CATEPT — Complex Action Theory and Entropic Proper Time

A Lean 4.29 integration workspace for formalization around Complex Action Theory and Entropic Proper Time (CAT/EPT) and connected domains.

This README is intentionally operational and conservative:

• It describes what is in the repository and how to run it.
• It references theorem names that are explicitly present in code.
• It avoids claims beyond machine-checkable formal artifacts.

This branch (feat/publication) is the curated publication-facing artifact.

Purpose

• Provide Lean 4 implementations and interfaces for CAT/EPT-related constructs.
• Host cross-domain bridge modules under explicit theorem/axiom boundaries.
• Keep build, audit, and dependency workflows reproducible.

Scope

• CATEPT core definitions and plugin constraints.
• Navier-Stokes and Sobolev/Fourier supporting modules.
• Quantum and quantum-information modules.
• GR tensor/ADM-oriented modules.
• Statistical mechanics and measure-theory support modules.

Requirements

• Lean toolchain from lean-toolchain (Lean 4.29 line on this branch).
• lake and git.
• Internet access for dependency fetch and cache warmup.

Quick Start

git clone https://github.com/jagg-ix/catept-main.git
cd catept-main
git checkout feat/publication
lake exe cache get                    # warm Mathlib olean cache (first run)
lake build CATEPT.Showcase.QMGRUnification

Common Build Targets

• Full integration surface:
  • lake build CATEPTMain
• Bridge aggregator:
  • lake build CATEPTMain.Bridges
• Core-only surface:
  • lake build CATEPT.CATEPT.Core
• Example focused surfaces:
  • lake build CATEPTMain.GravitasStandalone
  • lake build CATEPTMain.QuantumInfoStandalone

Quantum Mechanics ↔ General Relativity via entropic proper time

A single plugin slot carries actionIm, ℏ, and eptClock. The constraint

∀ x, actionIm(x) / ℏ = eptClock(x)

is proved on two instances:

Theory	Instance	Consistency theorem
Quantum Mechanics (n-level density matrices)	quantumCATEPTSlot n	qm_satisfies_catept_spine
General Relativity (Minkowski background)	gravitasMinkowskiSlot	gr_minkowski_satisfies_catept_spine

All proofs depend only on the Lean kernel axioms propext, Classical.choice, Quot.sound.

Machine-checking from the command line

cat > /tmp/catept_audit.lean <<'EOF'
import CATEPT.Showcase.QMGRUnification
#print axioms CATEPT.Showcase.QMGRUnification.qm_satisfies_catept_spine
#print axioms CATEPT.Showcase.QMGRUnification.gr_minkowski_satisfies_catept_spine
EOF
lake env lean /tmp/catept_audit.lean

Expected output (one line per theorem):

'…qm_satisfies_catept_spine' depends on axioms: [propext, Classical.choice, Quot.sound]
'…gr_minkowski_satisfies_catept_spine' depends on axioms: [propext, Classical.choice, Quot.sound]

Any other axiom appearing in the list is a regression. A CI gate at .github/workflows/axiom-gate.yml enforces this check on pushes and pull requests.

Significant solved artifacts (Lean4)

Examples of formalized artifacts with explicit theorem names:

• QM ↔ GR spine consistency (entropic proper time identity across two instances): CATEPT.Showcase.QMGRUnification.qm_satisfies_catept_spine, CATEPT.Showcase.QMGRUnification.gr_minkowski_satisfies_catept_spine
• Rigorous complex Feynman-Kac (entropically damped class): CATEPTMain.Integration.RigorousComplexFeynmanKac.complex_FK_rigorous
• Counterterm-free / no-renormalization UV certificate: CATEPTMain.Integration.PhysicalUVConvergenceCertificate.physical_uv_certificate_no_counterterm_needed

Verification and audits

• Lean axiom inspection:
  • #print axioms <theorem_name>
• NS semantic checks:
  • python3 tools/verification/check_ns_semantic_strictness.py --strict
• LeanMillennium bundle:
  • python3 tools/verification/run_ns_leanmillennium_conformance_bundle.py

Axiom-free theorems (no kernel axioms required)

A subset of the audited surface clears an even stronger bar than "kernel-only": 10 theorems in CATEPTMain/Domains/CoherenceShowcase.lean print

'…' does not depend on any axioms

These are fully structural term-mode witnesses (definitional level), with no propext, no Classical.choice, and no Quot.sound.

#	Theorem (fully qualified)	Sibling plugin	What it asserts
1	CATEPTPluginQuantumInfo.quantumInfo_integration_contract	catept-plugin-quantum-info	CPTP / Braket / von Neumann / Renyi / Shannon / capacity contract package
2	CATEPTPluginBochnerMinlos.bochnerMinlos_integration_contract	catept-plugin-bochner-minlos	Bochner-Minlos PD / Sazonov / Schur / abstract Minlos contract
3	CATEPTPluginGibbsMeasure.gibbsMeasure_integration_contract	catept-plugin-gibbs-measure	Kolmogorov extension / Gibbs-DLR / Giry monad witness
4	CATEPTPluginHopfLean.hopfLean_integration_contract	catept-plugin-hopf-lean	coalgebra / bialgebra / Hopf / Yang-Baxter / BMod-monoidal witness
5	CATEPTPluginKolmogorovComplexity.kolmogorovComplexity_integration_contract	catept-plugin-kolmogorov-complexity	AIT invariance / Chaitin Omega / incompressibility / Godel-2 via K
6	CATEPTPluginCarleson.carleson_integration_contract	catept-plugin-carleson	abstract Carleson / a.e. Fourier convergence contract
7	CATEPTPluginCarleson.concrete_witness_contract	catept-plugin-carleson	concrete Carleson witness (Dirichlet / Jackson / antichain)
8	CATEPTPluginCslib.cslib_integration_contract	catept-plugin-cslib	computability / automata / Ramsey integration
9	CATEPTPluginThermodynamicsLean.thermodynamicsLean_integration_contract	catept-plugin-thermodynamics-lean	Lieb-Yngvason axioms / entropy existence-uniqueness-continuity / Kelvin-Planck
10	CATEPTPluginVMLLandau.vml_landau_content_available	catept-plugin-vml-landau	VML-Landau collision-content marker

Axiom policy

• Kernel axioms should be inspectable via #print axioms.
• Domain assumptions not yet theoremized are declared explicitly.
• Mixed theorem/axiom status is tracked in worklogs and verification artifacts.

Repository layout (top level)

CATEPTMain/        Integration bridges and domain adapters
CATEPT/            Core CATEPT framework and DSL modules
NavierStokes/      NS strategy and bridge modules
NavierStokesClean/ Clean NS carrier/types/Sobolev interfaces
QuantumInfo/       Quantum information modules
ClassicalInfo/     Classical information modules
StatMech/          Statistical mechanics modules
GibbsMeasure/      Gibbs/DLR modules
Carleson/          Carleson formalization modules
LY/                Lee-Yang modules
BrownianMotion/    Brownian/Kolmogorov modules
tools/             Verification and automation scripts
verification/      Generated audit artifacts

Dependencies

• Dependencies are pinned through:
  • lakefile.lean
  • lake-manifest.json
• Main ecosystem dependency is Mathlib (version pinned by this branch).

Acknowledgments

Intellectual foundations

The framework builds on the entropic-dynamics research programme of Prof. Ariel Caticha (University at Albany, SUNY) — arielcaticha.com.

Ported libraries

• CATEPTMain/Gravitas/ — Lean 4 port of the Gravitas symbolic general-relativity package (original: Wolfram Language).
• CATEPTMain/QuantumOps/IsabelleMarresDirac/ — Lean 4 port from Isabelle/HOL.

Lean 4 mathematical-physics dependencies by Michael R. Douglas

Three analytic-functional pillars this repository depends on are contributed by mrdouglasny:

• bochner — Bochner-Minlos theorem.
• hille-yosida — Hille-Yosida theorem for C0-semigroups.
• pphi2 — phi4 scalar field theory infrastructure.

Other external Lean 4 dependencies

Thanks to the upstream authors and maintainers of Mathlib4, Physlib, pphi2N, GaussianField, LGT, cslib, DeGiorgi, spectralPhysics, DimensionalAnalysis, UnifiedTheory, aristotle, aqeiBridge, lean-inf, and QuantumAlgebra. Pinned revisions live in lakefile.lean.

Current status

• This repository contains both theoremized modules and scaffold modules.
• Read theorem statements and run #print axioms for exact assurance level.

License

Apache-2.0. See LICENSE.