Spectral Admissibility of Influence: Reputation, Autonomy, Fredholm Obstruction, and the Forbidden Laws

This manuscript establishes a rigorous operator-theoretic framework to mathematically distinguish admissible influence from forbidden manipulation. A core axiom of this architecture is the strict separation of reputational testimony from intrinsic truth: reputation is modeled merely as an external observer field dependent on coordinate frames, whereas true autonomy and manipulation are defined by absolute internal spectral invariants. The system encodes ten "Forbidden Laws" of influence—including autonomy preservation, non-coercion, reversibility, and truth integrity—into a binary admissibility predicate. To achieve maximal rigor, the framework is lifted from bounded operators to closed quadratic forms. Admissible influence is mathematically proven to correspond to a form-subcritical perturbation of a baseline autonomous dynamics, governed by Kato-Lions-Milgram-Nelson (KLMN) stability and coercivity bounds. Conversely, forbidden manipulation is formally identified as the loss of autonomy through spectral rupture. The manuscript demonstrates that this collapse can be explicitly detected via Fredholm index failure, Birman-Schwinger threshold crossing, or the vanishing of a regularized Hilbert-Schmidt determinant. Finally, the theory is extended into semigroup dynamics to encode temporal restoration and exponential contractivity, and into a pseudodifferential layer where the Connes Spectral Action and Zeta-regularized determinants extract legitimate, curvature-invariant geometry from the admissibility operators. Ultimately, this work establishes a terminal closure: testimony proves only perception, while exact spectral structure proves admissibility.