Complementary Phase Geometry: An Inverse Reconstruction of Intrinsic Rotation

Abstract

An inverse reconstruction of the intrinsic rotational structure underlying Complementary Phase Geometry (CPG) is developed. Starting from the phase--action relation $dS=-b\,d\theta_S$, the massive material phase is realized as an intrinsic rotor angle $\theta$, with $d\theta=d\theta_S$. Its canonical realization on $T^\ast S^1$ gives $J=b$, while extension to $T^\ast Spin(1,2)$ yields a normalized intrinsic rotor on the Casimir shell $\mathcal C=b^2$. The chosen free Casimir dynamics conserves the rotor-plane orientation, and the reduced ordinary--complementary transport coefficients are represented as components of a single intrinsic proper-time angular velocity. Particle dynamics is derived independently from a reparametrization-invariant action with prescribed connection. The reduced CPG-0 Hamiltonian is recovered, while preservation of the kinetic mass shell in the nontrivial mixed sector restricts the reduced dynamics to a mass-shell-compatible momentum-locked zero-mode family containing the propagating CPG-0 branch. Only after the particle equations and mass-shell condition have been obtained is the particle connection identified on shell with the vector representation of the intrinsic rotor angular velocity. Conditional on this constitutive identification and the proper-time conversion, the CPG-0 trajectory-level phase-closure relation is recovered rather than imposed independently. The construction applies to the massive one-particle sector; the strictly massless intrinsic-phase problem remains separate.