[4 Application] This paper offers a pivotal theoretical step in the development of the 0-Sphere electron model. It applies the previously established oscillatory two-point energy model (DOI: 10.5281/zenodo.16759284) to a longstanding puzzle in quantum mechanics: Tonomura’s observation that perfect 100% contrast is never achieved in electron double-slit experiments. This limitation had resisted theoretical explanation, leaving open whether it reflected a fundamental physical law or experimental imperfection. Building on the geometric framework of earlier works, the study proposes that the electron’s internal structure—two distinct kernels arranged along a 0-sphere geometry—explains this contrast limit. These kernels, which radiate and absorb Thermal Potential Energy (TPE), undergo temporal phase variations that alter the electron’s behavior during slit passage. When TPE condenses fully into a single kernel, the electron behaves as a point particle, and the interference pattern disappears entirely. This work forms a bridge between the initial geometric concepts of prior papers and the more developed internal-structure theories that followed. While the precise mathematical formulation of kernel dynamics was still under exploration, the paper demonstrates how the 0-Sphere model can resolve concrete experimental anomalies. The proposal that perfect contrast is fundamentally unattainable marks the first instance in which the framework directly explains puzzling laboratory observations—an important early step toward a complete internal-structure interpretation of quantum phenomena. Relation to Previous Works: 0-Sphere Model — DOI: 10.5281/zenodo.16759284 Coexistence of Dirac Positive/Negative States — DOI: 10.5281/zenodo.16817190 Anomalous Magnetic Moment via Lorentz Contraction — DOI: 10.5281/zenodo.16871305 Community: Satoshi-Hanamura-Papers
[4 Application] This paper offers a pivotal theoretical step in the development of the 0-Sphere electron model. It applies the previously established oscillatory two-point energy model (DOI: 10.5281/zenodo.16759284) to a longstanding puzzle in quantum mechanics: Tonomura’s observation that perfect 100% contrast is never achieved in electron double-slit experiments. This limitation had resisted theoretical explanation, leaving open whether it reflected a fundamental physical law or experimental imperfection. Building on the geometric framework of earlier works, the study proposes that the electron’s internal structure—two distinct kernels arranged along a 0-sphere geometry—explains this contrast limit. These kernels, which radiate and absorb Thermal Potential Energy (TPE), undergo temporal phase variations that alter the electron’s behavior during slit passage. When TPE condenses fully into a single kernel, the electron behaves as a point particle, and the interference pattern disappears entirely. This work forms a bridge between the initial geometric concepts of prior papers and the more developed internal-structure theories that followed. While the precise mathematical formulation of kernel dynamics was still under exploration, the paper demonstrates how the 0-Sphere model can resolve concrete experimental anomalies. The proposal that perfect contrast is fundamentally unattainable marks the first instance in which the framework directly explains puzzling laboratory observations—an important early step toward a complete internal-structure interpretation of quantum phenomena. Relation to Previous Works: 0-Sphere Model — DOI: 10.5281/zenodo.16759284 Coexistence of Dirac Positive/Negative States — DOI: 10.5281/zenodo.16817190 Anomalous Magnetic Moment via Lorentz Contraction — DOI: 10.5281/zenodo.16871305 Community: Satoshi-Hanamura-Papers
[5 Application] This paper presents the first systematic application of the geometric framework introduced in A Model of an Electron Including Two Perfect Black Bodies (DOI: 10.5281/zenodo.16759284). It marks a conceptual step toward linking the 0-Sphere electron model with stochastic processes in quantum mechanics. The analysis builds on the closed geometric equationE₀ = E₀ ( cos⁴(θ/2) + sin⁴(θ/2) + 1/2 sin²(θ) ),assigning it a quantum-mechanical interpretation. This allows the two-kernel model to account algebraically for random walk and Brownian motion phenomena. A key insight comes from the temporal phase dynamics between kernels A and B. During inter-kernel transitions, the photon sphere oscillates harmonically. At the π time phase, all electron energy transfers to kernel B, giving sin⁴(θ/2) = 1. At this instant, the photon sphere’s energy drops to zero, creating a unique geometric state in which motion becomes differentiable across all temporal domains. The next kernel is then chosen probabilistically, forming the stochastic basis of the model. The study advances from one-dimensional random walks (drunkard’s walk) to two-dimensional lattice configurations. This dimensional progression shows how instantaneous velocity nullification enables the coexistence of continuous motion with genuine randomness, requiring probabilistic landing points to maintain mathematical continuity. As the first exploration of stochastic applications within the internal-structure model, this work establishes groundwork for explaining how deterministic geometric equations can produce apparently random quantum behaviors, opening a new research direction in the 0-Sphere theoretical program. Relation to Previous Works: 0-Sphere Model — DOI: 10.5281/zenodo.16759284 Coexistence of Dirac Positive/Negative States — DOI: 10.5281/zenodo.16817190 Anomalous Magnetic Moment via Lorentz Contraction — DOI: 10.5281/zenodo.16871305 Community: Satoshi-Hanamura-Papers
[3 Supplement] This 2019 study represents an early and ambitious attempt to extend the two-kernel "oscillatory black-body electron" model (Zenodo:10.5281/zenodo.16759284), first introduced in *A Model of an Electron Including Two Perfect Black Bodies*, into the framework of complex analysis and Riemann surfaces. In the earlier model, the sum of the spinors' thermal potential energy (TPE) and the photon's kinetic energy remained constant and followed the algebraic form:E0 = E0 ( cos⁴(θ/2) + sin⁴(θ/2) + 1/2 sin²(θ) )This paper explores how the two spinor bases could be placed on separate sheets of a Riemann surface, leading to four possible composite spin states within a single electron: |↑↑⟩, |↑↓⟩, |↓↑⟩, |↓↓⟩. A magnetic field gradient or a measurement process is proposed to "disconnect" the analytic continuation between the sheets, collapsing the superposition into a restricted two-state system (|↑↑⟩ or |↓↓⟩). After this work, our research accepted a redefinition of spin via the reinterpretation of Thomas precession. As of 2025, the examination of the four possible composite spin states proposed here is no longer in active use.Within the author's series, this work marks a transitional phase: it preserves the black-body kernel mechanics from the first paper but begins to treat spin as part of a structure embedded in a two-sheeted analytic space. Although later studies refined the interpretation, this paper stands as the first to test the Riemann surface as a mathematical stage for the electron's internal state space. Its domain-disconnection mechanism remains an early foundation for later explorations of measurement and symmetry breaking in the 0-Sphere model. This paper serves as the logical foundation for a subsequently submitted work, Redefining Electron Spin and Anomalous Magnetic Moment Through Harmonic Oscillation and Lorentz Contraction (DOI: 10.5281/zenodo.16861903), which provides predictions that electron Zitterbewegung oscillates at approximately 4% of the speed of light. While some mathematical refinements remain, the work opens a new path for interpreting fundamental equations through internal electron dynamics rather than particle–antiparticle dualism. Relation to Previous Works: 0-Sphere Model — DOI: 10.5281/zenodo.16759284 Coexistence of Dirac Positive/Negative States — DOI: 10.5281/zenodo.16817190 Anomalous Magnetic Moment via Lorentz Contraction — DOI: 10.5281/zenodo.16871305 Community: Satoshi-Hanamura-Papers
[64 Foundational] The 0-Sphere Model treats the electron not as a structureless point but as a small internal geometric region whose phase trembles in time. This paper asks where one of the most stubborn features of that electron comes from: the fact that its spin has exactly two values, and that the particle must be turned through two full revolutions — not one — before it returns to itself. The proposed answer is that it comes from a single, very ordinary mathematical operation: taking a square root. — The idea in one breath — Picture the electron’s internal world as the surface of a four-dimensional ball — a closed, finite space with no edges. On such a closed space there is a natural “averaging” operation, and its allowed values come out evenly spaced and labelled by whole numbers (0, 1, 2, …). These whole numbers describe the orbital, direction-like part of the internal motion. Nothing surprising yet. The surprise appears when you take the square root of that averaging operation. A square root is always defined only up to a sign — √4 is both +2 and −2 — so its allowed values arrive in matched positive–negative pairs. Those two signs are exactly the two spin states: up and down. The same square root, applied historically by Dirac in 1928 to the equation of a relativistic particle, is what gave physics the electron’s spin in the first place. This paper’s claim is that one and the same operation shows up at two stages of the model and reduces the two-valuedness of spin to plain geometry. — Why this matters — Two structural facts fall out for free, where before they had to be put in by hand: Why the values are discrete. Because the internal space is closed and finite (compact), the allowed values can only be separated, whole-number-labelled steps — never a continuous smear. Why there are exactly two of them. Because the square-root step inevitably splits each value into a + and a − partner. Two signs, two spin sheets. The two sheets join smoothly into the double cover of the rotation group — the precise mathematical reason a spin one-half object needs two full turns to come home. In plain terms: walk once around the rotations and you land on the other sheet; walk around twice and you finally return to where you started. — What is new, and what is not — The square-root operation itself is classical and is claimed as no one’s discovery — it has been textbook material since 1928. What is new is the recognition that the very same operation appears at two stages of the model, and that together they reduce the two-valuedness of spin to the geometry of a closed surface. This replaces a picture of two states being secretly present at once with a cleaner, time-sequential one: the electron visits up, then down, in turn, rather than holding both simultaneously. — Key Contributions — Spin’s two values as a square root. The two spin sheets are the matched plus/minus values that any square root must take — here, the square root of the natural averaging operator on the closed internal space. Closedness gives discreteness. The fact that the allowed values are separated, whole-number steps follows from the internal space being compact (finite and edgeless). One operation across two stages. The same square root acts both on the geometric internal stage (giving a clean, discrete set of states) and on the historical relativistic stage (the one Dirac used in 1928), the two being linked by a standard mathematical bridge. The double cover as a branch structure. The two sheets are the two faces of the familiar “two-storey” surface that the square-root function naturally lives on; their joining is the rotation group’s double cover — the two-valuedness of the square root made visible. A firmer footing for earlier results. The half-angle and g = 2 results obtained earlier from the fiber-bundle picture are placed on an operator-theoretic footing. A correction that keeps the good part. The earliest spin paper’s reading of four simultaneous states is replaced by a time-sequential one, while its elegant two-storey (Riemann-surface) picture is kept — now on firmer ground. — Series Position — This paper continues the half-angle internal-energy programme and gives an operator-theoretic foundation to the geometric g = 2 result of Paper #48, consolidates the threads of Paper #50 and Paper #57, and corrects the four-state reading of Paper #3 while retaining its two-sheeted Riemann surface. Through a standard mathematical bridge it points toward the metric-emergence programme of Paper #51 and Paper #63 as the destination it does not itself develop. It belongs to the series founded by Paper #1. — References (Hanamura papers) — # Title (abbreviated) DOI 1 A Model of an Electron Including Two Perfect Black Bodies 10.5281/zenodo.16759284 3 Distinguishing Electron Spin via Riemann Surface Guidance 10.5281/zenodo.17759634 10 Redefining Electron Spin and the Anomalous Magnetic Moment 10.5281/zenodo.17764997 48 Geometric Origin of g=2: U(1) Fiber Bundle 10.5281/zenodo.19227518 50 Rotation from Scalar Oscillation: Photon-Sphere Angular Momentum 10.5281/zenodo.19482145 51 Helical Trajectory, Line Integrals, Emergence of the Metric (v2) 10.5281/zenodo.20388056 57 Spherical Harmonic Imprinting and Gravitational Coupling 10.5281/zenodo.19932176 59 Proton-Trapped Electron and Spin-2 Gravitational Radiation 10.5281/zenodo.19932907 60 The Berry–Synge–Bonnet–Myers Triangle 10.5281/zenodo.19932922 63 From Line Integral to Covariant Derivative: A Reader's Map 10.5281/zenodo.20767589 — Series Context — The 0-Sphere Model is an ongoing research programme (2018–present) that derives spin, anomalous magnetic moment, Zitterbewegung, and emergent spacetime from the geometry and thermodynamics of a two-kernel electron model. All papers in the series are archived on Zenodo: Zenodo search: Hanamura, Satoshi
Abstract One of the greatest mysteries that has puzzled and seemingly hoodwinked our very understanding of nature is the single electron double slit experiment – one that Feynman termed the central mystery of quantum mechanics. At the heart of this mystery is how can an electron that is a physical particle exhibit a wave-like property? We’ve proposed a non-counterintuitive way to explain this phenomenon called the randomly-jumping electron hypothesis, in which the electron randomly “jumps” in discrete succession to specific points only and not others which is what causes the electron to indeed be a classical particle only but is able to produce a wave-like interference pattern. To provide supporting evidence for our hypothesis, we’ve developed a computer program to simulate the single electron double slit experiment by treating the electron as a classical particle while still exhibiting wave-like fringes in agreement with real-world observed data of the single electron double slit experiment performed. Our interpretation does not set out to “solve” the mystery of why an electron can produce a wave-like interference pattern once and for all, however our hypothesis is a monism ontological interpretation that does address some of the crucial debates that have remained unsolved for 100 years since the formulation of quantum mechanics.
In the early 1900s more and more experimental evidence was accumulated indicating that microscopic particles show wave-like properties in certain situations. These particle-wave features are very evident, e.g., in measurements where electrons are diffracted from a double slit to propagate toward a screen where they are detected. Based on the classical averaging of particles discussed in Section 1.2.2, one expects that the double slit only modulates the overall intensity, not the spatial distribution. Experimentally, however, one observes a nearly perfect interference pattern at the screen implying that the electrons exhibit wave averaging features such as discussed in Section 2.1.2. This behavior, originally unexpected for particle beams, persists even if the experiment is repeated such that only one electron at a time passes the double slit before it propagates to the detection screen. Thus, the wave aspect must be an inherent property of individual electrons and not an ensemble effect. Another, independent argument for the failure of classical physics is that the electromagnetic analysis of atoms leads to the conclusion that the negatively charged electron(s) should collapse into the positively charged ion because the electron–ion system loses its energy due to the emission of radiation. As we will see, this problem can be solved by including the wave aspects of particles into the analysis. In particular, as discussed in Section 2.3.3, waves can never be localized to a point without increasing their momentum and energy beyond bounds.
This paper presents a minimal arc-geometric reconstruction of single-electron double-slit interference within the conceptual framework of Arc Theory. The central aim is not to reject the empirical success of standard quantum mechanics, but to show that a nonstandard geometric ontology can be expressed in a traditional mathematical-physical form, with explicit definitions, axioms, propositions, derivations, and stated limitations. Starting from the proposed arc-geometric constraint R = DA, where R denotes the curvature radius of the external zero-weight ring and DA denotes the intrinsic arc-radius of the electronic arc-helix, the paper derives the de Broglie relation lambdaA = h/p through phase closure and action quantization. The double-slit condition d sin(theta) = m lambdaA is then reconstructed as a consequence of arc-branch phase closure under boundary-induced splitting. The paper further formulates the double-slit process in terms of four structural operators: the arc-branching operator SAlpha, the inter-branch coherence kernel SigmaAlphaBeta, the arc-difference field DeltaAlphaBeta, and the measurement-localization operator LambdaD. In this formulation, the Born density is interpreted not as a primitive probabilistic substance, but as the observable projection density of an underlying high-dimensional arc trajectory onto a lower-dimensional detection interface. In compact form, the proposed interpretation may be written as: rhoArc(x) = |A(x)|^2 where A(x) denotes the projected arc amplitude and rhoArc(x) denotes the observable arc-projection density. A central part of the work is the quantitative formulation of topological friction. The proposed self-decoherence rate GammaSelf is expressed as a functional of curvature mismatch, torsion mismatch, and deviation from the R = DA constraint: GammaSelf = Integral over GammaA of [ gammaKappa * (DeltaKappaA)^2 + gammaTau * (DeltaTauA)^2 + gammaR * ((R - DA)/R)^2 ] dmuA This gives a precise geometric meaning to internal self-decoherence and separates it from ordinary external environmental decoherence. The paper also demystifies observational collapse by representing measurement as environment-induced arc localization. In the proposed formalism, the disappearance of interference is described by the suppression of off-diagonal coherence terms: rhoAlphaBeta -> muAlphaBeta * rhoAlphaBeta with muAlphaBeta -> 0 corresponding to complete which-path distinguishability or environmental trace formation. The scope of the paper is deliberately limited. It does not claim to derive the numerical values of Planck's reduced constant, electron mass, electric charge, magnetic moment, or the electron g-factor. Rather, it provides a mathematically disciplined reconstruction of the kinematics of single-electron double-slit interference and proposes a possible geometric foundation beneath the standard probabilistic formalism. The paper is intended as a preprint-level contribution to the foundations of quantum theory, mathematical physics, and alternative geometric models of microscopic phenomena.