Discrete Scale Invariance and the Hamiltonian Structure of Prime Gap Dynamics

Abstract

Paper I: We study the emergence of structure in prime gap dynamics under a logarithmic embedding and show that coherence appears through a multiscale transition rather than a sharp phase change. By treating consecutive prime gaps as deviations from the logarithmic baseline, we construct a coherence field whose dynamics reveal a four-term Hamiltonian structure: (1) a quartic confining potential, (2) an xp hyperbolic backbone, (3) Floquet modulation with period 2 in log-space, and (4) a regularization term involving a scale parameter δ. We identify an onset scale at which the Hamiltonian description becomes applicable, followed by a gradual completion regime where bulk statistical measures stabilize. A Bohr–Sommerfeld type semiclassical scaling heuristic produces effective spectral parameters exhibiting strong phenomenological correspondence with the first 100 Riemann zeta zeros. However, trace formula analysis reveals that the mapped spectrum lacks the explicit-formula prime-line signature present in the true zeros, indicating that the correspondence arises from density matching and GUE universality rather than arithmetic encoding—the Hamiltonian serves as a statistical shadow of the zeros, not a Hilbert–Pólya realization. Crucially, we observe that coherence and transport decouple in an intermediate regime: local structure emerges without predictive asymmetry. This coherence–transport decoupling is formally analogous to the pseudogap phase in superconductivity, where order exists without current. We propose this phenomenon as a general feature of bounded deviation systems and provide a falsifiable characterization of regimes where structure persists without transport.

CRL-0 · Stage-0 · observer-only · non-authoritative · no methods, thresholds, procedures, or operational guidance.

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 10, 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: Admissibility, Constraint Physics, Hamiltonian Systems, Floquet Theory, Semiclassical Analysis, Reimann Zeros