🎯 hexa-millennium

HEXA-Millennium family — millennium problems · math · Clay 7 n=6 closed-form candidates

BSD · Hodge · Navier–Stokes · P-vs-NP · Poincaré · Riemann · Yang–Mills · n=6 lattice · candidate-spec · falsifier-preregister

---

Clay 7 Millennium Problems — n=6 closed-form candidates (BSD / Hodge / Navier–Stokes / P-vs-NP / Poincaré / Riemann / Yang-Mills).

---

Why

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>.md candidate spec (the n=6 closed-form angle), and
• where present, a verify_millennium-<slug>.hexa lightweight verifier.

This is organizing hypothesis material, not a Clay-prize submission. See §Status below.

---

Status

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 canon/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).

---

n=6 invariant lattice

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.

---

Install

# 1. Install hexa-lang (gives you `hexa` + `hx` package manager)
/bin/bash -c "$(curl -fsSL https://raw.githubusercontent.com/dancinlab/hexa-lang/main/install.sh)"

# 2. Install hexa-millennium
hx install hexa-millennium

Run

hexa-millennium bsd             # Birch–Swinnerton-Dyer                  [CANDIDATE]
hexa-millennium hodge           # Hodge Conjecture                       [CANDIDATE]
hexa-millennium navier_stokes   # Navier–Stokes existence/smoothness     [CANDIDATE]
hexa-millennium p_vs_np         # P vs NP                                [CANDIDATE]
hexa-millennium poincare        # Poincaré (Perelman 2003 — n=6 verif)   [VERIFICATION SPEC]
hexa-millennium riemann         # Riemann Hypothesis                     [CANDIDATE]
hexa-millennium yang_mills      # Yang–Mills mass gap                    [CANDIDATE]
hexa-millennium status          # 7-verb status table + verdict + caveats
hexa-millennium selftest        # 7-verb sentinel sweep (directory presence)
hexa-millennium version         # print version
hexa-millennium help            # full --help (subcommands + env vars)

---

Verify

Spec-first closure verification (4-script SPEC_FIRST pattern, sibling of hexa-matter/hexa-rtsc/hexa-cern/hexa-fusion):

# Aggregate sweep (4/4 scripts):
hexa run verify/run_all.hexa

# Per-anchor:
hexa run verify/spec_presence.hexa         # 7/7 Clay-verb candidate-spec docs
hexa run verify/lattice_arithmetic.hexa    # n=6 self-consistency (aux only)
hexa run verify/real_limits_anchor.hexa    # LIMIT_BREAKTHROUGH anchors
hexa run verify/closure_consistency.hexa   # CLI · toml · README scoreboard

Closure shape (4 tiers):

#	Script	What it checks
1	spec_presence.hexa	7/7 millennium-<slug>.md candidate-spec files exist on disk
2	lattice_arithmetic.hexa	σ·φ = n·τ = 24, σ(6)=12, τ(6)=4, φ(6)=2, sopfr(6)=5 — aux, never sole evidence
3	real_limits_anchor.hexa	LIMIT_BREAKTHROUGH.md references Gödel, ZFC, Clay 2-yr review, Perelman 2003
4	closure_consistency.hexa	CLI ↔ hexa.toml ↔ README badge all agree on 7 verbs, 0 formal proofs

• PASS means 7 verbs ship a candidate spec + falsifier preregister. It does NOT mean any Clay Millennium Problem has been solved by this repo. Acceptance requires a formal Lean4/Coq proof artifact plus Clay-mandated 2-year community review.
• Gödel HARD_WALL acknowledged: no closed-form expression over the n=6 lattice substitutes for a formal proof in the host theory (ZFC + Peano + relevant axioms). ZFC's own consistency is unprovable in ZFC (Gödel 2nd, 1931).
• P vs NP open since Cook 1971. Riemann open since 1859. BSD open since 1965. Hodge open since 1950. Navier–Stokes open since Leray 1934. Yang–Mills mass gap open since 1954/1969.
• Poincaré is RESOLVED by Perelman 2003 (Ricci flow with surgery). The poincare/ verb is an n=6 verification-side spec that rides alongside Perelman's proof, not a re-solution.
• Clay Mathematics Institute, Perelman, and Fields medallists use their own proofs/methods — this repo imposes no lattice-fit on their work.

---

Repo layout

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
├── verify/                           # 4-script SPEC_FIRST closure (run_all + 3 anchors)
├── LATTICE_POLICY.md                 # real-limits-first policy (dancinlab-wide)
├── LIMIT_BREAKTHROUGH.md             # Gödel HARD_WALL + Clay/medium/lang walls
├── 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)

---

Provenance

• Verb directories extracted from canon/domains/physics/millennium-* at SHA c0f1f570 (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.

---

License

MIT.