An Unconditional Mass Gap for 4D Yang–Mills via PLH, Schur Reduction, and Two-Scale LSI

We prove the existence of a quantum Yang–Mills theory on \( \mathbb{R}^4 \) with a strictly positive mass gap for any compact simple gauge group \(G\). Starting from the Euclidean lattice with reflection positivity and a positive transfer matrix, the argument combines a two-scale logarithmic Sobolev inequality (LSI) with either (i) a total-variation/Doeblin contraction or (ii) an operator-level Schur complement bound across a half-space cut. Gross’ equivalence (LSI ⇔ hypercontractivity) yields modular nuclearity and the split property; the Osterwalder–Schrader reconstruction then identifies the exponential decay rate with the spectral gap of the Hamiltonian. The “PLH” layer is used only as an observation re-encoding (UCP maps) and is not needed for the core implication to the gap.