Radial Power Spectra: Quantifying the Pi Matrix Across Scales

By James Johan Sebastian Allen — PatternFieldTheory.com

The images captured through field ion microscopy (FIM), transmission electron microscopy (TEM), and later scanning tunneling/atomic force microscopy (STM/AFM) are more than static portraits of atoms. They contain hidden information in their frequency domain. By computing the radial power spectrum, we can quantify fractality, coherence, and resonance in the Pi Matrix across scales from sub-nanometer lattices to the cosmic microwave background radiation (CMBR).

How the Spectra Are Computed

The process begins with the 2D Fourier transform of the image intensity, I(x,y), giving the frequency-space representation F(kx,ky). The power spectrum is then: 

P(kx,ky) = |F(kx,ky)|²

To reduce this to a single curve, the power is averaged radially over all directions, producing: 

P(k) = ⟨ |F(k)|² ⟩θ

Plotted on log-log axes (Power vs. Spatial Frequency), the slope β of the spectrum relates to fractal properties: 

D = (7 - β) / 2

where D is the effective fractal dimension of the structure. For self-affine surfaces, this also connects to the Hurst exponent H: 

β = 2H + E

with E the Euclidean dimension of embedding space (typically 2 for images).

Pre-Correction Halo

Early TEM images showed halos and blur. To mainstream science, these were aberrations. In PFT, they represent the living cascade of resonance updating faster than the optics can capture. Their spectra reveal long-range fractality.

Radial spectrum of pre-correction halo
Pre-correction halo spectrum — halos as fractal resonance, not noise.

Corrected Lattice

Aberration correction froze the motion into a static lattice. The spectrum flattens at high frequency, reflecting suppression of fractal cascades. This is a Zeno Frame — a frozen slice of resonance.

Radial spectrum of corrected lattice
Corrected lattice spectrum — sharp atoms, but suppressed fractality.

Ghost Layer

Beneath the lattice, faint understructures appear as ghost layers. Their spectra retain scale-invariance and fractal slopes, evidence of the Logical Layer guiding atomic positions.

Radial spectrum of ghost layer
Ghost layer spectrum — hidden fractal structure beneath atomic order.

Blurry Pre-Correction

Before correction, blurred cascades were visible. Their spectra show high power across scales, consistent with active resonance. These were dismissed as imaging artefacts but are in fact signatures of the Pi Matrix.

Radial spectrum of blurry pre-correction
Blurry pre-correction spectrum — resonance across scales.

From Atoms to Cosmos

The same analysis applies to the Cosmic Microwave Background Radiation (CMBR). The Planck 2018 spectra reveal scale-invariance at cosmic scales, matching the fractal dynamics seen in atomic images. Together, they confirm that the Pi Matrix is universal — from the smallest to the largest scales.

Conclusion

Radial power spectra provide a bridge between visual evidence and quantifiable fractality. The slopes and scale-invariance measured here align with Pattern Field Theory’s claim: reality is not built from static atoms but from motion-sustained resonance fields. The Pi Matrix expresses itself in every spectrum, from ghost layers to the CMBR sky.