PG-04 — The object before the symmetry

Rewritten 2026-10-24 after the fenced reading. Earlier versions are in git. No saved query carries this reading, because a reading programme has no grain; it cites run files and readings.

Survived. - The happy/pariah line is real on the object side. It is a line about stabilisers, not about ownership. "Stabilises a substructure of the Leech lattice, the Golay code or V♮" places 26 of 26: all twenty happy-family groups, and none of the six pariahs (113). The pariahs are kept out by theorems the readers cited. - The readers' kinds replicate. Two independent sets of twenty readers put every group in the same kind (116): W = {M₂₄, Co₁, M}; V eight; R = {J₂, Suz}; F eight. - The prime-order centralisers of M with a sporadic factor. Direct: 3C → 3 × Th, 5A → 5 × HN, 7A → 7 × He, 11A → 11 × M₁₂ (118). Central: 2A → 2.B, 3A → 3.Fi₂₄′. GAP's class orders show that 7B, 17, 19, 23, 31 and 47 have centralisers whose orders no sporadic group divides (RESULTS-gap.md). 11 × M₁₂ is derived from |C_M(11A)| = 11·|M₁₂| and the normaliser. Three readers in a row found no source that states it. - The kinds, read on M's own data, are two chains and their pieces (fenced reading; formation readers). - The Leech chain. An order-p symmetry of Λ that fixes no vector makes Λ a free ℤ[ζ_p]-module of rank 24/(p−1). This needs (p−1) | 24, so p ∈ {2, 3, 5, 7, 13}. At exactly these primes M has a class pB whose centraliser is p^{1+24/(p−1)} over a quotient: Co₁, 2Suz, 2J₂, 2A₇ and a group at 13. In moonshine terms these pB classes are the non-Fricke genus-zero classes: T_{pB} = (η(τ)/η(pτ))^{24/(p−1)} + 24/(p−1) is the Hauptmodul of Γ₀(p), and Γ₀(p) has genus zero exactly when (p−1) | 12, which gives the same five primes. W and R were one chain. They looked like two because 2A₇ and the 13-group are not sporadic. - Pieces of the Leech chain. The V groups are maximal subgroups one level down. Co₂ and Co₃ are maximal in Co₁. M₂₃, HS and McL sit in maximal subgroups of both Co₂ and Co₃ (as HS.2 in Co₂, as McL.2 in Co₃). M₁₂ sits in Co₃ (2 × M₁₂) and in Fi₂₂. M₂₂ sits over 2¹⁰ in Co₂ and Fi₂₂, M₁₁ over 3⁵ in Co₃, M₁₂ over 3⁶ in Co₁ (as 3⁶:2M₁₂) (RESULTS-chief-fenced.md). - The Fischer/Fricke family. B, Fi₂₄′, HN and He are factors of the centralisers of 2A, 3A, 5A and 7A. Below them is the Fischer chain: Fi₂₂ in Fi₂₃ as 2.Fi₂₂, in Fi₂₄′ as 2.Fi₂₂.2, in B in S₃ × Fi₂₂.2; Fi₂₃ maximal in Fi₂₄′ and in B. - Th apart. 3C is neither: T_{3C} = j^{1/3}, centraliser 3 × Th. This is known in pieces (Conway–Norton 1979; Kondo 1985; Koike 1988). The chain as one statement was not found in one source. It closes as a record, not a seal.

Died. - "Owns an object in L" as the predicate. It marks five of the twenty happy-family groups (112). - The kinds as nested, R ⊆ V ⊆ F (115, withdrawn on a control that contradicted the definition of V). They do not nest. - M₁₂ as one group on two branches, and guesses 2 and 3 of 09:45 (CASE-33; both scored missed). The centralising M₁₂ has its order-3 elements in M's classes 3A and 3B. Any lift of the dodecad M₁₂ through 2^{1+24}·Co₁ has them in 3A and 3C. The outer automorphism cannot turn one set into the other. These are two subgroups with one isomorphism type. The forest's one overlap was an overlap of names. Under a composition-factor rule, Co₁ and Suz would also take a second kind, so the single overlap was also a product of the F rule's line at the centre. - The kinds as our own categories. W and R are one chain, and F against the chain is M's own class split. - The ATLAS letters as that split. The split is Fricke against non-Fricke. For p = 23, 31, 47, 59 and 71 both classes of each pair are Fricke, so "A = Fricke" fails there. - S1 as an object the GAP library can build. Only M₁₂ and Fi₂₃ reach M along stored fusions, and Maxes is stored for none of M's 46 maximal subgroups (RESULTS-subgroups.md). - "The Leech side enters the Fischer groups only p-locally." Inside B, Co₂ sits over 2^{1+22} and M₂₂ over 2³², but M₂₂ in A₅.2 × M₂₂.2, HS in 5:4 × HS.2 and M₁₁ as a maximal subgroup sit with nothing below them. Described, not sealed. - Guess 5 of run 116 (the V/F boundary moves between instruments). No group moved.

The one object. The Monster's prime-order classes, each with its centraliser and its Hauptmodul. The happy family is read off them as the pB Leech chain with its pieces as maximal subgroups down the chain, the Fricke centraliser factors with the Fischer chain below them, and Th at 3C. Two things in this object are still filed by isomorphism type and not by subgroup: the V pieces, and M₁₂'s two places. The frontmatter object: still names "the 26 sporadic groups read as symmetry groups of objects". The six pariahs are outside this object.

What the next seal is for, and why it is next. No C1 seal is named. The C1 thread's last three entries closed as records: the moonshine reading "closes as a record, not a seal", and the fenced reading is "described, not sealed". The fence around the maximal subgroups of Co₁, Co₂, Co₃, Fi₂₂, Fi₂₃, Fi₂₄′ and B was set to reserve them for a sealed test. They have now been read as data, so a C1 seal would need a domain the record does not yet name. The next seal the record names and has not run is R3, for line C3 (design doc §6). It enumerates |G(q)| for the sixteen Lie families at q ≤ 9 and rank ≤ 8, and finds the non-simple cells and the equal-order pairs. The bound, that the non-simple cells occur only at q ≤ 3 or rank ≤ 2, is sealed before enumeration. The pre-registered guess is that it holds, P = 0.85. It is next because C3 is the only line whose seal is named and unrun, it is integer arithmetic inside the programme's bound, and it needs no reader. Before any further C1 seal, the record names two inputs. One is the literature's subgroup classifications (ledger, §1b), which would let the kinds be read on conjugacy classes of subgroups, the question CASE-33 left. The other is the formation reader's two checks: the prime test, and the module test on 7^{1+4}:(3×2S₇) and 13^{1+2}:(3×4S₄). C2 is a reading, not a seal, and is unrun.

What would end the programme. C3 scored; C2 read; and C1's remaining question, whether the kinds hold on conjugacy classes of subgroups, either read from the literature or ruled outside the bound of eight choices and three conjectures. Then the programme writes back or is dissolved. Section 4 is empty after five scored runs (112, 113, 115, 116, 118). CASE-33 was closed from this line, and the formation design's dry run cites it, but neither is entered in Written back.

Vocabulary. 1. Words the programme used that the vocabulary did not have. kind, W, V, R, F; OWN⁺ and L; INH⁺ and "24-object"; HAPPY; roots and branches; Leech chain; Fischer chain; Fricke family; class signature; chief-factor reading; restated object. The candidates, which stand in a sealed clause of a scored run and in this Distilled: W, V, R, F (116, confirmed; 115's clauses were withdrawn), OWN⁺ (112), INH⁺ and "24-object" (113), HAPPY (112, 113). 2. Words the vocabulary had that the programme could not use or circled. The identity form and its mechanism test (rung-vocabulary §5): CASE-33 is filed form: identity, and "one subgroup or two" was settled by a fusion, not named as an identity question. Rigidity, the injectivity of an invariant (§8.2): the class signature was used as "a necessary match but not a proof", which is that axis, unnamed. "Right quantity, wrong place" (§8.2) describes isomorphism type against conjugacy class; the record does not use it. 3. Axis or rung that lacked each candidate. W, V, R, F: no axis for how a group is the symmetry of an object (whole, piece, coordinatisation, fixed set); §5's forms place the question, not the relation. OWN⁺ and INH⁺: §5's existence form has certificates and barriers but no word for owned against inherited. "24-object": the place axis (§4) has no stated domain to put it in. HAPPY: a named list, not a coordinate; no axis is missing. 4. What the vocabulary enforced. Run 115 was withdrawn because control (i) contradicted the definition of V. The phrase "M₁₂ the single two-parent vertex" was struck by CASE-33. It was a statement about a counting rule. 5. Symbols and collisions. W, V, R, F (kinds); L (the list of objects, 112); OWN⁺, INH⁺, HAPPY; K(G) (116); Λ, V♮, M, Co₀; pA, pB, 3C (ATLAS classes); T_{pB} (Hauptmodul); ζ_p. Collisions: - C1–C3, the programme's lines, against C1–C5, the filing choices in docs/design/2026-10-23-formation-and-filing.md (C3 is "discretisation makes the holes" in one and "up to centre" in the other; §1b cites the second). - S1, S1′, S2–S4, the design's steps, against the glossary's scaffold notation S1, … (PG-04 has no scaffolds). - T: the glossary already records three meanings; T_{pB} is a fourth. - kind: bare "kind" was resolved out of PG-02 on 2026-10-04, and is the frontmatter field and OUTTAKE's word for a record type. PG-04 reopens it. - R, a kind, beside ℤ[ζ_p], the ring that R groups coordinatise over. PG-04's §1 has no vocabulary block, so none of these has a home yet. 6. Definitions that drifted. - F: "the fixed set of an element of M on V♮" (115 seal, 2026-10-23); "for non-identity elements" (116); "the centraliser in M of an element or subgroup, up to its centre" (CASE-33, 2026-10-23); then factors of centralisers of Fricke classes (fenced reading, 2026-10-23 evening). - V: "a piece of Λ or the Golay code" (115); then maximal subgroups down the Leech chain (fenced reading). - W and R: two kinds (116); one chain (formation reader, 2026-10-23, 17:30). - The split: A against B (formation reader, 17:30; status line 81); Fricke against non-Fricke (moonshine reader, 20:30; status line 74). Both lines stand in §2 today. - The object: isomorphism types (runs 112–116); conjugacy classes with signatures (S1, 2026-10-23); chief factors of maximal subgroups (S1′, 16:52). 7. Prose-only terms. rooted forest, roots, branches, Leech chain, Fischer chain, Fricke family, restated object, class signature, resolution limit, two-parent vertex, moonshine coordinates. 8. Names that carry an open claim. "Leech chain": one chain is not found stated in one source, and the exponent = rank = extraspecial order link is "folklore unless a source is found". "Rooted forest": whether it is a spanning forest of a containment graph was left open (116). "Restated object, complete for the happy family" claims completeness on isomorphism types only. 9. Metaphors. Forest, roots, branches (the owner's thread of 2026-10-23, after renormalisation's forest): imports the matrix-forest theorem; untested. Parents and the two-parent vertex: imported a tree; struck by CASE-33. Chain: imports an order; the Leech chain is ordered by p and nothing has tested what the order means. Happy family and pariah (the literature's names, quoted in run 112's premise): import a social split; C1 tested it on the object side (112, 113). Moonshine: a name; imports nothing tested. 10. Quantifiers that weakened since the last rewrite. "M₁₂ the one group on two" became "two subgroups with one isomorphism type". "The split is A against B" became "Fricke against non-Fricke, apart from 3C". "11 × M₁₂" holds by derivation, not by a stated source. "The forest is forced" became a pattern seen on one data set that "suggested the hypothesis, so it cannot test it". "26 for 26" is unchanged.

Entry. No candidate enters docs/GLOSSARY.md at this rewrite. W, V, R, F, OWN⁺, INH⁺, "24-object" and HAPPY stand in sealed clauses of scored runs and in this Distilled. But no second programme uses them, and no panel has reached their object blind: CASE-33 closed before a panel. Ceiling. This is the first Distilled to carry the paragraph. Each candidate is at its first of two rewrites. "Two-parent vertex" died with its lead and leaves the block.