The Structural Necessity of TERM: From Difference to Geometry via Triadic Closure and Universal Regulation

This paper provides the definitive formal foundation of the TERM (Triadic Emergent Regulation Model) framework. It demonstrates that the structure of physical law is not an arbitrary collection of postulates, but the unique and necessary consequence of a single primitive requirement: the absence of unregulated differences (the No Free Difference Principle). Building on this principle, the paper derives: The Origin of Gravity: Local curvature R emerges as the mandatory regulatory response to relative field motion. Triadic Closure: A mathematical proof that stable physical interaction requires a triad (Φ,Ψ,Θ) to eliminate null‑modes and instability. Emergent Spacetime: The 3+1 dimensional Lorentzian manifold arises as the unique attractor compatible with hyperbolicity, marginality, and ghost‑freedom. Thermodynamic Consistency: Energy conservation follows directly from the convexity of the global consistency functional C. This work serves as the theoretical anchor for the eleven preceding TERM papers, unifying mass generation, gravitational dynamics, dimensional selection, and the dark sector into one logically forced architecture.