Title: A Machine-Checked Proof of the Strong-Coupling Mass Gap in U(1) Lattice Gauge Theory Author: Alex Novickis (alex.novickis@gmail.com) We give a complete, machine-checked proof, in the Lean 4 proof assistant on the Mathlib library, that pure U(1) lattice gauge theory in the strong-coupling regime has a positive mass gap — the lattice form of the Osterwalder–Seiler (1978) confinement result. The proof has two independent halves that meet at a single parameter. A convergent polymer (character) expansion bounds the anchored activity sum uniformly in the volume by a geometric series; a transfer-matrix spectral gap has the Fourier-diagonal transfer operator strictly contract the orthogonal complement of the vacuum. Both are governed by the small activity scale $\beta/2$: the quantity that makes the expansion converge is the subleading transfer-matrix eigenvalue that opens the gap $m = \log(2/\beta) > 0$. Every theorem depends only on the standard three axioms ($\texttt{propext}$, $\texttt{Classical.choice}$, $\texttt{Quot.sound}$), with no $\texttt{sorry}$ placeholders and no domain-specific axioms. This is the strong-coupling lattice theorem, not the continuum Yang–Mills mass gap; the continuum (Clay Millennium) problem lies outside its scope, and we state exactly what is and is not proved. Key results: A complete Lean 4 / Mathlib formalisation (standard-three axioms, no sorry) of the strong-coupling U(1) lattice gauge mass gap. A volume-independent lattice-animal entropy bound $K = (2d+1)^2$ from a covering-walk (spanning-tree) argument — the genuine origin of the gap's uniformity in the infinite-volume limit. The U(1) character-coefficient bound $I_n(\beta)/I_0(\beta) \le (\beta/2)^n$ for the modified Bessel functions, giving the strong-coupling activity decay. A Fourier-diagonal transfer operator with spectral gap $\log(2/\beta) > 0$, realised in both a finite-mode form and the full $\ell^2(\mathbb{Z})$ (infinite-dimensional) form, with exponential clustering of the Euclidean two-point function. Honest scope: this is the strong-coupling U(1) lattice theorem, not the continuum Yang–Mills (Clay Millennium) mass gap; the paper states precisely what is and is not proved. Keywords: [lattice gauge theory, mass gap, U(1) gauge theory, Osterwalder-Seiler, strong coupling, character expansion, cluster expansion, Kotecky-Preiss, polymer expansion, lattice animals, transfer matrix, spectral gap, exponential clustering, modified Bessel functions, Wilson lattice, confinement, Yang-Mills, reflection positivity, Hilbert space, Lean 4, Mathlib, formalization, machine-checked proof, interactive theorem proving, formal verification] Series: Paper CLXIII in the Hopf Soliton Programme (formalisation companion)