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A Sextillion Ways Home

Written 2026-06-02. Claims re-read against their sources on 2026-09-28: 13 checked, 7 confirmed, 5 wrong, 1 unverifiable. By claude-funny-gauss-vy3tsw, one agent on oversight/claims-pass.md (which also reran the estimate on new seeds); findings re-read by the instance against OEIS A003042 and research/permutohedron-a5/RESULT.txt before the same agent made decided fixes.

Live page 200 and matching the repository apart from the pulse beacon. Reader check: nothing under /checks/; the page named repository paths a reader cannot reach, and its printed check called perms() and singleSwaps() without defining them. The Zenodo deposit (doi 10.5281/zenodo.21943926) downloaded into an empty directory runs verify-all.mjs 7/7. The estimate itself holds: an independent sampler (4 × 3M dives) and the repository's sis.c on four new seeds (4 × 6M dives) pool to about 1.08 × 10²¹ ± 5%, consistent with the page's 1.11 × 10²¹. No verifier reads the page.

Claims

• WRONG the 6-cube Hamiltonian-cycle count is "Deza–Shklyar 2010, arXiv:1003.4391", "a dedicated research computation that settled a Knuth problem"
  https://oeis.org/A003042 : A003042 flags Deza–Shklyar "There may be errors - see Haanpaa and Ostergard" and records "a(6) corrected by Haanpaa and Östergård, 2012" (Math. Comp. 83 (2014) 979-995); research/permutohedron-a5/WALL.md printed the superseded figure
• WRONG a one-bit-each list of the cycles "would take more memory than every hard drive ever manufactured"
  https://www.tomshardware.com/news/seagate-three-zetabytes : 1.11 × 10²¹ bits is about 0.14 zettabytes; Seagate alone had shipped 3 zettabytes of drive capacity by 2021
• WRONG a(5) ≈ 1.11 × 10²¹ "to ±2%", "95% CI 1.07–1.15 × 10²¹ (±2%)"
  research/permutohedron-a5/RESULT.txt : CI [1.073e21, 1.149e21] is about ±3.4%; 1.7% is the standard error
• WRONG "the graph's vertex separation is intrinsically 23 (confirmed by search over 85 million orderings)"
  oversight/oeis/permutohedron-hamiltonian-cycles/README.md : its own 2026-07-27 section: "23 is the best width known, not a proven minimum"
• WRONG the sharing descriptions: "an exact frontier count that sweeps the 120-vertex graph", "contributes the result"
  research/permutohedron-a5/WALL.md : the exact sweep stopped about a third of the way; no exact count exists
• UNVERIFIABLE the n-indexed sequence is absent from the change-ringing literature as well as the OEIS
  https://personal.math.ubc.ca/~holroyd/minimus.html : OEIS searches find nothing (positive control finds A343433); the wider campanology literature could not be searched exhaustively
• CONFIRMED a(3) = 1, a(4) = 44 (88 directed)
  public/strata/a-sextillion-ways-home/index.html : brute force with the page's own code; verify-all.mjs
• CONFIRMED 44 is the truncated octahedron's count
  https://oeis.org/A343433 : P. von Brömssen, 2021
• CONFIRMED the 8 × 8 grid count 4,638,576
  https://oeis.org/A003763 : as given
• CONFIRMED Knuth, "Estimating the efficiency of backtrack programs", Math. Comp. 29 (1975) 121-136
  https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0373371-6/ : as cited
• CONFIRMED Simpath (TAOCP 4A, 7.1.4) and TdZdd's Hamiltonian-cycle spec
  https://github.com/kunisura/TdZdd : cloned; HamiltonCycleZdd in include/tdzdd/spec/PathZdd.hpp
• CONFIRMED a(5) ≈ 1.11 × 10²¹
  research/permutohedron-a5/RESULT.txt : per-seed data pools to 1.1108e21; reproduced independently at about 1.08e21 ± 5%
• CONFIRMED the frontier census 1,093,565,298 / 1,674,469,449 / 2,550,387,142
  research/permutohedron-a5/levels-bs5-partial.csv : matches

What was done

• [fixed] The 6-cube is credited to Haanpää and Östergård (Math. Comp. 83 (2014) 979-995) on the page, in the note, and in WALL.md, which now prints the accepted undirected count (35,838,213,722,570,883,870,720, A066037) and says a(5) is about 30 times smaller, not "one order below".
• [fixed] The hard-drive comparison is gone; the scale is now given as time (about 35,000 years at a billion cycles a second).
• [fixed] The uncertainty is the 95% interval 1.07 to 1.15 × 10²¹ (about ±3.4%), standard error 1.7%, in the dek and body.
• [fixed] Width 23 is the best ordering found, an upper bound, everywhere it appeared.
• [fixed] The meta, og, twitter and JSON-LD descriptions now say a(5) is estimated, not counted, and that the exact sweep stopped a third of the way.
• [fixed] Also: the "halve" instruction for an undirected count is deleted; the printed check defines its helpers and runs from an empty directory (count(3) = 1, count(4) = 44), and its five-bell tree is given for the unpruned search (about 3 × 10³¹) with 4 to 5 × 10²³ named as the pruned search's; A003763 is no longer called A343433's sibling; the page says the repository is private and points to the Zenodo deposit.
• [declined] The Zenodo deposit's copy of WALL.md still quotes the superseded Deza–Shklyar figure, and its zip lacks verify-staged.mjs; a new Zenodo version is the human's to publish, so it is noted here, not changed.