Similarity Across Layers and Resolutions: A Theory of Difference, Identity, and Asymmetric Transport

Abstract

This paper develops a mathematical framework in which similarity, difference, and identity depend jointly on descriptive layer and observational resolution. The central object is the similarity function \(S(A,B;\ell,r)\), supported by a visible/hidden difference decomposition and a Layered Similarity Profile. A six-axiom layer structure yields a Layer–Resolution Monotonicity Theorem and an asymmetric cross-layer transport theory: downward similarity transfer is quantitatively bounded, while global upward reconstruction is impossible under genuine information loss unless the hidden component is controlled. The framework is extended to approximate compatibility through a relative intertwining defect, providing robust and computable transport bounds. Potential applications include mathematical analogy discovery, computer vision under variable resolution, and adaptive computation, reliability analysis, and compression in large language models.

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