Is the world really more than four dimensional or for instance an eleven dimensional M-theory as proposed by Witten? The present Letter suggests that it is an eleven dimensional fractal. Based on this assumption it is shown that the probability of quantum entanglement is exactly equal to that predicted by L. Hardy’s well known criteria. It is further argued that this is an indirect but strong indication for the reality of the Cantorian fractal nature of micro spacetime.
Is the world really more than four dimensional or for instance an eleven dimensional M-theory as proposed by Witten? The present Letter suggests that it is an eleven dimensional fractal. Based on this assumption it is shown that the probability of quantum entanglement is exactly equal to that predicted by L. Hardy’s well known criteria. It is further argued that this is an indirect but strong indication for the reality of the Cantorian fractal nature of micro spacetime.
Building upon the pioneering work of J. Bell [1] and an incredible result due to L. Hardy [2] it was shown that the probability of quantum entanglement of two particles is a maximum of 9.0169945 percent [2]. This happens to be exactly the golden mean to the power of five (?5) [3-7]. Although it has gone largely unnoticed for a long time, this result was essentially established independently in a much wider context by the present author almost two decades ago [3-6]. The present work gives two fundamentally different derivations of Hardy’s beautiful result leading to precisely the same general conclusion, namely that by virtue of the zero measure of the underlying Cantorian-fractal spacetime geometry the notion of spatial separability in quantum physics is devoid of any meaning [7]. The first derivation is purely logical and uses a probability theory which combines the discrete with the continuum. The second derivation is purely geometrical and topological using the fundamental equations of a theory developed by the author and his collaborators frequently referred to as E-infinity or Cantorian spacetime theory [3-7].
Given the inverse of the Golden Mean $ \tau^{ -1} = \phi = { 1\over 2} (\sqrt 5 - 1)$, it is known that the continuous fraction expansion of $ \phi^{ -1} = 1 + \phi = \tau$ is $ ( 1, 1, 1, \cdots )$. Integer solutions for the pairs of numbers $ ( d_i, n_i ), i = 1, 2, 3, \cdots $ are found obeying the equation $ ( 1 + \phi)^n = d + \phi^n$. The latter equation was inspired from El Naschie's formulation of fractal Cantorian space time $ {\cal E}_\infty$, and such that it furnishes the continuous fraction expansion of $ ( 1 + \phi )^n ~= ~ (d, d, d, d, \cdots )$, generalizing the original expression for the Golden mean. Hardy showed that is possible to demonstrate nonlocality without using Bell inequalities for two particles prepared in nonmaximally entangled states. The maximal probability of obtaining his nonlocality proof was found to be precisely ϕ5. Zheng showed that three-particle nonmaximally entangled states revealed quantum nonlocality without using inequalities, and the maximal probability of obtaining the nonlocality proof was found to be $ 0.25 \sim \phi^3 = 0.236$. Given that the two-parameter $ p, q$ quantum-calculus deformations of the integers $ [ n ]_{ p, q} = F_n $ coincide precisely with the Fibonacci numbers, as a result of Binet's formula when $ p = ( 1 + \phi) = \tau, q = - \phi = - \tau^{ -1} ,weexplorefurthertheimplicationsoftheseresultsinthequantumentanglementoftwo−particlespin−s$ states. We finalize with some remarks on the generalized Binet's formula corresponding to generalized Fibonacci sequences.
Making sense of the nearly 100-hundred-year-old quantum nature of light and matter, with its — to name a few — wave-particle duality, measurement problem, and entanglement ‘spooky action at the distance’ — remains the greatest questions to physics. Since the 1980s, fractal geometry, a new and exciting field of mathematics — that the fathers of quantum mechanics did not have and has not been tested for quantum properties — has developed. Can the isolated fractal explain the quantum? An experiment was conducted on a simple — but isolated — fractal testing whether the geometry of fractals corresponds to quantum enigmas. It was found the isolated fractal emerges by a duality of propagation — of an oscillating sinusoidal wave — of ‘bits’ — of information — iterating in a superposition of time, scale, and symmetry, with a possible constant speed, demonstrating all the hypothesised. The quantum ‘measurement problem’ was addressed as being a problem of isolated scale-invariant fractals — or fractal landscapes — where position is only ‘known’ when additional (fractal) information is added — which ‘equally’ gives rise to a (quantum-like) ‘uncertainty’ problem. Also, quantum entanglement and other quantum features were explained by the fractal model. Consequences of the model were discussed, notably to its relevance to the nature of light (the speed of it etc.), the behaviour of the atom, time, knowledge and our reality, and the model’s direct inextricable connection with cosmological observations and conjectures. Finally, it was concluded that the model is preliminary but fundamental.
Title: Quantum Mechanics: The Interpretation of Barcelona Description: For a century, the Copenhagen interpretation has taught us that quantum mechanics is fundamentally probabilistic, that the wavefunction collapses upon measurement, and that there is no reality independent of observation. Einstein rejected this: "God does not play dice." This preprint presents a concrete, testable alternative: the Cosmic Mesh — a discrete, elastic, fractal tetrahedral lattice that constitutes spacetime itself. From this single physical substrate we recover all quantum phenomena without probability, without collapse, and without action at a distance: The wavefunction is a real vibration in the Mesh. Entanglement is the same vibration extended across space. Measurement is a local interaction that does not create reality; it only reveals it. Particles are standing waves trapped in topological knots; light is free vibration. We derive a single deterministic rule that governs all motion: At each discrete time step, a vibration migrates to the neighboring tetrahedron that maximizes the local entanglement gradient (Sk−Si). The fine-structure constant emerges geometrically as: α−1=135+2πϕ5≈137.034 where 135 is the ideal torsional-to-volumetric ratio emerging from the geometry of the tetrahedral lattice (90° + 45°), and 2πϕ/5 is the residual torsional stress from 120º cracks and fractal inheritance (golden ratio ϕ). The Mesh is local, deterministic, and testable. We predict that quantum entanglement decays with distance beyond a fundamental coherence length — a direct, falsifiable test of the Copenhagen interpretation. This concise summary is based on the full framework developed in "THE BREATHING UNIVERSE: A Torsional Solid Machine" (Zenodo, 2026). For complete mathematical derivations, including the mechanical origin of lepton masses and the emergence of general relativity, see that work. Keywords: quantum foundations, emergent quantum mechanics, discrete spacetime, topological defects, entanglement, Bell inequalities, Copenhagen interpretation, deterministic quantum theory, elastic lattice, tetrahedral mesh, Weyl curvature imprints, dark matter, cosmological constant, fine-structure constant, golden ratio License: Creative Commons Attribution 4.0 International (CC BY 4.0) Related works: Bertran, A. (2026). THE BREATHING UNIVERSE: A Torsional Solid Machine. Zenodo. Bertran, A., Jussà, E., Vila, M. (2025). The Universe's Geometric Inertia – The Jussà-Vila Ansatz. Zenodo.
The cardinal differences in the theoretical description of classical and quantum particles, the contradictions of the probabilistic (Copenhagen School) and deterministic (De Broglie - Bohm theory) interpretation of quantum mechanics, and the features of entangled quantum states are briefly discussed. It is assumed that the strange properties of the entangled pair can be explained by the localization of particles not in 3D physical space, but in the domain of a fractal space arising in the process of purely mathematical derivation of the Schrodinger and Pauli equations.
Partial entanglement may provide enough connections in the universe to satisfy the definition of a small world. To investigate this possibility, we define a network of particles on a given space-like hyper-surface with a long-range link between any two particles that are connected by a chain of exchanged particles involving a limited number of interactions. Considering the mean free paths of particles in different regions of space and the resulting probability distributions of entanglement connections vs. distance, we find evidence of small-world or random network structure on all but the smallest scales - corresponding to stars and planets - on which other types of connections would need to be added to complete the small world picture.