Phase-Flux Field (PFF): Axiomatic Substrate for Wave Confinement Theory Zero-Wave Invariance, Finite-k Lyapunov Band-Pass, Shell Quantization, and D4 to Continuum

Phase-Flux Field (PFF): Axiomatic Substrate for Wave Confinement Theory
Zero-Wave Invariance, Finite-k Lyapunov Band-Pass, Shell Quantization, and D₄ → Continuum
Richard J. Reyes - September 8, 2025

GitHub Repository: github.com/rickyjreyes/geometry_of_resonance

Description:
This paper introduces the Phase–Flux Field (PFF) as a minimal, wave-first substrate defined only by observables, energy density u(x,t), energy flux S(x,t), and phase θ(x,t). Two kinematic axioms, local conservation ∂t u + ∇·S = 0 and a null-flow constraint |S|² = u² (units c = 1), fix transport structure without invoking electromagnetic, particle, or gauge semantics. The uniform Zero-Wave state ZW₀ is isolated as the seed of organization.

A Lyapunov band-pass “rail” is derived that selects a finite wavenumber ring, with linear growth σ(k) = r + a k² − b k⁴ and saturating nonlinearity yielding IR/UV stability. Near onset, a triadic amplitude system and a strictly decreasing Lyapunov functional certify pattern selection at k⋆. A discrete D₄ cell model is constructed and shown to converge to the continuum PFF limit, providing a concrete discrete-to-continuum bridge. Phase winding yields a shell-quantization rule (integer topological charge), tying spectral selection to geometric organization. The manuscript includes well-posed Cauchy data, unit normalization, and a reproducible checklist for numerical validation.

Key results include:

• Axiomatization of PFF kinematics from observables: conservation and null-flow constraints fix transport without external field semantics.

• Zero-Wave invariance lemma (seed state persistence) and conditions for departure into finite-band growth.

• Lyapunov band-pass rail with σ(k) = r + a k² − b k⁴, finite-k selection k⋆, and a Lyapunov functional F with Ḟ ≤ 0.

• Near-onset amplitude equations with explicit triad selection rules and stability regions.

• Discrete D₄ cell → continuum ring derivation with matching flux and phase constraints.

• Shell quantization via phase winding m ∈ ℤ, linking topology to spectral shells.

• Reproducibility protocol: domain/grid/Δt specification, units table, and a validation checklist (finite-band spectrum, Lyapunov descent, phase-winding count).

Built upon the foundation established in:
The Geometry of Resonance: Wave Confinement Theory and the Emergence of Mass, Force, and Spacetime (Reyes, Zenodo 2025, DOI: 10.5281/zenodo.15356814).

Keywords:
Phase–Flux Field, wave confinement, Lyapunov band-pass, finite-k selection, D₄ discrete model, shell quantization, conservation laws, null flow, pattern formation, nonlinear dynamics, continuum limits.

Version: v1.0 (September 8, 2025)

Contact:
Richard J. Reyes — reyes.ricky30@gmail.com
ORCID iD: 0009-0005-5975-8718