    let scale = crate::solid_scale(source).max(1.0);
    let class = classify_point(witness.point, result, 1e-8 * scale).or_refuse(KernelStage::Select, "classify_point")?.class;
    if class == PointClass::In {
        return Err(KernelRefusal::unsupported(KernelStage::Select, "offset_shell_shadow_cascade", format!(
            "offset_shell: omitting offsets that read shadowed only after earlier omissions leaves material at \
             ({:.4}, {:.4}, {:.4}), which is {:.4} from every retained face, past the shell distance {distance}; \
             the cavity there could not be built — use a smaller shell distance",
            witness.point.x, witness.point.y, witness.point.z, witness.clearance
        )));
    }
    Ok(())
}

/// `BREP_OFFSET_SHELL_CAVITY_WITNESS` (default on; `=0` restores): [`cavity_witness`].
fn cavity_witness_enabled() -> bool {
    switch!("BREP_OFFSET_SHELL_CAVITY_WITNESS", true)
}

/// A point of the solid farther than the shell distance from every retained
/// face, and the faces nearest it, whose skins bound that cavity.
#[derive(Debug)]
pub(super) struct CavityWitness {
    pub(super) point: Vec3,
    pub(super) clearance: f64,
    pub(super) bounds: HashSet<u64>,
}

/// Looks for the cavity the shadow reading says is not there. A fragment's
/// single test point can read shadowed while most of the fragment is real skin
/// (a reported cylinder boss model at 4.8: the r 0.2 bore of a cylinder r 5, one uncut
/// fragment whose test point lies 4.72 from a boss wall), and a cavity that
/// thin falls between the 12³ cells of [`empty_cavity_shell`], which then
/// returned the source as a success.
///
/// Distance to the retained faces is 1-Lipschitz, so any cavity's maximum is
/// climbed to from the samples that read highest: the best cell centres start
/// a pattern search that stays inside the solid. It is a search, not a proof;
/// what it cannot miss is a cavity whose ridge those starts climb.
pub(super) fn cavity_witness(
    source: &BrepSolid,
    opening_face_ids: &[u64],
    distance: f64,
) -> Result<Option<CavityWitness>, KernelRefusal> {
    if !cavity_witness_enabled() || distance <= 0.0 || opening_face_ids.is_empty() {
        return Ok(None);
    }
    let faces = source.shells.iter().flat_map(|shell| &shell.faces).collect::<Vec<_>>();
    let retained = faces.iter().filter(|face| !opening_face_ids.contains(&face.id)).copied().collect::<Vec<_>>();
    if retained.is_empty() {
        return Ok(None);
    }
    // Each face's own edges: a foot off the trim falls back on the edges it is
    // given, and the whole solid's would read a wall as near as any rim.
    let edge_by_id = source.edges.iter().map(|edge| (edge.id, edge)).collect::<HashMap<_, _>>();
    let retained = retained
        .into_iter()
        .map(|face| {
            let edges = face
                .loops
                .iter()
                .flat_map(|loop_record| &loop_record.coedges)
                .filter_map(|coedge| edge_by_id.get(&coedge.edge_id).copied())
                .collect::<Vec<_>>();
            (face, edges)
        })
        .collect::<Vec<_>>();
    let Some((low, high)) = control_hull(source, &faces) else { return Ok(None) };
    let scale = crate::solid_scale(source).max(1.0);
    let classify_tolerance = 1e-8 * scale;
    let margin = 1e-6 * scale;
    let readings = |point: Vec3| -> Result<Vec<(u64, f64)>, KernelRefusal> {
        retained.iter().map(|(face, edges)| Ok((face.id, distance_to_trimmed_face(face, edges, point)?))).collect()
    };
    let clearance = |values: &[(u64, f64)]| values.iter().map(|(_, value)| *value).fold(f64::INFINITY, f64::min);
    let inside = |point: Vec3| -> Result<bool, KernelRefusal> {
        Ok(classify_point(point, source, classify_tolerance).or_refuse(KernelStage::Select, "classify_point")?.class == PointClass::In)
    };
    const CELLS: usize = 12;
    const STARTS: usize = 24;
    const MOVES: usize = 160;
    let span = high.sub(low);
    let mut starts = Vec::new();
    for i in 0..CELLS {
        for j in 0..CELLS {
            for k in 0..CELLS {
                let at = |index: usize| (index as f64 + 0.5) / CELLS as f64;
                let point = Vec3::new(low.x + span.x * at(i), low.y + span.y * at(j), low.z + span.z * at(k));
                if inside(point)? {
                    starts.push((clearance(&readings(point)?), point));
                }
            }
        }
    }
    starts.sort_by(|a, b| b.0.total_cmp(&a.0));
    starts.truncate(STARTS);
    const DIRECTIONS: [(f64, f64, f64); 14] = [
        (1.0, 0.0, 0.0), (-1.0, 0.0, 0.0), (0.0, 1.0, 0.0), (0.0, -1.0, 0.0), (0.0, 0.0, 1.0), (0.0, 0.0, -1.0),
        (1.0, 1.0, 1.0), (-1.0, -1.0, -1.0), (1.0, 1.0, -1.0), (-1.0, -1.0, 1.0),
        (1.0, -1.0, 1.0), (-1.0, 1.0, -1.0), (-1.0, 1.0, 1.0), (1.0, -1.0, -1.0),
    ];
    let first_step = span.x.max(span.y).max(span.z) / CELLS as f64;
    let mut footless = None;
    for (start_clearance, start) in starts {
        let (mut point, mut best, mut step) = (start, start_clearance, first_step);
        for _ in 0..MOVES {
            if step < 1e-6 * scale {
                break;
            }
            let mut moved = false;
            for (x, y, z) in DIRECTIONS {
                let length = (x * x + y * y + z * z).sqrt();
                let next = Vec3::new(point.x + step * x / length, point.y + step * y / length, point.z + step * z / length);
                let value = clearance(&readings(next)?);
                if value > best && inside(next)? {
                    (point, best, moved) = (next, value, true);
                    break;
                }
            }
            if !moved {
                step *= 0.5;
            }
        }
        if best > distance + margin {
            // The climb ends on the ridge, where every face around it ties.
            // Nudged off the ridge, still inside the cavity, each point has one
            // nearest face, which counts if its foot lands inside its trim (a
            // tie through a face's rim is that edge, not its offset). The faces
            // that wrap the cavity collect the votes; a fillet seen only
            // through the mouth of a boss does not.
            let nudge = 0.5 * (best - distance);
            let mut votes = HashMap::<u64, usize>::default();
            for (x, y, z) in DIRECTIONS {
                let length = (x * x + y * y + z * z).sqrt();
                let near = Vec3::new(point.x + nudge * x / length, point.y + nudge * y / length, point.z + nudge * z / length);
                let values = readings(near)?;
                let Some((id, _)) = values.iter().min_by(|a, b| a.1.total_cmp(&b.1)).copied() else { continue };
                let Some((face, _)) = retained.iter().find(|(face, _)| face.id == id) else { continue };
                if foot_inside_trim(face, near)? {
                    *votes.entry(id).or_default() += 1;
                }
            }
            let most = votes.values().copied().max().unwrap_or(0);
            let bounds = votes.iter().filter(|(_, count)| **count * 2 >= most).map(|(id, _)| *id).collect::<HashSet<_>>();
            let witness = CavityWitness { point, clearance: best, bounds };
            if !witness.bounds.is_empty() {
                return Ok(Some(witness));
            }
            footless = Some(witness);
        }
    }
    Ok(footless)
}

/// The box of the solid's vertices and face control points; `None` when a
/// control point cannot be read.
fn control_hull(source: &BrepSolid, faces: &[&FaceRecord]) -> Option<(Vec3, Vec3)> {
    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 vertex in &source.vertices {
        low = Vec3::new(low.x.min(vertex.point.x), low.y.min(vertex.point.y), low.z.min(vertex.point.z));
        high = Vec3::new(high.x.max(vertex.point.x), high.y.max(vertex.point.y), high.z.max(vertex.point.z));
    }
    for control in faces.iter().flat_map(|face| face.surface.control_points.iter().flatten()) {
        let point = control.point().ok()?;
        low = Vec3::new(low.x.min(point.x), low.y.min(point.y), low.z.min(point.z));
        high = Vec3::new(high.x.max(point.x), high.y.max(point.y), high.z.max(point.z));
    }
    Some((low, high))
}

/// `BREP_OFFSET_SHELL_EMPTY_CAVITY` (default on; `=0` restores): [`empty_cavity_shell`].
fn empty_cavity_enabled() -> bool {
    switch!("BREP_OFFSET_SHELL_EMPTY_CAVITY", true)
}


/// An INWARD shell whose every retained face's offset skin is shadowed: the
/// cavity, the points of the solid farther than `distance` from every retained
/// face, is empty, so the shell is the solid itself, every face its own
/// `Source` image (brep-bench shell/box-walls-meeting: the box 10 x 8 x 6
/// shelled by 4 open at its top, whose y walls' offsets meet in the plane
/// y = 4, was refused "no retained face keeps any offset skin ... there is no
/// cavity to shell"; the case reads the box, 480 and 376).
///
/// The shadow reading is checked before the solid is returned: on a 12³ grid
/// of cell centres over the solid's control hull, no point the solid
/// classifies Inside may lie farther than `distance` (plus 1e-9 of the
/// solid's size) from every retained face. A point that does means the
/// cavity is not empty, and the refusal stands.
fn empty_cavity_shell(
    source: &BrepSolid,
    opening_face_ids: &[u64],
    distance: f64,
) -> Result<Option<OffsetShellResultRecord>, KernelRefusal> {
    if !empty_cavity_enabled() || distance <= 0.0 || opening_face_ids.is_empty() {
        return Ok(None);
    }
    let faces = source.shells.iter().flat_map(|shell| &shell.faces).collect::<Vec<_>>();
    let retained = faces.iter().filter(|face| !opening_face_ids.contains(&face.id)).copied().collect::<Vec<_>>();
    if retained.is_empty() {
        return Ok(None);
    }
    let edges = source.edges.iter().collect::<Vec<_>>();
    let Some((low, high)) = control_hull(source, &faces) else { return Ok(None) };
    let scale = crate::solid_scale(source).max(1.0);
    let tolerance = 1e-9 * scale;
    const CELLS: usize = 12;
    let span = high.sub(low);
    let mut inside = 0usize;
    for i in 0..CELLS {
        for j in 0..CELLS {
            for k in 0..CELLS {
                let at = |index: usize| (index as f64 + 0.5) / CELLS as f64;
                let point = Vec3::new(low.x + span.x * at(i), low.y + span.y * at(j), low.z + span.z * at(k));
                let class = classify_point(point, source, tolerance * 10.0).or_refuse(KernelStage::Select, "classify_point")?.class;
                if class != PointClass::In {
                    continue;
                }
                inside += 1;
                let mut nearest = f64::INFINITY;
                for face in &retained {
                    nearest = nearest.min(distance_to_trimmed_face(face, &edges, point)?);
                    if nearest <= distance + tolerance {
                        break;
                    }
                }
                if nearest > distance + tolerance {
                    os_debug!(
                        "empty cavity declined: ({:.4}, {:.4}, {:.4}) lies {nearest:.6} from every retained face",
                        point.x, point.y, point.z
                    );
                    return Ok(None);
                }
            }
        }
    }
    if inside == 0 {
        return Ok(None);
    }
    os_debug!("empty cavity: every retained skin is shadowed and {inside} sampled interior points lie within the distance");
    let face_images = faces
        .iter()
        .map(|face| OffsetShellFaceImageRecord { role: OffsetFaceRole::Source, source_face_id: face.id })
        .collect();
    Ok(Some(OffsetShellResultRecord { solid: source.clone(), face_images }))
}

/// `BREP_OFFSET_THIN_SKIN_PAD` (default on; `=0` restores): [`thin_skin_pad`].
fn thin_skin_pad_enabled() -> bool {
    switch!("BREP_OFFSET_THIN_SKIN_PAD", true)
}


/// The pad of an INWARD planar carrier whose shell is thinner than the
/// fragmenter's graze band: `None` otherwise.
///
/// Cause (the 10 × 8 × 6 box shelled 0.001 open at its top, brep-bench
/// shell/box-paper-thin, read 0.1653 against 0.2959 with an open mesh): each
/// carrier is its source face moved 0.001, and its neighbours' skins cut it
/// 0.001 inside its rim, with both ends 0.001 from the carrier's corners. The
/// imprint's pcurves are right (v = 0.001), but `fragment_face_arranged` lets a
/// CUT endpoint join a boundary corner image within `arrangement_tolerance ·
/// 100` (1.2e-3 on a 10 × 8 chart) with no 3D check, so both ends moved onto
/// the corners, the cut lay on the rim and was dropped as its duplicate
/// ("drop cut chain 6 as duplicate of boundary chain 0"). Padded past its rims,
/// the carrier's corners are far from every section's ends; the surplus
/// strips beyond the sections are outside the source or shadowed by the
/// neighbours and are dropped by the ordinary selection.
///
/// Thin: `distance` under 2e-3 of the solid's size (the band is 1e-4 of a
/// chart's diagonal, a chart at most the solid's size). The pad is 4e-3 of
/// the size plus the distance.
///
/// The same switch holds the selection's skin band and the assembly's weld to
/// a quarter of an inward distance: both had a fixed 2e-3 floor, twice this
/// shell's thickness (see `offset_skin_tolerance` and `assembly_weld`).
fn thin_skin_pad(source: &BrepSolid, face: &FaceRecord, distance: f64) -> Result<Option<f64>, KernelRefusal> {
    if !thin_skin_pad_enabled() || distance <= 0.0 || !face.surface.is_affine().or_refuse(KernelStage::Refine, "is_affine")? {
        return Ok(None);
    }
    let size = crate::solid_scale(source).max(1e-9);
    if distance >= 2e-3 * size {
        return Ok(None);
    }
    Ok(Some(4e-3 * size + distance))
}