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}?

Statement

The reader prompt, fixed:

> You are a reader given one input: the finite simple group G. Four kinds of structure are defined for you. W, the whole: the entire extended binary Golay code (equivalently the Steiner system S(5,8,24)), the entire Leech lattice, or the entire moonshine module V♮. V, a piece: a vector, frame, triangle, point, pair of points, octad or dodecad of the Leech lattice, or of the 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 G is the group of symmetries preserving that structure. F, a symmetry's fixed set: the fixed points of a non-identity element, or a non-trivial subgroup, of the Monster acting on V♮, so that G is the centraliser or normaliser of that element or subgroup in the Monster. For each of W, 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 automorphism group or stabiliser — exactly, modulo its centre, or as a subgroup of index at most 2 — of a structure of that kind? If yes, name the structure and the exact relation. If the sources only say G "acts on", "is a subgroup of", or is a quotient of the stabiliser by a larger normal subgroup, say so and do not count it. If you find nothing for a kind, say so. Say whether you read pages or only search summaries. 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/116/readers116.md; the kinds table in RESULTS116.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 RESULTS116.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/116/RESULTS116.md. Control (i) held (the M₂₄ reader placed M₂₄ under W by the extended Golay code and S(5,8,24); the Co₁ reader placed Co₁ under W by Λ modulo ±1). No falsifier, no unanticipated case. (A)–(D) all held.

Faults in the seal's wording (the reader's eight, kept): "fresh" undefined (three readers had the repo's CLAUDE.md or memory index in context — workflow content, no group named); "read pages" undefined (a fetch tool returns a summary with quotes; one reader search-only; one extracted a PDF locally); control (ii) silent on URLs from summaries; "the Golay code" not pinned to the extended code (the M₂₃ reader's own decline kept the length-23 code out of W); centraliser vs normaliser and "modulo the element" not ruled (HN's evidence moved from one to the other); "substructures" reported-beside but not delivered as a column; the mapping from 115's kinds to 116's left to the codifier; no rule on readers' local computation.

Reading (the codifier's, after the score): the forest is a property of the happy family, not of one set of searches. Two independent sets of twenty readers, one in the cloud from search summaries and one on the M5 from pages, put every group in the same kind, and the only movement between them was in the evidence for one edge (HN, from normaliser to centraliser). The twenty are three roots that own the three whole 24-objects and three almost disjoint branches below them: cut a piece out (eight), put coordinates on Λ (two), fix by a symmetry of M (eight). "Subquotient of the Monster" — Griess's definition of the family — is the union of those three relations to those three objects, and the containment order of the groups (which group sits in which) is not the order of the kinds: M₁₁ and M₂₃ sit in M₂₄ and stabilise pieces of its code; J₂ and Suz sit in Co₀ and stabilise its coordinatisations, not its pieces; the Fischer groups, He, HN, Th and B sit in M and fix its symmetries. The single exception is M₁₂, which is both a dodecad stabiliser in M₂₄ and the centraliser of an 11-element in M — the one group that is a piece of the Golay code and a fixed set of the Monster at once. Pre-registered guess 5 (the V/F boundary moves for at least two groups between 115 and 116) did not hold: nothing moved. The forest's own question goes to PG-04's status: why M₁₂, and what an object with two roots would look like.

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