Generation Forcing in the Capacity-Constrained Dirac–Lambda Framework: The Capacity Box Construction

This paper establishes the capacity-box mechanism that forces generation structure within a capacity-constrained spectral framework. Working entirely at the operator level, the analysis couples a Dirac-type spectral carrier to a scale-dependent capacity inequality and demonstrates that admissible configurations are restricted to a discrete structural class. The central result is that spectral-load feasibility within a fixed global determinant scheme produces a finite “capacity box” in scale space. Within this box, the interplay between the Dirac-side entropy functional and the determinant-class budget enforces generation counting and excludes continuous degeneracy. The argument proceeds without phenomenological input. No external symmetry assumptions, ad hoc counting rules, or fine-tuned parameters are introduced. The forcing arises from: • The global capacity inequality• The determinant-class structure of the Lambda budget• Finite-support KKT saturation• Band-limited spectral envelope bounds The capacity box provides the first step in the structural closure program of the Dirac–Lambda system. Subsequent papers establish boundary-operator identification, mass determination, global uniqueness, and geometric selection. All conventions, spectral filter, determinant prescription, and normalization are fixed globally as part of the model definition. No per-background retuning is permitted. The framework is formulated in Euclidean signature and within spectral and variational operator theory.