Finite Resolution, Finite Bandwidth: Why Admissibility Cannot Be Infinitesimal

Abstract

Paper III: We establish that any system with finite resolution, bounded state space, and soft constraint enforcement necessarily admits a nonzero tolerance band separating recoverable from non-recoverable trajectories. The existence of such a band is structural, not empirical. In the context of the four-term Hamiltonian governing prime-gap dynamics, this tolerance band is represented by the parameter δ, which functions as a coherence bandwidth—a soft-constraint thickness parameter regulating admissible excursions from the phase-only constraint manifold. We introduce a thickness functional Θδ(q,p) := |qp|^δ and show that this term supplies normal penalty response without imposing singular hard constraints. This work distinguishes between the existence of a finite admissible bandwidth, which follows from finite resolution alone, and the embedding-dependent value that bandwidth may take in particular instantiations. The structural argument predicts an intermediate admissibility regime with asymmetric failure modes: in the thin regime dynamics exhibit brittle, ill-conditioned behavior, while in the leaky regime excessive excursion degrades structural rigidity and spectral statistics. Under the embeddings examined, the admissible bandwidth localizes near a characteristic value δ⋆, close to the minimal admissible transition scale. This interpretation frames δ as a domain regularizer consistent with viability of self-adjoint extensions under bounded enforcement.

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Revision v1.2 (January 2025) aligned the document to CRL-0 licensing posture and strengthened non-operational boundaries.

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 

• 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.

Keywords: constraint dynamics, coherence bandwidth, soft constraints, Hamiltonian systems, spectral statistics, Riemann hypothesis, Constraint Geometry, statistical physics, nonlinear dynamics, finite resolution effects