Structural Closure of the Coupled Dirac–Lambda Framework: Global Mass Determination and Scheme Rigidity

This paper establishes full structural closure of the coupled Dirac–Lambda framework by resolving the two remaining open problems: global mass determination without observational input, and scheme rigidity. For the fermion mass problem, we prove that the internal spectral action is strictly concave in the Yukawa mass variables, with diagonal negative-definite Hessian. The physical configuration is identified as the unique global maximizer of the spectral action subject to the capacity inequality. Using analytic dominance estimates, bounded feasible-domain arguments, and full-rank KKT structure, we establish unrestricted global uniqueness of the nine-parameter fermion mass vector. The previous anchor masses are eliminated by exhibiting an overdetermined self-consistent system. All four active scales are shown to be intrinsically determined by structural conditions, sensitivity extrema, Dirichlet-to-Neumann threshold crossing, and saturation-peak selection removing the final tolerance parameter. For the scheme rigidity problem, we prove: • The spectral filter is constrained to the Lorentzian decay family with exponent greater than or equal to six by shell-sum convergence and moment sign conditions.• The boundary budget functional is unique up to bounded correction.• The effective spectral band is determined by the filter itself.• All structural predictions, generation number, gauge coupling ratios, hierarchy forcing, cascade ordering, strong CP exclusion, and cosmological constant scaling are invariant under admissible scheme variation.• Exact fermion mass values vary smoothly and in a bounded manner under scheme changes. We introduce the dominance margin as an intrinsic manifold filter. For any compact Riemannian four-manifold with nonempty feasibility set and dominance margin greater than one, the framework achieves full closure: the fermion mass vector is uniquely and intrinsically determined with no free continuous parameters and no observational mass input. The only remaining structural input is the underlying Riemannian manifold. This work completes the structural closure program of the coupled Dirac–Lambda system.