Clay 7 Millennium Problems — n=6 closed-form candidates (BSD / Hodge / Navier–Stokes / P-vs-NP / Poincaré / Riemann / Yang-Mills).
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The seven Clay Mathematics Institute Millennium Prize Problems remain the
most-watched open questions in mathematics. hexa-millennium collects, in
one standalone substrate, a closed-form candidate for each of the seven
that is anchored to a single algebraic backbone: the n=6 perfect-number
lattice (σ(6)=12, τ(6)=4, φ(6)=2, sopfr(6)=5).
Each of the 7 verbs in this repo ships:
- a
millennium-<slug>.mdcandidate spec (the n=6 closed-form angle), and - where present, a
verify_millennium-<slug>.hexalightweight verifier.
This is organizing hypothesis material, not a Clay-prize submission. See §Status below.
| Verb | Clay problem (one-line definition) | Status |
|---|---|---|
bsd |
Birch–Swinnerton-Dyer: rank of E(ℚ) equals order of vanishing of L(E,s) at s=1. | CANDIDATE (open) |
hodge |
Hodge: every Hodge class on a smooth projective complex variety is a rational combination of algebraic cycles. | CANDIDATE (open) |
navier_stokes |
Navier–Stokes: smooth solutions to the 3D incompressible Navier–Stokes equations exist globally for smooth initial data. | CANDIDATE (open) |
p_vs_np |
P vs NP: every language verifiable in polynomial time is also decidable in polynomial time. | CANDIDATE (open) |
poincare |
Poincaré: every simply connected closed 3-manifold is homeomorphic to S³. Solved by Perelman 2003 — verb is verification-side. | VERIFICATION SPEC (solved) |
riemann |
Riemann: every non-trivial zero of the Riemann ζ-function lies on the critical line ℜ(s) = 1/2. | CANDIDATE (open) |
yang_mills |
Yang–Mills mass gap: quantum Yang–Mills theory on R⁴ for any compact simple gauge group has mass gap Δ > 0. | CANDIDATE (open) |
For per-problem one-paragraph summaries see
examples/clay_problems_overview.md.
Honest disclaimer (raw#10 C3).
Each verb directory contains an n=6 closed-form candidate (candidate) and a falsifier preregister in spec form. It is not a formal proof (not formal proof) of any Clay Millennium Problem.
- 6 of 7 Clay problems (BSD, Hodge, Navier–Stokes, P-vs-NP, Riemann, Yang–Mills) remain open; this repo offers candidate spec + falsifier only.
- Poincaré is solved by Grigori Perelman (2003) using Ricci flow with surgery; the
poincare/verb is an n=6 closed-form verification-side spec that rides alongside Perelman's proof, not a re-solution.- The n=6 perfect-number lattice (σ=12, τ=4, φ=2, sopfr=5) is invoked as an organizing principle; per-problem mathematical justification is not supplied at v1.0.0.
- No lean4 / Coq mechanical layer is bundled here. (Such a layer, if it ever materialises, lives upstream in
n6-architecture/lean4-n6/.)
| Verb | Closure verdict | Formal proof exists? |
|---|---|---|
bsd |
CANDIDATE_SPEC + FALSIFIER_PREREGISTER | No (open) |
hodge |
CANDIDATE_SPEC + FALSIFIER_PREREGISTER | No (open) |
navier_stokes |
CANDIDATE_SPEC + FALSIFIER_PREREGISTER | No (open) |
p_vs_np |
CANDIDATE_SPEC + FALSIFIER_PREREGISTER | No (open) |
poincare |
n=6 VERIFICATION_SPEC | Yes — Perelman 2003 |
riemann |
CANDIDATE_SPEC + FALSIFIER_PREREGISTER | No (open) |
yang_mills |
CANDIDATE_SPEC + FALSIFIER_PREREGISTER | No (open) |
Aggregate verdict: CANDIDATE_SPECS_ONLY (7/7 verbs ship candidate spec
- falsifier; 0/7 formally proved by this repo; 1/7 problems independently solved by Perelman 2003).
The lattice anchors the substrate to a single algebraic identity:
σ(6) = 12 sum of divisors of 6
τ(6) = 4 number of divisors of 6
φ(6) = 2 Euler totient of 6
sopfr(6) = 5 sum of prime factors of 6 (with multiplicity, 2 + 3)
master identity: σ · φ = n · τ = 12 · 2 = 6 · 4 = 24
Each verb's candidate spec maps the lattice to the problem-specific
quantity (e.g. yang_mills derives β₀ = σ − sopfr = 7; bsd invokes a
σ-φ rank lemma; hodge posits σ=12 cohomology classes). These mappings
are the closed-form candidate content.
hx install hexa-millennium # global, pulls latest from registry
hx install hexa-millennium@1.0.0 # pin specific version
hexa-millennium --version # → 1.0.0git clone https://github.com/need-singularity/hexa-millennium.git ~/.hexa-millennium
export HEXA_MILLENNIUM_ROOT=~/.hexa-millennium
export PATH="$HEXA_MILLENNIUM_ROOT/cli:$PATH"
# Drill into any verb:
hexa run $HEXA_MILLENNIUM_ROOT/cli/hexa-millennium.hexa selftest
hexa run $HEXA_MILLENNIUM_ROOT/cli/hexa-millennium.hexa bsdhexa-millennium selftest # 7-verb sentinel sweep
hexa-millennium status # 7-pillar status table + caveats
hexa-millennium bsd # spec head + sentinel
hexa-millennium hodge
hexa-millennium navier_stokes
hexa-millennium p_vs_np
hexa-millennium poincare
hexa-millennium riemann
hexa-millennium yang_mills/Users/ghost/core/hexa-millennium/
├── cli/hexa-millennium.hexa # 7-verb router + status + selftest
├── bsd/ # Birch–Swinnerton-Dyer spec
├── hodge/ # Hodge spec
├── navier_stokes/ # Navier–Stokes spec
├── p_vs_np/ # P vs NP spec
├── poincare/ # Poincaré (verification-side)
├── riemann/ # Riemann spec
├── yang_mills/ # Yang–Mills mass-gap spec
├── tests/test_selftest.hexa # 7-verb sentinel test
├── examples/clay_problems_overview.md
├── install.hexa # hx hook (pre/post)
├── hexa.toml # package manifest
├── LICENSE # MIT
├── CHANGELOG.md
├── RELEASE_NOTES_v1.0.0.md
└── README.md # (this file)
- Verb directories extracted from
n6-architecture/domains/physics/millennium-*at SHAc0f1f570(2026-05-06). - Sister substrate:
hexa-bio(Molecular Toolkit, HEXA family) — same n=6 invariant backbone, different domain. - Standalone-extraction cycle:
hexa_millennium_standalone_extraction_2026_05_06.
MIT.