TREP–Yang–Mills Mass Gap by Primitive Wilson Endpoint Persistence. A Non-Circular Conditional Transfer-Gap Criterion.

We formulate a non-circular TREP–Yang–Mills endpoint criterion for obtaining a positive gauge-invariant transfer/Hamiltonian gap in a locked Osterwalder–Schrader (OS) Hilbert space arising from four-dimensional Euclidean lattice Yang–Mills. The paper is organized around one primitive finite object: the Wilson conditional-variance ledger of exact block conditional expectations. The mass-gap conclusion is stated conditionally on an explicit Yang–Mills–TREP verification package rather than as an unconditional theorem. The package isolates the rows that cannot be hidden in the final contradiction: exact holonomic blocking by Markov conditional expectations, same-parent packing for the primitive Wilson variance, guarded product-branch extraction, uniform spin-network observer lower charge modulo registered ports, toral/reducible and high-Casimir port closure, exact infrared capture with a finite stopped parent budget, surface/OS endpoint lock, OS non-collapse and vacuum separation, slab-transfer contraction, and no-late-endpoint closure. Crucially, Peter–Weyl heat-kernel estimates serve only as a finite model row; they do not replace the necessity of exact Wilson blocking. Under this package, a port-free zero-gap endpoint would generate synchronized non-central Wilson witnesses on arbitrarily many physical slabs. Each witness carries a uniform primitive conditional-variance charge, while exact infrared capture supplies a finite parent budget. Same-parent packing yields a finite-budget contradiction, thereby forcing a strict transfer contraction. Thus, the paper provides an arXiv-safe conditional transfer-gap criterion. No area law, mass gap, OS positivity, continuum non-triviality, or exact infrared capture row is imported after the endpoint branch is selected or assumed in a form equivalent to the desired conclusion.