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

Abstract

Paper O28 established that the per-pair covariance ${\mathcal{C}_c}$ in ${\operatorname{End}}({H_{\mathrm{eff}}})$ has rank ${r_{\mathrm{eff}}} = 3$ with invariant eigenvalue structure $[1: \tfrac{1}{2}: \tfrac{1}{2}]$. The present paper gives the structural explanation of this result. The value ${r_{\mathrm{eff}}} = 3$ is real and stable across primes; the conjugation identity $\rho_{q-c} = \bar{\rho}_c$ (an O17 theorem, recalled in O18 Theorem 3.1 ) acts anti-linearly, so the conjugate-pair outer products $M_j = \pi_c(v_j) \otimes \pi_{q-c}(v_j)^{*}$ are (up to a small, $q$-decreasing defect) complex symmetric; and the O26 target ${r_{\mathrm{eff}}} = d_\rho^2 = 4$ is structurally inaccessible from conjugate-pair data. A direct characterisation on the O25/Q5a checkpoints shows that the trajectory $\{w_j = \pi_c(v_j)\}$ spans all three complex dimensions of ${H_{\mathrm{eff}}}$ (singular ratios $\sigma_3/\sigma_1 \simeq 0.7\text{--}1.0$), and that ${r_{\mathrm{eff}}} = 3$ is a hard $6 \to 3$ collapse of the Veronese (squaring) image: the symmetric squares $w_j \otimes w_j$ satisfy exactly three independent quadratic relations. The three constraint forms are traceless, and their commutators span exactly $\mathfrak{so}(3)$; the Schur commutant of the action on ${H_{\mathrm{eff}}}$ is one-dimensional, so ${H_{\mathrm{eff}}} = \mathbb{C}^3$ is irreducible. This identifies ${r_{\mathrm{eff}}} = 3$ as the invariant of the adjoint action ${H_{\mathrm{eff}}} \simeq {\mathfrak{su}}(2)$ of O23/O27 — the spin-$1$ (vector) representation. Since the adjoint is irreducible, ${H_{\mathrm{eff}}}$ contains no two-dimensional invariant subspace: the spin-$\tfrac{1}{2}$ space ${V_{\rho}}$ is the Schur target of the morphism $\bar\Phi: {\mathfrak{su}}(2) \to {V_{\rho}}$ (O27), not a subspace of ${H_{\mathrm{eff}}}$. Test 4 measures the adjoint carrier ${H_{\mathrm{eff}}} \simeq {\operatorname{Sym}}^2({V_{\rho}})$, not ${V_{\rho}}$ directly: the spin-$\tfrac{1}{2}$ coordinate ${V_{\rho}}$ is recovered only through the Veronese square-root structure, not as the rank measured by ${r_{\mathrm{eff}}}$. Consequently Test 4 confirms the carrier (spin-$1$ on ${H_{\mathrm{eff}}}$) and the inaccessibility of $d_\rho^2 = 4$, but does not by itself identify the spin-$\tfrac{1}{2}$ sector; a genuine spin-$\tfrac{1}{2}$ test requires an independent observable on ${V_{\rho}}$ (the locking-broken protocol of Section sec:results reaches the full $d_\rho^2 = 4$ and motivates it).