Corroboration · An audit of the record
The Classification RecordThe paper trail behind the Classification of Finite Simple Groups, censused with respect
The vanishing hedge
The record has a second life beyond the bibliographies: every time someone restates the theorem. We graded seventeen such restatements, from 1980 to 2024, against what was known to be open at the time. The question was simple: when a source says the classification is complete, does it carry the basis with it, or drop it.
Candour is not uniform, and it is not random. It concentrates where experts address experts and thins as the audience widens. In the Notices and the Bulletin the principals are precise and dated. Aschbacher's own 2004 statement carried the hedge "(for the moment)". By the time a general-audience venue compresses that event to a single line, the hedge is gone: not through anyone lying, but through summarisation. A claim with its basis attached becomes a bare assertion after one round of compression.
Where a source falls is graded on a plain ladder: candid (the basis is stated), stale (the basis was once stated and has since frozen out of date), or silent (the claim stands alone). Compliance is possible in one sentence; Aschbacher's 2004 statement is the model.
Each dot is one restatement of the theorem, placed by year and audience, coloured by whether it carried the basis. Candour thins as the audience widens.
| Source | Year | Audience | Grade |
|---|---|---|---|
| Collins (ed.), Finite Simple Groups II | 1980 | Textbooks | Candid |
| Gorenstein, Finite Simple Groups (preface) | 1982 | Textbooks | Silent |
| Feit, Bulletin of the AMS | 1983 | Experts to experts | Candid |
| Gorenstein, Scientific American | 1985 | Mass audience | Silent |
| Serre, Mathematical Intelligencer | 1986 | Experts to experts | Candid |
| Solomon, Notices of the AMS | 1995 | Experts to experts | Candid |
| Robinson, A Course in the Theory of Groups | 1996 | Textbooks | Candid |
| Maróti, J. Algebra | 2002 | Downstream use | Silent |
| Berry–Dubickas–Elkies–Poonen–Smyth, arXiv | 2003 | Downstream use | Silent |
| MathWorld (Wayback 2000 & 2004) | 2004 | Reference works | Silent |
| Wilson, The Finite Simple Groups | 2009 | Textbooks | Silent |
| nLab | 2018 | Reference works | Stale |
| Encyclopedia of Mathematics | 2020 | Reference works | Stale |
| Granville, arXiv 2305.02115 | 2023 | Experts to experts | Candid |
| MacTutor (St Andrews) | 2024 | Reference works | Candid |
| Quanta Magazine | 2024 | Mass audience | Silent |
| Wikipedia (accessed 2026) | 2026 | Reference works | Stale |
The second-generation write-up (the GLS project) has its own moving target, and it has slipped for two decades without a public correction anywhere we could find.
| Stated | The estimate | What happened |
|---|---|---|
| 2001 | the quasithin volumes "will probably be published in ... 2001 or 2002" (Solomon, Bulletin of the AMS) | published November 2004 |
| 2018 | roughly twelve volumes, by 2023 (stated in the Notices) | volume 10 appeared in 2023; the estimate expired unrevised |
| 2020 | expected to be completed by 2025 (Solomon, in a private email later quoted in a public paper) | 2025 reached; the project is open |
Michael Collins (ed.), preface to Finite Simple Groups II (Academic Press, 1980; Durham 1978 proceedings) · 1980
The classification "is now almost complete".
Basis carried: "Almost", accurate for 1980, before the announcement.
Pre-announcement baseline. The unqualified claims that followed were, oddly, less accurate than this earlier one.
reported, not independently verifiedDaniel Gorenstein, preface to Finite Simple Groups: An Introduction to Their Classification (Plenum, 1982) · 1982
"In February 1981, the classification of the finite simple groups was completed."
Basis carried: None.
An unpublished draft of the same preface said "In August, 1980", two unconditional completion dates, same author, same book. Silent on a knowable unpublished dependency (Mason's manuscript).
reported, not independently verifiedWalter Feit, review of Gorenstein's Finite Simple Groups, Bulletin of the AMS 8(1), January 1983 · 1983
The finite simple groups "have apparently been classified".
Basis carried: Hedges on the corpus itself: "No individual has gone through the whole proof and checked all the details. This is not an entirely satisfactory situation."
The best-behaved in-window claim found. Hedges on verifiability, though not on the specific gap, which was not knowable in 1983.
Daniel Gorenstein, "The Enormous Theorem", Scientific American 253(6), December 1985 · 1985
The finite simple groups are the 18 families and 26 sporadic groups "and no others!", the central problem "solved".
Basis carried: None. Mason is not named anywhere in the piece.
The widest-reaching completeness claim of the window, zero caveat, to the largest audience.
Jean-Pierre Serre, interview, Mathematical Intelligencer 8(4), 1986 · 1986
Asked whether he believed in the classification: "More or less — and rather more than less."
Basis carried: Relaxed, pre-gap.
Baseline. Pre-1989 Serre was relaxed; his 2004 sharpness was a response to the discovered gap, not a standing objection.
Ronald Solomon, "On Finite Simple Groups and Their Classification", Notices of the AMS 42(2), February 1995 · 1995
A vast literature of theorems ... combines to yield the classification, stated in full, with the debt disclosed in the same article.
Basis carried: Extensive and specific: names Mason's unpublished 800-page quasithin typescript, the 1989 discovery that subcases remained untreated, the 1992 Aschbacher patch distributed but unpublished, and two Stellmacher papers "also remain unpublished".
The exemplar. Corrects a piece of folklore: the gap was made a considerable deal of, in print, in 1995, not quietly left.
Derek J. S. Robinson, A Course in the Theory of Groups, 2nd ed. (Springer GTM 80, 1996) · 1996
"It is now generally believed that the classification ... is complete."
Basis carried: "However a complete proof has not yet been written down, and it would probably extend to several thousand printed pages."
Carry this loud: a mainstream textbook, in-window, got it exactly right. "Textbooks overclaimed" is false as a blanket claim; the textbook layer is mixed.
Attila Maróti, J. Algebra 258 (2002) · 2002
"In this paper we use the classification of finite simple groups to set the sharpest upper bounds possible ..."
Basis carried: None, flat tool-use, two years before the 2004 fix.
Black-box class. Expected behaviour; the community's own later "CFSG-free" badge convention is the answer to exactly this.
Berry, Dubickas, Elkies, Poonen, Smyth, arXiv math/0308069 (2003) · 2003
"... this result depends on the classification of finite simple groups."
Basis carried: None, flat dependency, one year before the fix.
Black-box class.
MathWorld, "Classification Theorem of Finite Groups", Wayback snapshots 31 May 2000 and 17 June 2004 · 2004
The finite simple groups "can be classified completely", identical wording in both snapshots, including the one five months before the Aschbacher–Smith fix.
Basis carried: None in either snapshot.
The star specimen: MathWorld's own reference list cites Solomon 1995, the document carrying the explicit caveat. The caveat was one citation-click away for at least four years and was not carried through. Not deception; non-transmission.
Robert Wilson, The Finite Simple Groups (Springer GTM 251, 2009) · 2009
With Aschbacher–Smith 2004, "we can reasonably regard the proof ... as complete."
Basis carried: A trace, in "reasonably regard"; no mention of the ongoing GLS write-up in the recovered passage.
The standard current graduate text. Passage reported from corroborated snippets; a clean library copy is the verify-at-source item.
reported, not independently verifiednLab, "classification of finite simple groups", last revised 27 March 2018 · 2018
"As of 2018 seven volumes had been published, out of an expected 11."
Basis carried: Carries the write-up status, but frozen since 2018, with a volume count and expected total now both wrong.
Right instinct, no maintenance.
Encyclopedia of Mathematics, "Simple finite group", last modified 6 June 2020 · 2020
"Although, as of 1990, some parts of the full proof have not yet appeared in official journals, the classification ... has been commonly accepted ever since 1982."
Basis carried: A 2020-stamped page carrying 1990-era phrasing; no 2004 fix, no GLS.
The phrasing concedes the whole point in passing: "commonly accepted ever since 1982" while "some parts ... have not yet appeared".
Andrew Granville, "Accepted Proofs: Objective Truth, or Culturally Robust?", arXiv 2305.02115, May 2023 · 2023
Records Aschbacher's own remark that at times he has believed the classification complete, at other times not, and is always certain there are minor errors throughout.
Basis carried: Preserves the oscillation, and notes the proof has not been computer-verified.
Current scholarly transmission that keeps the nuance intact.
MacTutor History of Mathematics, classification pages (updated March 2024) · 2024
Transmits Aschbacher's 2004 statement verbatim, including "(for the moment)" and "the GLS program is not yet complete".
Basis carried: Full and intact, because it quotes at length.
The hedge survives here precisely because the venue did not compress. It is 2004 wording redisplayed, not a fresh 2024 assessment.
Quanta Magazine, "'Groups' Underpin Modern Math ...", 6 September 2024 · 2024
"A new proof was finally published in 2004, finishing off the classification."
Basis carried: None, no GLS mention, hedge absent.
The freshest mass-audience specimen. The claim-with-basis defect, manufactured by compression, still happening in 2024.
Wikipedia, "Classification of finite simple groups" (live article, accessed July 2026) · 2026
The 2004 fix narrated flatly as "filling the last gap ... known at that time". GLS incompleteness is stated, but via a 2012 estimate.
Basis carried: GLS status present but stale (a 2012 estimate, no mention of any expired target). Aschbacher's "(for the moment)" hedge: full-text-searched and absent.
The most-read reference of all. Status is present but frozen; the hedge is gone under compression.