Let γ1, . . . , γq ⊢ n be the self-conjugate partitions, let δi be the distinct-odd partition formed by the principal hook lengths of γi, and write ResSn An χ γi = αi + βi. For λ ≠ λ', put ψλ = ResSn An χ λ. The classical Sn/An splitting argument used in the principal-hook criterion of Pak–Panova–Vallejo may be retained at the level of the two An constituents rather than recombined immediately. Doing so gives ⟨ψλαi, αj ⟩ = (g(λ, γi, γj ) + χ λ(δi)δij)/2, ⟨ψλαi, βj ⟩ = (g(λ, γi, γj ) − χ λ(δi)δij)/2. The purpose of this note is to record the quantitative and parity consequences of these identities. In particular, g(λ, γi, γi) ≥ |χ λ(δi)| with the same parity, whereas g(λ, γi, γj ) is even for i ≠ j. If Gλ = (g(λ, γi, γj )) and Xλ = diag(χ λ(δi)), then Gλ ≡ Xλ (mod 2); hence the rank, characteristic polynomial, determinant, and every principal minor of Gλ modulo 2 are determined by the principal-hook values. We also give an exact equality criterion for the diagonal bound and explicit computations in degrees 5 and 8. All Clifford-theoretic ingredients are classical; the contribution is the extraction and organization of these multiplicity and block-parity consequences.