[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"958wev1ZrA":3},"# Logos Library\n\n**Formally verified physics and mathematics in Lean 4 — from quantum mechanics \nto rotating black holes to rough path theory, built on Mathlib.**\n\n[![License: MIT](https://img.shields.io/badge/License-MIT-yellow.svg)](https://opensource.org/licenses/MIT)\n[![Lean 4](https://img.shields.io/badge/Lean-4-blue.svg)](https://leanprover.github.io/lean4/doc/)\n\n---\n\n## What This Is\n\nLogos Library is a Lean 4 formalisation built on \n[Mathlib](https://github.com/leanprover-community/mathlib4), spanning physics and \npure mathematics across ~125,000 lines of Lean in 354 files. The library contains \n4,034 machine-checked proofs, 114 explicit axioms, and 7 `sorry`.\n\nThe library has three parts:\n\n- **Core Physics Library** — quantum mechanics, information geometry, general \n  relativity, and quantum computing. The central thread runs from special relativity \n  through Kerr black holes, through a resolution of the Ott–Landsberg debate on \n  relativistic temperature, to a derivation of Einstein's field equations from \n  thermodynamics, and a proof that the Connes–Rovelli thermal time relation is the \n  unique Lorentz-covariant connection between temperature and time.\n\n- **Stochastic Calculus** — a formalisation of rough path theory from first \n  principles: the sewing lemma, Young integration, and rough path integration in \n  the range p ∈ [2,3), heading toward the Universal Limit Theorem. This has never \n  been formalised in any proof assistant.\n\n- **The Superior Method** — a methodology for using Lean's type-checker as a \n  structural comparator across independently formalised domains of physics.\n\n\nSee the [inner README](LogosLibrary/README.md) for detailed statistics, file \nlistings, and dependency information.\n\n---\n\n\n## Core Physics Library\n\nFour modules formalising quantum mechanics, information geometry, relativity, and \nquantum computing. The central thread: a dependency chain from special relativity \nthrough Kerr black holes, through a resolution of the 60-year Ott–Landsberg debate \non relativistic temperature, to a derivation of Einstein's field equations from \nthermodynamics, and a proof that the Connes–Rovelli thermal time relation is the \nunique Lorentz-covariant connection between temperature and time. Time dilation \nemerges as a theorem, not a postulate.\n\n| Module | What it covers |\n|--------|---------------|\n| [**Quantum Mechanics**](LogosLibrary/QuantumMechanics/) | Unbounded operators, Stone's theorem, spectral theorem (3 routes), Schrödinger equation, Bell/CHSH/Tsirelson, Dirac equation, uncertainty relations, unitarity ↔ first law, Tomita–Takesaki modular theory, Connes cocycle, KMS condition, thermal time |\n| [**Information Geometry**](LogosLibrary/InformationGeometry/) | Fisher–Rao metric from first principles, score functions, statistical manifolds, Cramér–Rao bound |\n| [**Relativity**](LogosLibrary/Relativity/) | Minkowski spacetime, Kerr black holes, Ott transformation (7 independent proofs), Jacobson's thermodynamic derivation of Einstein's equations, Connes–Rovelli thermal time uniqueness, SdS forbidden region (original) |\n| [**Quantum Computing**](LogosLibrary/QuantumComputing/) | Thermal Turing machine, Landauer compliance, conditional P ≠ NP from information geometry **(experimental)** |\n\nEach module has its own README with file-by-file descriptions, axiom inventories, \nand dependency information.\n\n### How the Core Modules Connect\n\n```\n            Information Geometry                Quantum Mechanics\n           +----------------------+         +-----------------------------+\n           | Fisher-Rao metric    |         | Stone's theorem             |\n           | Statistical manifold |         | Spectral theorem (×3)       |\n           | Cramer-Rao bound     |         | Bell / Tsirelson            |\n           +-------+------+-------+         | Dirac equation              |\n                   |      |                 | Unitarity \u003C-> 1st law       |\n                   |      |                 | Tomita-Takesaki / KMS       |\n                   |      |                 | Connes cocycle / Ott forced |\n                   |      |                 +-------+---------------------+\n                   |      |                         |\n        +----------+      +----------+              |\n        v                            v              v\n  Quantum Computing              Relativity\n  +------------------+        +---------------------------+\n  | Thermal TM       |        | Minkowski spacetime       |\n  | Fisher metric on |        | Kerr black holes          |\n  |  computation     |        | Ott transformation (x7)   |\n  |  paths           |        | Jacobson's derivation     |\n  | P != NP          |        | Thermal time uniqueness   |\n  |  (conditional)   |        | SdS forbidden region      |\n  +------------------+        +---------------------------+\n\nCross-module imports:\n  Info Geometry  → QM:         Cramér–Rao gives a second proof of Heisenberg\n  Info Geometry  → QC:         Fisher metric provides the geometric backbone\n  QM → Relativity:             Unitarity ↔ first law resolves Ott–Landsberg\n  QM → Relativity:             Robertson uncertainty used in SdS instability\n  QM → Relativity:             Modular theory / cocycle forces Ott covariance\n  Relativity → QM:             Lorentz structure feeds back into Dirac equation\n```\n\n---\n\n## Stochastic Calculus\n\nA formalisation of rough path theory from first principles, targeting a complete \nmachine-checked proof of the Universal Limit Theorem: the solution map \n**X** ↦ Y^**X** of a rough differential equation is well-defined and continuous \nin the p-variation rough path metric. This has never been formalised in any proof \nassistant.\n\nThe development is organised in five stages, each building on the previous:\n\n| Stage | What it covers | Status |\n|-------|---------------|--------|\n| [**Stage 0**](LogosLibrary/StochasticCalculus/Stage_0/) | Sewing lemma (three layers: interval control, cross-controlled, continuous control). p-variation theory. | Complete |\n| [**Stage 1**](LogosLibrary/StochasticCalculus/Stage_1/) | Young integration: the Young–Loève estimate, additivity, uniqueness, regularity, linearity, integration by parts, continuity. | Complete |\n| [**Stage 2**](LogosLibrary/StochasticCalculus/Stage_2/) | Truncated tensor algebra, group-like elements, p-rough paths, Gubinelli derivatives, the rough integral, closure theorem, Itô–Lyons continuity estimate. | Complete |\n| **Stage 3** | Picard iteration, local and global existence/uniqueness for rough differential equations. | Planned |\n| **Stage 4** | The Universal Limit Theorem. | Planned |\n\nClassical stochastic calculus builds integrals from probability — martingale theory, \nfiltrations, conditional expectations. Rough path theory takes a different approach: \nenhance the driving signal with sufficient algebraic data and the integral becomes a \ncontinuous function of the enhanced path. Probability enters only at the end, in \nidentifying which enhancement a particular stochastic process selects. Everything \nelse is deterministic analysis.\n\nSee the [Stochastic Calculus README](LogosLibrary/StochasticCalculus/) for the full \narchitecture, dependency flow, and references.\n\n\n---\n\n## The Superior Method\n\nThe Superior Method uses Lean's type-checker as a structural comparator across \nindependently formalised domains of physics. Two modules are built in separate \nfiles with no shared imports. A bridge file then asks: does Lean accept that the \nstructure built in Module A is definitionally equal to the structure built in \nModule B? If the bridge compiles, the two domains share algebraic structure. The \ncompiler is the comparator.\n\n| Cluster | What it covers |\n|---------|---------------|\n| [**CommonResources**](LogosLibrary/Superior/CommonResources/) | Shared foundations: Connes–Rovelli thermal time, Super-Causets, division algebras ℝ ↪ ℂ ↪ ℍ ↪ 𝕆, Hopf tower, Clifford periodicity, Fano plane, modular flow |\n| [**KnotTheory**](LogosLibrary/Superior/KnotTheory/) | QCD flux tubes, topological mass gap, string theory thermal structure, Hopf knot unification |\n| [**HotGravity**](LogosLibrary/Superior/HotGravity/) | Five thermodynamic routes to gravity: entropic gravity, loop quantum gravity, nanothermodynamics, shape dynamics, objective reduction |\n| [**SpectralTriples**](LogosLibrary/Superior/SpectralTriples/) | Noncommutative geometry, Geometric Unity and the observerse U⁹, spectral action |\n| [**SplitMechanics**](LogosLibrary/Superior/SplitMechanics/) | Subsystem decomposition, Bohmian mechanics, thermal Bell tests, Möbius gears, Riemann zeta via Weil positivity |\n| [**EconomicGauge**](LogosLibrary/Superior/EconomicGauge/) | Malaney–Weinstein economic index number problem, gauge-theoretic classification of price distortion |\n| [**Bridges**](LogosLibrary/Superior/Bridges/) | Lateral connections across clusters — zero axioms in every bridge file |\n\nSee the [Superior README](LogosLibrary/Superior/README.md) for the full \nmethodology, architecture, and the organising observation that makes the whole \nthing tick.\n\n\n## Library Statistics\n\n| | Proofs | Axioms | Sorry | Files |\n|---|:---:|:---:|:---:|:---:|\n| **Core Physics Library** | | | | |\n| Quantum Mechanics | | 59 | 0 | |\n| Information Geometry | | 0 | 0 | |\n| Relativity | | 7 | 0 | |\n| Quantum Computing | | 0 | 4 | |\n| **Stochastic Calculus** | | 0 | 1 | |\n| **Superior Method** | | 48 | 2 | |\n| **Library total** | **4,034** | **114** | **7** | **354** |\n\n124,623 lines of Lean. See the [inner README](LogosLibrary/README.md) for \nper-file breakdowns, and the [axiom audit](docs/axiom_audit.md) for the \ncomplete axiom catalogue.\n\n\n## What Is Proven vs. What Is Axiomatised\n\n**Fully machine-checked (no axioms, no sorry):** Minkowski spacetime. Lorentz \nboosts. Robertson, Schrödinger, and Heisenberg uncertainty (twice: algebraic and \ninformation-geometric). Stone's theorem (both directions). Schrödinger equation as \na corollary. Bell/CHSH inequality. Tsirelson's bound. Kerr horizon structure, \nergosphere, ring singularity. Seven independent refutations of Landsberg. \nUniqueness of Ott's temperature transformation. Uniqueness of the thermal time \nrelation. Time dilation as a theorem. Fisher–Rao metric construction. Sewing \nlemma (all three layers). Young integration (existence, uniqueness, regularity, \nlinearity, integration by parts). Rough path integration with Itô–Lyons \ncontinuity. All Superior bridge files.\n\n**Axiomatised (standard results not yet formalisable in Lean):** Bochner's theorem. \nDominated convergence applications. Adams' theorem on Hopf invariant one. Various \nspectral integration results requiring measure theory on infinite-dimensional \nspaces. These are textbook results whose formal proofs require infrastructure \nMathlib does not yet provide. See the [axiom audit](docs/axiom_audit.md) for the \nfull list — every axiom names a specific result.\n\n**Experimental:** The P ≠ NP development in Quantum Computing is conditional: *if* \nthe SAT phase transition creates an information-geometric bottleneck for sequential \nlocal computation, *then* P ≠ NP.\n\n\n## Design Decisions\n\n**Unbounded operators, honestly.** `UnboundedObservable H` models quantum \nobservables as genuinely partial functions. You cannot apply an operator to a \nvector without proving it lives in the domain. This is more work than pretending \neverything is bounded. It is also correct.\n\n**Physics axioms are features, not bugs.** The axioms are not gaps in proofs — \nthey are the physical content. A formalisation of physics must distinguish \nmathematical theorems (which the machine checks) from physical postulates (which \nthe machine takes as given). Making these assumptions visible and auditable is the \nentire point.\n\n**Multiple proof routes.** Heisenberg is proved twice. The spectral theorem is \nconstructed via three routes proved to agree. Relativistic thermodynamics is \napproached from two directions. When different branches of mathematics produce the \nsame result through different architectures, the agreement is evidence.\n\n**Independent construction, then bridging.** The Superior modules are built without \nimporting each other. Structural coincidence is discovered by bridge files, not \nassumed by shared imports. The bridges contain zero axioms.\n\n**Expository source files.** Several modules interleave Lean code with physical \nmotivation, historical context, and pedagogical explanation. The intent is that \nthese files serve as self-contained introductions to the physics for anyone with a \nworking knowledge of Lean.\n\n\n## Quick Start\n\n```bash\n# Clone\ngit clone https://gitlab.com/pedagogical/logos_library\ncd Logos_Library\n\n# Install Lean 4 (if needed)\ncurl https://raw.githubusercontent.com/leanprover/elan/master/elan-init.sh -sSf | sh\n\n# Fetch Mathlib cache\nlake exe cache get\n\n# Build everything (first build takes a while; 16 GB RAM recommended)\nlake build\n\n# Or build individual modules\nlake build LogosLibrary.QuantumMechanics\nlake build LogosLibrary.Relativity\nlake build LogosLibrary.StochasticCalculus\nlake build Superior\n```\n\n\n## References\n\nEach module README contains full references. The primary sources:\n\n- M. Reed, B. Simon, *Methods of Modern Mathematical Physics I*, Academic Press, 1980.\n- K. Schmüdgen, *Unbounded Self-adjoint Operators on Hilbert Space*, Springer, 2012.\n- B. C. Hall, *Quantum Theory for Mathematicians*, Springer, 2013.\n- M. Takesaki, *Theory of Operator Algebras I–III*, Springer, 1979–2003.\n- S. Amari, *Information Geometry and Its Applications*, Springer, 2016.\n- T. Jacobson, \"Thermodynamics of Spacetime,\" *Phys. Rev. Lett.* 75, 1260–1263, 1995.\n- A. Connes, C. Rovelli, \"Von Neumann algebra automorphisms and time-thermodynamics relation,\" *Class. Quantum Grav.* 11, 2899–2917, 1994.\n- P. Friz, M. Hairer, *A Course on Rough Paths*, 2nd ed., Springer, 2020.\n- T. Lyons, \"Differential equations driven by rough signals,\" *Rev. Mat. Iberoam.* 14, 215–310, 1998.\n\n\n## Author\n\nAdam Bornemann — [AdamBornemann@gmail.com](mailto:AdamBornemann@gmail.com)\n\n## License\n\nMIT. See [LICENSE](LICENSE).\n\n## Acknowledgements\n\nBuilt with [Lean 4](https://leanprover.github.io/lean4/doc/) and \n[Mathlib](https://github.com/leanprover-community/mathlib4). The Bell's theorem \ndevelopment is ported from Echenim and Mhalla's Isabelle/HOL formalisation with \nproper attribution and substantial extension.\n\n\n",1780846778977]