Yang–Mills Mass Gap
Unconditional Lattice Gap and an AF–free Continuum Construction
Recognition Science Institute, Austin, Texas, USA
Summary
The manuscript proves a positive mass gap for pure SU(N) Yang–Mills on the lattice via reflection positivity and a uniform two‑layer reflection deficit, and develops an AF‑free norm–resolvent convergence route to the continuum on fixed regions with all inputs proved inside the paper.
Abstract
We present an unconditional lattice proof of a positive mass gap for pure SU(N) Yang–Mills in four Euclidean dimensions. On finite 4D tori with Wilson action, Osterwalder–Seiler reflection positivity yields a positive self‑adjoint transfer operator; a uniform two‑layer reflection deficit on a fixed physical slab gives an odd‑cone one‑tick contraction with per‑tick rate \(c_{\rm cut}>0\), hence a slab‑normalized lower bound \(\gamma_0\ge 8\,c_{\rm cut}\), uniform in volume and \(N\ge 2\).
For the continuum, we give a precise, unconditional AF‑free norm–resolvent convergence (NRC) construction on fixed regions. The inputs are proved in this manuscript: UEI/equicontinuity and the U2 package (isometric embeddings, graph‑defect, and low‑energy projector control), together with a quantitative OS1 commutator/resolvent bound on fixed regions. Continuum mass‑gap statements are derived unconditionally with constants tracked and volume‑uniform on fixed slabs. An alternative Mosco/AF route is recorded in an appendix as a cross‑check only and is not used in the main chain.
Key Points
- Unconditional lattice mass gap via reflection positivity and a two‑layer reflection deficit.
- AF‑free norm–resolvent convergence to the continuum on fixed regions with proved UEI/OS1 inputs.
- Uniform lower bounds in volume and \(N\ge 2\); explicit constants tracked.