[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"MxczMDrzyJ":3},"[![CI](https://github.com/Project-Navi/cd-formalization/actions/workflows/lean_action_ci.yml/badge.svg)](https://github.com/Project-Navi/cd-formalization/actions/workflows/lean_action_ci.yml)\n![Lean v4.28.0](https://img.shields.io/badge/Lean-v4.28.0-blue)\n![Mathlib](https://img.shields.io/badge/Mathlib-dep-blue)\n![sorry-free](https://img.shields.io/badge/sorry--free-%E2%9C%93-brightgreen)\n![axiom boundary](https://img.shields.io/badge/axiom%20boundary-5%20classical-yellow)\n[![License: Apache 2.0](https://img.shields.io/badge/License-Apache_2.0-orange)](LICENSE)\n\n# Creative Determinant — Lean 4 Formalization\n\nFormal verification of the existence theory from:\n\n> N. Spence, \"The Creative Determinant: Autopoietic Closure as a Nonlinear Elliptic Boundary Value Problem with Lean 4-Verified Existence Conditions,\" 2026.\n\n## What is verified\n\nFifteen theorems and lemmas proved with zero `sorry`, organized in five dependency tiers:\n\n### Tier 1 — Pure mathematics (no domain axioms)\n\n| Result | Declaration | File | Topic |\n|--------|-------------|------|-------|\n| Spectral characterization (1D) | `spectral_characterization_1d` | `Theorems` | Algebra |\n| Scaling algebraic contradiction | `scaling_algebraic_contradiction` | `Theorems` | Algebra |\n| v^{p−1} ≤ b/c from PDE inequality | `rpow_le_of_mul_rpow_le` | `LinftyAlgebraic` | Real analysis |\n| v ≤ (b/c)^{1/(p−1)} | `linfty_bound_algebraic` | `LinftyAlgebraic` | Real analysis |\n| nextFixed ≤ super-fixed point | `OrderHom.nextFixed_le_of_le` | `MonotoneFixedPoint` | Order theory |\n| Fixed point between sub/super | `monotone_fixed_point_between` | `MonotoneFixedPoint` | Order theory |\n\n### Tier 2 — Operator lemmas (from abstract linearity/homogeneity)\n\n| Result | Declaration | File |\n|--------|-------------|------|\n| Laplacian of zero | `laplacian_zero` | `OperatorLemmas` |\n| Laplacian linearity | `laplacian_linear` | `OperatorLemmas` |\n| Gradient norm of zero | `gradNorm_zero` | `OperatorLemmas` |\n\n### Tier 3 — Coefficient bounds (from [0,1] field constraints)\n\n| Result | Declaration | File |\n|--------|-------------|------|\n| a(x) nonneg | `SemioticContext.a_nonneg` | `CoefficientLemmas` |\n| a(x) ≤ 1 | `SemioticContext.a_le_one` | `CoefficientLemmas` |\n| p − 1 > 0 | `SemioticContext.p_sub_one_pos` | `CoefficientLemmas` |\n\n### Tier 4 — PDE-level results (from `SemioticOperators` / `PDEInfra`)\n\n| Result | Declaration | File | Dependencies |\n|--------|-------------|------|--------------|\n| Scaling uniqueness (kΦ impossible for k > 1) | `scaling_uniqueness` | `ScalingUniqueness` | `SemioticOperators` axioms |\n| Existence of weak coherent configurations | `SemioticBVP.exists_isWeakCoherentConfiguration` | `Theorems` | `PDEInfra` |\n| Nontrivial configurations | `SemioticBVP.exists_pos_isWeakCoherentConfiguration` | `Theorems` | `PDEInfra` |\n\nAll definitions (semiotic manifold, BVP, operators, weak coherent configuration) are machine-checked against Mathlib.\n\nThe monotone fixed-point theorems depend only on `[propext, Quot.sound]` — no `Classical.choice` — making them candidates for constructive upstream contribution.\n\n## Axiom boundary\n\nThe `PDEInfra` typeclass in `CdFormal/Axioms.lean` packages five classical PDE results not yet in Mathlib:\n\n| Axiom | Classical source | Mathlib status |\n|-------|-----------------|----------------|\n| `T_compact` | Schauder estimates + Arzelà–Ascoli | Uses `IsVonNBounded` + `IsCompact` (bornological typing, [credit: Aaron Lin](https://leanprover.zulipchat.com)) |\n| `linfty_bound` | Maximum principle (Gilbarg–Trudinger) | No max. principle for manifolds |\n| `schaefer` | Schaefer 1955 | Not in Mathlib ([draft issue](drafts/mathlib_issue_schaefer.md)) |\n| `fixed_point_nonneg` | Strong maximum principle | No max. principle for manifolds |\n| `monotone_iteration` | Amann 1976 | No sub-/super-solution theory |\n\nThe existence theorems explicitly carry `[PDEInfra bvp solOp]` so the axiom surface is visible to Lean's kernel. Run `#print axioms` in `CdFormal/Verify.lean` to confirm no `sorryAx` — all theorems depend only on `[propext, Classical.choice, Quot.sound]` (monotone fixed-point theorems use only `[propext, Quot.sound]`).\n\n## Building\n\nRequires Lean 4 (v4.28.0) and Mathlib.\n\n```bash\nlake build\nlake build --wfail   # fail on any sorry or warning\n```\n\n## Project structure\n\n```\nCdFormal/\n  Basic.lean              — Definitions (manifold, coefficients, operators, BVP)\n  Axioms.lean             — PDEInfra typeclass (explicit axiom surface)\n  Theorems.lean           — Existence and algebraic theorems\n  OperatorLemmas.lean     — Laplacian/gradient-norm derived properties\n  CoefficientLemmas.lean  — Bounds from [0,1] field constraints\n  ScalingUniqueness.lean  — PDE-level scaling impossibility\n  MonotoneFixedPoint.lean — Knaster-Tarski fixed point between sub/super\n  LinftyAlgebraic.lean    — L∞ bound algebraic core (Paper Lemma 3.10)\n  Verify.lean             — #print axioms dashboard (17 declarations)\nartifacts/\n  aristotle/              — Proved outputs from the Aristotle theorem prover\ndrafts/                   — Mathlib issue drafts + Lean proof sketches\n  mathlib_issue_schaefer.md, BornologyBridge_sorry.lean\n  MonotoneFixedPoint_sorry.lean, LinftyAlgebraic.lean, OperatorLemmas.lean\n  ScalingUniqueness.lean, ScalingUniqueness_v2.lean, ScalingUniqueness_v3.lean\n```\n\n## Development Process\n\n**What the author did**: The original equations, proof strategy, and formalization\narchitecture — choosing to axiomatize via `PDEInfra`, identifying which five\nclassical PDE results to package, designing the typeclass hierarchy, and structuring\nthe Schaefer → L∞ bound → existence → sub/super-solution → nontriviality proof\nchain — are the core intellectual contribution. These are mathematical architecture\ndecisions that require understanding where the real difficulty lies. The underlying\nequations and theory are documented in the [paper](../paper/creative_determinant.pdf).\n\n**What AI tools did**: Claude Opus assisted with Lean 4 syntax, Mathlib API\nnavigation, and proof term synthesis. Aristotle (Harmonic.fun) automated proving\nof standalone algebraic lemmas. These roles are analogous to `omega`, `aesop`, and\nother proof automation — the strategy is human, the term-level search is machine-assisted.\n\n**Verification**: The final arbiter is the Lean compiler, not trust:\n```bash\nlake build --wfail   # type-checks or it doesn't — zero sorry\n```\nRun `#print axioms` in `CdFormal/Verify.lean` to confirm the axiom surface.\nEvery assumption is explicit in `PDEInfra`. Nothing is hidden.\n\n## License\n\nCopyright 2026 Nelson Spence. Licensed under [Apache 2.0](LICENSE).\n",1780846780292]