Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 118Settled

For which conjugacy classes pX of elements of prime order p in the Monster is the centraliser C_M(pX) a direct product p × S with S a sporadic simple group, and is p = 11 the only such prime whose S is a Mathieu group?

run_kind
reading
slug
prime-centralisers-of-the-monster
state
run
created
2026-10-23
sealed
2026-10-23 (committed before any reader is briefed)
builds_on
[116]
concepts
[classification, sporadic, centraliser, reading-run]
programme
PG-04
line
C1 — why M₁₂ (first guess)
from
run 116's reading (M₁₂ the single group on two branches of the happy family's forest) and guess 1 of the pre-registration "why M₁₂" (docs/walk/2026-10-21-pre-registration-symmetry.md, section of 2026-10-23, 09:45), committed before this seal; the owner's ruling of 2026-10-23, 09:48 ("get next steps going").
statistic
A reading run. Three readers, independent, each an agent with web access given the same one input (the fixed prompt below) and nothing else; none sees this file, the programme, the pre-registration, or another reader. Each returns, for every prime p dividing |M| and every class of elements of order p, the structure of C_M(pX) with a source (the ATLAS of Finite Groups, the GAP character table library, or a paper that tabulates the Monster's classes). The codifier records, per reader, the set P = {(p, X, S)} of classes whose centraliser the reader gives as p × S with S sporadic simple, and takes the majority (≥ 2 of 3) per class. Scored: the set of primes appearing in the majority P; which S at each. Reported beside: every centraliser each reader gives, disagreements between readers, sources reached (pages or summaries).
scored
2026-10-23
outcome
confirmed
outcome_note
by the prediction as written — majority P = {3C: 3 × Th, 5A: 5 × HN, 7A: 7 × He, 11A: 11 × M₁₂}, primes {3, 5, 7, 11}, a Mathieu group only at 11; controls pass. The confirmation rests on 11A, which two readers derived (normaliser (11:5 × M₁₂):2 and the centraliser order) and none found stated in a source; under a source-stated reading the primes would be {3, 5, 7} and the run falsified.
pre_seal_glimpse
the four centralisers above (runs 115–116 readers); the codifier's own memory of the Monster's class list, which is why the answer comes from readers and is scored blind; nothing looked up for this seal.
source
../lab/runs/118-prime-centralisers-of-the-monster.md

Prediction(A) the primes appearing in the majority P are exactly {3, 5, 7, 11}. (B) the S at p = 11 is M₁₂, and no other prime's S is a Mathieu group (M₁₁, M₁₂, M₂₂, M₂₃, M₂₄). Both scored; confirmed if both hold.

Falsifier(A) the set of primes differs from {3, 5, 7, 11}; (B) S at p = 11 is not M₁₂, or a Mathieu group appears at another prime. Unanticipated: the three readers disagree on more than two classes (the instrument cannot settle the table; recorded, and the question goes to the GAP library as a computation). Withdrawn if control (i) fails.

Statement

The reader prompt, fixed:

> You are a reader with web access. Do not read any file in a repository or on this machine; use only the web. Question: for every prime p dividing the order of the Monster group M, and every conjugacy class pX of elements of order p in M, what is the structure of the centraliser C_M(pX)? Give each as it is written in the ATLAS of Finite Groups or the GAP character table library (for example "2·B", "3 × Th", "2^{1+24}·Co₁"), with a source URL per class, and say whether you read the page or only a search summary. Then list separately the classes whose centraliser is a direct product p × S with S a sporadic simple group. No interpretation; under 400 words.

Reports verbatim in R28_reading/118/readers118.md; the table in RESULTS118.md.

Sealed 2026-10-23

As written. The scoring reader sees this file above "Not part of the seal" and RESULTS118.md.

Scored 2026-10-23 — confirmed

Scored by a blind reader given the seal above the "Not part of the seal" heading and R28_reading/118/RESULTS118.md. Controls held. (A) and (B) held on the majority: 3C, 5A, 7A by all three readers from stated sources; 11A by two of three, both derived. No falsifier, no unanticipated case (one class of disagreement on membership).

Faults in the seal's wording (the reader's five, kept): "the reader gives" did not say whether a structure a reader worked out counts, while the prompt asked for structures "as written" — the confirmation turns on that word, and under the stricter reading the run is falsified; "disagree on more than two classes" did not say on what (on structure, readers 1 and 3 derived ten classes where reader 2 gave orders only); control (ii) had no consequence and did not separate citing the structure from citing data it was derived from; a missing class (reader 2's 11A) had no rule; a Mathieu group in a minority set would be unscored. Instrument: every page reached through a summarising fetch; no reader saw the ATLAS or GAP text.

Reading (the codifier's, after the score): the guess held, and the record should carry its weight honestly: 11 × M₁₂ is a two-line derivation (the normaliser (11:5 × M₁₂):2 has order 110·|M₁₂|, the centraliser has index 10 in it and order 11·|M₁₂|, so it is 11 × M₁₂), which is why two readers made it, and it is not on any page the readers reached. The fact behind it is what matters for the forest: the four sporadic centralisers of prime-order elements are Th, HN, He, M₁₂, at 3, 5, 7, 11, and only the last is also a stabiliser in the Golay code. The next question (guesses 2 and 3 of 09:45) is whether the 11-element and the dodecad are one object; that needs the GAP library or an argument, not readers.

The codifier’s lead is kept for the private site.

generated 2026-10-25the lab index