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 fermion-mass determination without observational mass input, and rigidity of the admissible spectral scheme.

For the mass problem, we prove that the internal spectral action is strictly concave in the Yukawa mass variables. The physical mass configuration is therefore the unique global maximizer of the constrained spectral problem subject to the capacity inequality. Combining strict concavity, feasible-domain bounds, analytic dominance estimates, and full KKT regularity, we establish unrestricted global uniqueness of the nine-parameter fermion mass vector.

The previous anchor masses are then eliminated by constructing an overdetermined self-consistent closure system. All four active scales are shown to be intrinsically fixed by structural conditions, sensitivity extrema, Dirichlet-to-Neumann threshold crossing, and saturation selection, removing the final residual tolerance parameter.

For the scheme-rigidity problem, the paper proves that the admissible spectral filter is restricted to a Lorentzian decay family with exponent at least six by shell-sum convergence and moment-sign constraints. The boundary budget functional is unique up to bounded correction, the effective spectral band is fixed by the filter itself, and the main structural outputs of the framework remain invariant under admissible scheme variation. In particular, generation number, gauge-coupling ratios, hierarchy forcing, cascade ordering, strong-CP exclusion, and cosmological-constant scaling do not depend on per-background retuning, while exact fermion masses vary only smoothly and in a bounded way across admissible schemes.

The paper also introduces 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.