Helical Trajectory, Fixed-Endpoint Line Integrals, and the Emergence of the Spacetime Metric in the 0-Sphere Model

[51-v2 Essential] This paper presents a structural reconstruction of confinement and spacetime geometry within the 0-Sphere framework. The companion paper (#50) established that the orientation of the thermal geodesic in the internal phase space determines the electron/positron distinction as a Lorentz-invariant binary \( χ = sign\!(da/dt) \), confined to the transverse plane. The present paper extends that analysis to three dimensions, converts the planar rotation into a helix with two physically distinct velocity scales, establishes existence and uniqueness of the fixed-endpoint line integral via the Bonnet–Myers theorem, and proposes the spacetime metric as the second functional derivative of the resulting phase accumulation functional. — Revision Note (May 29, 2026) — This is a corrected version of the paper, originally released on April 12, 2026. Section V.C, on the radiation gradient as the helical driving force, has been revised. The helical driving force is now identified as the radiation gradient of the kernel potential difference, \[ F(θ) = d/dθ\!(Tₑ₂ - Tₑ₁) = E_0 sinθ, \] in agreement with the foundational Paper #1. The earlier expression — the gradient of the photon-sphere kinetic energy, proportional to \( sin(4θ) \) — is withdrawn. The energy-identity notation has been aligned to the half-angle (\( θ/2 \)) convention of Paper #1 throughout. The principal results of the paper — the helical trajectory, the two-velocity-scale decomposition, the Bonnet–Myers confinement theorem, and the metric-emergence proposal — are unchanged. — Core Results — Helical trajectory \[ x(τ) = vpitch \, τ, y(τ) = cos(ωZB τ), z(τ) = sin(ωZB τ) \] Two velocity scales (no free parameters) \[ vZB ≈ 0.04c (transverse, from γ = 1 + a_e) \] \[ vpitch ~ c (longitudinal, from Δ x ~ λ_C and ωZB) \] \[ {vpitch}{vZB} ≈ 25 \] Bonnet–Myers confinement \[ Ric ≥ K ~ λ_C⁻² \;\;⇒\;\; diam(photon sphere) ≤ π λ_C \] Phase accumulation functional \[ S(x_A, x_B) = ∫_{Γₑₓₜ} M_μ \, dx^μ \] Metric-emergence proposal \[ gμν(x) = . ∂Aμ\,∂Bν\,S_s(x_A, x_B) |x_A = x_B = x \] (Synge reconstruction theorem applied to the symmetrized world function) — Key Contributions — Two-velocity-scale helix. The transverse Zitterbewegung speed \( vZB ≈ 0.04c \) (from \( γ = 1 + a_e \)) and the longitudinal pitch speed \( vpitch ~ c \) (from \( Δ x ~ λ_C \)) are derived from the two-kernel architecture without free parameters. The ratio \( ≈ 25 \) breaks the velocity uniformity of the Dirac–Hestenes helical model. Arcwise-connectivity prerequisite. \( S^0 = \{A, B\} \) is not arcwise connected; no line integral can be defined within it alone. The photon sphere \( S^n \) (\( n ≥ 1 \)) supplies the minimal arcwise-connected embedding required. Confinement as a theorem. Applying the Bonnet–Myers theorem to the photon-sphere Ricci curvature \( Ric ≥ K ~ λ_C⁻² \) yields \( diam(S^n) ≤ π r ~ λ_C \), promoting the confinement domain \( D \) of Paper #33 from an assumption to a geometric theorem. Kernel separation bounds. Bonnet–Myers provides the upper bound \( π λ_C \); the energy floor \( E_0/2 \) of Paper #46 provides the lower bound. Together: \[ 0 < x_A - x_B ≤ π λ_C \] without free parameters. Phase accumulation functional as Synge world function. \( S(x_A, x_B) \) satisfies the structural axioms (coincidence vanishing, antisymmetry, correct mixed-derivative structure). Synge's reconstruction theorem then yields the metric directly. Metric-emergence proposal. \[ gμν = . ∂Aμ∂BνS_s |coincidence \] is verified in the flat-space limit (recovers \( ημν \) to leading order). Derivation chain: Berry connection \( → \) phase functional \( → \) symmetrized world function \( → \) Synge reconstruction. Internal vs. external observer. Co-rotating frame: \( gᵢₙₜₑᵣₙₐₗ = 2 \) (Dirac value, exact).Laboratory frame: \( gₑₓₜₑᵣₙₐₗ = 2(1 + a_e) = 2γ \). The anomalous magnetic moment is a special-relativistic observation effect, structurally identical to the muon lifetime extension. — Series Position — Paper #51 is the direct continuation of Paper #50 (companion) and builds on the line-integral programme of Papers #29–#31. It promotes a central assumption of Paper #33 to a theorem and descends one level below the derivative-order hierarchy of Paper #40, connecting the Berry connection to the spacetime metric via phase accumulation. The 0-Sphere Model series now spans 51 papers (2018–2026) deriving spin, anomalous magnetic moment, Zitterbewegung, confinement, and emergent spacetime from the geometry and thermodynamics of a two-kernel electron model. — Key References (this paper) — # Title (abbreviated) DOI #50 Rotation from Scalar Oscillation [companion] 10.5281/zenodo.19482145 #31 Line Integrals as Fundamental Observables 10.5281/zenodo.18203433 #30 From Curvature to Connection 10.5281/zenodo.18819682 #29 Spinorial Phase Accumulation along Thermal Geodesics 10.5281/zenodo.18067760 #33 Geometrical Confinement: Rest Mass and Zitterbewegung 10.5281/zenodo.18356895 #40 On the Derivative-Order Mismatch 10.5281/zenodo.18736670 #24 Thermal Geodesics in the 0-Sphere Model 10.5281/zenodo.17765349 #46 Geometric Origin of the One-Half Factor 10.5281/zenodo.19010945 #10 Redefining Electron Spin and AMM 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