Modular Forms — Hungerbühler–Wasem & Valence Formula in Lean🔗

This is the mathematical blueprint for the Lean 4 / Mathlib formalisation of the Hungerbühler–Wasem generalised residue theorem (Hungerbühler–Wasem 2018) and its application to the classical valence formula for modular forms on \operatorname{SL}_2(\mathbb{Z}), following the textbook proof of Diamond–Shurman.

The development proceeds in two arcs. The first builds the analytic machinery of closed piecewise-C^{1} immersions, the multi-point Cauchy principal value, and the generalised winding number, then states and discharges the paper-faithful form of the Hungerbühler–Wasem residue theorem under its two geometric side-conditions (A') and (B). The second computes the generalised winding numbers at the three elliptic points i, \rho, \rho+1 of the fundamental-domain boundary and feeds them through the residue theorem to obtain the valence formula \operatorname{ord}_\infty(f) + \tfrac12\operatorname{ord}_i(f) + \tfrac13\operatorname{ord}_\rho(f) + \cdots = k/12.

How to read this blueprint. Each node below is a definition or theorem with its mathematical statement and a paragraph-level proof sketch. The dependency graph records which results feed into which. A node is coloured green once the Lean declaration it references (lean := …) is fully proved, blue while it is stated but still contains sorry, and left uncoloured while it is roadmap-only. There is no manual status to maintain: Verso reads it from the Lean side directly.