integrity: a unified field theory

abstract

this paper introduces a unified mathematical framework in which behaviour is represented as a state field b : X × ℝ≥0 → ℝ^d evolving under stochastic gradient dynamics generated by an integrity functional F[b] = U[b] − ℵ S_I[b], where U[b] is a structural tension functional, S_I[b] is an integrity-dispersion functional, and ℵ ≈ e is an empirically estimated integrity constant. under admissible update rules the evolution is monotone, so F serves as a Lyapunov-style quantity governing structural relaxation. a conserved structural quantity emerges from the continuity relation of the integrity current, I = I* + H_norm, which remains approximately stable across epistemic, procedural, and institutional transformations under declared assumptions. the curvature spectrum of U[b] partitions systems into three operational coherence regimes, defined purely by the sign and magnitude of the principal curvatures. simulations confirm the model’s predictions: contradiction generates high structural tension, symmetry rules degrees of freedom, and institutional dynamics exhibit drift proportional to curvature imbalance. the resulting theory establishes integrity as a measurable structural invariant characterised by ℵ together with curvature, dispersion, and invariance constraints, providing a representation-invariant formalism for modelling coherence, deviation, and structural stability across heterogeneous decision systems. although ideas of coherence and contradiction can be traced back to Aristotle and Socrates, integrity has remained a philosophical concept for more than two millennia. this paper presents what appears to be the first quantitative framework that treats integrity as a measurable, dynamical quantity, complete with a critical threshold (“Aristo 1”) and an associated collapse phenomenon (“elenctic shock”).

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