TEBAC Yang–Mills Program II: Finite-Cutoff Gauge-Invariant Construction, Reflection Positivity, and Clay-Alignment Closure

This preprint is the second module of the TEBAC Yang--Mills program. It does not claim a completed proof of the Yang--Mills existence and mass gap problem. Its purpose is to close the finite-cutoff constructive layer required before Osterwalder--Schrader reconstruction and before any continuum mass-gap argument can be attempted. The module constructs a finite-cutoff lattice Yang--Mills system for compact gauge groups, with link variables \(U_∈ G\), plaquette variables \(U_p\), Wilson/heat-kernel admissible plaquette weights, Haar-product measure, gauge-invariant Wilson observable algebra, and a time-reflection structure. The central output is the finite-cutoff reflection-positivity theorem\[ Θ F\,FΛ,a≥ 0\]for positive-time gauge-invariant observables \(F\). This version internalizes the reflection-positivity closure mechanism through an admissible positive plaquette-kernel framework, records the Peter--Weyl positivity route, adds a cluster-readiness statement at finite cutoff, strengthens the \(9D→4D\) gauge-symmetry neutrality ledger, and includes a conditional Osterwalder--Schrader reconstruction handoff theorem. The module also contains a final closure audit distinguishing what is closed in YM-II from the downstream tasks left for YM-III, YM-IV, and YM-V. The closed output of this module is a reflection-positive finite-cutoff gauge-invariant Yang--Mills system ready for the next module:\[{YM-III}:Osterwalder--Schrader Reconstruction for Gauge-Invariant Observables.\] Claim-safety statement: YM-II is closed at the finite-cutoff reflection-positive lattice level for the admissible Wilson/heat-kernel plaquette layer. It does not by itself prove continuum existence, vacuum uniqueness, non-triviality, clustering in the continuum limit, or a positive Yang--Mills mass gap.