Relational Lysis and the Necessity of Geometric Solved States: A Structural Resolution of the Yang-Mills Mass Gap

We construct a nontrivial four-dimensional quantum Yang–Mills theory with compact gauge group and prove that its Hamiltonian possesses a strictly positive mass gap. The construction originates from Relational Lysis, a deterministic primitive framework in which admissible configurations admit finite solved states whose intrinsic relational couplings induce a canonical self-adjoint Laplacian. Finite connectivity of solved states yields uniform coercivity and a persistent spectral scale under refinement. Using Mosco convergence of quadratic forms and strong resolvent convergence of operators, we obtain a continuum covariant Laplacian associated with a smooth gauge connection on R4, whose holonomy recovers the Yang–Mills functional. A Gaussian measure constructed from the limiting operator satisfies reflection positivity, enabling Osterwalder–Schrader reconstruction of the quantum theory. The spectral scale originating from finite relational structure transfers to the continuum Hamiltonian, establishing a strictly positive mass gap. This construction yields a nontrivial Yang–Mills theory with a strictly positive mass gap under the stated hypotheses.