Geometric Origin of the Weak Equivalence Principle in the 0-Sphere Model: Force Classification and the λC Cancellation Theorem in the Helical Geometry

52 Supplement The present paper proposes a structural derivation of the weak equivalence principle (WEP) within the 0-Sphere model, identifying the geometric mechanism by which all particles fall with the same gravitational acceleration regardless of their mass. — Core Result: the \ (C \) Cancellation Theorem — The central result is the \ (C \) cancellation theorem: in the derivation of the gravitational force on a single electron, the Compton wavelength \ (C = mₑ c \) appears in two roles simultaneously, both governed by the same geometric object — the ZB axis unit vector \ (e \): Role I (lever arm) \ E = mₑ \, g \, C \, (e g) \ Role II (clock rate) \ ₙ₁2 = vₙ₁C \ Cancellation \ F = E (ₙ₁2) = [ mₑ g C (e) vₙ₁C = mₑ \, g \, ₙ₁ \] \ where ₙ₁ = vₙ₁c 0. 04 \ The cancellation is structural, not algebraic: both appearances of \ (C \) share a common geometric origin in \ (e \), exactly as the string wave speed \ (v = T/ \) reflects the shared physical origin of tension and linear density in the string itself. The WEP — universality of free fall — follows as a geometric identity; changing \ (mₑ \) changes \ (C \), but the change cancels because \ (C \) mediates both the energy scale and the time scale symmetrically. — Key Contributions — \ (C \) cancellation theorem (Theorem 2. 1): structural derivation of \ (F = mₑ g ₙ₁ \) with no free parameters; WEP as a geometric identity. Deterministic ZB axis drift: the anomalous magnetic moment \ (aₑ 0 \) causes the helical trajectory to not close (\ (= 2 aₑ \) per cycle), providing the single-particle scanning mechanism by which \ (e \) converges toward the gravitational direction — without ensemble averaging. Directional decomposition (three force sectors): the three coordinate directions of the photon-sphere helix carry three physically distinct interaction types: \ (eₓ \) (velocity, longitudinal) = gravitational sector; \ (eᵧ \) (acceleration, transverse) = electromagnetic sector; \ (eᵦ = eₓ eᵧ \) (Thomas precession) = spinorial sector. Two orthogonalities distinguished: the \ (L²0, 2 \) orthogonality of \ (⁴ (/2) \) and \ (⁴ (/2) \) in internal phase space is causally connected to, but distinct from, the physical-space orthogonality \ (v a \) in the helical geometry. Conflating them is identified as a category error. Reinterpretation of \ (aₑ \): within the 0-Sphere framework, the anomalous magnetic moment is not a computational correction but the essential physical mechanism by which a particle senses and responds to its environment. Without \ (aₑ 0 \), matter would consist of isolated rigid oscillators incapable of responding to gravity or electromagnetic fields. Weinberg-Witten theorem assessment (Appendix D): the theorem's hypotheses (fixed Minkowski background, perturbative Fock-space states) are non-trivially satisfied in the 0-Sphere emergent-metric setting; a potential structural escape route is identified without claiming evasion. — Position in the 0-Sphere Model Series — This paper uses the helical trajectory of Paper #51 (two velocity scales: \ (vₙ₁ 0. 04c \) transverse, \ (v₏₈ₓ₂₇ c \) longitudinal) as the geometric arena for the \ (C \) cancellation argument. It provides the direction-selecting mechanism that complements the photon-sphere ejection mechanism identified in Paper #49 as the gravitational mediator, and is consistent with the spacetime metric proposal of Paper #51 \ ( (g_ = ₀₁ Sₛ) \). The paper continues the deterministic single-particle program pursued throughout the series (Papers #5, #27, #38), working in the Lagrangian framework with definite hidden variables \ ( (, e) \) at each instant. — Primary References in the Series — # Title (abbreviated) DOI 51 Helical Trajectory / Fixed-Endpoint Line Integrals / Metric Emergence 10. 5281/zenodo. 19489126 50 Rotation from Scalar Oscillation / Chirality & Helicity 10. 5281/zenodo. 19482145 49 Photon-Sphere Fragmentation / Gravitational Mediator 10. 5281/zenodo. 19393391 48 Geometric Origin of g=2 / U (1) Fiber Bundle / Dual-Frame 10. 5281/zenodo. 19227518 47 Rotational Lorentz Contraction as Geometric Origin of AMM 10. 5281/zenodo. 19120057 10 Electron Spin and AMM via Harmonic Oscillation / Lorentz Contraction 10. 5281/zenodo. 17764997 1 A Model of an Electron Including Two Perfect Black Bodies 10. 5281/zenodo. 16759284 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