A Functional–Geometric Proof of the Mass Gap for Lattice SU(3) Yang–Mills

Mass Gap for Lattice SU(3) Yang–Mills: a Functional–Geometric Proof (no small parameter) This deposit presents a complete proof of the mass gap for pure SU(3) Yang–Mills on the lattice. The argument is organized around two scalar invariants that control the entire chain: a geometric intensity (“g-star”) extracted from a local non-abelian comparator, and an information–theoretic capacity (“sigma”) that measures mixing/entropy dissipation. With these two numbers and a renormalization-group corridor, we obtain an area law, exponential clustering, a positive spectral gap, and finally a positive Wightman mass—without invoking a small-parameter expansion. Core ideas (one screen): Master RP foundation. Reflection positivity is put into a single, reusable lemma that is stable under finite overlaps and the concrete locking/embedding maps used by the RG. Non-abelian “Villain” comparator. A local quadratic comparator on SU(3) simultaneously yields (i) Bakry–Émery curvature for a log-Sobolev inequality and (ii) Brascamp–Lieb domination. This single block powers two interchangeable routes to the area law. Two routes to the area law.Route A1: curvature ⇒ modified log-Sobolev ⇒ quasi-factorization of mutual information ⇒ area law.Route A2: RP chessboard inequality + Brascamp–Lieb ⇒ per-crossing free-energy cost ⇒ area law. RG “beta-corridor”. One-step multiplicative controls plus summable additive losses guarantee that the product beta × g-star stays above the entrance threshold along the entire RG trajectory to the continuum. No threshold “melts” as the lattice spacing goes to zero. Capacity as the currency. Capacity is monotone under CP layers (up to controlled errors) and transfers to the spectral gap; Osterwalder–Schrader reconstruction then transfers the spectral gap to a positive Wightman mass. What is in the paper: A one-page roadmap and a compact main proof line (“Main Theorem”). Appendices with full details: local chart calculus and the comparator, MLSI and quasi-factorization, RP chessboard exponents and per-crossing bounds, CP-monotonicity of capacity, RG budgets and corridor sums, OS reconstruction, and a failsafe KP/Dobrushin route (kept as a parachute, not as the main engine). A numerical “ledger” that freezes all thresholds, overlaps, and constants in one place, with conservative values chosen for transparency and auditability. An “OS audit” that checks the five Osterwalder–Schrader hypotheses one-by-one against the lemmas and theorems proved here. What this proves physically (pure gauge): Confinement via an area law with a strictly positive string tension. Exponential clustering in Euclidean space and a positive spectral gap. Existence of a Wightman theory with a positive mass gap in Minkowski space. Scope and limitations:The result covers the pure SU(3) gauge theory (no dynamical quarks). All constants are explicit and conservative; tightening them does not affect the logic. Extending the architecture to include fermions is discussed as future work. How to read:Start with the roadmap, then the Master RP lemma and the comparator. Choose either A1 or A2 to reach the area law, invoke the RG corridor to the continuum, and finish with the capacity-to-gap transfer and OS reconstruction. The ledger provides a single source of numerical thresholds used throughout.