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

Abstract

Paper II: We develop a constraint-geometry interpretation of the Riemann Hypothesis, characterizing why the critical line Re(s) = 1/2 emerges as the symmetry fixed manifold selected within this framework, compatible with three admissibility conditions: maximal symmetry under the functional equation, elimination of non-unitary (radial) degrees of freedom, and boundedness of the associated dynamical evolution. This framework unifies three mathematical languages—Hamiltonian mechanics, Hilbert space operator theory, and com plex analysis—into a single geometric picture. The exponential periodization w =exp(2π(s − 1 2 )) maps the critical line to the unit circle, making the ra dial/tangential decomposition literal rather than metaphorical. We introduce chiral symmetry as an operator-theoretic constraint, showing that the functional equation induces a chiral-like reflection structure that any admissible Hilbert–Pólya operator must respect. We do not prove the Riemann Hypothesis; rather, we characterize the space of admissible operators for which RH would be a necessary consequence. Any illustrative instantiation of these symmetry identities is implementation-dependent and intentionally omitted; the claims here rest on symmetry, boundedness, and admissibility 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.

Keywords: Constraint Geometry, Riemann Hypothesis, Periodization, Chiral Symmetry, Hilbert-Pólya Conjecture, Spectral Realization, Admissibility