Machine-Checked Haar Integration by Parts on Finite SU(2) Edge Spaces: Pauli Flows and Two-Generator Transfer

We formalize the finite-dimensional integration-by-parts mechanism used inlocal proofs of the Makeenko—Migdal equation. For a measure-preserving realflow on a finite measure space, Lean verifies differentiation under theintegral from an explicit locally uniform integrable majorant and proves thatthe integral of the generator vanishes. A bounded dominated product interfacethen yields integration by parts, and two successive applications transfer amixed pair of generators from a density to an observable with positive sign.These results are instantiated on normalized Haar probability measure ofconcrete SU(2) and on every finite product SU(2)^E, for left and rightmultiplication of one selected edge. To fix the representation normalization,we construct three explicit trigonometric curves in SU(2) and prove entrywisethat their tangents at the identity are i sigma_1/2, i sigma_2/2, and isigma_3/2. The producer contains 30 public definitions, structures, andtheorems in 519 physical lines; its audit contains no local sorry, admit, oraxiom, and the headline results depend only on propext, Classical.choice, andQuot.sound. This closes the Haar integration-by-parts layer. It does not claimthe full four-area Makeenko—Migdal identity: the remaining formal inputs arethe heat-density directional identity, extended gauge invariance at acrossing, and their geometric assembly.