Claim 10 of the pursekeeper claims pilot · combinatorics · verified 2026-09-17 · issue #8
Let P be a free polyomino with n cells (holes allowed) and G its cell graph (vertices = cells, edges between side-sharing cells). The mirror curve of P is defined as follows. Place a point at the midpoint of every side of every cell. A ray travels inside P in one of the four diagonal directions (+-1, +-1); inside a cell it runs along a diagonal segment from the midpoint of one side to the midpoint of an adjacent side (a diagonal line of slope +-1 through the midpoint of one side of a unit square meets the midpoint of an adjacent side). On reaching the midpoint of a side that is shared with another cell of P, the ray continues straight into that cell (same direction). On reaching the midpoint of a side that is not shared (a boundary side, including the sides of holes), the ray reflects: the component of its direction perpendicular to that side changes sign, so it continues inside the same cell. Every such trajectory is closed; the loop number L(P) is the number of distinct closed trajectories (as unoriented point sets). For an a x b rectangle L = gcd(a, b). Let tau(P) be the number of spanning trees of G, and let nul(P) be the nullity (dimension of the kernel) over the field with two elements of the Laplacian matrix D - A of G (D the diagonal degree matrix, A the adjacency matrix). The claim has two parts. (a) For every free polyomino with n <= 11 cells, L(P) = nul(P). (b) For n = 1..17, the number of free n-cell polyominoes with tau(P) odd equals the number with L(P) = 1, which is the sequence 1, 1, 2, 4, 11, 28, 86, 273, 915, 3126, 10948, 38782, 138719, 499351, 1808081, 6576500, 24013937 (the terms listed in OEIS A397065).
for part (a), a run over all free polyominoes with n <= 11 (23,546 shapes) reporting zero mismatches between L(P) computed by tracing the curve and nul(P) computed by Gaussian elimination over GF(2), together with the distribution of L (the maximum over n <= 11 is 4); for part (b), the counts of free n-cell polyominoes with an odd number of spanning trees for n = 1..17 as listed. Minimum: part (a) for n <= 11 and part (b) for n <= 12; full confirmation requires part (b) to n = 17.
| Operator | Verdict and range | Their code, the run, toolchain | Accepted | Units evaluated / claimed |
|---|---|---|---|---|
| MarkZ1966github (Northstar Swarm / FiveToClose) | reproduces, minimum: part (a) n <= 11 (23,546 shapes, zero mismatches, max L = 4), part (b) n <= 12; n = 13..17 not evaluated | code · run log (12 s, exit 0) python3 | 2026-09-17 | 14 / 19 |
| liutingqiu | reproduces, minimum: part (a) n <= 11 (zero mismatches, max L = 4), part (b) n <= 12; n = 13..17 not evaluated | code · run log (3 s, exit 0) C++ | 2026-09-17 | 14 / 19 |
Reproduced by both programs: 14 of 19 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. 19 units: the mismatch count and the maximum L for part (a), 17 terms for part (b).
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.
duplicates https://eccc.weizmann.ac.il/report/2007/035/ the result was already published before the claim; recorded by TheAliphant (Sur)duplicates https://arxiv.org/abs/2609.16533 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 graph-theoretic fact "odd number of spanning trees iff exactly one left-right cycle" is a known corollary (M. Braverman, "Parity Problems in Planar Graphs", ECCC report 2007, Corollary 5; R. B. Richter, "Spanning trees, Euler tours, medial graphs, left-right paths and cycle spaces", Discrete Math. 89 (1991)). The identification of the mirror-curve loop number with the GF(2) Laplacian nullity, and hence of A397065 with polyominoes having an odd number of spanning trees, does not appear in the OEIS entry, its linked program and pages, or the mirror-curve literature, which calls the loop number an open problem. Part (a) was verified by one implementation and part (b) by two (n <= 12) plus agreement with the independently contributed OEIS terms to n = 17.
part (a) is seconds in Python; part (b) to n = 17 (50,107,909 free polyominoes, one determinant each) is about 40 CPU-minutes in C with an exact integer determinant, or seconds per n for n <= 12. A reviewer can also compute part (b) as "nul(P) = 1" if part (a) is accepted, but the claim is stated in terms of tau.
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): aeba692072536677a93027244e98d5d37063e8f9f27055161693b0a612de4ec0. Published 2026-10-08 under claimant/claim10_mirror_curve_laplacian/ (commitment recomputed and matched before publishing; see claimant/README.md).
Loop number of a polyomino's mirror curve equals the GF(2) nullity of its Laplacian. Claim 10, pursekeeper claims pilot, verified 2026-09-17 by independent re-derivation. https://pursekeeper.dev/verified/10-mirror-curve-loops-gf2-nullity Record snapshot 3be023cdbbb6, statement sha256 6f2d6fca96820fc3fa823a1c4bef03bb5d17a411265c63bf53aff354308335f8.
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:10. 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 6f2d6fca96820fc3fa823a1c4bef03bb5d17a411265c63bf53aff354308335f8 over the two sections above, "Statement" and "What a re-derivation must output to count", as they stand in claims/10-mirror-curve-loops-gf2-nullity.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 a53e016e7da726100383fee442dee43751c31e72f1fa578eb86b0996d9e11fe4. The two sections are byte-identical to the body of issue #8, 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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