Paper II: Refinement Towers, Weighted Transfer, and Integral Separation of Operator Packets

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Included manuscript sections

• atlas_macros.tex
• 01_orientation.tex
• 02_coordinates.tex
• 03_weighted.tex
• 04_arithmetic.tex
• 05_defects.tex
• 06_models.tex
• 07_scope.tex
• current_edition.tex
• references.bib
• visual_atlas.tex

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129 \input{atlas_macros.tex}
130 \begin{document}
131 \KomorebiCover
132 \KomorebiOpeningNote{\textbf{Reading route.} Exact packet coordinates and residue selectors lead to weighted inversion and rational walk series. Lattice gluing then gives separation weights and precision delays. Smith congruences and operator defects connect these formulas to finite computation.
133 
134 \medskip
135 \textbf{Companions.} The worked visual atlas appears in this volume and as a separate guide. The Lean guide distinguishes the current formal development from the inherited 25-row walkthrough. Each guide records its evidence and scope.}
136 \begin{abstract}
137 
138 A complete family of idempotents gives exact coordinates, but a single family does not record scale or integral visibility.  We study towers in which every coarse idempotent is the sum of its fine descendants.  The coordinate changes are invertible and coherent, and every operator block is the sum of its refined blocks.  Residue selectors give a model over arbitrary rings.  When roots of unity are available they agree with character averages, and coprime moduli factor through the Chinese remainder theorem.  In a complete filtered algebra, a diagonal operator plus positive weight transfers has an inverse obtained by summing admissible weighted paths.  The formula truncates exactly at finite precision, while finite transfer graphs have rational path series.  For a rational spectral decomposition over a discrete valuation ring, resultants give denominator bounds.  The exact denominator is the largest Smith exponent of the lattice gluing module.  This yields a necessary and sufficient precision delay, monotonicity under refinement, and a full congruence solving formula.  We close with exact formulas for the propagation of a coarsening defect through powers, resolvents, holomorphic functions, exponentials, and multilevel towers.
139 \end{abstract}
140 \noindent This edition integrates the expanded mathematical narrative with the canonical v1.2.1 certificate level formulations.  It preserves the distinction between general written proofs, formal specifications, and finite computational examples.  The visual atlas and the separate Lean guide provide worked models and separate current and inherited formal crosswalks.
141 \tableofcontents\listoffigures\listoftables\clearpage
142 \pagenumbering{arabic}
143 \input{chapters/01_orientation.tex}
144 \input{chapters/02_coordinates.tex}
145 \input{chapters/03_weighted.tex}
146 \input{chapters/04_arithmetic.tex}
147 \input{chapters/05_defects.tex}
148 \input{chapters/06_models.tex}
149 \input{chapters/07_scope.tex}
150 \clearpage
151 \section{Visual atlas}\label{app:visual-atlas}
152 Twenty-four worked plates follow the main constructions through explicit vectors, matrices, weighted walks, and congruence fibres. Their labels V1 to V24 stay fixed across the integrated atlas and the standalone guide. Each plate names the corresponding statement and its printed page in this monograph.
153 
154 \begin{readerroadmap}[title={Three routes through the atlas}]
155 \textbf{Exact coordinates.} Read V1 to V8, then the finite filter in V20.
156 
157 \textbf{Weights and arithmetic.} Read V9 to V19. The first group follows weighted operator words and graph walks. The second follows denominator clearing and Smith congruences.
158 
159 \textbf{Realizations and defects.} Read V20 to V24. These plates separate the Smith impedance chart from Smith normal form, trace error propagation, and connect the examples to the formal contract.
160 \end{readerroadmap}
161 The matrices and finite ranges shown here are explicit models. The general proofs remain in the preceding sections. A picture or finite computation does not extend the hypotheses of a theorem or the coverage of a Lean declaration.
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167 \clearpage
168 \bibliographystyle{plain}\bibliography{references}
169 \end{document}

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