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)=Φ0cosΘ, so the Planck mass varies with the phase angle Θ 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. This paper does not introduce a new scalar-tensor EFT class as such; rather, it provides a controlled microphysical origin for a specific scalar stiffness law, Φ(Θ)∝cosΘ, and for the resulting logarithmic Einstein-frame canonical structure.