Universal Gluing and Contextual Choice: Categorical Logic and the Foundations of Analytic Approximation
Andreu Ballus Santacana

TL;DR
This paper develops a new categorical, constructive foundation for analytic approximation, enabling explicit, certifiable function approximations across broad spaces via a functorial, local-to-global logical framework.
Contribution
It introduces the Contextual Choice Principle and Universal Embedding and Linear Approximation Theorem, providing explicit, algorithmically verifiable approximations in a constructive, categorical setting.
Findings
Explicit approximation of functions in broad spaces with certificates
Categorical framework linking local logical probes to analytic models
Certificates remain stable under uniform convergence
Abstract
We introduce a new categorical and constructive foundation for analytic approximation based on a Contextual Choice Principle (CCP), which enforces locality and compatibility in the construction of mathematical objects. Central to our approach is the Universal Embedding and Linear Approximation Theorem (UELAT), which establishes that functions in broad spaces -- including C(K), Sobolev spaces W^{k,p}(Omega), and distributions D'(Omega) -- can be explicitly approximated by finite-rank linear projections, each with a constructive, algorithmically verifiable certificate of accuracy. These constructions are governed categorically by a functorial adjunction between local logical probes and analytic models, making analytic existence both formally certifiable and programmatically extractable. As a key result, we prove a uniform certificate stability theorem, ensuring that approximation…
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Taxonomy
TopicsLogic, programming, and type systems · Logic, Reasoning, and Knowledge · Formal Methods in Verification
