Finite Resolution, Finite Bandwidth: Why Admissibility Cannot Be Infinitesimal

Boundary and Scope Notice (CRL-0): This document is observer-only and non-authoritative. Any method-like descriptions, contrasts, figures, or illustrations are interpretive and non-normative; they are not offered as deployment guidance, certification criteria, or operational decision rules. No recommended calibration, thresholds, protocols, or implementation guidance are provided. Any evaluable instantiation belongs to downstream evaluation artifacts. Structural claims and results are unchanged. This paper establishes that finite resolution necessarily implies a finite admissible bandwidth in any constrained dynamical system. When constraint enforcement is bounded and resolution is non-singular, admissibility cannot be infinitesimal. A nonzero tolerance band must exist between recoverable and non-recoverable trajectories. This result is structural, not empirical. Using a constrained Hamiltonian framework, the paper introduces a thickness functional that regularizes constraint enforcement by penalizing transverse deviations from an admissible manifold. The resulting parameter δ is interpreted as a coherence bandwidth: a measure of how sharply admissibility can be enforced under finite resolution. The core contribution is the distinction between: the necessary existence of a finite admissible bandwidth, which follows from finite resolution alone, and the numerical value of that bandwidth under specific embeddings. Numerical experiments demonstrate a bounded admissible window with asymmetric failure modes. Below the window, dynamics become brittle and ill-conditioned; above it, excessive leakage degrades structural rigidity. Under the embeddings examined, the admissible region localizes near δ ≈ 0.062, close to the minimal admissible threshold. This paper does not claim universality of the numerical value, does not propose new physical laws, and does not assert results about the Riemann Hypothesis. Its contribution is to show that admissibility must have finite thickness in any finite-resolution system. Paper III provides the resolution-level necessity argument in the Constraint Geometry Series, explaining why admissibility must be finite before geometric, operator, or thermodynamic structure can be meaningfully addressed.