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1 \section{Correcting the global target before raising contact}\label{sec:correction}
2 \subsection{A missing input factor changes the theorem}
3 The September continuation begins with a correction, rather than a larger contact claim. Set
4 \[
5  R=\rad(abc),\qquad Q=abc/R.
6 \]
7 Section~\ref{sec:foundations} proved the exact identity $Q/(ab)=c/R$. A uniform subpolynomial bound on $Q$ alone is not equivalent to $abc$. It is false unconditionally.
8 
9 \begin{theorem}[Counterfamily for the unbalanced target]\label{thm:false-target}
10 For every fixed $0\le\eta<1$ and $A\ge0$, no constant $K$ makes
11 \[
12  Q\le K R^\eta(\log(3c))^A
13 \]
14 valid for all primitive positive triples.
15 \end{theorem}
16 \begin{proof}
17 Take $a=1$, $b=2^k$, $c=2^k+1$ for integers $k\ge1$. The triple is primitive and $c$ is odd. Thus
18 \[
19  R=2\rad(c)\le2c\le3\cdot2^k,
20  \qquad Q=2^{k-1}\frac c{\rad(c)}\ge2^{k-1}.
21 \]
22 Consequently
23 \[
24  \frac{Q}{R^\eta(\log(3c))^A}
25  \ge\frac{2^{(1-\eta)k-1}}{3^\eta(\log(3(2^k+1)))^A}\longrightarrow\infty.
26 \]
27 An exponential with positive exponent eventually exceeds every fixed power of $k$.
28 \end{proof}
29 
30 The correction retracts the discarded global target. It does not retract coefficient identities or local valuations proved independently of that target. Those valid statements are developed next, with their exact scope retained.
31 
32 \subsection{A necessary leading cost for powers of Q}
33 A stronger elementary obstruction measures the exponent cost of any all-triple bound on the unbalanced quantity.
34 \begin{theorem}[Necessary cost lower bound]\label{thm:necessary-cost}
35 Fix $s>0$. Suppose $B,A\ge0$ and $C>0$ satisfy
36 \[
37  Q^s\le C R^B(\log(3c))^A
38 \]
39 for every primitive positive triple. Then $B\ge2s$.
40 \end{theorem}
41 \begin{proof}
42 Fix an integer $d\ge2$ and take $a=t^d$, $b=(t+1)^d$, $c=t^d+(t+1)^d$ for $t\ge1$. The inputs are coprime. As $\rad(t^d)\le t$ and $\rad((t+1)^d)\le t+1$,
43 \[
44  R\le t(t+1)c,\qquad
45  Q=\frac{abc}{R}\ge[t(t+1)]^{d-1}.
46 \]
47 The assumed inequality implies
48 \[
49  [t(t+1)]^{s(d-1)}\le C[t(t+1)c]^B(\log(3c))^A.
50 \]
51 Take logarithms, divide by $\log t$ and send $t$ to infinity. Since $\log c/\log t\to d$ and $\log\log(3c)/\log t\to0$, this yields
52 \[
53  2s(d-1)\le B(d+2).
54 \]
55 It holds for every $d$. Sending $d$ to infinity proves $B\ge2s$.
56 \end{proof}
57 No contact multiplier can make the ratio $B/s$ tend to zero for this false target. A valid certificate must instead include $(ab)^s$ in its global size bound, or directly control $M_c$ relative to $R_aR_b$.
58 
59 \figplaceholder{09_balanced_defect}{The balanced defect is $c/R=Q/(ab)=M_c/(R_aR_b)$. Removing the input factor $ab$ changes the target into an unconditional false assertion.}
60 
61 \subsection{Two obligations that a new determinant must meet}
62 The proposed determinant program therefore has two separate obligations. It needs a nonzero contact expression, with an actual first nonzero coefficient and local divisibility. It also needs an arithmetic realization whose global size has the correct $(ab)^s$ normalization. The following local theorems solve parts of the first obligation. They leave the second as an explicit hypothesis.

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