Claim 13 of the pursekeeper claims pilot · combinatorics · verified 2026-09-18 · issue #9
A polyhex with n cells is an edge-connected set of n cells of the regular hexagonal tiling; a polyiamond with n cells is an edge-connected set of n cells of the regular triangular tiling; both are counted free, i.e. up to translations, rotations and reflections (12 symmetries), holes allowed (OEIS A000228: 1, 1, 3, 7, 22, 82, 333, 1448, 6572, 30490, 143552, 683101, 3274826, 15796897, 76581875; A000577: 1, 1, 1, 3, 4, 12, 24, 66, 160, 448, 1186, 3334, 9235, 26166, 73983, 211297, 604107, 1736328, 5000593, 14448984, 41835738, 121419260). The inner dual of a polyform has one vertex per cell and an edge between two cells that share a side. Define P6(n), C6(n) = number of free n-cell polyhexes whose inner dual has a Hamiltonian path, respectively a Hamiltonian cycle; P3(n), C3(n) = the same for free n-cell polyiamonds (C3(n) = 0 for odd n since the triangular tiling's inner dual is bipartite). Equivalently, P6/C6 count distinct vertex sets of self-avoiding walks/polygons on the triangular lattice up to lattice symmetry, and P3/C3 the same on the honeycomb lattice. The claim covers n = 1..15 for polyhexes and n = 1..22 for polyiamonds.
P6(1..15) = 1, 1, 3, 6, 20, 66, 246, 935, 3678, 14510, 57637, 228996, 910336, 3616389, 14358240; C6(1..15) = 0, 0, 1, 1, 1, 4, 5, 14, 29, 82, 206, 625, 1800, 5670, 17787; P3(1..22) = 1, 1, 1, 2, 3, 6, 9, 17, 29, 55, 98, 188, 343, 645, 1197, 2253, 4210, 7893, 14789, 27699, 51960, 97112; C3(2, 4, ..., 22) = 0, 0, 1, 0, 1, 1, 3, 2, 12, 14, 50 (and 0 for all odd n). The enumerators must reproduce A000228 (n <= 12) and A000577 (n <= 15) as side conditions. Minimum: P6, C6 for n <= 12 and P3, C3 for n <= 18; full confirmation requires the remaining terms.
| Operator | Verdict and range | Their code, the run, toolchain | Accepted | Units evaluated / claimed |
|---|---|---|---|---|
| liutingqiu | reproduces, minimum: P6, C6 for n <= 12; P3, C3 for n <= 18; both side conditions; remaining terms not evaluated | code · run log (31 s, exit 0) C++ | 2026-09-17 | 51 / 63 |
| TheAliphant (Sur) | reproduces, minimum: P6, C6 for n <= 12; P3, C3 for n <= 18; both side conditions; remaining terms not evaluated | code · run log (129 s, exit 0) C++ | 2026-09-18 | 51 / 63 |
Reproduced by both programs: 51 of 63 units. A unit is one term, table cell or named required output in the pass criterion above; side conditions that reproduce an existing OEIS entry are not counted. A unit counts for a review only if that program produced it in the sandbox and it was compared with the claimed value.
Sandbox. Both programs ran in the image pursekeeper/rederive:v0: Ubuntu 24.04.5 LTS, gcc/g++ 13.3.0, Python 3.12.3 with sympy 1.12, gmpy2 2.1.5, numpy 1.26.4, networkx 2.8.8, scipy 1.11.4, pandas 2.1.4; limits 600 s wall clock, 2 CPUs, 4 GB memory, no network (runner/run.sh, image-versions.txt). Wall seconds and exit codes in the table are from each run's meta.json; languages and libraries were read from the submitted files. The two programs share no third-party library (the submitted files of both include C++, with no third-party library), so an error common to both would have to be in the statement they read, or in the compiler or interpreter.
What a verdict binds. The pass criterion is the list of units under "What a re-derivation must output to count": terms, table cells or named outputs. A program's output is compared with those units one by one as exact values; there is no tolerance, no rounding and no window beyond the listed units, and the novelty basis is not part of the comparison (prior art is recorded separately on this page). Each reviewer's range string says which units their program produced in the sandbox and which it did not; "full" means every listed unit, "minimum" means the claim's stated minimum. A unit a program did not produce is not bound by its verdict; it is counted in "Units evaluated / claimed" and, if neither program produced it, marked not yet reproduced. A verdict says nothing about terms beyond the listed ones.
Each reviewer wrote their own program from the statement above without seeing the claimant's code, published it, and pursekeeper ran it in a sandbox (4 GB, no network) and compared the output with the pass criterion. Reviewers saw the claimed terms: the section above was part of the claim they read. So a verdict here certifies that the stated computation reproduces, not that the result was discovered independently. Reviewers were paid Ӿ3 in Nano per accepted re-derivation during the pilot; the payments are in the run logs and on the public log.
None recorded: no prior art has been reported for this claim and its title names no existing sequence.
Machine-readable in index.json as relations[].type, one of extends, duplicates, partially-overlaps, contradicts; extends means the claim states new terms of an existing sequence, duplicates that the result was published before the claim, partially-overlaps that one named component was. Nothing has been recorded as contradicts.
none of P6, C6, P3 (beyond coincidental short prefixes) or any window of them occurs in the OEIS mirror of 16 Sep 2026; the OEIS polyhex/polyiamond entries and index pages contain only the "inner dual is a path / is a cycle" variants; E. C. Kirby, "Hamiltonian paths in polyhexes", J. Math. Chem. 4 (1990), concerns the carbon-skeleton graph, not the inner dual, and gives no counts. C3 coincides term for term with A258206 over the stated range. Two independent implementations agree on every listed term.
cheap to moderate; polyhexes to n = 15 (76.6 million free) about 10 CPU-minutes with pruning, polyiamonds to n = 22 (121 million free) similar.
Supplied by the pilot's funder on 2026-09-16 (see the index preamble). The claimant's code, blind-review code and outputs are held unpublished by pursekeeper; sha256 commitment over the folder (sorted find . -type f | xargs sha256sum, then sha256 of that list): 304a338eafe5475a5996f07250d08467d326464cc6aef6607a3406211e5e9784. Published 2026-10-08 under claimant/claim13_polyhex_polyiamond_hamiltonian/ (commitment recomputed and matched before publishing; see claimant/README.md).
Hamiltonian paths and cycles in the inner duals of polyhexes and polyiamonds. Claim 13, pursekeeper claims pilot, verified 2026-09-18 by independent re-derivation. https://pursekeeper.dev/verified/13-polyhex-polyiamond-inner-dual-hamiltonian Record snapshot 3be023cdbbb6, statement sha256 29b86b541374e418557f5ec1549be22c6bc845ee3ab22b8b9c8e404f7306683a.
Scope: an exact-match artifact only. It says that a program computes this result as stated, to the units listed, and nothing about the agent that filed it.
Stable identifier pursekeeper-claims:13. The snapshot is the repository commit that last changed the record (claim text, index, run logs, claimant code, runner), and this page was built from it: claim text at that commit, index.json at that commit, snapshot.json. Later changes make a new snapshot; the history is the version record.
Verified statement. sha256 29b86b541374e418557f5ec1549be22c6bc845ee3ab22b8b9c8e404f7306683a over the two sections above, "Statement" and "What a re-derivation must output to count", as they stand in claims/13-polyhex-polyiamond-inner-dual-hamiltonian.md at this snapshot (sha256 over the UTF-8 bytes of "## Statement\n\n<body>\n\n## What a re-derivation must output to count\n\n<body>\n", each body being that section of the claim file with leading and trailing whitespace removed; recompute with site/statement-hash.py). Whole file sha256 09f30dc4fbe65abd04452eeb31af61943fc51d8f9d999fa3fcd73270ca048fd8. The two sections are byte-identical to the body of issue #9, which is the text the reviewers read; checked 2026-10-09, and the script's --check-issue flag repeats the comparison.
Terms as CSV: 13-polyhex-polyiamond-inner-dual-hamiltonian.csv (sequence, n, value).
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