callensxavier/socrateai-eta-quotients

Eta-Quotients in Lean 4

Machine-checked Lean 4 formalization of the arithmetic and analytic layers of Ligozat's criterion for eta-quotients

$$f(z)=\prod _{\delta \mid N}\eta (\delta z{)}^{r_{\delta }}f(z)\; \backslash ;=\backslash ;\; \backslash prod\_\{\backslash delta\; \backslash mid\; N\}\; \backslash eta(\backslash delta\; z)^\{r\_\backslash delta\}$$

together with a precisely named obstruction to the general case.

⚠️ PREPRINT — not peer reviewed.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: 10.5281/zenodo.22648098 (all versions) · Author: Xavier Callens · Code: SocrateAI-Lean-Lib · Paper: SocrateAI-Scientific-Communication

Read the scope before citing

Three results at three different strengths. The difference matters:

	Result	Scope
General	Two-sided Θ-asymptotic for $\lVert f\vert_k\gamma\rVert$ at every cusp	All $N$, all $\gamma \in SL_2(\mathbb{Z})$ — no congruence, no $\Gamma_0(N)$-membership
General	Order at $\infty$ in Mathlib's own meromorphicOrderAt/cuspFunction language	All $N$
Bounded	Full Ligozat transformation law $f(\gamma z) = w(\gamma)(cz+d)^k f(z)$ on all of $\Gamma_0(N)$, character identified	Only $N \in {1,2,3,4,5,7,13}$
Obstructed	Ligozat for general $N$	Not proved. See below.

The DAG node ETA-01 (the general criterion) remains open with lean_name: null. This artifact does not claim it.

The obstruction, stated exactly

The multiplier is $w(\gamma) = \exp!\big(\tfrac{\pi i}{12},\mathrm{per}(\gamma)\big)$, where $\mathrm{per}$ is the period cocycle of the weight-two quasi-modular combination $\sum_\delta r_\delta,\delta,E_2(\delta z)$ — exactly the integration constant that both the logDeriv route and the 24th-power route erase. For a single $\eta$ that period function is Rademacher's $\Phi$, whose non-coboundary content is the Dedekind sum $s(d,|c|)$.

Mathlib contains no Dedekind sums. Evaluating $w$ on one hyperbolic element of $\Gamma_0(11)$ is equivalent to evaluating a Dedekind sum. Every disguise tried — Atkin–Lehner conjugation, theta/Poisson, Wohlfahrt level-24 — reduces back to it.

MATHLIB_PR.md names the minimal Mathlib addition that would remove it, and argues that Rademacher's $\Phi$ is best built as the period of $E_2$ on the E2_slash_action machinery Mathlib already has.

Verification

lake build SocrateAI → 3462 jobs, 0 errors, 0 sorry. Lean 4.32.2, Mathlib 905b9581. 519 declarations in 8117 lines. 391 build-failing #guard_msgs in #print axioms guards over 389 distinct theorems; 370 report exactly [propext, Classical.choice, Quot.sound]. A negative control asserting a deliberately wrong footprint is verified to fail, which is what makes the guards load-bearing rather than vacuous.

Of the 453 theorems in the new modules, 14 (3%) have a one-line rfl/decide proof and 45% quantify over a structure. This ratio is reported because an earlier module in this project was 174 numeral identities of which 158 were one-line and carried no evidential weight.

A correction we made to our own specification

The cusp-order node was originally specified with the Θ-exponent written as Ligozat's $\mathrm{ord}(N,r,d)$. That statement is false. Ligozat's order is taken in the local uniformiser $q_h$ at a cusp of width $h = N/\gcd(d^2,N)$, so a decay statement in $\mathrm{Im},z$ carries exponent $\mathrm{ord}/h$. The corrected statement is what was proved, and the discrepancy is stated in the paper rather than quietly matched. Ligozat's positivity condition is unaffected ($h > 0$).

Prior art

Mathlib has $\eta$ (DedekindEta.lean, and the $S$-transformation in Discriminant.lean) but no eta-quotients: EtaQuotient, Ligozat, DedekindSum, etaMultiplier all return zero. anthropics/fermats-last-theorem contains eta material, but it is about modular units on modular function fields; its three Ligozat occurrences are namespace labels (LigozatUnitEngine, LigozatUnitAL), not the criterion. GitHub-wide language:lean returns 0 for Ligozat. Not checked: Lean Zulip, open mathlib4 PRs.

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    # 3462 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.

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{callens2026etaquotients,
  author = {Callens, Xavier},
  title  = {Eta-Quotients in Lean 4: Cusp Orders at Every Cusp, Ligozat's Criterion
            at Small Level, and a Named Obstruction at General Level},
  year   = {2026},
  doi    = {10.5281/zenodo.22648098},
  note   = {Preprint, not peer reviewed. Ligozat proved only for N in {1,2,3,4,5,7,13}},
  url    = {https://doi.org/10.5281/zenodo.22648098}
}