Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 113Settled

Does the inherited predicate — "G is the stabiliser, in Co₀ or the Monster, of a named substructure of a 24-object, or sits in a chain of such stabilisers" — reproduce the happy/pariah split of the 26 sporadic groups exactly, on the readers' reports of run 112, with no new reader?

run_kind
reading
slug
inherited-predicate
state
run
created
2026-10-22
sealed
2026-10-22 (committed before the predicate is applied)
builds_on
[112]
concepts
[classification, sporadic, stabiliser, reading-run]
programme
PG-04
line
C1, the object-side predicate (second form)
from
run 112's reading and the owner's reading of its table on paper (2026-10-22, 21:31) — "unless can say no own object in L because know for sure bc know whole space or just havent found one". Run 112's negative cells ("owns nothing in L") were absences in what 26 readers reached; this seal replaces the ownership predicate by one whose positive side is a published containment and whose negative side is a theorem.
statistic
Exact on existing files, no reader. Inputs: R28_reading/112/readers112.md and M11.md (26 reports, md5 below). For each group the codifier reads the report's section (2) and records every named containment G ≤ H or G a subquotient of H where the report names the object H preserves and the substructure G stabilises or centralises: a vector or triangle of the Leech lattice (Co₂, Co₃, HS, McL), a frame or dodecad (M₂₄, M₁₂), a ring structure on the Leech lattice (Suz, J₂), a point of S(5,8,24) (M₂₃, M₂₂), an element or small subgroup of the Monster acting on V♮ (a centraliser C_M(g) is the pointwise stabiliser of the fixed-point subalgebra V♮^g, which is L's "fixed-point subalgebra" clause), a transposition or S₃ in a Fischer group that itself sits in the Monster. INH(G) is true when such a containment is named, or when G is named inside a group H for which INH(H) is true (a chain, at most two steps, each step named by a reader). The negative side is not an absence: for J₄ and Ly the readers cite 37 ∤ |M| (an order argument), for J₁, J₃, Ru, O'N they cite Griess 1982 / Wilson that the group is not a subquotient of the Monster. Scored quantity: INH⁺ against HAPPY. Reported beside: the chain for each group (length 0, 1 or 2), the substructure named, the report's own wording; the groups whose containment is named without a substructure (a bare "subquotient of M").
scored
2026-10-22
outcome
confirmed
outcome_note
by the prediction as written — INH⁺ is the twenty happy-family groups and no pariah, 26 for 26 (chain lengths 0: 5, 1: 13, 2: 2); secondary held (Fi23, Fi24′, He, HN, Th placed by what they centralise in M with no object named; Fi22 through Fi23); controls pass; the reader's six wording faults kept, the first of which (a centraliser statement counted as a substructure in the statistic and as bare in the secondary) is a real ambiguity the seal left open.
pre_seal_glimpse
run 112's table and reading (the inherited column read once, at scoring, which is where this predicate came from); the owner's reading of the table on paper; the codifier has not yet gone through the 26 reports with the chain rule above.
source
../lab/runs/113-inherited-predicate.md

PredictionINH⁺ = HAPPY on all 26 — every happy-family group reaches a named substructure of a 24-object in at most two named steps, and no pariah does. Secondary, scored separately: at least three of the twenty reach it only by a bare "subquotient of the Monster" or a two-step chain with no substructure named at the last step (the readers' wording for Fi₂₂, Fi₂₃ and HN is the glimpse).

Falsifierany happy-family group with no named containment chain of length ≤ 2 ending at a named substructure or a bare Monster subquotient, or any pariah with one. Unanticipated: a happy-family group whose only chain runs through a group that is itself placed by a bare subquotient statement (chain length 2 with both steps bare) — recorded as the predicate's resolution limit, not scored. Withdrawn if control (i) fails.

Statement

Run 112 asked "owns an object in L" and found five of twenty. Its inherited column, read once at scoring, suggested the line the happy family actually follows: not owning a 24-object but stabilising something inside one. This seal makes that a predicate and applies it to the same 26 reports, so nothing new is read and the reports' absences are not asked to carry a negative: the pariah side rests on the theorems the readers cited, and the happy side on containments the readers named. What can fail is the chain rule — a happy-family group placed in the Monster by its reader with no substructure and no intermediate group named would fall outside the predicate as written.

The owner's three remarks of 21:31, carried in the seal: (1) an absence in what readers reached is not a negative — hence the theorem-backed pariah side; (2) whether the names encode anything — the names "happy family" and "pariah" are Griess's (1982) and encode containment in the Monster, nothing more; the Mathieu indices encode the object (the degree of the permutation action), and the Conway, Fischer and Janko names encode a discoverer; the predicate here uses none of them; (3) the Leech lattice and the octonions — Wilson's octonionic construction of Λ (three octonionic coordinates) is on the J₂ reader's source list; it is an instance of the same move as Suz and J₂ (a ring structure on Λ cutting Co₀ down to a stabiliser) and is recorded for PG-04's next status entry, not used here.

Sealed 2026-10-22

As written. The scoring reader sees this file above the "Not part of the seal" heading and R28_reading/112/RESULTS113.md.

Scored 2026-10-22 — confirmed

Scored by a blind reader given the seal above the "Not part of the seal" heading and R28_reading/112/RESULTS113.md. Withdrawal not triggered; falsifier not triggered (every happy-family row has a chain of length ≤ 2, no pariah has one); unanticipated not triggered (no chain with both steps bare). Primary held: 26 for 26. Secondary held: five placed by a bare centraliser or normaliser statement in the Monster (Fi23, Fi24′, He, HN, Th), six with Fi22 through the chain.

Faults in the seal's wording (the reader's six, kept): the statistic counts a centraliser C_M(g) as a named substructure (V♮^g) while the secondary clause and the table count the same statement as bare — the seal did not fix which reading governs, and the secondary held under the table's; "bare subquotient of the Monster" names a route no row uses (all bare rows are centraliser or normaliser statements); whether a substructure relation the reader marked unverified (B's VB♮) falls under control (ii) is unsaid; the pariah side is defined only for the citations the readers happened to give; length 0 is covered by control (i), not by the predicate's text. One carried from 112: the 26 reports are in two files.

Reading (the codifier's, after the score): what 112 and 113 say together is now exact on one body of evidence. The happy family is not the set of groups that own a 24-object (five do); it is the set of groups that are stabilisers of something inside the Leech lattice or V♮ — a vector, a triangle, a ring structure, a dodecad, or the fixed points of an element of the Monster — and every pariah is kept out by a theorem, not by an absence. "Subquotient of the Monster" was tracking "stabiliser of a substructure of V♮" all along; Griess's name for the twenty encodes the containment and nothing else, and the predicate that explains the containment is object-side. The resolution limit is the one the secondary counted: for six of the twenty the substructure is a fixed-point subalgebra V♮^g that no reader has ever seen named as an object, because the literature names the element and not the subalgebra. That is where the reading programme ends and a computation on V♮ would begin, and it is outside PG-04's bound. Two things carried forward for the status entry, from the owner's remarks: the ring structures on Λ (icosian for J₂, Eisenstein for Suz, and Wilson's octonionic Λ on the J₂ reader's source list) are a third kind of substructure — a coordinate ring, not a vector — and a predicate that distinguishes "stabilises a vector" from "stabilises a ring structure" from "stabilises an element's fixed points" would sort the twenty into three classes that the reports already separate; whether that sorting is forced or ours is the next thing to pre-register.

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

generated 2026-10-25the lab index