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 the coupled Dirac–Λ\LambdaΛ framework. Working entirely at the operator level, it couples a Dirac-type spectral carrier to a scale-dependent capacity inequality and proves that the resulting admissibility constraints uniquely force the generation number N=3N=3N=3.

The central result is that spectral-load feasibility within a globally fixed 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 Λ\LambdaΛ-budget yields a two-sided exclusion mechanism: N≤2N \le 2N≤2 is excluded by CP capacity, while N≥4N \ge 4N≥4 is excluded by a double-squeeze obstruction. Thus three generations are the unique record-admissible feasible stationary realization.

The analysis 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 Λ\LambdaΛ-budget
• Finite-support KKT saturation
• Band-limited spectral envelope bounds
• The transcendental capacity gradient and its induced scale-dependent root structure

In addition, the paper shows that the spectral-action stationarity polynomial alone cannot generate multiple generations, that a single active scale cannot produce full generation structure, and that multi-generation realizations are intrinsically multi-scale within the KKT support.

The capacity box provides the first step in the structural closure program of the coupled Dirac–Λ\LambdaΛ system. Subsequent papers establish boundary-operator identification, carrier-conditioned 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.