Universal Gluing and Contextual Choice: Categorical Logic and the Foundations of Analytic Approximation
Journal:
arXiv
Published Date:
Jun 28, 2025
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
certificates persist under uniform convergence.
The CCP avoids classical pathologies (e.g., non-measurable sets,
Banach--Tarski paradoxes) by eliminating non-constructive choice and replacing
it with a coherent, local-to-global semantic logic. Our framework strengthens
the foundations of constructive analysis while contributing tools relevant to
formal verification, type-theoretic proof systems, and computational
mathematics.
