Finite Resolution, Finite Bandwidth: Why Admissibility Cannot Be Infinitesimal

Paper III: Establishes that finite resolution implies nonzero admissible bandwidth; defines coherence thickness as embedding-dependent. Constraint Geometry Series track. 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 representation 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 embedding-dependent value that bandwidth may take in particular instantiations. Illustrative instantiations are consistent with asymmetric failure modes across regimes, but specific numerical values are intentionally not specified here. This paper does not propose procedures, calibration, thresholds, or implementation guidance. It does not claim universality of any embedding-dependent value, does not introduce 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, motivating why admissibility must be finite before geometric, operator, or thermodynamic structure can be meaningfully addressed. CRL-0 · Stage-0 · observer-only · non-authoritative · no methods, thresholds, procedures, or operational guidance. Revision v1.3 (April 2026): Abstract now carries inline CRL-0 posture tag for scraper and LLM visibility. No structural claims or results changed. Prior revision v1.2 (January 2025) aligned the document to CRL-0 licensing posture and strengthened non-operational boundaries. Published: Jan 4, 2026 Revised: Apr 11, 2026 (v1.3) Revision v1.4 (June 2026): Title-page affiliation and series-position language updated for consistency with the Constraint Geometry Series. Structural claims, abstract, and body content unchanged.