Carrier Identification via the Veronese Collapse: Effective Dimension of the Admissible Covariance in End(H_eff)

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 gives the structural explanation of this result. The value reff = 3 is real and stable across primes; the BornInfeld parity involution c ↔ q-c of O18 acts anti-linearly, so the conjugate-pair outer products Mⱼ = πc(vⱼ) ⊗ πq-c(vⱼ)* are (up to a small, q-decreasing defect) complex symmetric; and the O26 target reff = d_ρ² = 4 is structurally inaccessible from conjugate-pair data. A direct characterisation on the O25/Q5a checkpoints shows that the trajectory ⱼ = πc(vⱼ)\ spans all three complex dimensions of Heff (singular ratios σ₃/σ₁ 0.7--1.0), and that reff = 3 is a hard 6 → 3 collapse of the Veronese (squaring) image: the symmetric squares wⱼ ⊗ wⱼ satisfy exactly three independent quadratic relations. The three constraint forms are traceless, and their commutators span exactly so(3); the Schur commutant of the action on Heff is one-dimensional, so Heff = C³ is irreducible. This identifies reff = 3 as the invariant of the adjoint action Heff su(2) of O23/O27 — the spin-$1$ (vector) representation. Since the adjoint is irreducible, Heff contains no two-dimensional invariant subspace: the spin-1/2 space Vρ is the Schur target of the morphism Φ: su(2) → Vρ (O27), not a subspace of Heff. Test 4 measures the adjoint carrier Heff Sym²(Vρ), not Vρ directly: the spin-1/2 coordinate Vρ is recovered only through the Veronese square-root structure, not as the rank measured by reff. Consequently Test 4 confirms the carrier (spin-$1$ on Heff) and the inaccessibility of d_ρ² = 4, but does not by itself identify the spin-1/2 sector; a genuine spin-1/2 test requires an independent observable on Vρ (the locking-broken protocol of Section sec:results reaches the full d_ρ² = 4 and motivates it).