Born–Infeld Parity Forces an Equatorial Admissible Fibre: Analytical Derivation of the [1:½:½] Eigenvalue Ratio in Sym²(Vρ)

Abstract

Papers O28–O29 of the Cosmochrony spectral admissibility sub-programme established that the per-pair covariance $\mathcal{C}_c$ in $\operatorname{Sym}^2(V_\rho)$ has rank $r_{\mathrm{eff}} = 3$ with invariant eigenvalue structure $[\lambda_1: \lambda_2: \lambda_3] = [1: \tfrac{1}{2}: \tfrac{1}{2}]$, and identified this as an open problem requiring an analytical explanation. The present paper provides the complete derivation. This is the first analytical derivation of the invariant covariance spectrum observed in O28: the ratio $\lambda_1/\lambda_2 = 2$ is not dynamical, not statistical, but a purely structural consequence of three compatible structures—BI parity, the $\mathrm{Sym}^2(V_\rho)$ identification, and the Heisenberg grading. We prove that the anti-linear conjugation identity $\rho_{q-c} = \bar{\rho}_c$ (O17; recalled in O18 Theorem 3.1) forces every admissible square-root coordinate $w_j \in V_\rho \cong \mathbb{C}^2$ (with $\pi_c(v_j) = w_j \otimes w_j$, ${H_{\mathrm{eff}}} \cong {\operatorname{Sym}}^2(V_\rho)$) to satisfy $|\alpha_j| = |\beta_j|$, where $(\alpha_j, \beta_j)$ are the components in the canonical basis $\{v_+, v_-\}$ of $V_\rho$ aligned with the BI involution. This is the equatorial constraint: every $w_j$ lives on an equatorial $S^1 \subset S^3$ in $V_\rho$, admitting the canonical form $w_j = (r_j/\!\sqrt{2})(e^{i\phi_j} v_+ + e^{-i\phi_j} v_-)$. In the equatorial regime, the central component $\sqrt{2}\,\alpha_j \beta_j$ of $\operatorname{vec}(M_j)$ is phase-independent (equal to $r_j^2/\!\sqrt{2}$ for all $j$), while the two diagonal components $\alpha_j^2$ and $\beta_j^2$ oscillate as $e^{\pm 2i\phi_j}$. The covariance then diagonalises immediately under the single condition $\langle e^{2i\phi_j} \rangle = 0$ (phase decorrelation), yielding $\mathcal{C}_c = \operatorname{diag}\!\bigl(\tfrac{1}{4}, \tfrac{1}{2}, \tfrac{1}{4}\bigr) \langle r_j^4 \rangle$ and spectral ratio $[2:1:1]$, normalised $[1:\tfrac{1}{2}:\tfrac{1}{2}]$. The factor of $2$ is traced to the degree-$2$ Carnot weight of the central generator $Z = [X, Y]$ in $\mathrm{Heis}_3$: the bilinear term $\sqrt{2}\,\alpha_j\beta_j$ concentrates exactly twice the covariance of each pure-degree-$1$ term $\alpha_j^2$ or $\beta_j^2$. The correction to the equatorial condition is $O(q^{-1/2})$ from the universality rate of, consistent with the symmetry-ratio measurements of O29. This closes the open direction explicitly stated in O28 and O29.