The Universal Two-Prime Formula and Spectral Structure of Palindromic Distance Matrices

For any odd integer m ≥ 3, the palindromic distance ratio R(m) is the determinant ratio of the even and odd blocks of the symmetric distance matrix on the coprime residues modulo m. This paper proves the universal two-prime formula: for distinct odd primes p and q, R(pq) = [-3p(p-2)q^2 + 2p(3p-7)q + 3] / 6. The formula is verified against direct computation at 32 two-prime products with zero failures, and yields a complete denominator theorem (denominator equals 1 when 3 divides pq, and 3 otherwise), a symmetry identity (R(pq) = R(qp) as a polynomial identity), and a proof of the companion paper's linear recurrence as a special case. Beyond the determinant ratio, spectral analysis of the palindromic blocks across the primorial tower m = 30, 210, 2310, 30030, 510510 establishes: (1) the maximum eigenvalue of the odd block converges to a new arithmetic constant approximately -0.70704313, with deficit from -1/sqrt(2) exhibiting diverging algebraic complexity — no annihilating polynomial of degree at most 24 with bounded coefficients exists, as confirmed by 200-digit PSLQ search; (2) the Perron eigenvalue of the even block grows as (1/2) times the product of p(p-1) over odd prime factors, with base constant C = 1/2 exact; (3) the Double Helix Parity Law — in the FFT power spectrum of the eigenvalue ratio, the ancestor primorial shadow peak from the nearest same-parity ancestor (p mod 4) always dominates, verified at all three testable primorial levels with dominance factors from 2.3x to 17.5x. The paper proves the Permanent Irreconcilability Theorem: no coprime residue r satisfies both the palindromic involution (r maps to m-r) and multiplicative inversion (r maps to r⁻¹) simultaneously, because this would require r^2 = -1 mod m, which is impossible since 3 divides every primorial and -1 is not a quadratic residue mod 3. The Frobenius norm of the commutator [D, tau] normalized by the Perron eigenvalue converges to sqrt(2/3), a universal constant of circular geometry proved analytically via three independent limits: exact L^2 isometry of the inversion map, alignment converging to 3/4 by decorrelation, and spectral ratio converging to 2/sqrt(3) by equidistribution. The Chebyshev Cooling Law identifies the convergence rate mechanism: the inversion displacement variance retreats from its uniform limit at primorial level k if and only if the number of prime factors congruent to 1 mod 4 equals the number congruent to 3 mod 4 — when the Chebyshev prime race is exactly tied. Verified at 8 consecutive primorial levels with zero contradictions, after falsifying three alternative hypotheses (Grand Character, Kloosterman sums, k-parity orientability). The next predicted retreat occurs at k = 13 (p = 41). Companion paper: Primorial Thermodynamics: Gauss Sums, the IPB98 Scaling Law, and the Hierarchy Break at m = 30030 (DOI: 10.5281/zenodo.19188924).