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) q² + 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 r² = -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 L² 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).