Maxwell Electrodynamics as a Derived Low-Energy Limit of the 0-Sphere Line-Integral Ontology

[54 Application] This paper reformulates classical electromagnetism within the 0-Sphere Model by reversing the standard ontological priority: the electromagnetic vector potential is treated not as an auxiliary field defined pointwise on spacetime, but as a fundamental internal 1-form \(Π\) whose physical content is carried entirely by its integrals along open thermal geodesics. Maxwell's equations emerge as derived consequences of this single postulate, rather than as independent axioms. — Conceptual Shift: Geometry First, Spacetime Later — The central conceptual advance of this work is the explicit separation between geometric structure and physical dynamics. In this formulation, geometry precedes spacetime: the connection \(Π\), its line integrals, and the derived curvature \(F = dΠ\) are defined independently of any metric notion such as distance or time. These structures exist without assuming a predefined spacetime background. Spacetime enters only at the level of dynamics, when an action is introduced. The use of the Hodge dual \(\) in the action functional requires a metric, thereby introducing temporal and spatial structure as a secondary, derived framework. In this sense, spacetime is not a fundamental stage but a construct that emerges when physical evolution and measurement are defined. This viewpoint aligns with the principle of background independence: physical laws are not formulated on a fixed spacetime manifold, but instead give rise to spacetime structure as a consequence of deeper geometric relations. Furthermore, classical electromagnetism appears as an effective theory, arising from coarse-graining the underlying discrete process of phase transport along open paths. — Core Result — The primary observable of the electromagnetic sector is the open-path Wilson line traversed by the photon sphere between two consecutive kernel positions: \[ W[A → A'] = ∫A → A' Π, with rA' ≠ r_A \] The inequality of the endpoints is not an approximation but a geometric property of the radiation–recondensation cycle of the 0-sphere. From this one postulate, the full structure of classical electrodynamics follows: \[ F ≡ dΠ \] \[ dF = d^2 Π ≡ 0 \] \[ δ S = 0, S = -1/4μ_0 ∫ F F \] The inter-kernel radiation gradient \(Δ EAB\) = \(E_0 cos(ωZB t)\) drives the Wilson line to the resonant steady state \(W(t) = W_0 sin(ωZB t)\), with the electric and magnetic fields appearing respectively as the temporal derivative and the transverse variation of \(W\). — Key Contributions — Reversal of ontological priority: the internal 1-form \(Π\) is primary; the field-strength 2-form \(F = dΠ\) is derived. Maxwell's equations are not postulated; they emerge as an identity \(dF = 0\) and as a variational condition \(d F = J\). Geometry before spacetime: the geometric structure defined by \(Π\) exists independently of any metric, while spacetime enters only through the action via the Hodge dual. This establishes a clear hierarchy in which spacetime is not fundamental but introduced at the level of dynamics. Background independence in practice: the formulation does not assume a predefined spacetime manifold. Instead, relational phase transport along paths defines the underlying structure from which spacetime descriptions are later constructed. Effective theory interpretation: classical electromagnetism emerges as a macroscopic, coarse-grained description of discrete Wilson-line transport. The Maxwell field is therefore not fundamental but an averaged representation valid at scales larger than the internal kernel separation. Aharonov–Bohm phase as a derived object: the closed-loop observable is reconstructed as the difference of two open Wilson lines sharing their endpoints, turning the conventional interpretation of the effect on its head. Gauge freedom as boundary-relative phase reference: \(Π → Π + dχ\) shifts the Wilson line only by boundary terms \(χ(rA') - χ(r_A)\), in direct agreement with the clock-synchronization picture of U(1) gauge symmetry. Maxwell as low-energy continuum limit: at observation scales \( λZB ~ 3.8 × 10⁻¹⁰\,m\) the discrete Wilson-line structure smooths into classical electrodynamics; below this scale, sub-cycle signatures (anticorrelation of \(Eₚₕ\) and \(ε^2\) within each \(ωZB\) cycle) become in principle accessible to attosecond spectroscopy. Sector separation: the electromagnetic sector (open paths, \(2π\) periodicity) and the spinorial Berry-phase sector (closed loops, \(4π\) periodicity) are shown to be structurally distinct and are not conflated. Wave equation as integrability: \( Π = 0\) in the Lorenz gauge is not a dynamical postulate but the consistency condition ensuring smooth retarded interpolation of discrete Wilson-line transport. Topological origin of gauge invariance: U(1) gauge symmetry is induced by the \(S^0\) topology of the two-kernel source, rather than imposed as an external principle. — Position in the 0-Sphere Model Series — This paper is the direct application of the line-integral ontology program to classical electrodynamics. It continues the program initiated in Paper #1 (the 0-Sphere electron model) and developed in Paper #17 (U(1) from clock synchronization), Paper #29 (spinorial phase accumulation), Paper #30 (curvature from connection), and most directly Paper #31, which established line integrals as the unifying principle behind the Aharonov–Bohm effect, Berry phase, and Wilson loops. The open-path geometry \(rA' ≠ r_A\) used centrally here is the kernel-drift structure established in Paper #47. — Primary References within the Series — # Title (abbreviated) DOI 1 A Model of an Electron Including Two Perfect Black Bodies 10.5281/zenodo.16759284 17 From Clock Synchronization to Electromagnetism: U(1) Gauge Theory 10.5281/zenodo.17765136 24 Thermal Geodesics: Extending GR Through Internal Thermodynamic Structure 10.5281/zenodo.17765349 26 Spin from Geometry: Emergence of Spin via Internal Berry Phase 10.5281/zenodo.17765409 29 Geometric Structure of Spinorial Phase Accumulation along Thermal Geodesics 10.5281/zenodo.18067760 30 From Curvature to Connection: Geometric Origin of Conservation Laws 10.5281/zenodo.18135855 31 Line Integrals as Fundamental Observables in Physics 10.5281/zenodo.18203433 33 Geometrical Confinement: Rest Mass and Zitterbewegung 10.5281/zenodo.18356895 45 Open-Path Spinorial Transport: Holonomy Sufficiency 10.5281/zenodo.18950974 47 Rotational Lorentz Contraction as the Geometric Origin of the AMM 10.5281/zenodo.19120057 — 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