Higher-order constraints + exact arithmetic + certificate verification for the 87-year-old open problem
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║ ▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓ LIQUIDLEAN ▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓ ║
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║ Formal Verification Framework for the Jacobian Conjecture ║
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║ ├─ Four-Language Constitution (m4, HOC, Liquid Haskell) ║
║ ├─ 15 Immutable Architectural Decision Records (ADRs) ║
║ ├─ Claim Levels 0–9 (framework to full proof) ║
║ ├─ Independent Certificate Verifier ║
║ ├─ Exact Arithmetic (no floating-point) ║
║ └─ Apache 2.0 License (sovereign source) ║
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║ Status: Phase 10 Complete | Claim Level 8/9 | 5.4K LoC ║
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LiquidLean is a formal verification system for the Jacobian Conjecture (1939–present). It structures a proof attempt using:
- Four-language constitution enforced at build time (no Lean, no external SMT)
- Refinement types tracking proofs via
{v : T | P v}predicates - Exact arithmetic (rationals, no floating-point in proof code)
- Immutable governance via 15 architecture decision records (ADRs)
- Independent certificate checker (producer ≠ verifier)
- Thermal Monad with φ-decay energy bounds (Phase 4)
Current Status:
- ✅ Restricted cases proved: dimension-1, affine, triangular
⚠️ Full conjecture: reduced to one algebraic-geometric key lemma (Theorem 3)- 📊 25+ modules, ~5,400 lines Haskell, 10 phases, 15 ADRs
- GHC 9.4+ (or cabal 3.8+)
- Haskell toolchain
git clone https://github.com/SNAPKITTYWEST/liquidlean.git
cd liquidlean
cabal update
cabal build# Run all phases
cabal test
# Check status
cabal run liquidlean -- status
# View claim level
cabal run liquidlean -- claim| Language | Role | Scope |
|---|---|---|
| m4 | Deterministic code generation | Build-time only |
| HOC | Declarative theorem specifications | Phase 1 (parser, AST) |
| Liquid Haskell | SMT-verified refinement types | Phases 2–4 (core kernel) |
| Haskell | Trusted execution + verification | All phases (total functions) |
Enforcement: All modules must declare their language tier in module docstring. Build fails on violation.
| Phase | Module | Status | Claim |
|---|---|---|---|
| 0 | ADRs (1–15) + Foundation | ✅ Complete | — |
| 1 | HOC Parser & Printer | ✅ Complete | 1/9 |
| 2 | Refinement Kernel | ✅ Complete | 2/9 |
| 3 | Unification & Constraint Closure | ✅ Complete | 3/9 |
| 4 | Thermal Monad (φ-decay energy) | ✅ Complete | 4/9 |
| 5 | Certificate Kernel (independent) | ✅ Complete | 5/9 |
| 6 | Polynomial Kernel (exact arithmetic) | ✅ Complete | 6/9 |
| 7 | Jacobian Foundation (dim-1) | ✅ Proved | 6/9 |
| 8 | Restricted Theorems (affine, triangular) | ✅ Proved | 6/9 |
| 9 | Full Attempt (dependency graph) | ✅ Complete | 7/9 |
| 10 | Block Decomposition + Key Lemma | 8/9 |
Legend:
- ✅ = Complete and verified
- Proved = Mathematical theorem proved
⚠️ = Blocked on algebraic-geometric barrier (Theorem 3)
liquidlean/
├── README.md (this file)
├── LICENSE (Apache 2.0)
├── CONTRIBUTING.md (development guide)
├── RESEARCH.md (proof strategy + open questions)
│
├── src/LiquidLean/
│ ├── HOC/ (phases 1–2: parser, lexer, pretty-printer)
│ │ ├── Token.hs
│ │ ├── Lexer.hs
│ │ ├── Syntax.hs
│ │ ├── Parser.hs
│ │ ├── Pretty.hs
│ │ └── Diagnostic.hs
│ │
│ ├── Refinement/ (phase 2: predicates + subtyping)
│ │ ├── Predicate.hs
│ │ ├── Value.hs
│ │ ├── Subtyping.hs
│ │ ├── Environment.hs
│ │ └── Judgment.hs
│ │
│ ├── Constraint/ (phase 3: unification + closure)
│ │ ├── Syntax.hs
│ │ ├── Substitution.hs
│ │ ├── Unification.hs
│ │ ├── Closure.hs
│ │ └── Certificate.hs
│ │
│ ├── Thermal/ (phase 4: monad + laws)
│ │ ├── Energy.hs (golden ratio decay)
│ │ ├── Monad.hs
│ │ └── Laws.hs (three monad laws verified)
│ │
│ ├── Certificate/ (phase 5: independent verifier)
│ │ ├── Format.hs
│ │ ├── Parser.hs
│ │ ├── Validate.hs
│ │ └── Replay.hs
│ │
│ ├── Polynomial/ (phase 6: exact arithmetic)
│ │ ├── Natural.hs
│ │ ├── Rational.hs
│ │ ├── Monomial.hs
│ │ ├── Polynomial.hs
│ │ └── Jacobian.hs
│ │
│ └── Jacobian/ (phases 7–10: the conjecture)
│ ├── Statement.hs (exact conjecture formulation)
│ ├── DimensionOne.hs (PROVED: n=1)
│ ├── Affine.hs (PROVED: degree ≤ 1)
│ ├── Triangular.hs (PROVED: triangular structure)
│ ├── FullAttempt.hs (strategy orchestration)
│ ├── BlockDecomposition.hs (reduce to univariate)
│ ├── UnivariateInversion.hs (key lemma: x_n is polynomial)
│ ├── KeyLemmaAttempt.hs (four closure attempts)
│ ├── AlgebraicGeometry.hs (Theorem 3: genus-0 forcing)
│ └── MasterProof.hs (full orchestration + philosophy)
│
├── test/ (unit + property tests)
│ ├── HOC/
│ ├── Refinement/
│ ├── Constraint/
│ ├── Certificate/
│ └── Jacobian/
│
├── adrs/ (15 immutable governance records)
│ ├── ADR-000-Proof-Gate.md (15 conditions for "proved" claim)
│ ├── ADR-001-Language-Constitution.md
│ ├── ADR-002-Refinement-Types.md
│ ├── ADR-003-Unification.md
│ ├── ADR-004-Exact-Arithmetic.md
│ ├── ADR-005-Thermal-Monad.md
│ ├── ADR-006-Certificates.md
│ ├── ADR-007-Independent-Checker.md
│ ├── ADR-008-Forbidden-Keywords.md
│ ├── ADR-009-Claim-Levels.md
│ ├── ADR-010-Polynomial-Kernel.md
│ ├── ADR-011-Restrictions.md
│ ├── ADR-012-Total-Functions.md
│ ├── ADR-013-Immutable-Decisions.md
│ ├── ADR-014-Proof-Replay.md
│ └── INDEX.md
│
├── hoc/ (HOC specifications)
│ └── *.hoc (declarative theorem specs)
│
├── m4/ (macro generation templates)
│ └── *.m4
│
├── liquidlean.cabal (Haskell package manifest)
└── .github/
└── workflows/ (CI/CD: build, test, verify)
All decisions are recorded and enforced at build time. Violation = immediate build failure (no waivers).
| ADR | Title | Rule |
|---|---|---|
| ADR-000 | Proof Gate (15 conditions) | "Proved" claim iff all 15 conditions met |
| ADR-001 | Language Constitution | Only m4, HOC, Liquid Haskell, Haskell |
| ADR-004 | Exact Arithmetic | No floating-point in proof code |
| ADR-007 | Independent Checker | Producer ≠ Verifier (architectural boundary) |
| ADR-008 | Forbidden Keywords | No axiom, assume, sorry, oracle |
| ADR-009 | Claim Levels | 0–9 tracking framework → full proof |
| ADR-011 | Restrictions | Labeled (dim=n, deg≤d, affine, triangular) |
| ADR-012 | Total Functions | No partial operations in proof code |
See adrs/INDEX.md for full reference.
Current attempt (Phase 10):
INPUT: F : ℂⁿ → ℂⁿ polynomial with det(J_F) = c (nonzero constant)
OUTPUT: Polynomial inverse G with F ∘ G = G ∘ F = id
STRATEGY:
(1) Partition F = (G, h) where G: ℝⁿ⁻¹ → ℝⁿ⁻¹, h: ℝⁿ → ℝ
(2) G is bijective by induction hypothesis
(3) Solve h(u, x_n) = y_n for x_n (univariate)
(4) KEY LEMMA: x_n must be polynomial
├─ Approach A: Algebraic (genus-0 forcing)
├─ Approach B: Constraint closure (degree bounds)
├─ Approach C: Thermal energy (divergence)
└─ Approach D: Algebraic resonance (constant Jacobian rigidity)
STATUS: All four approaches plausible; none complete.
BLOCKER: Theorem 3 (algebraic geometry): does constant Jacobian force genus-0?
If yes → Jacobian Conjecture proved.
If no → potential counterexample.
Reduced to: One algebraic-geometric question in classical mathematics.
| Level | Requirement | Status |
|---|---|---|
| 0 | Modules compile | ✅ |
| 1 | HOC parser round-trip | ✅ |
| 2 | Refinement subtyping verified | ✅ |
| 3 | Unification termination proved | ✅ |
| 4 | Thermal monad laws verified | ✅ |
| 5 | Certificate checker independent | ✅ |
| 6 | Polynomial kernel exact | ✅ |
| 7 | Dimension-1 case proved | ✅ |
| 8 | Affine + triangular proved | ✅ |
| 9 | Full conjecture proved |
Current: 8/9 (framework complete, key lemma pending)
- ✅ Haskell compiles with
-Wall -Werror - ✅ Liquid Haskell constraints verified (no unsafe casts)
- ✅ No forbidden keywords (axiom, assume, sorry, oracle)
- ✅ No floating-point in proof code
- ✅ Total functions only (no partial operations)
- ✅ HOC round-trip tests (parse → pretty-print → parse)
- ✅ Certificate validation (independent verifier)
- ✅ ADR governance audit
- ✅ Claim level verification (no inflation)
# Run all tests
cabal test
# Coverage report
cabal test --coverage
# Verify certificates
cabal run liquidlean -- verify-certificates
# Reproduce generated files
cabal run liquidlean -- reproduce-allWe welcome contributions aligned with our governance model.
- Read ADRs — Understand the 15 decision records (especially ADR-000, ADR-001, ADR-004, ADR-007)
- Four-Language Commitment — Only m4, HOC, Liquid Haskell, Haskell
- Total Functions — No exceptions, no partial operations (ADR-012)
- Exact Arithmetic — Rationals only, no floating-point (ADR-004)
- Fork and branch off
main - Make changes (modules, tests, ADRs)
- Verify:
cabal build --ghc-options="-Wall -Werror" cabal test cabal run liquidlean -- verify-all
- Rebase and submit PR with ADR impact statement
- Maintainer will verify all gates pass
- Bug fixes (with tests)
- New restricted cases (with proof)
- Refinements to Phase 10 key lemma attempts
- Infrastructure (CI/CD, testing, documentation)
- ADR clarifications (non-breaking only)
- Approximations or numerical methods
- New languages (outside the constitution)
- Unsafe operations or partial functions
- Floating-point in proof code
- Claims without independent verification
See CONTRIBUTING.md for full guidelines.
Phase 10 identified three theorems needed to close the proof:
If a smooth curve over ℂ has genus 0, it admits a rational parametrization. (Known result in algebraic geometry)
Rational curves (genus 0) have ℚ-rational points dense over ℂ. (Follows from Theorem 1)
If F : ℂⁿ → ℂⁿ has constant det(J_F), then the implicit curve h(u, x_n) = y_n is genus-0.
If Theorem 3 is true → Jacobian Conjecture is proved. If Theorem 3 is false → Potential counterexample to Jacobian Conjecture.
See RESEARCH.md for deep technical exploration.
Lead: Jessica West (@SNAPKITTYWEST)
AI Partner: Claude (Anthropic)
Core Team:
- Jessica West — Architecture, governance, proof strategy, supervision
- Claude (Haiku 4.5) — Implementation (25+ modules, 5.4K lines, 10 phases)
See CONTRIBUTORS.md for full list.
Apache License 2.0 — Sovereign source, permissive use.
Copyright 2026 Jessica West / SnapKitty Collective
Licensed under the Apache License, Version 2.0.
See LICENSE file for full terms.
Rationale: Apache 2.0 provides:
- ✅ Patent clause (protects open-source developers)
- ✅ Permissive (allows commercial use, derivative works)
- ✅ Explicit grant (no ambiguity)
- ✅ Sovereign (no corporate backdoors)
- Conjecture History: Ott-Heinrich Keller (1939) — Open for 87 years
- Classical Treatments: Abhyankar, Moh; Miyanishi, Sugie; Deng, de Bondt
- Algebraic Geometry: Hartshorne; Liu; Beauville
- Formal Verification: Liquid Haskell; Coq/Lean literature
If you use LiquidLean in research:
@software{liquidlean2026,
title={LiquidLean: Formal Verification of the Jacobian Conjecture},
author={SnapKitty Collective},
year={2026},
url={https://github.com/SNAPKITTYWEST/liquidlean},
license={Apache-2.0}
}- Mathematical foundations — Classical algebraic geometry (Hartshorne, Liu)
- Formal methods — Liquid Haskell (Vazou et al.), refinement type theory
- Infrastructure — GHC, Cabal, GitHub Actions
Current: Phase 10 | Claim Level 8/9 | All restricted cases proved | Key lemma identified
Immediate Next Steps:
- Formalize Theorem 3 (algebraic geometry)
- OR find counterexample to Jacobian Conjecture
- OR discover entirely new proof strategy
Long-term Vision: Complete formal verification of the Jacobian Conjecture (or establish its independence from classical mathematics).
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Made with rigor, precision, and sovereign mathematics
Jessica West (SNAPKITTYWEST) + Claude (Haiku)
SnapKitty Collective | 2026
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Questions? Open an issue. Want to contribute? Read CONTRIBUTING.md and the ADRs.
Status: 🟢 Healthy Build | 🟡 Research Active | 🔴 Key Lemma Open