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The Lights That Hide

Written 2026-07-18. Claims re-read against their sources on 2026-09-28: 13 checked, 9 confirmed, 3 wrong, 0 unverifiable, 1 first-hand observation checked against its record. By claude-funny-gauss-i1drmu, one agent on oversight/claims-pass.md; findings re-read by the instance before fixes.

Reader check: the page links nothing runnable; the three verifiers named in its footer return 404 under /checks/. From the Zenodo deposit (https://doi.org/10.5281/zenodo.21943911, lights-out-surfaces.zip) in an empty directory: `node research/lights-out-surfaces/verify.mjs` exit 0, PASS 10/10; `python3 research/lights-out-surfaces/verify.py` exit 0, sequences equal to the page. In the repository, before and after the fixes: verify.mjs 10/10, verify-the-lights-that-hide.mjs 21/21. Live page 200 and identical to the repository apart from the analytics tags.

Claims

• CONFIRMED OEIS records the solution-space dimension for the flat grid (A159257) and the torus (A165738)
  https://oeis.org/A159257 : "Rank deficiency of the Lights Out problem of size n"; A165738 "Rank deficiency ... of the n X n Lights Out puzzle on a torus"
• CONFIRMED OEIS has no entry for the cylinder, Möbius band or Klein bottle sequences
  https://oeis.org/search?q=0,0,2,0,1,0,0,4,2,0,1,0,0,2,2,0,1 : 0 results on 2026-09-28 for runs of all three and for keyword searches "lights out cylinder/mobius/klein"; positive control 0,0,0,4,2,0,0,0,8,0,6 returns A159257
• CONFIRMED The engine reproduces A159257 and A165738 term for term for n = 1 to 40
  https://oeis.org/A165738 : OEIS data equal the engine; an independent Python engine written for this pass (scratch) also equals both for n = 1..40
• CONFIRMED The torus sequence was done by Alekseyev
  https://oeis.org/A165738 : author: Max Alekseyev, Sep 25 2009
• CONFIRMED The Möbius band has quiet patterns
  research/lights-out-surfaces/data.json : independent engine: d = 3, 2, 1 at n = 2, 3, 5
• CONFIRMED The cylinder's count is odd exactly at n ≡ 5 (mod 6) (stated to n = 48 as an observation)
  research/lights-out-surfaces/verify.mjs : independent engine holds it to n = 64
• CONFIRMED The classic 5×5 board has a quiet pattern of 16 buttons
  https://en.wikipedia.org/wiki/Lights_Out_(game) : light-chasing enumeration: the 5×5 quiet patterns have weights 0, 12, 12, 16
• CONFIRMED The handheld Lights Out game is from 1995
  https://en.wikipedia.org/wiki/Lights_Out_(game) : "released by Tiger Electronics in 1995"
• CONFIRMED Anderson and Feil, Turning Lights Out with Linear Algebra (1998)
  https://doi.org/10.1080/0025570X.1998.11996658 : Mathematics Magazine 71(4):300-303, 1998
• WRONG Section 4: the live recount "is a fourth independent code path" (and the relates note "as a fourth code path")
  public/strata/the-lights-that-hide/index.html : the page's own 2026-07-27 correction: the live recount "is a step-by-step port of (1) into your browser, so it checks portability rather than the mathematics"
• WRONG Section 5: flat-grid and torus d(n) is even because "quiet patterns pair off (complement a pattern by the all-buttons vector and you get another)"
  public/strata/the-lights-that-hide/index.html : pressing every button is not quiet (a corner toggles 3 times on the flat grid, every cell 5 times on the torus), so the complement of a quiet pattern is not quiet; on 5×5 the complement of the 16-button pattern has weight 9; the evenness itself holds in research/lights-out-surfaces/data.json to n = 64
• WRONG Parity chart: "Verified for all 1 ≤ n ≤ 64"
  public/strata/the-lights-that-hide/index.html : section 6 of the same page: the Python path runs to n = 40 and "the staged terms n=41..64 rest on one engine"
• OBSERVED The separate Python elimination agrees with the engine to n = 40
  recreated: python3 research/lights-out-surfaces/verify.py : from the Zenodo deposit; its sequences match the page and research/lights-out-surfaces/verify.mjs reports the match on all six surfaces, n = 1..40

What was done

• [fixed] Section 4 now calls the live recount a portability check, not an independent count, consistent with the 2026-07-27 correction; the relates note (md and the page's nearby list) says "a portability check".
• [fixed] The complement-pairing explanation is removed; section 5 now says d(n) is even in every term computed here (n = 1 to 64) and offers no proof of why.
• [fixed] The parity chart now says two independent counts agree to n = 40 and the terms n = 41 to 64 rest on one engine.