Introduction

For over a century, physicists have sought a unifying framework that seamlessly connects General Relativity (GR) and Quantum Field Theory (QFT). GR describes spacetime curvature at large scales, while QFT governs probabilistic quantum interactions at small scales. Conventional approaches, such as string theory and loop quantum gravity, have struggled to fully reconcile these two domains.

The Fractalverse interpretation presents a different perspective: rather than treating GR and QFT as separate frameworks, it views both as limiting cases of a single recursion-driven metric. This insight leads to a practical unification where quantum probability and classical determinism emerge from a recursion stability function that governs the transition between smooth curvature and statistical wave behavior.

A key result of this framework is the identification of a function \( \epsilon(x) \), which determines whether a system behaves classically (GR) or probabilistically (QFT). Unlike traditional interpretations that invoke measurement collapse or hidden variables, this function arises naturally from Fractal Diffraction Parastichy Interference (FDPI)—the accumulated interference of recursively structured wave patterns in spacetime.

This insight reveals that quantum randomness and classical chaos (such as in the three-body problem) share the same underlying interference-driven instability mechanism.

The Fractalverse interpretation proposes that:

  • Spacetime is a self-referential fractal structure, not fundamentally smooth.
  • Fractal Diffraction Parastichy Interference (FDPI) accumulates as recursion depth increases.
  • Recursion collapse occurs when FDPI exceeds a critical threshold, disrupting stable deterministic behavior and forcing a statistical description.

This suggests that the metric governing reality is dynamic:

\[ g_{\mu\nu}(x) = (1 - \epsilon(x)) g_{\mu\nu}^{\text{classical}}(x) + \epsilon(x) g_{\mu\nu}^{\text{effective}}(x). \]

where:

  • \( g_{\mu\nu}^{\text{classical}}(x) \) represents smooth spacetime curvature (GR).
  • \( g_{\mu\nu}^{\text{effective}}(x) \) is the effective quantum metric, derived as an expectation value over quantum states.
  • \( \epsilon(x) \) governs the transition between these two regimes.

Defining the Transition Function \( \epsilon(x) \)

The transition function \( \epsilon(x) \) is defined in terms of Fractal Diffraction Parastichy Interference (FDPI), which manifests as an increasing instability in geodesic motion:

\[ \epsilon(x) = \frac{1}{1 + e^{-\alpha (\lambda_{\text{FDPI}}(x) - \lambda_c)}} \]

where:

  • \( \lambda_{\text{FDPI}}(x) \) is the FDPI-induced Lyapunov exponent, which quantifies the rate at which small perturbations in recursive structures diverge.
  • \( \lambda_c \) is a chaos threshold, above which recursion collapse forces a probabilistic interpretation.
  • \( \alpha \) is a scaling factor that controls the transition sharpness.

This ensures that:

  • For stable recursion patterns (\( \lambda_{\text{FDPI}} \approx 0 \)): \( \epsilon(x) \approx 0 \), recovering GR.
  • For unstable recursion patterns (\( \lambda_{\text{FDPI}} \gg \lambda_c \)): \( \epsilon(x) \approx 1 \), requiring statistical interpretation (QFT).
  • For intermediate cases: The system exhibits partial recursion collapse, transitioning smoothly between deterministic and probabilistic behavior.

2. FDPI as the Driver of Recursion Collapse

Unlike previous models that treat quantum probability as fundamental, the Fractalverse approach identifies FDPI as the governing factor behind quantum randomness.

How FDPI Accumulates

  • In low-interference recursion layers, geodesics remain predictable, and space-qualia balance harmonically.
  • As recursion depth increases, overlapping wave structures interfere, creating diffraction parastichy patterns.
  • When interference amplifies beyond a stability threshold, recursion collapses, forcing a probabilistic description.

Geodesic Instability as a Secondary Effect

  • Traditional views associate quantum randomness with chaotic geodesic motion.
  • Fractalverse perspective: Geodesic chaos emerges as a symptom of accumulated FDPI rather than as its fundamental cause.

This means that:

  • Quantum mechanics emerges as a special case of FDPI-driven recursion collapse.
  • Decoherence corresponds to the loss of recursion stability over time.
  • Black hole interiors may transition into quantum behavior due to extreme FDPI accumulation.

3. Recovering GR and QFT as Limiting Cases

Since \( \epsilon(x) \) governs the transition between deterministic and probabilistic behavior, the Fractalverse framework naturally recovers both GR and QFT in their respective limits:

A. Einstein’s Field Equations (General Relativity)

When recursion remains stable (\( \lambda_{\text{FDPI}} \approx 0 \), so \( \epsilon(x) \approx 0 \)), the metric reduces to its classical form:

\[ g_{\mu\nu}(x) \approx g_{\mu\nu}^{\text{classical}}(x). \]

The governing action is the Einstein-Hilbert action:

\[ S_{\text{GR}} = \frac{c^4}{16\pi G} \int d^4x \sqrt{-g} R. \]

B. Schrödinger Equation (Quantum Mechanics)

When recursion collapses (\( \lambda_{\text{FDPI}} \gg \lambda_c \), so \( \epsilon(x) \approx 1 \)), the metric is governed by quantum expectations:

\[ g_{\mu\nu}(x) \approx g_{\mu\nu}^{\text{effective}}(x). \]

This leads to the quantum action:

\[ S_{\text{QM}} = \int d^4x \Psi^* \left( i \hbar \frac{\partial}{\partial t} + \frac{\hbar^2}{2m} \nabla^2 - V \right) \Psi. \]

which recovers the Schrödinger equation:

\[ i \hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \Psi + V \Psi. \]

Conclusion

The Fractalverse interpretation provides a practical, testable unification of GR and QFT by recognizing both as arising from a recursion-based metric. Unlike traditional approaches that assume quantum randomness as fundamental, this framework introduces a measurable transition function \( \epsilon(x) \) that determines when and where quantum vs. classical behavior emerges.

Key Results

  • Quantum randomness is not fundamental—it is FDPI-induced recursion collapse.
  • The FDPI Lyapunov exponent \( \lambda_{\text{FDPI}} \) provides a physical basis for quantum-to-classical transitions.
  • Recursion stability explains why GR and QFT appear separate but are actually limits of the same underlying process.

Next Steps

  1. Compute \( \lambda_{\text{FDPI}}(x) \) for known spacetimes (e.g., black holes, early universe).
  2. Test whether \( \epsilon(x) \) accurately predicts quantum-to-classical transitions in real-world systems.
  3. Extend this approach to dynamic systems where \( \lambda_{\text{FDPI}}(x, t) \) evolves over time.

🚀 This model represents a major step toward a practical, testable unification of gravity and quantum mechanics! 🚀