The Coupled Dirac–Λ Dynamical System: Unified Operator Equations for a Capacity-Constrained Spectral Action Framework

This paper formulates the coupled Dirac–Λ\LambdaΛ dynamical system, a unified operator framework combining a Dirac-type spectral carrier with an independent capacity-constrained scalar sector. The system is defined by a self-consistent pair consisting of a spectral-action stationarity equation and a global scale-dependent capacity inequality enforced through finite-support KKT constraints.

The work presents the complete unified field equations (E1–E8), together with the normalization bridge E9**, including:

• The spectral action and Dirac-side entropy functional
• The determinant-class Λ\LambdaΛ-budget and UV anchor normalization
• The global capacity inequality and KKT stationarity structure
• The finite active-scale saturation system governing generation structure, mass hierarchy, crossing scales, and parameter determination
• The boundary-to-coupling normalization bridge fixing the absolute gauge coupling scale

Low-energy recovery of Einstein–Hilbert gravity, Yang–Mills dynamics, and fermionic Dirac structure follows from the heat-kernel expansion of the spectral action. The capacity coupling introduces additional structural constraints absent in pure spectral-action models.

At the operator level, the framework yields:

• Finite-scale reduction via Carathéodory support
• Local mass uniqueness from Jacobian nondegeneracy
• A non-tunable spectral envelope bound
• Structural exclusion of the strong CP phase (θ=0\theta = 0θ=0) within the OS-reconstructible Euclidean formulation
• A curvature-rigidity mechanism selecting the Born exponent p=2p = 2p=2

The present paper establishes the unified dynamical architecture of the coupled Dirac–Λ\LambdaΛ system and defines the internal spectral invariants governing generation structure, mass scales, and gauge normalization. Full structural closure — including generation forcing, boundary-operator identification, carrier-conditioned mass determination, global uniqueness, and geometric fixed-point selection — is established rigorously in the companion closure series (Papers 1–5). On the adopted S3S^3S3 carrier, the framework yields α−1=137\alpha^{-1} = 137α−1=137 with zero free parameters.

The framework is formulated entirely in Euclidean signature and within spectral and variational operator theory. All determinant prescriptions, spectral filters, and normalization conventions are fixed globally as part of the model definition.