Realization Calculus

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24 \hypersetup{pdftitle={Packet Operator Algorithms, Exact Compilation, and GPU Realization: Final Traditional Edition},pdfauthor={Komorebi P.},pdfsubject={Finite packet operator calculus, exact compilation, structured transforms, precision allocation, and mathematical proof completions}}
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26 \fancyhead[L]{\small\textsc{Packet Operator Algorithms VIII}}
27 \fancyhead[R]{\small Komorebi P.}
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60 
61 % Authored modern monograph profile. All seven mathematical parts are unchanged.
62 \usepackage{komorebi-monograph}
63 \renewcommand{\KomorebiTitle}{Packet Operator Algorithms}
64 \renewcommand{\KomorebiSubtitle}{Exact compilation and GPU realization}
65 \renewcommand{\KomorebiSeries}{Core series VIII}
66 \renewcommand{\KomorebiKind}{Modern Monograph}
67 \renewcommand{\KomorebiEdition}{9 October 2026}
68 \renewcommand{\KomorebiShortTitle}{Paper VIII}
69 \renewcommand{\KomorebiTitleSize}{38}
70 \renewcommand{\KomorebiTitleLeading}{43}
71 \renewcommand{\KomorebiCoverTitle}{Packet Operator\\Algorithms}
72 \renewcommand{\KomorebiCoverBlurb}{Finite operator structure removes work before execution. Exact compilation makes the representation, error budget, output contract, and inherited hardware observations explicit.}
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115 \input{component-layout.tex}
116 \begin{document}
117 \KomorebiCover
118 \KomorebiOpeningNote{The complete finite manuscript comes first. Six supplementary proof developments follow as Appendices A--F. A worked visual atlas closes the volume. The separate visual and Lean companions provide another reading route and an explicit evidence register.}
119 
120 \section*{Reading this chapter}
121 \addcontentsline{toc}{section}{Reading this chapter}
122 An operator representation determines what a computation means before it determines how a device executes it. Chapter VIII develops that distinction through packet blocks, low rank corrections, graph support, preconditioning, precision, and finite compilation. Three exact examples introduce the main choices. The manuscript then gives their general statements and proofs.
123 
124 The main argument keeps its local section and item numbers. Appendices A--F give the additional constructions and proofs for exact semantics, compilation, structured transforms, scheduling and small-core recompression. These developments make assumptions and intermediate objects explicit when a shorter main-text argument leaves them implicit.
125 
126 The formal companion covers all 56 finite manuscript-proved items under their stated hypotheses. Its 392 supporting theorem declarations span 40 modules, including 19 reused quantum-packet declarations. The 66 construction audits and four classical inputs are tracked separately. The companion supplies the declaration map and the provenance of the recorded compiler evidence.
127 
128 The source already includes positive precision budgets and caps, nonempty allocation sets, positive physical denominators and transform orders, valid band widths, actual positive square roots, reflexive reachability, and the empty-graph depth-zero case. Its arithmetic counts describe declared event models. Compiler stability uses solved child scores, and local recompilation assumes correct unaffected sibling caches. These hypotheses are part of the results.
129 
130 The small-core recompression result starts from supplied ordered thin-SVD equations and handles arbitrary ambient competitors by compression. It does not establish SVD existence, a floating-point QR/SVD algorithm, or a canonical singular-values API identification. The other three classical inputs concern exact-arithmetic conjugate gradients, a stated floating-point model, and simultaneous spectral decomposition. A rounded action on varying inputs requires a uniform certificate before it can be treated as one fixed linear operator.
131 
132 \begin{keyidea}[title={Representation, output, model, observation}]
133 A compressed action and an explicit matrix are different outputs. An additive cost model and an observed execution time are different evidence. A finite compiler optimum and unrestricted structure discovery are different problems. Keep the representation, output contract, mathematical model, and recorded measurement together when using a result.
134 \end{keyidea}
135 
136 Every hardware time and speed ratio remains an inherited observation from the archived corpus. No fresh GPU run is reported. Missing source-value hashes and explicit correctness-reference fields limit the historical comparisons. Cross-scope CPU, graph-replay, and two-stream rows retain their qualifications. In the negative-control figure, ``matched baseline'' means the baseline recorded for that comparison. It does not certify the full fairness contract set out in the benchmark discussion.
137 
138 Appendices A--F complete the finite development under its stated hypotheses. Three broader targets remain open: unrestricted discovery, compilation with nonlocal overlapping models, and a packet-native communication theorem for a specified architecture. The exact examples and visual atlas illustrate finite mathematics. They add no hardware performance claim and do not replace the proofs.
139 \clearpage
140 \tableofcontents
141 \clearpage
142 \listoffigures
143 \clearpage
144 \pagenumbering{arabic}
145 \section*{Abstract}
146 A packet operator is a linear map written between the members of a finite direct sum. We study the resulting computational representation before choosing a processor or kernel. Exact formulas are given for block sparse application, block diagonal plus low rank algebra, Woodbury inversion, support propagation, block preconditioning, and packetwise perturbation. Products of low rank corrections admit optimal recompression through a small singular value problem, with explicit error propagation through a product chain. A rooted candidate decomposition supports an exact compiler when representation cost and error are additive. Dynamic programming gives the optimal Lagrangian pruning, an exact quantized error budget algorithm, and a local update rule. Discrete precision assignment becomes a multiple choice budget problem. Arithmetic, storage, data movement, launch count, discovery, transfer, and materialization are kept as separate costs. A preserved corpus of 154 recorded CPU and GPU measurements illustrates regular packet gains, precision frontiers, crossover behavior, and retained negative controls. Its largest reported structured ratio is 14.89 against a device resident dense GPU calculation, with archived matching-metadata limitations stated explicitly. The chapter proves finite algebraic and optimization statements. Hardware measurements remain specific to their recorded environment, and no generic dense matrix exponent or universal discovery algorithm is claimed.
147 \clearpage
148 \input{monograph_examples.tex}
149 \clearpage
150 \begin{readerroadmap}[title={The finite manuscript}]
151 The main reading path follows exact packet semantics, algebra, support, numerical certificates, and optimization before reaching the archived measurement corpus. The cost and error assumptions determine where each algorithm is exact.
152 \end{readerroadmap}
153 \input{parts/01_manuscript.tex}
154 \clearpage\FloatBarrier
155 \appendix
156 \input{parts/02_supplement_01.tex}
157 \input{parts/02_supplement_02.tex}
158 \input{parts/02_supplement_03.tex}
159 \input{parts/02_supplement_04.tex}
160 \input{parts/02_supplement_05.tex}
161 \input{parts/02_supplement_06.tex}
162 \clearpage\FloatBarrier
163 \section*{Worked visual atlas}
164 \addcontentsline{toc}{section}{Worked visual atlas}
165 \input{monograph_atlas.tex}
166 \clearpage\FloatBarrier
167 \section*{References}
168 \addcontentsline{toc}{section}{References}
169 \renewcommand{\refname}{}
170 \bibliographystyle{amsplain}
171 \bibliography{references}
172 \end{document}

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