# Catalogue audit of the learned GCTS lane **2026-09-21 follow-up:** [p9-48258 has a checked integer-grid non-tiling certificate](p9-48258-grid-obstruction.md), discovered through pair-corona learning and verified without learned assignments. The original screening below is historical; this result does not establish a true aperiodic monotile. The [next complete catalogue](3d-point-corona-sat.md#p9-42947-complete-classification-and-limits-of-the-pair-marking) for p9-42947 fits all 1,408 labels, but shows mixed warm growth and no infinite classification. The [viable-frontier follow-up](3d-viable-frontier-search.md) adds p10-346304's first ten positive pair classes and four proof-checked negative classes. Its catalogue remains incomplete; no marking has been activated. Follow-up: [geometric voxel checks](3d-voxel-marking-research.md) found overlaps in the six successful historical-polycube point windows and introduced an explicit center-and-corner research model. The point results below keep their original scope; they must not be interpreted as geometric packing certificates. The [v2.3.0 live catalogue](3d-research-catalogue.md) now exposes all fourteen polycube records directly, using the center-and-corner model in v2. Its 54-entry selector and five-case research suite are distinct from the historical sweep below. **Larger-window follow-up:** the [curated slab demonstrations](3d-gcts-demonstrations.md) use fresh tile-support-only markings and longer comparisons. The small-window measurements below remain historical; they do not describe those new presets. ## Purpose and scope This September 21, 2026 experiment tests the new pair-corona learner against the unmarked lane on every system in the deployed catalogue, then separately on the 14 historical polycube research records. The latter include 11 records marked inconclusive and three controls; they are imported from their actual voxel coordinates, not substituted with the cube fallback for unknown IDs. The discovery objective remains a geometric monotile that tiles space and forces aperiodicity. An exact finite point window, a successful learned classifier, and a bounded miss in periodic search each fall short of that objective. In particular, the current fractional vertex model has not been proved equivalent to nonoverlapping geometric polycubes. Results below do not replace the earlier voxel-corona or quotient certificates. ## Reproducible protocol ```sh node scripts/screen-3d-learned-catalog.mjs \ --time-ms=3000 --output=/tmp/gcts-catalog-screen node scripts/screen-3d-learned-catalog.mjs \ --catalog=polycubes --time-ms=3000 --output=/tmp/gcts-polycube-screen ``` Each tile receives one cold free-range run and one cold GCTS run: proper rotations, no reflected placements, seed 1, radius 1, 10,000 search attempts, 500 attempts per corona pair, and a three-second total lane budget. Full 3D windows have 27 required points; the Hat/Turtle slab windows have 14. Worker startup is included in `elapsedMs`, whereas `stats.totalMs` starts inside the engine and includes model construction, learning, graph construction and verification. This is a short diagnostic sweep, not a statistical speedup benchmark. Runs are sequential. The runner independently replays every reported finite solution, writes each full artifact as gzip JSON, and updates a summary after every case. A worker has a 1 GiB heap bound and a watchdog ten seconds beyond the requested lane budget. Worker failure or timeout is reported explicitly. Source hashes record the engine used by the run. Only aggregate results are checked into the repository; experimental markings remain in the local run directory. The aggregate receipts, including all per-system outcomes and source hashes, are in [the audit data](../../data/3d-learned-catalog-audit-2026-09-21.json). ## Defects exposed and corrected The first sweep exposed two implementation problems: 1. The hexagonal slab had three orientation records describing the same translated solid. Its point-group lookup was therefore noninvertible. Deduplication now compares exact weighted support and geometric vertices modulo permitted lateral translation, retaining one orientation. Both reflection settings have a bijective induced label action. 2. Counting candidate objects alone did not bound memory: each FCC/HCP candidate contains hundreds of point dependencies. Four GCTS workers hit their heap cap. The corona oracle now bounds stored dependency entries at one million and reports a resource-limited unresolved pair. It never turns an incomplete candidate graph into a negative label. The post-fix sweep is recorded separately from the original run. The existing Turtle and Hat reflected-slab regression scores remain unchanged. ## Research implications The corrected 40-system sweep gives: | Lane | Verified finite window | Unresolved | Unsupported exact model | Exhausted learned restriction | | --- | ---: | ---: | ---: | ---: | | Free-range | 16 | 15 | 9 | 0 | | GCTS | 4 | 25 | 9 | 2 | No worker crashed after the dependency limit was added. The four successful GCTS systems were cube, hexagonal slab, rhombic dodecahedron and truncated octahedron. On these small targets learning made the cold run slower: | Tile | Pair labels | Valid passed | Invalid blocked | Free total ms | GCTS total ms | | --- | ---: | ---: | ---: | ---: | ---: | | Cube | 26 | 26/26 | 0/0 | 5.6 | 115.7 | | Hexagonal slab | 24 | 6/6 | 18/18 | 5.5 | 101.2 | | Rhombic dodecahedron | 98 | 18/18 | 80/80 | 8.3 | 1028.0 | | Truncated octahedron | 146 | 14/14 | 132/132 | 12.4 | 2337.2 | The Hat and Turtle slabs with reflections **disabled** perfectly classify their resolved pair catalogues but exhaust the learned finite-window search. This is a concrete reason to retain the separate marked-growth check: fitting all pairs does not establish that a marking admits a tiling. These are different runs from the successful reflected-slab regressions, and neither result proves unmarked non-tiling. No new hard-case infinite tiling, non-tiling proof, or aperiodic monotile follows from this sweep. All 14 historical polycube records remain unresolved in the cold GCTS lane; the free lane finds six finite point windows and leaves eight unresolved. The GCTS runs label only 1–8 pairs each before stopping. None produces an accepted marking. These finite-window results do not upgrade or overturn any historical geometric classification, including the known non-tiler controls. Two larger-window follow-ups use radius 3 (343 required points) and a ten-second budget. Both lanes finish and independently verify both windows: | Tile | Free branches | GCTS branches | Marking eliminations | Free total ms | GCTS total ms | | --- | ---: | ---: | ---: | ---: | ---: | | Rhombic dodecahedron | 6 | 1 | 882 | 45.1 | 1095.1 | | Truncated octahedron | 3 | 1 | 610 | 55.1 | 2396.5 | This demonstrates candidate elimination through actual point markings, but still does not demonstrate a cold speedup. Both tiles are periodic controls. Pair labeling is the main bottleneck on the historical hard polycubes. A learner cannot reject their unresolved pairs, and increasing its equality solver speed will not resolve that labeling cost. The next useful experiments are deeper, resumable unmarked pair searches and comparisons on larger windows for systems whose pair catalogues already complete. Any optimization must preserve complete candidate incidence, global dead/forced checks, generation order, exact rollback, and the distinction between exhausted and unresolved. One concrete optimization to investigate is excluding candidates incompatible with the permanently fixed seed pair before allocating their dependency records. Such an exclusion would need an independent complete-domain audit; it must not exclude candidates merely incompatible with a reversible prefix. For a surviving geometric candidate, the subsequent proof work remains: - establish that its point model faithfully represents geometric placements; - produce an infinite construction, not just larger finite patches; - prove that all geometric tilings obey a recognizable hierarchy, or another condition excluding every nonzero translation; - if learned markings support that argument, prove their necessity or realize them in the geometry. A learned restriction alone cannot prove aperiodicity. A new external lead is Tsiokos's September 16 proof submission, [A Strongly Aperiodic Monotile in Three Dimensions](https://arxiv.org/abs/2609.19214). It claims a rational decorated chair with a forced hierarchy. Its [companion repository](https://github.com/ioannist/six-birds-tiles) is available, but this audit has not replayed or independently established its geometric or formal proof. It is not a discovery or verified result of this project.