A Two-Witness Criterion for Strong BSD in Positive-Density Non-Semistable Quadratic-Twist Families

Formally abstract the complete proof for the single curve 696.e1 from document 29 into a reusable sufficient criterion, with the status bar likewise stating "no claim of novelty or priority." Definition 2.1 defines the seven conditions of a "two-witness BSD certificate": (T1) 2-primary anchor; (T2) $S_3$ two-division field; (T3) controlled non-semistability (the only additive prime is $2$); (T4) first ramification witness (prime $3$, $v_3(\Delta_E)=1$); (T5) nonsplit Fouquet–Wan witness (there exists $\lambda\ne3$ such that $E$ is nonsplit multiplicative at $\lambda$ and $v_\lambda(\Delta_E)=1$); (T6) residual irreducibility; (T7) optimality after twisting—the document specifically notes that this last condition is deliberately added solely so that the period comparison does not rely on any auxiliary isogeny arguments. Theorem 4.1 proves: any curve satisfying T1-T7 has a positive-density prime family $\mathcal P_E$ ($\delta(\mathcal P_E)=\frac{[L_E\cap K_E:\mathbf Q]}{3[K_E:\mathbf Q]}>0$) such that every quadratic twist of it satisfies strong BSD. The proof structure is completely parallel to document 29 (seven steps covering the 2-part, residual irreducibility, additive twist primes, good ordinary, fixed multiplicative, good supersingular, and exhaustion), simply replacing the fixed $29$ with any $\lambda$ satisfying the conditions. 696.e1 is demoted here to Corollary 5.1—just a specific instance when $\lambda=29$, no longer the protagonist. Section 6, "What the criterion does and does not say," is the most honest part of the entire text: it explicitly admits that the valuation-one hypothesis is stronger than necessary, serving only to make the witness for every odd prime uniform and eliminate the need for exceptional prime auditing; and it explicitly points out two immediately actionable directions for generalization—a gcd-witness criterion (replacing valuation 1 with a looser condition like "gcd has no odd prime factors") and an odd-additive extension (allowing finitely many odd additive primes, verifying Fouquet–Wan's local conditions one by one). The document summarizes the overall architecture in one sentence: finite base certificate → positive-density Chebotarev support → all-prime BSD routing.