Constructive Yang–Mills in Four Dimensions: A Variational Path to the Mass Gap and a Conjectural Spectral Principle

This work develops a constructive framework for four-dimensional Yang–Mills theory that establishes a uniform spectral gap in three independent tracks. Track I (large-β) proves an unconditional lattice gap via finite-range decomposition, slice convexity, and heat-kernel tilt control. Track II (all-β) establishes a uniform gap unconditionally for SU(2), and conditionally for SU(3) pending improved group-analytic constants or via a coarse-Ricci route. Track III yields an unconditional continuum mass gap for all compact simple gauge groups using FRD locality, fixed-time block logarithmic Sobolev inequalities, and Osterwalder–Schrader reconstruction, with uniqueness ensured via Γ-convergence and EVI methods. The paper concludes with a conjectural Spectral Entropy Principle that, while not part of the proof, suggests a deeper variational structure behind the Yang–Mills spectrum. License: CC BY-NC-ND 4.0 International. Redistribution is allowed with attribution; commercial use and modifications are prohibited. All commercial rights are reserved to the author.