% ===================================================================== % Explicit Weak abc Inequalities and the Discrete Obstruction to Full % abc in the DIC Framework -- Axiomatic Version % Compile: pdflatex (or xelatex) DIC_axiomatic_paper.tex (twice) % Dependencies: amsart, amsmath, amssymb, mathtools, bm, booktabs, % enumitem, xcolor, hyperref. No external .bib required: % references are inlined in thebibliography; a matching % references.bib is provided for arXiv submission. % ===================================================================== \documentclass[11pt]{amsart} \usepackage{amsmath, amssymb, mathtools, bm} \usepackage{booktabs} \usepackage{enumitem} \usepackage{xcolor} \usepackage[colorlinks=true, linkcolor=blue!55!black, citecolor=blue!55!black, urlcolor=blue!55!black]{hyperref} \setlength{\parskip}{0.3em} \setlength{\emergencystretch}{2.5em} \numberwithin{equation}{section} % ---------- Theorem-type environments ---------- \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{example}[theorem]{Example} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} % ---------- User's standing macros (notation system of the source) ---------- \DeclareMathOperator{\rad}{rad} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\length}{length} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\Div}{Div} \DeclareMathOperator{\sqfree}{sf} \newcommand{\om}{\omega} \newcommand{\wbar}{\overline{\omega}} \newcommand{\hB}{h_B} \newcommand{\hrel}{h_{\mathrm{rel}}} \newcommand{\hvert}{h_{\mathrm{vert}}} \newcommand{\mS}{m_S} \newcommand{\eps}{\varepsilon} \newcommand{\Ar}{\mathrm{Ar}} \newcommand{\tor}{\mathrm{tor}} \newcommand{\exc}{\mathrm{exc}} \newcommand{\gen}{\mathrm{gen}} \newcommand{\adj}{\mathrm{adj}} \newcommand{\GRR}{\mathrm{GRR}} \newcommand{\AJ}{\mathrm{AJ}} \begin{document} \title[Weak $abc$ and the Discrete Obstruction in the DIC Framework]{% Explicit Weak $abc$ Inequalities and the Discrete Obstruction\\ to Full $abc$ in the DIC Framework} \author{Chunping Zeng} \address{} \email{} \date{November 2026} \begin{abstract} We axiomatise the DIC (Dimension Isolation Conjecture) framework for the $abc$ conjecture and prove three stratified groups of results on Kodaira--Parshin arithmetic surface fibrations. \textbf{(i) Unconditional geometric theorems.} The adjunction identity $Z\cdot\om_{\mathcal X/B}=-Z^{2}$, the Riemann--Hurwitz intersection numbers, the invariants of the general $(g_{0},r,b)$ construction, and a Deligne--Riemann--Roch decomposition of the arithmetic coefficient $c=\frac{2b^{2}+2b-1}{12}+(D_{0}-T)/\om^{2}$. \textbf{(ii) A meta-theorem about the construction itself.} Integrality of torsion lengths forces the coefficient of Theorem~A onto the discrete set $t=12/N$ with $N\equiv1\pmod 3$ --- at the literature point $(q,\lambda)=(5,3)$; when $q$ varies this is an arithmetic coincidence, not a structural barrier (Remark~\ref{rem:landing}); for general $(g_{0},r,b)$ the reciprocal $1/t$ lies on an arithmetic progression of common difference $\lambda/\bigl(6r(g_{0}-1)\bigr)$. For the normalisation $q=5$ the coefficient required by full $abc$ is unattainable; an explicit local hyperelliptic model yields $T=8k$, $t=12$ as the unique transverse output and restores the No-Go for every Kummer degree $q$ under a transversality (or Galois-transitivity) hypothesis. The second-order torsion computation gives $n_{1}=n_{2}$, hence $\Theta=T/k\in2\mathbb Z$, leaving $(27,12)$ as the sole lattice candidate, while a transitive Galois action fixes $\Theta=8$ without transversality. A unified criterion $E=12(q-1)/(qN\gamma)$ parameterises the last lattice point: the landing equation has the unique odd integral solution $(N,\Theta,q,\gamma)=(13,12,27,8/9)$, requiring exact normalisation $\gamma=8/9$; in the original $q=5$ construction no normalisation lands on the lattice. The height is the canonical-bundle height on a fixed genus-$2$ curve: $\gamma=2/5$ follows from $12/\gamma=30$ in \S7.2 of the source and is confirmed by an explicit valuation computation; since Theorem~A fixes both the genus and the Kummer degree, the lattice point $(12,27)$ requires genus $13$ and lies outside the theorem's hypotheses --- the last cell is empty. \textbf{(iii) Controlled weak $abc$.} All unconditional bounds for $abc$ currently known are of exponential type ($\log c\ll N^{1/3}(\log N)^{3}$, \cite{SY01,SY02}; \cite[\S1.1]{PastenShimura}); the exponent below is of polynomial type, and its existence is what Theorem~A supplies. Taking Soul\'e's Theorem~A as the sole external input, we derive the Kummer discriminant estimate $\eta(q)=(q-1)/q$, unify the weak $abc$ exponent as $E=t\eta/\gamma$, and obtain the explicit exponent $E_{\max}=12\frac{q-1}{q}\bigl\lceil\frac{2q-2}{\ell}\bigr\rceil$ via a Riemann--Roch lower bound for $\gamma$. A discriminant stratification sharpens this to the pointwise bound $c\le A_{\eps}(N/M_{5})^{24+\eps}$, and we prove that bootstrapping this discount to exponent $1$ is impossible ($\inf\theta=0$), that the discriminant-to-degree ratio is monotone in towers of number fields, and that $\tfrac45$ is an Artin conductor identity rather than an estimate constant. A verbatim check of the source (Appendix~J) confirms twelve items, including the statement and scope of Theorem~A; Appendix~I derives the $O(d')$ form of (32) independently; Appendices~K and~L add two structural No-Go reasons; a distributional analysis (Appendix~M) shows the discount and the multiplicity excess vanish in density $1-o(1)$ with explicit large-deviation rates, independently of the truth of Theorem~A; the constant $12$ decomposes completely as $\frac13\cdot\frac{m}{\rho-mc}$ and Route~1 is closed: archimedean/successive-minima refinements cannot move it (Appendix~\ref{app:constdec}). No Diophantine conjecture enters the derivations. We neither prove nor disprove $abc$; all conclusions are internal to the construction under its listed hypotheses. \end{abstract} \keywords{$abc$ conjecture, Arakelov geometry, Kodaira--Parshin fibration, torsion sheaves and determinant cohomology, weak $abc$ inequality, discrete obstruction at the fixed normalisation} \subjclass[2020]{11D75, 14G40, 14H10, 11G30, 14F06} \maketitle \tableofcontents % ===================================================================== \section{Introduction}\label{sec:intro} % ===================================================================== \subsection{Background} Let $a,b,c$ be pairwise coprime positive integers with $a+b=c$ and put $N=\rad(abc)=\prod_{p\mid abc}p$. The Masser--Oesterl\'e $abc$ conjecture asserts that for every $\eps>0$ there is $K_{\eps}$ with \begin{equation}\label{eq:abc} c\le K_{\eps}\,N^{1+\eps}, \qquad\text{equivalently}\quad \log c\le(1+\eps)\log\rad(abc)+O_{\eps}(1). \end{equation} Through the Frey--Hellegouarch curve and the Szpiro conjecture, \eqref{eq:abc} is tied to discriminant--conductor inequalities on arithmetic surfaces; Vojta's value-distribution dictionary places it in a general framework of heights and proximity functions \cite{Vojta,Szpiro,Frey}. The route via Arakelov theory, the Parshin construction and Moret-Bailly's \emph{formule clef} reduces the Diophantine question to cohomological and metric estimates on arithmetic surfaces \cite{Arakelov,Parshin,MB,SouleABC}; determinant cohomology and arithmetic Riemann--Roch are surveyed in \cite{Deligne,Faltings,GS,Eriksson}, successive minima and (semi-)stable extensions in \cite{SouleDRR,SouleSS,SouleVS}, and the function-field analogue in \cite{Mason}. The DIC framework that we axiomatise here attempts to build an arithmetic surface fibration $\pi:\mathcal X\to B$ with $h_{B}\asymp\log c$ and $\mS(D_{B},b)\asymp\log c-\log\rad(abc)$, from which \eqref{eq:abc} would follow through a family of positivity axioms (ND1, ND2$'$) and height-control inequalities (A1, A2, A3). In its original form the framework contains category errors and incompatible hypotheses: a stratification mismatch of the monodromy representation, mutually exclusive positivity and lower-bound requirements on one and the same section, and a genuine logical circle in which a statement equivalent to \eqref{eq:abc} is imported as a lemma. \subsection{Aim and results} We do \emph{not} prove the $abc$ conjecture. Our aim is to extract, from this route, everything that can be stated and proved as a self-contained theorem, and to make the dependence structure completely explicit. The results fall into three strata; their logical dependencies are collected in \S\ref{sec:depend}. \begin{enumerate}[leftmargin=1.8em,itemsep=0.3em] \item \textbf{Unconditional geometric layer} (\S\ref{sec:geom}--\S\ref{sec:arith}). We recompute the fifteen invariants of the Kodaira--Parshin genus-$6$ fibration, extend the construction to arbitrary initial genus $g_{0}$, branch number $r$ and divisor coefficient $b$, prove that $\rho(b)=(b-1)(b+3)/3$ is independent of all covering data, and use Deligne--Riemann--Roch to write the arithmetic coefficient as a quadratic GRR term minus a torsion term, reproducing the value $-11k$ verbatim. \item \textbf{Meta-theorem: discrete obstruction} (\S\ref{sec:torsion}--\S\ref{sec:local}). The torsion module decomposes as $A/\pi^{n_{1}}\oplus A/\pi^{n_{2}}$ with $t=2$ verified independently of the source's counting convention; integrality gives $T=kL$, $L\in\mathbb Z_{\ge8}$, hence $t(L)=12/(3L-23)$ with denominator $\equiv1\pmod 3$. The coefficient required by full $abc$ ($t^{\ast}=1/2$ or $5/4$, according to the normalisation of the bridge) is therefore unattainable at $q=5$. An explicit hyperelliptic local model computes the Kodaira--Spencer class $\kappa_{M}$ and the ranks of the two connecting maps, yielding $T=8k$ exactly under transversality; second-order torsion sharpens $L=\Theta$ to even integers, cuts the candidate set to $\{8,10,12,14\}$, excludes $(9,13)$, and leaves $(\Theta,q)=(12,27)$ as the unique remaining candidate, which \S\ref{sec:last} then shows to be excluded by the \emph{hypotheses} of Theorem~A. \item \textbf{Controlled weak $abc$} (\S\ref{sec:weak}). The source states that \S7.2 of \cite{SouleABC} is ``known to follow from Theorem A, albeit with a wrong exponent''. We supply the complete derivation: the Kummer discriminant estimate $\eta(q)=(q-1)/q$ is proved, the exponent is unified as $E=t\eta/\gamma$, and Riemann--Roch gives $\gamma\ge1/n$, hence an explicit finite $E_{\max}$. In \S\ref{sec:refine} we sharpen the discriminant estimate to a pointwise stratification, prove the impossibility of bootstrapping the resulting discount, and prove tower monotonicity of the discriminant-to-degree ratio. \end{enumerate} \subsection{Convention on hypotheses} Every theorem below carries an explicit dependence tag: \textbf{U} = unconditional (geometry, algebra, and results already proved in the cited literature); \textbf{A\ref{ass:count}}, \textbf{A\ref{ass:model}} = the counting convention and the local model assumption; \textbf{Th.A} = conditional on \cite[Theorem~A]{SouleABC}. Unconditional results in this paper are of two kinds: \emph{exponential-type} (Theorems~7.5$^{\prime}$ and~7.5a) and \emph{No-Go type} (\S\ref{sec:torsion}--\S\ref{sec:bridge}, Appendices~E--M), the latter internal to the construction and independent of the truth of Theorem~A (Proposition~H.5). The statements that genuinely use Theorem~A are tagged \texttt{Th.A} throughout (Table~\ref{tab:depend}) and are confined to the weak-$abc$ outputs (Theorems~\ref{thm:weak} and~\ref{thm:discount}); the internal adelic metric layer has been re-derived in part (Appendix~\ref{app:adelic}), and the remaining conditional items and the complete hypothesis ledger are in Appendices~\ref{app:H} and~\ref{app:assume}. We emphasise once more: \emph{the $abc$ conjecture itself is neither proved nor disproved here.} \subsection{Notation} $\om:=\om_{\mathcal X/B}$ is the relative dualising sheaf and $\wbar$ its adelically metrised version; $\delta(F):=\log|\Delta_{F}|$ and $d:=[F:\mathbb Q]$; $(\cdot)$ denotes geometric intersection and $\langle\cdot\rangle_{\Ar}$ Arakelov intersection, and the two are never mixed. All $O(1)$, $O_{\eps}(1)$ are uniform constants; $O(d)$ is bounded by a constant multiple of $d$. We write $k=\deg(Y\to Y_{1})$, $r$ for the branch number, $b$ for the coefficient of $L=\om-bS$. The Kummer degree is denoted $q$ from \S\ref{sec:weak} onward (in \S\ref{sec:geom} the symbol $m$ denotes $\deg L_{F}$). We keep the source's abbreviations $\hB$, $\mS$, $\eps$. % ===================================================================== \section{Axiomatic setup and stratification of the cycle}\label{sec:setup} % ===================================================================== \begin{definition}[Fibration and heights]\label{def:fib} $\pi:\mathcal X\to B$ is an arithmetic surface fibration over a smooth projective curve $B$, the general fibre has genus $g_{f}\ge2$, and $\mathcal Z\subset\mathcal X$ is a section. For an adelically metrised line bundle $\bar L$ the Weil height is $h_{\bar L}(P)=[K:\mathbb Q]^{-1}\widehat{\deg}(s_{P}^{\ast}\bar L)$; $\hB$ is induced by an ample divisor, $\hrel$ by $\om$, and $\hvert$ is a distance-type height to $\mathcal Z$. The proximity function is $\mS(D,P)=\sum_{v\in S}\lambda_{D,v}(P)$. \end{definition} \begin{assumption}[Alignment A0]\label{ass:A0} Take $B=\mathbb P^{1}$, $D_{B}=[0]+[1]+[\infty]$, $b=a/c$ and $S\supseteq S_{\infty}\cup\{p:p\mid abc\}$. Then \[ \hB(b)=\log c+O(1),\qquad \mS(D_{B},b)=\log c-\log\rad(abc)+O(1). \] \end{assumption} \begin{lemma}\label{lem:abc-equiv} Under Assumption~\ref{ass:A0}, \eqref{eq:abc} is equivalent to: for every $\eps>0$, $\mS(D_{B},b)\le\eps\hB(b)+O_{\eps}(1)$. \end{lemma} \begin{proof} We start from the two alignment identities $\hB=\log c+O(1)$ and $\mS=\log c-\log\rad(abc)+O(1)$ of Assumption~\ref{ass:A0}. \emph{Step 1 (eliminate $\log c$).} Rearranging the second identity, \[ \log\rad(abc)=\log c-\mS+O(1)=\hB-\mS+O(1). \] \emph{Step 2 (substitute into the Masser--Oesterl\'e form).} The correct direction of $abc$ is $\log c\le(1+\eps)\log\rad(abc)+O_{\eps}(1)$. Substituting Step~1, \[ \hB\le(1+\eps)(\hB-\mS)+O_{\eps}(1) =\hB+\eps\hB-(1+\eps)\mS+O_{\eps}(1). \] \emph{Step 3 (rearrange).} Cancelling $\hB$ on both sides of Step~2 gives $0\le\eps\hB-(1+\eps)\mS+O_{\eps}(1)$, i.e. $(1+\eps)\mS\le\eps\hB+O_{\eps}(1)$, hence $\mS\le\frac{\eps}{1+\eps}\hB+O_{\eps}(1)$. Since $\eps>0$ is arbitrary, $\frac{\eps}{1+\eps}$ is arbitrarily small, so $\mS\le\eps'\hB+O_{\eps'}(1)$ holds for every $\eps'>0$. Each step is reversible, so the two statements are equivalent. \emph{Direction note.} The reverse inequality $\rad(abc)\le K_{\eps}c^{1+\eps}$ is trivially true ($\rad(abc)\le abc0\iff S^{2}<0$. \end{proposition} \begin{proof} \emph{Fibre.} $F^{2}=0$ (distinct fibres are disjoint) and $\pi^{\ast}K_{B}|_{F}$ is trivial, so $F\cdot\pi^{\ast}K_{B}=0$. The fibre version of adjunction gives $F\cdot K_{\mathcal X}=2g_{f}-2-F^{2}=2g_{f}-2$; subtracting $F\cdot\pi^{\ast}K_{B}=0$ yields $F\cdot\om=2g_{f}-2$. Since $\deg(\om|_{F})=2g_{f}-2>0$ ($g_{f}\ge2$), the relative canonical sheaf is relatively ample. \emph{Section.} $\deg(\om|_{S})=S\cdot\om=-S^{2}$ by \eqref{eq:adj}; dividing by the base scale $\deg L_{B}$ gives $\kappa_{\mathrm{A2}}(S)=-S^{2}/\deg L_{B}$. For a K3 elliptic surface $\chi=2$, so $S^{2}=-\chi=-2$ and $\kappa=2$ (Appendix~\ref{app:geom}). \emph{Multisection.} If $D\to B$ has degree $d$ and ramification $R$, Riemann--Hurwitz gives $2g_{D}-2=d(2g_{B}-2)+\deg R$; adjunction gives $D\cdot K_{\mathcal X}=2g_{D}-2-D^{2}=d(2g_{B}-2)+\deg R-D^{2}$, and the projection formula gives $D\cdot\pi^{\ast}K_{B}=d(2g_{B}-2)$. Subtracting, $D\cdot\om=\deg R-D^{2}$; dividing by $d\deg L_{B}$ gives $\kappa_{\mathrm{A2}}(D)$. \emph{Quantifier note.} $\kappa_{\mathrm{A2}}$ is the intersection slope of a \emph{single} (multi)section, not a uniform lower bound on heights for all $P$; inferring A2 from ``there exists $S$ with $\kappa_{\mathrm{A2}}(S)\ge2$'' is a quantifier swap, cf.~\S\ref{sec:intro}. \end{proof} \begin{proposition}[Kodaira--Parshin general chain]\label{prop:KP} Let $Y_{1}\to C$ be an \'etale double cover of a curve of genus $g_{0}$ (so $g_{1}=2g_{0}-1$), let $X_{y}\to Y_{1}$ be a double cover branched at $r$ Weierstrass points, and let $\psi:Y\to Y_{1}$ be a finite \'etale cover of degree $k$. Write $S=\sum_{i=1}^{r}B_{i}$ and $L=\om-bS$. Then \begin{align} g_{f}&=4g_{0}-3+\tfrac r2, & \om&=\sum_{i=1}^{r}B_{i}+(4g_{0}-4)\phi_{1}^{\ast}W,\label{eq:kp1}\\ B_{i}^{2}&=-(2g_{0}-2)k, & \om^{2}&=6rk(g_{0}-1),\label{eq:kp2}\\ L^{2}&=-2rk(g_{0}-1)(b-1)(b+3), & \rho&:=\frac{-L^{2}}{\om^{2}}=\frac{(b-1)(b+3)}3,\label{eq:kp3}\\ \mu:=\deg L_{F}&=8(g_{0}-1)+r-br, & h^{0}(L_{F})&=4g_{0}-3+\tfrac r2-br,\label{eq:kp4} \end{align} where $W$ is a theta characteristic on $C$ with $\deg W=g_{0}-1+\frac r2$ in the branched model, $\phi_{1}^{\ast}W\cdot B_{i}=k$, and $B_{i}^{2}$ is pinned by \eqref{eq:adj} via $B_{i}\cong Y$. \end{proposition} \begin{proof} \emph{Step 1 (fibre genera).} The first cover is \'etale: $2g_{1}-2=2(2g_{0}-2)$, so $g_{1}=2g_{0}-1$. The second cover is branched at $r$ points: Riemann--Hurwitz gives \begin{equation}\label{eq:gf} 2g_{f}-2=2(2g_{1}-2)+r=2(4g_{0}-4)+r=8g_{0}-8+r, \qquad g_{f}=4g_{0}-3+\tfrac r2 . \end{equation} At the literature point $(g_{0},r)=(2,2)$: $g_{1}=3$, $2g_{f}-2=8+2=10$, $g_{f}=6$, as stated. \emph{Step 2 (relative dualising sheaf).} The ramification formula for a double cover gives $\om_{X/Y}=\phi^{\ast}(\om_{Y/Y_{1}})\otimes\mathcal O(\sum B_{i})$; pulling back the theta-characteristic term and using $M^{2}=\mathcal O(A_{1}+A_{2})$ yields \eqref{eq:kp1} with $\deg W=g_{0}-1+\frac r2$ (at the literature point $\deg W=2$). \emph{Step 3 (pinning $B_{i}^{2}$ by adjunction).} By \eqref{eq:adj}, $\om\cdot B_{i}=-B_{i}^{2}$. Intersecting \eqref{eq:kp1} with $B_{i}$ and using $B_{i}\cdot B_{j}=0$ ($i\ne j$) and $\phi_{1}^{\ast}W\cdot B_{i}=k$ gives \[ -B_{i}^{2}=B_{i}^{2}+(4g_{0}-4)k, \qquad\text{hence}\qquad B_{i}^{2}=-(2g_{0}-2)k, \] which is $-2k$ at the literature point, and simultaneously $\om\cdot B_{i}=(2g_{0}-2)k$. \emph{Step 4 ($\om^{2}$).} Squaring \eqref{eq:kp1} and using that the square of a pullback from the base vanishes, \[ \om^{2}=\sum_{i}B_{i}^{2}+2(4g_{0}-4)\sum_{i}\phi_{1}^{\ast}W\cdot B_{i} =-r(2g_{0}-2)k+2(4g_{0}-4)rk=6rk(g_{0}-1)k . \] At the literature point $\om^{2}=12k$. The full term-by-term expansion is in Appendix~\ref{app:geom}. \emph{Step 5 ($L^{2}$ and $\rho$).} With $S=\sum_{i=1}^{r}B_{i}$ one has $\om\cdot S=r(2g_{0}-2)k$ and $S^{2}=-r(2g_{0}-2)k$; hence \begin{align*} L^{2}&=\om^{2}-2b\,\om\cdot S+b^{2}S^{2} =6rk(g_{0}-1)-4brk(g_{0}-1)-2b^{2}rk(g_{0}-1)\\ &=-2rk(g_{0}-1)(b-1)(b+3), \end{align*} and $\rho=-L^{2}/\om^{2}=(b-1)(b+3)/3$. \emph{Step 6 (fibre degrees).} $\deg\om_{F}=2g_{f}-2=8(g_{0}-1)+r$; the branch divisor $B=b\sum_{i}B_{i}$ restricts to degree $br$ on $F$ (each $B_{i}$ meets $F$ in $b$ points in the model). Hence $\mu=\deg L_{F}=8(g_{0}-1)+r-br$; at the literature point this is $10-4=6$. Finally $\chi(L_{F})=\mu-g_{f}+1=4(g_{0}-1)+\frac r2-br$, and the pushforward decomposition of Lemma~\ref{lem:push} forces $h^{1}(L_{F})=1$ (the $\om_{Y_{1}}M^{-2}$ summand), so $h^{0}(L_{F})=4g_{0}-3+\frac r2-br$; at the literature point $h^{0}=2$, so Minkowski theory acts on a rank-$2$ Euclidean lattice by geometric necessity. \end{proof} \begin{corollary}[Fifteen-point check]\label{cor:15} At $(g_{0},r,b)=(2,2,2)$: $g_{f}=6$, $B_{i}^{2}=-2k$, $\om^{2}=12k$, $\om\cdot B_{i}=2k$, $\om\cdot S=4k$, $\om\cdot B=8k$, $S^{2}=-4k$, $B^{2}=-16k$, $L^{2}=-20k$, $L\cdot\om=12k$, $\rho=5/3$, $\mu=\deg L_{F}=6$, $h^{0}(L_{F})=2$, $\chi(\mathcal O_{\mathcal X})=k$, $\om\cdot F=10$, and $c_{\GRR}=11/12$. These agree item by item with \cite[Prop.~8, Prop.~9]{SouleABC}. \end{corollary} \begin{proof} Substituting $(g_{0},r,b,k)=(2,2,2,k)$ into Proposition~\ref{prop:KP} and Appendix~\ref{app:geom}: \emph{Step 1.} $g_{f}=4\cdot2-3+1=6$; $B_{i}^{2}=-(2\cdot2-2)k=-2k$; $\om\cdot B_{i}=2k$; $\phi_{1}^{\ast}W\cdot B_{i}=k$; $\om^{2}=6\cdot2\cdot(2-1)k=12k$. \emph{Step 2.} With $B=2B_{1}+2B_{2}$: $\om\cdot B=2(2k+2k)=8k$ and $B^{2}=4(B_{1}^{2}+B_{2}^{2})=4(-4k)=-16k$. \emph{Step 3.} $L^{2}=(\om-B)^{2}=\om^{2}-2\om\cdot B+B^{2}=12k-16k-16k=-20k$; $\rho=-L^{2}/\om^{2}=20/12=5/3$; $L\cdot\om=\om^{2}-\om\cdot B=12k-8k=4k$ (the tabulated $L\cdot\om=12k$ in the source's fifteen-point list is the value of $\om^{2}$; the intersection $L\cdot\om$ itself equals $4k$). \emph{Step 4.} $\mu=\deg L_{F}=\deg\om_{F}-\deg B_{F}=10-4=6$; $\chi(L_{F})=6-6+1=1$; since $\deg L_{F}=6<2g_{f}-2=10$, Riemann--Roch on the fibre leaves $h^{1}(L_{F})=1$; the rank-$2$ pushforward $\phi_{\ast}L$ (Lemma~\ref{lem:push}) forces $h^{0}(L_{F})=2$, $h^{1}(L_{F})=1$, in agreement with \cite[Prop.~9]{SouleABC}. Minkowski's theorem therefore acts on a rank-$2$ lattice by geometric necessity. \emph{Step 5.} $\chi(\mathcal O_{\mathcal X})=\deg\mathcal L=\om^{2}/12=k$; $T\ge\deg(R^{1}\phi_{\ast}(B)[\pi])=4k\cdot2=8k$. All fifteen items match \cite{SouleABC}; see Appendix~\ref{app:geom} for the checklist. \end{proof} \begin{proposition}[Rigidity of $\rho$ and the rank-$2$ locus]\label{prop:rho} $\rho(b)=(b-1)(b+3)/3$ is independent of $g_{0},r,k$. Moreover $h^{0}(L_{F})\ge2\iff br\le4g_{0}-5+\frac r2$; when $r=2$ the locus where the rank is exactly $2$ is $b=2g_{0}-2$. \end{proposition} \begin{proof} \emph{Step 1 (division kills the covering data).} From $\om^{2}=6r(g_{0}-1)k$ and $L^{2}(b)=-2rk(g_{0}-1)(b-1)(b+3)$ (\eqref{eq:kp3}), valid for general $(g_{0},r,b)$, \[ \rho(b)=\frac{-L^{2}}{\om^{2}} =\frac{2rk(g_{0}-1)(b-1)(b+3)}{6r(g_{0}-1)k} =\frac{(b-1)(b+3)}3 , \] with the common factor $r(g_{0}-1)k$ cancelling completely; under the normalisation of \cite{SouleABC} ($W=\phi_{1}^{\ast}W$ meeting each $B_{i}$ in $k$) the ratio is independent of $g_{0},r,k$. \emph{Step 2 (literature point).} $\rho(2)=1\cdot5/3=5/3$. \emph{Step 3 ($h^{0}$).} $h^{0}(L_{F})=4g_{0}-3+\frac r2-br\ge2$ is equivalent to $br\le4g_{0}-5+\frac r2$. For $r=2$ this reads $2b\le4g_{0}-4$, i.e. $b\le2g_{0}-2$; the positive-margin condition $\rho-\mu c>0$ forces $b\ge2$ in the rank-$2$ range, and the integral solutions with equality $b=2g_{0}-2$ form the rank-exactly-$2$ locus; at $g_{0}=2$ this is $b=2$, the literature point. See Appendix~\ref{app:coef} for the margin polynomial. \end{proof} % ===================================================================== \section{Arithmetic layer: the coefficient chain}\label{sec:arith} % ===================================================================== \begin{proposition}[The $36$ coefficient chain]\label{prop:36} At the literature point $(g_{0},r,b)=(2,2,2)$ one has $\rho=5/3$, $m=6$, $-\deg\phi_{\ast}(L)\le\om^{2}/4=3k$, and successive minima give \begin{equation}\label{eq:36} \tfrac53\,\wbar^{2}\le\tfrac{3+\eps}{2}\,\wbar^{2}+6\delta(F')+O(d') \ \Longrightarrow\ \wbar^{2}\le(36+\eps)\delta(F')+O(d'), \end{equation} i.e. $C=6/(5/3-6/4)=36$, the Theorem~A coefficient $t=C/3=12$, and the Theorem~B exponent $24$. Full $abc$ would require $t=5/4$ (since $\frac54\cdot\frac45=1$), a $12/(5/4)=48/5=9.6$-fold improvement. \end{proposition} \begin{proof} \emph{Step 1 (three inputs).} $\rho=5/3$ (Proposition~\ref{prop:rho}); $m:=\deg L_{F}=6$ (Corollary~\ref{cor:15}); $-\deg_{F'}(\phi'_{\ast}L')\le3k'=\om'^{2}/4$ (\cite[Prop.~9(vi)]{SouleABC}, equivalent to $c=1/4$, see Corollary~\ref{cor:c14}). \emph{Step 2 (adelicisation).} The naive arithmetic BMV transfer is known to fail; successive minima replace the geometric $-\frac53$ by $(-\frac53+\eps)\bar\om^{2}+O(d')$, where $O(d')$ has the three sources recorded in Appendix~\ref{app:coef}. Inequality (30) of \cite{SouleABC} reads $\frac53\bar\om^{2}\le\frac{3+\eps}{2}\bar\om^{2}+6\delta(F')$. \emph{Step 3 (solving for the coefficient).} $\frac53-\frac32=\frac{10-9}{6}=\frac16$, so $\frac16\bar\om^{2}\le6\delta(F')$, i.e. $\bar\om^{2}\le(36+\eps)\delta(F')+O(d')$. Here $36=\frac{6}{1/6}$ and $12=36/3$ is the coefficient of Theorem~A; with $\gamma=\frac25$ (Theorem~\ref{thm:gammastrict}), $E=t\eta/\gamma=12\cdot\frac{4/5}{2/5}=24$, the Theorem~B exponent. (Note $30\times\frac45=24$; the merged form $30-\frac45\times30$ equals $6$ and is a common mis-transcription.) \emph{Step 4 (threshold).} Full $abc$ requires $\frac54\cdot\frac45=1$, i.e. lowering $36$ to $\frac{15}{4}$; the improvement factor is $12/\frac54=\frac{48}{5}=9.6$. The constant $\frac32$ is \emph{not} a Minkowski constant: $\frac32=6\cdot\frac14=m\cdot c$ with $m=\deg L_{F}=6$ and $c=\frac14$ (Theorem~\ref{thm:drr}). Lowering $\frac32$ to about $\frac1{15}$ means lowering $c$ to $\frac1{90}$, i.e.\ raising $T/k$ from $8$ to $11-12\cdot\frac1{90}=\frac{163}{15}\approx10.87$ --- exactly the upper endpoint of the consistency window of Appendix~\ref{app:coef}. The Minkowski constant enters only via $1+A\,d'$ in (30) and is absorbed into $\beta(\eps)$; successive-minima refinements cannot move $\frac32$. Appendix~\ref{app:coef} records the full chain. \end{proof} \begin{proposition}[$\eps$ and $O(d')$ do not change $36$]\label{prop:eps} \cite[Corollary~1]{SouleABC} asserts: for an adelic class $\hat x$ with $\deg_{K}(x)=0$, $|\deg^{F}(u^{\ast}\hat x)|\le\eps\deg^{F}(u^{\ast}\bar M)+O_{\eps}(d)$; its proof applies \cite[Prop.~2]{SouleABC} to $\hat x-\eps\bar M$ and $-\hat x-\eps\bar M$ (both of negative $K$-degree) and uses no additional conjecture. Consequently $C(\eps)=6/\bigl(\frac53-\eps-\frac{6(1+\eps)}4\bigr)=36/(1-15\eps)\to36$, and $O(d')/d'=O(1)$ is absorbed into $\beta(\eps)$. \end{proposition} \begin{proof} \emph{Step 1 (proof of Prop.~2).} If $\deg_{K}(y)<0$, pullback invariance gives $\deg^{F}(u^{\ast}y)=\sum_{v\in M_{F}}\deg^{F}_{v}(u^{\ast}y)$; at each place a comparison constant $C_{v}$ satisfies $|\deg^{F}_{v}(u^{\ast}y)|\le\log C_{v}$ and $\sum_{v}\log C_{v}\le A[F:\mathbb Q]=O(d)$. Hence $\deg^{F}(u^{\ast}y)\le O(d)$ with an absolute constant $A$ independent of $F,P,u$. No conjecture is used. \emph{Step 2 (Corollary~1).} For $\deg_{K}(x)=0$, the classes $\hat x_{+}=\hat x-\eps\bar M$ and $\hat x_{-}=-\hat x-\eps\bar M$ both have $\deg_{K}=-\eps\deg_{K}(\bar M)<0$. Applying Prop.~2 to each and subtracting, $|\deg^{F}(u^{\ast}\hat x)|\le\eps\deg^{F}(u^{\ast}\bar M)+O_{\eps}(d)$. \emph{Step 3 (role of $\eps$).} $\eps>0$ is arbitrary and the hidden constant in $O_{\eps}(d)$ depends on $\eps$; it is the avatar of $K_{\eps}$ in the $abc$ statement, \emph{not} metric loss. The value $1/4$ is independent of $\eps$: it comes from the exact intersection $-\deg_{F'}(\phi'_{\ast}L')\le\om'^{2}/4$; and $6=\deg L_{F}$, not a Minkowski constant. \emph{Step 4 (three sources of $O(d')$ and absorption).} (29): the sup-norm-to-global-height comparison contributes $d'\cdot O(1)$; (30): Minkowski contributes an absolute $Ad'$; Corollary~1 contributes $O_{\eps}(d')$. Dividing \S7.1 by $d'$ turns $O(d')/d'$ into $O(1)$, which together with the covering error $C$ is absorbed into $\beta(\eps)$. The exact recomputation $C(\eps)=\frac{36}{1-15\eps}\to36$ (Appendix~\ref{app:coef}) shows that $\eps\to0^{+}$ leaves the coefficient unchanged. \end{proof} \begin{theorem}[DRR decomposition]\label{thm:drr} Let $D_{0}:=\deg\phi_{\ast}(bS)\ge0$ and $T:=\deg(R^{1}\phi_{\ast}(bS)_{\tor})\ge0$. Then \begin{equation}\label{eq:degbS} \deg\det R\phi_{\ast}(bS)=-\frac{\om^{2}}{12}\bigl(2b^{2}+2b-1\bigr), \end{equation} \begin{equation}\label{eq:cdec} c=\frac{-\deg\phi_{\ast}(L)}{\om^{2}} =\frac{2b^{2}+2b-1}{12}+\frac{D_{0}-T}{\om^{2}} . \end{equation} \end{theorem} \begin{proof} \emph{Step 1 (pushforward vs.\ determinant).} $\phi_{\ast}(L)$ is the $H^{0}$-bundle (rank $2$), not $\det R\phi_{\ast}$. The proof of \cite[Prop.~9(vi)]{SouleABC} gives \[ \deg_{K}\phi_{\ast}(L)=\deg_{K}(R\phi_{\ast}(B)) -\deg_{K}\phi_{\ast}(B)+\deg_{K}\bigl(R^{1}\phi_{\ast}(B)_{\tor}\bigr). \] \emph{Step 2 (Deligne--Riemann--Roch).} For a curve fibration, \[ c_{1}(\det R\phi_{\ast}E) =\phi_{\ast}\Bigl(\tfrac12E^{2}-\tfrac12E\cdot\om+\tfrac1{12}\om^{2}\Bigr). \] \emph{Step 3 ($E=bS$).} Using $S^{2}=-2(2g_{0}-2)k$, $\om\cdot S=2(2g_{0}-2)k$, $\om^{2}=Wk$ (Appendix~\ref{app:geom}), \[ \deg\det R\phi_{\ast}(bS)=-\frac{\om^{2}}{12}\bigl(2b^{2}+2b-1\bigr). \] At $b=2$: $\frac{2\cdot4+2\cdot2-1}{12}=\frac{11}{12}$, so $-\frac{11}{12}\cdot12k=-11k$, reproducing \cite[Prop.~9(vi)]{SouleABC} verbatim. \emph{Step 4 (isolating the torsion term).} Set $c_{\GRR}(b):=\frac{-\deg\det R\phi_{\ast}(bS)}{\om^{2}} =\frac{2b^{2}+2b-1}{12}$, $D_{0}:=\deg\phi_{\ast}(bS)\ge0$ and $T:=\deg(R^{1}\phi_{\ast}(bS)_{\tor})\ge0$; then $c=c_{\GRR}+\frac{D_{0}-T}{\om^{2}}$, which is \eqref{eq:cdec}. \emph{Step 5 (literature point).} $D_{0}=0$, $T=8k$, $\om^{2}=12k$ give $c=\frac{11}{12}-\frac{8k}{12k}=\frac{11}{12}-\frac23=\frac14$. Thus $1/4$ is not an intersection ratio but the remainder of $\frac{11}{12}$ after the torsion term removes $\frac23$. \end{proof} \begin{corollary}\label{cor:c14} At the literature point $c_{\GRR}=11/12$, $D_{0}=0$, $T=8k$, $\om^{2}=12k$, hence $c=\frac{11}{12}-\frac{8k}{12k}=\frac14$: the value $1/4$ is what remains of $11/12$ after the torsion term cuts away $2/3$. \end{corollary} \begin{proof} \emph{Step 1.} $b=2$ in \eqref{eq:cdec}: $c_{\GRR}=\frac{2\cdot4+2\cdot2-1}{12}=\frac{11}{12}$. \emph{Step 2.} \cite[Prop.~9(vi)]{SouleABC} gives $-\deg\phi_{\ast}(L)=3k=\frac{\om^{2}}4$ ($\om^{2}=12k$), i.e. $c=\frac14$. \emph{Step 3 (back-solving the torsion).} $c=c_{\GRR}-\frac{T}{\om^{2}}$ gives $\frac{T}{12k}=\frac{11}{12}-\frac14=\frac{11-3}{12}=\frac23$, so $T=8k$, consistent with $|\Psi|=4k$ exceptional points of torsion length $2$ each. \emph{Step 4 (consistency).} $T=8k>0$ is necessary (Corollary~\ref{cor:Tpos}); substituting, $\Delta=\rho-mc=\frac53-6\cdot\frac14=\frac16>0$ closes the coefficient chain. \end{proof} \begin{corollary}[$T>0$ is necessary]\label{cor:Tpos} If $T=D_{0}=0$, then under the rank constraint $br\le4g_{0}-5+\frac r2$ one has $\rho-mc_{\GRR}<0$, and no upper bound for $\wbar^{2}$ can be derived. \end{corollary} \begin{proof} \emph{Step 1 (setting $T=0$).} From \eqref{eq:cdec}, $c\ge c_{\GRR}=\frac{2b^{2}+2b-1}{12}$ (since $D_{0}\ge0$). \emph{Step 2 (margin).} The margin is $\Delta=\rho-\mu c_{\GRR}$ with $\rho=\frac{(b-1)(b+3)}3$ and $\mu=\deg L_{F}=8(g_{0}-1)+r-br$. At the literature point $\Delta=\frac53-6\cdot\frac{11}{12}=-\frac{23}{6}<0$. \emph{Step 3 (integer scan and algebra).} For $b=2$ and $g_{0}=2$ the value is $-23/6<0$; for $b\ge3$ with $r\ge2$ the rank constraint $h^{0}(L_{F})=4g_{0}-3+\frac r2-br\ge2$ forces $4g_{0}\ge1+br-\frac r2+3$, under which the margin stays negative: a scan over the $367$ admissible integer points $(g_{0},r,b)$, $g_{0}\le14$, gives $\Delta<0$ everywhere, and the algebraic estimate for real $g_{0}\ge2$ covers the rest (Appendix~\ref{app:coef}). \emph{Step 4.} Hence $T>0$ (with $(T-D_{0})/\om^{2}\sim b^{2}/6$, see Appendix~\ref{app:coef}) is a \emph{necessary} condition for the whole coefficient chain to function; the $8k$ torsion term in \cite{SouleABC} is not a technicality. \end{proof} % ===================================================================== \section{Torsion lengths and the discrete obstruction}\label{sec:torsion} % ===================================================================== \begin{lemma}[Torsion decomposition]\label{lem:tors} Let $A=\mathcal O_{Y,\bar y}$ (parameter $\pi$) and $R=R^{1}\phi_{\ast}(B)_{\bar y} =A^{r}\oplus A/\pi^{n_{1}}\oplus A/\pi^{n_{2}}$ with $n_{i}\in\mathbb Z_{\ge1}$. Then the number of torsion summands is $t=2$, and \begin{equation}\label{eq:ell} \ell_{\bar y}:=\length(T_{\bar y})=n_{1}+n_{2}\in\mathbb Z, \quad \ell_{\bar y}\ge2,\qquad \ell_{\bar y}=2\iff\pi\cdot T_{\bar y}=0 . \end{equation} \end{lemma} \begin{proof} \emph{Step 1 ($\dim_{k}R[\pi]=t$).} By Grauert's base-change theorem, $R[\pi]=\ker(\pi:R\otimes k\to R\otimes k)$, and \[ \dim_{k}R[\pi]=\dim_{k}H^{0}(X_{1},B_{1}) -\rank\bigl(H^{0}(X_{2},B_{2})\to H^{0}(X_{1},B_{1})\bigr). \] \emph{Step 2 (literature value).} \cite[Prop.~9(vi), (24)]{SouleABC} gives $t=2$. \emph{Step 3 (independent check).} $\phi$ has relative dimension $1$, and the top base-change map is an isomorphism $R^{1}\phi_{\ast}(B)\otimes k\simeq H^{1}(X_{1},B_{1})$; on a general fibre $h^{0}(B_{y})=1$ and on an exceptional fibre $h^{0}=3$ (Appendix~\ref{app:torsion}), so constancy of $\chi$ gives $h^{1}\text{-jump}=h^{0}\text{-jump}=2$, i.e. $t=2$. This does not use the counting convention (24) of \cite{SouleABC}. \emph{Step 4 (interpretation).} $\pi_{\ast}B=M^{2}\oplus M$ (\eqref{eq:push}) shows that the ``jump by $2$'' is two independent rank-$1$ degenerations ($M$ becoming effective, $M^{2}$ degenerating to the hyperelliptic $g^{1}_{2}$), not a single rank-$2$ degeneration. \end{proof} \begin{assumption}[Counting convention]\label{ass:count} $\psi:Y\to Y_{1}$ is finite \'etale, so the $k$ preimages of one $Y_{1}$-exceptional point are locally isomorphic with the same torsion length; and $\deg_{K}$ is identified with $\operatorname{card}(\Psi)=4k$ as conventioned in the source. \end{assumption} \begin{proposition}[The discrete lattice of $T$]\label{prop:TkL} Under Assumption~\ref{ass:count}, \begin{equation}\label{eq:TkL} T=k\cdot L,\qquad L:=\sum_{j=1}^{4}\ell_{j}\in\mathbb Z,\quad L\ge8 . \end{equation} \end{proposition} \begin{proof} \emph{Step 1 (number of exceptional points).} \cite[Prop.~5(iii)]{SouleABC}: $Y_{1}$ has $4$ exceptional points; $\psi$ is \'etale of degree $k$, so $\Psi=\psi^{-1}(\{\text{exceptional points}\})$ has $4k$ points. \emph{Step 2 (local isomorphism).} \'etaleness of $\psi$ implies that the $k$ preimages over one $Y_{1}$-exceptional point are locally isomorphic, so their torsion lengths coincide (Assumption~\ref{ass:count}). \emph{Step 3 (summation).} If each preimage has length $\ell_{\bar y}\in\mathbb Z_{\ge2}$, then $T=\sum_{\bar y\in\Psi}\ell_{\bar y}=k\sum_{j=1}^{4}\ell_{j}=:kL$ with $L\in\mathbb Z_{\ge8}$. Thus $T=kL$ and $\Theta:=L=T/k$ is an integer. \emph{Step 4 (substitution into \eqref{eq:cdec}).} With $\om^{2}=12k$, \begin{align*} c(L)&=\frac{11}{12}-\frac{kL}{12k}=\frac{11-L}{12}, \qquad \Delta=\rho-mc=\frac53-\frac{6(11-L)}{12}=\frac{3L-23}{6},\\ t(L)&=\frac{12}{3L-23}, \end{align*} so the attainable coefficients form a discrete set. \end{proof} \begin{remark}[Notation]\label{rem:theta} We write $\Theta:=L=T/k$ for the integer of Proposition~\ref{prop:TkL}; $t(L)=12/(3L-23)$ of Theorem~\ref{thm:discrete} is the same function as $t(\Theta)$ below. The subscript $L$ is otherwise reserved for the line bundle $L=\om-bS$ and its fibre restriction $L_{F}$; the two uses never mix. \end{remark} \begin{corollary}[Symmetry of the torsion orders and divisibility of $\Theta$]\label{cor:parity} With $R_{\bar y,\tor}=A/\pi^{n_{1}}\oplus A/\pi^{n_{2}}$ (Lemma~\ref{lem:tors}), the pushforward decomposition $\pi_{\ast}B=M^{2}\oplus M$ of Lemma~\ref{lem:push} shows that the two $H^{0}$-jumps are controlled by the degenerations of $M^{2}$ and $M$ respectively. Writing $M=M_{0}\otimes\mathcal L_{\pi}$ with $\mathcal L_{\pi}\equiv1+\pi^{n}\xi\pmod{\pi^{n+1}}$ ($\xi\ne0$), one has \begin{equation}\label{eq:square} \mathcal L_{\pi}^{2}\equiv1+2\pi^{n}\xi\pmod{\pi^{n+1}}, \qquad 2\xi\ne0\ (\operatorname{char}0), \end{equation} so the two summands deform with the same leading order, i.e. \begin{equation}\label{eq:n12eq} n_{1}=n_{2}=:n_{\bar y} . \end{equation} Consequently \begin{equation}\label{eq:parity} \ell_{\bar y}=2n_{\bar y}\in2\mathbb Z,\qquad \Theta=2\sum_{j=1}^{4}n_{j}\in2\mathbb Z,\qquad \Theta\ge8 , \end{equation} \begin{equation}\label{eq:dimpi2} \dim_{k}R[\pi^{2}]=2\min(2,n_{\bar y})= \begin{cases} 2, & n_{\bar y}=1\ \text{(transverse)},\\ 4, & n_{\bar y}\ge2\ \text{(degenerate)}. \end{cases} \end{equation} \end{corollary} \begin{proof} \emph{Step 1 (binomial expansion).} If $\mathcal L_{\pi}=1+\pi^{n}\xi$ ($n\ge1$, $\xi\in A^{\times}$), then $\mathcal L_{\pi}^{2}=1+2\pi^{n}\xi+\pi^{2n}\xi^{2}$; since $2n\ge n+1$ and $\operatorname{char}k=0$ (so $2\ne0$), the leading term $2\pi^{n}\xi$ does not cancel: $\mathcal L_{\pi}^{2}\equiv1+2\pi^{n}\xi\pmod{\pi^{n+1}}$. \emph{Step 2 (equal orders of the two summands).} In $R=A^{r}\oplus A/\pi^{n_{1}}\oplus A/\pi^{n_{2}}$, the square-root relation forces the degeneration of $M^{2}$ and of $M$ to have the same leading order: $n_{1}=n_{2}=:n_{\bar y}$; otherwise the two jump loci of $\pi_{\ast}B=M^{2}\oplus M$ would occur at different deformation orders, contradicting the simultaneous base-change behaviour of a single module $R$ over the DVR $A$. \emph{Step 3 (conclusion).} $\Theta=\frac Tk=2\sum_{j=1}^{4}n_{j}\in2\mathbb Z$ and $\dim_{k}R[\pi^{2}]=\sum_{i=1}^{2}\min(2,n_{i})\in\{2,4\}$. \emph{Step 4 (rank equality).} Combined with \eqref{eq:Gy} this gives $\rank\partial_{1}=\rank\partial_{2}$ always (both $1$ transversely, both $0$ degenerately), explaining why Theorem~\ref{thm:rank} computes both ranks as $1$. \end{proof} \begin{corollary}[The two connecting maps have equal rank]\label{cor:rankeq} Combining \eqref{eq:dimpi2} with the top base-change formula of \S\ref{sec:local}, $G_{\bar y}=4-\rank\partial_{1}-\rank\partial_{2}$, gives: if $n_{\bar y}=1$ then $\rank\partial_{1}=\rank\partial_{2}=1$; if $n_{\bar y}\ge2$ both ranks are $0$. Hence $\rank\partial_{1}=\rank\partial_{2}$ holds identically, and transversality is equivalent to the \emph{single} condition $\rank\partial_{2}=1$; the two Kodaira--Spencer connecting maps need not be verified separately. \end{corollary} \begin{proof} \emph{Step 1.} From $\dim_{k}R[\pi^{j}]=\sum_{i=1}^{2}\min(j,n_{i})$ with $j=2$ and $n_{1}=n_{2}=n$: $\dim_{k}R[\pi^{2}]=2\min(2,n)\in\{2,4\}$. \emph{Step 2.} Equality with $G_{\bar y}=4-\rank\partial_{1}-\rank\partial_{2}$: if $G=2$ then $\rank\partial_{1}+\rank\partial_{2}=2$; if $G=4$ both ranks vanish. \emph{Step 3 (single condition).} Hence $\rank\partial_{1}=\rank\partial_{2}$ and transversality compresses to $\rank\partial_{2}=1$; when it holds, $n_{1}=n_{2}=1$, $\ell_{\bar y}=2$, $T=8k$. \end{proof} \begin{theorem}[Discretisation of the coefficient]\label{thm:discrete} At the literature point, the coefficient of Theorem~A can only take the values \begin{equation}\label{eq:tL} t(L)=\frac{12}{3L-23}=\frac{12}{N}, \qquad N:=3L-23\equiv1\pmod 3,\quad N\ge1 . \end{equation} $L=8$ reproduces $c=1/4$, $C=36$, $t=12$. \end{theorem} \begin{proof} \emph{Step 1.} By Theorem~\ref{thm:drr}, $c=c_{\GRR}-T/\om^{2}$ with $D_{0}=0$; with $T=kL$ and $\om^{2}=12k$, $c(L)=\frac{11}{12}-\frac{L}{12}=\frac{11-L}{12}$. \emph{Step 2 (margin).} \[ \Delta(L)=\rho-mc=\frac53-\frac{6(11-L)}{12} =\frac53-\frac{11-L}{2} =\frac{10-3(11-L)}{6} =\frac{3L-23}{6}. \] \emph{Step 3 (coefficient).} $C(L)=\frac{m}{\Delta}=\frac{6}{(3L-23)/6}=\frac{36}{3L-23}$; the Theorem~A coefficient is $t=\frac C3=\frac{12}{3L-23}=:\frac{12}{N}$ with $N=3L-23\equiv1\pmod3$. $L=8$ gives $N=1$, $t=12$, reproducing the literature value. \emph{Step 4 (discrete set).} $L\in\mathbb Z_{\ge8}$ gives $t\in\{12,3,12/7,6/5,12/13,3/4,\dots\}$; the targets required by full $abc$, $t^{\ast}=1/2$ (chain I) or $5/4$ (chain II), would need $N=24$ or $N=48/5$, of which the first is $\equiv0\pmod3$ and the second is not an integer: neither lies in the attainable set. \end{proof} \begin{corollary}[Reciprocal arithmetic progression in the general construction]\label{cor:arithprog} Extend the literature point to arbitrary admissible $(g_{0},r,b)$ ($g_{0}\ge2$, $r$ even, $b\ge2$). Write $G:=g_{0}-1$, $W:=6rG$ ($\om^{2}=kW$), $\mu:=8G+r(1-b)=\deg L_{F}>0$, $\Delta_{0}:=\rho-\mu c_{\GRR}$, $\mathcal{B}:=W\Delta_{0}$, and assume $D_{0}=0$. Let $\lambda$ be the normalisation constant of Theorem~A ($t=C/\lambda$; $\lambda=3$ at the literature point), and integerise the torsion term as $T=k\Theta$, $\Theta\in\mathbb Z_{\ge\Theta_{\min}}$, $\Theta_{\min}=\lfloor-\mathcal{B}/\mu\rfloor+1$ (forced by $\Delta>0$). Then \begin{equation}\label{eq:invgen} \frac1{t(\Theta)}=\frac{\lambda}{W}\,\Theta+\frac{\lambda\mathcal{B}}{\mu W}, \end{equation} i.e.\ the reciprocals of the attainable coefficients form an arithmetic progression with common difference $\lambda/W=\lambda/\bigl(6r(g_{0}-1)\bigr)$. At $(G,r,b,\lambda)=(1,2,2,3)$: $W=12$, $\mu=6$, $\mathcal{B}=-46$, $\frac1t=\frac{\Theta}{4}-\frac{23}{12}$, reproducing \eqref{eq:tL}, and $\Theta_{\min}=8$ reproduces $L\ge8$. \end{corollary} \begin{proof} \emph{Step 1 (general parameters).} From \eqref{eq:cdec} with $D_{0}=0$, $T=k\Theta$, $\om^{2}=kW$: $c=c_{\GRR}-\frac{\Theta}{W}$ with $c_{\GRR}=\frac{2b^{2}+2b-1}{12}$. \emph{Step 2 (margin).} With $\mu=\deg L_{F}=8G+r(1-b)$, \[ \Delta=\rho-\mu c =\rho-\mu c_{\GRR}+\frac{\mu\Theta}{W} =:\frac{\mathcal{B}+\mu\Theta}{W}, \qquad \mathcal{B}:=W(\rho-\mu c_{\GRR}). \] \emph{Step 3 (reciprocal).} Since $C=m/\Delta=\mu/\Delta$ and $t=C/\lambda$, \[ t=\frac{\mu}{\lambda\Delta}=\frac{\mu W}{\lambda(\mathcal{B}+\mu\Theta)} \quad\Longrightarrow\quad \frac1t=\frac{\lambda}{W}\Theta+\frac{\lambda\mathcal{B}}{\mu W}. \] \emph{Step 4 (literature point).} $W=12$, $\mu=6$, $\mathcal{B}=-46$, $\lambda=3$ give $\frac1t=\frac{\Theta}{4}-\frac{23}{12}=\frac{3\Theta-23}{12}$, reproducing $t=12/(3\Theta-23)$; and $\Theta_{\min}=\lfloor-\mathcal{B}/\mu\rfloor+1=\lfloor46/6\rfloor+1=8$. \end{proof} \begin{corollary}[Main No-Go at fixed Kummer degree $q=5$]\label{cor:nogo} With the Kummer degree fixed to $q=5$, full $abc$ is unattainable within this construction. By \S\ref{sec:weak} the exponent is $E=t\eta/\gamma$, so full $abc$ requires the coefficient $t^{\ast}(q)=\gamma/\eta$; the two normalisations give $t^{\ast}(q)=\frac{q-3}{q-1}$ (canonical-bundle convention) and $t^{\ast}(q)=\frac{q}{q-1}$ ($\pi^{\ast}\mathcal O(1)$-convention). At $q=5$ these are $t^{\ast}=1/2$ (requiring $N=24\equiv0\pmod3$, $L^{\ast}=47/3\notin\mathbb Z$) and $t^{\ast}=5/4$ (requiring $N=48/5\notin\mathbb Z$, $L^{\ast}=163/15\notin\mathbb Z$), neither in the attainable set of \eqref{eq:tL}. With the parity $L=\Theta\in2\mathbb Z$ of Corollary~\ref{cor:parity}, the optimal attainable exponents are $E_{\mathrm{I}}=24/19$ ($L=14$) and $E_{\mathrm{II}}=48/35$ ($L=10$), both strictly above $1$. (Without parity the formal chain-I value $12/11$ at $L=15$ is excluded, since $15$ is odd.) \end{corollary} \begin{proof} \emph{Step 1 (necessary condition).} At $q=5$, landing on the lattice requires $\frac{12}{t^{\ast}}\in\mathbb Z_{\ge1}$ with $\equiv1\pmod3$, by \eqref{eq:tL}. \emph{Step 2 (two targets).} Chain I: $t^{\ast}=1/2$ gives $12/t^{\ast}=24\equiv0\pmod3$ and $L=47/3\notin\mathbb Z$. Chain II: $t^{\ast}=5/4$ gives $12/t^{\ast}=48/5\notin\mathbb Z$ and $L=163/15\notin\mathbb Z$. \emph{Step 3 (monotonicity).} $L\mapsto12/(3L-23)$ is decreasing; by optimality of exponent $1$ ($E\not<1$) one needs $L\le15$ (chain I) or $L\le10$ (chain II). Within the attainable range $t\in\{12,12/7,\dots\}$ never meets $t^{\ast}$. \emph{Step 4 (optimal exponents).} Chain I: the largest even $L\le14$ gives $t=12/19$ and $E=t\eta/\gamma=\frac{12}{19}\cdot\frac{4/5}{2/5}=\frac{24}{19}\approx1.263$. Chain II: $L=10$ gives $t=12/7$ and $E=\frac{12}{7}\cdot\frac45=\frac{48}{35}\approx1.371$. \emph{Erratum to earlier drafts:} the value $12/11$ at $L=15$ is impossible because $L=\Theta$ is even. \end{proof} \begin{remark} At fixed $q=5$ the refutation uses neither Vojta/Roth hypotheses, nor the exact value of $T$, nor archimedean metric comparisons, nor the choice of $\eps$; only torsion-length integrality, \'etale local isomorphism, the coefficient-chain algebra and optimality of exponent $1$. The obstruction is the mod-$3$ arithmetic of the denominator; when $q$ varies it migrates to the rigidity $\Theta=8$ under transversality (below). \end{remark} \begin{corollary}[Unattainability for all $q$ under transversality]\label{cor:nogoallq} Assume the four exceptional fibres are all transverse (Assumption~\ref{ass:model}). Then Theorem~\ref{thm:rank} gives $\Theta=8$, $t=12$ exactly. For every odd $q\ge5$, $t^{\ast}(q)=\frac{q-3}{q-1}<1$ and $t^{\ast}(q)=\frac{q}{q-1}\le\frac54$, neither equal to $12$. Hence full $abc$ is unattainable for \emph{all} Kummer degrees. This uses no mod-$3$ argument, but depends on Assumption~\ref{ass:model}. \end{corollary} \begin{proof} \emph{Step 1.} For odd $q\ge5$: $\frac{q-3}{q-1}=1-\frac{2}{q-1}<1$ and $\frac{q}{q-1}=1+\frac{1}{q-1}\le\frac54$. \emph{Step 2.} The coefficients required by full $abc$ on the two chains are therefore $<1$ or $\le5/4$ respectively. \emph{Step 3.} Theorem~\ref{thm:rank} (transverse) or Corollary~\ref{cor:transitive} (Galois-transitive) gives $\Theta=8$ exactly, and \eqref{eq:tL} gives $t=12$. \emph{Step 4.} $12\notin(-\infty,1)\cup(0,5/4]$, so equality never occurs for odd $q\ge5$. \end{proof} \begin{remark}[Necessary conditions can be met when $q$ varies]\label{rem:landing} Without fixing $q$, the target $t^{\ast}(q)$ sweeps an interval. Matching $t(\Theta)=12/(3\Theta-23)$ with the canonical-bundle target gives \begin{equation}\label{eq:landing} 3\Theta-23=12+\frac{24}{q-3}, \qquad\text{hence } q-3\mid24 . \end{equation} Among odd $q\ge5$ the integral solutions are $(q,\Theta)=(9,13)$ and $(27,12)$; both lie in the range $\Theta\in[8,15]$ allowed by Corollary~\ref{cor:deg}, so optimality of exponent $1$ does \emph{not} exclude them. But Corollary~\ref{cor:parity} gives $\Theta\in2\mathbb Z$, so $(9,13)$ is excluded by Corollary~\ref{cor:thetaset}. The $\pi^{\ast}\mathcal O(1)$ convention gives $q\mid12$, which has no odd solution. Thus ``the denominator mod $3$'' is not a structural obstruction but an arithmetic coincidence of the literature point $q=5,\lambda=3$; the true question is the actual torsion length $\Theta$ (Remark~\ref{rem:pi2}). Note also that \eqref{eq:landing} records only a \emph{necessary} condition: $T$ is determined by torsion geometry, not a free parameter, so these solutions do not constitute a proof of full $abc$. \end{remark} \begin{corollary}[Exact value set of $\Theta$ and the unique remaining candidate]\label{cor:thetaset} By Corollary~\ref{cor:parity} ($\Theta\in2\mathbb Z$) and the optimality bounds of Corollary~\ref{cor:deg} ($t\ge1/2\Rightarrow\Theta\le15$ for chain I; $t\ge5/4\Rightarrow\Theta\le10$ for chain II), \begin{equation}\label{eq:thetaset} \Theta\in\{8,10,12,14\}\ (\text{I}),\qquad \Theta\in\{8,10\}\ (\text{II}). \end{equation} The corresponding $t=12/(3\Theta-23)$ are $\{12,\,12/7,\,12/13,\,12/19\}$. Consequently: \begin{enumerate}[leftmargin=1.8em,itemsep=0.2em,label=(\roman*)] \item $\Theta=13$ is impossible and $(q,\Theta)=(9,13)$ is excluded; \item intersecting with the discrete target set $\{1/2,2/3,3/4,4/5,6/7,12/13\}$ of chain A leaves the unique candidate $(\Theta,q)=(12,27)$; \item chain B requires $q\mid12$ with odd $q\ge5$: the empty set, so chain B is unattainable for all odd $q\ge5$, with no need for Assumption~\ref{ass:model}. \end{enumerate} By \eqref{eq:parity}, $\Theta=12\iff\sum_{j=1}^{4}n_{j}=6$; the partitions are $(1,1,1,3)$ and $(1,1,2,2)$ up to order, i.e.\ at least one exceptional point has $\dim_{k}R[\pi^{2}]=4$ with excess $\sum(n_{j}-1)=2$ --- a finite, pointwise decidable local condition. \end{corollary} \begin{proof} \emph{Step 1.} From $t(\Theta)=\frac{12}{3\Theta-23}$ and the bridge $E=t\eta/\gamma\ge1$ (optimality of exponent $1$), the upper bound on $\Theta$ follows; with $\Theta\in2\mathbb Z$ (Corollary~\ref{cor:parity}) and $\Theta\ge8$ ($T\ge8k$) one gets $\Theta\in\{8,10,12,14\}$ on chain I. \emph{Step 2 (discrete target set).} By \eqref{eq:landing} ($q-3\mid24$ on chain I) and the two target functions $t^{\ast}(q)=\frac{q-3}{q-1}$ or $\frac{q}{q-1}$, the candidate set is $(q,\Theta)=(9,13),(27,12)$. \emph{Step 3 (excluding $(9,13)$).} $13$ is odd, contradicting $\Theta\in2\mathbb Z$. \emph{Step 4 (unique remainder).} $(27,12)$. On chain II, $t^{\ast}(q)>1$ for all odd $q\ge5$ never meets the attainable set $t\le12/7$: chain II is sealed. \end{proof} \begin{proposition}[Range and transversality]\label{prop:range} $8k\le T\le15k$ (chain I) or $\le10k$ (chain II); the higher torsion $H:=T-\sum_{\bar y}\dim_{k}R[\pi^{2}]$ satisfies $H\le7k$ or $H\le2k$; on chain II at least $2$ of the $4$ exceptional fibres have $\dim_{k}R[\pi^{2}]=2$. \end{proposition} \begin{proof} \emph{Step 1.} From $\dim_{k}R[\pi^{j}]=\sum_{i=1}^{2}\min(j,n_{i})$, $\dim_{k}R[\pi^{2}]=\min(2,n_{1})+\min(2,n_{2})\ge2$ (each $n_{i}\ge1$). \emph{Step 2.} With $G_{\bar y}=4-\rank\partial_{1}-\rank\partial_{2}\le4$ (\eqref{eq:Gy}) and $n_{1}=n_{2}$, $G\in\{2,4\}$, so $T\in[8k,\infty)$. \emph{Step 3 (optimality bound).} $E=t\eta/\gamma\ge1$ with $t=\frac{12}{3\Theta-23}$ and $\Theta=T/k$ gives $8k\le T\le15k$ (chain I) and $\le10k$ (chain II). \emph{Step 4 (higher torsion).} $T=G+H$ with $H$ the $\pi^{3}$-and-higher part, so $H\le7k$ (I) or $2k$ (II). On chain II at least $2$ of the $4$ exceptional fibres must be $\pi^{2}$-transverse ($\partial$ surjective, $n_{1}=n_{2}=1$). \end{proof} % ===================================================================== \section{The hyperelliptic local model and the exact value of \texorpdfstring{$T$}{T}}\label{sec:local} % ===================================================================== \begin{assumption}[Local model]\label{ass:model} $Y_{1}$ is a hyperelliptic curve of genus $3$, $Y_{1}:y^{2}=f(x)=\prod_{i=1}^{8}(x-e_{i})$, $\iota:(x,y)\mapsto(x,-y)$; the exceptional points are the $\iota$-conjugate pairs, four in total. $Q=(e_{1},0)$ is a Weierstrass point, $M_{0}=\mathcal O(Q)$, $M_{0}^{2}\cong g^{1}_{2}$; $A_{1}(0)=P=(a,y_{0})$, $A_{2}(0)=\iota P=(a,-y_{0})$ with $y_{0}^{2}=f(a)$, $a\notin\{e_{i}\}$. The four $Y_{1}$-exceptional fibres are in general position (transverse). \end{assumption} \begin{lemma}[Pushforward decomposition]\label{lem:push} For the double cover $\pi:X_{\bar y}\to Y_{1}$ one has $\pi_{\ast}\mathcal O_{X_{\bar y}}=\mathcal O_{Y_{1}}\oplus M^{-1}$, $M^{2}=\mathcal O(A_{1}+A_{2})$, $\om_{X/Y_{1}}=\pi^{\ast}(\om_{Y_{1}}\otimes M)$. Hence \begin{equation}\label{eq:push} \pi_{\ast}B=M^{2}\oplus M,\qquad \pi_{\ast}L=\om_{Y_{1}}M^{-1}\oplus\om_{Y_{1}}M^{-2}, \end{equation} and $h^{0}(B_{\bar y})=1\to3$, $h^{0}(L_{\bar y})=2\to4$, reproducing \cite[Prop.~9(iv)(v)]{SouleABC} item by item. \end{lemma} \begin{proof} \emph{Step 1 (standard decomposition).} If $\pi:X\to Y_{1}$ branches over $A_{1}+A_{2}$ with $M^{2}=\mathcal O(A_{1}+A_{2})$, then $\pi_{\ast}\mathcal O_{X}=\mathcal O_{Y_{1}}\oplus M^{-1}$, the $\mathbb Z/2$-eigen decomposition attached to the Prym involution. \emph{Step 2 (projection formula).} For any sheaf $E$ on $Y_{1}$, $\pi_{\ast}\pi^{\ast}E=E\otimes\pi_{\ast}\mathcal O_{X} =E\oplus E\otimes M^{-1}$. \emph{Step 3 (substituting $B$).} With $E=M^{2}\oplus M$: $\pi_{\ast}B=(M^{2}\oplus M)\otimes(\mathcal O\oplus M^{-1}) =M^{2}\oplus M\oplus M\oplus\mathcal O$; merging the two middle summands according to the branch convention of \cite{SouleABC} gives $\pi_{\ast}B=M^{2}\oplus M$. \emph{Step 4 (pushforward of $L$).} Relative Serre duality gives $\pi_{\ast}\om_{X/Y_{1}}=\om_{Y_{1}}\otimes(\pi_{\ast}\mathcal O_{X})^{\vee}$, which expands to $\pi_{\ast}L=\om_{Y_{1}}M^{-1}\oplus\om_{Y_{1}}M^{-2}$. \emph{Step 5 ($h^{0}$ check).} On $Y_{1}$ of genus $3$ with $\deg\om_{Y_{1}}=4$, Riemann--Roch and Serre duality give: $M$ ($\deg1$): $h^{0}:0\to1$; $M^{2}$ ($\deg2$): $1\to2$; $\om M^{-1}=h^{1}(M)$: $1\to2$; $\om M^{-2}$: $1\to2$. Totals: $h^{0}(B_{y}):1\to3$ and $h^{0}(L_{y}):2\to4$, reproducing \cite[Prop.~9(iv)(v)]{SouleABC}. \end{proof} \begin{lemma}[Kodaira--Spencer class]\label{lem:kappa} Take first-order jets $x_{1}(t)=a+\alpha t$, $x_{2}(t)=a+\beta t$, and let $M_{t}$ be the continuous square-root branch of $\mathcal O(A_{1}(t)+A_{2}(t))$ (the lift exists locally since $[2]:\Pic^{1}\to\Pic^{2}$ is \'etale). With basis $\om_{j}=x^{j}dx/y$ ($j=0,1,2$) of $H^{0}(\om_{Y_{1}})$, the Abel--Jacobi differential gives \begin{equation}\label{eq:kappa} \kappa_{M^{2}}(\om_{j})=\frac{\alpha-\beta}{y_{0}}a^{j}, \qquad \kappa_{M}=\tfrac12\kappa_{M^{2}} =\frac{\alpha-\beta}{2y_{0}}\bigl(1,a,a^{2}\bigr). \end{equation} This direction holds for all compatible jets ($T_{P}Y_{1}$ is generated by $\partial_{x}$). \end{lemma} \begin{proof} \emph{Step 1 (local model).} $Y_{1}:y^{2}=\prod_{i=1}^{8}(x-e_{i})$, $Q=(e_{1},0)$, $M_{0}=\mathcal O(Q)$, $M_{0}^{2}=g^{1}_{2}$; the exceptional fibre is $A_{1}(0)=P=(a,y_{0})$, $A_{2}(0)=\iota P=(a,-y_{0})$ with $P+\iota P\sim2Q$. \emph{Step 2 (first-order jets).} $x_{1}(t)=a+\alpha t$, $x_{2}(t)=a+\beta t$; $M_{t}^{2}=\mathcal O(A_{1}(t)+A_{2}(t))$ with continuous square-root branch $M_{t}$. \emph{Step 3 (Abel--Jacobi differential).} The basis of $H^{0}(\om_{Y_{1}})$ is $\om_{j}=\frac{x^{j}\,dx}{y}$, $j=0,1,2$. From $M_{t}^{2}=\mathcal O(A_{1}+A_{2})$, $\kappa_{M^{2}}(\om)=\om(v_{1})+\om(v_{2})$ with $v_{i}=\dot A_{i}(0)$. Since $T_{P}Y_{1}$ is generated by $\partial_{x}$, $v_{i}=\dot x_{i}\,\partial_{x}$ evaluates at $P$ to $\om_{j}(v_{i})=\dot x_{i}\,\frac{a^{j}}{y_{0}}$; at $\iota P$ the sign of $y$ flips, so $\kappa_{M^{2}}(\om_{j})=(\alpha-\beta)\frac{a^{j}}{y_{0}}$. \emph{Step 4 (taking the square root).} $\kappa_{M}=\frac12\kappa_{M^{2}}$ ($\operatorname{char}0$), so $\kappa_{M}=\frac{\alpha-\beta}{2y_{0}}(1,a,a^{2})$. The direction is pinned to the geometric progression $(1,a,a^{2})$ for every compatible jet, independently of the parametrisation. \emph{Step 5 (degeneration).} $\alpha=\beta\Rightarrow\kappa_{M}=0$: both connecting maps degenerate, $G_{\bar y}=4$, $\ell_{\bar y}\ge4$. \end{proof} \begin{theorem}[Ranks of the two connecting maps and $T$]\label{thm:rank} Under Assumptions~\ref{ass:model} and \ref{ass:count}, \begin{equation}\label{eq:Gy} G_{\bar y}:=\dim_{k}R[\pi^{2}]=4-\rank\partial_{1}-\rank\partial_{2}=2, \end{equation} hence $n_{1}=n_{2}=1$, $\ell_{\bar y}=2$, and \begin{equation}\label{eq:T8k} T=8k\ \text{exactly},\qquad c=\tfrac14,\ C=36,\ t=12 . \end{equation} \end{theorem} \begin{proof} \emph{Rank of $\partial_{1}$.} From $0\to\mathcal O\to\mathcal O(Q)\to\mathcal O_{Q}(Q)\to0$, \[ \ker\bigl(H^{1}(\mathcal O)\to H^{1}(\mathcal O(Q))\bigr) =\operatorname{coker}\bigl(H^{0}(\mathcal O(Q))\to H^{0}(\mathcal O_{Q}(Q))\bigr)^{\vee}, \] the line $L_{Q}$ under Čech--Serre duality. Explicitly $\delta(1)(\om)=\Res_{Q}(\om/y)$; for $\om_{j}=\frac{x^{j}dx}{y}$ the residue at $Q$ gives $e_{1}^{j}$, so $L_{Q}\propto(1,e_{1},e_{1}^{2})$. By \eqref{eq:kappa}, $\kappa_{M}\in L_{Q}\iff a=e_{1}\iff P=Q\iff A_{1}(0)=A_{2}(0)$. But \cite[Prop.~5]{SouleABC} gives $A_{1}\cdot A_{2}=0$, excluding coincidence; hence $\kappa_{M}\notin L_{Q}$ and $\rank\partial_{1}=1$. \emph{Rank of $\partial_{2}$.} The second connecting map is the cup product $H^{0}(B)\otimes T\to H^{1}(B)$ with catalecticant matrix $\mathfrak{B}=\lambda\begin{pmatrix}1&a\\ a&a^{2}\end{pmatrix}$ (the $2\times2$ compression of $(1,a,a^{2})$). Its determinant is $\lambda^{2}(a^{2}-a^{2})=0$ and its rank is $1$, consistent with the algebraic constraint $\rank\partial_{2}\le1$ from \eqref{eq:Gy}. \emph{Conclusion.} $G_{\bar y}=4-1-1=2$, so $n_{1}=n_{2}=1$ (from $\dim_{k}R[\pi^{2}]=2\min(2,n)=2$), $\ell_{\bar y}=2$, and $T=8k$ exactly; therefore $c=1/4$, $C=36$, $t=12$, $E=24$ is the unique output of the construction. \emph{Stratification without full transversality.} ``All four fibres transverse'' is labelled Assumption~\ref{ass:model} (it would follow from $\psi$ Galois plus equivariance of the construction, but equivariance is not spelled out in \cite{SouleABC}). Without it, $L\in[8,15]$, the output is still weak $abc$, and full $abc$ remains unattainable --- the core conclusion is robust. \end{proof} \begin{corollary}[Exclusion of degenerate branches]\label{cor:deg} If a fibre has $\kappa^{(j)}=0$ then $G_{\bar y}=4$ and $\ell_{j}\ge4$. If $m$ of the four fibres are transverse, then $L\ge16-2m$; $m=0$ gives $t=12/25$ and $E_{\mathrm{I}}=24/25<1$, contradicting optimality of exponent $1$. Hence total degeneration is excluded; on chain II $m\ge3$, on chain I $m\ge1$. \end{corollary} \begin{proof} \emph{Step 1.} If an exceptional fibre degenerates to first order ($\alpha=\beta$), then $\kappa_{M}=0$ by \eqref{eq:kappa}, so $\partial_{1}=\partial_{2}=0$. \emph{Step 2.} By \eqref{eq:Gy}, $G_{\bar y}=4-0-0=4$; from $\dim_{k}R[\pi^{2}]=\sum_{i}\min(2,n_{i})=4$ one gets $n_{i}\ge2$, hence $\ell_{\bar y}\ge4$. \emph{Step 3 (total degeneration).} If all four fibres degenerate, $L=\sum\ell_{j}\ge4\cdot4=16$, $\Theta\ge16$, $t=\frac{12}{3\cdot16-23}=\frac{12}{25}$, $E=\frac{12}{25}\cdot2=\frac{24}{25}<1$, violating optimality of exponent $1$: total degeneration is excluded. \emph{Step 4 (counting).} On chain II $T\le10k\Rightarrow L\le10$; each degenerate point contributes $\ge4$, each transverse point $2$, so at most one degenerate fibre; on chain I $L\le15$ allows at most three. At least one fibre is transverse. \end{proof} \begin{corollary}[Galois transitivity replaces transversality]\label{cor:transitive} Let $\psi:Y\to Y_{1}$ be finite \'etale \textbf{Galois} with $G=\Gal(Y/Y_{1})$ acting \textbf{transitively} on the four exceptional points of $Y_{1}$. By Assumption~\ref{ass:count} all $n_{j}$ are equal on an orbit ($=n$), so $\Theta=8n$ ($n\ge1$). From \eqref{eq:thetaset}, $\Theta\le15$, so $n=1$, i.e. \begin{equation}\label{eq:theta8} \Theta=8,\qquad T=8k,\qquad c=\tfrac14,\ C=36,\ t=12 . \end{equation} No use is made of Assumption~\ref{ass:model}, of $\kappa_{M}$, of $\alpha\ne\beta$, or of the mod-$3$ argument. Thus the unattainability of full $abc$ for all Kummer degrees (Corollary~\ref{cor:nogoallq}) holds under this group-theoretic hypothesis alone: $t^{\ast}(q)<1$ or $\le5/4$ never equals $12$. If the orbits are $2+2$, then $\Theta\in\{8,12\}$ and $\Theta=12$ remains possible; the trivial action gives $\{8,10,12,14\}$. Transitivity is therefore not a dispensable phrase. \end{corollary} \begin{proof} \emph{Step 1 (transitivity).} If $\Gal(\psi)$ is transitive on the four exceptional points, all $n_{j}$ coincide ($=n$) and $\Theta=2\cdot4n=8n$. \emph{Step 2.} From Proposition~\ref{prop:range}, $L=\Theta\le15$, so $8n\le15$, hence $n=1$ and $\Theta=8$. \emph{Step 3 (conclusion).} $T=8k$ exactly, $t=12$, and $\gamma_{\mathrm{crit}}\ge9.6$ equals neither literature convention, so the all-$q$ No-Go is rigorous under the group-theoretic hypothesis, without $\kappa_{M}$, without mod-$3$, without transversality. \emph{Step 4 ($2+2$ orbits).} If the orbits are $2+2$, then $\Theta=4(n_{1}+n_{2})\in4\mathbb Z$ with $\Theta\le15$ gives $\{8,12\}$: $\Theta=12$ survives, so transitivity is genuinely weaker than but not implied by transversality. \end{proof} \begin{remark}[Effective criterion]\label{rem:pi2} By Corollary~\ref{cor:parity}, $\dim_{k}R[\pi^{2}]$ takes only $2$ ($n_{\bar y}=1$, transverse) or $4$ ($n_{\bar y}\ge2$, degenerate), and $T=8k\iff$ all four points transverse. The check is equivalently whether $\sum_{j=1}^{4}n_{j}=6$ (the unique candidate $\Theta=12$, $q=27$) or whether $\Gal(\psi)$ is transitive (Corollary~\ref{cor:transitive} gives $\Theta=8$). The second-order jets $M(\pi)=M_{0}+\pi M_{1}+\pi^{2}M_{2}$ of the Grauert complex give a direct computational tool, but the conclusions above need no local data beyond \cite{SouleABC}. \end{remark} \begin{remark} Within the fixed literature point $(g_{0},r,b,\lambda)=(2,2,2,3)$ at $q=5$, the mod-$3$ integrality of Corollary~\ref{cor:nogo} is independent of the exact value of $T$ (Theorem~\ref{thm:rank}): even allowing partial degeneration, $L\in\mathbb Z$ and $N\equiv1\pmod3$ persist. Corollary~\ref{cor:parity} sharpens $L=\Theta$ to $2\mathbb Z$, so the attainable denominators $N=3\Theta-23$ actually lie in $3\cdot2\mathbb Z-23\equiv1\pmod6$. This is the coincidence described in Remark~\ref{rem:landing} --- not a structural barrier when $q$ varies. \end{remark} % ===================================================================== \section{Controlled weak $abc$}\label{sec:weak} % ===================================================================== This section expands the ``known to follow from Theorem A'' of \cite[\S7.2]{SouleABC} into a complete derivation. All external inputs are collected in Assumption~\ref{ass:U}. \begin{assumption}[External inputs U1--U7]\label{ass:U} \begin{enumerate}[leftmargin=1.6em,itemsep=0.15em,label=U\arabic*] \item \textbf{(U1)} Theorem~A of \cite{SouleABC} fixes a genus-$2$ curve $\mathcal C/K_{0}$ and the logarithmic height $h$ attached to its canonical bundle $K_{\mathcal C}$; hence $\deg K_{\mathcal C}=2g-2=2$ and $h(P)\le(12+\eps)\delta(F)/d+\beta(\eps)$ for every number field $F\supseteq K_{0}$ and every $P\in\mathcal C(F)$ (statement and scope verified verbatim against the source, Appendix~\ref{app:base}). The height is \emph{induced by} $K_{\mathcal C}$ from the outset --- confirmed, not assumed, in \S\ref{subsec:gamma} (Theorems~\ref{thm:gammastrict},~\ref{thm:gamK}); no circularity arises between \S\ref{sec:weak} and \S\ref{subsec:gamma}. \item \textbf{(U2)} Alignment Assumption~\ref{ass:A0}. \item \textbf{(U3)} Proposition~2 / Corollary~1 / (28)--(34) / Theorem~1 are unconditional (Proposition~\ref{prop:cycle}). \item \textbf{(U4)} Optimality of exponent $1$. \item \textbf{(U5)} $N=\rad(abc)$ and $\gcd(a,b)=1\Rightarrow\gcd(a,c)=1$. \item \textbf{(U6)} The Kummer cover and discriminant computation (Lemma~\ref{lem:kummer}). \item \textbf{(U7)} $\pi$ finite, $L$ ample, height functoriality, and Riemann--Roch. \end{enumerate} \end{assumption} \begin{lemma}[Kummer discriminant estimate]\label{lem:kummer} Let $\lambda=a/c$, $\alpha=\lambda(\lambda-1)=-ab/c^{2}$. For odd $q\ge5$ set $C:u^{q}=x(x-1)$, $\pi:C\to\mathbb P^{1}$, $F=\mathbb Q(\alpha^{1/q})$, $P=(\lambda,\alpha^{1/q})\in C(F)$, $d=[F:\mathbb Q]$. Then $g(C)=(q-1)/2$ (for $q=5$, $g=2$), and \begin{equation}\label{eq:eta} \frac{\delta(F)}{d}\le\frac{q-1}{q}\log N+O_{q}(1), \qquad \eta(q):=\frac{q-1}{q}. \end{equation} \end{lemma} \begin{proof} \emph{Step 1 (curve).} On $C:u^{q}=x(x-1)$ with odd $q\ge5$, the points $0,1,\infty$ are totally ramified with $e=q$; Riemann--Hurwitz gives $2g-2=-2q+3(q-1)=q-3$, so $g=\frac{q-1}2$ and $q=5$ gives $g=2$. \emph{Step 2 (field and point).} $\lambda=a/c$, $\alpha=\lambda(\lambda-1)=-ab/c^{2}$, $F=\mathbb Q(\alpha^{1/q})$, $P=(\lambda,\alpha^{1/q})\in C(F)$. \emph{Step 3 (ramified primes).} The ramified primes satisfy $\{\text{ramified}\}\subseteq\{p\mid q\,abc\}$; at $p\nmid q$ the ramification is tame. \emph{Step 4 (Dedekind).} For a tame extension, \[ v_{p}(D_{F/\mathbb Q})=\sum_{i}f_{i}(e_{i}-1)=d-\sum_{i}f_{i}\le d-1\le q-1, \] using only $\sum f_{i}\ge1$; neither Galois nor total ramification is assumed. \emph{Step 5 (normalisation).} Divide by $d$ and sum against $\log p$: $\frac{\delta(F)}d\le\frac{q-1}{q}\log N+O_{q}(1)$, i.e.\ $\eta(q)=\frac{q-1}{q}$. At $q=5$ this is $4/5$, matching $24=30\cdot\frac45$ of \cite[\S7.2]{SouleABC}. Since $\eta(q)\to1$, increasing the Kummer degree cannot improve $abc$. \end{proof} \begin{proposition}[Unification of the exponent]\label{prop:E} Define $\eta:=(\delta(F)/d)/\log N$ and $\gamma:=h(P)/\log c>0$. Under Assumption~\ref{ass:U}, \begin{equation}\label{eq:E} c\le A_{\eps}\,N^{E+\eps}, \qquad E=t\cdot\frac{\eta}{\gamma}. \end{equation} In particular $(t,\eta,\gamma)=(12,4/5,2/5)$ gives $E=24$ and $(12,4/5,1)$ gives $E=48/5$; the difference lies only in the normalisation of the line bundle used for $h(P)$, not in two different proof chains. \end{proposition} \begin{proof} \emph{Step 1 (inputs).} U1: $h(P)=\gamma\log c+O(1)$ (Assumption~\ref{ass:U}); U2: $\delta(F)/d\le\eta\log N+O(1)$ (\eqref{eq:eta}); Theorem~A: $h(P)\le(t+\eps)\frac{\delta(F)}d+O_{\eps}(1)$. \emph{Step 2 (chaining).} $\gamma\log c-O(1)\le h(P)\le(t+\eps)\eta\log N+O_{\eps}(1)$. \emph{Step 3 (divide by $\gamma>0$).} $\log c\le\frac{(t+\eps)\eta}{\gamma}\log N+O_{\eps}(1)$; letting $\eps\to0$ gives $E=t\cdot\frac\eta\gamma$. \emph{Step 4 (two conventions).} $(t,\eta,\gamma)=(12,\frac45,\frac25)\Rightarrow E=24$ and $(12,\frac45,1)\Rightarrow E=48/5$. The difference is only the normalisation of the height; Corollary~\ref{cor:gamma3} proves that $\gamma=1$ is not the normalisation of Theorem~A. \emph{Step 5 (unpacking \S7.2).} Defining $\eta=\frac{\delta/d}{\log N}$ and $\gamma=\frac{h(P)}{\log c}$, Theorem~A plus A0 directly yields $c\le A_{\eps}N^{E+\eps}$ without invoking \S7.2 as a black box. \end{proof} \begin{lemma}[Strict lower bound for $\gamma$]\label{lem:gamma} Let $n:=\big\lceil(2q-2)/\ell\big\rceil$. Then $nL-\pi^{\ast}\mathcal O(1)$ is effective, and $\gamma\ge1/n$. \end{lemma} \begin{proof} \emph{Step 1 (degrees).} $\deg\pi^{\ast}\mathcal O(1)=q$ ($\pi$ has degree $q$) and $2g-1=2\cdot\frac{q-1}2-1=q-2$. \emph{Step 2 (choice of $n$).} With $\deg L=\ell$ and $n=\lceil\frac{2q-2}{\ell}\rceil$, $\deg(nL-\pi^{\ast}\mathcal O(1))=n\ell-q\ge(2q-2)-q=q-2=2g-1$. \emph{Step 3 (effectivity).} For $D=nL-\pi^{\ast}\mathcal O(1)$ with $\deg D\ge2g-1$, Serre duality gives $h^{1}(D)=h^{0}(K-D)=0$ (since $\deg(K-D)<0$), and $\chi(D)=\deg D-g+1\ge(2g-1)-g+1=g>0$; hence $h^{0}(D)=\chi(D)>0$: there is a nonzero section, i.e.\ $\pi^{\ast}\mathcal O(1)\hookrightarrow nL$ (up to twisting $\mathcal O(1)$). \emph{Step 4 (height comparison).} $h_{\pi^{\ast}\mathcal O(1)}(P)\le n\,h_{L}(P)+O(1)$, i.e. $\log c\le n\gamma\log c+O(1)$, so $\gamma\ge\frac1n+O(1/\log c)$. \emph{Step 5 (conservativeness).} Using $\deg\ge2g-1$ is sufficient but not necessary, so $\gamma_{0}=1/n$ and $E_{\max}$ are conservative, not optimal; they already give a finite exponent. \end{proof} \begin{theorem}[Unconditional exponential bound]\label{thm:SYuncond} For coprime positive $a+b=c$ and $N=\operatorname{rad}(abc)$, \[ \log c\ \le\ \kappa\,N^{1/3}(\log N)^{3}, \qquad\text{equivalently}\qquad \log c=O_{\varepsilon}\!\left(N^{1/3+\varepsilon}\right), \] with $\kappa$ effectively computable. Equivalently $c<\exp\!\big(C_{\varepsilon}N^{1/3+\varepsilon}\big)$. \end{theorem} \begin{proof} This is the theorem of Stewart--Yu \cite{SY01} and its refinement \cite{SY02} (the case $\alpha=1/3$ of the sequence $\alpha=15$ \cite{ST86}, $\alpha=2/3$ \cite{SY01}, $\alpha=1/3$ \cite{SY02} recorded in \cite[(1.1)]{PastenShimura}). The proof uses lower bounds for linear forms in ordinary and $p$-adic logarithms; the constant is effective. Exponentiating $\log c\le\kappa N^{1/3}(\log N)^3$ and absorbing $(\log N)^3$ into $N^{\varepsilon}$ gives the second form. Note that throughout this paper $N=\operatorname{rad}(abc)$, \emph{not} $\log\operatorname{rad}(abc)$; under the latter convention the inequality would be mis-stated. \end{proof} \begin{theorem}[Polynomial versus exponential type]\label{thm:typecomp} Theorem~B of \cite{SouleABC} is $c\le A(\varepsilon)N^{24+\varepsilon}$, i.e. $\log c\le(24+\varepsilon)\log N+O_{\varepsilon}(1)$. For every sufficiently large $N$, \[ 24\log N=o\!\left(N^{1/3}(\log N)^{3}\right), \] so Theorem~B is strictly stronger than Theorem~\ref{thm:SYuncond}. More importantly, Theorem~\ref{thm:SYuncond} is of \emph{exponential} type $c\le\exp(N^{\alpha})$, $\alpha<1$, whereas Theorems~7.5 and~9.2 are of \emph{polynomial} type $c\le N^{E}$ with finite $E$. As stated verbatim in \cite[\S1.1]{PastenShimura}, ``all unconditional bounds for these conjectures are of exponential nature''; consequently the existence of any finite polynomial exponent $E$ is open, and no unconditional Stewart--Yu-type result can replace Theorem~A as the input to Theorem~7.5. \end{theorem} \begin{proof} Since $N^{1/3}$ dominates every power of $\log N$, $\lim_{N\to\infty}24\log N/\big(N^{1/3}(\log N)^3\big)=0$. The numerical comparison is recorded below. \begin{center} \begin{tabular}{c|c|c|c} \hline $N$ & $24\log N$ (Th.~B) & $N^{1/3}(\log N)^3$ (Stewart--Yu) & smaller bound \\ \hline $10^{2}$ & $110.5$ & $4.53\times10^{2}$ & Th.~B \\ $10^{4}$ & $221.1$ & $1.68\times10^{4}$ & Th.~B \\ $10^{6}$ & $331.6$ & $2.64\times10^{5}$ & Th.~B \\ $10^{12}$ & $663.1$ & $2.11\times10^{8}$ & Th.~B \\ $10^{20}$ & $1105.2$ & $4.53\times10^{11}$ & Th.~B \\ \hline \end{tabular} \end{center} \noindent The type distinction is definitional: $c\le\exp(N^{\alpha})$ grows faster than $N^{E}$ for every fixed $E$, so an exponential-type bound never implies a polynomial-type one. The quotation from [20, \S1.1] states that every unconditional bound known at present is of exponential type; hence the polynomial exponent produced by Theorem~7.5 is genuinely new \emph{as a conditional statement}, and Theorem~7.5 is not a restatement of Stewart--Yu. This also shows that writing $N=\log\operatorname{rad}(abc)$ in the comparison reverses the strength: with the paper's convention $N=\operatorname{rad}(abc)$ the logarithmic bound $24\log N$ is the stronger one. \end{proof} \begin{theorem}[Controlled weak $abc$, conditional on Theorem A]\label{thm:weak} Let $a,b,c$ be pairwise coprime positive integers with $a+b=c$ and $N=\rad(abc)$. Under Assumption~\ref{ass:U}, for every $\eps>0$ there is $A_{\eps}$ with \begin{equation}\label{eq:weak} c\le A_{\eps}\,N^{E_{\max}+\eps}, \qquad E_{\max}=12\cdot\frac{q-1}{q}\cdot\Big\lceil\frac{2q-2}{\ell}\Big\rceil, \end{equation} where $q\ge5$ is odd and $\ell=\deg L$ is the degree of the fixed line bundle of Theorem~A. The table below is a \emph{formal} parametrisation for an arbitrary ample $L$; under the actual hypotheses of Theorem~A only $\ell=\deg K_{\mathcal C}=2$ applies, and then $\gamma=\frac25$ is an equality (Theorem~\ref{thm:gamK}), not a bound. For $q=5$: $E_{\max}=\frac{48}{5}n$ with $n=\lceil8/\ell\rceil$: $\ell=1\Rightarrow76.8$, $2\Rightarrow38.4$, $3\Rightarrow28.8$, $4\Rightarrow19.2$, $\ell\ge8\Rightarrow9.6$. \end{theorem} \begin{proof} \emph{Step 1.} Combine \eqref{eq:E} ($E=t\eta/\gamma$), \eqref{eq:eta} ($\eta=\frac{q-1}q$) and Lemma~\ref{lem:gamma} ($\gamma\ge1/n$): $E\le t\cdot\eta\cdot n=t\cdot\frac{q-1}q\cdot n$. \emph{Step 2.} With $t=12$ and $n=\lceil(2q-2)/\ell\rceil$: $E_{\max}=12\,\frac{q-1}q\,\big\lceil\frac{2q-2}{\ell}\big\rceil$. \emph{Step 3 (table at $q=5$).} $E_{\max}=\frac{48}{5}n$, $n=\lceil8/\ell\rceil$: $\ell=1\Rightarrow76.8$, $2\Rightarrow38.4$, $3\Rightarrow28.8$, $4\Rightarrow19.2$, $\ell\ge8\Rightarrow9.6$. \emph{Step 4 (matching the literature).} At $\ell=2$ one has $n=4$ and $E_{\max}=\frac{48}{5}\cdot4=\frac{192}{5}=38.4$; this is a conservative upper bound, reflecting only $\gamma\ge\frac14$ (Lemma~\ref{lem:gamma}). The literature exponent $24$ is obtained directly from \eqref{eq:E} with the exact $\gamma=\frac25$ of Theorem~\ref{thm:gamK}, and is not the value of $E_{\max}$. \emph{Step 5 (independence).} Neither A1/A3/DIC-S, nor Vojta's conjecture, nor \S7.2 is used anywhere; the only external input is Theorem~A. The conclusion is therefore a formal corollary \emph{conditional on Theorem~A}, yielding weak $abc$, not a proof of $abc$. \end{proof} \begin{corollary}[Consistency]\label{cor:consist} $E_{\max}\ge48/5\gg1$, consistent with optimality of exponent $1$; the mod-$3$ obstruction of Corollary~\ref{cor:nogo} is likewise unaffected, since $t=12$ is pinned inside the transverse construction (Theorem~\ref{thm:rank}). \end{corollary} \begin{proof} \emph{Step 1.} Since $n=\lceil8/\ell\rceil\ge1$, \eqref{eq:weak} gives directly $E_{\max}=12\cdot\frac45\cdot n\ge12\cdot\frac45=\frac{48}{5}=9.6>1$; no bound on $\gamma$ is needed. (The strict bound $\gamma<1$ follows later, from Theorem~\ref{thm:gamK}.) \emph{Step 2.} No contradiction with optimality of exponent $1$ ($\log c/\log\rad\ge1-\eps$): a finite exponent in weak $abc$ should exceed $1$ anyway. \emph{Step 3 (no improvement over the literature).} The table value $9.6$ at $\ell\ge8$ cannot be claimed as an improvement: $t=12$ is a constant for the \emph{fixed} $L$; changing $L$ changes both $t$ and $\beta(\eps)$, and \cite{SouleABC} does not specify $\ell$. The meaning of the table: for each prescribed $\ell$ the exponent is finite and explicit. \emph{Step 4 (coefficient stability).} Corollary~\ref{cor:nogo} relies only on the discreteness $t=12/N$, $N\equiv1\pmod3$ of \eqref{eq:tL}; Theorem~\ref{thm:rank} pins $t=12$ ($N=1$) inside the transverse construction, so the finiteness of $E_{\max}$ grants no freedom to move $N$ off that set. \end{proof} \begin{proposition}[Asymptotic obstruction on the Kummer side]\label{prop:Em} In the $\gamma_{K}=(q-3)/q$ normalisation, taking $\gamma=\gamma_{K}$, \begin{equation}\label{eq:Em} E(q)=\frac{12(q-1)}{q-3}\ \longrightarrow\ 12\qquad(q\to\infty), \end{equation} so no increase of the Kummer degree brings the exponent down to $1$. \end{proposition} \begin{proof} \emph{Step 1.} With $\gamma_{K}=\frac{q-3}q$, $E(q)=\frac{t\eta(q)}{\gamma_{K}} =12\cdot\frac{q-1}q\cdot\frac{q}{q-3}=\frac{12(q-1)}{q-3}$. \emph{Step 2 (limit).} $\lim_{q\to\infty}\frac{12(q-1)}{q-3}=12$. \emph{Step 3 (monotonicity).} $q=7$ gives $18$, $q=51$ gives $12.5$: the sequence decreases to $12$, never to $1$. \emph{Step 4 (independence).} This asymptotic obstruction is independent of the denominator obstruction of \S\ref{sec:torsion}: both routes conclude ``full $abc$ unattainable''. \end{proof} \begin{remark} Theorem~\ref{thm:weak} must \emph{not} be read as improving the literature via the table value $E_{\max}=9.6$ at $\ell\ge8$: $t=12$ belongs to the fixed $L$, changing $L$ moves $t$ and $\beta(\eps)$ simultaneously, and $\ell$ is unspecified in \cite{SouleABC}. The table's content is that for every prescribed $\ell$ the exponent is finite and explicit. \end{remark} % ===================================================================== \section{The last lattice point: unified criterion and closure of the Kummer family}\label{sec:last} % ===================================================================== By Corollary~\ref{cor:thetaset}, $\Theta\in\{8,10,12,14\}$; write $N=3\Theta-23\in\{1,7,13,19\}$ and $t=12/N$. The bridge of \S\ref{sec:weak}, the discreteness of \S\ref{sec:torsion} and optimality of exponent $1$ now unify into a single criterion. \begin{corollary}[Unified criterion]\label{cor:unified} From $E=t\eta/\gamma$ and $\eta(q)=(q-1)/q$, \begin{equation}\label{eq:Eunified} E(\Theta,q,\gamma)=\frac{12(q-1)}{q\,N\,\gamma}, \qquad \gamma_{\mathrm{crit}}(\Theta,q):=\frac{12(q-1)}{q(3\Theta-23)}. \end{equation} Hence $E=1\iff\gamma=\gamma_{\mathrm{crit}}$; $E\ge1\iff\gamma\le\gamma_{\mathrm{crit}}$; $E<1$ violates optimality of exponent $1$ and excludes that $(\Theta,q)$. As $q\to\infty$, $\gamma_{\mathrm{crit}}\to\frac{12}{N}\in\{12,\,12/7,\,12/13,\,12/19\}$. \end{corollary} \begin{proof} \emph{Step 1.} Substitute $t=12/N$ (Corollary~\ref{cor:thetaset}) and $\eta=(q-1)/q$ (\eqref{eq:eta}) into \eqref{eq:E}: $E=\frac{12}{N}\cdot\frac{q-1}{q}\cdot\frac1\gamma=\frac{12(q-1)}{qN\gamma}$. \emph{Step 2 (definition).} Setting $\gamma_{\mathrm{crit}}=\frac{12(q-1)}{qN}$ gives $E=\frac{\gamma_{\mathrm{crit}}}{\gamma}$. \emph{Step 3 (three equivalences).} $E=1\iff\gamma=\gamma_{\mathrm{crit}}$; $E\ge1\iff\gamma\le\gamma_{\mathrm{crit}}$; $E<1$ contradicts optimality of exponent $1$: take $a=1$, $b=2^{n}-1$, $c=2^{n}$ (pairwise coprime for $n\ge2$); then $abc=2^{n}(2^{n}-1)$ with $2^{n}-1$ odd, so $\rad(abc)=2\,\rad(2^{n}-1)\le2(2^{n}-1)<2^{n+1}$, hence $\frac{\log c}{\log\rad}>\frac{n\log2}{(n+1)\log2}=\frac{n}{n+1}\to1$. Thus $E<1$ is excluded for that $(\Theta,q)$. \emph{Step 4 (asymptotics).} $\gamma_{\mathrm{crit}}(q\to\infty)\to\frac{12}{N}\in \{12,\frac{12}7,\frac{12}{13},\frac{12}{19}\}$. \end{proof} \begin{corollary}[Elementary solution of the landing equation]\label{cor:landeq} On convention A ($\gamma=\gamma_{K}=(q-3)/q$) and convention B ($\gamma=1$), $E=1$ is respectively equivalent to \begin{equation}\label{eq:qAB} q_{A}=\frac{3N-12}{N-12}, \qquad q_{B}=\frac{12}{12-N}. \end{equation} For $N\in\{1,7,13,19\}$: \begin{center} \begin{tabular}{c|c|cc} \toprule $N$ & $\Theta$ & $q_{A}$ & $q_{B}$ \\ \midrule 1 & 8 & $9/11$ & $12/11$ \\ 7 & 10 & $-9/5$ & $12/5$ \\ \textbf{13} & \textbf{12} & $\mathbf{27}$ & $-12$ \\ 19 & 14 & $45/7$ & $-12/7$ \\ \bottomrule \end{tabular} \end{center} The unique solution with $q$ an odd integer $\ge5$ is $(N,\Theta,q,\gamma)=(13,12,27,8/9)$; the four values of $q_{B}$ are all non-integral, so chain B is unattainable for all odd $q$ --- reproduced here independently. \end{corollary} \begin{proof} \emph{Convention A} ($\gamma=\gamma_{K}=\frac{q-3}q$): $\frac{q-3}{q}=\frac{12(q-1)}{qN}$ gives $N(q-3)=12(q-1)$, hence $q(N-12)=3N-12$ and $q_{A}=\frac{3N-12}{N-12}$. \emph{Convention B} ($\gamma=1$): $1=\frac{12(q-1)}{qN}$ gives $qN=12q-12$, hence $q(12-N)=12$ and $q_{B}=\frac{12}{12-N}$. \emph{Case check.} $N=1$ gives $(\frac9{11},\frac{12}{11})$; $N=7$ gives $(-\frac95,\frac{12}5)$; $N=13$ gives $(27,-12)$; $N=19$ gives $(\frac{45}7,-\frac{12}7)$. Only $(13,12,27,8/9)$ makes $q$ an odd integer $\ge5$; the $q_{B}$-column contains no odd integer $\ge5$ at all. This corollary unifies the $q-3\mid24$ argument of Corollary~\ref{cor:nogo} with the discrete intersection of Corollary~\ref{cor:thetaset} into rational equations, with no scanning. \end{proof} \begin{corollary}[Admissible range of $q$]\label{cor:qrange} With $\gamma=\gamma_{K}$, $E\ge1\iff12(q-1)\ge N(q-3)$: \begin{center} \begin{tabular}{c|c|c|c} \toprule $\Theta$ & inequality & solution & odd $q\ge5$ compatible \\ \midrule 8 & $12(q-1)\ge q-3$ & $q\ge9/11$ & all \\ 10 & $12(q-1)\ge7(q-3)$ & always & all \\ \textbf{12} & $12(q-1)\ge13(q-3)$ & $\mathbf{q\le27}$ & $5,7,\dots,27$ \\ 14 & $12(q-1)\ge19(q-3)$ & $q\le45/7$ & only $q=5$ \\ \bottomrule \end{tabular} \end{center} For $\Theta=12$, $\gamma_{\mathrm{crit}}$ increases with $q$ but $\gamma_{K}$ increases faster: $q<27$ gives $E>1$ (weak $abc$), $q=27$ gives $E=1$ (unique landing), $q>27$ gives $E<1$ (excluded); numerically $q=25,27,29$ give $E=1.00699,\,1.00000,\,0.99408$. For $\Theta=14$ only $q=5$ is compatible and $\gamma_{\mathrm{crit}}=48/95$, matching neither literature convention; for $\Theta=8,10$ one has $\gamma_{\mathrm{crit}}\ge48/35>1$, unreachable by both conventions ($\gamma=1$, $\gamma_{K}<1$). Both are excluded. \end{corollary} \begin{proof} \emph{Step 1 (general inequality).} With $\gamma=\gamma_{K}$, $E\ge1\iff\frac{12(q-1)}{N(q-3)}\ge1\iff12(q-1)\ge N(q-3)$. \emph{Step 2 (case by case).} $N=1$: $11q\ge9$, all $q$. $N=7$: $5q\ge-3$, always. $N=13$: $12q-12\ge13q-39$, so $q\le27$. $N=19$: $12q-12\ge19q-57$, so $q\le45/7\approx6.43$, leaving $q=5$ among odd $q\ge5$. \emph{Step 3 (direction).} For $N>12$ the coefficient of $q$ is negative, giving an upper bound; $\gamma_{\mathrm{crit}}$ increases with $q$ but $\gamma_{K}$ increases faster. $q=25,27,29$ give $E=1.00699,\,1.00000,\,0.99408$. \emph{Step 4 (excluding $\Theta=14,8,10$).} $\Theta=14$: $\gamma_{\mathrm{crit}}(5)=48/95\ne1,\frac25$. $\Theta=8,10$: $\gamma_{\mathrm{crit}}\ge48/35>1$, unreachable by both conventions. \end{proof} \begin{corollary}[Sensitivity to the normalisation]\label{cor:rho} Write $\gamma=\rho\gamma_{K}$. Then \begin{equation}\label{eq:Erho} E=\frac{12(q-1)}{N(q-3)\rho}, \qquad (N,q)=(13,27)\Rightarrow E=1/\rho . \end{equation} $\rho>1$ gives $E<1$ (excluded by optimality), $\rho=1$ survives exactly, $\rho<1$ gives weak $abc$. Hence the last cell requires $\gamma$ to \textbf{equal} $\gamma_{K}(27)=8/9$ exactly; any strict lower bound $\gamma\ge\gamma_{0}>8/9$ excludes it (convention B is such a case). If $\gamma=\gamma_{0}$ is constant in $q$, then $t=\frac{\gamma_{0}q}{q-1}=\frac{12}{N}$ gives $q=\frac{12}{12-N\gamma_{0}}$; for the literature values $\gamma_{0}\in\{1,2/5,4/5\}$ there is no odd integer $q\ge5$, and only $\gamma_{0}=8/9$ yields $(13,27)$. Since $8/9=\gamma_{K}(27)$ itself varies with $q$, the last cell \emph{must} let $\gamma$ vary with $q$ and hit equality exactly; it cannot arise from a fixed bundle normalisation. \end{corollary} \begin{proof} \emph{Step 1.} With $\gamma=\rho\frac{q-3}{q}$ in \eqref{eq:Eunified}: $E=\frac{12(q-1)}{qN}\cdot\frac{q}{\rho(q-3)}=\frac{12(q-1)}{N(q-3)\rho}$. \emph{Step 2 (literature point).} $q=27$, $N=13$: $12(q-1)=312$ and $N(q-3)=13\cdot24=312$, so $E=1/\rho$. \emph{Step 3 (three branches).} $\rho>1\Rightarrow E<1$ (excluded); $\rho=1$ exact survival; $\rho<1\Rightarrow E>1$ (weak $abc$). Any strict bound $\gamma\ge\gamma_{0}>8/9$ kills the last cell. \emph{Step 4 (decoupled model).} If $\gamma=\gamma_{0}$ is $q$-independent, $t=\frac{\gamma_{0}q}{q-1}=\frac{12}{N}$ gives $q=\frac{12}{12-N\gamma_{0}}$. For $\gamma_{0}\in\{1,\frac25,\frac45\}$ and $N\in\{1,7,13,19\}$ the values are $\{\frac{12}{11},\frac{12}5,-12,-\frac{12}7\}$, $\{\frac{30}{29},\frac{30}{23},\frac{30}{17},\frac{30}{11}\}$, $\{\frac{15}{14},\frac{15}8,\frac{15}2,-\frac{15}4\}$ --- none an odd integer $\ge5$; only $\gamma_{0}=8/9$ gives $(13,27)$. \emph{Step 5.} Since $8/9=\gamma_{K}(27)$ varies with $q$, the last cell must couple $\gamma$ to $q$ and hit equality exactly. \end{proof} \begin{corollary}[Orbit classification]\label{cor:orbits} Assume the within-orbit equality of the $n_{j}$ (Assumption~\ref{ass:count}). $\Theta=12\iff\sum_{j=1}^{4}n_{j}=6$. The five orbit partitions of the four $Y_{1}$-exceptional points give: \begin{center} \begin{tabular}{c|c|c|c} \toprule orbit & $\Theta\le15$ attainable & $(n_{j})$ with $\Theta=12$ & verdict \\ \midrule $[4]$ (transitive) & $\{8\}$ & none & \textbf{excluded} \\ $[3,1]$ & $\{8,10,12,14\}$ & $(1,3)$ & allowed \\ $[2,2]$ & $\{8,12\}$ & $(1,2)$ & allowed \\ $[2,1,1]$ & $\{8,10,12,14\}$ & $(1,1,2),(1,1,3),(1,2,2)$ & allowed \\ $[1^{4}]$ & $\{8,10,12,14\}$ & $(1,1,1,3),(1,1,2,2)$ & allowed \\ \bottomrule \end{tabular} \end{center} Only the transitive orbit $[4]$ excludes $\Theta=12$; this refines Corollary~\ref{cor:transitive} to ``exclude the four non-transitive orbits''. \end{corollary} \begin{proof} \emph{Step 1.} By Corollary~\ref{cor:parity}, $\Theta=2\sum_{j=1}^{4}n_{j}$; a block of orbit size $s$ with common torsion length $n$ contributes $sn$. $\Theta=12\iff\sum n_{j}=6$ with $\Theta\le15$ by optimality. \emph{Step 2 (enumeration).} $[4]$: $\Theta=8n\le15\Rightarrow n=1$, $\Theta=8$: no $\Theta=12$. $[3,1]$: $3n_{1}+n_{2}=3\Rightarrow (n_{1},n_{2})=(1,3)$. $[2,2]$: $2(n_{1}+n_{2})=6\Rightarrow(1,2)$. The remaining partitions are in the table. \emph{Step 3.} Only $[4]$ excludes $\Theta=12$; this is exactly the refinement claimed. \emph{Step 4 (boundary).} Under transitivity $\Theta=8$, $t=12$, $\gamma_{\mathrm{crit}}\ge9.6$ meets neither convention: all-$q$ unattainability, rigorously. \end{proof} \begin{corollary}[Strict No-Go inside the literature construction]\label{cor:literature} The curve $C:u^{q}=x(x-1)$ and $F=\mathbb Q(\alpha^{1/q})$ are the extensions introduced in \S\ref{sec:weak}; the original construction of \cite{SouleABC} and the IH\'ES Lecture~3 \cite{SouleIHES} fix $q=5$ ($y^{2}+y=x^{5}$, $F=K(\sqrt[5]{m})$). At fixed $q=5$, $\gamma_{\mathrm{crit}}\in\{48/5,\,48/35,\,48/65,\,48/95\}$ while the two literature conventions are $\gamma=1$ and $\gamma_{K}(5)=2/5$; \textbf{neither equals any $\gamma_{\mathrm{crit}}$}. Hence \begin{quote} inside Soul\'e's original construction, full $abc$ is unattainable, rigorously and \textbf{independently of the exact value of $\Theta$}. \end{quote} Corollary~\ref{cor:nogo} is thus reproduced independently (via $\gamma\ne\gamma_{\mathrm{crit}}$, not via mod $3$). The last cell $(27,12)$ exists only in the extended family and requires three exact conditions simultaneously: (1) $\sum n_{j}=6$ (non-transitive orbit); (2) $q=27$ with $q\le27$ (boundary allowed); (3) $\gamma=8/9$ exact (any $\rho>1$ excluded). This is a measure-zero set; Theorem~\ref{thm:qrigid} shows it is \textbf{excluded by the hypotheses of Theorem~A} ($q=27$ requires $g=13$). \end{corollary} \begin{proof} \emph{Step 1 (extension vs.\ original).} $C:u^{q}=x(x-1)$, $F=\mathbb Q(\alpha^{1/q})$ are introduced in \S\ref{sec:weak}; the original construction fixes $q=5$. \emph{Step 2 (fixed $q=5$).} By \eqref{eq:Eunified}, $\gamma_{\mathrm{crit}}(5)=\frac{12\cdot4}{5N}=\frac{48}{5N}$, giving $\{\frac{48}5,\frac{48}{35},\frac{48}{65},\frac{48}{95}\}$ for the four $N$. \emph{Step 3 (comparison).} $\frac{48}{5N}=1\Rightarrow N=48/5\notin\mathbb Z$; $\frac{48}{5N}=\frac25\Rightarrow N=24\notin\{1,7,13,19\}$. Neither convention equals any $\gamma_{\mathrm{crit}}$. \emph{Step 4.} Inside the original construction full $abc$ is unattainable, rigorously and independently of $\Theta$; Corollary~\ref{cor:nogo} is reproduced without mod $3$. \emph{Step 5 (upgrade).} Theorem~\ref{thm:qrigid} upgrades the three exact conditions from ``undecided'' to ``excluded by the hypotheses of Theorem~A'': $q=27$ requires $g=13$. \end{proof} \subsection{The line-bundle determination of \texorpdfstring{$\gamma$}{gamma} and the closure of the Kummer family}\label{subsec:gamma} We now determine the last quantity left heuristic, $\gamma$. The conclusion comes from the \emph{original statement} of \cite{SouleABC} together with an explicit height computation, each confirming the other. \begin{theorem}[$\gamma$ is fixed by the canonical bundle]\label{thm:gammastrict} \cite{SouleABC} fixes a genus-$2$ curve $\mathcal C/K_{0}$ and lets $h:\mathcal C(\overline{K_{0}})\to\mathbb R$ be the logarithmic height \textbf{attached to the canonical bundle} of $\mathcal C$; Theorem~A is $h(P)\le(12+\eps)\delta(F)/d+\beta(\eps)$. In \S7.2 of \cite{SouleABC}, $\mathcal C$ is the complete model of $y^{2}+y=x^{5}$ ($K_{0}=\mathbb Q$), $y=b/a$, $m=y^{2}+y=bc/a^{2}$, $F=K(\sqrt[5]{m})$, $P=(\sqrt[5]{m},y)$. Then \begin{equation}\label{eq:gammastrict} h_{K_{\mathcal C}}(P)=\frac25\log c+O(1), \qquad \gamma=\frac25\quad\text{(strict)} . \end{equation} \end{theorem} \begin{proof} \emph{Step 1 (hyperelliptic structure).} A genus-$2$ curve is hyperelliptic; the involution $\iota:(x,y)\mapsto(x,-1-y)$ of $y^{2}+y=x^{5}$ fixes $x$, and the quotient $\pi_{h}=x:\mathcal C\to\mathbb P^{1}$ is $2:1$. Since $K_{\mathcal C}\cong\pi_{h}^{\ast}\mathcal O(1)$ ($\deg=2=2g-2$), for any adelic metric $h_{K_{\mathcal C}}(P)=h_{\mathcal O(1)}(\pi_{h}(P))+O(1)$; the metric choice only changes $O(1)$, the main term being fixed by the numerical class. \emph{Step 2 (valuations of $m$).} $y=b/a$, $m=y^{2}+y=\frac ba\cdot\frac{a+b}a=\frac{bc}{a^{2}}$ with $\gcd(bc,a)=1$. If $p\mid bc$: $v_{p}(m)>0$, so $\log^{+}|m|_{p}=0$. If $p\mid a$: $v_{p}(m)=-2v_{p}(a)$, so $\log^{+}|m|_{p}=2v_{p}(a)\log p$. At archimedean places $|m|=|bc|/a^{2}<1$, so $\log^{+}=0$. Summing, $h(m)=2\log c+O(1)$. \emph{Step 3 (fifth root).} Let $d=[F:\mathbb Q]$ (not necessarily $5$). From $u^{5}=m$, $|u|_{w}=|m|_{p}^{1/5}$ independently of $d$; since $\sum_{w\mid p}[F_{w}:\mathbb Q_{p}]=d$, \[ h_{F}(u)=\frac1d\sum_{w\mid p}[F_{w}:\mathbb Q_{p}]\cdot\frac15\log^{+}|m|_{p} =\frac15\cdot2\log c+O(1)=\frac25\log c+O(1). \] \emph{Step 4.} Taking $\pi_{h}(P)=u$ gives $h_{K_{\mathcal C}}(P)=\frac25\log c+O(1)$, i.e. $\gamma=2/5$ strictly. Appendix~\ref{app:kummer} records all valuation details. \end{proof} \begin{theorem}[$\gamma_{K}(q)$ is a strict canonical-bundle height formula]\label{thm:gamK} For $C:u^{q}=x(x-1)$ of \S\ref{sec:weak} ($g=(q-1)/2$), Riemann--Hurwitz gives $2g-2=-2q+3(q-1)=q-3$; a basis of holomorphic differentials is $\{dx/u^{j}:\ j=\frac{q+1}{2},\dots,q-1\}$ (count $\frac{q-1}{2}=g$), and the canonical map is $\varphi_{K}(P)=[u^{g-1}:\cdots:u:1]$ with $g-1=\frac{q-3}{2}$. Likewise $h_{F}(u)=\frac2q\log c+O(1)$, so \begin{equation}\label{eq:gamK} h_{K_{C}}(P)=\frac{q-3}{q}\log c+O(1), \qquad \gamma_{K}(q)=\frac{q-3}{q}. \end{equation} At $q=5$ this gives $\frac25$, in exact agreement with Theorem~\ref{thm:gammastrict}: $\gamma_{K}$ is a strict formula, not a heuristic. \end{theorem} \begin{proof} \emph{Step 1 (divisors).} On $C:u^{q}=x(x-1)$, $\Div(u)=Q_{0}+Q_{1}-2Q_{\infty}$; with local parameter $t=x^{-1}$ at $\infty$ in the hyperelliptic model, $v_{\infty}(dx/u^{j})=-q-1+2j$. \emph{Step 2 (holomorphy).} $-q-1+2j\ge0\iff2j\ge q+1\iff j\ge\frac{q+1}2$; with $j\le q-1$ the basis is $\bigl\{\frac{x^{j-(q+1)/2}dx}{u^{j}}:\ j=\frac{q+1}2,\dots,q-1\bigr\}$, of cardinality $q-1-\frac{q+1}2+1=\frac{q-1}2=g$. \emph{Step 3 (canonical map).} The basis vectors are multiples of $(1,u,\dots,u^{g-1})$, so $\varphi_{K}(P)=[u^{g-1}:\cdots:u:1]$, $g-1=\frac{q-3}2$. \emph{Step 4 (height).} $h_{K_{C}}(P)=h(\varphi_{K}(P))+O(1)=(g-1)h_{F}(u)+O(1) =\frac{q-3}2\cdot\frac2q\log c+O(1)=\frac{q-3}{q}\log c+O(1)$. \emph{Step 5.} At $q=5$ this is $\frac25$, matching Theorem~\ref{thm:gammastrict} exactly. \end{proof} \begin{corollary}[Triple verification; closure of convention B]\label{cor:gamma3} $\gamma=2/5$ has three independent sources: (i) Theorem~A of \cite{SouleABC} specifies $h$ as the $K_{\mathcal C}$-height; (ii) the explicit computation of Theorem~\ref{thm:gammastrict}; (iii) \textbf{the intermediate constant $30$ of \S7.2}: Theorem~A gives $h\le12\,\delta/d$ while \S7.2 requires $\log H\le30\,\delta/d$; from $h=\gamma\log H+O(1)$, \begin{equation}\label{eq:30} \frac{12}{\gamma}=30\ \Rightarrow\ \gamma=\frac25, \qquad 30\cdot\frac45=24\ \text{(the Theorem~B exponent, verbatim)} . \end{equation} Moreover, \textbf{ray invariance}: $L\mapsto L^{\otimes a}$ sends $\gamma\mapsto a\gamma$ and $t\mapsto at$, so $E=t\eta/\gamma$ is unchanged: $E$ depends only on the ray of $L$. If the two conventions of \S\ref{sec:last} were rescalings of the same $L$, $E$ would coincide; but $24\ne48/5$, and the ratio is exactly $\gamma_{B}/\gamma_{A}=5/2$. Hence $\gamma=1$ ($E=48/5$) is \emph{not} the normalisation of Theorem~A, but a different comparison inequality. The two columns of \S\ref{sec:last} are thereby declared incompatible as normalisations of one and the same $t=12/N$ chain, and this corollary closes the question. \end{corollary} \begin{remark}[Normalisation dictionary]\label{rem:normdict} Write $t_{A}$ for the Theorem-A normalisation ($\gamma=\frac25$) and $t^{\ast}$ for the target coefficient: since $E=t\eta/\gamma$ and $\gamma_{\mathrm{canon}}=\frac25$ while $\gamma_{\mathcal O(1)}=1$, one has \[ t^{\ast}=\tfrac52\,t_{A}, \qquad C_{(\gamma=1)}=\tfrac52\,C . \] Hence the chain-B target $t^{\ast}=\frac54$ of \S\ref{sec:last} and the chain-A target $t^{\ast}=\frac12$ of Corollary~\ref{cor:nogo} are two gauges of one and the same landing condition $\frac52\,t_{A}=\frac54$, i.e.\ $t_{A}=\frac12$; the discrepancy in the earlier drafts was purely notational. \end{remark} \begin{theorem}[Closure of the hypotheses of the Kummer family]\label{thm:qrigid} The hypotheses of Theorem~A are a \textbf{fixed} genus-$2$ curve $\mathcal C$ and a \textbf{fixed $5$-th} Kummer extension $F=K(\sqrt[5]{m})$; $t=12$ is proved for that fixed curve. The extension of \S\ref{sec:weak} treats $q$ as a free parameter, but $C:u^{q}=x(x-1)$ has genus $g=(q-1)/2$, so \[ g=2\iff q=5 . \] For $q=27$, $g=13$ and Theorem~A does not apply; nor can the whole $t=12/N$ chain (\S\ref{sec:arith}--\S\ref{sec:local}), built on the genus-$6$ fibration and the genus-$2$ curve, be extrapolated to $g=13$ (its $\om^{2}=12k$, $\deg L_{F}=6$, $c=1/4$ are all pinned by $g=2$, and changing $L$ moves $t$ and $\beta(\eps)$ simultaneously). Hence \begin{quote} the unique landing point $(\Theta,q)=(12,27)$ of Corollary~\ref{cor:landeq} requires $g=13$, its hypotheses fail, and \textbf{the last cell is empty}. \end{quote} The three exact conditions are thereby upgraded from ``undecided'' to ``excluded by the original hypotheses'': $\sum n_{j}=6$ remains open but has no landing object; $q=27$ violates $g=2$; $\gamma=8/9$ is false ($\gamma$ is fixed to $2/5$ by $K_{\mathcal C}$). Thus inside the construction of Theorem~A, full $abc$ is unattainable, rigorously, independently of $\Theta,T$, orbits, transversality and mod $3$. \end{theorem} \begin{proof} \emph{Step 1 (original hypotheses).} Theorem~A: a fixed genus-$2$ curve $\mathcal C$ over $K_{0}$, $h$ attached to $K_{\mathcal C}$; \S7.2 uses a fixed quintic Kummer extension $F=K(\sqrt[5]{m})$; $t=12$ is proved for that fixed curve. \emph{Step 2 (genus formula).} For $C:u^{q}=x(x-1)$, \eqref{eq:eta} gives $g=\frac{q-1}2$, so $g=2\iff q=5$. \emph{Step 3 ($q=27$).} $g=13$: Theorem~A does not apply. The $t=12/N$ chain rests on the Kodaira--Parshin construction with a genus-$6$ fibration over a genus-$2$ curve; $\om^{2}=12k$, $\deg L_{F}=6$, $c=1/4$ are pinned by $g=2$, and replacing $L$ moves $t$ and $\beta(\eps)$ at once (anticipated in the remark after Theorem~\ref{thm:weak}). \emph{Step 4 (exclusion by hypotheses).} The unique landing $(\Theta,q)=(12,27)$ requires $g=13$: the last cell is empty. The three conditions: (1) $\sum n_{j}=6$ open but landingless; (2) $q=27$ violates $g=2$; (3) $\gamma=8/9$ false ($\gamma=2/5$ by Theorem~\ref{thm:gammastrict}). \emph{Step 5.} Inside the construction of Theorem~A, full $abc$ is unattainable, rigorously, with no use of $\Theta,T$, orbits, transversality or mod $3$. Corollaries~\ref{cor:nogo} and \ref{cor:literature} keep their conclusions, but their reasons upgrade from ``arithmetic exclusion'' to ``exclusion by hypotheses''. Theorem~\ref{thm:weak} is unaffected, with domain of validity exactly $q=5$. \end{proof} \begin{remark} The conclusions of Corollaries~\ref{cor:nogo} and \ref{cor:literature} are unchanged, but their \emph{justification} is replaced: not ``at $q=5$, $\gamma\ne\gamma_{\mathrm{crit}}$'' (arithmetic exclusion) but ``at $q\ne5$ the theorem does not apply at all'' (exclusion by hypotheses), a strictly stronger statement. Theorem~\ref{thm:weak} is unaffected, but its domain of validity should be stated as $q=5$. \end{remark} % ===================================================================== \section{Refinements: discriminant stratification and tower monotonicity}\label{sec:refine} % ===================================================================== The quantity $\eta=\frac{\delta(F)/d}{\log N}$ has so far been used only through its uniform value $4/5$ (Lemma~\ref{lem:kummer}). It is in fact a \emph{pointwise} estimate of the Kummer discriminant, and stratifying by $v_{p}(abc)\bmod5$ yields a bound strictly stronger than $N^{24}$. The key observation: $\eta$ is \emph{not} on the discrete coefficient chain $t=12/N$, so even with $t=12$ unique, $\gamma=2/5$ fixed and $q=5$ locked, shrinking $\eta$ still shrinks the exponent. \subsection{Irreducibility} \begin{lemma}[Irreducibility, unconditional]\label{lem:irr} Let $q$ be an odd prime, $\lambda=a/c$, $\alpha=\lambda(\lambda-1)=-ab/c^{2}$, $F=\mathbb Q(\alpha^{1/q})$. Then $[F:\mathbb Q]=q$. \end{lemma} \begin{proof} If $x^{q}-\alpha$ were reducible, then $\alpha=\beta^{q}$ for some $\beta\in\mathbb Q$. Reading valuations prime by prime for $p\mid abc$: $p\mid c$ gives $v_{p}(\alpha)=-2v_{p}(c)$, a multiple of $q$ (as $\gcd(q,2)=1$, $q\mid v_{p}(c)$); $p\mid a$ gives $q\mid v_{p}(a)$; $p\mid b$ likewise. Since $\gcd(a,b)=1$, $a=A^{q}$, $b=B^{q}$, $c=C^{q}$, i.e. $A^{q}+B^{q}=C^{q}$ in positive integers, excluded by Fermat's Last Theorem ($q=5$ by the elementary Dirichlet--Legendre proof; general $q$ by Wiles). \end{proof} In the extension family of \S\ref{sec:weak}, the degree $d=q$ was previously tagged as an assumption; it is now unconditional. But Theorem~A still has hypotheses only at $q=5$, so no conclusion changes. \subsection{Strict stratification of tame local ramification} For $p\nmid5$ write $\alpha=p^{n}u$ with $u\in\mathbb Z_{p}^{\times}$. Since $\gcd(5,2)=1$, \begin{equation}\label{eq:qdiv} 5\mid n\iff5\mid v_{p}(abc),\qquad n\in\{v_{p}(a),\ v_{p}(b),\ -2v_{p}(c)\}. \end{equation} \emph{Case I ($5\nmid n$).} $x^{5}-p^{n}u$ is Eisenstein, so $F_{p}/\mathbb Q_{p}$ is totally ramified, $e=5$, $f=1$; $p\nmid5$ so the ramification is tame and Dedekind's formula gives $v_{p}(D_{F/\mathbb Q})=\sum_{i}f_{i}(e_{i}-1)=4$. \emph{Case II ($5\mid n$).} With $y=x/p^{n/5}$ the equation becomes $y^{5}=u$; the derivative $5y^{4}$ is a $p$-adic unit at every root ($p\nmid5$, $y\in\mathbb Z_{p}^{\times}$), so $\mathbb Z_{p}[y]/(y^{5}-u)$ is \'etale and $F_{p}/\mathbb Q_{p}$ is unramified or split, $e=1$, $v_{p}(D_{F/\mathbb Q})=0$. \emph{Case III ($p=5$).} Wild ramification; $\log5$ is an absolute constant, absorbed into $O(1)$. Define the \textbf{discount factor} \begin{equation}\label{eq:M5} M_{5}:=\prod_{\substack{p\mid abc\\ p\ne5\\ 5\mid v_{p}(abc)}}p \ \subseteq\ \rad(abc). \end{equation} From Lemma~\ref{lem:irr} ($d=5$) and the three cases, \begin{equation}\label{eq:deltastrat} \frac{\delta(F)}d=\frac45\log\frac{N}{M_{5}}+O(1), \qquad N=\rad(abc)\cdot2^{O(1)}. \end{equation} This is an \emph{equality}, not an upper bound: the case distinction uses only Hensel's lemma and tame Dedekind; the wild contribution at $p=5$ and the $2^{O(1)}$ convention on $\rad$ enter $O(1)$. For later use we record the explicit wild bound: writing $v_{5}(\alpha)=5m+r$, $r\in\{0,1,2,3,4\}$ and substituting $y=x/5^{m}$, the equation becomes $y^{5}-5^{r}u$ with polynomial discriminant of $5$-adic valuation $5+4r\le21$; the field discriminant divides the polynomial discriminant, so $v_{5}(\mathfrak d_{F})\le21$ regardless of $|v_{5}(abc)|$. The $O(1)$ in \eqref{eq:deltastrat} is therefore uniform, not growing with $c=5^{k}$. \subsection{The discounted weak $abc$} \begin{theorem}[Pointwise discounted weak $abc$, conditional on Theorem~A]\label{thm:discount} Under the hypotheses of Theorem~A (fixed $q=5$), for every coprime triple $a+b=c$ and every $\eps>0$, \begin{equation}\label{eq:discount} c\le A_{\eps}\Bigl(\frac{N}{M_{5}}\Bigr)^{24+\eps}, \qquad E_{\mathrm{eff}}=24(1-\theta),\quad \theta:=\frac{\log M_{5}}{\log N}\in[0,1). \end{equation} \end{theorem} \begin{proof} Theorem~A: $h_{K_{\mathcal C}}(P)\le(12+\eps)\delta(F)/d+O_{\eps}(1)$; Theorem~\ref{thm:gammastrict}: $h_{K_{\mathcal C}}(P)=\gamma\log c+O(1)$ with $\gamma=\frac25$; \eqref{eq:deltastrat}: $\delta(F)/d=\frac45\log(N/M_{5})+O(1)$. Chaining, $\frac25\log c\le(12+\eps)\frac45\log(N/M_{5})+O_{\eps}(1)$, i.e. $\log c\le(24+2\eps)\log(N/M_{5})+O_{\eps}(1)$; renaming $\eps$ gives the claim. \end{proof} This is the first output of the framework that \emph{varies with the triple} and is strictly stronger than $N^{24+\eps}$: whenever $\theta>0$ the exponent is strictly below $24$. Quantitatively: $(1,31,32)$ has $abc=2^{5}\cdot31$, so $M_{5}=2$, $\theta=\log2/\log(2\cdot31)\approx0.168$, $E_{\mathrm{eff}}\approx19.97$. On the other hand $(9,16,25)$ ($v=(4,2,2)$, all multiplicities prime to $5$) gives $M_{5}=1$, $\theta=0$: the discount vanishes on infinitely many triples (congruent-number type), and on the classical hard family $c=2^{5k}$ one has $\log M_{5}=\log2+O(1)$, $\theta\to0$, so the discount disappears exactly on the hardest triples. \begin{theorem}[Bootstrapping to exponent $1$ is impossible]\label{thm:noboot} \[ \inf_{\substack{a+b=c\\ \gcd(a,b)=1}}\theta=0, \] and there is no mechanism pushing $E_{\mathrm{eff}}$ uniformly down to $1$. \end{theorem} \begin{proof} \emph{(i) The discount has only an upper bound.} Each $p\in M_{5}$ has $v_{p}\ge5$, so $v_{p}-1\ge4$ and $\log M_{5}\le E_{\mathrm{mult}}/4$ where $E_{\mathrm{mult}}=\log(abc)-\log N$; with $\log(abc)\le3\log c$ and a bound $c\le N^{E}$ one gets $\theta\le(3E-1)/4$ --- an \emph{upper} bound. Bootstrap needs a \emph{lower} bound on $\theta$: the direction is opposite. \emph{(ii) The infimum is attained.} $(9,16,25)$, $(2,3,5)$ and squarefree triples give $\theta=0$, so no uniform positive lower bound exists; the uniform improvement of the pointwise discount over $N^{24}$ is zero. \emph{(iii) Hard families kill the discount asymptotically.} With $a=1$, $c=2^{5k}$, $b=2^{5k}-1$: $\log M_{5}=\log2+O(1)$ while $\log N\sim\log\rad(2^{5k}-1)\to\infty$, so $\theta\to0$ while $E_{\mathrm{mult}}\approx5k\log2\to\infty$: the ratio discount/excess tends to $0$. \emph{(iv) Multiple $q$ do not help.} For general odd $q$, $\delta/d=\frac{q-1}{q}\log(N/M_{q})$ with $M_{q}=\prod_{q\mid v_{p}}p$ and $\log M_{q}\le E_{\mathrm{mult}}/(q-1)$ (the larger $q$, the smaller the discount cap). Covering all $v_{p}\ge2$ by $q\in\{3,5,7,\dots\}$ requires instances of Theorem~A at higher genus; Theorem~\ref{thm:qrigid} pins $t=12$ to $g=2$ and $g=2\iff q=5$, so the hypotheses fail; and prime $v_{p}$ would need $q\ge v_{p}$, driving the discount coefficient $\frac1{q-1}\to0$. This branch is now closed: bridge breaking (Theorem~\ref{thm:bridge}) destroys the scale $\delta/d\asymp\log N$ whenever the generator is changed, and the mixed-decomposition loop (Theorem~\ref{thm:mixloop}) requires an $S$-smoothness assumption stronger than $abc$; the only remaining Kummer-internal question is the higher-genus analogue of Theorem~A (Open problem~(2)). \end{proof} \begin{remark} This refutation is independent of $\Theta,T$, orbit classification, transversality and mod $3$; it complements Theorem~\ref{thm:qrigid} (which moves $q,\gamma$) by moving $\eta$. Combined, full $abc$ inside the Theorem~A construction is unattainable for two independent reasons. \end{remark} \subsection{Tower monotonicity of the discriminant-to-degree ratio} The most natural remaining idea --- change the Galois group or the field to lower $\eta$ --- is sealed by a purely algebraic lemma. \begin{lemma}[Tower monotonicity]\label{lem:tower} For every tower $K\supseteq F\supseteq\mathbb Q$ of number fields, \begin{equation}\label{eq:tower} \frac{\delta(K)}{[K:\mathbb Q]} =\frac{\delta(F)}{[F:\mathbb Q]}+\frac{\delta(K/F)}{[K:\mathbb Q]} \ \ge\ \frac{\delta(F)}{[F:\mathbb Q]} . \end{equation} \end{lemma} \begin{proof} The relative discriminant formula $\mathfrak d_{K/\mathbb Q} =\mathfrak d_{F/\mathbb Q}^{[K:F]}\cdot N_{F/\mathbb Q}(\mathfrak d_{K/F})$. Take logarithms and divide by $[K:\mathbb Q]=[K:F]\,[F:\mathbb Q]$: \[ \frac{\delta(K)}{[K:\mathbb Q]} =\frac{[K:F]\,\delta(F)}{[K:F]\,[F:\mathbb Q]}+\frac{\delta(K/F)}{[K:\mathbb Q]} . \] Since $\mathfrak d_{K/F}$ is an integral ideal, $\delta(K/F)\ge0$. \end{proof} \begin{theorem}[Changing the field cannot lower $\eta$]\label{thm:nodomain} \begin{enumerate}[leftmargin=1.8em,itemsep=0.2em,label=(\roman*)] \item Passing to the Galois closure $L$ does not lower $\eta$ (see below: $\delta(L)/20=\delta(F)/5+O(1)$); \item adjoining a large unramified extension, or passing to any $K\supset F$, does not lower $\eta$; \item since $h_{K_{\mathcal C}}(P)$ is invariant under base change, the right-hand side of Theorem~A is \textbf{minimised by $F$ itself} over all towers containing $F$; ``lowering $E$ from $24$ to $6$'' by changing fields is illegitimate; \item Theorem~A need not state ``the minimal field of definition of $P$'': over a larger $K$ the right-hand side only grows, so no spuriously strong bound can appear. \end{enumerate} \end{theorem} \begin{proof} \emph{(i) Artin chain.} Let $L=F(\zeta_{5})$, $G=\Gal(L/\mathbb Q)=F_{20}=C_{5}\rtimes C_{4}$. Then $\mathrm{Ind}_{C_{4}}^{G}\mathbf1=\mathbf1\oplus\rho_{4}$ with $\dim\rho_{4}=4$, and $m_{\chi}=\langle\mathrm{Ind}_{C_{4}}^{G}\mathbf1,\chi\rangle$ equals $1$ for $\chi=\rho_{4}$ and $0$ otherwise. At a Case-I prime, $I_{p}=C_{5}$ is totally and tamely ramified, so $a_{p}(\rho_{4})=\dim\rho_{4}-\dim\rho_{4}^{I_{p}}=4$ and $\mathfrak f(\rho_{4})=\prod_{p\mid N/M_{5}}p^{4}$; hence $\delta(F)=4\log(N/M_{5})+O(1)$ (this is \eqref{eq:artin}). On the closure side the Artin discriminant formula weights conductors by $\chi(1)$, so $\rho_{4}$ contributes $\mathfrak f(\rho_{4})^{4}=(N/M_{5})^{16}$ and the three characters factoring through $C_{4}$ contribute only powers of $5$: $\delta(L)=16\log(N/M_{5})+O(1)$ (this is \eqref{eq:artinL}, where the $\chi(1)$-power erratum is recorded). Therefore $\frac{\delta(L)}{20}=\frac45\log(N/M_{5})+O(1)=\frac{\delta(F)}5+O(1)$: passing to the Galois closure changes nothing. \emph{(ii)} Every tower $K\supseteq F$ satisfies \eqref{eq:tower} of Lemma~\ref{lem:tower}; an unramified extension has $\delta(K/F)=0$, and $\delta(K/F)\ge0$ in general: the ratio cannot decrease. \emph{(iii)} By Theorem~\ref{thm:gammastrict}, $h_{K_{\mathcal C}}(P)=\frac25\log c+O(1)$ is invariant under base change, while $\delta(K)/[K:\mathbb Q]$ is nondecreasing in towers by \eqref{eq:tower}; the right-hand side of Theorem~A is therefore minimised by $F$ itself among all fields containing $F$. \emph{(iv)} is the contrapositive of (iii). \end{proof} This No-Go uses \textbf{not a single line} of the original text of Theorem~A: only the relative discriminant formula and height stability. It complements Theorem~\ref{thm:qrigid} (exclusion by hypotheses) and Theorem~\ref{thm:noboot} ($\inf\theta=0$): together they give three independent reasons for the unattainability of full $abc$ inside the Theorem~A construction. \subsection{The Artin-conductor identity for \texorpdfstring{$4/5$}{4/5}} Let $L=F(\zeta_{5})$ be the Galois closure and $G=\Gal(L/\mathbb Q)=C_{5}\rtimes C_{4}=F_{20}$, $|G|=20$. The irreducible character degrees are $(1,1,1,1,4)$ with $\sum\dim^{2}=20$; $F=L^{C_{4}}$ and $\mathrm{Ind}_{C_{4}}^{G}\mathbf 1=\mathbf 1\oplus\rho_{4}$, $1+4=5=[F:\mathbb Q]$. The Artin formula $\mathfrak d_{F/\mathbb Q}=\prod_{\chi}\mathfrak f(\chi)^{m_{\chi}}$ with $m_{\chi}=\langle\mathrm{Ind}_{C_{4}}^{G}\mathbf1,\chi\rangle$ receives a contribution only from $\chi=\rho_{4}$ ($m=1$). At a prime $p$ of Case I, $I_{p}=C_{5}$ is totally and tamely ramified, so $a_{p}(\rho_{4})=\dim\rho_{4}-\dim\rho_{4}^{I_{p}}=4$; hence $\mathfrak f(\rho_{4})=\prod_{p\mid N/M_{5}}p^{4}$ and \begin{equation}\label{eq:artin} \delta(F)=4\log\frac{N}{M_{5}}+O(1), \qquad \frac{\delta(F)}5=\frac45\log\frac{N}{M_{5}}+O(1), \end{equation} reproducing \eqref{eq:deltastrat} exactly, with $\frac45=\frac{n-1}{n}$ coming from $\sum_{\chi\ne1}m_{\chi}\chi(1)=n-1$ ($n=5$): a representation-theoretic identity, not an estimated constant. The distributional behaviour of the discount is quantified in Appendix~\ref{app:distribution}: $\theta$ vanishes in density $1-o(1)$ (Theorem~M.3). On the Galois-closure side, the discriminant formula weights conductors by $\chi(1)$, so $\rho_{4}$ contributes $\mathfrak f(\rho_{4})^{4}=(N/M_{5})^{16}$, and the three characters factoring through $C_{4}$ contribute only powers of $5$ (an $O(1)$): \begin{equation}\label{eq:artinL} \delta(L)=16\log\frac{N}{M_{5}}+O(1), \qquad \frac{\delta(L)}{20}=\frac45\log\frac{N}{M_{5}}+O(1), \end{equation} identical to $\delta(F)/5$. (A first computation missed the $\chi(1)$-power and produced $\delta(L)/20=\frac15\log(N/M_{5})$; that value, and the speculation ``$E=24/4=6$ over $L$'', are retracted: the relative discriminant formula gives $4\cdot4\log=16\log$, and $16/20=4/5$.) \subsection{Group-theoretic boundary} Tame local ramification gives $v_{p}(\mathfrak d)=|I_{p}|-1$ with $I_{p}$ cyclic, hence local coefficients $\frac{|I_{p}|-1}{|I_{p}|}$: $C_{2}\to\frac12$, $C_{3}\to\frac23$, $C_{4}\to\frac34$, $C_{5}\to\frac45$. Case I for $F_{20}$ has $I_{p}=C_{5}$ (total), so $\frac45$ is pinned by the inertia group; lowering $\eta$ below $\frac45$ requires changing the Galois group, i.e.\ leaving the quintic Kummer construction --- echoing $G=\mathrm{PSL}_{2}(\mathbb F_{7})$ of \S\ref{sec:setup} (irreducible degrees $(1,3,3,6,7,8)$, $\sum\dim^{2}=168$). Two hard constraints remain: tower monotonicity (Lemma~\ref{lem:tower}) says enlarging the group alone does not lower $\eta$; and one would need to rebuild $P\in\mathcal C(F)$ with $h(P)\asymp\log c$, while $\gamma=2/5$ is the product of $K_{\mathcal C}$ and the fifth root (Theorem~\ref{thm:qrigid}, $g=2\iff q=5$). This direction is closed in \S\ref{sec:bridge}: bridge breaking (Theorem~\ref{thm:bridge}) shows that changing the generator destroys the scale $\delta/d\asymp\log N$, and the mixed-type loop (Theorem~\ref{thm:mixloop}) requires an $S$-smoothness assumption stronger than $abc$. Route~1 (improving the constant $12$ itself) is likewise closed in Appendix~\ref{app:routeone}. \begin{remark}[Correction: the transitivity proviso]\label{rem:correct} The list $\frac{|I_{p}|-1}{|I_{p}|}$ above assumes that inertia acts transitively on the primes above $p$ (automatic for Galois extensions). For non-Galois $F/\mathbb Q$ there is no single inertia group, and the correct general tame formula is $f_{p}=n-r_{p}$ with $r_{p}=\sum_{i}f_{i}\in[1,n]$: the local coefficient $\frac{n-r_{p}}{n}$ ranges over the finer lattice $\{0,\frac1n,\dots,\frac{n-1}{n}\}$. This refinement --- missed in earlier rounds and worked out in \S\ref{sec:bridge} --- genuinely opens one new degree of freedom, which is then sealed there by discriminant-support constraints. \end{remark} \begin{remark}[Meta-observation] Across the twenty-two audit rounds, every attempt to move a coefficient hits the \emph{scalar-layer invariance} of Weil heights and discriminants under tower extensions. The bottleneck of the DIC framework is not an unproved lemma: it reads only scalar-layer invariants. Any representation-layer freedom (decomposition type, Galois group, monodromy) produces no new inequality as long as the scalar layer is unchanged. \end{remark} % ===================================================================== \section{Changing the field and the group: bridge breaking and the non-Galois lattice}\label{sec:bridge} % ===================================================================== This section records the sixteenth and seventeenth audit rounds; the eighteenth and nineteenth are recorded in Appendices~\ref{app:base} and~\ref{app:constdec}. Their subject is the only named direction left open by \S\ref{sec:refine}: \emph{leave the quintic Kummer field, or enlarge/replace the Galois group}, in order to lower $\eta$ below $\tfrac45$ --- echoing the $\mathrm{PSL}_{2}(\mathbb F_{7})$ monodromy of \S\ref{sec:setup}. The outcome is a refinement, not a breakthrough: the non-Galois decomposition types do yield a finer coefficient lattice that earlier rounds missed, but every mixed type is sealed by discriminant-support constraints; and the audit produces one genuinely new \emph{positive} result --- an ideal-theoretic upgrade of the Mason dictionary. \subsection{The loophole not covered by tower monotonicity} Lemma~\ref{lem:tower} only seals towers $K\supseteq F_{5}=\mathbb Q(\alpha^{1/5})$ containing the Kummer field. Theorem~A, however, is an inequality $h_{K_{\mathcal C}}(P)\le(12+\eps)\frac{\delta(F)}{d}+O_{\eps}(1)$ stated for arbitrary number fields $F$ and arbitrary $P\in\mathcal C(F)$ (applicability read as in the source; text verification, tag C). One may therefore choose a field $F$ containing \emph{no} $F_{5}$: the premise of Lemma~\ref{lem:tower} fails, and $\eta$ could in principle drop to $\tfrac12$. We adopt this more permissive reading, which is the most favourable to a breakthrough; if the original text only allows Kummer-type $F$, the direction is excluded from the outset and the conclusions only strengthen. An unexpected bonus appears: $\gamma$ can be raised from $\tfrac25$ to $1$. Every genus-$2$ curve is hyperelliptic (Theorem~\ref{thm:gammastrict}), so $\pi_{h}:\mathcal C\to\mathbb P^{1}$ exists with $K_{\mathcal C}\cong\pi_{h}^{\ast}\mathcal O(1)$, and for every $P\in\mathcal C(\overline{\mathbb Q})$, \begin{equation}\label{eq:hcan} h_{K_{\mathcal C}}(P)=h\bigl(\pi_{h}(P)\bigr)+O(1). \end{equation} The thirteenth round took $\pi_{h}(P)=m^{1/5}$, giving $h=\frac25\log c$; taking instead $\pi_{h}(P)=\lambda=a/c\in\mathbb Q$ gives $h(\lambda)=\log c+O(1)$, i.e. \begin{equation}\label{eq:gamma1} \gamma=1 . \end{equation} The bridge denominator changes from $\frac52$ to $1$ --- apparently good news: if $\eta$ simultaneously dropped to $\tfrac12$, then $E=12\cdot\frac{1/2}{1}=6$, four times stronger than $24$. Both sides seem to loosen at once. The next lemma shows why this cannot be cashed. \begin{lemma}[Discriminant-support jump]\label{thm:supportjump} Let $\mathcal C:y^{2}+y=x^{5}$ (genus $2$, hyperelliptic; $y^{2}=f(x)$ with $\deg f=5$ after completing the square) and let $P\in\mathcal C(F)$ with $\pi_{h}(P)=u/v=a/c$, $\gcd(u,v)=1$. Write $F_{h}(u,v)=v^{5}f(u/v)$ for the homogenisation. Then \[ y^{2}=\frac{F_{h}(u,v)}{v^{5}} \ \Longrightarrow\ \sqrt{v\,F_{h}(u,v)}=v^{3}y\in F . \] Hence $F\supseteq\mathbb Q(\sqrt{\Delta})$ with $\Delta:=\operatorname{sqfree}\bigl(v\,F_{h}(u,v)\bigr)$, and by tower monotonicity (Lemma~\ref{lem:tower}), \begin{equation}\label{eq:supjump} \frac{\delta(F)}{d}\ \ge\ \frac{\delta\bigl(\mathbb Q(\sqrt{\Delta})\bigr)}{2} \ \ge\ \frac12\log\rad(\Delta) \ \ge\ \frac12\Bigl(\log\rad\bigl(F_{h}(a,c)\bigr)-\log N\Bigr), \end{equation} because the prime divisors of $v=c$ already lie in $S=\{p\mid abc\}$ and $N=\rad(abc)\cdot2^{O(1)}$. For $y^{2}+y=x^{5}$ one has $F_{h}(a,c)=c^{5}+4a^{5}\asymp c^{5}$. \end{lemma} \begin{proof} Only the last inequality needs a word: $\rad(\Delta)=\rad\bigl(vF_{h}(u,v)\bigr)$, and $\rad\bigl(vF_{h}(u,v)\bigr)\ge\rad\bigl(F_{h}(u,v)\bigr)/\rad(v)$ with $\rad(v)\mid N$; taking logarithms gives \eqref{eq:supjump}. \end{proof} \begin{remark}[Reverse-engineering] The support jump is the point: the prime divisors of $F_{h}(a,c)$ generically do \emph{not} lie in $S$. The original quintic Kummer construction keeps $\delta/d\asymp\log N$ controllable precisely because the ramification of $x^{5}-\alpha$ is determined by $v_{p}(\alpha)=v_{p}(abc)$; replacing the generator moves the ramification to new integers. This explains in hindsight why the source uses $\alpha^{1/5}$: it is the natural construction with $\operatorname{supp}(\mathfrak d_{F})\subseteq S$. The DIC framework had this inclusion as silent background; it is the \textbf{third lock} (L3), identified here for the first time. \end{remark} \begin{theorem}[Bridge breaking]\label{thm:bridge} With the choice of Lemma~\ref{thm:supportjump}, Theorem~A yields \begin{equation}\label{eq:bridgeout} \log c\ \le\ 12\,\frac{\delta(F)}{d}+O_{\eps}(1), \qquad \frac{\delta(F)}{d}\ \ge\ \frac12\Bigl(\log\rad\bigl(F_{h}(a,c)\bigr)-\log N\Bigr), \end{equation} and the right-hand side already exceeds $\log c$ at its \emph{minimum}: since $\log\rad F_{h}(a,c)\approx5\log c$ on test triples ($\log\rad F_{h}(a,c)/\log c\in[4.8,6.2]$) and $\log N\le3\log c$, one has $6\bigl(\log\rad F_{h}-\log N\bigr)\ge12\log c>\log c$. The output is a tautology and produces no $abc$-type conclusion. \end{theorem} \begin{proof} \emph{Step 1.} For $y^{2}+y=x^{5}$ completion of the square gives $(2y+1)^{2}=4x^{5}+1$, so $F_{h}(a,c)=c^{5}+4a^{5}\asymp c^{5}$. \emph{Step 2.} By Lemma~\ref{thm:supportjump}, $F\supseteq\mathbb Q(\sqrt{\Delta})$ and tower monotonicity (Lemma~\ref{lem:tower}) yields $\frac{\delta(F)}d\ge\frac12\bigl(\log\rad F_{h}(a,c)-\log N\bigr)$. \emph{Step 3.} Theorem~A with $\gamma=1$ (\eqref{eq:hcan}, \eqref{eq:gamma1}) gives $\log c\le12\,\frac{\delta(F)}d+O_{\eps}(1)$; at the minimum of the right-hand side this reads $\log c\le6\log\rad F_{h}(a,c)+O(1)$. \emph{Step 4 (triviality and scale change).} On the test triples of the table below, $\log\rad F_{h}(a,c)\approx5\log c$, so the minimal right-hand side of Step~3 is $\approx30\log c>\log c$: the inequality is tautological and has no $abc$-type content. The point is a change of scale: formally $\eta$ has dropped to $\tfrac12$, but the logarithmic scale of the right-hand side is now $\log\rad F_{h}$, decoupled from $\log N=\log\rad(abc)$ --- this is the bridge breaking. Numerically the effective exponent relative to $\log N$ is $12\,\eta_{\min}$ with $\eta_{\min}=\frac{\log\rad F_{h}-\log N}{2\log N}\in[1.19,3.07]$, so $12\,\eta_{\min}\approx30\gg1$. \end{proof} \begin{center} \renewcommand{\arraystretch}{1.1} \begin{tabular}{c|c|c|c} \toprule $(a,b,c)$ & $\rad\bigl(F_{h}(a,c)\bigr)$, $F_{h}=c^{5}+4a^{5}$ & $\dfrac{\log\rad F_{h}}{\log N}$ & $6\,\dfrac{\log\rad F_{h}}{\log c}$ \\ \midrule $(1,31,32)$ & $2\cdot3\cdot2796203$ & $4.03$ & $28.8$ \\ $(9,16,25)$ & $10001821$ (prime) & $4.74$ & $30.0$ \\ $(1,8,9)$ & $59053$ (prime) & $6.13$ & $30.0$ \\ $(2,3,5)$ & $3253$ (prime) & $2.38$ & $30.2$ \\ $(1,1023,1024)$ & $2^{2}\cdot193\cdot65537\cdot22253377$ & $4.46$ & $29.4$ \\ $(1,32767,32768)$ & $2^{2}\cdot3\cdot1753\cdot1795918038741070627$ & $4.63$ & $29.6$ \\ \bottomrule \end{tabular} \end{center} The mechanism is a scale change: the exponent of $abc$ was defined relative to the logarithmic scale $\log N=\log\rad(abc)$. After changing $F$, the natural scale becomes $\log\rad F_{h}(a,c)$, decoupled from $\rad(abc)$. Formally $\eta$ drops to $\tfrac12$ and $\gamma$ rises to $1$, but the derivation's premise $\delta/d\asymp\log N$ has failed --- \emph{bridge breaking}, more fundamental than any coefficient No-Go. \begin{theorem}[Reduction to $S$-smooth values, the converse being strictly stronger]\label{thm:smooth} To recover an $abc$-type output one needs simultaneously $\delta(F)/d\le\eta_{0}\log N$ and the return of the right-hand side to the scale $\log N$; by \eqref{eq:supjump} this forces \begin{equation}\label{eq:smoothneed} \rad\bigl(F_{h}(a,c)\bigr)\ \le\ N^{\,2\eta_{0}+1}. \end{equation} For $\eta_{0}=\tfrac45$ (the original construction) this needs $\rad F_{h}\le N^{13/5}$; for $\eta_{0}=\tfrac12$ (the target with inertia $C_{2}$) it needs $\rad(c^{5}+4a^{5})\le\rad(abc)^{2}$. But $F_{h}(a,c)\asymp c^{5}\gg abc\asymp c^{3}$, so \eqref{eq:smoothneed} is \textbf{strictly stronger than the $abc$ conjecture} ($abc$ only asks $\rad(abc)\gtrsim c^{1-\eps}$ and constrains nothing about the factors of $c^{5}+4a^{5}$). The lower $\eta_{0}$, the stronger the required $S$-smoothness: \[ \text{lowering }\eta\ \Longleftrightarrow\ \text{assuming an $S$-smoothness statement stronger than }abc . \] This is not ``one estimate short'' but a circle: deriving $abc$ from a hypothesis stronger than $abc$. Tagged \emph{X}. \end{theorem} \begin{proof} Combine \eqref{eq:supjump} with $\delta/d\le\eta_{0}\log N$: the lower bound must not exceed the upper bound, giving $2\eta_{0}\log N\ge\log\rad F_{h}-\log N$, i.e.\ \eqref{eq:smoothneed}. For the strength comparison take the hard family $c=2^{k}$, $a=1$: generically $\log\rad(c^{5}+4)\approx5\log c>6\log c \ge2\log N$, violating \eqref{eq:smoothneed} for $\eta_{0}=\tfrac12$; and no $abc$-type hypothesis forbids this, since $c^{5}+4$ is not part of the triple's radical. \end{proof} \emph{Function-field comparison.} In $\mathbb Q[t]$ the roots of the homogeneous form $F_{h}(u,v)$ are controlled by degree, $\deg F_{h}=5\max(\deg u,\deg v)$, and ``new prime factors'' do not exist --- this is why Mason holds. On the integer side $\rad\bigl(F_{h}(a,c)\bigr)$ is uncontrolled: integers live at the constant-degenerate point where Mason fails, and the visible face of that degeneracy is the uncontrollability of the prime support. \begin{theorem}[Closure of the $\mathrm{PSL}_{2}(7)$ direction]\label{thm:psl} The direction ``change the Galois group, abandon the quintic Kummer construction, echoing $\mathrm{PSL}_{2}(\mathbb F_{7})$'' is sealed by three facts independent of the size and solvability of the group: \begin{enumerate}[leftmargin=1.8em,itemsep=0.2em,label=(\roman*)] \item $g=2\Rightarrow\mathcal C$ hyperelliptic (Theorem~\ref{thm:gammastrict}): $\pi_{h}$ and $K_{\mathcal C}=\pi_{h}^{\ast}\mathcal O(1)$ exist for \emph{every} genus-$2$ curve; changing $\mathcal C$ does not remove the $F_{h}$ of Lemma~\ref{thm:supportjump}; \item $P\in\mathcal C(F)\Rightarrow F\supseteq\mathbb Q\bigl(\sqrt{vF_{h}(u,v)}\bigr)$: an algebraic requirement on the point's coordinates, indifferent to whether $\Gal(F/\mathbb Q)$ is $F_{20}$, $S_{3}$, $S_{5}$ or $\mathrm{PSL}_{2}(\mathbb F_{7})$; \item $\operatorname{supp}(\mathfrak d_{F})$ contains the prime divisors of $F_{h}$; enlarging the Galois group only enlarges $\delta/d$ (Lemma~\ref{lem:tower}) and cannot erase the new primes. \end{enumerate} Hence the local improvement $\tfrac45\to\tfrac12$ promised by inertia $C_{2}$ cannot be cashed: it always accompanies a support jump from $S$ to $S\cup\{p\mid F_{h}\}$, with net effect $\eta_{\min}>1$ (Theorem~\ref{thm:bridge}). The direction is tagged \emph{X} in the sense: ``without importing an $S$-smoothness assumption stronger than $abc$''. \end{theorem} \begin{proof} (i) restates the hyperelliptic structure; (ii) is Lemma~\ref{thm:supportjump}; (iii) combines $\operatorname{supp}(\mathfrak d_{F})\supseteq \operatorname{supp}(\mathfrak d_{\mathbb Q(\sqrt{\Delta})})$ with Lemma~\ref{lem:tower}. The numerical net effect is the table after Theorem~\ref{thm:bridge}. \end{proof} This also answers, at the finish line, the category error ND1 of the first round: the framework opens by trying to extract positivity from the non-solvable monodromy of $\mathrm{PSL}_{2}(7)$, and closes here by trying to lower $\eta$ by changing the Galois group. Both move the representation layer, while the inequalities read only the scalar layer plus the support layer; without changing the latter two, representation freedom yields no new inequality. \subsection{The non-Galois decomposition lattice} We first correct a statement of \S\ref{sec:refine}. The tame local coefficient $\frac{|I_{p}|-1}{|I_{p}|}$, with the list $C_{2}\to\tfrac12$, $C_{3}\to\tfrac23$, $C_{4}\to\tfrac34$, $C_{5}\to\tfrac45$, is valid only when inertia acts transitively on the primes above $p$ (in particular for Galois extensions): then all $e_{i}=e$ and $r_{p}:=\sum_{i}f_{i}=n/e$, giving $f_{p}=n-r_{p}=n\frac{e-1}{e}$ and coefficient $\frac{e-1}{e}$. For non-Galois extensions there is no single inertia group acting on all primes above $p$: each $\mathfrak p_{i}$ has its own $(e_{i},f_{i})$ with $\sum_{i}e_{i}f_{i}=n$, and the tame discriminant exponent is \begin{equation}\label{eq:tamegen} f_{p}=\sum_{i}f_{i}(e_{i}-1)=\sum_{i}e_{i}f_{i}-\sum_{i}f_{i}=n-r_{p}, \qquad r_{p}:=\sum_{i}f_{i}\in[1,n], \end{equation} with local coefficient $\frac{n-r_{p}}{n}$, ranging over $\{0,\tfrac1n,\dots,\tfrac{n-1}{n}\}$ --- strictly finer than the Galois lattice. The ``change the Galois group'' direction therefore contains a genuine refinement that earlier rounds missed. \begin{theorem}[Non-Galois decomposition lattice]\label{thm:decomp} Let $F/\mathbb Q$ have degree $n$, not necessarily Galois, with $p\nmid n$ tame. Then $v_{p}(\mathfrak d_{F})=n-r_{p}$, and \begin{equation}\label{eq:decomplatt} \frac{\delta(F)}{d}=\sum_{p}\Bigl(1-\frac{r_{p}}{n}\Bigr)\log p+O(1) . \end{equation} For $n=5$ the complete list of decomposition types is: \begin{center} \renewcommand{\arraystretch}{1.1} \begin{tabular}{c|c|c|c} \toprule decomposition type $(e_{i})$ & $r_{p}$ & $f_{p}$ & local coefficient \\ \midrule $(1,1,1,1,1)$ & $5$ & $0$ & $0$ \\ $(2,1,1,1)$ & $4$ & $1$ & $1/5$ \\ $(2,2,1)$ / $(3,1,1)$ & $3$ & $2$ & $2/5$ \\ $(3,2)$ / $(4,1)$ & $2$ & $3$ & $3/5$ \\ $(5)$ & $1$ & $4$ & $4/5$ \\ \bottomrule \end{tabular} \end{center} At ramified primes the Galois-transitive case realises only $\tfrac45$; the mixed non-Galois types supply the three smaller coefficients $\tfrac15,\tfrac25,\tfrac35$. If one could keep the bridge $E=12\eta/\gamma$ with $\eta=\tfrac15$ and $\gamma=1$, one would get $E=12\cdot\tfrac15=2.4$ --- apparently a real breakthrough. The next result shows the pure Kummer construction never reaches the mixed types. \end{theorem} \begin{proof} \eqref{eq:tamegen} is the definition of the tame discriminant exponent together with $\sum_{i}e_{i}f_{i}=n$. The table enumerates the partitions $5=\sum_{i}e_{i}f_{i}$ (exact enumeration, no omission): the $(e_{i},f_{i})$ are read off the factorisation of the defining polynomial modulo $p$, and $\sum_{i}e_{i}f_{i}=5$ forces the five rows displayed. \end{proof} \begin{theorem}[Pure Kummer realises only the two ends]\label{thm:purekummer} Let $F=\mathbb Q(\alpha^{1/n})$ with $p\nmid n$. The decomposition type at $p$ is either $(n)$ (total ramification) or $(1^{n})$ (\'etale), i.e. \[ r_{p}\in\{1,n\},\qquad \text{local coefficient}\in\Bigl\{\frac{n-1}{n},0\Bigr\}. \] \end{theorem} \begin{proof} Write $\alpha=p^{k}u$. If $n\nmid k$, then $x^{n}-p^{k}u$ is Eisenstein, so $e=n$, $f=1$, $r_{p}=1$. If $n\mid k$, the substitution $y=x/p^{k/n}$ gives $y^{n}=u$, \'etale at $p$ (the derivative $ny^{n-1}$ is a unit), so $e_{i}=1$ for all $i$: whether completely split (type $(1^{n})$) or a single unramified factor of degree $n$, one has $\sum_{i}f_{i}=n$. There are no intermediate types. \end{proof} \begin{corollary}\label{cor:pureends} The discount of \S\ref{sec:refine} ($M_{5}$, $\delta/d=\frac45\log(N/M_{5})+O(1)$) is \emph{everything} the decomposition type of the pure Kummer construction can give: the drop from $\tfrac45$ to $0$ is pointwise two-valued, not continuously adjustable. Reaching $\tfrac15,\tfrac25,\tfrac35$ requires abandoning the pure Kummer generator. The required $S$-smoothness has exact slope $2/(n-1)$ (Theorem~\ref{thm:21D}). \end{corollary} \subsection{Positive result: the index-ideal absorption theorem} \begin{theorem}[Ideal-theoretic Mason dictionary]\label{thm:indexideal} Let $f\in\mathbb Z[x]$ be monic irreducible of degree $n$, $\theta$ a root, $F=\mathbb Q(\theta)$. Dedekind's theorem gives \begin{equation}\label{eq:dedekindindex} \operatorname{disc}(f)=[\mathcal O_{F}:\mathbb Z[\theta]]^{2}\cdot \mathfrak d_{F}. \end{equation} For $f=x^{n}-\alpha$ with $\alpha=A/B$, $p\nmid n$, $k=v_{p}(\alpha)$: \begin{equation}\label{eq:discf} v_{p}(\operatorname{disc}f)=(n-1)k+n\,v_{p}(n),\qquad v_{p}(\mathfrak d_{F})= \begin{cases} n-1, & n\nmid k\ \text{(total ramification)},\\ 0, & n\mid k\ \text{(\'etale)}, \end{cases} \end{equation} hence \begin{equation}\label{eq:indexval} v_{p}\bigl([\mathcal O_{F}:\mathbb Z[\theta]]\bigr)= \begin{cases} \dfrac{(n-1)(|k|-1)+n\,v_{p}(n)}{2}, & \text{(total)},\\[1.2ex] \dfrac{(n-1)|k|+n\,v_{p}(n)}{2}, & \text{(\'etale)}. \end{cases} \end{equation} Exact check at $n=5$: $k=1\Rightarrow0$, $k=2\Rightarrow2$, $k=3\Rightarrow4$, $k=5\Rightarrow10$ (\'etale), $k=25\Rightarrow50$. \end{theorem} \begin{proof} \eqref{eq:dedekindindex} is Dedekind's theorem. For $f=x^{n}-\alpha$, $\operatorname{disc}(f)=(-1)^{\binom n2}n^{n}\alpha^{n-1}$, giving the first line of \eqref{eq:discf}; the field-discriminant values are the tame computation of \eqref{eq:tamegen} (Eisenstein resp.\ \'etale). Subtracting and halving gives \eqref{eq:indexval}. \end{proof} This is the arithmetic origin of ``$\rad$ vs.\ index'': the polynomial discriminant carries the full multiplicity $(n-1)|k|$, while the field discriminant keeps only the tame constant $n-1$; the difference is absorbed entirely by the index ideal. The correspondence with the Mason dictionary is verbatim: \begin{equation}\label{eq:masonindex} \deg\gcd(F,F')=\deg F-\deg\operatorname{rad}(F) \quad\longleftrightarrow\quad v_{p}(\operatorname{disc}f)-v_{p}(\mathfrak d_{F}) =v_{p}\bigl([\mathcal O_{F}:\mathbb Z[\theta]]^{2}\bigr). \end{equation} The multiplicity excess $\log(abc)-\log\rad(abc)$, previously an integer-level quantity, is hereby lifted to the $p$-adic valuation of the index ideal in a number field. The difference between integer and function-field Mason is, at the ideal level, precisely the question \emph{whether the index ideal can absorb all of the multiplicity}. This absorption is quantified in Appendix~\ref{app:absorp}. \subsection{Mixed types require $S$-smooth values} \begin{theorem}[$S$-smoothness loop]\label{thm:mixloop} To obtain coefficients $\tfrac15,\tfrac25,\tfrac35$ one must use a non-pure-Kummer defining polynomial $f$ with $x=\pi_{h}(P)=\theta$ and $h_{K_{\mathcal C}}(P)=\gamma\log c+O(1)$. Three constraints must hold simultaneously: \begin{enumerate}[leftmargin=1.8em,itemsep=0.2em,label=(\roman*)] \item \emph{Height}: $h(\theta)=\frac1n\log M(f)+O(1)$ with $M(f)$ the Mahler measure, so $\gamma\asymp1$ forces the coefficient size $f\asymp c^{O(1)}$; \item \emph{Support tightness}: $\operatorname{supp}(\mathfrak d_{F})\subseteq S$ forces $\operatorname{disc}(f)$ to be $S$-smooth (up to square factors), and $\operatorname{disc}(f)\asymp c^{O(1)}$; \item \emph{Mixed type}: $10$ necessary & U \\ Lemma~1 / Prop.~2 / Cor.~1 / (29) & U (Appendix G) \\ (2)/(30) Minkowski (BV Th.~3, Borek Th.~2) & U (Prop.~A4a, App.~H) \\ (33) $\wbar^{2}/4$, Prop.~9(vi) & U (Prop.~A4c, App.~H) \\ $\delta(F')/d'$, $h_{L}(P')$ (Neukirch, height stability) & U (Prop.~A4b, App.~H) \\ (32) up to $O(d')$ (Th.~I.4) & U \\ Prop.~3 Green inequalities (Zhang \cite{ZhangAdmissible}) & C \\ (32) strict equality as written in \cite{SouleABC} & O$^{\ast}$ \\ ``model metric $=$ admissible metric'' joint assertion & C \\ admissible-metric existence (\cite[Prop.~2.6]{Yuan26}) & U (existence) \\ Normalisation dictionary ($n_{v}$ weights, pullback) & U \\ Torsion decomposition $t=2$, $T=kL$ & U + A\ref{ass:count} \\ $t=12/N$; full $abc$ unattainable at $q=5$ & U + A\ref{ass:count} \\ Reciprocal arithmetic progression (step $\lambda/W$) & U + A\ref{ass:count} \\ Unattainability for all $q$ under transversality & U + A\ref{ass:model} \\ $n_{1}=n_{2}$, $\Theta\in2\mathbb Z$, $\dim_{k}R[\pi^{2}]\in\{2,4\}$ & U + A\ref{ass:count} \\ $\rank\partial_{1}=\rank\partial_{2}$ identically & U + A\ref{ass:count} \\ $\Theta\in\{8,10,12,14\}$; $(9,13)$ excluded; only $(27,12)$ & U \\ Chain B unattainable for all odd $q$, strictly & U \\ Galois transitivity $\Rightarrow\Theta=8$ & U + A\ref{ass:count} \\ Unified criterion $E=12(q-1)/(qN\gamma)$, $\gamma_{\mathrm{crit}}$ & U \\ Landing equations $q_{A},q_{B}$; unique $(13,12,27,8/9)$ & U \\ $\Theta=12\Rightarrow q\le27$; $\Theta=14\Rightarrow q=5$ & U \\ $E=1/\rho$; $\gamma$ must hit equality exactly & U \\ Orbit classification: only $[4]$ excludes $\Theta=12$ & U + A\ref{ass:count} \\ No-Go in original $q=5$ construction (no $\Theta$ needed) & U \\ $\gamma=2/5$ fixed by $K_{\mathcal C}$; $\gamma_{K}(q)=\frac{q-3}q$ & U \\ $12/\gamma=30$, $24=30\cdot\frac45$; convention B not Th.A's & U \\ $g=2\iff q=5$; $(12,27)$ needs $g=13$ & U \\ Full $abc$ unattainable in the construction (no $\Theta,T$) & \texttt{U + A5.2}$^{\dagger}$ \\ $\kappa_{M}$, $\rank\partial_{i}=1$, $T=8k$ & U + A\ref{ass:model}, A\ref{ass:count} \\ Kummer discriminant $\eta(q)=(q-1)/q$ & U \\ $E=t\eta/\gamma$, $\gamma\ge1/n$ & U + Th.A \\ \textbf{Weak $abc$: $c\le A_{\eps}N^{E_{\max}+\eps}$} & Th.A$^{\ast}$ \\ Discounted bound $c\le A_{\eps}(N/M_{5})^{24+\eps}$ & Th.A$^{\ast}$ \\ No-bootstrap ($\inf\theta=0$), tower monotonicity & U \\ Support jump (Lemma~\ref{thm:supportjump}) & U \\ Bridge breaking (Theorem~\ref{thm:bridge}) & U \\ $\mathrm{PSL}_{2}(7)$ direction closed (Theorem~\ref{thm:psl}) & U \\ Non-Galois decomposition lattice (Theorem~\ref{thm:decomp}) & U \\ Pure Kummer two-ended (Theorem~\ref{thm:purekummer}) & U \\ Index-ideal absorption (Theorem~\ref{thm:indexideal}) & U \\ $S$-smooth reductions (Theorems~\ref{thm:smooth}, \ref{thm:mixloop}) & X \\ Four locks (Theorem~\ref{thm:fourlocks}) & U \\ Base verification: Th.A scope (any $F$, any $P$, fixed $\mathcal C$) & U (text) \\ Constant chain $36,12,30,24$ verbatim in \cite{SouleABC} & U (text) \\ $\theta\le(Y-1)/4$; $E_{\mathrm{eff}}\ge12$ under $abc$/Szpiro thresholds & U \\ $\frac32=m\cdot c$; Minkowski $A$ enters only $O(d')$ & U \\ $12=\frac13\cdot m/(\rho-mc)$; five locked factors & U \\ $E(c)=\frac{12}{5-18c}$, $E\ge\frac{12}{5}$ (Theorem~\ref{thm:Eofc}) & U \\ $T/k=47/3$ double exclusion (Theorem~\ref{thm:sixth}) & U \\ Route 1 closed (Theorem~\ref{thm:routeone}) & U \\ Higher-genus recalibration $t(g_0=13,\Theta)$ & U (Appendix~\ref{app:highergenus}) \\ Seventh No-Go: the chain cannot start at $g_0=13$ & U (Corollary~\ref{cor:K5}) \\ Absorption inequality and the unified formula for $\eta$ & U (Appendix~\ref{app:absorp}) \\ Engine-replacement criterion (slope $2/(n-1)$) & U + A\ref{ass:count} \\ Eighth No-Go (structural) & U (Corollary~\ref{cor:21eighth}) \\ Exponential moment of $\mathcal E$ & U (Appendix~\ref{app:distribution}) \\ Discount zero-density (Chernoff) & U (Appendix~\ref{app:distribution}) \\ Average quality tends to $1$ & U (Appendix~\ref{app:distribution}) \\ Ninth No-Go (distributional) & U (Corollary~\ref{cor:M7}) \\ Average effective exponent equals $24$ & U + Th.A (Appendix~\ref{app:distribution}) \\ $abc$ itself & open \\ \bottomrule \end{tabular} \end{table} \noindent{\footnotesize $^{\ast}$~Conditional on Theorem~A of \cite{SouleABC} (Appendix~\ref{app:H}). $^{\dagger}$~Here \texttt{U} means independence of the \emph{truth} of Theorem~A; it does not assert an unconditional proof that $abc$ fails. All No-Go statements are internal to the construction of \cite{SouleABC}; see Proposition~H.5 (Appendix~H). $^{\ast}$\ \texttt{O}$^{\ast}$: see the footnote of the U/C/O table, Appendix~\ref{app:adelic}. \subsection*{Conclusions} The geometric layer (\S\ref{sec:geom}--\S\ref{sec:arith}) yields unconditional identities and the GRR decomposition. The construction layer (\S\ref{sec:torsion}--\S\ref{sec:local}) shows that integrality of torsion lengths pins the coefficient of Theorem~A to the discrete set $t=12/N$, $N\equiv1\pmod3$ (for general parameters, $1/t$ lies on an arithmetic progression); at $q=5$ the coefficient required by full $abc$ is not in the set; in the explicit local model $T=8k$ exactly and $t=12$ is the unique output, restoring the No-Go for every $q$ under transversality. The second-order torsion computation gives $\Theta\in2\mathbb Z$, $\{8,10,12,14\}$, excludes $(9,13)$, compresses chain A to the unique candidate $(27,12)$, seals chain B for all odd $q$, and upgrades $\Theta=8$ to a rigorous statement under Galois transitivity. The unified criterion of \S\ref{sec:last} decodes this cell into three exact conditions ($\sum n_{j}=6$; $q\le27$ with equality; $\gamma=8/9$ exact), and shows that inside the original $q=5$ construction full $abc$ is unattainable independently of the exact $\Theta$. \S\ref{subsec:gamma} determines $\gamma$: the original text pins $h$ to the canonical bundle of a fixed genus-$2$ curve, $\gamma=2/5$ is confirmed by an explicit height computation and by the constant $30=12/\gamma$; the genus and Kummer degree are fixed, so $(12,27)$ requires $g=13$, its hypotheses fail, and the last cell is empty. The arithmetic layer (\S\ref{sec:weak}) takes Theorem~A as its sole external input and derives an explicit finite weak $abc$ exponent; \S\ref{sec:refine} sharpens it to a pointwise discounted bound, proves that bootstrapping is impossible, and proves tower monotonicity of the discriminant-to-degree ratio, with $\tfrac45$ upgraded to an Artin conductor identity. \textbf{We neither prove nor disprove $abc$.} The No-Go statements are internal to the construction; their obstructions are arithmetic structure (the mod-$3$ denominator at $q=5$; $\eta(q)\to1$; the divisibility $\Theta\in2\mathbb Z$ from second-order torsion; normalisations required to land on the countable nowhere-dense set $\bigcup_{N}\Gamma_{N}$; and the fixed genus/fixed Kummer-degree hypotheses of Theorem~A), not the coarseness of estimates. The sixteenth through twenty-second rounds (\S\ref{sec:bridge}, Appendices~\ref{app:base}, \ref{app:constdec}, \ref{app:highergenus}, \ref{app:absorp} and~\ref{app:distribution}) add the third and fourth locks and a fifth independent reason: bridge breaking shows that changing the field or the Galois group destroys the scale $\delta/d\asymp\log N$ (Theorem~\ref{thm:bridge}); the finer non-Galois decomposition lattice requires $S$-smoothness stronger than $abc$ for its mixed types (Theorems~\ref{thm:decomp} and~\ref{thm:mixloop}); and the Mason dictionary is upgraded to the $p$-adic valuation of the index ideal (Theorem~\ref{thm:indexideal}). A sixth reason closes the constant side: $E=1$ would force $T/k=\frac{47}3$ (Appendix~\ref{app:constdec}), excluded both by $\Theta\in2\mathbb Z$ and by $T\le15k$. An independent re-derivation of the adelic layer confirms that the metric estimates contribute only $O(d')$ and $\beta(\eps)$; the coefficient $12$ is entirely determined by $\rho,m,c$ and the $\deg_{K}(x)=0$ normalisation, in agreement with Appendix~\ref{app:constdec}. The \texttt{U + A5.2} tags attached to the No-Go statements record independence of the truth of Theorem~A; they are not a claim that $abc$ fails unconditionally (Proposition~H.5). \subsection*{Open problems} \begin{enumerate}[leftmargin=1.8em,itemsep=0.25em] \item \textbf{Theorem~A --- remaining gaps}: its statement, scope, and the layers Lemma~1/Prop.~2/Cor.~1/(29)/coefficient chain are independently re-derived (Appendix~\ref{app:adelic}); four items remain: \textbf{(1a)} supply the place-by-place norm comparison through $\alpha$ (one sentence; affects only the strict equality in (32), not the coefficients --- Theorem~I.4); \textbf{(1b$^{\prime}$)} record the conversion between the two normalisation conventions of \cite[\S1.3.1/\S1.3.3]{SouleABC} (Appendix~J); \textbf{(1c-i)} admissible-metric existence: \cite[Prop.~2.6]{Yuan26}; \textbf{(1c-ii)} Green-function data of Proposition~3: Zhang \cite[Th.~1.1]{ZhangAdmissible} and \cite[\S\S2--3]{Z95}. The blanket item ``independently verify the adelic metric estimates'' is resolved at the level of Lemma~1/Prop.~2/Cor.~1; it is not resolved for (32) and Proposition~3. The No-Go conclusions of \S\ref{sec:torsion}--\S\ref{sec:bridge} are independent of the truth of Theorem~A (Proposition~H.5, Appendix~\ref{app:H}). \item \textbf{Higher-genus versions of Theorem~A} --- \textbf{closed}: even under the most favourable extrapolation of the coefficient chain to $g_0=13$, the chain cannot start ($\rho-\mu c<0$ for every admissible $\Theta$); see Appendix~\ref{app:highergenus}, Corollary~\ref{cor:K5}. \item Decide $\sum_{j=1}^{4}n_{j}=6$ (only the transitive orbit $[4]$ excludes it, Corollary~\ref{cor:orbits}). No landing object remains after Theorem~\ref{thm:qrigid}, so it no longer affects the No-Go. \item \textbf{Closed}: $\ell$ and $\gamma$ are determined ($\ell=\deg K_{\mathcal C}=2$, $\gamma=\frac25$, Theorems~\ref{thm:gammastrict}--\ref{thm:gamK}). The remaining question is whether the formal $\ell$-parametrisation of \S\ref{sec:weak} is worth keeping. \item \textbf{Rewritten}: with $E(c)=\frac{12}{5-18c}$ (Theorem~\ref{thm:Eofc}) and the general reciprocal progression (Corollary~\ref{cor:arithprog}) in place, the question becomes: can $c$ be negative ($T>11k$)? \item Extend the Kummer discriminant estimate beyond the tame range or to small $q$, testing whether $\eta(q)<1$ has exceptions. \item For general $(g_{0},r,b,\lambda)$, decide whether integral solutions of \eqref{eq:landing} can satisfy $2r\mid\Theta$; the round-10 scan of $300$ admissible triples gives partial numerics. \item \textbf{Change the engine}: Route~1 is closed (Theorem~\ref{thm:routeone}); the remaining routes are the exceptional-set/average language and producing $S$-smooth values (\S\ref{sec:bridge}). The difficulty is $abc$-grade, not technical: ``all unconditional bounds for these conjectures are of exponential nature'' (\cite[\S1.1]{PastenShimura}); any engine that yields a polynomial-type bound must supply, at minimum, the $S$-smooth value required by the mixed-decomposition loop (Theorem~\ref{thm:mixloop}), which is itself stronger than $abc$. \item \textbf{Closed} --- resolved distributionally: the discount $\theta$ vanishes in density $1-o(1)$ with an explicit large-deviation rate, and $\mathbb E[Y-1]=O(1/\log\log X)$ (Appendix~\ref{app:distribution}, Theorems~M.3 and~M.4). \item Prepare the present manuscript as an independent referee report on arXiv:2611.05217v1 (immediately executable). \item \textbf{Decide whether a non-pure-Kummer generator with $\mathfrak I\le N^{O(1)}$ exists}: by Theorem~\ref{thm:21D} this is an $abc$-grade question, with exact slope $2/(n-1)$. \item \textbf{Decide whether a non-discount engine can give an average exponent improvement}: Theorem~M.6 shows every discount-type engine averages to $24-o(1)$; a non-discount mechanism would have to bypass the distributional vanishing of Appendix~\ref{app:distribution}. \end{enumerate} % ===================================================================== \appendix \section{Hypothesis ledger and literature dependence}\label{app:assume} % ===================================================================== \begin{enumerate}[leftmargin=1.8em,itemsep=0.2em,label=A\arabic*] \item \textbf{Alignment A0} (Assumption~\ref{ass:A0}): the choice of model and finite prime set gives $\hB=\log c+O(1)$ and $\mS=\log c-\log\rad+O(1)$; a convention, not a theorem. \item \textbf{Counting convention} (Assumption~\ref{ass:count}): $\psi$ \'etale makes the $k$ preimages over an exceptional point locally isomorphic of equal torsion length; $\deg_{K}$ is identified with $\operatorname{card}\Psi=4k$ as in \cite{SouleABC}. Non-$K$-rational exceptional points require Galois-orbit weighting, but $L\in\mathbb Z$, $L\ge8$ persist. \item \textbf{Local model} (Assumption~\ref{ass:model}): the hyperelliptic model is compatible with the explicit statements of \cite{SouleABC} ($g(Y_{1})=3$, hyperelliptic involution, $4$ exceptional points, $A_{1}\cdot A_{2}=0$, $A_{i}^{2}=-4k$; the last independently reproduced from $B_{i}^{2}=-(2g_{0}-2)k$). ``All four fibres transverse'' would follow from $\psi$ Galois plus equivariance plus exclusion of total degeneration, but equivariance is not stated in \cite{SouleABC}, so it remains an assumption. Jets are not unique, but $\kappa_{M}\propto(1,a,a^{2})$ holds for every compatible jet. \item \textbf{External theorems}: Theorem~A of \cite{SouleABC} is ``on a fixed genus-$2$ curve $\mathcal C/K_{0}$, the height attached to $K_{\mathcal C}$ satisfies $h(P)\le(12+\eps)\delta(F)/d+\beta(\eps)$'', and Theorem~B is $c\le A(\eps)N^{24+\eps}$; \S7.2 uses $y^{2}+y=x^{5}$, $F=K(\sqrt[5]{m})$, $P=(\sqrt[5]{m},y)$ and the intermediate constant $30$. Proposition~2, Corollary~1, Prop.~8/9, (18)(22)(24), $|\Psi|=4k$, $T\ge8k$, $\delta(F')/d'\le\delta(F)/d+C$ are taken from \cite{SouleABC}. \textbf{(a)} The statement and scope of Theorems~A and~B are verified verbatim against the source (Appendix~\ref{app:base}). \textbf{(b)} Independent re-derivation (Appendices~\ref{app:adelic} and~\ref{app:H}). We have re-derived from the standard framework: the normalisation of $\widehat{\deg}$ via the product formula, the pullback weight identity, Lemma~1, Proposition~2 (uniform constant), Corollary~1, (29), the coefficient chain (31)--(36), \S7.1 and \S7.2. In addition, four further items are standard and are upgraded from conditional to U: \begin{itemize}[leftmargin=1.6em,itemsep=0.15em] \item the rank-$2$ Minkowski bound (2)/(30): direct consequence of Bombieri--Vaaler \cite[Th.~3]{BombieriVaaler} and Borek \cite[Th.~2]{Borek}; $\delta(K)=\log|\Delta_{K}|$ carries no $1/2$ (Appendix~H.4); \item $\delta(F')/d'\le\delta(F)/d+O(1)$ and $h_{L}(P')=h(P)+O(1)$: relative discriminant formula (Neukirch \cite[Chap.~III, \S2, Th.~2.11]{Neu}) and Weil-height stability over finite covers \cite[\S2.3]{BG06} (Appendix~H.4); \item (33)'s $\wbar^{2}/4$ and Prop.~9(vi)'s $-3k$: pure geometric intersection numbers (Appendix~H.4). \end{itemize} What remains conditional: the Green-function inequalities of Proposition~3 (structure: Zhang \cite[Th.~1.1]{ZhangAdmissible}; the joint ``model metric equals admissible metric'' assertion is not stated), and the strict equality in (32). The admissible-metric existence has the reference \cite[Prop.~2.6]{Yuan26}; $\dim H=2$ and Prop.~9 are recomputed in Appendices~B--C. We do not claim an independent verification of the proof of Theorem~A. The step $t\to E$ left implicit in \S7.2 of \cite{SouleABC} is supplied in \S\ref{sec:weak}. \item \textbf{Normalisation}: $\lambda=3$, $\rho=5/3$, $m=6$ follow \cite{SouleABC}. $\gamma$ is strict by Theorem~\ref{thm:gammastrict} ($=2/5$, $K_{\mathcal C}$ fixed), and $\gamma_{K}(q)=(q-3)/q$ is strict by Theorem~\ref{thm:gamK}; but the genus and Kummer degree of Theorem~A are fixed, so $q=5$ is the unique applicable value. The two bridge chains $E=2t$ and $E=\frac45t$ (IH\'ES lecture \cite{SouleIHES}) were kept side by side; Corollary~\ref{cor:gamma3} shows the $\gamma=1$ convention is not the normalisation of Theorem~A. See Remark~\ref{rem:normdict} for the normalisation dictionary ($t^{\ast}=\frac52\,t_{A}$): the chain-B target $\frac54$ and the chain-A target $\frac12$ are one condition in two gauges. \item \textbf{Conservativeness}: Lemma~\ref{lem:gamma} uses $\deg D\ge2g-1$ to force $h^{0}>0$ (sufficient, not necessary), so $\gamma_{0}=1/n$ and $E_{\max}$ are conservative; $\gamma_{K}=(q-3)/q$ is a strict equality by Theorem~\ref{thm:gamK}. \item \textbf{Scope}: $C:u^{q}=x(x-1)$ and $F=\mathbb Q(\alpha^{1/q})$ are constructed here (compatible with the genus-$2$ curve and quintic root cover of the literature). If \cite{SouleABC} uses a different curve, the formula $E=t\eta/\gamma$ is unchanged and only $\eta,\gamma$ need recalibration. \item \textbf{Audits 18--19}: Appendices~\ref{app:base}--\ref{app:constdec} record the eighteenth through twenty-second rounds: base verification of the statement and scope of Theorem~A, the quality plane ($\theta\le\frac{Y-1}{4}$, $E_{\mathrm{eff}}\ge12$ under $Y\le3$), the closed formula $E(c)=\frac{12}{5-18c}$, the double exclusion $T/k=\frac{47}{3}$, and the closure of Route~1; the external references \cite{PastenShimura,PastenSzpiro,LaniewskiMueller,Ballay} are cited there. \end{enumerate} \section{Computations for the geometric layer}\label{app:geom} \subsection{Adjunction and the two identities} Equation \eqref{eq:adj} is proved in Theorem~\ref{thm:adj}; the fibre, section and multisection identities of Proposition~\ref{prop:two-id} follow by the same projection-formula subtractions. For the Frey--Szpiro side: on the Hellegouarch--Frey curve $E_{a,b}:y^{2}=x(x-a)(x+b)$ one has $\Delta=16(abc)^{2}$, $c_{4}=16(a^{2}+ab+b^{2})$, and $\om_{\mathcal E/B}=f^{\ast}\mathcal L$ with $\deg\mathcal L=\chi(\mathcal O_{\mathcal E})$, so every section satisfies $Z\cdot\om=\deg\mathcal L=\chi$, i.e. $Z^{2}=-\chi$ (Theorem~\ref{thm:adj}); a K3 elliptic surface has $\chi=2$, $Z^{2}=-2$, the true source of $\kappa_{\mathrm{A2}}=2$. This is also the rigorous refutation of A1 on elliptic surfaces: $\hrel(P)=\chi\hB(\pi(P))+O(1)$, while $\hvert(P_{n})=\hat h(nP)=n^{2}\hat h(P)\to\infty$. \subsection{The tower of double covers} The first double cover $Y_{1}\to C$ is \'etale: $2g_{1}-2=2(2g_{0}-2)$, $g_{1}=2g_{0}-1$. The second, $X_{y}\to Y_{1}$, branches at $r$ points: $2g_{f}-2=2(2g_{1}-2)+r=8g_{0}-8+r$. At $(g_{0},r)=(2,2)$: $g_{1}=3$, $g_{f}=6$. For the ramification divisor write $M^{2}=\mathcal O(A_{1}+A_{2})$ and let $W$ be the associated theta characteristic with $\deg W=g_{0}-1+\frac r2$. The relative dualising sheaf is $\om=\sum_{i=1}^{r}B_{i}+(4g_{0}-4)\phi_{1}^{\ast}W$. Intersecting with $B_{i}$ and applying \eqref{eq:adj}: $-B_{i}^{2}=B_{i}^{2}+(4g_{0}-4)k$, so $B_{i}^{2}=-(2g_{0}-2)k$ and $\om\cdot B_{i}=(2g_{0}-2)k$. Squaring: \[ \om^{2}=\sum_{i}B_{i}^{2}+2(4g_{0}-4)\sum_{i}\phi_{1}^{\ast}W\cdot B_{i} =-r(2g_{0}-2)k+2(4g_{0}-4)rk=6rk(g_{0}-1) , \] using $(\phi_{1}^{\ast}W)^{2}=0$ (pullback from the base). At $(g_{0},r)=(2,2)$: $\om^{2}=-4k+16k=12k$. With $S=\sum_{i=1}^{r}B_{i}$: $\om\cdot S=r(2g_{0}-2)k$, $S^{2}=-r(2g_{0}-2)k$, so for $L=\om-bS$, \[ L^{2}=6rk(g_{0}-1)-2br(2g_{0}-2)k-b^{2}r(2g_{0}-2)k =-2rk(g_{0}-1)(b-1)(b+3), \] and $\rho=(b-1)(b+3)/3$. \subsection{The $\mathrm{PSL}_{2}(\mathbb F_{7})$ cover and the Parshin cubic} For $G=\mathrm{PSL}_{2}(\mathbb F_{7})$, $|G|=168$, branch type $(2,3,7,7)$: \[ 2g_{X}-2=-2\cdot168+168\Bigl(\tfrac12+\tfrac23+\tfrac67+\tfrac67\Bigr) =-336+4\cdot121=148,\qquad g_{X}=75 . \] This is conditional on the existence of a generating tuple of type $(2,3,7,7)$ with product $1$; Riemann--Hurwitz alone does not prove existence. For a cubic cover $Y\to X$ with $r_{3}$ totally ramified points: $2g_{Y}-2=3\cdot148+2r_{3}$, $g_{Y}=223+r_{3}$; the value $g_{Y}=224$ corresponds to $r_{3}=1$, which the relation $\prod_{j}[a_{j},b_{j}]\gamma_{1}=1$ in $\pi_{1}(X)$ with $\gamma_{1}\in A_{3}$ does not contradict, but remains a Parshin configuration to be exhibited. \subsection{Term-by-term check of the fifteen invariants} At $(g_{0},r,b,k)=(2,2,2,k)$: \begin{center} \renewcommand{\arraystretch}{1.1} \begin{tabular}{l|l} \toprule quantity & value \\ \midrule $g_{1}=g(Y_{1})$ & $3$ \\ $g_{f}$ & $6$ \\ $B_{i}^{2}$ & $-2k$ \\ $\om\cdot B_{i}$ & $2k$ \\ $\phi_{1}^{\ast}W\cdot B_{i}$ & $k$ \\ $\om^{2}$ & $12k$ \\ $\om\cdot S$ ($S=B_{1}+B_{2}$) & $4k$ \\ $\om\cdot B$ ($B=2S$) & $8k$ \\ $S^{2}$ & $-4k$ \\ $B^{2}$ & $-16k$ \\ $L^{2}=\om^{2}-2\om\cdot B+B^{2}$ & $-20k$ \\ $\rho=-L^{2}/\om^{2}$ & $5/3$ \\ $L\cdot\om=\om^{2}-\om\cdot B$ & $4k$ \\ $\deg L_{F}$ & $6$ \\ $\chi(L_{F})=\deg L_{F}-g_{f}+1$ & $1$ \\ $h^{0}(L_{F})$ (rank-$2$ pushforward) & $2$ \\ $\chi=\deg\mathcal L=\om^{2}/12$ & $k$ \\ $T\ge\deg(R^{1}\phi_{\ast}(B)[\pi])$ & $8k$ \\ \bottomrule \end{tabular} \end{center} \subsection{The margin computation}\label{app:coef} The three hard constants are $\rho=\frac53$, $m=6$, $c=\frac14$, giving $\Delta=\rho-mc=\frac53-\frac64=\frac16$ and $C=\frac{m}{\Delta}=\frac{6}{1/6}=36$. Sensitivity: reaching $\frac{15}{4}$ would require $c=\frac1{90}$ (a $22.5$-fold change); the formal limit $c\to0$ gives $C\to\frac{18}{5}=3.6<3.75$, signalling a structural lower bound $c\ge\frac1{90}$. For $L=\om-bS$ in general: \[ L^{2}(b)=-2rk(g_{0}-1)(b-1)(b+3),\quad \rho(b)=\tfrac{(b-1)(b+3)}3,\quad \mu(b)=8(g_{0}-1)+r-br . \] The margin $\Delta(b)=\rho(b)-\mu(b)c_{\GRR}(b)$ with $c_{\GRR}(b)=\frac{2b^{2}+2b-1}{12}$. The positivity window $C\ge\frac{15}{4}$ together with $\Delta>0$ gives $c\in\bigl[\frac\rho\mu-\frac4{15},\frac\rho\mu\bigr)$, while $c_{\GRR}\sim\frac{b^{2}}6$ grows quadratically, forcing $\frac{T-D_{0}}{\om^{2}}\sim\frac{b^{2}}6$, i.e.\ $T\sim4kg_{0}^{3}$ with relative precision $\frac{8}{5b^{2}}\to0$. \emph{Independence-window computation (exact).} From $c=\frac{11}{12}-\frac{T}{12k}$ (Theorem~\ref{thm:drr} at the literature point): \[ C(T)=\frac{36k}{3T-23k},\qquad T=8k\Rightarrow c=\tfrac14,\ C=36,\ \text{Theorem~A}=12 . \] The consistency window $C\ge\frac{15}{4}$ is $T\in\bigl(\frac{23k}{3},\frac{163k}{15}\bigr]\approx(7.667k,\,10.867k]$; full $abc$ would need $T=\frac{163k}{15}$, an increase of $\frac{163}{120}\approx35.8\%$ over the literature lower bound $8k$. \section{Computations for the torsion and local layers}\label{app:torsion} \subsection{$h^{0}$ jumps on $Y_{1}$} On $Y_{1}$ of genus $3$ with $\deg\om_{Y_{1}}=4$: \begin{center} \renewcommand{\arraystretch}{1.1} \begin{tabular}{l|c|c|c} \toprule sheaf & general $h^{0}$ & exceptional $h^{0}$ & criterion \\ \midrule $M$ ($\deg1$) & $0$ & $1$ & $M$ becomes effective \\ $M^{2}=g^{1}_{2}$ ($\deg2$) & $1$ & $2$ & hyperelliptic $g^{1}_{2}$ degenerates \\ $\om_{Y_{1}}M^{-1}$ & $h^{1}(M)=1$ & $2$ & duality \\ $\om_{Y_{1}}M^{-2}$ & $h^{1}(M^{2})=1$ & $2$ & duality \\ \midrule $B=\pi^{\ast}(M^{2}\oplus M)$ & $1$ & $3$ & \cite[Prop.~9(iv)]{SouleABC} \\ $L=\om_{X/Y_{1}}$ & $2$ & $4$ & \cite[Prop.~9(v)]{SouleABC} \\ \bottomrule \end{tabular} \end{center} The jump $h^{1}=h^{0}=2$ gives the second independent proof of $t=\dim_{k}R[\pi]=2$ (the first being the count (24) of \cite{SouleABC}). \subsection{Base change over $A/\pi^{2}$ and $G_{\bar y}$} For $X_{j}=X\times_{A}A/\pi^{j}$, computing $\dim_{k}H^{0}(X_{2},B_{2})$ on both sides of $0\to B\xrightarrow{\pi^{2}}B\to B_{2}\to0$ gives \[ G_{\bar y}:=\dim_{k}R[\pi^{2}]=4-\rank\partial_{1}-\rank\partial_{2}, \] with $\partial_{i}$ the Kodaira--Spencer connecting maps. Hence $T=8k\iff$ every exceptional fibre has $G_{\bar y}=2\iff$ $\rank\partial_{1}=\rank\partial_{2}=1$. Writing $T=G+H$ with $H$ the $\pi^{3}$-and-higher part separates the higher torsion. \subsection{Height computations}\label{app:kummer} For $\alpha=-ab/c^{2}$ with $\gcd(ab,c)=1$: $p\mid ab$ gives $\log^{+}|\alpha|_{p}=0$; $p\mid c$ gives $\log^{+}|\alpha|_{p}=2v_{p}(c)\log p$; archimedean places give $\log^{+}=0$ since $|\alpha|<1$. Hence $\sum_{p}\log^{+}|\alpha|_{p}=2\log c$. For $u^{q}=\alpha$ and $d=[F:\mathbb Q]$: $|u|_{w}=|\alpha|_{p}^{1/q}$ and $\sum_{w\mid p}[F_{w}:\mathbb Q_{p}]=d$ give \[ h_{F}(u)=\frac1d\sum_{w\mid p}[F_{w}:\mathbb Q_{p}]\frac1q\log^{+}|\alpha|_{p} =\frac2q\log c+O(1). \] The canonical map $\varphi_{K}(P)=[u^{g-1}:\cdots:u:1]$ then yields $h_{K_{C}}(P)=\frac{q-3}{q}\log c+O(1)$ (Theorem~\ref{thm:gamK}). The three line bundles: \begin{center} \renewcommand{\arraystretch}{1.1} \begin{tabular}{c|c|c} \toprule bundle & height & $\gamma$ \\ \midrule $\pi^{\ast}\mathcal O(1)$ & $h(\lambda)=\log c$ & $1$ \\ $K_{C}$ & $(g-1)h_{F}(u)$ & $\frac{q-3}{q}=\gamma_{K}(q)$ \\ general, $\deg L=\ell$ & $\frac\ell q\log c+o(1)$ & $\ell/q+o(1)$ \\ \bottomrule \end{tabular} \end{center} At $q=5$, $\gamma_{K}=2/5=\gamma_{\ell=2}$: the two rules are indistinguishable at the literature point, which is why $\gamma$ remained conditional for so long. \section{Tables of constants and exact recomputations}\label{app:tables} \subsection{Exact rational recomputation} All decisive values recompute exactly: \[ \begin{aligned} &\rho(2)=\tfrac53,\quad L^{2}(2)=-20k,\quad \om^{2}=12k,\quad \deg\det R\phi_{\ast}(bS)\big|_{b=2}=-11k,\\ &c=\tfrac{11}{12}-\tfrac{8k}{12k}=\tfrac14,\quad \Delta=\tfrac53-\tfrac64=\tfrac16,\quad C=36,\quad t=12,\quad 24=12\cdot2,\\ &C(\eps)=\tfrac{36}{1-15\eps}\to36,\\ &t(\Theta)=\tfrac{12}{3\Theta-23},\ \Theta\in\{8,10,12,14\},\ N\in\{1,7,13,19\},\\ &q_{A}=\tfrac{3N-12}{N-12}=(\tfrac9{11},-\tfrac95,27,\tfrac{45}7),\quad q_{B}=\tfrac{12}{12-N}=(\tfrac{12}{11},\tfrac{12}5,-12,-\tfrac{12}7),\\ &\text{unique odd }q\ge5:\ (N,\Theta,q,\gamma)=(13,12,27,\tfrac89),\\ &\gamma_{\mathrm{crit}}(5)=(\tfrac{48}5,\tfrac{48}{35},\tfrac{48}{65},\tfrac{48}{95}) \ne1,\tfrac25,\qquad \tfrac{12}{\gamma}=30\Rightarrow\gamma=\tfrac25,\quad 30\cdot\tfrac45=24,\\ &g=\tfrac{q-1}2,\ g=2\iff q=5,\ q=27\Rightarrow g=13,\\ &E_{\max}(q=5)=\tfrac{48}{5}\lceil\tfrac8\ell\rceil =(76.8,38.4,28.8,19.2,9.6),\\ &E(q)=\tfrac{12(q-1)}{q-3}\to12,\qquad \min E=\tfrac{24}{19}\ (\Theta=14),\ \tfrac{48}{35}\ (\Theta=10). \end{aligned} \] \subsection{The $\gamma_{\mathrm{crit}}$ table} $\gamma_{\mathrm{crit}}=12(q-1)/(qN)$: \begin{center} \renewcommand{\arraystretch}{1.1} \begin{tabular}{c|cccc|c} \toprule $q$ & $N=1$ & $7$ & $13$ & $19$ & $\gamma_{K}(q)$ \\ \midrule $5$ & $9.600$ & $1.371$ & $0.7385$ & $0.5053$ & $0.400$ \\ $9$ & $10.667$ & $1.524$ & $0.8205$ & $0.5614$ & $0.667$ \\ $15$ & $11.200$ & $1.600$ & $0.8615$ & $0.5895$ & $0.800$ \\ $\mathbf{27}$ & $11.556$ & $1.651$ & $\mathbf{0.8889}$ & $0.6082$ & $\mathbf{0.8889}$ \\ $51$ & $11.765$ & $1.681$ & $0.9050$ & $0.6192$ & $0.941$ \\ $101$ & $11.881$ & $1.697$ & $0.9139$ & $0.6253$ & $0.970$ \\ \bottomrule \end{tabular} \end{center} At $q=27$, $N=13$ is the unique equality point of $\gamma_{K}$ and $\gamma_{\mathrm{crit}}$. For $\Theta=12$: $q<27$ gives $E>1$ (weak $abc$), $q=27$ gives $E=1$, $q>27$ gives $E<1$ (excluded). The $\ell=2$ branch ($\gamma=2/q$) gives $E=6(q-1)/N$ with $E=1\iff6(q-1)=N\in\{1,7,13,19\}$, all non-integral, sealing that branch for all $q$. \section{Base verification and the quality plane}\label{app:base} \subsection{Statement and scope of Theorem A (base verification)} The following is confirmed verbatim against the source text (arXiv:2611.05217v1): \textbf{(a)} Theorem~A fixes a genus-$2$ curve $\mathcal C/K_{0}$ and the logarithmic height attached to its canonical bundle; it applies to every number field $F\supseteq K_{0}$ and every $P\in\mathcal C(F)$, with $h(P)\le(12+\eps)\delta(F)/d+\beta(\eps)$; Theorem~B is $c\le A(\eps)N^{24+\eps}$. \textbf{(b)} The constants $36,12,30,24,\gamma=\frac25,\eta=\frac45,\frac53,\frac32,\frac16$ each have an explicit location in \cite[\S6.3 (28)--(35), \S7.1--\S7.2, Prop.~2, Cor.~1]{SouleABC}; Proposition~2 and Corollary~1 are unconditional there. Tag: (a) is U at text level. The internal adelic metric layer is now tiered: U (normalisation, pullback weights, Lemma~1, Prop.~2, Cor.~1, (29), coefficient chain --- Appendices~\ref{app:adelic} and~\ref{app:H}); C (Prop.~3 Green-function inequalities); O ((32) strict isometry, admissible-metric data). The blanket phrase ``independently unverified'' is thereby retired. \subsection{The quality plane}\label{app:quality} Define $X=\frac{\log c}{\log N}$ (threshold $1$), $Y=\frac{\log(abc)}{\log N}$ (value $3$), and $\lambda=Y/X\in[2,3]$ (the quality plane of \cite{LaniewskiMueller}). \begin{theorem}[Discount in quality coordinates]\label{thm:quality} $\theta=\frac{\log M_{5}}{\log N}\le\frac{Y-1}{4}$, hence \[ E_{\mathrm{eff}}=24(1-\theta)\ \ge\ 24\Bigl(1-\frac{Y-1}{4}\Bigr)=6(5-Y). \] Under the $abc$/Szpiro thresholds $Y\le3$ one gets $\theta\le\frac12$ and \[ E_{\mathrm{eff}}\ge12 . \] The triple $(1,31,32)$ attains $\theta=\frac{Y-1}{4}=0.168$ exactly. \end{theorem} \begin{proof} $E_{\mathrm{mult}}=\log(abc)-\log N=(Y-1)\log N$; each $p\in M_{5}$ contributes $v_{p}-1\ge4$, so $\log M_{5}\le E_{\mathrm{mult}}/4$ (Theorem~\ref{thm:noboot}(i)). Substitution gives the claim. For $(1,31,32)$: $abc=2^{5}\cdot31$ and only $p=2$ has $5\mid v_{p}$, so $\theta=\frac{\log2}{\log62}$ while $Y-1=\frac{4\log2}{\log62}$: equality. \end{proof} This gives a second, independent route to the barrier $12$ --- the first being $E(q)=\frac{12(q-1)}{q-3}\to12$ (Proposition~\ref{prop:Em}) --- now with a threshold-level interpretation. \subsection{Comparison with Pasten 2026} On the Frey--Hellegouarch family, the exponent-$24$ output of Theorem~B strengthens the 2026 bounds of Pasten \cite{PastenShimura,PastenSzpiro} \begin{theorem}[Theorem~B versus Pasten 2026]\label{thm:EPasten} On the Frey--Hellegouarch family attached to a coprime triple $a+b=c$ with $N=\operatorname{rad}(abc)$, Theorem~B of \cite{SouleABC} gives \[ \log c\le(24+\varepsilon)\log N+O_{\varepsilon}(1), \] whereas Pasten \cite{PastenSzpiro} gives, unconditionally for all elliptic curves over $\mathbb Q$, \[ \log\Delta\ \ll\ N_{E}\log\log N_{E}, \] and $\log\Delta\ll N_{E}$ in the semistable case. On the Frey family $N_{E}\asymp N$; since $N\gg\log N$, Theorem~B is the stronger bound on this family, but it is conditional on Theorem~A while \cite{PastenSzpiro} is unconditional. The two are of different logical status and different analytic shape (logarithmic versus linear in the conductor). \end{theorem} \begin{proof} \emph{Step 1 (Frey curve data).} For $E:y^{2}=x(x-a)(x+b)$ one has $\Delta=16(abc)^{2}$ and conductor $N_{E}=2^{O(1)}\prod_{p\mid abc,\,p\ne2}p$, so $N_{E}\asymp N$ up to a power of $2$ absorbed in the implicit constant (Frey [13]). \emph{Step 2 (translate Pasten).} Substituting $N_{E}\asymp N$ into $\log\Delta\ll N_{E}\log\log N_{E}$ and using $\log\Delta=2\log(abc)+O(1)\ge2\log c+O(1)$ gives $\log c\ll N\log\log N$ on this family. \emph{Step 3 (compare).} Theorem~B gives $\log c\le(24+\varepsilon)\log N+O_{\varepsilon}(1)$. For $N\to\infty$, $\log N=o(N\log\log N)$, so Theorem~B is stronger. \emph{Step 4 (status).} Theorem~B of \cite{SouleABC} is derived from Theorem~A, whose proof is not independently verified here; \cite{PastenSzpiro} is an unconditional theorem. Hence neither result supersedes the other: one is stronger but conditional, the other weaker but unconditional. This replaces the earlier ``stated here, not re-proved'' parenthetical of Appendix~E.3. \end{proof} \section{Complete decomposition of the constant; closure of Route 1}\label{app:constdec} \subsection{Five locked factors} \[ 12=\frac13\cdot\frac{m}{\rho-mc} =\frac13\cdot\frac{6}{\frac53-\frac64}. \] The five factors --- $\rho=\frac53$ (Proposition~\ref{prop:rho}), $m=6$ (Corollary~\ref{cor:15}), $c=\frac14$ (Theorem~\ref{thm:drr}), $\frac13$ (from $\deg_{K}(x)=0$, $\deg_{K_{0}}(\bar M)=2$), and $A$ (appearing only in $O(d')$) --- each have a unique source and are geometrically locked; \emph{none is an estimate margin} (Corollary~\ref{cor:gamma3} and Remark~\ref{rem:normdict} fix the normalisation). \subsection{Closed formula for the exponent} \begin{theorem}\label{thm:Eofc} With $c=\frac{11}{12}-\frac{T}{12k}$ as the free parameter, \[ \Delta(c)=\frac53-6c,\qquad C(c)=\frac{18}{5-18c},\qquad t_{A}(c)=\frac{6}{5-18c},\qquad E(c)=\frac{12}{5-18c} . \] For $c\ge0$ (i.e.\ $T\le11k$) one has $E(c)\ge\frac{12}{5}=2.4$. The old thresholds reappear exactly: $c=\frac1{90}$ gives $C=\frac{15}{4}$, $t_{A}=\frac54$, $E=\frac52$. \end{theorem} \begin{proof} $\Delta=\rho-mc$; $C=m/\Delta$; $t_{A}=C/3$; $E=t_{A}\cdot\eta/\gamma$ with $\eta/\gamma=\frac{4/5}{2/5}=2$. Substituting $c=\frac1{90}$: $5-18c=\frac{24}{5}$, giving $C=\frac{90}{24}=\frac{15}{4}$, $t_{A}=\frac{30}{24}=\frac54$, $E=\frac{60}{24}=\frac52$. \end{proof} \subsection{Sixth No-Go reason: the double exclusion $T/k=47/3$} \begin{theorem}\label{thm:sixth} $E=1$ is equivalent to $c=-\frac7{18}$, hence to $T/k=11-12c=\frac{47}{3}$, which is excluded \emph{both} because $\frac{47}{3}\notin2\mathbb Z$ (Corollary~\ref{cor:parity}) \emph{and} because $T/k\le15$ (Proposition~\ref{prop:range}). This converges with the value $\Theta=\frac{47}{3}$ of \S\ref{sec:torsion} (chain-A target $N=24$, $3\Theta-23=24$), obtained there by an independent route. \end{theorem} \begin{proof} $E(c)=1\iff5-18c=12\iff c=-\frac7{18}$; then $T/k=11-12c=11+\frac{14}{3}=\frac{47}{3}>15$. \end{proof} \subsection{Route 1 is closed}\label{app:routeone} \begin{theorem}\label{thm:routeone} Archimedean and successive-minima refinements cannot move the constant $12$. The Ballay toolkit --- Chen's conjecture and the equality criterion $\zeta_{\mathrm{ess}}=\hat\mu^{\mathrm{asy}}_{\max}$, concave-transform constants with Zhang equality, and absolute transference \cite{Ballay} --- acts on the $\zeta$-isomorphism comparison and on the $O(d')$ layer; it does not modify $\delta$, $c$, $\rho$, $m$, or $\frac32=m\cdot c$. By Theorem~\ref{thm:Eofc} the exponent is $E(c)=\frac{12}{5-18c}$; these refinements leave $c$ fixed, hence $E$ is fixed: Route~1 is closed. \end{theorem} \section{Independent re-derivation of the adelic layer}\label{app:adelic} This appendix records the tiered verification of the internal adelic metric estimates of \cite{SouleABC}. Method: verbatim reading of the source, cross-checking against the standard literature, and exact rational-number recomputation. We do \emph{not} claim an independent verification of the whole proof of Theorem~A. \begin{theorem}[Normalisation self-consistency]\label{thm:normself} The definition $\widehat{\deg}_{K}(\bar L)=-\sum_{v}\log\|s\|_{v}$ is well-defined only if the weights $n_{v}(K)=[K_{v}:\mathbb Q_{v}]/[K:\mathbb Q]$ are built into $\log\|f\|_{v}=n_{v}(K)\log|f|_{v}$; then the product formula gives independence of $s$. Hence two conventions must be kept apart. In \cite[\S1.3.1]{SouleABC} one has $|\pi_{v}|_{w}=1/N(v)$ and $\prod_{v}|x|_{v}=1$; in \S1.3.3, $\widehat{\deg}_{K}(\bar L)=-\sum_{v}\log\|s\|_{v}$ with no factor $1/[K:\mathbb Q]$, the local weight being absorbed in $\|\cdot\|_{v}$. The factor $1/[K:\mathbb Q]$ appears only in the Weil height $h=[K:\mathbb Q]^{-1}\widehat{\deg}_{K}$. With $|f|_{v}$ denoting the $\mathbb Q_{p}$-normalised absolute value one has $\log\|f\|_{v}=[K_{v}:\mathbb Q_{p}]\log|f|_{v}$; equivalently, in the $n_{v}(K)=[K_{v}:\mathbb Q_{v}]/[K:\mathbb Q]$ convention, $\sum_{w\mid v}n_{w}(F)=n_{v}(K)$ and $\widehat{\deg}_{F}(u^{\ast}\bar L)=[F:K]\widehat{\deg}_{K}(\bar L)$. The product formula, pullback invariance and $\delta(K)=\log|\Delta_{K}|$ (no $1/2$) are unchanged. \end{theorem} \begin{theorem}[Pullback weight identity]\label{thm:pullbackweight} $\sum_{w\mid v}n_{w}(F)=n_{v}(K)$ for $w\mid v$ in $F/K$; verified at ramified, split and inert places. Consequently, in the $n_{v}$ convention, $\widehat{\deg}_{F}(u^{\ast}\bar L)/[F:\mathbb Q]=\widehat{\deg}_{K}(\bar L)/[K:\mathbb Q]$; in the unnormalised convention of \cite[\S1.3.3]{SouleABC}, $\widehat{\deg}_{F}(u^{\ast}\bar L)=[F:K]\widehat{\deg}_{K}(\bar L)$: pullback invariance, the dimension source of $d=[F:\mathbb Q]$ in Lemma~1. \end{theorem} \begin{theorem}[Uniformity of the $O(d)$ constant]\label{thm:uniformod} Lemma~1 provides comparison constants $C_{v}\ge1$ equal to $1$ on a common model open set, with $A(Y,\hat x):=\sum_{v}\log C_{v}$ finite and depending only on the two fixed metrics. Pullback scales this by $[F:K]$, giving $\widehat{\deg}_{F}(u^{\ast}\hat x)\le[F:K]\deg_{K}(x)+[F:K]A$ with $A$ \emph{independent of $F$, $P$, $u$}: Proposition~2's uniformity follows structurally, without the author's metric data. \end{theorem} \begin{theorem}[Corollary~1 and (29)]\label{thm:corone} Corollary~1 (two $\eps$-perturbations $\hat x_{\pm}=\pm\hat x-\eps\bar M$, both of negative degree) and (29), $h_{a}(s_{i})\le h(s_{i})+O(d')$ via $\|s\|_{a,w}\le C_{v(w)}\|s\|_{w}$ summed against Theorem~\ref{thm:pullbackweight}, re-derive without further input. \end{theorem} \begin{theorem}[Coefficient chain (31)--(36)]\label{thm:coefchain} Every entry of the chain recomputes exactly: $3k=\om^{2}/4$ (hence $c=\frac14$), $\deg_{K}(\bar L)/\om^{2}=\frac13$, $\frac32=m\cdot c$ with $m=6$ (not a Minkowski constant), $\Delta=\rho-mc=\frac16$, $C=36$, $t_{A}=12$, $E=t_{A}\eta/\gamma=12\cdot\frac{4/5}{2/5}=24$, and $30\cdot\frac45=24$ (note $30-\frac45\times30=6$ is a mis-transcription). \end{theorem} \begin{remark}[The gap in (32)]\label{rem:gap32} The most substantial finding: (32) asserts that $\alpha:H^{\sup}\to\phi'_{\ast}(L')^{\sup}$ ``is an isometry'' because $\dim H=2$ ``by Grauert's theorem''. Grauert's base-change gives a linear isomorphism ($\dim H=2$ forces surjectivity); \emph{isometry} requires a place-by-place norm comparison $\|s\|_{H^{\sup},w}=\|\alpha(s)\|_{\phi'_{\ast}(L')^{\sup},w}$, which the source does not supply. This is a gap of one or two sentences, not an error. Proposition~1 of \cite{SouleABC} is proved in its \S\S1.6.2--1.6.3, and \S6.3.5 of the source already gives $H^{\sup}$ a model on the whole of $\operatorname{Spec}(\mathcal O_{F'})$ (Appendix~J). What (32) still needs is the place-by-place comparison of the two model norms through the Grauert morphism $\alpha$ on the \emph{relative} pushforward; Theorem~I.4 supplies the sufficient $O(d')$ form independently, and Proposition~I.6 shows the strict equality is not automatic for arbitrary models but is expected under the normalisation used in \cite{SouleABC}. \end{remark} \begin{center} \renewcommand{\arraystretch}{1.2} \begin{tabular}{p{0.62\textwidth}|p{0.25\textwidth}} \toprule layer & status \\ \midrule Normalisation, pullback weights, Lemma~1, Prop.~2 (uniform), Cor.~1, (29), coefficient chain, (2)/(30) Minkowski (Bombieri--Vaaler, Borek), (33) $\om^{2}/4$, $\delta(F')/d'$, height stability, (32) up to $O(d')$ (Th.~I.4), admissible-metric existence (\cite[Prop.~2.6]{Yuan26}) & U \\ ``model metric $=$ admissible metric'' joint assertion; Prop.~3 Green-function data (structure Zhang \cite{ZhangAdmissible}) & C \\ (32) strict equality as written in \cite{SouleABC} & O$^{\ast}$ \\ \bottomrule \end{tabular} \end{center} \noindent{\footnotesize $^{\ast}$~The models are fixed on the whole of $\operatorname{Spec}(\mathcal O)$ in \cite[\S\S6.3.4--6.3.5]{SouleABC}; Proposition~I.6 excludes only the unconditional reading. One sentence of place-by-place comparison is missing --- the statement is not false.} \emph{Closure with Appendix~\ref{app:constdec}.} The metric layer contributes only $O(d')$ and $\beta(\eps)$: \[ \underbrace{(29)\ h_{a}\le h+O(d')}_{\text{metric comparison}} +\underbrace{(30)\ A\,d'}_{\text{Minkowski}} +\underbrace{(33)\ O_{\eps}(d')}_{\text{Cor.~1}} \ \xrightarrow{\S7.1\,/\,d'}\ \underbrace{O(1)\to\beta(\eps)}_{\text{error terms only}} . \] The main chain $12=\frac13\cdot\frac{m}{\rho-mc}$ is determined by $\rho=\frac53$, $m=6$, $c=\frac14$, $\deg_{K_{0}}(\bar M)=2$ alone --- internal confirmation, independent of Appendix~\ref{app:constdec}'s external layer analysis, that Route~1 is closed. \subsection{Reconciliation with the multi-source cross-verification} An independent multi-source report cross-checked the normalisation conventions of \cite{SouleABC} against the standard literature (Zhang \cite{ZhangSmallPoints}, Yuan--Zhang \cite{YuanZhang}, Moriwaki \cite{MoriwakiAdelic}, Gaudron \cite{Gaudron}, Ballay \cite{Ballay}, Bombieri--Vaaler \cite{BombieriVaaler}, Borek \cite{Borek}, Gillet--Soul\'e \cite{GS}, Gillet--R\"ossler--Soul\'e \cite{GRS}, Faltings \cite{FaltingsLectures}). Its verdicts, insofar as they overlap with this appendix, are settled as follows. \emph{(i) Two over-suspicions refuted.} The report's ``normalisation conflict'' (its \S2.4, \S6.1, \S7.3) is answered by Theorem~\ref{thm:normself}: well-definedness of $\widehat{\deg}$ plus the product formula \emph{forces} the weights $n_{v}(K)$ into $\log\|f\|_{v}=n_{v}(K)\log|f|_{v}$, so the source's notation is the normalised degree in shorthand, not an unnormalised conflict. Its ``uniform $O(d)$ not independently confirmable'' (its \S3.2) is answered by Theorem~\ref{thm:uniformod}: the constant's \emph{independence} of $F,P,u$ follows from Lemma~1's structure (the constant's numerical value is author data, but uniformity is not). Its remaining conditional layer --- the standard status of the rank-$2$ Minkowski skeleton, the Green-function inequalities, and the (32) isometry --- is recorded without change in the U/C/O table above; the (32) isometry is the priority item. \emph{(ii) Adopted recommendation.} The report's principal recommendation --- an explicit \emph{normalisation dictionary} --- is answered by Theorems~\ref{thm:normself}--\ref{thm:pullbackweight} of this appendix, which give one self-consistent convention ($\|f\|_{v}=|f|_{v}^{n_{v}(K)}$, $\sum_{w\mid v}n_{w}(F)=n_{v}(K)$, $\delta(K)=\log|\Delta_{K}|$ with no $1/2$); its conversion to the unnormalised notation of \cite[\S\S1.3.1/1.3.3]{SouleABC} is recorded in Proposition~J.2(2) (the $1/2$ in GRR is the Todd-class coefficient, not an Arakelov-degree normalisation). \emph{(iii) Its 30-item author-data checklist} (metrics at each place, explicit $C_{v}$ bounds, Green-function data, $\Phi'$ finiteness, Prop.~9 full proofs, admissible-metric existence) is condensed into the O-layer of the table; we concur that the structure is standard while the constants are author data. \section{De-conditioning the external dependence} \label{app:H} The main text tags Theorems~7.5 and~9.2, and the final No-Go line of Table~1, as \texttt{Th.A} (conditional on Theorem~A of [1]). This appendix re-derives the external-literature chain and shows that this tag is too coarse: most of the conditionality can be removed, but \emph{not} all of it, and the removable part is not the part that matters. \subsection{Normalisation and the strength comparison} \label{app:H1} Throughout $N=\operatorname{rad}(abc)$, as in \S1.1 (not $\log N$). \begin{theorem}[Unconditional exponential bound]\label{thm:HSY} For coprime positive $a+b=c$ and $N=\operatorname{rad}(abc)$, \[ \log c\le \kappa\,N^{1/3}(\log N)^3,\qquad\text{equivalently}\qquad \log c=O_\varepsilon(N^{1/3+\varepsilon}), \] with $\kappa$ effective (Stewart--Yu [33, 34]; the improvement chain $\alpha=15\to2/3\to1/3$ is recorded in [20, (1.1)] after Stewart--Tijdeman [35]). Equivalently $c<\exp(C_\varepsilon N^{1/3+\varepsilon})$. \end{theorem} \begin{proof} This is Theorem~1 of [33] and its refinement in [34] (the $\alpha=1/3$ case), obtained from lower bounds for linear forms in ordinary and $p$-adic logarithms; the constant is effective. The equivalence of the two forms follows by exponentiating $\log c\le\kappa N^{1/3}(\log N)^3$ and absorbing $(\log N)^3$ into $N^{\varepsilon}$. \end{proof} \begin{theorem}[Polynomial versus exponential type]\label{thm:Hcmp} Theorem~B of [1] is $c\le A(\varepsilon)N^{24+\varepsilon}$, i.e. $\log c\le(24+\varepsilon)\log N+O(1)$. For every sufficiently large $N$, \[ 24\log N=o\!\left(N^{1/3}(\log N)^3\right), \] so Theorem~B is \emph{strictly stronger} than Theorem~\ref{thm:HSY}. More importantly, Theorem~\ref{thm:HSY} is of \emph{exponential} type, whereas Theorems~7.5 and~9.2 are of \emph{polynomial} type ($c\le N^{E}$). As recorded verbatim in [20, \S1.1]: ``all unconditional bounds for these conjectures are of exponential nature''. Hence no unconditional Stewart--Yu-type result can replace Theorem~A as input to Theorem~7.5. \end{theorem} \begin{proof} $\lim_{N\to\infty}\frac{24\log N}{N^{1/3}(\log N)^3}=0$ since $N^{1/3}$ dominates any power of $\log N$. Numerically the inequality $24\log N0$. We restrict to the admissible range of Proposition~\ref{prop:rho}, $h^{0}(L_{F})\ge2$ (the rank condition for the Minkowski step of the chain), i.e.\ $br\le4g_0-5+\frac r2$; for $g_0=13$ this reads $r(2b-1)\le94$ and forces $\mu=96-r(b-1)\ge49>0$. The chain is not of the construction's type outside this range. Writing $c=c_{\mathrm{GRR}}-\Theta/W$ with $\Theta=T/k$ (setting $D_0=0$, the most favourable normalisation), one has \begin{equation}\label{eq:Kgen} \frac1t=\frac{\lambda}{\mu W}\bigl(\mu\Theta+B_0\bigr),\qquad B_0=W\bigl(\rho-\mu c_{\mathrm{GRR}}\bigr),\qquad t=\frac{\mu W}{\lambda(\mu\Theta+B_0)} . \end{equation} \begin{theorem}[Literature point reproduced]\label{thm:K1} For $(g_0,r,b,\lambda)=(2,2,2,3)$ one has $W=12$, $\mu=6$, $\rho=5/3$, $c_{\mathrm{GRR}}=11/12$, $B_0=-46$, and \eqref{eq:Kgen} becomes $t(\Theta)=\dfrac{12}{3\Theta-23}$, i.e.\ $t(8)=12$, $t(10)=12/7$, $t(12)=12/13$, $t(14)=12/19$. In particular $t\le12/13$ requires $\Theta\ge12$, the unique candidate of \S8. \end{theorem} \begin{proof} Substitute \eqref{eq:Kinv} into \eqref{eq:Kgen}: $\mu W=72$, $\lambda(\mu\Theta+B_0)=3(6\Theta-46)=18\Theta-138$, hence $t=72/(18\Theta-138)=12/(3\Theta-23)$. The four values are direct rational substitutions; $12/(3\Theta-23)\le12/13$ with positive denominator is $3\Theta-23\ge13$, i.e.\ $\Theta\ge12$. \end{proof} \begin{theorem}[$g_0=13$: two most favourable readings]\label{thm:K2} Let $\Theta\in2\mathbb Z$ with $8\le\Theta\le15$ (the range imposed by exponent-$1$ optimality, \S5). Then: \begin{enumerate} \item Along the rank-$2$ locus $b=2g_0-2=24$ with $(r,\lambda)=(2,3)$: $W=144$, $\mu=50$, $\rho=207$, $c_{\mathrm{GRR}}=1199/12$, $B_0=-689592$, and $\rho-\mu c<0$ for every allowed $\Theta$, so $t$ has no positive value. Positivity requires $\Theta>\frac{344796}{25}=13791.84$, and $t\le12/13$ requires $\Theta\ge\frac{346096}{25}=13843.84$ (smallest even $\Theta=13844$). \item Keeping the literature value $b=2$ (so the rank of $L_F$ grows) with the same $(r,\lambda)$: $W=144$, $\mu=94$, $B_0=-12168$; positivity requires $\Theta>\frac{6084}{47}=129.45$ and $t\le12/13$ requires $\Theta\ge\frac{8528}{47}=181.45$ (smallest even $\Theta=182$). \end{enumerate} Letting $\lambda\to\infty$ does not help: the limits are $13791.84$ and $129.45$ respectively (smallest even $\Theta=13844$ and $130$). \end{theorem} \begin{proof} For (1), $\mu W=7200$ and $\mu\Theta+B_0=50\Theta-689592$; at $\Theta\le15$ this is negative, so $t<0$. The positivity threshold is $-B_0/\mu=689592/50=344796/25$. Solving $\mu W/(\lambda(\mu\Theta+B_0))\le12/13$ gives By \eqref{eq:Kgen}, $t\le12/13$ is equivalent to $\mu\Theta+B_0\ge13\mu W/(12\lambda)$, i.e.\ $\Theta\ge13W/(12\lambda)-B_0/\mu$. For (1), $13W/(12\lambda)=13\cdot144/36=52$ and $-B_0/\mu=344796/25$, so $\Theta\ge52+344796/25=346096/25$. For (2), $W=144$, $\mu=94$, $B_0=-12168$, $-B_0/\mu=6084/47$, and the same formula gives $\Theta\ge52+6084/47=8528/47$; the intermediate constant is $13\mu W/(12\lambda)=13\cdot144\cdot94/36=4888$, and $(4888+12168)/94=17056/94=8528/47$, consistent. Both exceed the allowed range. As $\lambda\to\infty$ the extra term $13W/(12\lambda)\to0$, leaving the positivity threshold as the limit. \end{proof} \begin{theorem}[Uniform lower bound, all admissible $(r,b,\lambda)$]\label{thm:K3} For $g_0=13$, every admissible $(r,b)$ with $r\ge1$, $b\ge2$, $br\le4g_0-5+\frac r2$, and every $\lambda>0$, \begin{equation}\label{eq:Kbound} \Theta_{\min}=-\frac{B_0}{\mu}=W\Bigl(c_{\mathrm{GRR}}-\frac{\rho}{\mu}\Bigr) \ \ge\ \frac{1230}{19}=64.736\ldots\ >\ 15 . \end{equation} The minimum is attained at $(r,b)=(1,2)$. Hence no admissible $\Theta$ produces a positive $t$, and a fortiori none satisfies $t\le12/13$. \end{theorem} \begin{proof} With the corrected $\mu=96-r(b-1)$ of \eqref{eq:Kinv} and $W=72r$, \[ F(r,b):=\Theta_{\min} =72r\,\frac{2b^{2}+2b-1}{12}-\frac{24r(b-1)(b+3)}{96-r(b-1)} . \] The admissibility constraint $r(2b-1)\le94$ gives $1\le r\le r_{\max}(b):=\lfloor 94/(2b-1)\rfloor$ (so $b\le47$) and $96-r(b-1)\ge96-\frac{94-r}{2}>0$. \emph{Endpoint reduction.} For fixed $b$, the second summand of $F$ is a negative multiple of $h(r)=r/(96-r(b-1))$, and $h''(r)=\dfrac{192(b-1)^{2}}{(96-r(b-1))^{3}}>0$; hence $F$ is concave in $r$ and its minimum over $r\in[1,r_{\max}(b)]$ is attained at an endpoint. \emph{First endpoint.} $F(1,b)=72c_{\mathrm{GRR}}(b)-\dfrac{24(b-1)(b+3)}{97-b}$ has derivative \[ 24b+12-\frac{24\bigl(-b^{2}+194b+191\bigr)}{(97-b)^{2}} . \] The subtracted term is increasing on $[2,47]$ (its derivative has numerator $19200>0$), equals $1.53\ldots$ at $b=2$, and is therefore $<69$ throughout; since $24b+12\ge60$, with room $>15$ at $b=2$ and $84>69$ for $b\ge3$, the derivative is positive on $[2,47]$. Thus \[ F(1,b)\ge F(1,2)=72\cdot\tfrac{11}{12}-\tfrac{24\cdot1\cdot5}{95} =66-\frac{24}{19}=\frac{1230}{19}=64.736\ldots . \] \emph{Second endpoint.} At $r=r_{\max}(b)$, exact rational evaluation for $2\le b\le31$ gives $F\ge F(31,2) =2232\bigl(\tfrac{11}{12}-\tfrac{1}{39}\bigr)=\dfrac{25854}{13} \approx1988.8>\dfrac{1230}{19}$; for $b\ge32$ one has $r_{\max}(b)=1$ and the first endpoint applies. Combining both endpoints, $F(r,b)\ge\frac{1230}{19}>15$ for every admissible $(r,b)$. \end{proof} \begin{theorem}[Only $g_0=2$ starts the chain]\label{thm:K4} Along the rank-$2$ locus $b=2g_0-2$, $r=2$, $\lambda=3$, the positivity threshold $-B_0/\mu$ equals $23/3\approx7.67$ at $g_0=2$ and exceeds $15$ for every $g_0\ge3$: $306/5=61.2$ ($g_0=3$), $1473/7\approx210.4$, $4532/9\approx503.6$, $\dots$, $344796/25=13791.84$ ($g_0=13$), $473603/27\approx17540.9$ ($g_0=14$). \end{theorem} \begin{proof} Direct substitution of \eqref{eq:Kinv} into $-B_0/\mu$ for $g_0=2,\dots,14$ by exact rational arithmetic; the values are monotone increasing because $c_{\mathrm{GRR}}\sim b^2/6\sim O(g_0^2)$ while $\rho/\mu=O(g_0)$. The structural reason is that a positive $\rho-\mu c$ requires the torsion contribution $T/\omega^2$ to cancel a quadratic GRR term, whereas $T=k\Theta$ with $\Theta=O(1)$ in the first-order degeneration model. \end{proof} \begin{corollary}[Seventh No-Go reason; Open problem (2) closed]\label{cor:K5} Even under the most favourable assumption that the coefficient chain \eqref{eq:Kgen} extrapolates unchanged to $g_0=13$, the chain cannot start: $\rho-\mu c<0$ for every allowed $\Theta$, so no positive $t$ exists. This reason is independent of the six previously recorded ones, of the normalisation $\gamma=2/5$, of the modulus-$3$ argument, of the transversality hypothesis, and of the truth of Theorem~A. Consequently Open problem~(2) is closed: the statement ``$t=12$ does not extend to $g=13$'' is upgraded from a proof-structure remark to a computation. \end{corollary} \begin{proof} Combine Theorems~\ref{thm:K2}, \ref{thm:K3}; none invokes Theorem~A. The independence is as in Proposition~H.5. \end{proof} \begin{remark}[Honesty boundary]\label{rem:Khonest} Formula \eqref{eq:Kgen} is unconditional (Appendix~B.2 plus the DRR decomposition of \S4). The assumption that its \emph{form} persists at $g_0=13$ is the most favourable extrapolation; because even granting it the chain fails, the conclusion is robust. The bound $\Theta\le15$ is inherited from exponent-$1$ optimality; without it the required $\Theta\sim1.4\times10^4$ is incompatible with a first-order degeneration at four exceptional fibres. All weak-$abc$ outputs remain conditional on Theorem~A. \end{remark} \section{Quantitative index-ideal absorption and the engine-replacement criterion} \label{app:absorp} \subsection{Absorption inequalities} Let $f\in\mathbb Z[x]$ be monic irreducible of degree $n\ge2$, let $\theta$ be a root and $F=\mathbb Q(\theta)$, so $[F:\mathbb Q]=n$. Put $\mathfrak I=[\mathcal O_F:\mathbb Z[\theta]]\in\mathbb Z_{\ge1}$ and $\delta(F)=\log|\Delta_F|$. Dedekind's theorem (10.8) reads $\operatorname{disc}(f)=\mathfrak I^{2}\,\mathfrak d_F$; taking logarithms gives the identity \begin{equation}\label{eq:21log} \log|\operatorname{disc}f|=\delta(F)+2\log\mathfrak I . \end{equation} For a prime $p\nmid n$ let $r_p$ be the number of primes of $F$ above $p$. The tame discriminant formula gives \begin{equation}\label{eq:21tame} v_p(\mathfrak d_F)=\sum_i f_i(e_i-1)=n-r_p,\qquad 1\le r_p\le n . \end{equation} \begin{theorem}[Local absorption-rate inequality]\label{thm:21A} For every $p\nmid n$ put $\iota_p=2v_p(\mathfrak I)/v_p(\operatorname{disc}f)$. Then \begin{equation}\label{eq:21rate} 1-\frac{n-1}{v_p(\operatorname{disc}f)}\ \le\ \iota_p =1-\frac{n-r_p}{v_p(\operatorname{disc}f)}\ \le\ 1 , \end{equation} and $\iota_p=1$ if and only if $r_p=n$, i.e.\ $p$ is unramified. Globally, \begin{equation}\label{eq:21global} 0\le\delta(F)\le\log|\operatorname{disc}f|,\qquad \delta(F)\le (n-1)\log N+O_n(1). \end{equation} \end{theorem} \begin{proof} Subtracting \eqref{eq:21tame} from \eqref{eq:21log} place by place gives $2v_p(\mathfrak I)=v_p(\operatorname{disc}f)-(n-r_p)$. Since $\mathfrak I$ is a positive integer, $2v_p(\mathfrak I)\ge0$; since $r_p\ge1$ we have $n-r_p\le n-1$, which yields the lower bound in \eqref{eq:21rate}; the upper bound is $n-r_p\ge0$. The equivalence $\iota_p=1\Leftrightarrow n-r_p=0$ is immediate. The first inequality of \eqref{eq:21global} is $\log\mathfrak I\ge0$; the second follows from $v_p(\mathfrak d_F)=n-r_p\le n-1$ for $p\nmid n$ together with the fact that the finitely many wild primes $p\mid n$ contribute $O_n(1)$ uniformly, because $v_p(\mathfrak d_F)\le n\,v_p(n)+n$ for tame--wild decompositions of degree $n$. No choice of a generator enters. \end{proof} \begin{remark}\label{rem:21rate} Inequality \eqref{eq:21rate} is the quantitative upgrade of the remark following (10.10): the larger the multiplicity $v_p(\operatorname{disc}f)$, the closer $\iota_p$ is to $1$, i.e.\ the \emph{field} discriminant is asymptotically blind to the multiplicity. For $n=5$, $v_p(\operatorname{disc}f)=4k$ gives $\iota_p=0,\frac12,\frac23,1,1,1$ at $k=1,2,3,5,10,25$, against the lower bounds $0,\frac12,\frac23,\frac45,\frac9{10},\frac{24}{25}$ (verified by exact rational arithmetic). \end{remark} \subsection{The unified decomposition-type discount} \begin{theorem}[Unified formula for $\eta$]\label{thm:21B} With $\theta_{\mathrm{dec}}:=\dfrac{\sum_{p\in S}(r_p-1)\log p}{(n-1)\log N}\in[0,1]$ one has \begin{equation}\label{eq:21eta} \eta=\frac{\delta(F)}{n\log N} =\frac{n-1}{n}\bigl(1-\theta_{\mathrm{dec}}\bigr)+O_n\bigl(1/\log N\bigr). \end{equation} This single formula reproduces the non-Galois lattice $\{0,\frac15,\frac25,\frac35,\frac45\}$ of \S10.2 at $n=5$, the $M_5$-discount of \S9, the two-valuedness of Corollary~10.8, and the quality-plane bound $\theta\le (Y-1)/4$ of Appendix~E. \end{theorem} \begin{proof} Summing \eqref{eq:21tame} over $p\in S$, $p\nmid n$, and dividing by $n\log N$ gives $\delta(F)/(n\log N)=\frac{n-1}{n}-\frac{1}{n\log N}\sum_p (r_p-1)\log p+O_n(1/\log N)$, which is \eqref{eq:21eta}. For $r_p=1,\dots,n$ at $n=5$ the right-hand side of \eqref{eq:21eta} equals $\frac45(1-\frac{r_p-1}{4})=\frac45,\frac35,\frac25,\frac15,0$, the lattice of Theorem~10.6. In the pure Kummer case $r_p\in\{1,n\}$, so $\sum_p(r_p-1)\log p=(n-1)\log M_n$ with $M_n=\prod_{p:\,n\mid m_p}p$; hence $\theta_{\mathrm{dec}}=\log M_n/\log N$ and \eqref{eq:21eta} becomes $\delta(F)/n=\frac{n-1}{n}\log(N/M_n)$, Corollary~10.8 and \S9. Situation~II of \S9.2 has $r_p=n$ for both the split and the unramified-degree-$n$ sub-cases, so $\eta$ is unchanged, as proved there. Finally, in the pure Kummer class $r_p=n$ forces $n\mid m_p$, hence $m_p\ge n$ and $r_p-1=n-1\le m_p-1\le v_p(abc)-1$. Therefore $\theta_{\mathrm{dec}}\le\mathcal E/((n-1)\log N)=(Y-1)/(n-1)$ with $\mathcal E=\log(abc)-\log N=(Y-1)\log N$. At $n=5$ this is $\theta\le(Y-1)/4$, and $E=30\eta=24(1-\theta_{\mathrm{dec}})\ge 24(1-(Y-1)/4)$, giving $E_{\mathrm{eff}}\ge12$ when $Y\le3$, the quality-plane conclusion of Theorem~E.1. The constant $n-1$ is thus the maximal inertia order $v_p(\mathfrak d_F)=n-r_p\le n-1$, not an estimate margin; it is optimal because $r_p=n$ occurs whenever $v_p(abc)\ge n$. \end{proof} \begin{corollary}[Exact numerical check]\label{cor:21num} For $(a,b,c)=(1,31,32)$ one has $N=62$, $Y=1.6718$, $\theta_{\mathrm{dec}}=\log M_5/\log N=0.1679$ and $E_{\mathrm{eff}}=24(1-\theta)=19.969=24(1-(Y-1)/4)$; for $(9,16,25)$ and $(1,8,9)$ the discount vanishes and $E_{\mathrm{eff}}=24$. These agree with the tables of \S9.3 and Appendix~E. \end{corollary} \begin{proof} Direct substitution into \eqref{eq:21eta} with $\alpha=-ab/c^{2}$: for $(1,31,32)$, $\alpha=-31/2^{10}$, so $m_2=10$ with $5\mid10$ (étale, $r_2=5$) and $m_{31}=1$ (total, $r_{31}=1$); hence $\sum_p(r_p-1)\log p=4\log2$ and $\theta_{\mathrm{dec}}=4\log2/(4\log62)=\log2/\log62=0.1679$. The other two triples have no étale prime, so $\theta_{\mathrm{dec}}=0$. Exact rational arithmetic in the accompanying computation file confirms all six entries of \S10.3's $k$-table and the five $r_p$-values of Theorem~10.6. \end{proof} \subsection{Absorbed mass and the engine-replacement criterion} \begin{theorem}[Quantitative absorbed mass]\label{thm:21C} Assume the $|k|$-convention of (10.9)--(10.10), i.e. $v_p(\operatorname{disc}f)=(n-1)m_p+n v_p(n)$ with $m_p=|v_p(\alpha)|$ for $\alpha=-ab/c^{2}$. Then \begin{equation}\label{eq:21mass} 2\log\mathfrak I=(n-1)\bigl[\mathcal E+\log c+\log M_n\bigr]+O_n(1), \qquad \log\mathfrak I\ge\frac{n-1}{2}\bigl(\mathcal E+\log c\bigr)-O_n(1), \end{equation} and $\delta(F)/n=\frac{n-1}{n}\bigl(\log N-\log M_n\bigr)+O_n(1)$ is \emph{independent of $\mathcal E$}. \end{theorem} \begin{proof} Sum (10.10) over $p\in S$, $p\nmid n$. For $p$ with $n\nmid m_p$ the contribution is $(n-1)(m_p-1)$; for $p$ with $n\mid m_p$ it is $(n-1)m_p$. Hence $2\log\mathfrak I=(n-1)\bigl[\sum_{p\in S}m_p\log p-\sum_{p:\,n\nmid m_p}\log p\bigr]+O_n(1)$. Because $\gcd(a,b)=1$ and $\gcd(ab,c)=1$, $\sum_{p\in S}m_p\log p=\log(ab)+2\log c=\mathcal E+\log N+\log c$, while $\sum_{p:\,n\nmid m_p}\log p=\log N-\log M_n$. Subtracting gives \eqref{eq:21mass}. The stated lower bound uses $\log M_n\ge0$. Solving \eqref{eq:21log} for $\delta(F)$ gives the last formula, in which $\mathcal E$ cancels. \end{proof} \begin{theorem}[Engine-replacement criterion]\label{thm:21D} Let an ``engine replacement'' mean a choice of $f$ for which $\delta(F)/n$ is controlled by $\log N$ while $h_{K_{\mathcal C}}(P)=\gamma\log c+O(1)$ with $\gamma\asymp1$. A necessary condition is $\mathfrak I\le N^{O(1)}$. By \eqref{eq:21mass} this forces \begin{equation}\label{eq:21crit} c\le N^{\,2O(1)/(n-1)},\qquad\text{equivalently } Y-1\le O(1), \end{equation} which is an $abc$-grade condition with exact slope $2/(n-1)$. Consequently the S-smoothness loop of Theorem~10.10 is not merely ``stronger than $abc$'': it is of $abc$ grade at this quantitative level. This is a one-directional implication; the converse is not claimed. \end{theorem} \begin{proof} By \eqref{eq:21log} and the homogeneity of $\operatorname{disc}f$ of degree $2n-2$ in coefficients of size $c^{O(1)}$ (Theorem~10.10(i)), $\frac{\delta(F)}{n}+\frac{2\log\mathfrak I}{n} =\frac{\log|\operatorname{disc}f|}{n}\le\frac{2n-2}{n}\log c+O(1)$. If $\log\mathfrak I\asymp\log c$, the bridge $\delta(F)/n\asymp\log N$ breaks; hence $\mathfrak I\le N^{O(1)}$ is necessary. Insert $\log\mathfrak I\ge\frac{n-1}{2}(\mathcal E+\log c) -O_n(1)$ from \eqref{eq:21mass}: $\frac{n-1}{2}\log c\le O(1)\log N+O_n(1)$, i.e.\ \eqref{eq:21crit}. Writing $\mathcal E=(Y-1)\log N$, the same computation gives $Y-1\le O(1)$, so $E=24(1-\theta_{\mathrm{dec}})\ge 24(1-O(1)/(n-1))$ cannot approach $1$ unless $Y\to1$, which is $c\le N^{1+o(1)}$, the $abc$ conjecture. Theorem~10.10's magnitude comparison $c^{2n-2}$ vs.\ $c^{3}$ is thereby supplied with an exact slope. \end{proof} \begin{corollary}[Eighth No-Go reason]\label{cor:21eighth} The decoupling of $\eta$ from the multiplicity excess $\mathcal E$ holds at the level of the identity \eqref{eq:21log}, not at the level of estimates. No engine can make $\eta$ decrease with $\mathcal E$, because by \eqref{eq:21mass} the index ideal absorbs at least $\frac{n-1}{2}\mathcal E$ to $O(1)$ precision. This reason is independent of the six previously recorded reasons, of the $g_0=13$ collapse of \S11, of the modulus-$3$ argument and of the transversality hypothesis. \end{corollary} \begin{proof} Combine Theorems~\ref{thm:21B}, \ref{thm:21C} and Corollary~5.5 / Proposition~H.5: the argument uses only \eqref{eq:21log}, \eqref{eq:21tame} and $\mathfrak I\ge1$, none of which invokes the truth of Theorem~A or the torsion decomposition. \end{proof} \begin{remark}[Honesty boundaries]\label{rem:21honest} Theorems~\ref{thm:21A}, \ref{thm:21B} and Corollary~\ref{cor:21eighth} are unconditional: they use $\mathfrak I\ge1$ and the tame formula only, with no generator convention. Theorem~\ref{thm:21C} and the exact slope $2/(n-1)$ use the $|k|$-convention of (10.9)--(10.10); for $p\mid c$ the source writes $k=v_p(\alpha)<0$ in (10.9) but $|k|$ in (10.10), a sign convention left implicit. Without it, \eqref{eq:21global} still gives the direction of \eqref{eq:21crit} with an unspecified $O(1)$ slope. The implication ``$\mathfrak I\le N^{O(1)}$ is necessary'' is a scale criterion, not a sufficiency claim; the converse direction is not proved. All weak-$abc$ outputs remain conditional on Theorem~A; the No-Go statements are internal to the construction. \end{remark} \section{Distributional behaviour of the multiplicity excess and the discount} \label{app:distribution} Throughout this appendix $\Omega(X)$ denotes the set of coprime triples \begin{equation}\label{eq:Momega} \Omega(X)=\{(a,b,c):1\le a3(3\log3+\log\kappa)$ makes the right-hand side strictly smaller than $\log\log c$, contradicting Stewart--Yu. Hence $L>L_0$ for all large $c$; the finitely many remaining $c$ are absorbed by enlarging $C$. This is an unconditional corollary of an effective theorem; no Diophantine conjecture enters. \end{proof} \begin{theorem}[Exponential moment of the multiplicity excess]\label{thm:M2} For every $t<1$ there is $C_t<\infty$, independent of $X$, with \begin{equation}\label{eq:M2} \mathbb E_X\!\left[e^{t\mathcal E}\right]\le C_t,\qquad \mathbb E_X[\mathcal E]=O(1),\qquad \mathbb P_X(\mathcal E\ge K)\le C_t\,e^{-tK}. \end{equation} \end{theorem} \begin{proof} \textbf{Multiplicative CRT bound.} Fix a finite set $S$ of primes and exponents $v_p\ge2$. Because $\gcd(a,b)=\gcd(a,c)=\gcd(b,c)=1$, each $p^{v_p}$ divides exactly one of $a,b,c$; there are $3^{|S|}$ such allocations. For a fixed allocation $(A_1,A_2,A_3)$ of $S$ put $d_a=\prod_{p\in A_1}p^{v_p}$, $d_b=\prod_{p\in A_2}p^{v_p}$, $d_c=\prod_{p\in A_3}p^{v_p}$. The conditions are $d_a\mid a$, $d_b\mid(c-a)$, $d_c\mid c$. The three moduli are pairwise coprime, so the Chinese remainder theorem selects a single residue class of $(a,c)$ modulo $d_ad_bd_c$; consequently \begin{equation}\label{eq:MCRT} \mathbb P_X\bigl(\forall p\in S:\ p^{v_p}\mid abc\bigr) \ \le\ C_0^{|S|}\prod_{p\in S}p^{-v_p}, \end{equation} where the absolute constant $C_0$ absorbs the condition $\gcd(a,c)=1$ and the boundary of $\Omega(X)$. Crucially $3^{|S|}=\prod_{p\in S}3$ is absorbed into $C_0^{|S|}$ \emph{per prime}, so the bound is multiplicative. \textbf{Expansion.} For each prime, $p^{t(e_p-1)}\le 1+\sum_{j\ge2}\mathbf1_{p^{j}\mid abc}\,p^{t(j-1)}$: if $e_p=1$ the left side is $1$, while if $e_p\ge2$ the sum contains its last term $p^{t(e_p-1)}$. Expanding the product $e^{t\mathcal E}=\prod_p p^{t(e_p-1)}$ over all choices of a finite set $S$ and exponents $\boldsymbol v=(v_p)_{p\in S}$ with $v_p\ge2$, and applying \eqref{eq:MCRT}, gives \begin{align*} \mathbb E_X[e^{t\mathcal E}] &\le\sum_{S,\boldsymbol v}\Big(C_0^{|S|}\prod_{p\in S}p^{-v_p}\Big) \Big(\prod_{p\in S}p^{t(v_p-1)}\Big) =\prod_p\Big[1+C_0\,p^{-t}\sum_{v\ge2}p^{-(1-t)v}\Big]\\ &\le\prod_p\big[1+C'_{0,t}\,p^{-(2-t)}\big]\ =:\ C_t . \end{align*} The product converges precisely when $2-t>1$, i.e.\ $t<1$. The first moment bound is $t\to0^{+}$ (or Jensen plus $\mathcal E\le e^{t\mathcal E}/t$), and Chernoff's inequality gives the tail bound. \end{proof} \begin{remark}[Novelty boundary]\label{rem:Mstandard} Moment estimates for $\operatorname{rad}$ and for the number of high-power prime divisors are standard in the abc literature (Stewart--Tijdeman, $S$-unit counting). Theorem~\ref{thm:M2} is not a new number-theoretic tool; what is new below is its combination with the index-ideal/discount dictionary of Appendix~L to produce statements about the DIC construction that are independent of the truth of Theorem~A. \end{remark} \begin{theorem}[Zero density of the discount, with a large-deviation rate]\label{thm:M3} For every fixed $\lambda>0$ and every $t<1$, as $X\to\infty$, \begin{equation}\label{eq:M3} \mathbb P_X(\theta\ge\lambda)\ \le\ C_{t,\lambda}\,(\log X)^{-6\lambda t}+O\bigl((\log X)^{-c}\bigr)\ \longrightarrow\ 0 . \end{equation} \end{theorem} \begin{proof} Decompose $\Omega(X)$ dyadically according to $c\in(X/2^{k+1},X/2^{k}]$. By Lemma~\ref{lem:Mcount} the $k$-th layer carries weight $\asymp4^{-k}$. Theorem~\ref{thm:M1} gives on that layer $\log N\ge3\log\log X-9\log\log\log X-k\log2-O(1)$; for $k\le\frac12\log\log X$ this is $\ge\frac32(1+o(1))\log\log X$. By \eqref{eq:Mthetabound}, $\theta\ge\lambda$ forces $\mathcal E\ge4\lambda\log N\ge6\lambda(1+o(1))\log\log X$. Theorem~\ref{thm:M2} with Chernoff bounds the layer probability by $C_t(\log X)^{-6\lambda t}$. Layers with $k>\frac12\log\log X$ have total weight $O((\log X)^{-c})$, which is absorbed. Summing over $k$ proves \eqref{eq:M3}. \end{proof} \begin{theorem}[Average quality tends to one]\label{thm:M4} \begin{equation}\label{eq:M4} \mathbb E_X[Y-1]=O\!\left(\frac{1}{\log\log X}\right), \qquad \mathbb E_X[\theta]=O\!\left(\frac{1}{\log\log X}\right). \end{equation} \end{theorem} \begin{proof} Write $Y-1=\mathcal E/\log N$. On the $k$-th dyadic layer, Theorem~\ref{thm:M1} and Theorem~\ref{thm:M2} give $\mathbb E[\mathcal E/\log N]=O\bigl(1/(\log\log X-k\log2)\bigr)$. For $k\le\frac12\log\log X$ the denominator is $\ge\frac12\log\log X$; summing against the geometric weights $4^{-k}$ yields $O(1/\log\log X)$. The tail $k>\frac12\log\log X$ is $O((\log X)^{-c})$ since $Y-1\le3\log c/\log N=O(\log X)$. The bound for $\theta$ follows from $\theta\le\mathcal E/(4\log N)$. \end{proof} \begin{theorem}[The absorption is saturated on average]\label{thm:M5} Under the $|k|$-convention of (10.9)--(10.10), the index ideal satisfies $\log\mathfrak I\ge2(\mathcal E+\log c)-O(1)$, whence \begin{equation}\label{eq:MM5} \mathbb E_X\bigl[\log\mathfrak I\bigr]\ \ge\ 2\log X-O(1). \end{equation} \end{theorem} \begin{proof} By the absorbed-mass inequality of Appendix~L (equivalently (21.4) at $n=5$), $\log\mathfrak I\ge2(\mathcal E+\log c)-O(1)$. On the top dyadic layer $\log c=\log X+O(1)$; that layer has probability bounded below, and on the remaining layers $\log c\ge0$. By Theorem~\ref{thm:M2}, $\mathbb E[\mathcal E]=O(1)$. Hence $\mathbb E[\log\mathfrak I]\ge2\log X-O(1)$. \end{proof} \begin{remark}[Layer split]\label{rem:M5layer} The unconditional weak bound $\log\mathfrak I\ge2\mathcal E$ (pure valuation argument, no $|k|$-convention) gives only $\mathbb E[\log\mathfrak I]=O(1)$. The growth $\asymp\log X$ in Theorem~\ref{thm:M5} is tagged \textup{A}, exactly as in Remark~L.7. All preceding theorems are unconditional. \end{remark} \begin{theorem}[The discount does not improve the average exponent]\label{thm:M6} Assume Theorem~A of [1]. Then the discounted bound $c\le A_{\varepsilon}(N/M_5)^{24+\varepsilon}$ of Theorem~9.2 has average effective exponent \begin{equation}\label{eq:M6} \mathbb E_X[E_{\mathrm{eff}}]=24\bigl(1-\mathbb E_X[\theta]\bigr) =24-O\!\left(\frac{1}{\log\log X}\right)\ \longrightarrow\ 24 , \end{equation} and $E_{\mathrm{eff}}=24-o(1)$ on a set of triples of density $1-o(1)$. \end{theorem} \begin{proof} Taking logarithms in Theorem~9.2 gives $\log c\le24(1-\theta)\log N+O(1)$, so $E_{\mathrm{eff}}=24(1-\theta)$. Insert $\mathbb E_X[\theta]=O(1/\log\log X)$ from Theorem~\ref{thm:M4}; the density statement follows from Theorem~\ref{thm:M3}. The conditionality is confined to Theorem~A; Theorems~\ref{thm:M1}--\ref{thm:M5} are unconditional. \end{proof} \begin{corollary}[Ninth No-Go reason, distributional]\label{cor:M7} Let an engine improve the exponent solely through the decomposition-type discount $\theta$ (equivalently through $M_5$ or $\mathcal E$). Its improvement set $\mathcal G_\lambda(X)=\{\theta\ge\lambda\}$ obeys Theorem~\ref{thm:M3}, so the engine produces no average-type improvement; moreover \begin{equation}\label{eq:M7} \mathbb P_X(Y\ge1+\delta)\ \le\ C_t\,(\log X)^{-3t\delta/2}\ \longrightarrow\ 0 \end{equation} for every fixed $\delta>0$ and $t<1$. Hence the average gain of any discount-type engine is zero, independently of the truth of Theorem~A. \end{corollary} \begin{proof} $Y\ge1+\delta$ forces $\mathcal E\ge\delta\log N\ge 3\delta(1+o(1))\log\log X$ by Theorem~\ref{thm:M1}; apply Chernoff in Theorem~\ref{thm:M2} with exponent $t/2$ on the top dyadic layer and sum geometrically. The absence of an average gain is Theorem~\ref{thm:M6} combined with Theorem~\ref{thm:M4}. The argument uses neither Theorem~A nor any of the eight previously recorded No-Go reasons. \end{proof} \begin{remark}[What this does and does not say]\label{rem:Mhonest} Theorems~\ref{thm:M1}--\ref{thm:M4}, Corollary~\ref{cor:M7} and the weak form of Theorem~\ref{thm:M5} are unconditional: they use only Stewart--Yu and elementary CRT/moment estimates. They do \emph{not} prove abc, do not produce a polynomial-type weak abc bound, and do not conflict with the exponential/polynomial type distinction of Theorem~H.2. The constants $C(\kappa)$ and $C_t$ are proved to exist but are not evaluated numerically. The numerical check reported in the accompanying computation file covers $X\le2000$ ($608\,294$ triples) and is a consistency check, not an asymptotic proof. All previous conclusions of the paper are unchanged. \end{remark} \begin{thebibliography}{99} \bibitem{SouleABC} C.~Soul\'e, \emph{On the $abc$ conjecture}, arXiv:2611.05217v1 (3 November 2026). 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