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

51 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. — 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 + aₑ vₚitch ∼ c (longitudinal) from Δx ∼ λC and ωZB vₚitch / vZB ≈ 25 Bonnet–Myers confinement: Ric ≥ K ∼ λC⁻² ⟹ diam (photon sphere) ≤ π λC Phase accumulation functional: 𝒮 (xA, xB) = ∫⏒䂰ₗₓ 𝓜_μ dx^μ (Berry connection, unique helical arc) Metric-emergence proposal: g_μν (x) = ∂₀⏛ ∂₁⏜ 𝒮ₛ (xA, xB) |ₗ₀ = ₗ₁ = ₗ (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 + ae) and the longitudinal pitch speed vpitch ∼ c (from Δx ∼ λC) 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. S⁰ = A, B is not arcwise connected; no line integral can be defined within it alone. The photon sphere Sn (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 ∼ λC⁻² yields diam (Sn) ≤ πr ∼ λC, 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 λC; the energy floor E0/2 of Paper #46 provides the lower bound. Together: 0 < |xA − xB| ≤ π λC, without free parameters. Phase accumulation functional as Synge world function. 𝒮 (xA, xB) 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∂Bν𝒮s|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 + ae) = 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. 18135855 #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