Veronese Collapse in the Measured Trajectory Span: What the O28 Covariance Does and Does Not Identify

Abstract

Paper O28 reported effective covariance rank ${r_{\mathrm{eff}}}=3$ in the supplied three-coordinate space ${H_{\mathrm{eff}}}$. This paper audits what that finite-data result identifies. In an extended recomputation, rank three remains modal but is not invariant: it occurs in $32/50$ stored samples at $q=101$ and $103/105$ at $q=211$. Three spaces must be separated. The measured trajectory span lies in ${H_{\mathrm{eff}}}$; its leading two-dimensional PCA truncation ${U^{(2)}_{c}}$ is data-dependent; and the spin-$\tfrac{1}{2}$ module ${V_{\rho}}$ is supplied elsewhere. No calculation here identifies these objects. The exact Weil-block identity $\rho_{q-c}=\bar\rho_c$ motivates, but does not prove, the additional projected-coordinate relation $\pi_{q-c}(v)=\overline{\pi_c(v)}$. Under that relation the outer products are complex symmetric; the stored data satisfy it only approximately. At $q\in\{29,61\}$, the trajectory fills the three measured coordinates while its symmetric-square evaluation matrix has numerical rank three rather than six. The fitted constraints are numerically traceless; their commutators have numerical span three in the antisymmetric basis, and the computed commutant is one-dimensional. These finite-precision diagnostics are compatible with the supplied adjoint reading of ${H_{\mathrm{eff}}}$, but they do not derive that carrier or prove exact irreducibility. The tested conjugate-pair samples do not attain the O26 target rank four. Breaking the per-vector lock restores rank four only in ${\operatorname{End}}({U^{(2)}_{c}})$, not in ${\operatorname{End}}({V_{\rho}})$. Interpretive outlook. The collapse diagnoses geometry within measured coordinates; it does not select a physical representation. A genuine spin-$\tfrac{1}{2}$ test still requires an independent observable and a typed bridge to ${V_{\rho}}$.