Fractal Admissibility at the Edge of Stability

Recent work on deep learning theory, especially the GATES framework introduced in Generalization at the Edge of Stability, suggests that modern neural networks trained with large learning rates may generalize not because they converge to isolated static minima, but because stochastic optimization explores lower-dimensional fractal attractors at the edge of stability. This paper develops a high-rigor theoretical integration of that result into the Fractal Consistency Law (FCL) and its Principle of Minimum Inconsistency (PMI). The aim is not to infer physical validation from machine learning, but to extract a transferable mathematical architecture: local instability, global confinement, dimensional reduction, and emergent coherence. We introduce the Hessian of Inconsistency as the FCL analogue of the curvature of a loss landscape, define the admissibility corridor between trivial rigidity and divergent inconsistency, formalize a Fractal Confinement Condition for persistent configurations, and reinterpret grokking as a late-stage transition from high-dimensional memorization to lower-dimensional structural compression. The paper proposes that physical reality should not be understood as a perfectly static minimizer of inconsistency, but as a dynamically confined regime in which inconsistency remains bounded, processable, and geometrically organized. In this formulation, fractal dimension is not the microscopic cause of persistence; it is the observable macroscopic trace of PMI-driven dimensional economy. The resulting framework strengthens the FCL by converting a broad ontological intuition into a sharper variational and spectral program, while explicitly preserving the distinction between formal analogy, bridge evidence, and physical validation.