The Framework Boundary: How Local-Field and Directed-Path Descriptions Diverge in the 0-Sphere Model —with Application to Microscopic Time Reversibility—

[56 Conceptual — Companion to Paper #55] Is the arrow of time real at the level of a single electron, or only a statistical illusion of large numbers? Since Loschmidt's objection of 1876, the standard answers have shared one premise — that microscopic dynamics is time-reversal symmetric — and have located irreversibility elsewhere, in statistics (Boltzmann) or in cosmology (Penrose). This paper challenges the premise itself, through a structural feature of physical description it calls the framework boundary: a single mathematical identity, read within two different frameworks for what counts as primary physical information, can yield opposite physical conclusions. In the 0-Sphere Model — where the electron is two thermodynamic kernels joined by a circulating, light-like energy structure carrying a real internal Zitterbewegung — one such identity makes the electron's internal arrow of time invisible under one reading and topologically protected under the other. Microscopic time-reversal symmetry, the paper argues, is itself a framework artifact rather than a fact of physical reality. — Core Equations — The branching identity — one equation, two physical readings:\[ \sum_{k=0}^{n-1}\sin(4 b_k) = \sum_{k=0}^{n-1}\sin\!\left(\frac{4\pi k}{n}\right) = 0 \qquad (n \geq 2) \] This zero net torque on the global phase of the 0-Sphere \(n\)-oscillator system admits a local-field reading — the internal modes decouple and are unobservable — and a directed-path reading — the directed advance of the internal phase is topologically protected from modal disruption. The two readings reach opposite conclusions from the same identity. The Aharonov–Bohm effect is the canonical empirical case of the same divergence: both frameworks compute the identical phase shift, but disagree on what physical content it reveals. The protected directed-path observable:\[ \chi = \mathrm{sign}\!\left(\frac{da}{dt}\right) \in \{+1,\,-1\} \] The chirality \(\chi\) — the sign of the internal phase advance — is invariant under proper Lorentz transformations and odd under time reversal. The physical state space splits into two topologically disconnected sectors \(\chi = \pm 1\), separated by a dynamically unreachable degenerate state, so the time-reversed trajectory cannot be reached by any continuous evolution. Empirical commitment of the directed-path framework:\[ \frac{\Delta\nu_{\text{ZB}}}{\nu_{\text{ZB}}} \approx 5.29\times10^{-10} \] The gravitational redshift of the internal Zitterbewegung clock between satellite-borne and surface-bound electron populations — invisible within the local-field reading, structurally natural within the directed-path reading — is the falsifiable test by which the framework choice is to be settled. — Key Contributions — The framework boundary as a structural feature of physical description: the local-field framework (the value of a field at a point is primary) and the directed-path framework (the phase accumulated along a directed path is primary) compute identical numbers from identical equations, yet can assign opposite physical meaning to the same result. The Aharonov–Bohm effect is the canonical empirical demonstration. The branching identity \(\sum_k\sin(4 b_k) = 0\): the zero net torque on the 0-Sphere \(n\)-oscillator's global phase reads as decoupling (the internal modes are unobservable and ignorable) in the local-field framework, and as topological protection (the directed phase advance is shielded from modal disruption) in the directed-path framework. Topological chirality of the single electron: the chirality \(\chi = \mathrm{sign}(da/dt)\) is a Lorentz-invariant, time-reversal-odd directed-path observable; the physical state space splits into two topologically disconnected sectors \(\chi = \pm 1\), so the time-reversed state is dynamically unreachable. Microscopic time reversibility as framework-dependent: time-reversal symmetry of the microscopic dynamics holds in the local-field reading and fails in the directed-path reading. Both readings are mathematically self-consistent; they describe different physical worlds. A third response to the Loschmidt objection: alongside Boltzmann's statistical response and Penrose's cosmological response, the directed-path framework offers a structural one — irreversibility is a single-particle, topologically protected feature, so the Loschmidt premise of microscopic time-reversal symmetry becomes framework-dependent rather than empirically established. A falsifiable empirical commitment: the directed-path framework predicts a gravitational redshift of the internal Zitterbewegung clock, \(\Delta\nu/\nu \approx 5.29\times10^{-10}\) between satellite-borne and surface-bound electron populations — a test invisible to the local-field reading and accessible to satellite-based experimental geometries. — Series Position — This paper is the conceptual companion to Paper #55. Where Paper #55 develops the structural decomposition of the open Wilson line into four functions, the present paper articulates the framework boundary as a general structural feature of physical description and applies it through three applications within the 0-Sphere \(n\)-oscillator system — the zero integral torque identity, the topological chirality, and conservation laws read as directional phase-consistency conditions — converging on microscopic time reversibility as the paradigm case. Together the two papers form the structural and conceptual completion of the 0-Sphere line-integral programme as it stood in early 2026. The directed-path framework rests on the line-integral trilogy — Papers #29, #30 (v2), and #31. The chirality structure central to the argument was developed in Paper #50, the underlying U(1) fiber bundle in Paper #48, and the open-path versus closed-loop distinction in Paper #45. The empirical commitment traces to the gravitational-redshift prediction of Paper #22, organized within the metric sector of Paper #55, and the foundational two-kernel architecture is established in Paper #1. The paper is explicit about its scope: the choice between the local-field and directed-path frameworks is empirical, not conceptual, and the directed-path reading is not asserted to be uniquely correct — its empirical commitments are what future experiment will test. — References (Hanamura papers) — # Title (abbreviated) DOI 1 A Model of an Electron Including Two Perfect Black Bodies 10.5281/zenodo.16759284 22 Gravitational Redshift of Internal Quantum Clocks 10.5281/zenodo.17765318 29 Geometric Structure of Spinorial Phase Accumulation along Thermal Geodesics 10.5281/zenodo.18067760 30 (v2) From Curvature to Connection: Geometric Origin of Conservation Laws 10.5281/zenodo.18819682 31 Line Integrals as Fundamental Observables in Physics 10.5281/zenodo.18203433 45 Open-Path Spinorial Transport: Holonomy Sufficiency 10.5281/zenodo.18950974 48 Geometric Origin of \(g=2\): U(1) Fiber Bundle and Dual-Frame Phase Decomposition 10.5281/zenodo.19227518 50 Rotation from Scalar Oscillation: Chirality and Helicity 10.5281/zenodo.19482145 51 Helical Trajectory, Fixed-Endpoint Line Integrals, and Spacetime Metric Emergence 10.5281/zenodo.19489126 55 Four Functions of the Open Wilson Line: Dual-Sector Hierarchy 10.5281/zenodo.19800797 — 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