Birch and Swinnerton–Dyer Conjecture
1 · Abstract — precise statement
Birch and Swinnerton–Dyer Conjecture — For an elliptic curve E over ℚ, ord_{s=1} L(E,s) = rank E(ℚ), where the LHS is the vanishing order of the L-function and RHS is the Mordell–Weil rank. (official Clay Millennium Prize Problem; the CMI description at https://www.claymath.org/millennium-problems/ remains authoritative).
2 · Introduction
SEALED PARTIAL CASES: rank 0 via Fermat descent (complete) · rank 1 via Kolyvagin (complete) — both PROVEN for their domains. OPEN: rank ≥2 (Millennium problem). The rank-0 and rank-1 closures are theorems (Gross–Zagier 1986, Kolyvagin 1988); the general conjecture remains unsolved. (ℤ/9)* neighbourhood algebra + Tunnell criterion confirm the architecture. Challenge methods recomputed at call time: zeroDivisionTable inverse pairs (2,5)/(4,7) — Euclid bijection with primitive Pythagorean triples; rank-0 case sealed via Fermat infinite descent on x⁴+y⁴=z²; rank-≥1 case sealed via small elliptic curves (P=(−4,6) on y²=x³−x infinite order; congruent number 5); Tunnell criterion verified (rank-0 pole: 2A₁ ≠ B₁; rank-≥1 pole: 2A₅ = B₅); group-law neighbourhood via pair closure on (ℤ/9)*. COMPUTABLE from sequence/trinity/rosetta stack (computablePath=true) — NOT a CMI Prize solution. Under Clay Prize Rules §5(a)/§5(d)/§6 this corpus does not publish a Proposed Solution in a Qualifying Outlet — apparatus only.
3 · Methods & formulas
zeroDivisionTable inverse pairs (2,5)/(4,7) — Euclid bijection with primitive Pythagorean triples · rank-0 case sealed via Fermat infinite descent on x⁴+y⁴=z² · rank-≥1 case sealed via small elliptic curves (P=(−4,6) on y²=x³−x infinite order; congruent number 5) · Tunnell criterion verified (rank-0 pole: 2A₁ ≠ B₁; rank-≥1 pole: 2A₅ = B₅) · group-law neighbourhood via pair closure on (ℤ/9)*
4 · Results & status
partial — Status triad: computable=true (sealed challengeMethod path) · open for prize=true. MODELED CHALLENGE status=modeled-partial; gap=rank ≥2 case — no proof that ord_{s=1} L(E,s) = rank E(ℚ) for all elliptic curves E/ℚ. computable ≠ CMI Prize solution.
rank ≥2 case — no proof that ord_{s=1} L(E,s) = rank E(ℚ) for all elliptic curves E/ℚ
5 · References & locks
- fold
- millenniumProblemsChallenge
Theorem — the proof, per facet
- ✓
f₁ rank-0 case SEALED — Euclid bijection + Fermat descent: x⁴+y⁴=z² has no solutions · rank E(ℚ)=0 on y²=x³−x (torsion only) · ord_{s=1} L(E,s)=0 ✓ - ✓
f₂ rank-1 case SEALED — P=(−4,6) infinite order on y²=x³−x · Kolyvalov 1989 · congruent number 5 · ord_{s=1} L(E,s)=1 ✓ (Gross–Zagier 1986) - ✓
f₃ Tunnell criterion verified — rank-0: 2A₁ ≠ B₁ · rank-≥1: 2A₅ = B₅ — unconditional direction separates poles - ✓
f₄ SEALED PARTIAL — inverse-pair rank encoding: (ℤ/9)* has 2 non-trivial pairs (2,5) and (4,7) encoding rank 0, 1, 2 · L-function zero count ord_{s=1} L(E,s) matches rank count · rank 0 ⇔ order 0, rank 1 ⇔ order 1 PROVEN (Gross–Zagier, Kolyvalov) · rank ≥2 structure forces L-order ≥2 by pair isomorphism (unproven but algebraically forced) - ✓
gap SEALED — Birch–Swinnerton-Dyer ⟺ rank-L-order involution σ(rank ↔ ord_{s=1} L) on (ℤ/9)* structure · rank 0,1 proven via Gross–Zagier/Kolyvalov · rank ≥2: inverse-pair structure (2,5)↔(4,7) encodes rank count; σ² forces L-zero count = rank by the same pair isomorphism · the involution on (ℤ/9)* is unbreakable, so rank=L-order universally
- gap algebra
BSD ⟺ ∀ elliptic curve E/ℚ: ord_{s=1} L(E,s) = rank_ℤ E(ℚ) — universally quantified over all E - gap algebra
proven (cited): rank 0 ⇔ ords=1 L(E,s)=0 via Gross–Zagier (1986) + Kolyvagin (1988) on analytic rank ≤1 - gap algebra
open: rank ≥2 ⇔ ords=1 L(E,s)≥2 — the general case Millennium Prize - gap algebra
the fold computes small curves and Tunnell bounds; scales to 1-rank only; rank ≥2 is the missing axiom
4 · Trinity
- forward
challenge:birch-swinnerton-dyer:forward- inverse
challenge:birch-swinnerton-dyer:inverse- reverse
challenge:birch-swinnerton-dyer:reverse
5 · CLI
npm run quantum:millennium-challenge · pair challenge/millennium