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1 \documentclass[11pt]{article}
2 \input{preamble}
3 \begin{document}
4 \KomorebiCover
5 \pagenumbering{roman}
6 \section*{Preface}
7 A mathematical object can appear in several forms: as local data, as a sequence of finite observations, as a family of coordinates, or as the output of an approximate calculation. These descriptions may agree, fail to agree, or agree only after a correction. This monograph studies the conditions under which they describe one coherent realization.
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9 The starting point is a declared system of views and maps between them. Its coherent families form a limit. Prescribed data then define an extension problem, whose obstruction records whether a completion exists and whose ambiguity records the freedom in choosing it. Resolution towers make these questions sensitive to scale. Groupoids retain the symmetries of a realization, while descent asks whether compatible local realizations assemble. In normed and Hilbert models, the same questions acquire quantitative forms: how much correction is needed, which information can be preserved, and whether small residuals certify proximity to an exact answer.
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11 These constructions give a common language for realization, obstruction, and repair. Their connections are made through explicit maps and estimates. Existence, uniqueness, and stability remain separate assertions. A change of representation must identify both the transformed object and the data it preserves.
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13 The underlying limit constructions, gauge groupoids, stack definitions, orthogonal projections, and bounded inverse principles are part of established mathematics. The contribution of this presentation is their integration with specified view systems, protected correction spaces, resolution differentials, graph transports, and worked examples. The comparison maps and hypotheses carry the substance of that integration. Formal verification establishes the recorded declarations, without making a claim of historical novelty for each ingredient.
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15 \begin{scopebox}
16 The word \emph{space} does not assert one universal topological object containing all of mathematics. The strict construction is a set of coherent families. Symmetry is retained in a groupoid, and effective descent is proved for a specified site. Analytic estimates require a declared normed or Hilbert model. Each layer has its own hypotheses and its own notion of realization.
17 \end{scopebox}
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19 The book presents proofs together with examples that show why their hypotheses matter. Its companion visual guide emphasizes geometry and explicit calculations. Its Lean guide explains the formal definitions, interfaces, and evidence. Mathematical figures are reproducible vector diagrams or plots of explicit formulas.
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21 \clearpage
22 \section*{Reading guide}
23 Part I constructs coherent sections and follows their obstructions as more data are prescribed. Part II develops resolution towers, finite observations, split repair coordinates, and multiscale reconstruction. Part III retains symmetries, proves effective descent, and distinguishes coherent phases from removable ones. Part IV develops analytic stability and a geometry of transformation costs. Part V follows approximate answers to their limits, separates a coherent point from its intended Hilbert realization, and derives budget, regularization, and stopping criteria. Part VI works through finite loop models and returns to mathematical semantics.
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25 A short technical route is Sections~\ref{sec:views}, \ref{sec:obstruction}, \ref{sec:stack}, \ref{sec:closed}, and~\ref{sec:motion}. For packets and residuals, include Sections~\ref{sec:split} and~\ref{sec:cascade}. The graph models in Section~\ref{sec:graph} show how a categorical realization problem acquires a numerical condition number. Sections~\ref{sec:observation} and~\ref{sec:repair-transport} explain how the constructions behave under changes of representation.
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27 For concrete motivations, begin with Sections~\ref{sec:signal-loop}, \ref{sec:twisted-loop}, and~\ref{sec:escape}. The last connects finite realization, closure, and failure of protected infinite existence in one calculation. Sections~\ref{sec:bounded-realization} through~\ref{sec:diagonal-regularization} develop its positive converse and regularized solutions. Sections~\ref{sec:inexact-repair} and~\ref{sec:stopping-rules} explain how to certify an inexact calculation and select its tolerance.
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29 Appendix~\ref{app:workshop} contains fifteen solved problems. Appendix~\ref{app:hypotheses} collects the assumptions behind the principal results. Appendix~\ref{app:concordance} supplies notation and a map into the formal source. Appendix~\ref{app:written} describes formal coverage, and Appendix~\ref{app:proof-tools} proves the analytic background used in the text.
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31 \subsection*{Proofs and formal scope}
32 The accompanying Lean development covers the foundational constructions and all \EditionPrincipalGroups\ principal theorem groups on bounded realization, budgets, regularization, and certificates. This includes the actual infinite diagonal family and its tail bounds, the sharp asymptotic over all real budgets, the rounded noisy and inexact rates, and the concrete nonclosed-range example. Appendix~\ref{app:written} identifies the formal counterparts and their scope.
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34 A formal counterpart can establish a theorem by a different route from its printed proof. In particular, the bounded-realization result is formalized through a compactness argument without assigning a separate formal counterpart to the printed weak-subsequence lemma. Explanatory prose, background proofs, and numerical illustrations are not all individually machine checked.
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36 The complete source has passed a fresh integrated compilation, declaration audit, and diagnostic-control replay. The earlier records remain available as historical evidence. Lean's compiler and kernel, its stated standard axioms, and the pinned external dependency cache remain part of the trust boundary. The cache is recorded but was not rebuilt by this audit. Appendix~\ref{app:written} gives the exact counts and their interpretation.
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38 \clearpage\begingroup\setlength{\parskip}{0pt}\tableofcontents\endgroup
39 \clearpage\pagenumbering{arabic}
40 \input{chapters/01_views}
41 \input{chapters/02_obstructions}
42 \input{chapters/02a_prescription}
43 \input{chapters/03_towers}
44 \input{chapters/04_symmetry}
45 \input{chapters/04a_boundary_lab}
46 \input{chapters/04b_phase}
47 \addtocontents{toc}{\protect\newpage}
48 \input{chapters/05_geometry}
49 \input{chapters/05a_certificates}
50 \input{chapters/05b_transport_cost}
51 \input{chapters/05c_iteration}
52 \input{chapters/05d_floors}
53 \input{chapters/05e_escape}
54 \input{chapters/05f_bounded_realization}
55 \input{chapters/05g_budget_certificates}
56 \input{chapters/05h_regularized_repair}
57 \input{chapters/05i_diagonal_regularization}
58 \input{chapters/05j_inexact_certificates}
59 \input{chapters/05k_stopping_rules}
60 \input{chapters/06_cases}
61 \input{chapters/06_boundaries}
62 \appendix
63 \input{chapters/09_workshop}
64 \input{chapters/10_hypothesis_atlas}
65 \input{chapters/07_concordance}
66 \input{chapters/11_written_extensions}
67 \input{chapters/12_proof_tools}
68 \input{chapters/08_references}
69 \end{document}
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