A Renormalization-Based Approximation Theory for the Temporal

This work develops a unified mathematical framework for understanding hierarchical computation in the temporal lobe through renormalization-based approximation theory. The model formalizes neural representations as metric–measure spaces and describes hierarchical transformations as scale-dependent coarse-graining operations combined with group-theoretic invariance projections. The theory introduces a general renormalization operator that integrates three fundamental sources of approximation error—compression, geometric coarse-graining, and invariance breaking—providing a principled way to analyze how neural systems maintain stable, abstract representations under finite computational resources. Conditions for inverse reconstruction (inverse RG) are established, clarifying when fine-scale information can be recovered from coarse representations. This framework explains why certain nonlinear transformations preserve deep hierarchical structure while others collapse it, offering insight into long-standing issues such as the vanishing gradient problem in deep learning. By grounding neural computation in Banach-space approximation theory rather than Hilbert-space assumptions, the model captures the inherently non-linear, non-commutative, and resource-limited nature of biological and artificial hierarchical systems. The result is a general, mathematically rigorous foundation for studying abstraction, invariance, and bidirectional processing in the temporal lobe, with implications for neuroscience, machine learning, and the theory of hierarchical computation.