From the Principle of Least Action to the Principle of Co-Adaptation

We introduce the co-adaptation principle (CAP), a fix-free bivariational framework in which dynamics arise from a reciprocal balance between two labeled modes of variation. Unlike the principle of least action, which enforces stationarity of a single functional, CAP imposes a zero-sum reciprocity condition on directional variations of an interaction functional. The theory is formulated axiomatically, without assuming configuration spaces, metrics, or analytic structure. Integrability—commutativity of mixed variations—is treated as an optional structural condition rather than an axiom. We obtain a structural classification of co-adaptive systems and show that classical variational and geometric frameworks, including Hamiltonian, Riemannian, Kähler, and metriplectic formalisms, arise as integrable or degenerate special cases. The non-integrable regime exhibits intrinsic curvature induced solely by the non-commutativity of reciprocal co-variations.