Contemporary machine learning is overwhelmingly framed as an optimization problem. Gradient descent and its variants define learning as a trajectory in parameter space, governed by step sizes, penalty weights, and convergence criteria. In such formulations, the solution is not assumed it is approached. This work challenges that framing in the linear regime. When a linear system satisfies the compatibility condition A W = b, equilibrium is not produced by optimization. It is already encoded in the algebraic structure of the system. Both L₁ and L₂ objectives attain zero residual at the same structural solution. The minimum does not emerge from descent dynamics; it exists as a consequence of determinacy. Optimization, in this setting, is epistemic — it reveals a solution. Equilibrium is structural it precedes the algorithm. Regularization does not create equilibrium. It modifies the geometry surrounding an equilibrium that may already be present. It stabilizes degeneracy, suppresses ill-conditioning, and reshapes curvature. But when the system is structurally solvable, the equilibrium is not a product of penalty tuning. Principal Component Analysis exemplifies this distinction. Princi
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