Papers O28–O29 of the Cosmochrony spectral admissibility sub-programme established that the per-pair covariance Cc in Sym² (V_) has rank r₄₅₅ = 3 with invariant eigenvalue structure ₁: ₂: ₃ = 1: 12: 12, 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 ₁/₂ = 2 is not dynamical, not statistical, but a purely structural consequence of three compatible structures—BI parity, the Sym² (V_) identification, and the Heisenberg grading. We prove that the Born–Infeld parity anti-linearity ₐ-₂ = c (O18) forces every admissible square-root coordinate wⱼ V_ C² (with c (vⱼ) = wⱼ wⱼ, H₄₅₅ Sym² (V_) ) to satisfy |ⱼ| = |ⱼ|, where (ⱼ, ⱼ) are the components in the canonical basis \v_+, v_-\ of V_ aligned with the BI involution. This is the equatorial constraint: every wⱼ lives on an equatorial S¹ S³ in V_, admitting the canonical form wⱼ = (rⱼ/\!2) (e^iⱼ v_+ + e^-iⱼ v_-). In the equatorial regime, the central component 2\, ⱼ ⱼ of vec (Mⱼ) is phase-independent (equal to rⱼ²/\!2 for all j), while the two diagonal components ⱼ² and ⱼ² oscillate as e^ 2iⱼ. The covariance then diagonalises immediately under the single condition e^2iⱼ = 0 (phase decorrelation), yielding Cc = diag\! (14, 12, 14) rⱼ⁴ and spectral ratio 2: 1: 1, normalised 1: 12: 12. 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\, ⱼⱼ concentrates exactly twice the covariance of each pure-degree-1 term ⱼ² or ⱼ². 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.
Jérôme Beau (Mon,) studied this question.