An Explicit Non-Semistable Quadratic-Twist Family Satisfying the Strong BSD Conjecture

The entire argument built up along the main thread from documents 00-28 is written here as a complete, formal theorem note with numbered Lemmas/Propositions and real literature citations. The author is listed as Neo.K, and the status bar states: "Records an explicit derived conclusion from cited theorems, claims no priority, and should be independently reviewed before submission." Theorem 1.1 states: For the prime family $\mathcal P=\{q: q\equiv1\pmod{24},\ (\frac q{29})=1,\ f_2\bmod q\text{ irreducible}\}$, the density is $\delta(\mathcal P)=\frac1{24}$, and $\forall q\in\mathcal P$, $\operatorname{BSD}(E^{(q)})$ holds. Sections 3-6 use six numbered lemmas to close each link one by one: Lemma 3.1/3.2 (splitting and good ordinary properties of support primes), Proposition 4.1 (Chebotarev density, with Remark 4.2 listing the first ten concrete primes 241, 313, 457, 673, 937, 1009, 1153, 1753, 2017, 2089), Proposition 5.1 (2-part), Lemma 6.1-6.5 (irreducibility, additive twist prime witness $\ell=29$, good ordinary, fixed multiplicative $3/29$ acting as mutual witnesses, good supersingular using FW Theorem 1.7 + Corollary 1.10 and closing the period issue via the Manin constant results of Česnavičius–Neururer–Saha), and Proposition 6.6 (exhaustion of the four classes). Section 8 reports a rare real numerical check for the entire series: scanning $q<10^7$ found $27{,}667$ support primes, compared to $\pi(10^7)=664{,}579$, yielding an empirical ratio of $\approx0.041630867$, which highly matches the theoretical density of $\frac1{24}\approx0.041666667$ — the document explicitly states this is merely a sanity check and not part of the proof. The Concluding remarks summarize the structure of the entire result in one sentence: $2$-part + additive twist prime + ordinary/multiplicative parts + supersingular part, and pose a clearly unaddressed extension problem: finding other non-semistable rank-zero curves that possess two odd multiplicative ramification reservoirs (at least one nonsplit) and a compatible 2-primary twist family. Appendix A provides a complete finite data list for independent reproduction in SageMath/Magma, specifically designed so that if any input is incorrect, the error will be localized to a single lemma rather than hidden within the overall argument.