Universal Tame Semistability of Goldbach–Frey Jacobians: A Proof of fᵣ = 2 at All Odd Primes

We prove that for the Goldbach–Frey curve C: y² = x (x² − p²) (x² − q²) with p ≠ q distinct odd primes and p + q = 2N, the conductor exponent of Jac (C) at every bad odd prime r equals exactly 2. The proof classifies bad odd primes into four mutually exclusive cases determined by root collision patterns modulo r, and shows that in each case the identity component of the Néron model's special fibre satisfies 2a + t = 2. Two geometrically distinct mechanisms conspire to produce the same conductor exponent: a single A₂ cusp yielding (a, t, u) = (1, 0, 1), or two A₁ nodes yielding (a, t, u) = (0, 2, 0). This establishes the exact odd conductor formula Condₒdd (Jac (C) ) = radₒdd (p·q·N· (p−q) ) ² and completes the theoretical foundation of the Band Shifting Law.