root 112 · 6 records · depth 6 · 4 confirmed · 1 withdrawn · full density (§5)
112 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?
113 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?
115 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?
116 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}?
118 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?
CASE-33 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?