Undecidability, Fractal Geometry and the Unity of Physics

Stephen Wolfram

Johanna L. Miller

Lawrence B. Crowell

Quantum mechanics places limits on what can be observed. Further, while it is a deterministic wave theory, outcomes of specific measurements are purely stochastic. The geometry of quantum entanglements, or quantum phases in general, describes a topological obstruction to unitary transformations between GHZ and W states or bipartite and tripartite entanglements. The momentum map defined on Kirwan polytopes is in the framework of a fractal sets in state space has p-adic measure. Axiomatic incompleteness of any p-adic algorithm illustrates how these obstructions define these obstructions. This is similar to how the Euclid 5th axiom is undecidable, and geometry has different model systems. This is then argued to connect with quantum gravitation in how spacetime is an epiphenomenology from entanglement.

Andrew Stewart Caldin

FINDING: The spectral gap problem is undecidable in general — no algorithm can determine whether an arbitrary quantum many-body Hamiltonian has a gap, linking computational limits to physical phase structure. | MATH: Spectral gap Δ = E₁ − E₀ > 0 (gapped) vs. Δ → 0 as N → ∞ (gapless). Undecidability proven via reduction from the halting problem (Cubitt, Perez-Garcia, Wolf, Nature 2015). The almost Mathieu operator H = λ cos(2παn) + shift + shift† exhibits a metal-insulator transition at λ = 2, with fractal spectrum (Hofstadter butterfly) when α is irrational. | CONNECTION: The golden mean renormalization for the almost Mathieu operator uses α = (√5−1)/2 = 0.618... as the critical irrational rotation number. The renormalization group flow converges to a universal fixed point governed by the golden ratio's continued fraction [0;1,1,1,...] — the slowest-converging continued fraction, making it the most stable (KAM tori survive longest at golden-mean frequencies). This is the same 0.618 tha Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Rodolfo Gambini, Jorge A. Pullin

Jincheng Zhang

This paper investigates the potential of fractal geometry to facilitate stable and controllable quantum entanglement over macroscopic distances. Traditional entanglement protocols often rely on physical transmission of particles, limiting scalability and practicality. We propose a novel approach utilizing fractal geometry as a channel for information transfer, demonstrating a method for establishing entanglement across distances that leverages the inherent properties of fractal patterns to circumvent distance limitations. The core mechanism involves constructing a "quantum branching network" – a self-organizing system of interconnected fractal structures – which allows for the distribution and manipulation of entangled particles without requiring physical travel. This work explores the theoretical implications and initial experimental investigations supporting this concept, highlighting its potential for revolutionizing quantum information transfer.