Triadic Closure as the Necessary and Sufficient Condition for TERM Stability: Rigorous Foundations in the Dual-Field Regulator Sector

The TERM (Triadic Equilibrium Regulation Model) framework establishes a mathematically rigorous foundation for emergent physics, unifying stability theory, variational principles, and dual‑field dynamics through the principle of triadic closure. This work proves that triadic interaction structure is the necessary and sufficient condition for Hessian positive‑definiteness in both discrete networks and continuum field theories, eliminating all non‑trivial null‑modes and ensuring isolated equilibria. The analysis introduces the TERM Stability Theorem, demonstrating that dyadic (pairwise) systems are structurally incapable of producing stable physical laws, while triadic regulation provides the minimal architecture required for consistent field dynamics, emergent constants, and regulated spacetime structure. The results connect mathematical physics, network theory, scalar‑tensor models, renormalization‑group monotonicity, and emergent geometry, positioning TERM as a generative meta‑framework underlying unified field theories and theories of everything. This paper provides a foundational stability criterion for any consistent physical theory, offering deep implications for quantum gravity, cosmology, gauge unification, and the emergence of physical constants.