The Unified Gradient Invariant (IGU): A Formal Principle for the Coherence of Change in Dynamics, Cognition, and Culture

The Unified Gradient Invariant (IGU) is proposed as a formal, conditional, and falsifiable principle that identifies a shared differential structure in adaptive dissipative systems across heterogeneous domains —non-equilibrium physics, computational learning, cognition, and cultural dynamics— under verifiable operational conditions and with an explicit criterion of inapplicability. The principle holds that, whenever a system admits an operationally definable potential V(x, t) and a corresponding mobility matrix M(x, t), its coherent evolution takes the form of a gradient descent flow modulated by that matrix. Three differential invariants are identified —gradient-guided direction of change, stability determined by the curvature of the potential (Hessian), and critical reorganizations associated with Hessian collapse— across non-equilibrium physics, machine learning, predictive processing, and symbolic-normative dynamics. A formal domain decomposition distinguishes the operative manifold of the IGU from non-gradient regimes through a continuous dominance index κ(x) and a non-IGU residual R(x). Unlike unifying frameworks that can accommodate any trajectory through parametric adjustment, κ renders the IGU's conditionality operationally measurable: when κ ≪ 1, the principle declares its own inapplicability rather than reframing the evidence. The principle is equipped with a strict geometric falsification criterion: the IGU is locally refuted if the observed trajectory systematically ascends the gradient of the potential. Philosophically, the IGU aligns with structural realism as a principle of form rather than substance: it characterizes the structural conditions under which coherent change is scientifically representable, without committing to a shared ontology across domains. The implications for explanatory unification, comparison with the Free Energy Principle (FEP), and limits of applicability are discussed in detail. Unlike the Free Energy Principle and other minimization frameworks, the IGU postulates neither agency nor an internal generative model: it operates with a kinematic falsification criterion directly verifiable from observational time series, and declares its own inapplicability when the index κ falls below the coherent regime threshold.