We first isolate an index-two split-fusion principle for a normal subgroup H ◁ G of index two: whenever a family of split restrictions has orthogonal difference characters ∆i that are eigenvectors for multiplication by a restricted irreducible character, the two fusion channels are exactly the half-sum and half-difference of the ambient tensor multiplicity and the corresponding difference-sector eigenvalue. Specializing to Sn/An, let γ ⊢ n be self-conjugate, let δ(γ) be its distinct-odd principal-hook partition, and write ResSn An χ γ = αγ + βγ. For a non-self-conjugate partition λ, set ψλ = ResSn An χ λ and ∆γ = αγ − βγ. We prove that ∆γ is an exact eigenvector for multiplication by ψλ, with eigenvalue χ λ(δ(γ)). Resolving the two fusion channels between split constituents gives aij = (g(λ, γi, γj ) + χ λ(δi)δij)/2, bij = (g(λ, γi, γj ) − χ λ(δi)δij)/2. Consequently g(λ, γi, γi) ≥ |χ λ(δi)| with the corresponding parity congruence, while off-diagonal coefficients g(λ, γi, γj ) are even. We identify the exact defect from equality as twice the smaller diagonal fusion multiplicity, so equality holds precisely when one diagonal channel vanishes. Moreover the self-conjugate Kronecker block satisfies Gλ ≡ Xλ (mod 2), which determines its rank, characteristic polynomial, and all principal minors modulo 2 from the principal-hook values. In matrix form the split-fusion operator decomposes as Gλ ⊕ Xλ, where Gλ is the self-conjugate Kronecker block and Xλ is diagonal with entries χ λ(δi). The An splitting mechanism already underlies the principal-hook nonvanishing criterion of Pak–Panova–Vallejo; the refinement here is to retain both split fusion channels simultaneously, yielding exact multiplicity identities, magnitude bounds, and parity constraints.