Constrained Horizon Expansion Fractal (HEF) as a Mathematical Framework for Space-Time

Abstract This research presents the Quantum Constrained Horizon Expansion Fractal (HEF), which integrates a quantum-inspired energy threshold into its recursive fractal expansion. Using zero-point energy En = (1/2) hf, only fractal points satisfying the quantum escape condition En > E_escape = (h f c) / 2 continue expanding. The fractal demonstrates discrete quantum-like growth, where expansion occurs in localized bursts rather than a continuous process. This model preserves recursive growth, introduces spatial randomness to mimic quantum uncertainty, and connects to Loop Quantum Gravity (LQG) through its structured expansion constraints. 1. Introduction The Horizon Expansion Fractal (HEF) was initially developed as a recursive fractal model tied to cosmic expansion. This modified version introduces a quantum-inspired constraint where the escape condition is determined by Planck energy principles. Unlike traditional fractals, which expand continuously, the Quantum-Constrained HEF exhibits discrete growth zones, aligning with quantum mechanics and potential space-time quantization effects. 2. Mathematical Formulation The fractal expansion is governed by the energy threshold: En = (1/2) h f where h is Plancks constant, and f is a frequency term linked to cosmic evolution:f(t) = H(t) / (2) where H(t) is the time-dependent Hubble parameter. The quantum escape condition is: E_escape = (h f c) / 2 A fractal node expands only if En > E_escape. This introduces quantum discreteness into the expansion model. 3. Algorithmic Implementation The fractal's recursive expansion follows these steps: 1. Initialize a fractal grid with complex-numbered coordinates. 2. Compute the local quantum energy En = (1/2) h f at each iteration. 3. Compute the escape threshold E_escape, incorporating a dynamic scaling function. 4. Apply regional variations to introduce stochastic behavior mimicking quantum fluctuations. 5. Nodes satisfying En > E_escape continue expanding in the next recursion step. 6. Iterate recursively, preserving structured expansion patterns within quantum constraints. The Quantum Constrained HEF is governed by: 1. Quantum energy constraint: En = (1/2) h f 2. Escape energy threshold: E_escape = (h f c) / 2 3. Frequency evolution linked to the Hubble parameter: f(t) = H(t) / (2) 4. Escape scaling function to adjust probability: S(t) = 0.2 + 0.8 * sin(0.1 t) 5. Regional variability to simulate quantum fluctuations:V(x, y) ~ N(1.0, 0.3) 6. Escape probability condition: If En * V(x, y) > E_escape * V(x, y), then expand 7. Recursive weighting: Iterations are weighted as (time_steps - t + 1) to maintain hierarchical expansion. 4. Results & Visualization The quantum-constrained HEF exhibits: - Discrete quantum-like fractal growth. - Localized expansion zones following wavefunction-like probability distributions. - Recursive self-organization consistent with quantum space-time discretization. The visualization below demonstrates this behavior.