callensxavier/socrateai-fricke-involution

Fricke Involution on Γ₀(N) — Lean 4 Formalization

Machine-checked Lean 4 formalization of the Fricke involution

on the congruence subgroup Γ₀(N), and of the operator it induces on modular forms — together with the papers written around it and the record of a prior-art audit that changed what those papers claim.

DOI: 10.5281/zenodo.22542571 · Author: Xavier Callens · Code: SocrateAI-Lean-Lib (release fricke-v1) · Papers: SocrateAI-Scientific-Communication

Read this first: no priority is claimed

An earlier draft of the main paper claimed this was the first Lean 4 formalization of the Fricke involution. That claim is false and has been withdrawn. anthropics/fermats-last-theorem — a repository the paper itself cites — already contains Fricke and Atkin–Lehner material in Lean 4, including atkinLehnerLin, a ℂ-linear operator on ModularForm covering the whole Atkin–Lehner family. Its CohCarrier.frickeMat is close to a verbatim match for the conjugation lemma here.

What remains is an independent, Mathlib-idiomatic treatment with an explicit axiom-audit discipline. That is a real but modest contribution. PRIOR_ART_FINDING.md documents how the claim was refuted and what survived. The recommended path for this material is a Mathlib PR, not a venue submission — the internal review verdict was REJECT as a standalone paper.

What is verified

lake build SocrateAI → 3769 jobs, 0 errors, 0 sorry (library-wide; this module's own 16 headline theorems are sorry-free). Lean 4.32.2, Mathlib 905b9581.

Every headline theorem is pinned by a build-failing axiom guard:

/-- info: 'SocrateAI.ModularForms.frickeW_mem_GLPos' depends on axioms:
    [propext, Classical.choice, Quot.sound] -/
#guard_msgs in #print axioms SocrateAI.ModularForms.frickeW_mem_GLPos

16 such guards live in FinalCheck.lean, so a widened axiom footprint breaks compilation rather than quietly drifting. A deliberately-wrong negative control was verified to fail, which is what establishes the guards are load-bearing rather than vacuous.

A negative result: an attempted T-duality bridge

A follow-up module, lean/TDualityBridge.lean, attempted to formalize that $W_N$ is the Lean incarnation of T-duality's action on a $T^2$ complex-structure modulus. One lemma survives and is reusable: frickeW_smul_coe, $(W_N\cdot\tau:\mathbb{C}) = -1/(N\tau)$, proved and sorry-free. The bridge theorem itself does not: an adversarial review found it a relabelling of three already-proved lemmas with no added content, citing a physics reference (Giveon–Porrati–Rabinovici) that only supports level $N=1$. It stays sorry under an inverted axiom-guard tripwire and is reported as an open, actively contested question — the correct physical home for a level-$N$ Fricke-type map appears to be Persson–Volpato's S-duality on the heterotic axio-dilaton in CHL orbifold models, a different modulus entirely. See §A negative result in the main paper.

Contents

Path	What it is
lean/FrickeInvolution.lean	W_N ∈ GL(2,ℝ)⁺, W_N² = −N·I, frickeConj, W γ W⁻¹ ∈ Γ₀(N)
lean/FrickeSlash.lean	slash-invariance preserved → operator on SlashInvariantForm
lean/FrickeModular.lean	holomorphy + cusp-boundedness transported → operator on ModularForm
lean/FrickeComposite.lean	(f∣W)∣W = (−1)^k N^(k−2) f
lean/TDualityBridge.lean	frickeW_smul_coe (proved); the T-duality bridge theorem (contested, sorry)
lean/FinalCheck.lean	1122 #guard_msgs in #print axioms guards, library-wide
dag/theorems.jsonl	statement-dependency graph, 104 nodes, 95 proved, acyclic
dag/check_dag.py	validator: proved ⟹ the named declaration exists and ⟹ all deps proved
dag/PROOF-PATH.md	classical argument ↔ Lean name table
papers/Lean4_Fricke_Involution.{tex,pdf}	the main paper (10pp, English): the involution, its operator, the T-duality negative result, and a note on AI-assisted method
papers/Lean4_Involution_Fricke_FR.{tex,pdf}	French version, for deposit on HAL (hal.science)
papers/Lean4_GL2_Poincare.{tex,pdf}	superseded earlier dry-run paper, kept for the record
PRIOR_ART_FINDING.md	the audit that refuted the priority claim

Method note

The DAG layer follows the architecture described in Anthropic's Formalizing Fermat's Last Theorem and the Prove2Me platform paper: statements carry a natural-language description for search and reuse, status: claimed is the concurrency primitive, and a validator refuses to let a node be marked proved unless it names a declaration that actually exists. That last check is the one that caught a Tier-A claim in this project pointing at a nonexistent theorem.

The paper's closing section, "A note on method: AI assistance and human responsibility," follows the disclosure norm argued for by Terence Tao (Mathematics in the age of AI, arXiv:2608.16753) and the Leiden Declaration on Artificial Intelligence and Mathematics: the Lean formalization, literature search, and drafting were produced with LLM assistance under human direction, the Lean kernel — not the model — admits or rejects every mathematical claim, and the author is responsible for every claim in the paper, including the two retracted ones recorded above and in the T-duality section.

Reproducing this

git clone https://github.com/xaviercallens/SocrateAI-Lean-Lib
cd SocrateAI-Lean-Lib
lake exe cache get      # ~5 GB of prebuilt Mathlib oleans
lake build SocrateAI    # 0 errors; FinalCheck's axiom guards run as part of the build

The build configuration is portable and distributed: lakefile.lean requires Mathlib from git at the pinned revision 905b95818eb3…, lean-toolchain matches it, and no tracked file contains a machine-specific path. See BUILDING.md.

Earlier versions of this card said the artifact was not third-party buildable. That was true, and understated — the published lakefile.lean declared no Mathlib dependency at all and its lean-toolchain named a different compiler version. Both are fixed.

Citation

@misc{callens2026fricke,
  author = {Callens, Xavier},
  title  = {A Mathlib-Idiomatic Formalization of the Fricke Involution
            and its Operator on Modular Forms},
  year   = {2026},
  doi    = {10.5281/zenodo.22542571},
  note   = {Lean 4 formalization; no priority claimed, see PRIOR_ART_FINDING.md},
  url    = {https://doi.org/10.5281/zenodo.22542571}
}