TauLib
Mechanized Formalization of Category τ

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TauLib is a 125,000-line Lean 4 formalization of Category τ — a categorical framework built from 7 axioms (K0–K6) on 5 generators (α, π, γ, η, ω) with a single primitive iterator ρ. Starting from these axioms alone, TauLib derives arithmetic, analysis, geometry, physics, biology, and philosophy as earned consequences — all compiled and verified by Lean's kernel with zero sorry in Books I–VI.

Companion to the 7-book Panta Rhei series by Thorsten Fuchs and Anna-Sophie Fuchs (2nd Edition, 2026).

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At a Glance

Metric	Value
Source files	450
Lines of Lean 4	125,771
Theorems & lemmas	4,332
Definitions	3,542
Structures, classes & inductives	1,685
Instances	28
Examples	350
Computations (#eval)	3,721
Axioms	4 (3 conjectural, 1 structural)
Sorry	3 (methodological, Book VII only)
Books I–VI sorry	0
Mathlib usage	Tactics only — all mathematics from scratch

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What Makes TauLib Unique

• Everything from scratch. TauLib does not import mathematical content from Mathlib or any other library. All arithmetic, algebra, analysis, topology, category theory, quantum mechanics, and cosmology are derived within the framework, from the 5 generators and 7 axioms. Mathlib is used for proof tactics only (simp, omega, ring, decide, linarith, norm_num).

• 3,721 live computations. Every quantitative claim is backed by #eval statements that execute in the Lean kernel, producing concrete numerical values from the master constant ι_τ = 2/(π + e).

• Physics predictions at ppm accuracy. The library formalizes predictions for 9 electroweak quantities, CMB first peak position (+69 ppm), 20 galaxy rotation curves, baryon asymmetry, and more — all derived from a single constant with zero free parameters.

• Transparent foundations. The 4 axioms and 3 sorry are explicitly documented, with precise classification (conjectural, structural, methodological). The conjectural axioms follow a "compute-then-axiomatize" pattern: finite checks pass computationally; the axiom asserts the infinite extension.

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Quick Start

Prerequisites

• Lean 4 via elan (version managed by lean-toolchain)

Clone and Build

git clone https://github.com/Panta-Rhei-Research/taulib.git
cd TauLib
lake build

The first build downloads Mathlib (for tactics) and compiles ~1,256 lake jobs. Subsequent builds use the cache and are fast.

Use as a Dependency

Add to your lakefile.lean:

require TauLib from git
  "https://github.com/Panta-Rhei-Research/taulib.git" @ "main"

Or in lakefile.toml:

[[require]]
name = "TauLib"
git = "https://github.com/Panta-Rhei-Research/taulib.git"
rev = "main"

Build API Documentation

TauLib includes a full doc-gen4 documentation pipeline with Panta Rhei theming, a statistics dashboard, and an interactive dependency graph:

# First build (initializes doc-gen4):
cd docbuild && MATHLIB_NO_CACHE_ON_UPDATE=1 lake update doc-gen4 && cd ..

# Generate full documentation:
./scripts/build_docs.sh

# Preview locally:
cd docbuild/.lake/build/doc && python3 -m http.server 8000

The generated documentation includes:

• API pages for all 450 modules with rendered docstrings and cross-linked types
• Statistics dashboard with per-book formalization coverage from the registry
• Interactive dependency graph visualizing 4,500+ mathematical registry entries across 7 books

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Start Here

Open these files in VS Code with the Lean 4 extension and step through line by line:

Tour	Time	What You'll See
Tour/VerifyItYourself.lean	15 min	Start here. 5 surprising claims, verified live — the skeptic’s tour
Tour/Foundations.lean	10 min	5 generators, 7 axioms, ρ operator, master constant, rigidity
Tour/CentralTheorem.lean	10 min	Split-complex ring, τ³ fibration, O(τ³) ≅ Aspec(ℒ)
Tour/Physics.lean	15 min	EW synthesis, 3 generations, CMB, rotation curves, baryogenesis
Tour/OneConstant.lean	10 min	Full constants ledger: α, h, ℓ₁, ωb, r — all from ιτ
Tour/MillenniumProblems.lean	15 min	GRH, BSD, Poincaré, Hodge, Navier-Stokes through the τ-lens
Tour/LifeFromPhysics.lean	10 min	4+1 life sectors, genetic code, neural architecture, crossing limit
Tour/MindAndEthics.lean	15 min	Categorical Imperative, consciousness, free will, Logos, the 3 sorry

Pick Your Path

Audience	Start With	Then Explore
Skeptic / Reviewer	Tour/VerifyItYourself	Tour/OneConstant → any module you doubt
Mathematician	Tour/Foundations → Tour/CentralTheorem	Tour/MillenniumProblems → BookIII/Doors/GrandGRH
Physicist	Tour/Physics → Tour/OneConstant	BookIV/Electroweak/EWSynthesis → BookV/Cosmology/CMBSpectrum
Biologist	Tour/LifeFromPhysics	BookVI/Source/GeneticCode → BookVI/Consumer/Neural
Philosopher	Tour/MindAndEthics	BookVII/Ethics/CIProof → BookVII/Final/Boundary
Lean user	Tour/Foundations → lakefile.lean	Browse any module — all 450 files have 30+ line docstrings

See the Architecture Guide for detailed reading paths, dependency graphs, and per-book start files.

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Module Architecture

All 450 modules are organized under seven book namespaces. The dependency order is strict: each book builds only on what came before.

TauLib
 ├── BookI    Categorical Foundations    94 files   20,554 lines
 ├── BookII   Categorical Holomorphy     65 files   18,069 lines
 ├── BookIII  Categorical Spectrum       70 files   16,807 lines
 ├── BookIV   Categorical Microcosm      89 files   29,730 lines
 ├── BookV    Categorical Macrocosm      80 files   28,394 lines
 ├── BookVI   Categorical Life           30 files    5,221 lines
 ├── BookVII  Categorical Metaphysics     7 files    4,278 lines
 └── Tour     Interactive Guides          3 files      674 lines

Book I — Categorical Foundations (94 files, 20,554 lines)

The foundation builds everything from the 5 generators, using 12 module families:

Family	Files	Content
Kernel	4	5 generators, axioms K0–K6, ρ operator
Orbit	8	Orbit generation, closure, iterator ladder, rigidity: Aut(τ) = {id}
Denotation	9	τ-Idx (earned natural numbers), rank transfers, program monoid
Coordinates	10	Normal form, ABCD chart, hyperfactorization, Chebyshev coordinates
Polarity	14	Prime Polarity Theorem, lemniscate ℒ, CRT, bipolar algebra
Boundary	14	Split-complex ring ℍ[j], number tower, ιτ, characters
Sets	8	Internal set theory, Cantor refutation, counting
Logic	3	Truth₄ logic, explosion barrier, Boolean recovery
Holomorphy	9	ω-germ transformers, Global Hartogs, presheaf essence
Topos	7	Earned arrows, functors, sites, sheaves
MetaLogic	7	Proof-theoretic mirror, linear discipline, structural exclusion
CF	1	Continued fraction window algebra

Books II–VII

Book	Files	Lines	Key Content
II Holomorphy	65	18,069	τ³ = τ¹ ×f T², Central Theorem: O(τ³) ≅ Aspec(ℒ)
III Spectrum	70	16,807	8 spectral forces, Millennium Problems, τ-Turing machine
IV Microcosm	89	29,730	Electroweak synthesis, 3 generations, Majorana, strong CP, Higgs
V Macrocosm	80	28,394	Gravity, CMB (+69 ppm), rotation curves, baryogenesis, lensing
VI Life	30	5,221	5-condition life predicate, origin of life, consciousness
VII Metaphysics	7	4,278	Saturation, archetypes, Categorical Imperative, social ontology

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Key Results

Formalized results with their Lean entry points:

Result	Lean Identifier	ppm	Book
Central Theorem: O(τ³) ≅ Aspec(ℒ)	central_fwd_3_15	—	II
Rigidity: Aut(τ) = {id}	rigidity_non_omega	—	I
Three Generations: H₁(τ³; ℤ) ≅ ℤ³	gen_count_three	exact	IV
9 EW Quantities from ιτ + mn	nine_ew_quantities	sub-ppm	IV
Majorana Neutrinos: σ = Cτ	all_neutrinos_majorana	—	IV
Strong CP: θQCD = 0	theta_qcd_zero_from_sa_i	exact	IV
CMB First Peak: ℓ₁ (NLO)	first_peak_holonomy_thm	+69	V
20 Galaxy Rotation Curves	flat_rotation_theorem	RMS 0.067 dex	V
Baryogenesis: ηB = α · ιτ15 · (5/6)	sakharov_reduction	~1%	V
S₈ Tension Resolved: S₈ = 0.783	s8_tau_value	within 1σ	V
Categorical Imperative as j-closed fixed point	ci_j_closed_fixed_point	—	VII
Saturation: Enrich(E₃) = E₃	saturation_theorem	—	VII

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Axioms and Sorry

TauLib is maximally transparent about its foundations.

4 Axioms

Axiom	Module	Classification	Pattern
bridge_functor_exists	BookIII/Bridge/BridgeAxiom	Conjectural	Finite checks pass; axiom asserts ∀ n
spectral_correspondence_O3	BookIII/Doors/SpectralCorrespondence	Conjectural	Finite checks pass; axiom asserts ∀ n
grand_grh_adelic	BookIII/Doors/GrandGRH	Conjectural	Finite checks pass; axiom asserts ∀ n
central_theorem_physical	BookIV/Arena/BoundaryHolonomy	Structural	References Central Theorem (Book II)

The 3 conjectural axioms follow a "compute-then-axiomatize" pattern: a decidable finite check is verified computationally via native_decide, then an axiom asserts the property holds universally. This makes the conjectural boundary maximally sharp and auditable.

3 Sorry (Book VII only)

Theorem	Module	Assertion
omega_point_theorem	BookVII/Logos/Sector	True := sorry
science_faith_boundary	BookVII/Logos/Sector	True := sorry
no_forced_stance	BookVII/Final/Boundary	True := sorry

All three are typed True := sorry — the goal is trivially True; the sorry marks a philosophical boundary where formalization itself is the content under discussion. They involve the ω-generator, which is non-diagrammatic by design. Books I–VI contain zero sorry.

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Dependency Policy

TauLib uses Mathlib for proof tactics only:

Allowed	Not Allowed
simp, omega, ring, aesop	Mathlib.Order.*
decide, native_decide, norm_num	Mathlib.Algebra.*
linarith, positivity, field_simp	Mathlib.CategoryTheory.*
constructor, exact, apply	Mathlib.Topology.*

All mathematical content — natural numbers, rings, fields, groups, topology, analysis, category theory — is built from scratch within TauLib, derived from the 5 generators and 7 axioms. This is a deliberate design choice: the framework claims to generate all of mathematics from its foundation, and TauLib proves this claim mechanically.

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The Books

TauLib formalizes content from the Panta Rhei series (2nd Edition, 2026):

#	Title	Chapters	LaTeX Pages
I	Categorical Foundations	83	461
II	Categorical Holomorphy	66	484
III	Categorical Spectrum	104	415
IV	Categorical Microcosm	83	455
V	Categorical Macrocosm	97	504
VI	Categorical Life	45	412
VII	Categorical Metaphysics	74	521
	Total	552	3,252

Available at panta-rhei.site.

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Documentation

Document	Description
Architecture Guide	Module dependency graph, reading paths by audience, per-book start files
Scope Labels	The 4-tier scope discipline: established, τ-effective, conjectural, metaphorical
Glossary	Key terms, symbols, constants, and registry ID format
Formalization Status	Per-book formalization coverage and sorry/axiom inventory
Contributing	Issue reporting, code style, citation, and fork guidelines

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Building and Development

# Build the full library (~1,256 lake jobs)
lake build

# Build a specific book
lake build TauLib.BookI

# Check Lean version
cat lean-toolchain

Continuous Integration

Every push to main triggers a full lake build via GitHub Actions. The CI badge at the top of this README reflects the latest build status.

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Citation

If you use TauLib in academic work, please cite:

@software{taulib2026,
  author    = {Fuchs, Thorsten and Fuchs, Anna-Sophie},
  title     = {{TauLib}: Mechanized Formalization of Category $	au$},
  year      = {2026},
  version   = {2.0.0},
  url       = {https://github.com/Panta-Rhei-Research/taulib},
  note      = {450 modules, 125{,}771 lines of Lean 4, 4{,}332 theorems},
  license   = {Apache-2.0}
}

Or use GitHub's "Cite this repository" feature, which reads from CITATION.cff.

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License

Copyright 2025–2026 Thorsten Fuchs and Anna-Sophie Fuchs.

Licensed under the Apache License, Version 2.0.