Realization Calculus: twin edition

Download original file

This is the original TeX manuscript source. Use the paper’s PDF for typeset equations.

1 \clearpage
2 \chap{Background and references}\label{app:references}
3 The references identify standard sources for limits, descent, Hilbert projection, bounded inverse theorems, and regularization. Appendix~\ref{app:proof-tools} gives the analytic arguments used repeatedly in the text. The accompanying Lean source and its pinned mathematical-library revision determine the formal interfaces used in this edition. Online library documentation is a navigation aid and may describe a different revision. The consultation dates below are retained from the recorded source checks.
4 
5 \begin{thebibliography}{99}
6 \bibitem{stacks-limits} The Stacks Project Authors. \emph{Limits and colimits over preordered sets}. Section 4.21, Tag 002Z, including the finite-set inverse-limit theorem in Lemma 4.21.7. \url{https://stacks.math.columbia.edu/tag/002Z}. Accessed 10 October 2026.
7 \bibitem{stacks-stack} The Stacks Project Authors. \emph{Stacks}. Section 8.4, Tag 0268. In particular the effective descent and equivalence-of-descent-categories formulations. \url{https://stacks.math.columbia.edu/tag/0268}. Accessed 10 October 2026.
8 \bibitem{mathlib-stack} The mathlib Community. \emph{CategoryTheory.Sites.Descent.IsStack}. Standard pseudofunctor stack interface. \url{https://leanprover-community.github.io/mathlib4_docs/Mathlib/CategoryTheory/Sites/Descent/IsStack.html}.
9 \bibitem{mathlib-projection} The mathlib Community. \emph{Analysis.InnerProductSpace.Projection.Basic}. Orthogonal projection onto complete subspaces. \url{https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/InnerProductSpace/Projection/Basic.html}.
10 \bibitem{mathlib-banach} The mathlib Community. \emph{Analysis.Normed.Operator.Banach}. Banach open mapping and continuous inverse constructions. \url{https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/Normed/Operator/Banach.html}.
11 \bibitem{clason-inverse} Christian Clason. \emph{Regularization of Inverse Problems}. Graduate lecture notes, arXiv:2001.00617v2, 2021. Sections 2, 4, and 6 provide standard background on weak convergence, parameter choice, and Tikhonov regularization. \url{https://arxiv.org/abs/2001.00617v2}. Consulted 12 October 2026.
12 \end{thebibliography}
13 

Back to source controls

Back to Realization Calculus: twin edition