physics: horizons, collapse, and the integrity field

abstract

this paper develops the dynamical theory governing the evolution of integrity fields. behaviour is represented as a behaviour density b : X × ℝ≥0 → ℝ whose evolution is generated by the integrodynamic functional F[b] = U[b] − ℵ S_I[b], with ℵ = e the empirically observed integrity constant. the resulting dynamics are monotone under admissible flows, so F acts as a Lyapunov-style quantity governing structural dissipation, interaction, and curvature-driven drift. a unified diagnostic operator emerges from the continuity equation for the integrity current, under which contradiction, divergence, and curvature coincide as a single structural quantity. these relations define the collapse manifold, the region of state space on which curvature intensifies to form finite-time singularities. renormalisation analysis identifies a fixed point at ℵ ≈ e, yielding scale-stable behaviour near the coherence horizon and separating recoverable from unrecoverable regimes. simulations confirm the theoretical predictions: curvature imbalance generates accelerating drift, contradiction concentrates into localised tension zones, and collapse occurs when information load exceeds the system’s dispersion capacity. the resulting physics specifies the governing equations, critical thresholds, and stability conditions of integrity fields across epistemic, procedural, and institutional domains. although collapse phenomena are familiar in physics — from gravitational singularities to critical phase transitions — integrity collapse has remained unquantified. this paper provides the first dynamical account of integrity collapse, identifying a critical Aristotelian threshold (Aristo 1, the Ar = 1 limit of the integrity field), an associated elenctic-shock collapse surface, and a unified diagnostic structure linking curvature, contradiction, divergence, and dispersion.

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