Local–Global Defects C: Fine Theory of the Defect Factor——Spectral Characterization, Central Limit, and Unboundedness

This paper systematically develops the fine structure of the defect factorDr(G) = ∥H1(Fr,G)∥|G|r−1of finite groups, where H1(Fr,G) is the set of conjugacy orbits of representationsof the free group of rank r. Six theorems are proved. First, the Burnside closedform: ∥H1(Fr,G)∥ = (1/|G|)∑︁ g |CG(g)|r. Second, the spectral characterization:Dr(G) = ∑︁ gρG(g)r (with ρG(g) = |CG(g)|/|G|), whence Dr is strictly monotone inr (strictly decreasing for non-abelian groups, with equality characterizing abelianity), D1 is the number of conjugacy classes, the limit is |Z(G)|, and Dr ≡ |G|if and only if G is abelian. Third, the central limit sandwich: Dr converges to|Z(G)| at a geometric rate governed by maxg/∈Z ρ(g) ≤ 1/2, with explicit rationalinequalities on both sides. Fourth, the duality inequality: the commuting probability cp(G) and D2 satisfy |G| · cp2 ≤ D2 ≤ |G| · cp, with equality characterizingabelianity; combined with Gustafson’s theorem (cp ≤ 5/8) this yields D2 ≤ 5|G|/8,but this paper simultaneously proves that no absolute constant upper bound ofGustafson type exists on the defect side——the generalized dihedral family has D2diverging linearly with D2/n → 1/4; and the projective special linear group family gives a self-contained estimate showing that the infimum of D2 is 1 and is notattained. Fifth, the exact formula for the generalized dihedral family: for odd n,Dr(Dih(Cn)) = 1+(n−1)/2r+n(1/n)r, and for even n, Dr = 2+(n−2)/2r+n(2/n)r.Sixth, the representation-theoretic reading:D2(G) = 1 |G| ∑︂ χ,ψ ⟨|χ|2, |ψ|2⟩,that is, the mean over all pairs of the inner products of squared character moduli.Six groups of machine checks (two-path mutual verification, asymptotic sandwich,dihedral family closed form, character inner products, duality inequality, asset verification) close exactly on ten families of finite groups and on the full dihedral family.Negative asset registration: the naive extrapolation that “the minimum defect ofnon-abelian groups is 11/6” is refuted by D2(A5) = 77/60 (the minimum within thecovering set; the global infimum is 1 and is not attained, see Section 5.3); semidirectproducts do not preserve multiplicativity; the equality family is completely characterized. Keywords: defect factor; non-abelian cohomology; commuting probability; conjugacy classes; Gustafson’s theorem; finite group representations