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 = ρeiθ. 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 ΘT = 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 = NΦdθ and completing e0 with three independent coframe legs supplied by the parent response. The reconstructed Lorentzian metric g = ηabea ⊗ 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.

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Keywords: emergent time, problem of time, scalar field, scalar clock, internal time, relational time, Lorentzian geometry, coframe reconstruction, emergent spacetime, Axis Model