Spin-1/2 Sector Identification via the Symmetric Rank Formula: Effective Dimension of the Admissible Covariance in End(Vρ)

Paper O28 established that the per-pair covariance Cc in End(Heff) has rank reff = 3 with invariant eigenvalue structure [1 : 1/2 : 1/2]. The present paper provides the complete analytical explanation of this result and carries out the sector identification required for O26. We prove that the Born{}Infeld parity involution c ↔ q-c of O18, which implies πq-c(v) = πc(v)̄ in the canonical basis of Heff (up to a slowly varying phase factor, as discussed in Section 6), forces every outer product Mⱼ = πc(vⱼ) ⊗ πq-c(vⱼ)* to lie in the complex symmetric subspace Sym(Vρ, C), where Vρ = spanC\πc(vⱼ)\ ⊂ Heff is the trajectory subspace. This is an algebraic identity that holds independently of the specific values of πc(vⱼ) (the symmetry structure requires no genericity assumption; the rank formula of Theorem 3.3 additionally requires the trajectory to span Vρ, which is verified numerically){equation*}reff = {dρ(dρ+1)}{2},{equation*}where dρ = C Vρ is the complex dimension of the trajectory subspace. The observed value reff = 3 has the unique positive integer solution dρ = 2, confirming the spin-1/2 sector candidate without ambiguity. This result holds in both End(Heff) (proxy computation of O28) and End(Vρ) (correct O26 test space): restricting to End(Vρ) does not change the rank, since the symmetric constraint is intrinsic to the pair structure. We show that the O26 Criterion~5.4 target of reff = dρ² = 4 is structurally inaccessible from conjugate-pair data and must be reformulated as reff = dρ(dρ+1)/2 for the present measurement setting. The reformulated criterion is satisfied uniquely by dρ = 2, and the numerical computation from Q5a-O5 checkpoints at q ∈ \61, 101, 151, 211\ confirms reff = 3 for ≥ 98% of conjugate pairs at each prime, with zero inter-pair variance at q ∈ \61, 151\.