Arc from 112

root 112 · 6 records · depth 6 · 4 confirmed · 1 withdrawn · full density (§5)

The chain is read from builds_on: (docs/OUTTAKE.md, Cross-record fields). Every id, question, sealed prediction, verdict and count below comes from the records at a354234.

Concept not yet on the map. centraliser, classification, forcing, moonshine, reading-run, sporadic, stabiliser appear in concepts: on records in this arc and are not among the 58 ids domain-math.js defines, so the map has nowhere to place this arc yet. Reported here rather than dropped; it does not fail the build.

112 object-side-predicate

Does the object-side predicate "G owns an object in the 24-component" reproduce the symmetry-side split of the 26 sporadic groups into the happy family (20) and the pariahs (6) exactly, group for group?

Sealed 2026-10-22 (committed before any scored reader is briefed) OWN⁺ ≠ HAPPY — at least one of the 26 groups is misclassified by the predicate.

CONFIRMEDbuilds on root

113 inherited-predicate

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?

Sealed 2026-10-22 (committed before the predicate is applied) INH⁺ = 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.

CONFIRMEDbuilds on 112

115 three-kinds-of-substructure

On the twenty happy-family sporadic groups, do the three kinds of substructure they stabilise — V (a piece of the Leech lattice or Golay code), R (a ring or coordinate structure on the Leech lattice), F (the fixed points of an element or small subgroup of the Monster on V♮) — nest as R ⊂ V ⊂ F, group for group?

Sealed 2026-10-23 (committed before any reader is briefed) (A) R⁺ ⊆ V⁺ ⊆ F⁺ with no exception (the kinds nest).

WITHDRAWNbuilds on 113 · 112

116 rooted-forest

Under four kinds — W (the whole Golay code, Leech lattice or V♮), V (a piece of Λ or the Golay code), R (a coordinatisation of Λ), F (the fixed set of a non-identity element or subgroup of M on V♮) — does every happy-family group have at least one kind, with the roots exactly {M₂₄, Co₁, M}?

Sealed 2026-10-23 (committed before any reader is briefed) (A) no group has K(G) empty.

CONFIRMEDbuilds on 115 · 113 · 112

118 prime-centralisers-of-the-monster

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?

Sealed 2026-10-23 (committed before any reader is briefed) (A) the primes appearing in the majority P are exactly {3, 5, 7, 11}.

CONFIRMEDbuilds on 116

CASE-33 case

Why is M₁₂ the one happy-family group read both as a piece of the Golay object and as a centraliser factor in the Monster, and is that one fact or two?

builds on 112 · 113 · 115 · 116 · 118

6 records, longest path 6. The vertical bar in the gutter marks a record with more than one parent. A withdrawn or retired record keeps its place: struck id, retired ink, still on the page. A probe or note has no question:, prediction: or verdict: field, so its row carries its title and nothing is invented to fill the gap. A row shows a whole sentence or none. Print variant (Paper Pro, DESIGN-SYSTEM §7).

generated 2026-10-25the lab index