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?

Statement

The reader prompt, fixed:

> You are a reader given one input: the finite simple group G. Three kinds of substructure are defined for you. V, a piece: a vector, frame, triangle, point, octad or dodecad of the Leech lattice, or of the binary Golay code and its Steiner systems. R, a coordinatisation: a ring or coordinate structure on the whole Leech lattice — quaternionic (icosian), complex (Eisenstein), octonionic — so that the group in question is the group of symmetries preserving that structure. F, a symmetry's fixed set: the fixed points of an element or small subgroup of the Monster group acting on the moonshine module V♮, so that the group in question is the centraliser or normaliser of that element or subgroup in the Monster. For each of V, R, F, answer from sources you can reach on the web (cite one or two URLs per affirmative): does the literature document G as the full stabiliser (exactly, modulo centre, or as a subgroup of index at most 2) of a substructure of that kind? If yes, name the substructure and the exact relation; if the sources only say G "acts on" or "is a subgroup of" something, say so and do not count it. If you find nothing for a kind, say so. Under 300 words; no interpretation.

The twenty inputs: M₁₁, M₁₂, M₂₂, M₂₃, M₂₄, J₂, Co₁, Co₂, Co₃, Fi₂₂, Fi₂₃, Fi₂₄′, HS, McL, He, Suz, HN, Th, B, M. Reports verbatim in R28_reading/115/readers115.md; the kinds table in RESULTS115.md.

Sealed 2026-10-23

As written. The readers see the prompt and the group name only. The scoring reader sees this file above the "Not part of the seal" heading and RESULTS115.md.

Scored 2026-10-23 — withdrawn

Scored by a blind reader given the seal above the "Not part of the seal" heading and R28_reading/115/RESULTS115.md. Control (i) failed: it required the M₂₄ and Co₁ readers to place their groups under V, and V as the readers were given it is "a piece"; both readers named the whole code or lattice and correctly declined it. The control was written from run 112's owned column (whole objects) and the definition from run 113's inherited column (pieces), and the codifier did not check that the two agreed before sealing — OPS instance 9 of "a premise is computed before it is sealed". Clauses after the withdrawal are reported, not scored: (A) would fail (J₂ and Suz have R without V; seven groups have V without F); (B) and (C) would fail; (D) would hold (R⁺ = {J₂, Suz} exactly); the unanticipated clause fires (M₂₄, Co₁, M have no kind).

Faults in the seal's wording (the reader's eight, kept): control (i) and prediction (C) contradict the prompt's definition of V; the V counting rule has no entry for a whole object's automorphism group; F does not exclude the identity, and the codifier excluded the Monster's trivial fixed set by an unstated rule; a quotient by a non-central normal subgroup (Co₁ in C_M(2B)) is not addressed; no precedence between the unanticipated clause and the falsifier; no control on whether readers read pages (most worked from search summaries); "run 112's owned column" is undefined for a reader of the seal alone.

Reading (the codifier's, after the score; not a result — the run is withdrawn): the guess "the kinds nest" was wrong in the most informative way available. The twenty sort into a rooted forest: three roots that own the whole objects (M₂₄, Co₁, M), and three kinds of branch that are almost disjoint — pieces (eight), coordinatisations (two), fixed sets (eight), with M₁₂ the single group on two branches. Nothing nests. What the run shows, read as a picture rather than scored, is that "stabiliser of a substructure" is not one ladder but three, one per kind of thing you can do to a 24-object: cut a piece out of it, put coordinates on it, or fix it by a symmetry. That is close to, and not the same as, Griess's three generations of the happy family (the Mathieu groups; the Leech-lattice groups; the Monster's sections) — here the Mathieu groups and the Conway chain share the V branch, J₂ and Suz split off as R, and M₁₂ crosses into F. The next seal under C1, if it is worth one, is whether the forest is forced: the definitions rewritten so the roots are their own kind (W, the whole object), the guess pre-registered that the four kinds W, V, R, F partition the twenty with exactly one overlap, and the same twenty reports re-read under the new definitions with no new reader. Before that, the instrument's limit goes on record: readers reached search summaries, not pages, and a kind not found is a kind not found by search.

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