Born–Infeld Parity Forces an Equatorial Admissible Fibre: Analytical Derivation of the [1 : 1/2 : 1/2] Eigenvalue Ratio in Sym^2(Vρ)

Papers O28--O29 of the Cosmochrony spectral admissibility sub-programme established that the per-pair covariance Cc in Sym²(V_ρ) has rank reff = 3 with invariant eigenvalue structure [λ₁ : λ₂ : λ₃] = [1 : 1/2 : 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 λ₁/λ₂ = 2 is not dynamical, not statistical, but a purely structural consequence of three compatible structures---BI parity, the Sym²(V_ρ) identification, and the Heisenberg grading. We prove that the Born--Infeld parity anti-linearity ρq-c = ρ̄c (O18) forces every admissible trajectory vector wⱼ = πc(vⱼ) ∈ V_ρ C² to satisfy |αⱼ| = |βⱼ|, where (αⱼ, βⱼ) are the components in the canonical basis ₊, v₋\ of V_ρ aligned with the BI involution. This is the equatorial constraint: every wⱼ lives on an equatorial S¹ ⊂ S³ in V_ρ, admitting the canonical form wⱼ = (rⱼ/\!√2)(eiφⱼ v₊ + e-iφⱼ v₋). In the equatorial regime, the central component √2\,αⱼ βⱼ of vec(Mⱼ) is phase-independent (equal to rⱼ²/\!√2 for all j), while the two diagonal components αⱼ² and βⱼ² oscillate as e± 2iφⱼ. The covariance then diagonalises immediately under the single condition e2iφⱼ = 0 (phase decorrelation), yielding Cc = diag\!(1/4, 1/2, 1/4) rⱼ⁴ and spectral ratio $[2:1:1]$, normalised [1:1/2:1/2]. The factor of $2$ is traced to the degree-$2$ Carnot weight of the central generator $Z = [X, Y]$ in Heis₃: the bilinear term √2\,αⱼβⱼ concentrates exactly twice the covariance of each pure-degree-$1$ term αⱼ² or βⱼ². The correction to the equatorial condition is O(q-1/2) from the universality rate of U1, consistent with the symmetry-ratio measurements of O29. This closes the open direction explicitly stated in O28 and O29.