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Stochastic Rupture: Numerical Evidence from Entanglement Percolation, Modular Variance Scaling, and Threshold Dynamics

GUILHERME ZAMBUZI

Published May 26, 2026• 0 citations
Publisher

Abstract

We present numerical evidence for the structural predictions of the Stochastic Rupture (SR) framework [1], focusing on three independent simulation programs. First, direct computation of modular variance Var(Jz) confirms the GHZ/Bell scaling theorem: ISR[GHZN ] = N2/4 and ISR[BellN/2] = N/2, giving τBell/τGHZ ∼ N/2 for systems of equal total mass. Second, percolation simulations on entanglement networks reveal a universal scaling law for the Erd˝os– R´enyi topology: pc ·N = 1+√η, confirmed numerically for N = 50 to N = 1000 and converging to the theoretical value 1+√η as N → ∞. Third, these results are shown to depend on network topology in a structured way—reflecting the bond percolation critical exponents of each graph class—and are robust to random site defects up to 50%, with the universal form surviving after replacing N with Neff. Connections to the superfluid 4He λ-transition and to measurementinduced phase transitions in random quantum circuits are identified as consistency anchors, providing plausibility support without constituting independent tests of SR. The simulations suggest that η is experimentally calibrable from the entanglement percolation threshold of a physical platform, provided the underlying network topology is known.

DOI
10.5281/zenodo.20387154

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We present numerical evidence for the structural predictions of the Stochastic Rupture (SR) framework [1], focusing on three independent simulation programs. First, direct computation of modular variance Var(Jz) confirms the GHZ/Bell scaling theorem: ISR[GHZN ] = N2/4 and ISR[BellN/2] = N/2, giving τBell/τGHZ ∼ N/2 for systems of equal total mass. Second, percolation simulations on entanglement networks reveal a universal scaling law for the Erd˝os– R´enyi topology: pc ·N = 1+√η, confirmed numerically for N = 50 to N = 1000 and converging to the theoretical value 1+√η as N → ∞. Third, these results are shown to depend on network topology in a structured way—reflecting the bond percolation critical exponents of each graph class—and are robust to random site defects up to 50%, with the universal form surviving after replacing N with Neff. Connections to the superfluid 4He λ-transition and to measurementinduced phase transitions in random quantum circuits are identified as consistency anchors, providing plausibility support without constituting independent tests of SR. The simulations suggest that η is experimentally calibrable from the entanglement percolation threshold of a physical platform, provided the underlying network topology is known.
Stochastic Rupture (SR) proposes that objective wavefunction collapse is triggered when local von Neumann entropy saturates a fraction eta of the holographic Bousso bound on causal-diamond surfaces. The SR network is concretely realized as a simple cubic lattice — square holographic sheets stacked at Planck-scale intervals — whose exact Laplacian spectrum derives the dimensional flow d s: 2 ( UV, holographic ) to 3 ( stacking ) to 4 ( Lorentzian, open ) without free parameters. The framework has two logically independent sectors. The collapse sector is self-contained and makes parameter-free falsifiable predictions. The general SR decoherence rate Gamma SR = Gm squared /(4 h bar sigma x squared ) produces a sigma x to the minus 2 scaling that discriminates SR from CSL (sigma x to the minus 3) and Diósi–Penrose (flat), independently of any free parameters; the definitive test is at MAQRO (10 to the 9–10 to the 10 amu). For GHZ/Bell states on NISQ hardware, the SR-native prediction is tau Bell / tau GHZ is similar to N squared, with a finite-size correction: reliable collapse onset at N stable is approximately equal to 1.25/(1 minus eta ) rather than the theoretical threshold 1/(1 minus eta ). The geometric sector — Newton's law, Einstein equations, spin-2 graviton — is conditional on the IR limit d s to 4 and is presented explicitly as such. Two new structural results reduce the number of free parameters. The bond-percolation threshold of the simple cubic lattice, p c = 0.2488, matches the cosmological structure-suppression parameter e 0 = 0.249 plus or minus 0.035 to within 0.1 %, linking the connectivity transition of the SR network to the S8 tension without a free fit. The restoration rate lambda q, previously unconstrained, is shown to satisfy lambda q/ gamma 0 is approximately equal to 1/ the square root of pi through a three-step geometric argument: vibrating holographic sheets drive saturation fluctuations to the 2D square interface critical point p c=1/2; Bernoulli variance at criticality gives sigma = the square root of 3 /2; three independent propagation channels in 3D then fix lambda q/ gamma 0 = C/ the square root of pi with C is approximately equal to 0.995. The remaining open step — showing the SR interface dynamics belongs to the Model A universality class with additive noise — reduces to a single one-loop RG calculation. The framework makes fourteen falsifiable predictions spanning NISQ processors, MAQRO, LISA, and Euclid, explicitly distinguishing derived results from calibrated parameters and open problems.
The evolution of connectivity in quantum networks under decoherence and link degradation is a central problem in quantum information, calling for further understanding of the nature of its transition during structural network degradation. By diluting each link with probability 1−f, we focus on connectivity strength transitions in diluted hierarchical scale-free quantum networks, the (u,v) flowers, which are analytically tractable through two adjustable path-length parameters, u≤v. Incorporating quantum concurrence percolation and comparing it with classical percolation, we analyze the transitions of critical thresholds for various values of f and v from analytical, numerical, and simulation perspectives. The results demonstrate that quantum percolation exhibits consistently lower critical thresholds than classical percolation, even under various topologies and dilution levels. Our work implies that quantum multipath entanglement provides an intrinsic compensatory mechanism against structural degradation and that the hierarchical scale-free topology contributes to the failure resistance and robustness of quantum networks with multipath coupling.
Quantum walks have been used as simple models of quantum transport phenomena, applicable to systems as diverse as spin chains [1] and bio-molecules [2]. Here we investigate the properties of quantum walks on percolation lattices, disordered structures appropriate for modelling biological and experimentally realistic systems. Both bond (edge) and site percolation have similar definitions: with independent randomly chosen probability p the bond or site is present in the lattice. In two and higher dimensions, percolation lattices exhibit a phase transition from a set of small disconnected regions to a more highly connected structure (“one giant cluster”). On 2D Cartesian lattices, the critical probability pc = 0.5 (bond) and pc = 0.5927... (site). Below pc, the quantum walk will not be able to spread. Approaching pc from above, the spreading slows down completely, as the number of long-distance connected paths reduces to zero. For p = 1, the lattice is fully connected, and the standard quantum walk spreading applies (linear in t). In between, we find the quantum walks show fractional scaling of the spreading, i.e., proportional to tα for (0.5 < α < 1). The numerically determined scaling exponents α are shown in fig.
We estimate the critical thresholds of bond and site percolation on nonplanar, effectively two-dimensional graphs with chimeralike topology. The building blocks of these graphs are complete and symmetric bipartite subgraphs of size 2n, referred to as K_{n,n} graphs. For the numerical simulations we use an efficient union-find-based algorithm and employ a finite-size scaling analysis to obtain the critical properties for both bond and site percolation. We report the respective percolation thresholds for different sizes of the bipartite subgraph and verify that the associated universality class is that of standard two-dimensional percolation. For the canonical chimera graph used in the D-Wave Systems Inc. quantum annealer (n=4), we discuss device failure in terms of network vulnerability, i.e., we determine the critical fraction of qubits and couplers that can be absent due to random failures prior to losing large-scale connectivity throughout the device.
The realization of entanglement distribution (ED) in large-scale quantum networks (QNs), as a fundamental theoretical framework of quantum communication, faces significant challenges. Percolation theory, drawn from statistical physics, has been identified as a potential solution to this problem. Nevertheless, when addressing ED, percolation theory primarily relies on numerical simulations and approximate algorithms. This approach often lacks interpretability and controllability for many phenomena occurring during the ED process in large-scale networks, including emergent phenomena near the threshold. To address these limitations, in this paper, we {focus on} leveraging operator theory to construct a mathematical framework for entanglement percolation in series-parallel QNs. Through the application of operator-theoretic methodologies, we analyze several critical issues within the entanglement percolation process.