Constraint Geometry of the Critical Line: A Geometric Framework for Admissible Spectral Realizations

Paper II: Isolates the geometric substrate of admissible manifolds and critical loci, independent of enforcement or resolution. Constraint Geometry Series track. This paper establishes the constraint geometry underlying admissible dynamics, independent of enforcement, resolution, or cost. It isolates the geometric structure of admissible manifolds using a minimal Hamiltonian formulation, deliberately excluding thickness, stabilization, or dissipation terms. The analysis identifies critical lines and surfaces that partition state space into admissible and inadmissible regions as geometric features of the constraint topology, without invoking tolerance bands or enforcement mechanisms. In this setting, admissibility is treated as a structural property of the manifold itself, not as an operational or numerical criterion. By restricting attention to the invariant geometric skeleton, the paper separates constraint geometry from later questions of finite resolution, admissible bandwidth, and irreversibility. No assumptions are made about how constraints are enforced, how sharply boundaries are applied, or how trajectories are stabilized. This document is observer-only and non-procedural. It provides no diagnostic method, calibration, thresholds, or implementation guidance, and it does not present evaluable numerical validation. Its role in the Constraint Geometry Series is foundational: to define the geometric substrate that later papers enrich with enforcement, resolution, and cost, without retroactively altering the underlying structure. CRL-0 · Stage-0 · observer-only · non-authoritative · no methods, thresholds, procedures, or operational guidance. Revisionv1.2: This revision aligns the document to CRL-0 licensing posture (observer-only, non-authoritative, non-procedural) and strengthens non-operational boundaries. Structural claims and results unchanged. Published: Jan 7, 2026 Revised: Jan 25, 2026 (v1.2) Revision v1.3 (June 2026): Title-page affiliation and series-position language updated for consistency with the Constraint Geometry Series. Structural claims, abstract, and body content unchanged.