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

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 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 purely by geometric constraint, 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 cleanly 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 paper does not propose mechanisms, procedures, or domain-specific models. 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.