A Solution to the Yang–Mills Existence and Mass-Gap Problem for the Gauge Group SU(3), via the Pattern Algebra Framework

This work presents a construction bearing on the Yang–Mills existence and mass-gap problem for the gauge group SU(3), developed within Pattern Algebra, a framework built from a small set of primitives and axioms from which the four normed division algebras, the SU(3) lattice gauge theory, and its coupling β = 3/π are forced rather than chosen. The construction proceeds at a fixed, physically motivated lattice spacing, where existence and a positive, volume-uniform spectral gap are established by two independent routes: an algebraic fence theorem pricing the mandatory orientation event, and the standard constructive route via reflection positivity and the convergent strong-coupling expansion (Osterwalder–Seiler). The gap scale (320.6 MeV), the confinement scale, and a terminating glueball tower are derived parameter-free and checked against lattice determinations. The continuum is reached through an explicitly constructed correlator-level limit object on ℝ⁴, exactly Euclidean-invariant and carrying the gap; the identification of that limit with the continuum theory the Clay problem intends is stated as the work's one defended reading. Every load-bearing claim carries an explicit grade — Theorem, Derived, Conditional, or Interpretive — and the document closes with the full theorem registry: identifiers, axiom dependencies, and proofs. The document is offered for verification, criticism, and attempted refutation.