Stokes' Theorem for Smooth Singular Cubes in Lean 4: True Pullback, Bridges to mathlib4, and Chain-Level d^2=0

TL;DR

This paper formalizes Stokes' theorem for smooth singular cubes in Lean 4, incorporating true pullback, chain-level properties, and bridges to existing mathlib4 structures, advancing formal differential geometry.

Contribution

It introduces a comprehensive Lean 4 formalization of Stokes' theorem with true pullback and chain-level results, connecting to mathlib4 and extending previous work.

Findings

Formalization of Stokes' theorem in Lean 4 for arbitrary dimensions

Proof of d^2=0 for singular cubical chains

Bridge established between Lean 4 and mathlib4's extDeriv

Abstract

We present a sorry-free Lean 4/mathlib4 formalization of Stokes' theorem for smooth singular cubes in arbitrary dimension, using true differential-form pullback via the Frechet derivative. The development also includes a bridge to mathlib4's abstract extDeriv, chain-level Stokes extended by Z-linearity, d^2=0 for singular cubical chains, box Stokes for axis-aligned cubes, dimensional specializations, and a structured comparison with Harrison's HOL Light formalization.