the smallest curves witness Birch–Swinnerton-Dyer — finite kernel, open bridge
Theorem.the smallest curves witness Birch–Swinnerton-Dyer — finite kernel, open bridge — rank E(ℚ) vs ord_{s=1} L(E, s) agree on the smallest curves — a finite WITNESS toward BSD, the conjecture stays open.
Proof.the Clay description's finite kernel recomputed exactly, the conjecture claimed nowhere: (1) Euclid COMPLETE — the (m,n) parametrization is a verified bijection with the brute-forced primitive Pythagorean triples to hypotenuse 100; (2) the rank-0 pole — x⁴+y⁴=z² has no nontrivial solution in the sweep, Fermat's infinite descent (cited, elementary, complete) extends it to ALL sizes, so 1 is not a congruent number and y²=x³−x has only its four torsion points; (3) the rank-≥1 pole — P=(−4,6) lies on y²=x³−25x and 2P computes to a NON-INTEGRAL point in exact arithmetic, so by Nagell–Lutz P has infinite order: infinitely many rational points, and the (3/2, 20/3, 41/6) right triangle of area exactly 5 realizes the congruent number; (4) Tunnell's counts (unconditional direction) separate the poles: 2A₁ ≠ B₁ refuses n=1, 2A₅ = B₅ admits n=5; (5) the associated L-function is a computed object — aₚ from exact 𝔽ₚ point counts, Hasse |aₚ| ≤ 2√p and the CM vanishing aₚ = 0 ⟺ p ≡ 3 (mod 4) verified across the prime range. Both curves have CM, so for THEM the zeta-sees-the-points bridge is a theorem (Coates–Wiles 1977; Gross–Zagier 1986, Kolyvagin 1988) — the GENERAL conjecture stays OPEN and UNCLAIMED (Millennium); Hilbert's tenth is unsolvable (Matiyasevich 1970, cited).
Checked by exact arithmetic over the stated finite range — a verified witness, evidence toward the claim, not a ∀-proof.
src/7/3/index.ts#theSmallestCurvesWitnessBirchSwinnertonDyer
1 · Classification
bounded-witness — computed witness within a cited frame (the unbounded form leans on the cited literature)
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"the smallest curves witness Birch–Swinnerton-Dyer — finite kernel, open bridge" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation theSmallestCurvesWitnessBirchSwinnertonDyer.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (theSmallestCurvesWitnessBirchSwinnertonDyer @ src/7/3) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes theSmallestCurvesWitnessBirchSwinnertonDyer (src/7/3/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.