Temporal Co-Orientation from Compact Scalar Phase: A Coframe-Reconstruction Criterion
Andrew Morton · Public preprint · Conditional reconstruction criterion
States the conditions under which a compact scalar phase can supply temporal co-orientation and support reconstruction of a Lorentzian coframe.
Abstract
We formulate a scalar-clock coframe-reconstruction criterion in which the phase of a parent-selected compact scalar coherence channel supplies a premetric phase co-orientation that is later promoted to the temporal leg of a reconstructed Lorentzian coframe. Let a parent theory select a single local scalar coherence channel with order coordinate Φ = ρe^(iθ). On a scalar-coherent patch where ρ > 0, the compact phase admits a smooth local lift, dθ ≠ 0, and the parent certificate selects a unique light compact scalar clock channel, the one-form Θ_T = dθ defines a premetric phase co-orientation. Its kernel is an integrable codimension-one distribution, and its positive ray orders the associated phase hypersurfaces. This ordering is defined at the smooth-manifold level, prior to any reconstructed causal cone.
Metric reconstruction promotes this premetric co-orientation to the temporal coframe leg by setting e⁰ = N_Φ dθ and completing e⁰ with three independent coframe legs supplied by the parent response. The reconstructed Lorentzian metric g = η_ab eᵃ ⊗ eᵇ assigns e⁰ the positive-signature coframe direction and recovers proper time along scalar-comoving worldlines. The criterion identifies the finite parent-response data required for a compact scalar phase to serve as the clock co-orientation in a reconstructed Lorentzian geometry: scalar-channel selection, phase admissibility, coframe completion, normalization, and current-branch admissibility. Constant-phase branches, phase defects, competing light scalar channels, degenerate coframe completions, and nonadmissible physical current branches are classified as non-clock domains for the one-clock reconstruction criterion.
Scope
The criterion requires a selected scalar clock channel, an admissible branch, a complete coframe and its normalization. Dynamical selection of a physical metric, existence of a global time function and predictions of clock rates require additional construction.
Paper and supporting files
The manuscript is available on Zenodo.
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