A Complex Tension Origin for Dilaton Gravity: Jordan Stiffness and Logarithmic Einstein Dynamics

We propose a microphysical completion for the scalar sector of dilatonic gravity by identifying the dilaton with the coarse--grained stiffness mode of a constrained complex tension field defined on a discrete relational network. Under a controlled ordered--regime coarse--graining, the real projection of the tension scales as Φ(Θ) = Φ0 cos Θ, so the Planck mass varies with the phase angle $\Theta$ and the Einstein--frame canonical scalar becomes φ ∝ ln[Φ(Θ)/Φ0]. This logarithmic structure emerges naturally from the Weyl map and provides the correct canonical variable for vacuum models inspired by the Logarithmic Schrödinger Equation (LogSE). We outline how this scalar--tensor interface can satisfy Solar--System constraints through environmental locking and discuss avenues for laboratory and astrophysical tests based on stiffness--coherence coupling.