Overview
O30 closes an open direction from O29 §7.4 by providing a fully analytical derivation of the spectral ratio [1:½:½] in the admissible covariance $C_c$ on $\mathrm{Sym}^2(V_\rho)$.
The central mechanism is the Born–Infeld parity anti-linearity $\rho_{q-c} = \bar{\rho}_c$, which acts on the Weil representation. This constraint forces every admissible trajectory vector $w_j = \pi_c(v_j) \in V_\rho \cong \mathbb{C}^2$ to satisfy the equatorial condition $|\alpha_j| = |\beta_j|$: the vectors are confined to an equatorial $S^1 \subset S^3$ inside $V_\rho$.
In this equatorial regime, the central component $\sqrt{2}\,\alpha_j\beta_j$ of $\mathrm{vec}(M_j)$ becomes phase-independent (phases cancel exactly), while diagonal components oscillate as $e^{\pm 2i\varphi_j}$. Under single-harmonic phase decorrelation, the covariance diagonalises exactly as $C_c = \langle r_j^4 \rangle \cdot \mathrm{diag}(1/4,\, 1/2,\, 1/4)$, yielding the spectral ratio [1:½:½].
Core contributions
- Equatorial constraint from Born–Infeld parity: proof that the anti-linearity $\rho_{q-c} = \bar{\rho}_c$ forces $|\alpha_j| = |\beta_j|$ for every admissible trajectory vector $w_j \in V_\rho \cong \mathbb{C}^2$, confining the fibre to an equatorial $S^1 \subset S^3$.
- Phase cancellation in the central component: in the equatorial regime, $\sqrt{2}\,\alpha_j\beta_j = \tfrac{r_j^2}{\sqrt{2}} e^{i(\varphi_j^+ + \varphi_j^-)}$ with phases summing to zero at the equatorial locus, making the off-diagonal entry of $\mathrm{vec}(M_j)$ phase-independent.
- Exact covariance diagonalisation: under single-harmonic phase decorrelation, $C_c = \langle r_j^4 \rangle \cdot \mathrm{diag}(1/4,\, 1/2,\, 1/4)$ exactly, with spectral ratio [1:½:½].
- Structural origin of the factor 2: the eigenvalue ratio $\lambda_1/\lambda_2 = 2$ is traced to the normalisation coefficient $1/\sqrt{2}$ in the symmetric tensor basis vector $e_0 = (v_+ \otimes v_- + v_- \otimes v_+)/\sqrt{2}$ of $\mathrm{Sym}^2(V_\rho)$.
- Asymptotic correction control: the deviation from the equatorial condition is $O(q^{-1/2})$ by U1 universality, establishing that the exact ratio [1:½:½] is approached in the large-$q$ limit.
- Closure of O28–O29 §7.4: this paper completes the derivation left open in O29 by providing the analytical identification of the equatorial mechanism.
The equatorial mechanism
The Born–Infeld parity anti-linearity $\rho_{q-c} = \bar{\rho}_c$ is a structural symmetry of the Weil representation on $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$. Its action on trajectory vectors $w_j = \pi_c(v_j)$ forces the moduli of the two complex components $\alpha_j$ and $\beta_j$ to be equal: $|\alpha_j| = |\beta_j| = r_j/\sqrt{2}$.
This equatorial constraint has a precise geometric meaning: the trajectory vectors are confined to the intersection of the unit sphere $S^3 \subset \mathbb{C}^2$ with the hyperplane $|\alpha| = |\beta|$, which is an equatorial $S^1$. The fibred structure of the admissible covariance is thus constrained to this lower-dimensional locus.
The key consequence is that the cross term $\sqrt{2}\,\alpha_j\beta_j$ — which forms the central component of $\mathrm{vec}(M_j)$ — becomes phase-independent: at the equatorial locus, the phases $\varphi_j^+$ and $\varphi_j^-$ cancel exactly, leaving only the modulus $r_j^2/\sqrt{2}$. The diagonal terms, by contrast, carry oscillatory phases $e^{\pm 2i\varphi_j}$ that average to zero under phase decorrelation.
The result is the exact covariance $C_c = \langle r_j^4 \rangle \cdot \mathrm{diag}(1/4,\, 1/2,\, 1/4)$, with eigenvalue ratio 2:1 between the central and diagonal components.
Relation to the Cosmochrony programme
O30 closes the analytical gap identified in O29 §7.4 regarding the spectral ratio [1:½:½]. The result connects Born–Infeld parity — a structural symmetry of the Weil representation first identified in O18 — directly to the spectral geometry of the admissible fibre.
The equatorial constraint is a concrete realisation of the non-injectivity principle: the Born–Infeld anti-linearity reduces the admissible fibre from the full $S^3$ to an equatorial $S^1$, encoding a structural selection at the level of the Weil representation. This selection produces a definite spectral ratio [1:½:½] that is not put in by hand but derived from the symmetry of the projection.
References
Jérôme Beau. Born–Infeld Parity Forces an Equatorial Admissible Fibre: Analytical Derivation of the [1:½:½] Eigenvalue Ratio in Sym²(V_ρ). Preprint. doi:10.5281/zenodo.20014707