Paper O28 established that the per-pair covariance Cc in End (H₄₅₅) has rank r₄₅₅ = 3 with invariant eigenvalue structure 1: 12: 12. The present paper gives the structural explanation of this result. The value r₄₅₅ = 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 = c (vⱼ) ₐ-₂ (vⱼ) ^* are (up to a small, q-decreasing defect) complex symmetric; and the O26 target r₄₅₅ = d_² = 4 is structurally inaccessible from conjugate-pair data. A direct characterisation on the O25/Q5a checkpoints shows that the trajectory \wⱼ = c (vⱼ) \ spans all three complex dimensions of H₄₅₅ (singular ratios ₃/₁ 0. 7--1. 0, not < 10^-2), and that r₄₅₅ = 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 H₄₅₅ is one-dimensional, so H₄₅₅ = C³ is irreducible. This identifies r₄₅₅ = 3 as the invariant of the adjoint action H₄₅₅ su (2) of O23/O27 — the spin-1 (vector) representation — not as the symmetric-rank inversion 3 = d_ (d_+1) /2 for d_ = 2. Since the adjoint is irreducible, H₄₅₅ contains no two-dimensional invariant subspace: the spin-12 space V_{} is the Schur target of the morphism: su (2) V_{} (O27), not a subspace of H₄₅₅, and is invisible to Test 4. Consequently Test 4 confirms the carrier (spin-1 on H₄₅₅) and the inaccessibility of d_² = 4, but does not by itself identify the spin-12 sector; a genuine spin-12 test requires an independent observable on V_{} (the locking-broken protocol of Section sec: results reaches the full d_² = 4 and motivates it).
Jérôme Beau (Sun,) studied this question.