ABC: source and evidence reader

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1 \section{The correctly normalized global bridge}\label{sec:bridge}
2 \subsection{A sufficient certificate with every hypothesis visible}
3 The jet block supplies formal and local expressions. The next theorem states exactly how a suitable global arithmetic witness would imply $abc$. Its hypotheses have not been established for the present determinant candidate.
4 
5 \begin{theorem}[Conditional integral witness criterion]\label{thm:global-certificate}
6 Let $\mathcal R$ be an unbounded set of positive integer ranks, and put $s_r=r(r+1)/2$. Suppose that for each $r\in\mathcal R$ there are constants $C_r>0$, $A_r\ge0$ and $E_r\ge0$, independent of the primitive triple, such that every primitive positive triple admits a nonzero integer $W_r$ with
7 \[
8  Q^{s_r}\mid W_r,
9  \qquad |W_r|\le C_r(ab)^{s_r}R^{E_r}(\log(3c))^{A_r}.
10 \]
11 If $E_r/s_r\to0$ along $\mathcal R$, then Conjecture~\ref{conj:abc} follows.
12 \end{theorem}
13 \begin{proof}
14 Nonzero integer divisibility gives $Q^{s_r}\le|W_r|$. Divide by $(ab)^{s_r}$ and use $Q/(ab)=c/R$:
15 \[
16  (c/R)^{s_r}\le C_rR^{E_r}(\log(3c))^{A_r}.
17 \]
18 Taking positive roots yields
19 \[
20  c\le D_rR^{1+\eta_r}(\log(3c))^{a_r},
21 \quad D_r=C_r^{1/s_r},\quad \eta_r=E_r/s_r,\quad a_r=A_r/s_r.
22 \]
23 Fix the target $\eps>0$. Choose one rank with $\eta_r<\eps$. Next choose
24 \[
25  0<\delta<\frac{\eps-\eta_r}{1+\eps}.
26 \]
27 For the fixed exponent $a_r$, there is a constant $L_{r,\delta}$ such that $(\log(3c))^{a_r}\le L_{r,\delta}c^\delta$ for all $c\ge2$. The elementary fact follows because a fixed logarithmic power divided by any positive power of $c$ tends to zero, and the remainder is bounded on a finite initial interval. Thus
28 \[
29  c^{1-\delta}\le D_rL_{r,\delta}R^{1+\eta_r}.
30 \]
31 Raise to $1/(1-\delta)$. The chosen inequality for $\delta$ gives
32 \[
33  \frac{1+\eta_r}{1-\delta}<1+\eps.
34 \]
35 Absorb the fixed constants into $K_\eps$ and use $R\ge2$ to obtain the desired bound. No prior power bound relating $c$ to $R$ was assumed.
36 \end{proof}
37 Only one rank is selected for each target $\eps$. Its constants may be arbitrarily large but must be uniform in the triple. A rank that depends on the triple without uniform constant control would not meet the theorem.
38 
39 \subsection{The contact determinant has not met these hypotheses}
40 For the proposed Hankel realization, all of the following remain open:
41 \begin{enumerate}
42 \item An actual global arithmetic object whose local realization is the chosen determinant.
43 \item A nonzero algebraic scalar, or a nonzero section of a rigorously metrized determinant line.
44 \item The full local divisibility in the correct sector charts, with every comparison factor included.
45 \item Bounds on every finite denominator and every archimedean conjugate or norm.
46 \item A global size estimate with the required $(ab)^{s_r}$ normalization and vanishing relative radical cost.
47 \end{enumerate}
48 Real positivity proves nonvanishing of a real analytic value. It does not prove that value algebraic. The depleted exact valuations belong to prime-dependent functions. The raw determinant supplies a common expression but not an authenticated product formula across its analytic realizations. The rational truncation constructs an algebraic number but fails the output-prime contact gate.
49 
50 A product formula applies to the places of one nonzero algebraic number. It does not apply to an arbitrary list of formally related complex and $p$-adic periods. A determinant-line alternative would likewise need its own proved arithmetic-degree identity and metric comparisons.
51 
52 \figplaceholder{24_global_bridge}{The local determinant results reach a precise global interface. The missing realization, nonvanishing, norm comparison and uniform height hypotheses are shown as obligations, rather than supplied by the diagram.}
53 
54 \subsection{The coherence route has a parallel endpoint}
55 The inherited exact factorization is
56 \[
57  M_c=\epsilon_2KWC,\qquad\epsilon_2\le2,\quad K\le\log_2c.
58 \]
59 Here the factor denoted $W$ is the local logarithmic defect, distinct from the integer witness $W_r$ in Theorem~\ref{thm:global-certificate}. The letters have separate domains.
60 
61 If for every $\eta>0$ one proves
62 \[
63  WC\le C_\eta R_aR_bR^\eta,
64 \]
65 then $c\le C'_{\eta}R^{1+\eta}\log(3c)$. The same absorption argument gives $abc$. Conversely $WC\le M_c$, so the all-$\eps$ output-sector $abc$ inequality immediately supplies this bound. The uniform combined $WC$ estimate is therefore the exact endpoint. A bound on $C$ alone suffices only under independent control of $W$, for example in a stated class where $W=1$.
66 
67 \subsection{What a useful next rank-two result would be}
68 The strongest immediate target is an explicit arithmetic realization of
69 \[
70  \mathcal H_2(t)=\Li_1(t)\Li_3(t)-\Li_2(t)^2,
71 \]
72 with every local comparison and global norm recorded. A motivic or iterated-integral construction is a possible research direction, not an input assumed here. A different nonzero algebraic determinant could also be useful if it passes the same gates.
73 
74 A new block should demonstrate a quantitative gain on $c/R$ or $M_c/(R_aR_b)$. Another local valuation, formal transformation or finite successful example is evidence within its declared scope. It does not establish that global gain.

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