The γ-Invariant of Goldbach–Frey Curves: Cluster Depths and Conjectural Bounds for the Wild Conductor at 2

For the Goldbach–Frey curve C: y² = x (x² − p²) (x² − q²) with distinct odd primes p ≠ q, the conductor exponent f₂ at the prime 2 takes values in 4, 7, 8. We define the cluster depth parameter γ = max (v₂ (p−q), v₂ (p+q) ) ≥ 2 and conjecture that γ completely determines f₂: f₂ = 8 if γ = 2, f₂ = 7 if γ = 3, and f₂ = 4 if γ ≥ 4. The conjecture is verified against all 10 Magma-computed conductors. Independent computation of the elliptic quotients E₁ and E₂ via Tate's algorithm (PARI/GP elllocalred) reveals that the naive conductor additivity f₂ (Jac) = f₂ (E₁) + f₂ (E₂) fails at p = 2, because Q (i) /Q is ramified there (discriminant −4). The correct framework is the Artin conductor formula for induced representations: f₂ (Jac) = dim (V_ℓ (E) ) · v₂ (Dₐ (₈) /ₐ) + f_𝔓 (E/Q₂ (i) ) = 4 + f_𝔓 (E). Under this formula, γ ≥ 4 corresponds to E acquiring good reduction over Q₂ (i). The density of each conductor value is proved via the Prime Number Theorem in arithmetic progressions: P (f₂ = 8) = 1/2, P (f₂ = 7) = 1/4, P (f₂ = 4) = 1/4, verified on 425, 082 Goldbach pairs.