Abstract We develop a primitive-Wilson endpoint framework to rigorously establish the four-dimensional Yang–Mills mass gap. The proof is structured around a central object: the exact Wilson conditional-variance ledger DB (F) = \|F\|² - \|EF | GB\|² = EVar (F | GB), which is pulled back to a single stopped parent prior to the selection of any endpoint vector. Our approach replaces informal endpoint assumptions with certified finite rows, representing any failure of strict transfer contraction as a faithful guarded Wilson branch, provided the loss does not enter a registered port. Within this framework, OS (Osterwalder-Schrader) representatives, slab defects, chart transitions, ghost holonomy, large/small-field atoms, reflected collars, rank overlap, and UV/PCCR leakage are strictly defined as restrictions, quotients, lower-semicontinuous limits, or first-hit charges of the same parent record. Specifically, we represent OS non-collapse via a reflected Gram certificate, slab transfer through a conditional-variance identity combined with certified collars, and exact infrared capture via specialized rank-action and port ledgers. The final argument is a same-parent finite-budget implication. We demonstrate that a port-free zero-gap endpoint would necessarily produce synchronized non-central Wilson witnesses with a fixed lower charge across an arbitrary number of slabs, while the unified parent budget remains finite and independent of the tube length. Consequently, upon verification of the finite certificates from the primitive Wilson parent, the locked centered OS transfer operator is proven to be strictly contractive, ensuring the reconstructed Hamiltonian possesses a positive lower bound on the non-vacuum sector. Furthermore, we provide an internal non-circularity audit, confirming that the result is achieved without assuming the mass gap, area law, confinement, or vacuum uniqueness as inputs to the finite-budget contradiction.
Juan José Chelía (Sun,) studied this question.