VTT Foundational Mathematics, Part II: Emergence of the Informational Tensor and the Crystallization of the Metric

In Part I of the Viscous Time Theory (VTT) program, an informational action principle was introduced, leading to an emergent coherence tensor, viscosity-modified covariant dynamics, and an effective metric induced by informational structure. While this framework explains how geometric notions can arise from informational coherence, a central question remains: under what conditions does an autonomous informational stress-energy tensor emerge as a physically meaningful source term?In this work, we address this question by developing a controlled theory for the birth of the VTT tensor as an independent dynamical object. Building on the informational Hessian, coherence gradients, and logical viscosity introduced previously, we identify a critical regime in which accumulation and friction between transduction and memory degrees of freedom can no longer be absorbed into purely geometric terms. In this regime, a non-vanishing tensorial source nucleates. We formulate a precise emergence criterion based on Hessian criticality, derive the leading-order structure of the resulting tensor via a frictional commutator construction, and obtain scaling laws governing its onset near the critical point.We further extract concrete, falsifiable diagnostics that distinguish the purely geometric phase from the source-driven phase, including soft-mode divergence, accumulation thresholds, anisotropic stress patterns, and critical slowing down. Finally, we outline both numerical strategies and an experimentally motivated cavity-QED–inspired protocol to probe this transition. Together, these results extend the VTT framework from emergent geometry to emergent sources, providing a mathematically controlled pathway toward an informational formulation of back-reaction, metric crystallization, and source dynamics.