A Positive Mass Gap for Z₂ Lattice Gauge Theory in 2+1 Dimensions: Exact Lemmas, Uniform Bounds, and the Road to Yang–Mills

Preprint and research-program record concerning the mass-gap demand of the Clay Mathematics Institute Yang–Mills Millennium Prize problem. Headline results at explicit grades: the test-bed theorem (proven) — for the Z₂ lattice gauge theory in 2+1 dimensions in its exact transfer form, for every plaquette weight v ≤ 0.4858, the infinite-volume limit of the strip family exists and every local gauge-invariant correlator decays exponentially with box-independent constants, the decay floor bounded below by 0.514; Theorem B4 (proven) — the same constants uniform in the coupling on every certified one-sided window; the center theorem (proven) — the quantum superselection sectors are the characters of the gauge group's center, one statement unifying Z₂, SU(2), and SU(3), with S₃ the verified null case. Measured columns with machine receipts: certified mass floors 0.514 / 3.946 / 8.667 on Z₂ / SU(2) / SU(3); the deconfinement transition bracketed at [0.19, 0.23]; the dimensionless gap bounded in [1.96, 3.88] with the exact free-loop anchor 2. The paper does not claim the Clay statement itself; the remaining distance is mapped as three named gaps. This is version 2: the public title adopted for searchability; all results and claims unchanged from version 1 (DOI 10.5281/zenodo.23736552). The deposit includes the program's dated research ledger (64 version entries, 2026-10-20 to 2026-10-28) and a SHA-256 manifest of the program's working corpus, fixing its state as of 28 October 2026.