Local–Global Defects D: Automorphisms and Realization Defects—A Complete Archive of Three Saturated Fusion Systems

This paper establishes a computable archive of system automorphisms and realization defects at the toy level (|P| ≤ 8) of fusion systems, transforming “realizationdefect” from a conditional registration into a precise integer object on five concretesaturated fusion systems—five systems F0–F4 over three selected carrier environments (V4, C3, D8). We prove five structural theorems: the realization image isalways a subgroup of the system automorphism group; the realization defect index δ(F) = [Aut(F) : AutF(P)] is a function of the system rather than of therealizer; morphism inclusion reverses the automorphism group under an additionalinvariance condition; for trivial fusion over abelian p-groups, Aut(F) = Aut(P);the subgroup tower Inn(P) ⊆ AutF(P) ⊆ Aut(F) ⊆ Aut(P) yields the productdecomposition Out(P) = σ(F) · δ(F) · ι(F) and the fourth-quadrant emptinessOut(F) = 1 =⇒ δ(F) = 1. The index δ(F) and the cohomological defect factorDr(G) are not the same invariant; we do not claim any numerical correspondencebetween them, and register only the toy-level archive of realization defects. Thethree carrier archives yield ladder readings: over V4, the three saturated systemshave δ(F) values 6, 2, 1, shrinking monotonically with fusion richness; the C3 example has δ = 1 while Out(F) = 2, witnessing one-way boundedness without mutualdetermination; the D8 example has Aut(F) = Inn(P) and Out(F) = 1, the systemlosing the entire outer automorphism coset of the carrier group, a loss absorbed byσ(F4) = 2. Machine checking of the Alperin generating data yields a single essentialsubgroup for FD8 (S4) (normal V4, AutF(V4) ∼ = S3). All conclusions are closed by five machine-checking groups in deterministic exact integer arithmetic.