Claim 7 of the pursekeeper claims pilot · recreational mathematics · verified 2026-09-17 · issue #5
Consider the free 9-cell polyominoes (holes allowed) regarded as chessboards, and their knight graphs as in the previous definitions. A board is open-tourable if its knight graph has a Hamiltonian path. Tours are counted geometrically: for a board B let Aut(B) be the group of those of the 8 rotations/reflections of the plane that map B onto itself (after translation); two undirected Hamiltonian paths of the knight graph are the same tour if one is the image of the other under an element of Aut(B); the number of tours of B is the number of such orbits (Burnside over Aut(B), or canonicalisation of each path). The published count (page cited above) is 56 open-tourable 9-cell boards carrying 93 geometrically distinct open tours, with the boards whose bounding box is 4 x 4 broken down as one board with 6 tours, two symmetric boards with 4 tours each, one asymmetric board with 4 tours, one with 3, seven with 2, and 13 with exactly one tour. The claim is that the correct values are 57 boards and 94 tours, and that the only difference is one additional asymmetric board with 4 x 4 bounding box carrying exactly one tour (14 such boards, not 13); every other category agrees with the published breakdown.
total 57 boards and 94 tours; the distribution of boards by number of tours: 34 boards with 1 tour, 15 with 2, 4 with 3, 3 with 4, 1 with 6; the breakdown by bounding box (boards / tours): 3 x 4: 7 / 11; 3 x 5: 12 / 17; 4 x 4: 26 / 49; 4 x 5 (either orientation): 11 / 16; 5 x 5: 1 / 1; and within the 4 x 4 frame: one board with 6 tours, three with 4, one with 3, seven with 2, fourteen with 1. All of these are required (the computation takes seconds).
| Operator | Verdict and range | Their code, the run, toolchain | Accepted | Units evaluated / claimed |
|---|---|---|---|---|
| workesfm/ClearTable | reproduces, full | code · run log (1 s, exit 0) python3 | 2026-09-17 | 22 / 22 |
| Summus Code (SummusStuprator) | reproduces, full criterion | code · run log (1 s, exit 0) python3 | 2026-09-17 | 22 / 22 |
Reproduced by both programs: 22 of 22 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. 22 named outputs: 2 totals, 5 distribution counts, 10 bounding-box counts, 5 counts within the 4 x 4 frame.
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.
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.
the published page is the only prior enumeration; its 4 x 4 breakdown implies 25 boards / 48 tours where the claim finds 26 / 49, all other frames agreeing exactly. Two independent implementations (C++ with Burnside counting, Python with path canonicalisation) agree on 57 / 94 / 14. The published per-frame totals for 3 x 4 plus 3 x 5 (19 boards, 28 tours), 4 x 5 (11 boards, 16 tours) and 5 x 5 (1 / 1) agree with the claim.
trivial (1285 free 9-ominoes; seconds in Python).
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): e81f9c541355576dff79c96cd137a5373cceabb024848afcae80fa0f8223b9ac. Published 2026-10-08 under claimant/claim07_knight_9cell_correction/ (commitment recomputed and matched before publishing; see claimant/README.md).
Correction to the published count of open-tourable 9-cell boards. Claim 7, pursekeeper claims pilot, verified 2026-09-17 by independent re-derivation. https://pursekeeper.dev/verified/07-nine-cell-open-tour-boards-correction Record snapshot 3be023cdbbb6, statement sha256 0cc2a9f44176adac1a76c1b3fcf6112edb8a3cda32a9dbbaccb3489083bdd072.
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:7. 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 0cc2a9f44176adac1a76c1b3fcf6112edb8a3cda32a9dbbaccb3489083bdd072 over the two sections above, "Statement" and "What a re-derivation must output to count", as they stand in claims/07-nine-cell-open-tour-boards-correction.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 950c1b77b099fe9b2943054b01dde3bb57f1efca09e6ee643a549b0a84ce8ded. The two sections are byte-identical to the body of issue #5, which is the text the reviewers read; checked 2026-10-09, and the script's --check-issue flag repeats the comparison.
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