45 Structural Clarification Note The 0-Sphere model describes the electron as a two-kernel system in which energy is exchanged by radiation between spatially separated points, giving rise to Zitterbewegung, anomalous magnetic moment, and gravitational redshift of internal clocks. This note examines whether the geometric structure already established in the model — in particular the 4π spinorial periodicity, the prohibition of same-point radiation, and the ordered-path constraint A→B→A′ ≠ A→A′→B — is sufficient to force the appearance of geometric torsion in the sense of Cartan. Three distinct mechanisms are identified as candidates for generating non-zero torsion: (i) the anti-symmetry of the connection implied by the ordered-path physical constraint, (ii) the Thomas precession contribution to the spin connection and its spatial gradient, and (iii) the directed nature of the spinorial arc-length Lspin(A,B) under the 4π periodicity of SU(2). None of these mechanisms, however, yields torsion as a logical necessity from the currently established equations alone. The 4π periodicity implies non-trivial holonomy but not by itself a non-symmetric connection; the exponential kernel remains invertible and thus admits a group (rather than semigroup) structure; and the Thomas precession spin connection can in principle cancel the exterior derivative of the vierbein in the Cartan torsion equation. The central unresolved question is whether the internal arc-length Lspin is a true scalar distance or a directed quantity whose reversal is physically inequivalent. The note formulates this question precisely, outlines what additional structure would promote each candidate mechanism to a theorem, and explains how the answer determines whether the model belongs to the class of SU(2) holonomy theories or to a genuinely new class of causal torsion geometries. Key results The 4π spinorial periodicity is fully consistent with a torsion-free SU(2) holonomy description; it does not by itself force torsion. The ordered-path asymmetry is most naturally encoded in the curvature of the non-Abelian SU(2) connection, not in a non-symmetric affine connection. The exponential kernel Kint(A,B) = exp(iω0Lint/vZB) remains group-invertible: K(B,A) = K(A,B)−1. The torsion question reduces entirely to the directed arc-length conjecture: whether Lspin(A,B) is a proper-time scalar or a genuinely directed quantity incompatible with standard Lie group fiber bundle structure. The phase Δφ(A,B) is identified as most likely an internal gauge connection (supported by papers #17, #32, #40), which would displace the torsion question from spacetime geometry into the algebra of the internal gauge group. Relation to previous work This note is a structural clarification supplement to the 0-Sphere Model series, directly extending the line-integral trilogy (#29, #30, #31) and the derivative-order mismatch supplement (#40). The kernel structure and 4π periodicity are drawn from the geometrical confinement papers (#33, #35) and the SU(2) interference paper (#38).
Satoshi Hanamura (Wed,) studied this question.