The Square Root of the Hyperspherical Laplacian: A Geometric Foundation for Spin Two-Valuedness on S³ in the 0-Sphere Model

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