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…