Corpus record: PFT:MAIN_4
Availability: patternfieldtheory patternfieldtheory zenodo academia
Publication check: pending database verification. Public approval: no.
main
This paper develops the Prime-Indexed Curvature Equation (PICE), a mathematical relation describing how prime-weighted resonance determines curvature, torsion, and rotational stability across natural systems. It extends the Pattern Field Theory (PFT) framework by formalizing the transition from discrete prime-indexed lattice geometry within the Allen Orbital Lattice (AOL) to continuous rotational flow governed by the Pattern Field (PF). The equation demonstrates that helical and vortex structures—from DNA helices and plasma filaments to tornadoes and galactic arms—arise from the same prime-resonant coherence law that underlies all stable curvature phenomena.
Pattern Field Theory
The Universal Architecture Beneath Reality
Prime Indexed Curvature Equation
Paper III: Applied Pattern Field Theory
Equations
From DNA helices, tornadoes, plasma filaments, vortex rings,
and galaxy arms
James Johan Sebastian Allen
Theoretical Physicist, Systems Analyst & Developer
Founder — Pattern Field Theory
https://patternfieldtheory.com
Introduction
Background on DNA helical periodicity and statistical signals; prior literature; open questions. We report prime-indexed phase resonance on DNA helices across parameter grids (pitch \(p\in\{10.4,10.5,10.6\}\); window \(\pm\{25^\circ,35^\circ,45^\circ\}\)) with BH/FDR-corrected minima and odds ratios. Results suggest a universal hex-coherence constraint consistent with the Allen Orbital Lattice and the Riemann link.
Theory: AOL–Riemann Connection
Prime-weighted helical phase functional \(\mathcal{A}_p(\phi)\); mapping \(t=\alpha(p)\phi\); stationary phases \(\phi^\*\mapsto s=1/2+it^\*\) (constructive statement).
Methods
Genome sources; phase-scan pipeline; BH/FDR correction; odds ratio computation; reproducibility notes.
Results
Tables of BH-min phases, OR\(_{\max}\); phase-vs-OR plots; parameter grid figures; cross-clade test plan.
Discussion
Interpretation as field coherence on hex substrate; implications for regulation, evolution, and universality.
Statement of Originality and Theoretical Novelty
Author: James Johan Sebastian Allen Framework: Pattern Field Theory (PFT) — Allen Orbital Lattice (AOL) and Riemann Field Model
Conceptual novelty.
The Allen Radius Equation \(r=k\,s_n\) defines observable radii as field amplitudes on a hexagonal coherence substrate, unifying atomic, crystalline, and geological radii under one law.
Mathematical novelty.
The radius law derives from stationary-phase rings on a prime-seeded hex field and incorporates charge-dependent phase contraction \(r_{\mathrm{ion}}=r_{\mathrm{atom}}(1-\Delta\phi/\phi_0)\). Links to the Basel constant \(\pi^2/6\) indicate number-theoretic structure in material coherence.
Structural and geological novelty.
Crystal systems and mineral structures arise as AOL projections or deformations; geological phenomena (polymorphs, solid solutions, facies, textures, and ore systems) map to variations in \((k,n)\) and deformation/tilt.
Interdisciplinary scope.
Solid-state physics, crystallography, quantum chemistry, mineralogy, and petrology are integrated under a single generative field ontology; nucleation zones align with high-coherence regions of the AOL field.
Conclusion.
This report provides a unified field description of radii and structure across scales, representing a step toward a comprehensive scientific unification.
Acknowledgements
Pattern Field Theory (Allen Orbital Lattice).
9 Allen, J.J.S. (2025). Prime-Indexed Helical Phasing in Genomic Structures.