This work introduces a self-consistent training framework for learning residue–residue interaction parameters in matrix-based coarse-grained protein models. By combining Direct Coupling Analysis (DCA) with a physics-based design model, we construct a feedback loop in which the interaction matrix is iteratively updated from sequences generated by the model itself. We provide a mathematical foundation for this behavior by showing that, for a given iterative map satisfying explicit contact-locality and spectral assumptions, the training process is a contraction, guaranteeing convergence to a unique fixed point independent of the initial interaction matrix. This fixed-point property is general for coarse-grained models in which the residue–residue energy is represented by a distance-dependent term modulated by a type-dependent coupling matrix, while the learned interaction parameters remain specific to the underlying model physics. We then examine the empirical consequence of this framework in Caterpillar, a coarse-grained potential for folding and design. Independent training trajectories, defined by different target structures and stochastic sampling, converge to statistically equivalent interaction matrices. Each trajectory recovers the full interaction solution without requiring ensemble averaging, and the averaged trajectory reaches the same result. The resulting framework provides a link between sequence statistics and physically modeled interaction energies. Applied to protein design, it yields interaction matrices associated with improved structural fidelity and accurate contact prediction, while clarifying how learned sequence patterns relate to the interactions that stabilize protein structure.