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. We integrate Coherence Evaluations as atemporal relational evaluations that determine when recursive updates can stabilize, and Equilibrion commits provide verification/lock-in.

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. Stabilization occurs when configurations qualify as Coherence Evaluations and are verified by the Equilibrion.

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

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

2.4 Coherence Evaluations (Formal Role)

We formalize Coherence Evaluations as atemporal relational evaluations that determine when a recursive update can stabilize. Let $\Delta_n$ denote relational variation introduced by the Differentiat on the π-field, and let $E$ denote the Equilibrion commit operator.

// Evaluability predicate (atemporal)
𝔼(Δ_n) ∈ {true, false}   // "is the configuration evaluable for coherence?"

// Commit/verification (atemporal in logic, projected locally as sequence)
S_{n+1} = E(S_n, Δ_n)   if   𝔼(Δ_n) = true

Interpretation. 𝔼 is not a temporal step; it is the precondition that must hold for local continuity to exist. Projection through the π-field translates this logic into experienced sequence.

3. Metacontinuum Curvature Emergence

3.1 Definition

The Metacontinuum is a pre-dimensional motion domain characterized by localized tensions and propagating motion. It is not a container; it is a modeling aspect of expression when the π-field geometry is not yet locked.

3.2 Curvature Emergence Function

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

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

3.3 Dimensional Thresholding

$$\kappa \geq \kappa_c$$

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 a testable scientific framework: recursive propagation governed by curvature and energy thresholds, with stabilization defined by Coherence Evaluations and verified by Equilibrion commits. Dimensional behavior emerges from motion–curvature interaction rather than singularities.

References

• Allen, J. “Pattern Field Theory Archive,” PatternFieldTheory.com/articles/
• OpenTimestamps Project
• Planck 2018 results
• Sloan Digital Sky Survey
• WMAP Seven-year Observations
• Tegmark, Multiverse Ensemble Theory