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 \( λ_C \) Cancellation Theorem — The central result is the \( λ_C \) cancellation theorem: in the derivation of the gravitational force on a single electron, the Compton wavelength \( λ_C = /m_e c \) appears in two roles simultaneously, both governed by the same geometric object — the ZB axis unit vector \( ê \): Role I (lever arm) \[ δ E = m_e \, g \, λ_C \, (ê · ĝ) \] Role II (clock rate) \[ {ωZB}{2π} = {vZB}{λ_C} \] Cancellation \[ F = δ E × ({ωZB}{2π}) = [ m_e g λ_C (ê̂) ] × [ {vZB}{λ_C} ] = m_e \, g \, βZB \] \[ where βZB = {vZB}{c} ≈ 0.04 \] The cancellation is structural, not algebraic: both appearances of \( λ_C \) share a common geometric origin in \( ê \), exactly as the string wave speed \( v = √T/μ \) 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 \( m_e \) changes \( λ_C \), but the change cancels because \( λ_C \) mediates both the energy scale and the time scale symmetrically. — Key Contributions — \( λ_C \) cancellation theorem (Theorem 2.1): structural derivation of \( F = m_e g βZB \) with no free parameters; WEP as a geometric identity. Deterministic ZB axis drift: the anomalous magnetic moment \( a_e ≠ 0 \) causes the helical trajectory to not close (\( Δ φ = 2π a_e \) per cycle), providing the single-particle scanning mechanism by which \( ê \) 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: \( ê_x \) (velocity, longitudinal) = gravitational sector; \( ê_y \) (acceleration, transverse) = electromagnetic sector; \( ê_z = ê_x × ê_y \) (Thomas precession) = spinorial sector. Two orthogonalities distinguished: the \( L^2[0,2π] \) orthogonality of \( cos^4(θ/2) \) and \( sin^4(θ/2) \) 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 \( a_e \): 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 \( a_e ≠ 0 \), 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: \( vZB ≈ 0.04c \) transverse, \( vpitch ~ c \) longitudinal) as the geometric arena for the \( λ_C \) 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 \( (gμν = ∂Aμ∂Bν S_s) \). 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 \( (θ, ê) \) 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