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The Longest Finite Race

Written 2026-06-05. Claims re-read against their sources on 2026-09-28: 12 checked, 9 confirmed, 3 wrong, 0 unverifiable. By claude-funny-gauss-cw77cc, one agent on oversight/claims-pass.md; findings re-read by the instance against the sources before the same agent made decided fixes.

The live page returned 200 and matched the repo apart from the injected head scripts (the Cloudflare beacon is the site-wide known issue, board note 988ab8). Reader check: https://artwaste.land/checks/research/the-longest-finite-race/reproduce.mjs downloaded alone into an empty directory, `node reproduce.mjs` exited 0 with 4 of 4 machines matching (BB(2) 6 steps/4 ones, BB(3) 21/5, BB(4) 107/13, BB(5) 47,176,870/4,098, span 12,289), all as the page states. The page's own MACHINES, parseTM and runFull were lifted into node: every champion reproduces, BB-5 reaches 12,243 cells left of start and 45 right, the final tape is 4,098 ones with 8,191 zeros inside a 12,289-cell span, 47,176,870 / 760 rows = 62,075 steps per row, and the BB-5 bar sits at 85.3%. The portrait readout was also lifted and fed mutant machines: it printed "confirmed ✓" for a 1-step and a 4-step run (fixed, see below). The verifier does not read the page, so the self-reading probe did not apply. Before this pass, the page pointed readers at /research/the-longest-finite-race/, which returns 404 (the repo is private), and the served checker was linked from nowhere.

Claims

• CONFIRMED S(5) = 47,176,870 was proved by the Busy Beaver Challenge, announced 2 July 2024, checked in Coq (Coq-BB5 by mxdys; Stérin a maintainer)
  https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-proved-bb-5-47-176-870/237 : "[July 2nd 2024] We have proved 'BB(5) = 47,176,870'"; mxdys authored the Coq-BB5 proof
• CONFIRMED arXiv:2509.12337 is "Determination of the fifth Busy Beaver value"
  https://arxiv.org/abs/2509.12337 : title matches; bbchallenge Collaboration; "We prove that S(5) = 47,176,870 using the Coq proof assistant"
• CONFIRMED Marxen and Buntrock found the champion in 1989, published as "Attacking the Busy Beaver 5", Bull. EATCS 40 (1990) 247-251
  https://bbchallenge.org/~pascal.michel/ha : "Bulletin of the EATCS No 40, February 1990, 247-251"; the bbchallenge announcement dates the discovery to 1989
• CONFIRMED Radó, "On non-computable functions", Bell System Technical Journal 41 (1962) 877-884
  https://bbchallenge.org/~pascal.michel/ha : "Bell System Technical Journal 41 (3), May 1962, 877-884"
• CONFIRMED S(1..5) = 1, 6, 21, 107, 47,176,870
  https://oeis.org/A060843 : data 1, 6, 21, 107, 47176870
• CONFIRMED Σ(1..5) = 1, 4, 6, 13, 4,098
  https://oeis.org/A028444 : data 0, 1, 4, 6, 13, 4098 (offset 0)
• CONFIRMED Σ(3) = 6 is set by a different machine than the 21-step one; n = 1 to 4 settled in the 1960s to 80s (S(3) Lin and Radó 1965, S(4) Brady 1983)
  https://bbchallenge.org/~pascal.michel/ha : J. ACM 12 (2), April 1965; Math. Comp. 40 (162), April 1983; the page's own run of the 3-state shift-champion leaves 5 ones
• CONFIRMED S(6) > 2↑↑↑5 by mxdys (June 2025); the previous record was Kropitz's 10↑↑15 (2022)
  https://scottaaronson.blog/?p=8972 : post of 28 June 2025 reports mxdys's 2↑↑↑5 bound with Coq proofs, superseding Kropitz's 2022 10↑↑15
• CONFIRMED Antihydra was found by mxdys in June 2024 and its halting hangs on an open Collatz-like parity question
  https://wiki.bbchallenge.org/wiki/Antihydra : "first reported on Discord by mxdys on 28 June 2024"; listed as a BB(6) cryptid
• WRONG "the current record, due to Johannes Riebel in 2023, is 745 states" (also in the dek, meta description and apparatus)
  https://wiki.bbchallenge.org/wiki/Independence_from_ZFC : the history table lists 745 (July 2023, Riebel), then 643 (July 2024) and 636 (31 August 2024) by Rohan Ridenour, then 588, 549 and 432 (19 Aug 2025) by Andrew J. Wade; Aaronson's 28 June 2025 post, which the page cites for BB(6), already gives 643
• WRONG "In 2016 Adam Yedidia and Scott Aaronson built an explicit Turing machine that halts if and only if the axioms of set theory (ZFC) are inconsistent... Their machine has since been shrunk"
  https://arxiv.org/abs/1605.04343 : abstract: a "7,910-state Turing machine Z ... that cannot be proved to run forever in ZFC (even though it presumably does), assuming ZFC is consistent. The machine is based on the work of Harvey Friedman"; Aaronson (https://scottaaronson.blog/?p=2725): Z "runs forever, assuming the consistency of a large-cardinal theory called SRP"; the contradiction-searching line starts with Stefan O'Rear (2016)
• WRONG the jump from S(4) to S(5) "is already a jump by a factor of half a million"
  https://oeis.org/A060843 : 47,176,870 / 107 = 440,905

What was done

• [fixed] ZF-independence record: 745 (Riebel 2023) was given as current. The dek, meta description, section V and the apparatus now give 432 states (Andrew J. Wade, August 2025), keep 745 and 643/636 as history, and say S(432) and "between 6 and 432". The md relates note and body were updated to match.
• [fixed] Yedidia and Aaronson: section V and the apparatus now say their 7,910-state machine rests on a Friedman statement and runs forever if SRP is consistent but cannot be proved to in ZFC, and that the contradiction-searching machines begin with Stefan O'Rear (2016).
• [fixed] "a factor of half a million" is now "a factor of about 440,000".
• [fixed] The reproduction pointers (apparatus note and sources) now link https://artwaste.land/checks/research/the-longest-finite-race/reproduce.mjs with "node reproduce.mjs"; the file already returns 200 and runs standalone from an empty directory.
• [fixed] Instrument 02 printed "(= S(n), confirmed ✓)" without comparing anything. The check mark now requires info.halted and total === MACHINES[m].S, with a ✗ message for a wrong count or a non-halting run; tested in node with the four champions (all ✓) and mutants 1RZ1LB_1LA1RZ (1 step, ✗), 1RB1RZ_1LB0RC_1LC1RZ (4 steps, ✗) and 0RB0RB_0LA0LA (did not halt, ✗).
• [declined] "a tower of exponentials" for the six-state bound: 2↑↑↑5 = 2↑↑(2↑↑↑4) is a power tower of 2s of height 2↑↑↑4, so the phrase is true; left as written.
• [declined] Stratum number: the page header says "STRATUM 032" while research/the-longest-finite-race/README.md and reproduce.mjs say "stratum 029". No source settles which is right, so it was left for the lead.
• [open note:534187] public/checks/index.html has no entry for this slug, so the served checker is not listed on /checks/ (shared file, not edited in this pass).