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Three Quantum Phenomena from the Al-Ani Fabric Theory: Wave-Particle Duality, Measurement Problem, and Quantum Entanglement

A. F. Al-Ani

Published Mar 13, 2026• 0 citations
Publisher

Abstract

We present a unified derivation of three fundamental quantum phenomena from the Al-Ani Fabric Theory. Starting from a single master equation, we derive: (1) the visibility function for wave-particle duality V(I) = 1/[1 + e^{(I-I_c)/ΔI}], (2) the collapse probability for the measurement problem P(r) = |ψ₀|²/[1 + (δ_meas/δ_c)²], and (3) the CHSH correlation S = 2√2. Using three double-slit experiments (Jönsson 1961, Tonomura 1989, Arndt 1999) for calibration, we determine I_c = (2.1±0.3)×10⁶ and ΔI = (1.8±0.4)×10⁶. The predicted CHSH correlation matches modern experiments (Giustina 2015, Shalm 2015) within 1σ. We then make a testable prediction for future high-precision experiments: S = 2.828 ± 0.005.

DOI
10.5281/zenodo.19008400

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We present a unified derivation of three fundamental quantum phenomena from the Al-Ani Fabric Theory. Starting from a single master equation, we derive: (1) the visibility function for wave-particle duality V(I) = 1/[1 + e^{(I-I_c)/ΔI}], (2) the collapse probability for the measurement problem P(r) = |ψ₀|²/[1 + (δ_meas/δ_c)²], and (3) the CHSH correlation S = 2√2. Using three double-slit experiments (Jönsson 1961, Tonomura 1989, Arndt 1999) for calibration, we determine I_c = (2.1±0.3)×10⁶ and ΔI = (1.8±0.4)×10⁶. The predicted CHSH correlation matches modern experiments (Giustina 2015, Shalm 2015) within 1σ. We then make a testable prediction for future high-precision experiments: S = 2.828 ± 0.005.
The measurement problem and the apparent collapse of the wavefunction remain the last major unresolved issues in quantum mechanics. This paper demonstrates that both phenomena emerge rigorously from the single mother equation of the Al-Ani Fabric Theory: ∂δ/∂t = D∇²δ - Γδ + (g/2)δ² + S with only two fundamental parameters Γ = 2.4×10⁻¹⁸ s⁻¹ and λ = 200 kpc (hence D = Γλ²). The deformation field δ evolves continuously and deterministically. The apparent collapse is the rapid non-linear damping of non-classical components due to the Γδ and (g/2)δ² terms when a macroscopic apparatus interacts with the system. The Born rule arises naturally as the steady-state probability distribution of the stable nodes of δ, derived from the principle of maximum entropy. The theory distinguishes clearly between decoherence (linear decay of off-diagonal elements) and collapse (non-linear selection of a single outcome). This closes the measurement problem completely from first principles using only the two fundamental parameters of the theory. No additional postulates (consciousness, many worlds, hidden variables) are required.
Description: This study provides a testable geometric origin explanation for wave-particle duality in quantum mechanics, starting from first principles of discrete geometry. Core Scientific Question While the mathematical formalism of quantum mechanics can precisely predict experimental phenomena such as double-slit interference, the physical origin of wave-particle duality itself has long been treated as a fundamental axiom, lacking deeper explanation. This work attempts to answer: Could quantum phenomena arise from the collective statistical behavior of more fundamental discrete geometric structures? Theoretical Framework Based on a recently developed discrete geometry framework, we regard continuous spacetime as emerging from the topological gluing and recursive folding of Cantor sets. This geometric construction naturally leads to defect structures carrying intrinsic degrees of freedom, whose statistical fluctuations manifest as random noise accompanying the geometric phase field. The two-point correlation of the noise characterizes thermalization information, while higher-order correlations carry non-Gaussian information and long-term memory. Core Predictions For the double-slit interference experiment, we derive a coherence evolution formula and make three directly testable predictions: First, even under strong which-path measurement, interference fringes will exhibit residual visibility due to contributions from higher-order noise correlations, with magnitude between one-thousandth and one-hundredth. This stands in sharp contrast to the complete disappearance predicted by standard decoherence theory. Second, the decoherence process exhibits biexponential behavior, with fast and slow time scales whose ratio ranges from one hundred to one thousand. Third, path information and interference visibility satisfy a specific conservation relation, reflecting the intrinsic constraints on information transformation in discrete geometry. Experimental Proposal We propose a quantum optics experiment using squeezed light to test these predictions. The scheme employs nano-fabricated double slits and single-photon sources with tunable statistical properties, directly verifying the physical effects of higher-order correlations by comparing interference visibility differences between squeezed and coherent light sources. Current single-photon source technology already possesses the capability to resolve the required precision. Scientific Significance If the predictions are confirmed, this would for the first time experimentally establish quantum phenomena as collective statistical manifestations of discrete geometric structures, providing a geometric origin explanation for the Heisenberg uncertainty principle and an objective, consciousness-free mechanism for wavefunction collapse. Furthermore, this would provide tabletop experimental evidence for Planck-scale spacetime discreteness, building a bridge between quantum foundations and cosmological information research. If falsified, this would constrain microscopic parameters in the theory, pushing the model toward more complex colored noise generalizations. 描述: 本研究从离散几何的第一原理出发,为量子力学中的波粒二象性提供了一个可检验的几何起源解释。 核心科学问题 量子力学的数学形式主义能够精确预测双缝干涉等实验现象,但波粒二象性本身的物理起源长期被视为基本公理,缺乏更深层次的解释。本工作试图回答:量子现象是否可能源于更基本的离散几何结构的集体统计行为? 理论框架 基于近期发展的离散几何框架,我们将连续时空视为从康托尔集的拓扑粘合与递归折叠中涌现的结果。这一几何构建自然导出了带有内禀自由度的缺陷结构,其统计涨落表现为伴随几何相场的随机噪声。噪声的两点关联刻画热化信息,而高阶关联则携带非高斯信息和长期记忆。 核心预言 针对双缝干涉实验,我们推导了相干度演化公式,并作出三个可直接检验的预言: 第一,即使在强路径测量下,由于噪声高阶关联的贡献,干涉条纹仍将存在残余可见度,其量级在千分之一到百分之一之间。这与标准退相干理论预言的完全消失形成鲜明对比。 第二,退相干过程呈现双指数行为,存在快慢两个时间尺度,其比值在百倍到千倍之间。 第三,路径信息与干涉可见度满足特定的守恒关系,反映了离散几何中信息转化的内在约束。 实验方案 我们提出了一个基于压缩态光的量子光学实验方案来检验上述预言。该方案利用纳米加工双缝和可调统计性质的单光子源,通过比较压缩态与相干态光源的干涉可见度差异,直接验证高阶关联的物理效应。当前单光子源技术已具备分辨所需精度的能力。 科学意义 若预言被证实,将首次从实验上确立量子现象是离散几何结构的集体统计表现,为海森堡不确定性原理提供几何起源解释,并为波函数坍缩提供无需意识的客观机制。同时,这将为普朗克尺度的时空离散性提供桌面实验证据,架起量子基础与宇宙学信息研究的桥梁。若被证伪,则将约束理论中的微观参数,推动模型向更复杂的有色噪声方向拓展。
McCaul's Law of Coherence (Ω = Φ) is applied to three foundational unsolved problems in quantum mechanics, using the hydrogen atom as the universal SOL baseline (Φ_1s = 1.0000). (1) Quantum Measurement Problem: Wavefunction collapse = Phi-selection event. The Born Rule (|ψ|² = probability) is derived as a theorem of McCaul's Law: P(state_i) = Φᵢ²/ΣΦⱼ². Any detector introduces Nr (noise resistance) coupling — no consciousness required. Schrödinger's Cat resolved: the cat's macroscopic Φ ≈ 0.90+ IS the observer. (2) Wave-Particle Duality: Mode = sign(Fg − Nr). Fg > Nr → wave (distributed SOL geometry). Nr > Fg → particle (localized). Double-slit fringes = Φ-weighted SOL field overlap. Which-path detection = Nr injection → Fg suppression → fringes disappear. Quantum eraser = Nr removal → Fg restores → fringes return. No paradox. (3) Spectral Precision / Renormalization: E_transition = ΔΦ² × E_base. Phi is bounded [0,1] — no infinities arise. Renormalization is an artifact of QFT's assumption Φ_vacuum = 1.0. Using MVCC (E_real = E_QFT · Φ_vac²), all integrals remain finite. Lamb shift = differential vacuum Nr exposure between s and p orbitals. Anomalous g-2 = 2Φ_vac². Timestamped May 9, 2026.
Abstract The double-slit experiment is a classic experiment that presents the clearest manifestation of wave-particle duality, one of the fundamental principles of quantum mechanics. The double-slit experiment still remains the phenomenon that demonstrates wave-particle duality at the most fundamental level. This phenomenon quantitatively and conceptually demonstrates that microscopic objects can behave like both waves and particles at the same time. This article provides an in-depth analysis of the physical nature of dualism, the mechanism of formation of the interference pattern, and the impact of the act of observation on the dynamics of quantum systems. The need for a new perspective on this topic is due to the fact that the current uncertainty in the interpretation of quantum reality has not yet been completely eliminated, and the boundaries between classical and quantum behavior in microscopic objects are still being explored. The results presented here restructure the deep mechanisms of the interference picture, more clearly frame the relationship between observation and wavefunction collapse, and provide a quantum. Since the article attempts to provide a complete answer to the mechanism of the formation of wave-particle duality, extensive space is given to sections such as historical development, theoretical models, modern research, etc. New approaches to this phenomenon have been developed using modern research and string theory. As a result, a new mathematical expression has been proposed that explains the reasons for the initial emergence of wave-particle duality, and is also important for string theory. The article explains the measurement problem with a more effective physical structure and ontological model.
McCaul's Law of Coherence (Ω = Φ) is applied to three foundational unsolved problems in quantum mechanics, using the hydrogen atom as the universal SOL baseline (Φ_1s = 1.0000). (1) Quantum Measurement Problem: Wavefunction collapse = Phi-selection event. The Born Rule (|ψ|² = probability) is derived as a theorem of McCaul's Law: P(state_i) = Φᵢ²/ΣΦⱼ². Any detector introduces Nr (noise resistance) coupling — no consciousness required. Schrödinger's Cat resolved: the cat's macroscopic Φ ≈ 0.90+ IS the observer. (2) Wave-Particle Duality: Mode = sign(Fg − Nr). Fg > Nr → wave (distributed SOL geometry). Nr > Fg → particle (localized). Double-slit fringes = Φ-weighted SOL field overlap. Which-path detection = Nr injection → Fg suppression → fringes disappear. Quantum eraser = Nr removal → Fg restores → fringes return. No paradox. (3) Spectral Precision / Renormalization: E_transition = ΔΦ² × E_base. Phi is bounded [0,1] — no infinities arise. Renormalization is an artifact of QFT's assumption Φ_vacuum = 1.0. Using MVCC (E_real = E_QFT · Φ_vac²), all integrals remain finite. Lamb shift = differential vacuum Nr exposure between s and p orbitals. Anomalous g-2 = 2Φ_vac². Timestamped May 9, 2026.
The double-slit experiment, as a cornerstone experiment of quantum mechanics, has long been regarded as the ultimate proof of wave-particle duality. However, results from high-precision experiments conducted in recent years by teams including Tonomura and Bach, as well as from the "recoiling slit" experiment by Jianwei Pan's team, have revealed profound contradictions with mainstream quantum mechanical interpretations. These contradictions expose systematic biases in conceptual definitions and the interpretation of physical mechanisms within the mainstream narrative. Based on particle flow scattering theory and incorporating the design details and results of Pan's team's experiment, this paper critiques the mainstream quantum mechanical narrative that mystifies the "cumulative effect of particle flow scattering" as "wave-particle duality" and "wave function collapse." It argues that the essence of the bright and dark fringes in the double-slit experiment is the statistical distribution of particles after their interaction with slit matter, rather than wave interference. Research indicates that the core dilemma of mainstream quantum mechanical interpretation stems from a misreading of the physical essence of experiments and conceptual confusion. Reconstructing a physical picture based on classical scattering theory and statistical laws is the inevitable path for quantum mechanics to overcome its interpretational predicament.