This study presents the Cekirge σ-regularized learning framework as a deterministic, closed-form alternative to iterative optimization in artificial intelligence systems. Learning is formulated as an explicit equilibrium solution obtained from a σ-regularized inverse of the representation operator, yielding a unique and numerically stable mapping without gradient descent, random initialization, or iterative updates. Attention-style Query, Key, and Value mappings are treated as deterministic encoding operators, while decoding is resolved in a single algebraic step. To characterize intrinsic stability, a deterministic perturbation protocol is introduced in which controlled fractional disturbances of amplitude ε are applied directly to the encoding operators, and the resulting change in loss ΔL is evaluated without retraining. Because the solution is closed-form, the perturbation response reflects the structural sensitivity of the equilibrium itself rather than adaptation dynamics. The observed near-linear ΔL–ε relationship with a gently decreasing slope indicates bounded energy response and first-order stability under perturbation. This behavior contrasts with the path-dependent and
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