Mathematical Formalism of the MID/QC Framework

This paper presents the mathematical foundation of the MID/QC framework, in which physical geometry, relativistic behavior, and quantum‑like coherence emerge from a substrate tension field. The formalism replaces geometric curvature as a primitive with a scalar tension field T (x), whose gradients, minima, and thresholds generate the structures traditionally attributed to spacetime geometry. The paper introduces the core mathematical objects of the framework, including the substrate tension field, the coherence threshold, the Tension Sphere, the Tension Horizon, and the emergence map that links substrate dynamics to observable geometric behavior. Symbolic expressions for time dilation, spatial deformation, microstate encoding, and fluctuation‑driven radiation are developed. This formalism establishes the minimal mathematical scaffolding required for MID/QC, providing a unified substrate‑level description of gravitational, relativistic, and coherence‑driven phenomena. It serves as the foundation for future work on the full PDE governing tension flow, the emergence of geometry from substrate dynamics, and the thermodynamic behavior of Tension Horizons.