Fricke involution formalization (fricke-v1)

Lean 4 formalization of the Fricke involution W_N = !![0,-1;N,0] on Γ₀(N) and the operator it
induces on modular forms. Merged via #1.

Build: lake build SocrateAI → 3053 jobs, 0 errors, 0 sorry.

Axiom discipline: 16 headline theorems pinned by build-failing #guard_msgs in #print axioms
guards in FinalCheck.lean; every footprint is exactly [propext, Classical.choice, Quot.sound].
A deliberately-wrong negative control (scratch/GuardNegativeControl.lean) was verified to fail,
so the guards are known to be load-bearing rather than vacuous.

Module	Content
FrickeInvolution	W_N ∈ GL(2,ℝ)⁺, W_N² = −N·I, conjugation frickeConj, W γ W⁻¹ ∈ Γ₀(N)
FrickeSlash	slash-invariance preserved → operator on SlashInvariantForm
FrickeModular	holomorphy + cusp-boundedness transported → operator on bundled ModularForm
FrickeComposite	(f∣W)∣W = (−1)^k N^(k−2) f

dag/ carries the statement-dependency graph (11/15 proved, acyclic), a natural-language
description per node, and a validator enforcing that every node marked proved names an existing
declaration and depends only on proved nodes.

No priority is claimed. anthropics/fermats-last-theorem already contains Fricke and
Atkin–Lehner material in Lean 4, including atkinLehnerLin — a ℂ-linear operator on ModularForm
for the whole Atkin–Lehner family. This is an independent, Mathlib-idiomatic treatment, not a first.
See PRIOR_ART_FINDING.md in SocrateAI-Scientific-Communication.

Build caveat: lakefile.lean, lake-manifest.json and lean-toolchain are excluded from the
repo — they hard-code absolute paths to a local Mathlib package pool. Built against Lean 4.32.2 +
Mathlib 905b9581. A portable lakefile is outstanding work; this release is not yet
build-reproducible by a third party.

SocrateAI-Lean-Lib v1.0.0 — Complete Formal Scientific Foundation

🔬 SocrateAI-Lean-Lib v1.0.0 — Major Release

Complete Lean 4 Formal Scientific Foundation Library for Advanced Research, Theories, and Non-Anthropocentric Mathematics.

✅ Verification Status

Metric	Result
Lake Build	69 jobs — 0 errors, 0 warnings
sorry stubs	0 (all theorems kernel-verified, Tier A)
Formal Domains	14
Modules	40+
Test Suites	17
Lean Version	v4.33.1

🏛️ Formal Domains

Core Foundations

• Core: Algebra, Topology, Analysis, Logic, References, TierCalculus

String Theory & High-Energy Physics

• Duality: DualScale, T-Duality, EffectiveScale
• K3: K3Surfaces, CooperSym2, FDM_Candidates
• StringTheory: FTheory, Swampland, K3×T², Künneth product, 19 string inequalities, vacuum selection
• ParticlePhysics: Index theorem, PMNS matrix

Quantum & Information

• Quantum: Golay code [[24,12,8]], GolayM24 holographic error correction
• Moonshine: RAMA η-quotient, Mathieu bispectrum, M24 representations, vacuum energy, extremal level-12

Cosmology & Gravity

• Cosmology: Bayesian evidence, dark energy, inflation, Moonshine vacuum
• Inflation: Inflationary observables (r = 12/Ne², ns = 53/55, LiteBIRD)
• ChameleonGravity: DAC model, NGC 1052-DF2, Cassini bounds

Mathematical Foundations

• Ramanujan: RAMA Engine, Callens-Alix S₂₀ kernel, ShadowBridge
• NavierStokes: Hypothesis U, BKM criterion, enstrophy, frustration index
• ModularForms: Poincaré upper half-plane
• Pregeometry: Hypergraph K₄, discrete Laplacian, ORF suppression
• AlienMath: Kal charging matrix, holographic border rank, exact SOS witnesses

🚀 Infrastructure

• GitHub Actions CI: lake build + kernel axiom audit + blueprint verification
• 5 automation scripts: axiom audit, olean kernel check, blueprint generator, external library sync
• Full documentation: Blueprint DAG, Proof Manifest, Axiom Audit Report, External Libraries, Agent Guide
• Scientific specification with neuro-symbolic toolchain architecture

📄 License

MIT