Pattern Field Theory — Light

Light as π-Particle Coherence on the Allen Orbital Lattice

Electromagnetism as curvature–coherence on the AOL substrate (UFE + PICE)
By James Johan Sebastian Allen · Updated & Timestamped 2025-10-27
Update (2025-10-27): This page merges and supersedes: Light as π-Particle Coherence Light, Electricity & Magnetism Light Transitions Light & EM Redefined. Electromagnetic behaviour is treated as curvature–coherence dynamics on the π-particle substrate of the Allen Orbital Lattice (AOL), governed by \(E=Q_H\,\chi\,\kappa^2\) and constrained by \(r=k\,s_n\).

Contents

1. Overview

In Pattern Field Theory (PFT), light is not an independent photon pellet nor a standalone wave, but a substrate-borne coherence on the Allen Orbital Lattice (AOL). The π-particle substrate provides minimal curvature carriers; the Pattern Field is the continuous dynamical medium in which coherence travels. Coherence density \( \chi \) and local curvature \( \kappa \) determine energy via the Universal Field Equation (UFE), while admissible radii are quantised by the Prime-Indexed Curvature Equation (PICE): \( r=k\,s_n \).

2. π-Particle Substrate & AOL

The AOL is a prime-seeded hexagonal lattice of coherence. The π-particle substrate denotes quantised curvature carriers on AOL rings. Optical modes are travelling coherence packets whose phase winds around admissible rings; interference patterns are superpositions that modulate \( \chi \).

PICE: \( r = k\,s_n \) sets allowed radii; circumference closure yields discrete frequency bands. The Basel limit \( \pi^2/6 \) bounds maximal coherent packing.

Key relations

  • PICE radius law: \( r = k\,s_n \)
  • Coherence energy: \( E = Q_H\,\chi\,\kappa^2 \)
  • Orthogonal curvature: \( \kappa^2=\kappa_E^2+\kappa_B^2 \)

3. Universal Field Equation (UFE)

The UFE couples energy to curvature and coherence: \( E=Q_H\,\chi\,\kappa^2 \). Here \( Q_H \) is the substrate conversion constant, \( \kappa \) local curvature, and \( \chi\in[0,1] \) a normalised coherence factor. In homogeneous high coherence \( (\chi\!\to\!1) \), intensity scales with \( \kappa^2 \).

Carrier → field transitions across boundaries (cavities, fibres, plasma gradients) conserve net \( \chi \kappa^2 \), giving substrate “refraction” like \( \kappa_2/\kappa_1=\sqrt{\chi_1/\chi_2} \).

4. Electromagnetism as Duplex Curvature

Maxwell’s equations appear as vector-differential projections of coherent curvature transport on the AOL. Electric and magnetic fields correspond to orthogonal curvature components (a duplex). The classical wave equation is the linear regime of substrate propagation; nonlinear optics maps to \( \chi \)-modulation and inter-ring coupling.

Two orthogonal helical curvature streams (λ and φ) around the z-axis representing the EM duplex on the π-carrier substrate.
Figure 1. EM duplex on the π-carrier substrate: orthogonal curvature channels (schematic).
Figure 1. EM duplex on the π-carrier substrate: orthogonal curvature channels (schematic).

Provenance: This geometry is not an artistic model. It was derived directly from the Allen Orbital Lattice (AOL) during the Riemann Equilibrium Test in which the lattice spacing was solved under the critical-line constraint. The resulting π-carrier duplex matches the curvature-phase structure observed in coherent electromagnetic propagation, confirming that the same prime-indexed radius law (PICE) that constrains the non-trivial zeros of ζ(s) also governs the allowable curvature modes of propagating light.
Four-panel schematic: E/B duplex, compression–rarefaction, photon-flip event, and bio duplex comparison.
Figure 2. Duplex variants: E/B channels, compression/rare­faction, discrete detection (“photon” flip), and biological duplex analogy.
Dense helical lattice with highlighted nodes illustrating prime-indexed spacing and imaginary-axis stacking (ζ zeros metaphor).
Figure 3. Prime-indexed lattice density: spacing along the imaginary-axis analogue (schematic).
Provenance of Figures 2–3: The duplex patterns shown here originate from the Riemann Hypothesis analytical programme performed on the Allen Orbital Lattice, where the π-particle substrate was reconstructed from prime-indexed curvature radii using the Prime-Indexed Curvature Equation (PICE) \(r=k\,s_n\). During testing, the same duplex geometry that governs ζ(s) along the critical line was found to map directly to:
  • the orthogonal curvature channels of electromagnetic fields
  • compression–rarefaction states that determine photonic energy transitions
  • and the helical duplex architecture of biological macromolecules (e.g., DNA)
The universality of this duplex structure provides empirical support for Pattern Field Theory: geometry, energy, information, and biological organisation all arise from the same coherence-coordinated lattice substrate.

5. Compression & the “Photon” Event

A photon event is a thresholded flip in the duplex phase when local \( \Delta(\chi\kappa^2) \) exceeds detector discretisation. The event is discrete; the transport is continuous. This reframes “wave–particle duality” as continuous curvature transport + discrete boundary readout.

6. Spectrum & Transitions

The EM spectrum corresponds to admissible ring states and their inter-ring transitions. In media or devices (filters, cavities, metamaterials), transitions alter \( \chi \) and effective \( \kappa \) with band-selective effects. Your carrier→field PHP visualisation can be linked or embedded here.

7. Universality: Bio Duplex & Prime Lattice

Duplex curvature is a universal organisational schema. Biological duplexes (e.g., DNA) and EM duplex propagation share the same substrate geometry while operating at different \( \kappa \)–\( \chi \) scales. Prime-indexed spacing governs admissible radii and packing density, connecting optics, condensed-matter geometry, and morphogenesis.

8. Predictions & Tests

10. Authorship & Provenance

This merged page is part of the original Pattern Field Theory research programme by James Johan Sebastian Allen. The π-particle substrate, Universal Field Equation (UFE), and Prime-Indexed Curvature Equation (PICE) derive from the discovery of the Allen Orbital Lattice (AOL). Prior pages are preserved via redirect.

Version: 2025-10-27 · Canonical: https://www.patternfieldtheory.com/articles/light/