The Fermion Mass Prediction Problem in the Coupled Dirac–Lambda Framework: Balance Equations and Multi-Scale KKT Forcing

This paper reduces the fermion mass determination problem in the coupled Dirac–Lambda framework to a finite-dimensional system of balance equations and solves it explicitly on a round three-sphere background. On product geometries of the form M × F, KKT saturation separates into an internal saturation function depending only on the nine Standard Model fermion masses and a geometric deficit function depending only on the Riemannian background. At each active scale, the equality between Dirac-side spectral load and boundary dissipative budget yields a nonlinear balance equation linking the internal mass spectrum to geometric spectral data. We establish the following structural results: • The full 9 × 9 Jacobian of the balance system has full rank at the Standard Model mass point, giving local uniqueness by the implicit function theorem.• The Jacobian is approximately lower-triangular when masses are ordered from lightest to heaviest, producing a hierarchical cascade of determination.• Heavy fermion masses generate boundary budget deficits that require geometric compensation, forcing the observed pattern of many light masses and few heavy ones.• Optimal active scales span eleven decades in proper-time parameter space.• A single-scale spectral cubic admits at most one positive root per sector, requiring a multi-scale finite-support KKT mechanism for three-generation structure. For the round S3 background, we compute the geometric deficit exactly using representation-number shell sums and identify a unique electroweak-scale capacity crossing. Four active scales determined by sensitivity extrema, a Dirichlet-to-Neumann threshold condition, and a plateau-onset criterion together with KKT complementarity conditions determine all nine charged fermion masses using three anchor inputs and at most one residual continuous parameter. The resulting framework produces four genuine mass predictions spanning five orders of magnitude, with numerical agreement at or below the few-percent level. The analysis clarifies the precise scope of a Tier-1 no-go theorem: exact Yukawa ratios are not determined by algebraic data alone, but once the geometric deficit function is specified, the mass vector is uniquely determined locally and exhibits forced hierarchy independent of background choice. The framework is formulated in Euclidean signature using fixed spectral filters, determinant prescriptions, and global normalization conventions.