Diagrammatic Dyson–Magnus Admissibility

This manuscript constructs a fully explicit operator-theoretic and diagrammatic calculus that unifies time-ordered evolution, logarithmic Magnus closure, regularized Fredholm admissibility, and spectral invariants into a single, exact mathematical framework. The architecture introduces a strict perturbative splitting (L = L0 + Omega + Sigma), where Omega represents a raw transport or deformation sector, and Sigma represents an admissibility-restoring control sector. Through this splitting, the paper translates the heat-kernel Dyson expansion into an exact loop calculus, the Magnus expansion into a connected commutator-tree calculus, and the regularized Birman-Schwinger determinant into a connected spectral-loop functional. Under precise Schatten-class hypotheses, the manuscript rigorously derives trace-class closure, resolvent expansions, and exact odd-sector heat coefficients (a3, a5, a7) for the local scalar-potential model. In non-self-adjoint settings, the formalism proves that pseudospectral amplification can be precisely controlled. By mapping spectral loops to eigenvalue obstructions and pseudospectral chains to transient amplification geometry, computable observables—such as resolvent norms and minimal singular values—are derived to exactly distinguish destabilizing perturbations from restorative control interventions. Crucially, the manuscript extends this baseline theory to the absolute noncommutative horizon via two terminal constructions: The Unified Generating Functional: The paper establishes a single, trace-class heat-resolvent generating functional, Z(t,z), proving that heat traces, resolvent bounds, zeta functions, determinant loops, and odd local invariants are all simply exact contour or Mellin projections of this single parent object. The Dirac Lift & Spectral Action Closure: The framework is lifted to self-adjoint Dirac-type operators. The paper proves that the Omega + Sigma transport/control split rigorously deforms the Weitzenbock connection and endomorphism. Consequently, the Connes Spectral Action is shown to not be an external heuristic, but rather a direct, asymptotic contour-Laplace projection of the unified generating functional. Ultimately, this work completely closes the Dyson-Magnus-Fredholm-Pseudospectral chain, establishing that transport geometry, determinant obstruction, and the Spectral Action all belong to a single, irreducible operator-theoretic generating structure.