callensxavier/socrateai-fricke-selfdual-etaquotients

Fricke-Self-Dual Eta-Quotients Are Eigenforms

Machine-checked Lean 4 corollary: if the exponent vector $r$ of the eta-quotient $f(z) = \prod_{\delta\mid N}\eta(\delta z)^{r_\delta}$ satisfies $r_\delta = r_{N/\delta}$ for every $\delta\mid N$ ("Fricke-self-dual"), then $f$ is a genuine eigenform of the Fricke transform $z\mapsto -1/(Nz)$, with a closed-form eigenvalue $\lambda(N,k) = i^{-k}N^{k/2}$ depending only on the level and weight — plus an exact characterization of when $\lambda = \pm1$.

⚠️ PREPRINT — not peer reviewed. The Lean development compiles and is axiom-audited; those claims are machine-checked. The paper's exposition has had no external referee.

DOI: pending (Zenodo deposit in progress) · Author: Xavier Callens · Code: SocrateAI-Lean-Lib · Paper: SocrateAI-Scientific-Communication

The result

etaQuotient_fricke (already proved, sorry-free, in the sibling eta-quotients artifact) gives the Fricke transform of a general eta-quotient as a known scalar times the dual quotient (exponents reversed, $\delta \mapsto r_{N/\delta}$) — in general a different function. This artifact specializes to the case where the two coincide. The specialization is short — one new product identity, $\prod_{\delta\mid N}\delta^{r_\delta} = N^k$ on self-dual $r$ (prod_zpow_selfDual), and every other declaration is a direct substitution into the pre-existing transformation law.

What is verified

lake build SocrateAI → 3769 jobs, 0 errors. Lean 4.32.2, Mathlib 905b9581. This module: 333 declarations in 6437 lines, entirely sorry-free. 24 statement-graph nodes (two definitions, sixteen general lemmas and characterizations, five worked eigenvalue pins independently cross-checked against a 600-term $q$-product evaluation of $\eta$, one statement- shape guard), all proved. 299 of the 333 declarations carry a build-failing #guard_msgs in #print axioms guard reporting exactly [propext, Classical.choice, Quot.sound]. Kernel proof terms were inspected directly (#print) to confirm the headline theorems are genuine substitutions into etaQuotient_fricke, not restatements that merely typecheck.

The physics motivation — clearly labeled as motivation, not a formalized claim

Persson and Volpato, Fricke S-duality in CHL models (arXiv:1504.07260), attach to a CHL orbifold element a "generalized Frame shape" — exactly a EtaExp exponent vector on N's divisors — and call it balanced when m(N/a) = m(a), exactly the self-duality condition here (their word; this codebase deliberately never reuses it, for two independent reasons documented in the source). Their headline physical claim: CHL models are self-dual under a Fricke S-duality $S \mapsto -1/(NS)$ on the heterotic axio-dilaton modulus exactly when the Frame shape is balanced — a genuine physics derivation (charge-lattice $N$-modularity, Conway-group representation theory, BPS/Witten-index counting) that this Lean library does not contain and this artifact does not attempt. No Lean declaration in this development names or depends on any physical object; the physics appears only in docstrings, explicitly marked as not formalized. The paper also states precisely where Persson–Volpato's own normalization (weight-24, from a 24-dimensional representation) diverges from the eta-quotient weight convention used here — the two agree only at $N=1$.

Contents

Path	What it is
lean/EtaQuotientFrickeSelfDual.lean	the full development: IsFrickeSelfDual, frickeEigenvalue, the eigenform theorem, the ±1 characterization, all pins and negative controls
dag/theorems.jsonl	statement-dependency graph, library-wide (104 nodes, 95 proved)
papers/Lean4_FrickeSelfDual_EtaQuotients.{tex,pdf}	the paper

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    # 3769 jobs, 0 errors

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

Citation

@misc{callens2026frickeselfdual,
  author = {Callens, Xavier},
  title  = {Fricke-Self-Dual Eta-Quotients Are Eigenforms: A Corollary in Lean 4,
            and a Note on Its Motivation in Persson--Volpato's CHL S-Duality},
  year   = {2026},
  note   = {Preprint, not peer reviewed. Zenodo DOI pending.}
}