Triadic closure and structural equilibrium: Conditions for Peristence Under Transformation

This work investigates the minimal structural conditions required for persistence under transformation. Rather than assuming time, dynamics, or probabilistic laws, the analysis begins from two basic principles: triadic closure (the requirement that transformations be internally verifiable) and structural equilibrium (the requirement that identity-preserving and identity-changing contributions remain simultaneously operative). From these principles, the paper shows that any system capable of persistence must admit a representation of deviation with specific structural properties. Composability of transformations leads to a cocycle structure, which induces a vector space of deviations. The equilibrium condition then constrains this space to carry an inner product, ultimately selecting a complex Hilbert space as the minimal representational framework. Within this setting, deviation admits a canonical operator formulation. A unique scalar measure of deviation is derived from structural requirements of positivity, invariance, and additivity. This measure is necessarily quadratic. The Born rule then emerges as the probabilistic interpretation of this quadratic structural density. The result provides a unified structural account of both the Hilbert space framework of quantum theory and its probability rule, derived from a single chain of reasoning grounded in the conditions required for persistence. This version is a revised preprint currently under review.