Two-Clock Gravity from Internal Time τ: Stationary Weak-Field Sector of Union Dipole Theory and Its Precision-Metrology Falsifiable Structure

This work derives the stationary weak-field gravitational sector of Union Dipole Theory (UDT) as a direct physical consequence of the experimentally extractable internal time τ. In this framework, physical evolution is fundamentally parameterised by intrinsic time τ, while the laboratory coordinate time t is not primary but emerges operationally through a measurable scalar projection field χ = dt/dτ. Gravity is therefore not introduced as spacetime curvature at the fundamental level; it appears as the spatial structure of the τ→t projection field. Starting from an intrinsic variational principle in τ, we identify the gravitational potential as Φ = c² lnχ and obtain a sourcing equation ∇² lnχ = 4πGρ / c² which represents the stationary field law of the projection mechanism. This construction produces a complete and internally consistent two-clock kinematic, dynamical, and metrological framework with the following properties: — Exact recovery of Newtonian gravity and all first post-Newtonian observables — γ = β = 1 without parameter tuning — Full empirical compatibility with all tested weak-field regimes Thus General Relativity appears as an effective geometric limit of a deeper two-clock structure rather than a competing theory. The framework introduces a single controlled higher-order parameter ε governing all deviations beyond the tested post-Newtonian sector, yielding three explicit and quantitatively falsifiable predictions: A universal second-order gravitational redshift term A calculable 2PN correction to light deflection and perihelion advance A global optical-clock network Poisson reconstruction of the mass distribution from clock-rate data alone The third prediction constitutes the primary near-term experimental test, and is structurally independent of relativistic geodesy because it tests the sourcing law of the projection field itself rather than the Einstein field equations. Within UDT the logical hierarchy is inverted relative to one-clock physics: intrinsic evolution in τ → projection field χ → effective metric → post-Newtonian observables. This establishes internal time as a configuration-invariant physical degree of freedom, empirically grounded in precision metrology and not introduced as a mathematical reparameterisation. Conceptually, gravitational acceleration becomes motion toward regions of lower clock-rate, and gravitational time dilation becomes a direct measurement of the spatial structure of the τ-projection. The theory is strictly falsifiable: If the clock-network reconstruction fails, the χ-field sourcing law is rejected. If all higher-order coefficients converge to the GR values, the deviation sector collapses. If ε ≠ 0 is detected, a new gravitational regime beyond the standard PPN structure is established. This work therefore provides: — a metrologically operational definition of gravity in terms of internal time — a structurally closed weak-field completion of UDT — a quantitatively testable pathway from precision clocks to fundamental physics and represents a shift from a one-clock description of nature to a two-clock physical ontology in which observable time is a projection of intrinsic temporal dynamics.