Galashin's Conjecture 4.3 via a convex gradient

This paper presents a proof candidate for the loopless case of P. Galashin's Conjecture 4.3 (arXiv:2102.13339): for every loopless bounded affine permutation f, the critical measurement map from Galashin's gauge slice to the critical variety is a homeomorphism, uniformly in (k,n), with a constructive inverse. After the Muller-Speyer left twist and passage to logarithmic Grassmann-necklace coordinates, the measurement map becomes the gradient of an explicit potential. Its Hessian is positive semidefinite with kernel exactly the crossing-component shifts, because the crossing edges split into winding-one cycles and a cotangent cycle inequality applies to each. Convexity on a gauge slice then gives an injective local diffeomorphism and the inverse as the unique minimizer of a strictly convex function.

Applications: a homeomorphism between the critical electrical and critical Ising loci of the same generalized region, and a sharp single-cycle convexity modulus. The results are offered as proof candidates pending external review. The research and this text were produced with AI assistance; see the section "A note on process" in the paper. Before release the paper was reviewed internally in a 'reading edition' web application, which sets each formal statement beside what checks it and each step without one beside its source. The Lean sources and two finite checks are in the source supplement, 10.5281/zenodo.23676891.