Non-Existence of the Continuum Limit for Confining Non-Abelian Gauge Theory under Endpoint-Only Ontology

The Clay Mathematics Institute formulation of the Yang–Mills mass gap problem requires a rigorous construction of quantum Yang–Mills theory on R⁴ within the Wightman or Osterwalder–Schrader axioms. This paper proves a non-existence-or-triviality theorem under the Endpoint-Only Ontology, where physical content is exhausted by countable interaction events and conditional probability distributions over them.

The argument proceeds in three steps. First, coarse-graining of the discrete event set yields only discrete translation invariance, never continuous Poincaré invariance at finite scale. Second, the continuum limit either violates countability (densification route) or, under countable persistence, produces tempered distributions whose physical content on a countable subset is forced to a free-field generalized limit. Third, this conclusion is independently confirmed by the Aizenman–Fröhlich triviality result for φ⁴₄, extended to Yang–Mills under the endpoint constraint. The result accommodates causal set theory rather than attacking it: any causal-set construction converging to a Poincaré-invariant continuum limit with interaction recovers only a free theory, providing a structural explanation for the persistent null result of the program in 4D.

The cosmological constant problem dissolves as an artifact of the continuous-vacuum premise. Together with the companion paper (Hadronic Mass Generation through Three-Dimensional Magnetic Locking, DOI 10.5281/zenodo.19827953), which establishes Δ = 3·ε_lock as a geometric mass gap in a discrete locking mechanism, the juxtaposition implies that the Clay problem is not unsolved but misformulated: the object to be constructed does not exist as a continuous field theory.

The paper is part of a broader Sincere Science program applying the Endpoint-Only methodology to four Clay-class problems: Yang–Mills, Navier–Stokes, P versus NP, and the Riemann Hypothesis.