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(DQ(i)/Q) + 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.