The Unified Field of Time-Driven Dynamics: A Dynamic Extension of General Relativity with an Endogenous Time Scalar (DAG)

We present a \emph{complete} (no-omission) unification that: (i) couples optical refractive
structure to gravitational mass via a flux-certified field equation; (ii) endogenizes time through
a scalar clock \(U\) with source weighting \(W(U)\), yielding dynamic time--energy exchange and
exact total conservation; and (iii) stitches arithmetic microstructure to geometry via a zeta--\(U\)
bridge with reciprocity \(g=1/K\) (Eqs.\ (4$'$), (6)).

\textbf{Strong-field evidence (from \(\kappa\)-maps).}
Lensing reconstructions of Abell2744, MACS0416, and Abell370 \emph{strongly favor}
a nonlinear refractive correction \( \beta\,N(|\nabla\phi|^2) \) (Eq.\ (16$'$)), decisively exceeding
the GR weak-field Poisson limit (\(\beta=0\)). Two clusters prefer a soft-saturation kernel with a
finite threshold \(\Lambda\), and cross-validation / multi-scale scans / phase-scramble controls
confirm robustness. These results are established under the \(\kappa\)-only approximation
with \(W(U)\!=\!1\); flux consistency is compatible with the three-term certificate (Eq.\ (19))
in this setting.

\textbf{Temporal algebra (SCI) outlook.}
The temporal sector predicts windowed Structure--Complexity indices \(K\!\sim\!0.6\!-\!0.7\)
(\(g=1/K\!\sim\!1.4\!-\!1.7\)), consistent with the zeta-phase reciprocity and enabling falsifiable
Phase--Poisson morphology tests.

The pure-theory core operates without Einstein--Hilbert terms and requires no zeta prior; a
covariant EH toggle restores GR compatibility. The present \(\kappa\)-based tests thus support the
unified field framework ((4$'$), (6), (16$'$), (19)) and the UTH variational sector, with full
source-weighted validations (X-ray/SZ/dynamics for \(\rho\) and \(W(U)\)) left to future work.