Claim 14 of the pursekeeper claims pilot · graph theory · verified 2026-09-18 · issue #10
Model the honeycomb (hexagonal) lattice as the graph whose vertices are the points (a, b, c) of Z^3 with a + b + c in {0, 1}, two vertices adjacent when they differ in exactly one coordinate by exactly 1; this is the 3-regular bipartite honeycomb graph. For a finite vertex set S consider the induced subgraph H[S]. The claim has two parts. (a) For every S with |S| <= 36 such that H[S] has a Hamiltonian cycle, H[S] has exactly one Hamiltonian cycle (as an undirected cycle). (b) The set S38 of 38 vertices (0,0,0) (1,0,0) (1,0,-1) (2,0,-1) (2,0,-2) (2,1,-2) (2,1,-3) (2,2,-3) (1,2,-3) (1,3,-3) (1,3,-4) (1,4,-4) (0,4,-4) (0,5,-4) (-1,5,-4) (-1,5,-3) (-2,5,-3) (-2,5,-2) (-2,4,-2) (-2,4,-1) (-2,3,-1) (-1,3,-1) (-1,3,-2) (-1,4,-2) (-1,4,-3) (0,4,-3) (0,3,-3) (0,3,-2) (0,2,-2) (1,2,-2) (1,1,-2) (1,1,-1) (0,1,-1) (0,2,-1) (-1,2,-1) (-1,2,0) (-1,1,0) (0,1,0) induces a subgraph with 49 edges (16 vertices of degree 2, 22 of degree 3) that has exactly two Hamiltonian cycles, and up to the symmetries of the lattice S38 is the only 38-vertex set with more than one Hamiltonian cycle. Equivalently, in terms of polyiamonds: every polyiamond with at most 36 cells whose inner dual is Hamiltonian has a unique Hamiltonian cycle, and the 38-cell polyiamond corresponding to S38 is the smallest with two.
for part (b), the numbers 49 (edges), the degree multiset, and 2 (undirected Hamiltonian cycles of H[S38]); for part (a), an enumeration (of polyiamonds with a Hamiltonian inner dual, or of honeycomb self-avoiding polygons and their vertex sets) reporting that the maximum number of Hamiltonian cycles over all Hamiltonian vertex sets is 1 for every size up to 36, and that exactly one vertex set (up to symmetry) at size 38 has more than one. Minimum: part (b) in full and part (a) up to 30 vertices; full confirmation requires part (a) to 36 and the uniqueness at 38. The vertex-set counts by size that arise on the way (sizes 24..36: 97, 312, 744, 2291, 6186, 18714, 53793) may be reported for comparison with A258206 but are not part of the pass criterion.
| Operator | Verdict and range | Their code, the run, toolchain | Accepted | Units evaluated / claimed |
|---|---|---|---|---|
| liutingqiu | reproduces, minimum: part (b) in full (49 edges, 16+22 degrees, 2 cycles), part (a) max 1 cycle through 30 vertices; 36 and uniqueness at 38 not evaluated | code · run log (2 s, exit 0) C | 2026-09-17 | 3 / 5 |
| TheAliphant (Sur) | reproduces, minimum: part (b) in full (49 edges, 16+22 degrees, 2 cycles), part (a) max 1 cycle through 30 vertices; 36 and uniqueness at 38 not evaluated | code · run log (30 s, exit 0) C++ | 2026-09-18 | 3 / 5 |
Reproduced by both programs: 3 of 5 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. 5 named outputs: 3 for part (b), 2 for part (a); part (a) to 30 vertices is a partial of one output and does not count.
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 and are written in different languages, 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 uniqueness phenomenon and its first failure at 38 vertices were found in the original run (one implementation verified uniqueness to 30 vertices, a second independently to 36 and found S38); the 38-vertex example was re-verified with a separate embedding and unpruned cycle count. No statement of either fact was found in OEIS or the literature; the question appears not to have been asked.
part (b) is instant; part (a) to 36 vertices requires enumerating honeycomb self-avoiding polygons to length 36 (or polyiamonds to 36 cells, which is heavier) and testing cycle counts only for vertex sets with chords, about 10-15 minutes in C.
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): 8e64caf0e880cba217dce6b54d6c0fcdf35816ee00b4be060c8096d8a8b9f299. Published 2026-10-08 under claimant/claim14_honeycomb_unique_hamiltonian_cycle/ (commitment recomputed and matched before publishing; see claimant/README.md).
Hamiltonian induced subgraphs of the honeycomb lattice with more than one Hamiltonian cycle. Claim 14, pursekeeper claims pilot, verified 2026-09-18 by independent re-derivation. https://pursekeeper.dev/verified/14-honeycomb-unique-hamiltonian-cycle Record snapshot 3be023cdbbb6, statement sha256 61db5c9a250e1e305f8779f8a60cfe0f631ab3a9e58bca90e3a800b9dd67e971.
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:14. 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 61db5c9a250e1e305f8779f8a60cfe0f631ab3a9e58bca90e3a800b9dd67e971 over the two sections above, "Statement" and "What a re-derivation must output to count", as they stand in claims/14-honeycomb-unique-hamiltonian-cycle.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 634bac921e54a9e1272eb4d0fdccca28ba7dad35a591610a8c69567298e99c36. The two sections are byte-identical to the body of issue #10, 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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