Introduction

DNA exhibits helical periodicity with an average pitch of approximately 10.4–10.6 base pairs per turn. This periodicity reflects stacking interactions, hydration shell ordering, and electrostatic constraints of the phosphate backbone. Previous work has identified periodic signals in nucleosome positioning, transcription factor binding regions, and replication initiation zones. However, statistical coherence linked to prime-number indexing of helical rotation phases has not been systematically examined.

Pattern Field Theory proposes that coherence in physical and biological systems reflects alignment with a hexagonal substrate field, the Allen Orbital Lattice (AOL). If this field is present, rotational or spiral structures such as DNA helices should show phase-stability points corresponding to prime-weighted angular intervals.

This study performs prime-indexed phase scanning across multiple genomes to test whether specific angular positions display statistically significant recurrence, using multiple pitch values and false discovery rate correction. The goal is to determine whether DNA helices participate in a broader coherence structure that also governs organization in crystals, atmospheric vortices, plasma filaments, and galactic rotation patterns.

Theory: AOL–Riemann Connection

Let \(\phi\) denote angular position along the DNA helix: \[\phi = \frac{2\pi n}{p},\] where \(n\) is the base index and \(p\) is the helical pitch in base pairs per turn.

Let \(p_k\) denote the \(k\)th prime. Define the prime-weighted phase sum: \[\mathcal{A}(\phi) = \sum_{k=1}^{K} e^{i p_k \phi}.\]

Stationary-phase points occur where: \[\frac{d}{d\phi}\mathcal{A}(\phi)\bigg|_{\phi=\phi^{*}} = 0.\]

These stationary-phase points correspond to coherence shells on the Allen Orbital Lattice.

Map angular phase to a Riemann field coordinate: \[t = \alpha \phi,\] which places coherence alignment near points on the critical frequency line \[s = \frac{1}{2} + it.\]

In Pattern Field Theory, this correspondence reflects shared field geometry rather than a number theoretic assumption.

Methods

Genome Data

GRCh38 (human), R64-3-1 (yeast), and NC_000913 (E. coli) genome assemblies were obtained from NCBI.

Phase Scan

Genomic regions were scanned using sliding windows of 120 to 240 base pairs. Angular phase was computed using \(p = 10.4\), \(p = 10.5\), and \(p = 10.6\) base pairs per turn.

Prime Phase Weighting

Prime index positions were computed using prime sequences up to \(K = 5000\) where applicable.

Statistical Tests

Odds ratios were calculated for enrichment at each angular offset. Benjamini-Hochberg FDR correction was applied with \(\alpha = 0.05\).

Software

Python 3.11 with NumPy, SciPy, and pandas was used for data processing and statistical evaluation.

Results

Peak enrichment occurred at discrete angular intervals corresponding to prime-indexed rotational positions. The recurrence across species indicates a shared coherence condition rather than species-specific tuning.

Discussion

Prime-indexed phase alignment in DNA helices suggests that biological structure formation occurs on a coherence substrate consistent with the Allen Orbital Lattice. The same geometric organization is observed in hexagonal mineral lattices, atmospheric vortex radii, plasma filament structures, and galactic arm curvature.

The recurrence of these patterns across scale domains indicates a common generative field structure. DNA participates in this structure in a manner consistent with stable resonance and minimal energy deformation on a hexagonal field.

Further study should expand the genome set and include chromatin loop domain geometry to test multilayer field alignment effects.

Curvature Law: Radius and Prime-Indexed Coherence

Let \(s_n\) denote the \(n\)th stationary-phase shell on the Allen Orbital Lattice (AOL). Each shell corresponds to a stable resonance orbit where constructive coherence occurs. The physical radius associated with a shell is given by

\[r = k\, s_n,\]

where \(k\) is a scale factor determined by the material system. The value of \(k\) differs across domains (atomic, crystalline, biological, geological) but the shell index \(s_n\) is field-invariant.

Stationary Shell Index

The shell index is determined by prime-indexed stationary phases: \[s_n = \frac{p_n}{\pi},\] where \(p_n\) is the \(n\)th prime. This relates radius formation to discrete coherence conditions rather than continuous variation.

Atomic and Ionic Radii

For ions, charge contraction modifies the radius through a phase-adjustment term: \[r_{\text{ion}} = r_{\text{atom}}\left(1 - \frac{\Delta\phi}{\phi_0}\right),\] where \(\Delta\phi\) measures coherence displacement from the nearest stable shell, and \(\phi_0\) is the local phase period.

Geometric Interpretation

The quantity \(\pi^2/6\) appears as the normalization constant for shell density on the hexagonal lattice. This links the Basel constant to shell distribution: \[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.\]

The curvature law therefore ties three structures together:

• prime number indexing (\(p_n\)),

• shell spacing on a hexagonal resonance field (\(s_n\)),

• physical radii across matter domains (\(r = k s_n\)).

This relationship provides a single generative rule underlying observed radii in atoms, crystals, biological helices, vortex rings, and spiral galaxies.

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).