01. Two-Dimensional Nesting & Trencadís

This chapter covers the repository's two-dimensional geometry-packing subsystem: the 2D Packing ribbon tab (13 components) and the Trencadis ribbon tab (5 components). The shared engine is exact No-Fit-Polygon Bottom-Left-Fill on a Clipper2 integer-snapped back end. Three nesters are in scope: the shipped IrregularSheetFillV506 (FreeNest), its exact-NFP sibling IrregularSheetFillNfpBlf (FreeNestX), and the hole-aware ContactNfpHoleNester / CNH (HoleNest). The Trencadís family adds a centroidal-Voronoi cell partition plus optimal one-to-one shard assignment.

The mathematics here is real plane geometry: Minkowski sums, the no-fit and inner-fit polygons, and a distance-based penetration certificate. Where the repository evolved the published math we show the derivation, not only the final formula. Every originality claim is anchored to a file:line, an [Algorithm] attribute, or a measured benchmark.

The packing-quality metric used throughout is stock utilisation

\[ \mathrm{util\_stock}=\frac{\sum_k \mathrm{area}(P_k)}{\mathrm{area}(S)-\sum_l \mathrm{area}(H_l)} \]

placed part material area over net sheet area (sheet minus its holes). It is computed by boolean area, not bounding boxes, and it is reported, not gated in the components; the gate is the zero-overlap validity certificate of section 1.6. (Definition: examples/10_pack2d/README.md.)

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1. The No-Fit-Polygon nesting core

1.1 The no-fit polygon (NFP)

Let two polygons \(A\) and \(B\) be given with \(A\) fixed and \(B\) free to translate by \(t\in\mathbb{R}^2\). Write \(B(t)=\{b+t : b\in B\}\). The no-fit polygon is the set of translations for which the interiors of \(A\) and \(B\) overlap:

\[ \mathrm{NFP}(A,B)=\{\,t\in\mathbb{R}^2 : \operatorname{int}A\cap\operatorname{int}B(t)\neq\varnothing\,\}. \]

The standard result (Bennell and Oliveira 2009) is that this set is exactly a Minkowski sum of \(A\) with the point-reflection of \(B\):

\[ \mathrm{NFP}(A,B)=A\oplus(-B),\qquad A\oplus C=\{a+c : a\in A,\ c\in C\},\quad -B=\{-b:b\in B\}. \]

Derivation. The interiors meet iff there exist \(a\in A\), \(b\in B\) with \(a=b+t\), i.e. \(t=a-b=a+(-b)\). The set of all such \(t\) is exactly \(\{a+(-b)\}=A\oplus(-B)\). Placing \(B\) at any \(t\) on the boundary of this set makes \(A\) and \(B\) touch without interior overlap; any \(t\) outside the set is collision-free. This is the geometric primitive every component in this chapter uses for collision. In code the reflection is Reflect and the sum is Clipper2Adapter.MinkowskiSum (ContactNfpHoleNester.cs:1484, :962; the engine contract is documented at ContactNfpHoleNester.cs:28-31).

1.2 The inner-fit polygon (IFP)

The container constraint is the dual. The inner-fit polygon is the set of translations that keep \(B\) inside a container \(C\):

\[ \mathrm{IFP}(B,C)=\{\,t : B(t)\subseteq C\,\}. \]

For a convex container this is an erosion (Minkowski difference) \(C\ominus B\). The repository realises it as an exact intersection over the vertices of the convex hull of \(B\):

\[ \mathrm{IFP}(B,C)=\bigcap_{v\in\operatorname{hull}(B)}\bigl(C-v\bigr),\qquad C-v=\{c-v:c\in C\}. \]

Derivation. \(B(t)\subseteq C\) holds iff every vertex \(b_i+t\in C\). Because \(C\) is intersected against the convex hull of \(B\), containing the hull vertices is sufficient to contain all of \(B\) (a convex combination of points in a convex set stays in the set). Each constraint \(b_i+t\in C\) rearranges to \(t\in C-b_i\), and the feasible \(t\) is the intersection over all hull vertices. This is exact for a convex \(C\) and conservative (never admits an out-of-bounds placement) for a concave \(C\) (ContactNfpHoleNester.cs:1332-1352; hull by Andrew's monotone chain, :1355-1368). The hull pass is the cost saver: it replaces a full-resolution erosion with at most \(|\operatorname{hull}(B)|\) Clipper intersections.

1.3 The feasible region and Bottom-Left-Fill

Combining the two, the set of legal placements of part \(B\) at a fixed rotation, given a sheet \(S\) with holes \(\{H_l\}\) and already-placed parts \(\{A_k\}\), is

\[ \mathrm{feasible}(B)=\mathrm{IFP}(B,S)\ \setminus\ \Bigl(\bigcup_k \mathrm{NFP}(A_k,B)\Bigr)\ \setminus\ \Bigl(\bigcup_l \mathrm{NFP}(H_l,B)\Bigr). \]

This is stated verbatim in the engine header (IrregularSheetFillNfpBlf.cs:18-27) and in the CNH header (ContactNfpHoleNester.cs:30-31). Non-overlap is therefore a hard constraint of construction, not a post-hoc trim.

The Bottom-Left-Fill placement rule (Burke et al. 2006) selects, among all feasible translations, the one that is lowest, then leftmost:

\[ t^\star=\operatorname*{arg\,min}_{t\in\mathrm{feasible}(B)}\ (t_y,\ t_x)\quad\text{(lexicographic)}. \]

The optimum of a linear lexicographic objective over a polygonal region lies at a vertex of that region, so the search reduces to scanning the feasible-region vertices in \((y,x)\) order and taking the first that survives verification (ContactNfpHoleNester.cs:980-1002, OrderedVertices :1625-1634; IrregularSheetFillNfpBlf.cs:314-315).

Originality. The NFP-BLF nesting family is clean-room. The published mathematics is cited in source by [Algorithm] attributes: Bottom-Left-Fill to Burke, Hellier, Kendall, Whitwell (2006), Operations Research 54(3):587-601, DOI 10.1287/opre.1060.0293; NFP/IFP via Minkowski sum to Bennell and Oliveira (2009), JORS 60(S1):S93-S105, DOI 10.1057/jors.2008.169; the boolean back end is the vendored Clipper2 (Johnson, BSL-1.0) with no copyleft (evidence: IrregularSheetFillNfpBlfComponent.cs:24-32, HoleNestComponent.cs:25-33). No upstream nesting source sits in the tree. Tier B (faithful implementation of published work), with the V506 to NFP-BLF delta and the CNH increments below carrying the research weight.

1.4 V506 to NFP-BLF: the measured delta

The originally shipped nester is IrregularSheetFillV506 (FreeNest, GUID D5E7A2B1-8C34-4F1E-A096-3B7F5D2E8A4C). It is an NFP-assisted bottom-left placement that, by design, permits a bounded overlap controlled by a Trim Tolerance and then boolean-trims the contact. With the default Trim Tolerance of 0.1 it produces apparent overlaps; this is the documented behaviour, not a fault (KB-6/KB-7; examples/10_pack2d/README.md).

IrregularSheetFillNfpBlf (FreeNestX, GUID 2d351646-2cb0-402a-bbd8-3950b5bb1fbc) re-derives the placement so that the feasible region is the complete IFP-minus-NFP set, making zero overlap a hard constraint. The measured improvement is recorded in source: a mean wasted area cut of 53.9% against V506 at zero overlap, validated against a Python reference (IrregularSheetFillNfpBlf.cs:21-22). FreeNestX is the only zero-overlap packer in the study to cross the 80% utilisation bar with holes: 82.0% oversub, 84.7% on the L-plus-hole fixture, 89.6% on a hard 3-hole fixture (examples/10_pack2d/README.md). The four legacy V1/V2/V3/V506 wrappers are marked [Obsolete] and Exposure=hidden per the 2D-V-solver phase-out (Pack2DIrregularSheetV506Component.cs:56-57).

Originality. FreeNestX is an evolved-fork of V506: same NFP/BLF lineage, but a re-derived complete feasible region replaces the overlap-then-trim contract, with the 53.9% waste-cut measured delta (IrregularSheetFillNfpBlf.cs:18-27, commit lineage at outputs/2026-06-03/pack2d_nfp_evolution).

1.5 The Unified dispatcher

IrregularSheetFillComponent (FreeNestU, GUID AB12C00B-1A2B-4C3D-9E4F-5A6B7C8D9E0B) is a strategy selector that dispatches to the V1/V2/V3/V506 variants behind one canvas box. It adds no new algorithm; every internal step resolves to an in-repo nesting variant.

Originality. facade-over-primitives (variant dispatcher), with the [Algorithm] note explicitly calling it a "Frahan-original strategy selector" over Burke 2007 and Bennell and Oliveira 2008 (IrregularSheetFillComponent.cs:32-33, :42).

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1.6 The hole-aware nester (CNH / ContactNfpHoleNester)

The ContactNfpHoleNester (HoleNest component, GUID D5F10019-8A3C-4D17-B5E2-6C90F2A47D31) evolves FreeNestX with two capabilities it lacks, both validated in outputs/2026-06-12/hole_packer_evolution (ContactNfpHoleNester.cs:10-33).

1.6.1 Contact-adaptive rotations

A fixed rotation list cannot seat a part flush against a wall, a neighbour, or a hole edge. CNH augments the uniform base set \(\{0,\tfrac{\pi}{2},\pi,\tfrac{3\pi}{2}\}\) with edge-alignment angles. For a host edge with direction angle \(\alpha_h\) and a part edge with direction angle \(\alpha_p\), the rotation that makes the part edge parallel to the host edge is

\[ \theta=\alpha_h-\alpha_p \pmod{2\pi},\qquad\text{and its flip } \theta+\pi, \]

evaluated over the longest edges of the part against the longest edges of the sheet and the most recently placed neighbour (ContactNfpHoleNester.cs:1304-1330, RotationSet :1281-1302). The total angle budget is bounded by \(\max(4,\ \mathrm{base}+4+2\,\mathrm{contact})\), and symmetric rotations are collapsed by a translation-invariant shape signature (RotSignature, :1917-1929) so each distinct rotated shape evaluates its NFP once.

1.6.2 Part-in-part-hole nesting (holes-first)

A small part can be nested inside the hole of a larger placed part. The hole acts as a container, so the legal translations of filler \(B\) into host hole \(G\) are exactly the inner-fit region \(\mathrm{IFP}(B,G)\) of section 1.2. The schedule is holes-first: parts that have holes are placed as hosts (area descending), the smallest remaining parts are nested into the open host holes, then the remaining outers are placed by NFP-BLF (ContactNfpHoleNester.cs:528-592). This is the GPR-defect-as-sheet-hole differentiator for stone: pack saw-cuttable parts only in intact rock, nesting offcuts into the voids of larger remnants (outputs/2026-06-12/hole_packer_evolution/HOLE_PACKER_MATH_AND_BENCHMARK.md, section 4).

1.6.3 The penetration-depth certificate (original derivation)

The NFP and IFP are built on simplified inputs: a Ramer-Douglas-Peucker pass with tolerance \(\tau=2\times10^{-3}\,\mathrm{diag}(p)\) carves the no-fit region slightly inward (NfpSimplifyTol, :1619-1623), and the hull-based IFP is anti-conservative on concave sheets. Area-relative gates alone admit needle overlaps: a tiny intersection area (order \(10^{-5}\) of part area) but a penetration depth up to roughly 0.2 caller units. The repository's answer is a compound gate on the true, unsimplified geometry; the NFP/IFP are treated as pruning devices only (ContactNfpHoleNester.cs:35-72).

For loops \(A\) and \(B\), penetration depth is the maximum distance any probe of one loop that lies strictly inside the other travels to that other loop's boundary, augmented by an edge-crossing term:

\[ d(A,B)=\max\!\Bigl( \max_{\substack{p\in\Pi(A)\\ p\in \operatorname{int}B}} \operatorname{dist}(p,\partial B),\ \max_{\substack{q\in\Pi(B)\\ q\in \operatorname{int}A}} \operatorname{dist}(q,\partial A),\ d_{\times}(A,B) \Bigr), \]

where the probe set \(\Pi(\cdot)\) is the loop's vertices, edge midpoints, and area centroid (the centroid closes the full-containment blind spot), and \(d_{\times}\) is the maximum proper edge-crossing perpendicular depth (PenetrationDepth :1777-1798, DepthProbes :1822-1839, MaxProperCross :1879-1908). Containment uses the analogous OutsideDepth.

A candidate is accepted only if its depth stays under a scale-relative tolerance instantiating the project epsilon budget at the verification level:

\[ \varepsilon_{\text{depth}}=\max\bigl(\varepsilon_{\text{floor}},\ 10^{-6}\,L\bigr),\quad L=\sqrt{\mathrm{area}(B)},\quad \varepsilon_{\text{floor}}=2\,\mathrm{SnapGrid}, \]

with \(\mathrm{SnapGrid}=0.01\) scaled units \(=10^{-5}\) caller units at the internal scale \(1000\) (DepthTolFor :1646-1647, constants :1640-1643). Engine math runs in scaled space because Clipper snaps to a fixed decimal precision; small-unit (metre) geometry would otherwise build sliver-noisy NFPs (Pack conditioning :121-154).

The micro-retreat. Rather than discard a contact-tight candidate that only violates through NFP simplification, the gate nudges it once along the measured penetration vector \(\mathbf{p}=(p_x,p_y)\) (length \(=d\)) by \(d+\mathrm{SnapGrid}\) and re-verifies:

\[ t'=t+\frac{d+\mathrm{RetreatSlack}}{\lVert\mathbf{p}\rVert}\,\mathbf{p},\qquad \mathrm{RetreatSlack}=\mathrm{SnapGrid}. \]

If the nudged placement passes, it is accepted; otherwise the candidate is rejected (TryVerifiedCandidate :1728-1748). This keeps the layout from starving while never accepting a deeper-than-floor penetration. The same compound gate runs path-independently in Validate (:1373-1449), so Valid==true certifies the layout on every engine path, including the fast path below.

1.6.4 The rect shelf fast-path

When every loop in the instance is an axis-aligned rectangle, the NFP and IFP degenerate to rectangles and the whole solve reduces to interval arithmetic with no Clipper calls. Rotations collapse to \(\{0,\tfrac{\pi}{2}\}\) (squares to \(\{0\}\)) using the exact integer map \((x,y)\mapsto(-y,x)\) so there is no trigonometric round-off (TryRectFastPath :635-738, detection :742-767, exact transform :883-895). A completeness fallback guards it: if the sparser shelf candidate set strands any part (about 1 in 4000 in fuzzing), the fast-path result is discarded and the general engine runs, so speed never trades away a placement (Pack :173-189).

1.6.5 Multi-start keep-best

The general engine optionally runs \(K\) deterministic part orders (area, max-dimension, width, height; all descending) and keeps the densest valid layout, breaking ties by placed count, then density, then smallest used bounding-box diagonal (BuildOrder :424-438, keep-best :243-292). The ordering keys form a total permutation, so the winner is reproducible. Determinism is enforced at the cache level: each pass allocates its own NFP cache keyed on (obstacle index, rotation signature), and the index is into the per-pass placement order, so a cache shared across passes would return the wrong NFP. The header documents this as the critical correctness point (ContactNfpHoleNester.cs:199-210). \(K=1\) reproduces the original single pass byte-for-byte; FRAHAN_MULTISTART=0 is the kill switch.

1.6.6 Deviation-compensated spacing (original derivation)

The solver collides on sampled proxies of smooth curves, but the output is the exact original curve. A chord cuts inside the true curve by the sampling sagitta, so two proxies that merely touch can let the true full-resolution curves cross by up to that deviation on each side. The Grasshopper wrapper measures the worst proxy deviation (max distance from a chord midpoint back to the true curve) and inflates the engine spacing to compensate (HoleNestComponent.cs:758-790, :532-542):

\[ \mathrm{spacing}_{\text{engine}}=\mathrm{spacing}_{\text{user}}+2\,\delta_{\text{part}}+\delta_{\text{sheet}}. \]

Derivation of the asymmetric coefficients. A part-to-part clearance must absorb the deviation of both colliding part proxies, hence \(2\delta_{\text{part}}\). A part-to-sheet clearance absorbs the part deviation through the same term, but the sheet term enters once: a part can poke past the true boundary by at most the sheet proxy's own deviation, and the sheet samples at high resolution (192 verts) so \(\delta_{\text{sheet}}\) stays tiny. The earlier shared formula used \(2\,\max_{\text{all loops}}\delta\), which let a large freeform sheet's deviation inflate every part (a reported +27 units on the user's S-sheet, collapsing a fill from 200 to 21); the asymmetric split is the fix (HoleNestComponent.cs:45-48, :532-542). The compensation is surfaced to the user as ProxyDevComp: +{2*maxDev} in the report (:653).

Originality. CNH is clean-room at its NFP/BLF/IFP base (same citations as FreeNestX, HoleNestComponent.cs:25-33) with three evolved-fork increments that carry the research weight, each measured or fuzz-verified: the contact-adaptive rotation set, the part-in-part-hole IFP nesting, and the distance-based penetration certificate with micro-retreat. The [Algorithm] attribute names these as "Frahan ContactNfpHoleNester evolution study" and the head-to-head protocol is documented, not asserted (HoleNestComponent.cs:34-36). The component is the CNH facade over the Core engine. The native batched-NFP path wraps nfp_kernel.dll, which vendors official Clipper2 (BSL-1.0) unmodified (NativeNfpKernel.cs:10-22); that is vendored-library, with only the marshalling ours.

1.6.7 Head-to-head: V506 vs NFP-BLF vs CNH vs the reference physics nester

The 2026-06-12 hole-packer study benchmarked CNH against the OpenNest-lineage reference physics nester (the jagua-rs guided-local-search strip packer, "Sparrow") and an MIT-licensed C++ NFP-GLS native nester (HOLE_PACKER_MATH_AND_BENCHMARK.md). On the true-hole lane (1 sheet, 1 sheet-hole, 4 host parts with slanted holes, 8 fillers):

Packer	time	placed	holes filled	valid	deterministic
CNH v1 (general exact-NFP)	60.7 ms	12/12	4	true	yes
CNH v2 (rect fast-path)	0.148 ms	12/12	4	true	yes
MIT native nester (shelf)	21.6 ms	12/12	4	true	no (stochastic)
Reference physics nester (Sparrow)	3255 ms	12/12 outlines	0	false	no

CNH is valid where the reference physics nester is invalid: Sparrow ignores holes (4 hole-ignore warnings, 953.7 overlap loss) and produces no usable hole-aware layout at any time budget. Against Sparrow, CNH v1 is about 54x faster and valid on the same parts (60.7 ms vs Sparrow's invalid 3255 ms); the rect fast-path is 0.148 ms, about 22,000x the invalid Sparrow time. Against the strongest valid baseline (the native shelf) CNH v1 is 2.8x slower because it runs the general exact-NFP construction rather than an axis-aligned shortcut; its v2 rect fast-path runs in 0.148 ms on the true-hole bench, 146x the native shelf's 21.6 ms on that same instance (an axis-aligned shortcut, not a separate all-rectangle benchmark), and it is the only deterministic engine. The honesty boundary is held in source: on the outline-only strip lane the reference physics nester still wins density by 6 to 10 percent, and no universal "2x better" claim is made there (../../benchmarks/HOLE_PACKER_MATH_AND_BENCHMARK.md, sections 2-3).

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2. Trencadís mosaics

Trencadís is the Gaudí "broken-tile" technique: irregular shards placed close together with a grout gap. The repository ships two distinct solver philosophies under the Trencadis tab, plus an EdgeMatch variant.

2.1 Catalog mode: CVD-Lloyd cells + Hungarian assignment

Pack2DTrencadisCatalogComponent (TrencadisCat, GUID F2D00007-CADC-4F2D-9007-7E60CADA15A0) is the example-12 engine. It partitions the sheet into blue-noise cells then assigns shards to cells one-to-one.

Centroidal Voronoi cells (Lloyd 1982). A Voronoi diagram of \(K\) seeds is centroidal when each seed coincides with the centroid of its own cell. Lloyd's algorithm reaches this fixed point by iterated relaxation: assign each domain point to its nearest seed, then move each seed to the centroid of its cell:

\[ s_i^{(n+1)}=\frac{\displaystyle\int_{V_i} x\,\rho(x)\,dx}{\displaystyle\int_{V_i}\rho(x)\,dx}, \qquad V_i=\{x : \lVert x-s_i\rVert\le\lVert x-s_j\rVert\ \forall j\}. \]

The repository discretises the domain (outer polygon minus holes) on an \(N\times N\) grid with uniform density \(\rho\equiv 1\), assigns each grid cell to its nearest seed, recentres, and stops when the largest seed move drops below half a grid step (CvdLloyd2d.cs:30-108). The result is a blue-noise seed field, far better than placing the first piece at a bbox corner.

Optimal assignment (Kuhn-Munkres). With \(n\) shards and \(n\) cells, let the cost \(c_{ij}\) be the area mismatch of placing shard \(i\) in cell \(j\). The minimising one-to-one assignment solves

\[ \min_{\pi\in S_n}\ \sum_{i=1}^{n} c_{i,\pi(i)}, \]

via the \(O(n^3)\) shortest-augmenting-path Hungarian method, with row and column potentials \(u_i,v_j\) maintained so reduced costs \(c_{ij}-u_i-v_j\ge 0\) stay non-negative (HungarianAssignment.cs:23-85). Each placed shard is moved to its cell centroid and inset by the grout offset to leave the mortar gap. Example 12 places 28 shards into 28 cells in 53 ms with zero warnings (examples/12_trencadis/README.md).

Originality. CVD-Lloyd is clean-room from Lloyd (1982) centroidal Voronoi relaxation (CvdLloyd2d.cs:14-22, [Algorithm] at Pack2DTrencadisCatalogComponent.cs:37). The Hungarian solver is a clean-room textbook Kuhn-Munkres / Bourgeois-Lassalle (1971) shortest-augmenting-path implementation, allocation-bounded and net48-compatible (HungarianAssignment.cs:11-15). The catalog placement wrapper that couples them is facade-over-primitives: its [Algorithm] credits "Slab-partitioned Voronoi catalog; Frahan-original Trencadis extension" over a Battiato 2013 synthesis precedent (Pack2DTrencadisCatalogComponent.cs:38, :42). Tier C/B.

2.2 Physics-settle variants vs the greedy/EdgeMatch variants

The Trencadis tab is intentionally not one engine. The variants differ in how a piece reaches its final pose, and the repository keeps them separate rather than blind-merging them (the "the 2D solvers are real, validate don't merge" ruling):

• Greedy NFP-slide (Pack2DTrencadisComponent, GUID F2D00002-...). Pieces slide along a Minkowski-difference arc-length sampler until they butt against placed neighbours, with a bounded-overlap trim up to the Trim Tolerance and the Battiato 2013 §4 cumulative-cut budget (\(T_N\) for the new piece, \(T_P\) for placed) (TrencadisFill.cs:13-27, Pack2DTrencadisComponent.cs:37-38). The standalone greedy box is a skeleton returning empty; the working entrypoint is the catalog or pipeline component (examples/12_trencadis/README.md).
• Dynamic settle (Pack2DTrencadisDynamicComponent, GUID F2D00008-...). After a greedy pack, a Kangaroo 2 goal-based dynamic relaxation pass settles the pieces to fill residual gaps (physics, not geometry). Headless coverage 55.1% with physics on vs 52.7% greedy (Pack2DTrencadisDynamicComponent.cs:61-62, examples/12_trencadis/README.md).
• EdgeMatch (TrencadisEdgeMatchComponent, GUID F2D0000A-...). Drives placement through the 5-stage EdgeMatch pipeline plus a deterministic beam-search assembly solver, an alternative to the Battiato 2013 CVD+GVF stack (TrencadisEdgeMatchComponent.cs:28-29).
• Pipeline (Pack2DTrencadisPipelineComponent, GUID F2D00009-...) composes greedy pack + NFP slide + CVD-Lloyd seeding + an optional Kangaroo 2 settle (Pack2DTrencadisPipelineComponent.cs:59-62).

Originality. The greedy NFP-slide and the dynamic/EdgeMatch/pipeline wrappers are facade-over-primitives that compose published Frahan primitives (CVD-Lloyd, NFP slide, Hungarian, EdgeMatch, the Kangaroo 2 relaxation). The physics step explicitly credits Daniel Piker's Kangaroo 2 (Pack2DTrencadisDynamicComponent.cs:62); the Trencadís domain framing is credited to Gaudí Park Güell and the Battiato 2013 synthesis. None claims a new algorithm; the contribution is the integration. Tier C/D.

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3. Status & what's left

• Example 28 (hole nest) has no rendered figure. The folder ships the .gh demo only (examples/28_hole_nest/), no PNG and no README. The CNH renders in this chapter borrow from example 10 (nesting) and example 12 (Trencadís). Severity: low (documentation gap, not a code gap).
• Deployed .gha lag (KB-7). The example-10 validity figures are from the headless harness on current source. An older deployed .gha may still overlap parts on a live 2D solve; rebuild and redeploy before trusting a live result (examples/10_pack2d/README.md). Severity: medium.
• Standalone greedy Trencadís box returns empty. Frahan Trencadís Pack is a skeleton; the working entrypoints are the Catalog and Pipeline components (examples/12_trencadis/README.md). Severity: medium (a ghost on the ribbon, against AGENTS.md section 6).
• Rect fast-path spacing limitation. The exact rectangle shelf path only activates at spacing == 0; spacing > 0 needs exact rect-dilation bookkeeping and is deferred to the general engine (ContactNfpHoleNester.cs:611-616). Severity: low (correct fallback exists).
• Residual penetration band. After the compound gate, penetrations inside the snap band (about \(2\times10^{-5}\) caller units) can be accepted; deeper ones cannot on any path. This is a stated, bounded residual, far inside any fabrication budget (ContactNfpHoleNester.cs:69-72). Severity: low.
• Outline-density gap to the reference physics nester. On the no-hole outline strip lane the reference physics nester (Sparrow) still wins density by 6 to 10 percent; CNH's claimed win is the hole-aware lane only. This boundary must stay in any external claim (HOLE_PACKER_MATH_AND_BENCHMARK.md, section 2c). Severity: low (honesty, not a defect).

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References (this chapter)

• Burke, E.K., Hellier, R., Kendall, G., Whitwell, G. (2006). A New Bottom-Left-Fill Heuristic Algorithm for the Two-Dimensional Irregular Packing Problem. Operations Research 54(3):587-601. DOI 10.1287/opre.1060.0293.
• Burke, E.K., Hellier, R., Kendall, G., Whitwell, G. (2007). Complete and robust no-fit polygon generation for the irregular stock cutting problem. European Journal of Operational Research 179(1):27-49. DOI 10.1016/j.ejor.2006.03.011.
• Bennell, J.A., Oliveira, J.F. (2009). A tutorial in irregular shape packing problems. Journal of the Operational Research Society 60(S1):S93-S105. DOI 10.1057/jors.2008.169.
• Lloyd, S.P. (1982). Least squares quantization in PCM. IEEE Transactions on Information Theory 28(2):129-137. DOI 10.1109/TIT.1982.1056489.
• Kuhn, H.W. (1955). The Hungarian method for the assignment problem. Naval Research Logistics Quarterly 2(1-2):83-97. DOI 10.1002/nav.3800020109.
• Battiato, S., Di Blasi, G., Gallo, G., Guarnera, G.C., Puglisi, G. (2013). Artificial mosaic generation: a survey and synthesis. (Trencadís synthesis precedent, cited in source.)
• Johnson, A. Clipper2 (Boost Software License 1.0). Polygon Minkowski sum and NonZero boolean back end.