Poincaré Conjecture
1 · Abstract — precise statement
Poincaré Conjecture — Every simply-connected closed 3-manifold is homeomorphic to the 3-sphere: π₁(M) = 0 with M a closed 3-manifold ⟹ M ≅ S³. (Proved: Perelman 2003, Ricci flow with surgery.) (official Clay Millennium Prize Problem; the CMI description at https://www.claymath.org/millennium-problems/ remains authoritative).
2 · Introduction
SOLVED EXTERNAL (Perelman 2003, Ricci flow with surgery) — this corpus does not re-solve it. Challenge only verifies the documented solved status plus the genus-2 homology analogy (H₁=ℤ⁴). NOT a new proof. Challenge methods recomputed at call time: theSevenMillenniumProblemsDefinedFormallyUnclaimed; homology rank 4 structural analogy. 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
every simply connected closed 3-manifold is homeomorphic to S³4 · Results & status
solved-external — Status triad: computable=true (sealed challenge path) · open for prize=false (solved-external, Perelman 2003) — documented status only; this fold does not re-solve or claim the CMI Prize.
5 · References & locks
- claySolvedByThisFold
- 0
- physicalFtlClaim
- 0
- fold
- millenniumProblemsChallenge
Theorem — the proof, per facet
- ✓
f₁ solved (external) — π₁(M)=0, M closed 3-manifold ⊢ M ≅ S³ (Perelman 2002–03, Ricci flow with surgery, completing Hamilton); this fold verifies the DOCUMENTED status, it does not re-prove - ✓
f₂ homology analogy — H₁(Σ₂)=ℤ⁴ recomputes (432/108=4); frontier.solved=1 — the single solved core among the seven
- gap algebra
No open gap: proved externally (Perelman 2002–03) — Ricci flow with surgery drives every simply-connected closed 3-manifold to S³; this corpus records the status, it does not re-solve it
4 · Trinity
- forward
challenge:poincare:forward- inverse
challenge:poincare:inverse- reverse
challenge:poincare:reverse
5 · CLI
npm run quantum:millennium-challenge · pair challenge/millennium