The Discriminant Structure of Frey Curve Isogeny Classes: Crater-Peak Duality, the Volcano Law, and a Mass Criterion

Let $a,b,c$ be coprime positive integers with $a+b=c$, and let $E: y^2=x(x-a)(x+b)$ be the associated Frey curve. We determine the structure of minimal discriminants across the Q-isogeny class of E by closed formulas in (a,b,c). The three 2-torsion points of E are rational, so E is always the centre of its class; the first-level quotients via Velu's formula have monomial model discriminants {16(abc)^2, 256a^4bc, 256ab^4c, 256abc^4}. We prove the Crater-Peak Law: among the first-level vertices the minimal and maximal discriminants are min{16(abc)^2, 256 min(a,b)^4 max(a,b)c} and 256abc^4; for star classes these are the class extrema, while on square or quartic triggers the extrema are the minimum and maximum over ten explicit candidates. The class size |V| in {4,6,8} is decided by the square and quartic character of ac and bc (107 samples, zero failures). When ac or bc is a square the class extends to depth 2 with model discriminants 2^10 AC u^8 v^2 and -2^10 AC u^2 v^8, and to depth 3 on quartic triggers with discriminants 2^11 pq u^16 v^4 e and 2^11 pq u^4 v^16 e. On the multiplicative bad-prime set we prove the Volcano Law: each vertex carries an integer height vector k(V) with v_p(Delta^min_V) = 2^{k_p(V)} v_p(Delta^min_crater), and along every 2-edge k maps to k+sigma with sigma in {+-1}^pi independently per prime (578 edge-prime samples, zero exceptions), via the Tate parameterization. For odd primes p dividing min(a,b) we have v_p(j) = -2 v_p(min). The peak curve is explicit (y^2 = x^3 - 2(b-a)x^2 + c^2 x), and the j-denominator exponent is a theorem: log den(j)/log num(j) = (alpha+2)/3 + O(1/log c), with the exact discrepancy identity verified to machine precision on 2185 samples; we also show the Pasten mechanism is information-free on the Frey family. As an application, for A^x + B^y = C^z we give an explicit two-branch formula for the crater ratio rho_c; and with alpha = log min / log c we prove sigma_max -> (5+alpha) q, so Szpiro is equivalent on the Frey family to the mass bound q <= 6/(5+alpha), sharp at alpha = 1 where Szpiro and abc coincide, while the residual gap lives at alpha = 0, the Pillai/S-unit corner. A Sage-verified table of 17 classes gives sigma_max > 6 in 14 of 17 rows (non-refutations: Szpiro allows O(1)/log N slack). We prove no conjecture: abc, Szpiro, Beal and the height conjecture remain open.