Fractal Recursion and Metacontinuum Formalization — Pattern Field Theory

Abstract

Pattern Field Theory (PFT) proposes a unified framework connecting quantum mechanics, relativity, biology, and cosmology through recursive motion, curvature emergence, and coherent pattern dynamics. This article formalizes two foundational mechanisms: (1) Fractal Recursion, governing pattern emergence and scale-invariant structure, and (2) Metacontinuum Curvature Emergence, explaining dimensional formation and gravitational effects without singularities.

1. Introduction

PFT identifies the universe as a recursively self-rendering structure where patterns emerge through motion coherence. Time and gravity arise from motion-curvature interactions, photons are reframed as traversal resonance states, and fractal recursion governs both cosmic and biological architectures.

2. Fractal Recursion Formalism

2.1 Recursive Pattern Propagation Function

$$R_{n+1} = F(R_n, C_n, E_n)$$

  • $R_n$: Pattern at recursion layer $n$
  • $F$: Propagation function
  • $C_n$: Local curvature field impact
  • $E_n$: Energy coherence threshold

2.2 Curvature-Modulated Propagation

$$F(R_n, C_n, E_n) = R_n \times (1 + \alpha C_n) \times \Theta(E_n)$$

  • $\alpha$: Curvature coupling constant
  • $\Theta(E_n)$: Energy threshold activation function

2.3 Self-Similarity Across Scales

$$D_s = \frac{\log(N)}{\log(S)}$$

Fractal recursion visualization

3. Metacontinuum Curvature Emergence

3.1 Definition

Metacontinuum is a pre-dimensional motion field defined by localized tensions and propagating motion.

3.2 Curvature Emergence Function

$$\kappa = \frac{dM}{dP}$$

  • $M$: Net motion vector magnitude
  • $P$: Gradient of potential (tension)

3.3 Dimensional Thresholding

$$\kappa \geq \kappa_c$$

Metacontinuum curvature emergence

4. Predictive Claims

  • Fractal dimensions in biological structures align with cosmic filament patterns.
  • AI networks optimized via fractal recursion increase efficiency.
  • CMB low multipole anomalies arise from metacontinuum interference.
  • Gravitational lensing exhibits curvature drift beyond General Relativity predictions.

5. Conclusion

This formalization elevates PFT into scientific theory territory, providing structured derivations and testable predictions suitable for academic challenge.

References