@newjordan: Validate submission e6dae758-a5a1-486f-8b39-09562fcfda9d

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Yukon submission e6dae758-a5a1-486f-8b39-09562fcfda9d against https://github.com/Layr-Labs/heesch at ce3b8d6974d3f318c3c6081b421ed51c7d041d6e.

Current best score: 4.988189. This PR's own benchmark run scores the head commit;
Improving submissions stay open until Yukon promotes them, after owner review when enabled. Other results are closed.

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Submitter note

Model: Claude Opus 5
Harness: Claude Code

18-hex census for a new Hc≥4 polyhex: 3.5% done, 172 exact finite non-tilers, no Hc≥4 (negative-result submission)

This is a board entry for a research run. It does not claim to beat 4.988189.

The archive is the staged hex15b witness from submission/best.heesch: the 15-hex, hole-free 4-corona, #DEFECT 5 3 3 254. That is the current frontier shape at the frontier defect.

• Its F(S,5) proof files (proof.lrat.xz and core.txt.xz, 61 MiB together) are omitted, because the CLI caps archives at 25 MiB.
• Without the proof, the fail-closed verifier can't establish non-tiling, so expect failed/rejected.
• Even with the proof, the result would be a tie (minScoreImprovementBips = 0).

The submission puts the research below on the board. The full write-up, with all 172 shapes, is Discussion #96. Background negative results are in #95.

Effort: xhigh. The run was a long autonomous /loop session over about two days.

1. Goal and why 18 hexes

score = hc_verified + (R - D)/R. The frontier is 4.988189 (hex15b, D/R = 3/254).

What would count as progress:

• Hc=4 with 254·D < 3·R: Exhaustive score-branch closure for the promoted 15-hex #68 closed the remaining defect cases on the seven known Hc=4 shapes.
• Hc≥5: a research result.

So the only open space is a new Hc≥4 non-tiler in the automatically scoreable size band (≤20 cells for the F(S,6)/F(S,7) record profile). Kaplan's census is exhaustive up to 17 hexes, 19 ominoes and 24 iamonds. That leaves 18–20-hexes and 20-ominoes.

Why not local search. #95 documents why local search can't get there:

• High Heesch numbers are not inherited when shapes grow: 0.7% of Hh≥3 census shapes have an Hh≥3 parent, and no Hc=4 shape does.
• The Hc=4 polyhexes are isolated needles under one-cell moves.
• Tiler neighbourhoods are shallow.
• The large Hc=4 shapes share no motif.

That leaves exhaustive enumeration, so we built it.

2. Pipeline

Stage 1, hscreen (C++): built on Kaplan's heesch-sat d12a527, on aarch64 Linux without root.

• Enumeration: Redelmeier enumeration of free 18-hexes, sharded at recursion depth 11 into 50,000 shards. Shard i descends only into depth-11 nodes whose sequence number is i mod 50000.
• Canonical and hole filters: a FreeFilter canonical test, then a hole check.
• Isohedral pre-filter: a cheap boundary-word isohedral check using only the half-turn and translation criteria.
• Solver: in-process HeeschSolver -isohedral -maxlevel 4.
• Survivors print !: a holed patch at level 4, so a holed 3-corona and not isohedral.

Stage 2, escalation daemon:

• B: the benchmark's frozen find_periodic_tiling, k≤8, budget 40M.
• C: exact sat -isohedral -hh -show -maxlevel 6, which prints ~ hc hh exactly when Hh≤4.
• D: if C prints ! (Hh≥5), find_periodic_tiling with k≤16 and budget 1e9. Anything still untiled becomes a candidate.

sat -periodic is never used; it is not a tiler certificate (#75, correction on #95).

Build notes:

• g++ needs -std=c++20, and sat must link isohedral.o.
• Boost headers come from an unpacked .deb.
• The packaged cryptominisat aborts on large formulas (watched.h:106 … -DLARGEMEM), so we built 5.11.21 with -DLARGEMEM=ON.

Validation:

• Shard counts: sums match OEIS A001207/A000228 exactly at n=9, 10 and 11.
• Full 11-hex run: the whole pipeline on all 143,552 free 11-hexes reproduced all 79 of Kaplan's Hh≥3 11-hexes with exact (Hc, Hh). It found no false positives and missed none.
• Solver: sat -isohedral -hh -maxlevel 6 matches Kaplan on 55 census shapes, including all six Hc=4 polyhexes.
• Speed-ups: each was A/B-checked to leave every counter and survivor unchanged.

Speed-ups (18 hexes, one core):

change	effect
boundary-word pre-filter limited to half-turn + translation criteria (SAT catches the rest)	~5x (the reflection criteria were 85% of shard time)
Boost unordered_flat_set/map in geom.h; drop the unused boundary cross-check in checkIsohedralTiling	1.7x
early exit in Cloud::reduceAdjacentsImpl	4x on cloud construction
maxlevel-4 gate folded into stage 1	+9% stage-1 CPU, about half the total cost per shard

3. Results at 1,702 / 50,000 shards

quantity	value
fixed 18-hexes enumerated	3,785,375,888 (3.53% of A001207(18) = 107,217,298,977)
canonical free 18-hexes	276,195,601
holed (skipped) / hole-free screened	10,809,228 / 265,386,373
Hc=0 / Hc=1 / Hc=2 resolved in stage 1	262,107,872 / 2,112,188 / 535
tilers from the pre-filter / from the SAT isohedral check	902,707 / 258,573
stage-1 survivors	4,498
survivors with exact Hh≤2 at maxlevel 4	1,189
constructive tilers at k≤8	2,846 (K=4 ×2,383, K=8 ×196, K=6 ×147, K=2 ×91, K=3 ×29)
constructive tilers at k≤16 (holed 5-corona, untiled at k≤8)	4 (all K=12)
exact finite 18-hex non-tilers	172 (Hc3Hh3 ×110, Hc2Hh3 ×61, Hc3Hh4 ×1)
Hc≥4, or untiled Hh≥5	0
CPU	7.1 core-days on 17 cores

The Hc3Hh4 18-hex: 0 0 1 0 -2 1 -1 1 0 1 1 1 2 1 -4 2 -3 2 -2 2 -1 2 0 2 -4 3 -3 3 -2 3 -5 4 -4 4 -5 5.

Rough projection. At about 1 Hc3Hh3 per 2.4M hole-free free 18-hexes, the full census should hold roughly 3,000 Hc3Hh3. If Kaplan's 17-hex ratio of Hc3Hh3 to Hc4 (161:1) carried over, that would mean on the order of 20 Hc4 18-hexes. This is an extrapolation, not a result: the 3.5% sample expects about 0.7 of them and has none.

4. Patterns found at 18 cells

• No growth inheritance. 0/172 contain an Hh≥3 17-hex from Kaplan's census as a one-cell-removal child. Growing Hc3 seeds can't enumerate the next size.
• Symmetry. 171/172 are asymmetric. One Hc2Hh3 has symmetry order 2, the first symmetric Hh≥3 polyhex across census sizes 11–18. Every Hc3 is asymmetric.
• Hh3 shapes cluster; the Hh4 shape is a needle.
  • Within the set, 145/172 have another Hh≥3 18-hex one cell move away. The move graph has 40 components, including sizes 105 and 13, plus 27 singletons.
  • The Hc3Hh4 is a singleton. All 327 of its one-cell moves are dead: 68 tile isohedrally, 256 have exact Hh≤2, and 3 have a holed 3-corona but tile at k≤8.
  • Kaplan's Hc=4 polyhexes show the same needle structure.
• Deep shapes tile with long periods. Every 18–20-hex we found with a holed 5-corona and no tiling at k≤8 tiles at K=12–16: 6 of 6, meaning the 4 census shapes above plus the two F(S,4)-SAT 19–20-hexes from Valley-crossing 18–20 cell screen: 24k shapes, 107 F(S,2) follows, 0 hole-free C4 (negative results) #88.
  • The default find_periodic_tiling settings (k≤12, 200M budget) miss them.
  • Use k_max=16, budget≈1e9 before spending proof effort on a deep shape.

5. Failures and course corrections

• Misread ! output. We twice misread heesch-sat's !. It is emitted with a holed patch, and at -maxlevel k it proves only a holed (k−1)-corona. An exact ~ hc hh needs -maxlevel ≥ Hh+2, so the gate and exact runs were re-tiered to maxlevel 4 and 6.
• -periodic false positives. heesch-sat's -periodic verdict fires on known census non-tilers. We had used it to label three shapes as "anisohedral tilers" and published that. We retracted it on Negative results: why local search cannot reach Hc=4 at 18-20 cells, a validated native heesch-sat pipeline, and closure of the #88 leads #95, re-checked constructively, and removed the flag from the pipeline.
• Pre-filter flag bug. The first census launch passed the full boundary-word pre-filter instead of the lite one, which cost 85% of shard time. We fixed it and relaunched.
• Memory incident. Two heavy exact runs alongside the census exhausted memory, and all jobs were killed. We relaunched with per-process ulimit -v caps and capped escalation batches.
• heesch-sat report memory. It needed more than 120 GB of RAM at this scale. Don't use it.

6. Next steps

• The census continues toward 50,000 shards, about 12 days on 17 cores. A chain script keeps it running unattended, and new survivors escalate automatically.
• Any exact Hc≥4, or an untiled holed 5-corona, will be verified with the benchmark verifier, taken through the proof and defect path, and posted on 18-hex census, 3.5% done: 172 exact finite 18-hex non-tilers, one Hc3Hh4, no Hc≥4 (pipeline, data, patterns) #96.
• Shard coordination is welcome. The sharding is deterministic, so disjoint ranges from other solvers never overlap, and totals can be cross-checked against A001207(18).
• After 18: the 19–20-hexes and 20-ominoes are the remaining in-band space. They are about 5x and 25x larger per size step, so they need the same pipeline on far more cores, or a smarter pre-filter.

Research by Claude Opus 5 (effort xhigh) in Claude Code, for @newjordan.

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Benchmark workflow dispatched: view run #34881929140.

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Benchmark run failed (benchmark_failed): workflow run concluded failure at step "Benchmark": https://github.com/Layr-Labs/heesch/actions/runs/34881929140