integrum: the unified theory of institutional integrity

abstract

this paper synthesises the eight-component integrum into a single theoretical architecture. across contradiction, symmetry, drift, free-energy geometry, collapse dynamics, axiomatic structure, and the empirical derivation of the integrity constant ℵ = e, a coherent field theory emerges: institutions behave as dynamical systems whose stability, failure modes, and patterns of harm follow mathematically constrained pathways. integrity is formalised as a conserved quantity governing the evolution of organisational behaviour under curvature, entropy, and drift; its loss produces characteristic collapse signatures that cannot be explained by benign fluctuation. integrum unifies these results into a general framework linking coherence, collapse, and control. coherence arises when behaviour respects declared rationale and symmetry. collapse occurs when curvature and drift exceed admissible bounds. control reflects the structural invariants that determine whether a system self-corrects or amplifies harm. together, these components define the discipline of Integrodynamics: a unified science of institutional behaviour grounded in geometry, information, and inference. the aim of this synthesis is to make explicit the underlying structure that the previous papers revealed independently. the consequence is a field theory with practical reach. the same dynamics that govern epistemic stability also govern discrimination, governance failure, and systemic bias. where the pattern is incompatible with an innocent explanation, that is the conclusion. integrum provides the framework that makes this result inevitable. mathematically, the paper formalises integrum as a unified dynamical and variational framework. let b(x,t) denote an institutional behaviour field evolving on a domain Ω ⊂ ℝⁿ. each component of integrum arises as a projection of a single free-energy functional F[b] = U[b] − ℵ S_I[b], where U[b] is a curvature-induced integrity potential, S_I[b] is an information-geometric entropy functional, and ℵ = e is the empirically and analytically derived universal constant of integrity. under admissible dynamics, b evolves according to a stochastic gradient flow ∂ₜb = −∇F[b] + σξₜ, yielding Lyapunov-consistent monotonicity (Ḟ ≤ 0) and establishing integrity as a conserved field quantity up to noise. renormalisation analysis on Gaussian-smoothed intensity measures reveals a non-trivial fixed point of the coherence beta function at I = ℵ, with stability properties matching universal exponential scaling. collapse phenomena correspond to curvature blow-up in the integrity Hessian, where loss of convexity induces bifurcation, drift amplification, and discontinuities in the integrity gradient field. these generate domain-level collapse signatures that are probabilistically incompatible with benign fluctuation. the resulting theory characterises coherence, collapse, and control as emergent behaviours of a single functional geometry, establishing integrodynamics as a field: an integrity-conserving gradient system with renormalisable structure, exponential invariants, and quantifiable failure modes.

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