Viscous Time Theory as a Variational Source Geometry

In this paper, we formulated a source layer for Viscous Time Theory (VTT) inside a regularized anchored variational model. Finite anchors are represented by mollified averaging functionals on a bounded Lipschitz domain, so the field problem remains compatible with weak convergence. The passive Hessian of the anchored action is coercive; therefore, source criticality is introduced through an active Hessian obtained by memory-feedback loading. When the active Hessian develops an isolated soft sector, scalar soft modes are lifted to symmetric tensor modes, and the memory-friction tensor is projected onto this tensorial soft space. A resolvent-weighted response defines the VTT source tensor, with an additional coherence-shear contribution. Under an explicit operator-norm bound, the tensor induces a positive definite metric-type tensorrelative to a reference metric. With sufficient regularity, the metric determines a curvature response, and a model-internal effective mass functional is defined as the positive metric trace of the source tensor. The paper proves well-posedness, tensor birth, metric positivity, mass positivity, stability, inverse-design existence, and finite-difference diagnostic criteria. The result is not a derivation of physical spacetime, but a rigorous variational-to-geometric mechanism for conditional source formation.