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1 \section*{How to read this piece}
2 \addcontentsline{toc}{section}{How to read this piece}
3 The equation $a+b=c$ joins two kinds of structure. Multiplication describes prime factors and their repeated powers. Addition forces those three factorizations to fit one exact relation. The $abc$ conjecture says that this compatibility limits how large the sum can be compared with the distinct primes that appear.
4 
5 This edition develops the conjecture and the existing Operation Komorebi research in one mathematical order. The opening sections explain the radical, the quantifiers, the polynomial analogue and the consequences. The middle sections give the prime contact calculus, including its exact divisibility, lattice, signal, filtered and logarithmic constructions. The final sections develop the September jet determinant block and identify the global theorem it still needs.
6 
7 \begin{center}
8 \begin{tabularx}{\linewidth}{@{}p{0.21\linewidth}X@{}}
9 \toprule
10 Reading path & Sections to emphasize\\
11 \midrule
12 First encounter & Foundations, examples, the polynomial theorem, consequences, and the final global bridge\\
13 Proof spine & Prime packets through coherence, then the jet filter, Hankel determinant, local valuation and truncation proofs\\
14 Research boundary & The first-order barrier, exact synthesis costs, integral projectors, coherence endpoint, corrected defect and failed rational truncation\\
15 Visual companion & The separate guide follows the same objects with exact figures, short reading instructions and worked examples\\
16 \bottomrule
17 \end{tabularx}
18 \end{center}
19 
20 The statement labels have precise roles. A theorem has a proof under its stated hypotheses. A finite calculation has a declared tested domain. A conditional theorem proves an implication from a hypothesis that remains unproved. A structural limitation closes a specified method or target, rather than all possible approaches.
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22 \BoundaryBox{\textbf{Present boundary.} This work does not prove the general $abc$ conjecture. The strongest new determinant statements are formal, real analytic and $p$-adic local theorems. A nonzero global arithmetic realization with a uniform radical-scale height bound has not been constructed.}
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24 The research has also made a substantive correction. If $Q=abc/\rad(abc)$, then the defect relevant to the conjecture is $Q/(ab)=c/\rad(abc)$. The target $Q=R^{o(1)}$ is false. Its missing factor $ab$ is restored explicitly in every final certificate.
25 
26 The elementary and finite proofs below are self-contained where practical. References identify external theories and the authenticated literature used to place this work. No claim of historical priority follows from a newly organized derivation.
27 \clearpage

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