Source supplement to Galashin's Conjecture 4.3 via a convex gradient

Lean 4 / Mathlib sources accompanying the paper Galashin's Conjecture 4.3 via a convex gradient (October 2026 revision, DOI 10.5281/zenodo.23676807), with two standalone finite checks. The library proves 123 frozen statements with only the standard axioms (propext, Classical.choice, Quot.sound): the combinatorial spine (cotangent cycle inequality, full row-difference identity, winding-one partition and its cycle count, coloop facts, the partition for decorated permutations), the analytic steps (assembled Hessian with kernel exactly the component-constant vectors, hinge identity, potential and gradient, strict convexity and an injective open gradient on the gauge slice, the variational inverse), the sharp cotangent cycle inequality with its constant, the two-sided Laplacian bound and the inverse-Lipschitz bound modulo component shifts, the first form of the cycle count, the combinatorial half of the shift corollary, the three facts of loop deletion used in the paper's remark on loops, the paper's Lemma 14 and the lemmas of its window induction as stated, the component reduction of its Lemma 5, a conditional homeomorphism theorem whose four hypotheses stand for Propositions 4.2, 5.1 and 5.12 and Lemma 5.13 of Galashin and the Muller-Speyer twist, a second conditional form of that theorem and of the paper's Corollary 32 in the ordinary necklace minors, whose hypotheses do not mention the twist, and an interface with Lean transcriptions of Galashin's definitions (bounded affine permutations, necklaces, the refined boundary, crossings, alignments and the exponent of his Proposition 5.12). Among them is the paper's gradient-preimage corollary: every point of the gradient's image on a gauge slice has exactly one preimage there, the unique minimizer of the tilted potential; its statement, unlike the others, was frozen after its proof. Lean constructs neither the measurement map nor the twist, so this is not a formal proof of the paper's main theorem. The archive contains the exact sources and dependency pins, a source manifest with digests, a standalone NumPy checker of the measurement-to-gradient convention with its retained receipt, a standalone exhaustive check of the arithmetic behind the window-induction proof, and a retained definition-check report. Its README lists what is proved, what is assumed, and what is not formalized.

The paper's results are offered as proof candidates pending external review: the mathematics awaits external verification, and these sources are released to solicit verification, corrections, or counterexamples. The text was produced with AI assistance; see the paper's note on process.