Mathematical and Theoretical Foundations of the Basic Phase-Transport Equation: A Unified Differential, Covariant, and Path-Functional Theory

Abstract

This paper studies phase formation, continuous comparison, and cross-state transport when the observation map varies with the state of an observer. Starting from the classical model y = h(x), we prove that a complete representation of a nontrivial family of observation maps requires an independent variable position for the current map. State realization and its canonical quotient then yield an observer O = (SO, ΓO) and an observation map FO : X × SO → Y , establishing a three-layer paradigm of the physical world, the observer, and the observation world. For a selected scalar coherent complex readout, the observation map induces the response g = r ◦ (FO|M). On the nonzero domain B = {m : g(m) = 0}, the observed-phase one-form is \( A_g = \operatorname{Im}\left( \frac{\mathrm{d}g}{g} \right). \) Amplitude–phase reduction of the full complex-response bundle produces a principal U(1) phase bundle. Given g and the standard phase normalization, we prove the existence and uniqueness of the compatible connection \( \Omega_g = p r_{\mathrm{U}(1)}^* \omega_{\mathrm{U}(1)} - \pi_{\mathrm{ph}}^* A_g, \) whose horizontal condition reproduces the local phase law. The first-order differential functional equation on lifted path space, \( \widetilde{c}^* \Omega_g = 0 \), is established as the basic phase-transport equation. It is equivalent to the group-valued differential form, the covariant-derivative form, the continuously lifted real-phase form, and the finite path-transport functional. The equation is uniquely induced by the complex response, is equivalent to horizontal lifting under the unique compatible connection, uniquely determines finite transport and path composition for prescribed initial data, is covariant under local phase-gauge transformations, and is invariant under positive amplitude rescaling. Inverse same-source double-state phase transport follows directly on fixed-physical-state slices. Finally, three representation levels are constructed: the full complex-response level G0, the phase-differential and transport level G1, and the task-quotient observation level G2. Their relative completeness is proved for their respective problem classes, providing a unified mathematical structure from variable observation maps, complex responses, and phase connections to phase transport, maximal invariants, and target recoverability.