From Zero and One to the Electron: Path-Connectedness, the Double Cover, and the Geometric Origin of the Quartic Energy in the 0-Sphere Model

Abstract

[65 Essential] This paper follows one thread from the arithmetic of \(0\) and \(1\) to electric charge in the 0-Sphere model, where the electron is two separated kernels exchanging a captured photon. The two energy identities of the model share the right-hand side \(1\), but they are solved by different sets: a segment \(D^1\) and its two endpoints \(S^0=\partial D^1\). Joining the endpoints needs path-connectedness, and closing the path gives an integer winding number that the model identifies with charge. The paper derives the integer of charge, but not its size: the fine-structure constant is not derived. The paper also shows that the energy exchange between the two kernels is exactly the interference of the positive- and negative-energy Dirac branches, so that zitterbewegung is read as a real internal oscillation and negative energy as the absorbing half of a paired emission and absorption, without antiparticles. — Core Equations — One normalization, two solution sets (Proposition III.1): \[ n_A+n_B=1 \;\Rightarrow\; D^1, \qquad n_A^2+(1-n_A)^2=1 \;\Leftrightarrow\; n_A\in\{0,1\} \;\Rightarrow\; S^0=\partial D^1 \] Charge as a protected winding number (Wronskian of the two kernel amplitudes): \[ Q=\frac{2}{\omega}\left(a\,\dot b-b\,\dot a\right)=\pm1, \qquad |Q|\le 1, \qquad W=\chi_4(q)\ \text{for detuned pairs} \] Kernel rotor, unitarily equivalent to the Dirac rest solutions (Theorem VI.1): \[ i\,\partial_t\chi=\frac{\omega}{2}\,\sigma_y\,\chi, \qquad \chi^\dagger\chi=1 \] Zitterbewegung as the interference of the two Dirac branches: \[ n_A=\underbrace{\tfrac14+\tfrac14}_{\text{each branch alone}}+\underbrace{\tfrac12\cos\omega t}_{\text{cross term}},\qquad \dot n_A=-\dot n_B \] Internal time and distance from the clock: \[ \sqrt{1-\beta^2}=\frac{1}{1+a_e/\sqrt2},\quad \beta=0.0404720, \qquad \nu_{\mathrm{tr}}=\frac{\beta mc^2}{h}=5.0007\times10^{18}\,\mathrm{Hz}, \qquad \lambda_C=\beta c\,T_{\mathrm{tr}} \] — Key Contributions — Zero is the boundary of one: The integers are generated by \(1\), the idempotents are \(\{0,1\}\) and the units are \(\{\pm1\}=S^0\). The bosonic identity is solved by the segment \(D^1\), the kernel part of the fermionic identity only by its boundary \(S^0\); the cross term \(2n_An_B\) is the seat that carries energy between the separated points. Charge as a protected winding number: Lock and normalization of the two kernels make the amplitude \(z=a+ib\) wind around the origin. The Wronskian is conserved (Abel), quantized (\(\pm1\)) and bounded (Cauchy–Schwarz); its sign is the sign of charge, and detuned pairs give the mod-4 Dirichlet character \(\chi_4(q)\), read as the neutral neutrino. The integer and its size come from different places: As with quantized circulation in superfluids and flux quantization in superconductors, the integer is topological while the quantum is a separate input. The paper states explicitly that \(\alpha\) is not derived. Quartic energy as a Hopf pullback: The rotor equation is unitarily equivalent to the Dirac rest solutions, the complex unit is the symplectic structure of the internal oscillator, and the quartic energies are quadratic Bloch forms pulled back through the double cover \(S^3\to S^2\). Negative energy without antiparticles: The sign of an energy eigenvalue is the sign of a frequency; a real oscillation contains both, and the cross term between the two Dirac branches is exactly the energy exchange between the kernels. The negative branch is read as the absorbing half of a paired emission and absorption, and the sign factor of the degree match with the Dirac polynomial becomes structural. The exchange is drawn as a Riemann surface: the energy flow \(\dot n_A=\mp\omega\sqrt{n_An_B}\) has one sheet for the cycle in which kernel A radiates and B absorbs and one for the reverse cycle, with branch points at the kernel states \(\{0,1\}=S^0\) and the cut along \(D^1\). Four solutions, spin and hidden variables: The four Dirac rest solutions are read as the radiating or absorbing role of a kernel together with spin, assigned to the direction in which radiation and absorption progress in space; charge is the internal winding, and the reversed rotor \(H_4=\mathrm{diag}(H,-H)\) is identified as its charge partner. A winding number is the integer-valued closed line integral of a phase, and its meaning is set by the path: the sign of charge around the amplitude origin, the energy level around a closed orbit. The model posits no local hidden variable; its hidden variable is global, and an entangled pair is read as two exchanges of opposite order whose drives cancel. Internal time as the transport cycle: The internal clock runs at \(\beta mc^2/\hbar\), exactly \(\beta\) times slower than the Dirac branch phase, is proportional to proper time, and shows the same gravitational redshift as every atomic clock. Distance is built from it: \(\lambda_C=\beta c\,T_{\mathrm{tr}}\). Where the metric enters: \(dF=0\) and \(dG=J\) need no metric; the Hodge star, the constitutive relation and \(\alpha\) enter together with transport, when two electrons compare clocks. — Empirical Prediction — Internal transport speed \(\beta c\approx0.04\,c\): fixed by the measured \(a_e\) with no free parameter; unmeasured. A measurement would confirm or exclude the internal clock, while the topological results remain as mathematics. — Series Position — This paper continues the spin two-valuedness construction on \(S^3\) of Paper #64, the harmonic-oscillation reading of the anomalous magnetic moment of Paper #10, which fixes \(\beta\), and the path-connectedness chain of Paper #51. It replaces the unpublished working draft of #65 and retains its two results, the rest-frame Dirac equivalence and the Hopf pullback of the quartic energy. A numerical verification script (verify65.py) is included in this record. — References (Hanamura papers) — # Title (abbreviated) DOI 1 A Model of an Electron Including Two Perfect Black Bodies (foundational) 10.5281/zenodo.16759284 10 Redefining Electron Spin and AMM 10.5281/zenodo.17764997 17 From Clock Synchronization to Electromagnetism 10.5281/zenodo.17765136 20 Emergent Conservation Laws from Internal Geometry 10.5281/zenodo.17765244 33 Geometrical Confinement of Energy 10.5281/zenodo.18356895 38 Electron Interference from Internal Geometry 10.5281/zenodo.18718174 50 Rotation from Scalar Oscillation 10.5281/zenodo.19482145 51 Helical Trajectory and the Emergence of the Metric 10.5281/zenodo.20388056 62 The Bridge Equation \(\gamma=1+a\) from First Principles 10.5281/zenodo.20091680 64 The Square Root of the Hyperspherical Laplacian 10.5281/zenodo.20820646 — 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