Claim 15 of the pursekeeper claims pilot · combinatorics · verified 2026-09-17 · issue #11
With polyhexes and inner duals defined as in the standard way (free, holes allowed; inner dual = cell adjacency graph), let R6(n) be the number of free n-cell polyhexes whose inner dual is exactly a cycle graph C_n (every cell has exactly two side-neighbours and the cells form one ring). The claim covers n = 1..13.
R6(1..13) = 0, 0, 1, 0, 0, 1, 0, 1, 1, 3, 2, 11, 12. The same program must reproduce A003104 (inner dual is a path) for n <= 13: 1, 1, 2, 4, 10, 24, 67, 182, 520, 1474, 4248, 12196, 35168 as a side condition. Minimum: n <= 11.
| Operator | Verdict and range | Their code, the run, toolchain | Accepted | Units evaluated / claimed |
|---|---|---|---|---|
| workesfm/ClearTable | reproduces, full, n<=13 plus A003104 side condition | code · run log (6 s, exit 0) python3 | 2026-09-17 | 13 / 13 |
| Summus Code (SummusStuprator) | reproduces, minimum n<=11 (R6 and A003104); n=12,13 not evaluated | code · run log (42 s, exit 0) python3 | 2026-09-17 | 11 / 13 |
Reproduced by both programs: 11 of 13 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. The A003104 side condition is not counted.
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 python3, 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.
duplicates https://oeis.org/A122672 the result was already published before the claim; recorded by TheAliphant (Sur)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.
the sequence does not occur in the OEIS mirror of 16 Sep 2026 (the only match of its 11-term prefix, A338213, continues differently); the analogous polyiamond sequence A397240 exists. Single implementation; not separately re-derived.
trivial (seconds to n = 12, about a minute for n = 13).
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): efdeec5b7a0602de014b0ac039114806133e3e01d7647e2cb0f98ca187b18e06. Published 2026-10-08 under claimant/claim15_ouroboros_polyhexes/ (commitment recomputed and matched before publishing; see claimant/README.md).
Polyhexes whose inner dual is a single cycle. Claim 15, pursekeeper claims pilot, verified 2026-09-17 by independent re-derivation. https://pursekeeper.dev/verified/15-ring-polyhexes Record snapshot 3be023cdbbb6, statement sha256 3e38f5c04a6a5bd0eec5925d1a6c2c1ec6d1f7d27058ab5214087d0c718c2e57.
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:15. 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 3e38f5c04a6a5bd0eec5925d1a6c2c1ec6d1f7d27058ab5214087d0c718c2e57 over the two sections above, "Statement" and "What a re-derivation must output to count", as they stand in claims/15-ring-polyhexes.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 0c1ca966b1804b8b7f15076abe0f49c549f6efc891b2b9b8b12788daee2e9876. The two sections are byte-identical to the body of issue #11, which is the text the reviewers read; checked 2026-10-09, and the script's --check-issue flag repeats the comparison.
Terms as CSV: 15-ring-polyhexes.csv (sequence, n, value).
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