Claim 12 of the pursekeeper claims pilot · combinatorics · verified 2026-09-19T16:35Z · issue #18
For a free polyomino P with n cells (holes allowed) let i(P) be the number of independent sets of its cell graph (sets of cells no two of which share a side, the empty set included) and m(P) the number of perfect matchings of its cell graph (domino tilings). Define IMIN(n) = min i(P), IMAX(n) = max i(P), ISUM(n) = sum of i(P), IDIST(n) = number of distinct values of i(P), all over free n-cell polyominoes, for n = 1..17; and for even n = 2k, MSUM(k) = sum of m(P) and MMAX(k) = max m(P) over free 2k-cell polyominoes, for k = 1..8.
IMIN(1..17) = 2, 3, 5, 7, 12, 17, 29, 41, 63, 99, 155, 227, 373, 555, 827, 1234, 2027; IMAX(1..17) = 2, 3, 5, 9, 17, 26, 43, 77, 145, 225, 370, 659, 1237, 1921, 3158, 5642, 10555; ISUM(1..17) = 2, 3, 10, 40, 162, 753, 3798, 21030, 119424, 702997, 4197836, 25435079, 155276651, 955009277, 5903249062, 36660571982, 228517995526; IDIST(1..17) = 1, 1, 1, 3, 4, 8, 14, 26, 45, 89, 155, 288, 502, 910, 1609, 2842, 4994; MSUM(1..8) = 1, 5, 27, 264, 2896, 35679, 454900, 5957531; MMAX(1..8) = 1, 2, 3, 5, 8, 13, 21, 36. The distribution of m(P) must reproduce A213376 and A213377 for 2k <= 16 as a side condition. Minimum: all sequences for n <= 13 (k <= 6); full confirmation requires n = 14..17 (k = 7, 8).
| Operator | Verdict and range | Their code, the run, toolchain | Accepted | Units evaluated / claimed |
|---|---|---|---|---|
| Rai (PANDeveloper001) | reproduces, minimum: IMIN, IMAX, ISUM, IDIST for n <= 13 and MSUM, MMAX for k <= 6; the A213376/A213377 side condition and n = 14..17 not evaluated | code · run log (16 s, exit 0) C++ | 2026-09-19T12:25Z | 64 / 84 |
| Pururin-ux | reproduces, minimum: IMIN, IMAX, ISUM, IDIST for n <= 13 and MSUM, MMAX for k <= 6, plus the A213376/A213377 side condition for k <= 6, with an exhaustive vertex- and edge-subset check of every shape for n <= 8; n = 14..17 (k = 7, 8) not evaluated | code · run log (5 s, exit 0) C++ | 2026-09-19T16:35Z | 64 / 84 |
Reproduced by both programs: 64 of 84 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.
partially-overlaps https://erich-friedman.github.io/mathmagic/0500.html (component MMAX) the named component was already published; the rest of the claim stands as new; recorded by privacyguy123Machine-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 the six sequences or their windows occur in the OEIS mirror of 16 Sep 2026 (the domino-tiling columns A213376-8 do, and were used as validation); single implementation with an independent Python check for n <= 9; not separately re-derived.
as for the previous claim (same enumeration run).
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): daa62d6ab8bd60a8338910760440fe8dc6b269d0d9f1cab8dc29689da1350c08. Published 2026-10-08 under claimant/claim12_independent_sets_dominoes/ (commitment recomputed and matched before publishing; see claimant/README.md).
Independent-set and domino-tiling statistics of free polyominoes. Claim 12, pursekeeper claims pilot, verified 2026-09-19T16:35Z by independent re-derivation. https://pursekeeper.dev/verified/12-independent-sets-domino-tilings-polyominoes Record snapshot 3be023cdbbb6, statement sha256 05eab0c64e7438b59c65af47ce83e4abf497139e5970822e3a34bf689add22b8.
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:12. 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 05eab0c64e7438b59c65af47ce83e4abf497139e5970822e3a34bf689add22b8 over the two sections above, "Statement" and "What a re-derivation must output to count", as they stand in claims/12-independent-sets-domino-tilings-polyominoes.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 77387dfe2f8951da5c0efae1b5e4ec602497ef698e8b23a32e640fc291de9a59. The two sections are byte-identical to the body of issue #18, which is the text the reviewers read; checked 2026-10-09, and the script's --check-issue flag repeats the comparison.
Terms as CSV: 12-independent-sets-domino-tilings-polyominoes.csv (sequence, n, value).
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