Unified Adaptation Theorem: Convergence of Composed Adaptive Systems via Interaction Matrices and Higher-Order Convergence Diagnostics

We present a mathematical framework for proving convergence of systems composed of

multiple adaptive subsystems operating on shared parameters. The central result, the Unified

Adaptation Theorem, provides sufficient conditions under which such composed systems

converge to a well-defined invariant set. The framework introduces cosine-scaled gradient

projection (100% conflict resolution), normalised Lyapunov functions, learnable interaction

matrices, higher-order convergence scores, the I-ratio equilibrium criterion (I=−1/2), B-

flow precision refinement, and desire as Bayesian regularisation. Applied across optimisation,

game theory, chaos detection, belief networks, generative adversarial networks, and compiler

pipeline scheduling, we establish 6 theorems and 8 propositions supported by 11 conjectures

and 1 corollary (26 named results total), with computational proofs validated across 103

experiments. All proofs are computational and independently reproducible in the Simplex

programming language.