Viscous Time Theory (VTT) Foundational Mathematics (Part 1): Variational Origin of Informational Geometry, Coherence Curvature, and the VTT Lagrangian

We propose a mathematical framework for Viscous Time Theory (VTT), exploring how spacetime geometry could emerge from discrete informational events. We formulate a variational principle for an informational coherence field Φ, constrained by discrete informational anchors, and derive a Lagrangian governing coherence propagation and dissipation. From its second variation, we introduce an informational coherence tensor DeltaC_{mu nu} that is symmetric and positive-semidefinite, enabling the definition of a candidate emergent metric g_{mu nu}^{(C)} without presupposing geometry. Within this framework, curvature is interpreted as a response to coherence gradients, and a Raychaudhuri-type focusing identity is obtained, parameterized by informational viscosity η. A preliminary discrete-to-continuum numerical demonstration supports convergence of the coherence field and the appearance of non-trivial curvature patterns under sparse informational constraints. These results suggest that classical geometric structure may be understood as a stable organizational property of informational interactions. Possible implications – including informational interpretations of curvature phenomena and future coupling to biological and physical systems – are outlined as directions for further investigation.