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 nonlinear balance equations and solves that system on a round three-sphere background.

For product geometries of the form M x F, the KKT saturation conditions separate 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, equality between Dirac-side spectral load and boundary dissipative budget yields a balance law linking the internal mass spectrum to geometric spectral data.

The paper establishes several structural results. The full 9 x 9 Jacobian of the balance system has full rank at the Standard Model mass point, giving local uniqueness by the implicit function theorem. When the masses are ordered from lightest to heaviest, the Jacobian is approximately lower triangular, producing a hierarchical cascade of determination. Heavy fermion masses create boundary-budget deficits that require geometric compensation, forcing a spectrum with many light masses and few heavy ones. The analysis also shows that a single active scale cannot generate full three-generation structure, so the solution is necessarily multi-scale and realized through finite-support KKT saturation.

For the round S3 background, the geometric deficit is computed explicitly from shell-sum spectral data and a unique electroweak-scale capacity crossing is identified. Four active scales, selected by intrinsic spectral criteria together with KKT complementarity, determine the full charged-fermion mass spectrum from a small anchor set with at most one residual continuous parameter.

The resulting framework yields four genuine fermion-mass predictions spanning five orders of magnitude, with numerical agreement at the few-percent level or better.

More generally, the analysis clarifies the scope of the Tier-1 no-go theorem: exact Yukawa ratios are not fixed by algebraic data alone, but once the geometric deficit function is specified, the fermion mass vector is locally unique and its hierarchical structure is forced independently of the detailed background.

The framework is formulated in Euclidean signature using fixed spectral filters, determinant prescriptions, and global normalization conventions.