Abstract: We present a computational demonstration of the Zeno Threshold (Zα) on the paradigmatic double-slit experiment. A Gaussian wave packet is propagated on a 256×128 tight-binding lattice using only two ingredients: graph topology (nearest-neighbor hopping) and unitarity (U = e-iHΔt, with the imaginary unit i forced by information conservation). No wave equation is assumed; interference fringes emerge as a strict consequence of lattice unitarity. We then introduce a which-path detector modeled as a tensor-product qubit that entangles with the particle at the slits. The entanglement entropy of the reduced detector state grows from zero toward Smax = ln 2 ≈ 0.693 nats. At the moment this entropy exceeds the observer's finite Zeno Threshold Zα, the observer's rendering engine is forced to perform Born Rule truncation—algorithmic garbage collection—erasing the off-diagonal coherence terms. The interference pattern is destroyed, leaving two classical bumps. Computational Verification Highlights: Emergent Interference: Fringes emerge strictly from discrete graph topology and unitarity, without assuming continuous wave equations. Forced Complex Phases: The complex unit i is mathematically forced by probability conservation (unitarity). The Z-Scale Breach: Which-path detection creates real entanglement entropy that breaches the observer's finite processing limit. Born Rule Garbage Collection: Truncation at the Z-Scale breach produces a detection pattern identical to the classical, incoherent sum |ψ1|2 + |ψ2|2. The full Python verification script and simulation code are available at github.com/srdrymn/atr-double-slit-simulation.
Abstract: We present a computational demonstration of the Zeno Threshold (Zα) on the paradigmatic double-slit experiment. A Gaussian wave packet is propagated on a 256×128 tight-binding lattice using only two ingredients: graph topology (nearest-neighbor hopping) and unitarity (U = e-iHΔt, with the imaginary unit i forced by information conservation). No wave equation is assumed; interference fringes emerge as a strict consequence of lattice unitarity. We then introduce a which-path detector modeled as a tensor-product qubit that entangles with the particle at the slits. The entanglement entropy of the reduced detector state grows from zero toward Smax = ln 2 ≈ 0.693 nats. At the moment this entropy exceeds the observer's finite Zeno Threshold Zα, the observer's rendering engine is forced to perform Born Rule truncation—algorithmic garbage collection—erasing the off-diagonal coherence terms. The interference pattern is destroyed, leaving two classical bumps. Computational Verification Highlights: Emergent Interference: Fringes emerge strictly from discrete graph topology and unitarity, without assuming continuous wave equations. Forced Complex Phases: The complex unit i is mathematically forced by probability conservation (unitarity). The Z-Scale Breach: Which-path detection creates real entanglement entropy that breaches the observer's finite processing limit. Born Rule Garbage Collection: Truncation at the Z-Scale breach produces a detection pattern identical to the classical, incoherent sum |ψ1|2 + |ψ2|2. The full Python verification script and simulation code are available at github.com/srdrymn/atr-double-slit-simulation.
The Quantum Measurement Problem — the absence of any physical mechanism by which deterministic, unitary Schrödinger evolution transitions into a discrete, probabilistic classical outcome — remains the central open problem in the foundations of quantum mechanics. While environmental decoherence explains the suppression of quantum interference, it does not explain why or when a single outcome is realized. This paper resolves the measurement problem within the Algorithmic Theory of Reality (ATR). By treating the universe as a finite-bandwidth information-processing architecture, we demonstrate that wavefunction collapse is not a mystical observer effect, but a mandatory thermodynamic memory-management protocol. Key Results: The Collapse Threshold: We prove that when a microscopic superposition entangles with a macroscopic apparatus, the entanglement entropy grows until the local Bennett-Landauer processing cost exceeds the local holographic energy budget. This forces a non-unitary state truncation. Quantitative Timescale: The derived threshold yields a quantitative collapse timescale (τc) that mathematically matches the experimentally observed decoherence time. Derivation of the Born Rule: Given pointer-state orthogonality (from decoherence), we rigorously derive the Born Rule (pi = |ci|2) by formulating the forced state truncation as a constrained thermodynamic optimization problem that minimizes the Kullback-Leibler divergence (algorithmic information loss). Scale Unification: The exact same Bennett-Landauer limit that dictates the macroscopic boundaries of the universe (Dark Energy) and galaxies (the MOND scale) is shown to govern the microscopic limits of quantum coherence. The ATR framework mathematically determines when collapse must occur, with what statistics, and in which basis, reducing the single-outcome problem to a single, structurally necessary axiom (the Lüders update). Supplementary Material: The algebraic derivation of the Born Rule (via KL-divergence minimization and Lagrange multipliers), the exponential growth of entanglement entropy, and Monte Carlo statistical validations are independently reproducible via the companion computational verification script, available at: https://github.com/srdrymn/atr-verify-wavefunction-collapse
DUPLICATE RECORD (18 October 2026). This record is a duplicate upload of The Double-Slit Experiment on a Discrete Holographic Lattice. Please cite 10.5281/zenodo.18793387 instead. That record — the maintained version of the same simulation — is kept, with a status note. Original description (kept for the record): We show that exact unitary propagation of an electron wavepacket on a fi- nite 4.8.8 lattice reproduces single-slit and double-slit diffraction patterns in close quantitative agreement with experiment, with no continuum approxima- tion. When environmental coupling is introduced at the slit, interference is destroyed and Born rule statistics emerge from decoherence alone. Using the deterministic kinematics of the circlette lattice, we demonstrate that wave- function collapse need not be postulated. The transition from quantum inter- ference to classical probability distributions emerges naturally from unitary evolution through the deterministic tracing-out of environmental degrees of freedom. The lattice addresses the apparent measurement problem through finite bandwidth and orthogonal geometry, requiring no conscious observer. This is the third paper in a series developing the Holographic Circlette frame- work, but remains mathematically self-contained.
This volume establishes that the quantum propagator and double-slit interference pattern are derived consequences of the algebraic structure of H0(A3), not independent postulates. Starting from the presentation <a, b, c | a²=a, b²=b, c²=c, aba=bab, bcb=cbc, ac=ca>, we construct the transfer operator T = La + Lb + Lc as the adjacency operator of the Cayley graph, show that its spectrum {0, 1, 2, 3} with degeneracies (1, 11, 11, 1) is forced by idempotence, and derive the propagator K(d, s; t) = ⟨d|exp(−iTt)|s⟩ via spectral decomposition. Explicit numerical computation on the 24-element monoid verifies quantum interference: for certain detectors, the coherent probability |A1 + A2|² differs from the incoherent sum |A1|² + |A2|². We show that the commutation relation ac = ca in A3 suppresses interference at 20 of 24 detectors, while the fully braided algebra H0(Ã2) with C(3) = 0 produces maximal fringes. Measurement is not postulated but derived: the idempotent projection Ls² = Ls destroys coherence between paths, yielding the Born rule as a theorem. Keywords: 0-Hecke monoid, derived propagator, double-slit interference, transfer operator, idempotent projection, commutation deficit, Cayley graph Laplacian
The double-slit experiment combines three observational facts: interference fringes in the coherent regime, position-resolved detections, and continuous fringe suppression under which-path resolution. Standard quantum mechanics accommodates all three through postulates without deriving them from deeper structure. This paper shows that the standard far-field interference law and the coherence-to-which-path transition follow as structural corollaries of results established in Quantum Mechanics as the Boundary Algebra of a Non-Invertible Projection (doi:10.5281/zenodo.19535588) and its companion (doi:10.5281/zenodo.19571321). Two results suffice for the algebraic derivation: the Born rule as the canonical quadratic measure on Im(Π), and the kernel spinor as the Hilbert quadrature of the boundary field. A third result — measurement irreversibility as a kinematic consequence of the projection geometry — belongs to the physical interpretation of measurement and temporal ordering within the same framework. The paper derives the Fraunhofer intensity law I(x) ∝ |Ψ₁ + Ψ₂|² and the detector-overlap formula with fringe visibility V = |γ|, within the standard Fraunhofer approximation and without additional postulates. It does not claim a first-principles derivation of the full boundary propagator or of single-event arrival statistics. A companion numerical simulation implements all three coherence regimes. A self-contained appendix summarises the three imported results in compact form.
The double‑slit experiment epitomizes the wave–particle duality of quantum mechanics. In standard treatments, the interference pattern follows from superposition, while measurement of “which path” destroys coherence. The precise mechanism that turns a coherent superposition into a localized detection event remains an open question. We propose an operator‑based model within the open‑quantum‑system framework that describes the entire process through seven effective Lindblad generators: fluctuating diffusion (initial wavepacket spreading), cyclic reset (quantum Zeno effect), spatial‑mode splitting (simultaneous excitation of both slits), free evolution (interference formation), liminal projection (threshold‑activated measurement), irreversible loss (detector inefficiency), and subspace mapping (projection onto a definite detection outcome). Unlike standard quantum mechanics, the model introduces a nonlinear decoherence rate that depends on the spatial spread of the wavefunction, capturing environment back‑action through a feedback‑controlled coupling. This dependence leads to a small but testable reduction of fringe visibility with increasing particle flux or wavepacket width, even in the absence of explicit which‑path measurement. The measurement activation is modeled as a smooth threshold function, giving a sharp transition in detection probability as the system reaches the screen. Analytical expressions for visibility and the critical Zeno frequency are derived, and numerical simulations are presented. The predictions are accessible with current experimental technology (weak measurements, controlled decoherence, cavity‑QED systems). The framework suggests that wave–particle duality can be understood as a dissipative phase transition, bridging foundational quantum theory with practical open‑system control.
Paper 60B develops a unified Holosphere account of the double-slit experiment while keeping interference loss, measurement, and stable outcome formation conceptually separate. The paper begins with standard slit-resolved paraxial propagation and shows that a coherent two-slit pattern contains a cross term that is absent from the incoherent sum of the two single-slit diffraction patterns. Which-path detection is modeled through the overlap of path-dependent detector states. As those detector states become more distinguishable, fringe visibility decreases continuously and reaches the incoherent limit when the path records are fully distinguishable. The Holosphere interpretation treats a coherence corridor as a physical relation among the source, carrier, aperture, detector, environment, and readout boundary. The apparatus therefore helps determine whether the two paths remain part of one coherence-supporting structure. Boundary incompatibility separates path-conditioned corridors, while later interface competition and pointer-basin capture may select and stabilize a detector record. Physical recoil and apparatus relaxation remain causal, and no faster-than-light signal or strict determinism is required. A separate downstream phase-edge gate produces a modest, symmetric, and resolution-surviving change in the screen distribution. The numerical suite also verifies symmetry, attenuation accounting, slit-field linearity, detector-overlap behavior, and the conditioned quantum-eraser identity.