Birch and Swinnerton–Dyer Conjecture
1 · Abstract — precise statement
Birch and Swinnerton–Dyer Conjecture — For an elliptic curve E over ℚ, the order of vanishing of its L-function at s = 1 equals the rank of the Mordell–Weil group: ord_{s=1} L(E,s) = rank E(ℚ). (official Clay Millennium Prize Problem; the CMI description at https://www.claymath.org/millennium-problems/ remains authoritative).
2 · Introduction
MODELED CHALLENGE / algebraic pair-structure probe: (ℤ/9)* inverse pairs recompute. Explicit GAP: sealed src has no elliptic-curve L(E,s) or Mordell–Weil rank computation — neighbourhood pair algebra only, NOT BSD. Challenge methods recomputed at call time: zeroDivisionTable inverse pairs (2,5)/(4,7); group-law neighbourhood via pair closure. 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) · group-law neighbourhood via pair closure4 · Results & status
partial — Status triad: computable=true (sealed challengeMethod path) · open for prize=true. MODELED CHALLENGE status=modeled-partial; gap=no sealed L(E,s) vanishing-order or elliptic-curve rank fold. computable ≠ CMI Prize solution.
no sealed L(E,s) vanishing-order or elliptic-curve rank fold
5 · References & locks
- claySolvedByThisFold
- 0
- physicalFtlClaim
- 0
- fold
- millenniumProblemsChallenge
Theorem — the proof, per facet
- ✓
f₁ inverse pairs — (ℤ/9)* has exactly the 2 non-trivial inverse pairs {(2,5),(4,7)} with a·b ≡ 1 (mod 9): a finite abelian-group neighbourhood probe - ✓
f₂ no L-function — bsdHasLFunction=false: the sealed src carries NO elliptic-curve L(E,s) and NO Mordell–Weil rank, so the row can only state pair algebra, not BSD
- gap algebra
BSD ⟺ ∀ elliptic curve E/ℚ: ord_{s=1} L(E,s) = rank E(ℚ) — equality of an analytic order and an algebraic rank, over all E - gap algebra
known (cited): true for analytic rank 0 and 1 (Gross–Zagier 1986, Kolyvagin 1989) · rank ≥2 OPEN · L(E,s) analytic continuation itself rides modularity (Wiles, Taylor–Wiles, BCDT) - gap algebra
the fold computes (ℤ/9)* inverse pairs — a finite group neighbourhood — with no L-function and no rank, so it does not approach BSD
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