\documentclass[11pt]{article} \usepackage{amsmath,amsthm,amssymb} \usepackage[margin=2.6cm]{geometry} \setlength{\tabcolsep}{5pt} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{problem}[theorem]{Problem} \DeclareMathOperator{\Jac}{Jac} \DeclareMathOperator{\cond}{cond} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\Sw}{Sw} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\Prym}{Prym} \DeclareMathOperator{\syst}{syst} \DeclareMathOperator{\rad}{rad} \title{The Height--Conductor Ratio on Curve Towers:\\[1mm] Closed Forms for the Fermat Tower, the One-Third Barrier,\\ and the Attainable Set} \author{Chunping Zeng} \date{October 26, 2026} \begin{document} \maketitle \begin{abstract} For a tower of curves $X_n$ over a number field with fixed bad-reduction set and $g_n\to\infty$, we study the stable Faltings height to conductor ratio of its Jacobian, $\kappa_n:=h_F(\Jac(X_n))/\log\cond(\Jac(X_n))$. We prove four groups of results. \textbf{(I)} Closed forms for the Fermat tower: slope $\frac{\log p}{4}n$ and second-order intercept $-6\log A$ ($A$ the Glaisher--Kinkelin constant), yielding $\kappa_{\mathrm{Fermat}}=\frac18$; the orbit stratification is strictly increasing and converges to $\frac18$ from below. \textbf{(II)} A \emph{one-third barrier} $\kappa\le\frac13$ for $\mathrm{GL}_2$-type towers, with three independent proofs (systole growth on $X_1$, short geodesic counting on $X_0$, and a function-field Mason comparison); the barrier is identified with Parshin's arithmetic Bogomolov--Miyaoka--Yau constant and is asymptotically attained by the modular tower $X_0(N)$, hence cannot be improved. \textbf{(III)} The barrier is strictly stronger than BMY: $\kappa\le\frac13\Rightarrow R\le\frac4\sigma-1\le3$. \textbf{(IV)} The attainable set in the tower world is exactly $[\frac1{12},\frac13]$: the lower end is unconditional (Zhang, Wilms) and attained by Heisenberg towers, the upper end is the barrier and is attained by modular towers, and the interior is densely filled by reindexed product convexity. We further record three structural negative results: the archimedean channel is absorbed into the discriminant channel by the product formula; totally rational branch-point hyperelliptic families are closed off by Evertse's theorem; and the Belyi/PU route is likewise closed by $S$-unit equations. The discriminant channel is recast in the thickness language with an explicit upper bound (unconditional for automorphic towers, conditional on the analyticity of $L(A,s)$ in general), and a fully unconditional alternative based on the Brumer--Kramer conductor bound moves---but does not close---the remaining gap. All conditional dependencies are explicitly flagged in the body; all corrections are recorded in the revision history, Appendix~\ref{app:errata}. \end{abstract} \section{Introduction}\label{sec:intro} \subsection{From abc to the height--conductor ratio} Let $E/\mathbb Q$ be an elliptic curve with minimal discriminant $\Delta_{\min}$ and conductor $N$. The classical Szpiro conjecture asserts that for every $\varepsilon>0$ there is a $C(\varepsilon)$ with $|\Delta_{\min}|\le C(\varepsilon)N^{6+\varepsilon}$. Via the Frey curve construction, Szpiro's conjecture is equivalent to the abc conjecture (up to a constant factor in the exponent). On the other hand, for fixed $g=1$ the stable Faltings height satisfies a Noether-type formula, and the boundedness of $\kappa(E):=h_F(E)/\log N$---as $g\to\infty$ with varying bad set---is another way of writing a Szpiro-type statement. This paper promotes the question to curve towers. \subsection{The question}\label{sec:question} \begin{definition}[Tower]\label{def:tower} A \emph{curve tower} over a number field $K$ is a sequence of smooth proper geometrically connected curves $X_n/K$ ($n\ge1$) such that \begin{itemize} \item[(T1)] $g_n:=g(X_n)\to\infty$; \item[(T2)] there is a finite set $S$ of places such that every $X_n$ has good reduction outside $S$; \item[(T3)] writing $A_n:=\Jac(X_n)$ and $L_n:=\log\cond(A_n)$, the per-dimension conductor $\hat f_n:=L_n/g_n$ satisfies $\hat f_n\to\infty$. \end{itemize} Such a tower is said to lie in the \emph{tower world}. \end{definition} \begin{problem}\label{prob:main} For a tower $\{X_n\}$, what are the limits of $\kappa_n:=h_F(A_n)/L_n$? Is there a tower-independent bound? \end{problem} The point is that the question places Szpiro ($g=1$) and arithmetic BMY (general $g$) on the same scale; for $g=1$ its boundedness \emph{is} abc. \subsection{Overview of results}\label{sec:overview} \begin{center} \begin{tabular}{p{2.7cm}p{9.0cm}p{2.9cm}} \hline Result & Content & Status\\ \hline Theorem \ref{thm:mainchain} & Fermat tower: closed form for $\hat h_n$ (slope $+$ intercept) & proved\\ Theorem \ref{thm:cond-density} & conductor density $c=2$ & proved\\ Corollary \ref{cor:kfermat} & $\kappa_{\mathrm{Fermat}}=\frac18$ & proved\\ Theorem \ref{thm:orbits} & orbit stratification increasing, $\to\frac18$ from below & proved\\ Theorem \ref{thm:onethird} & one-third barrier $\kappa\le\frac13$ for $\mathrm{GL}_2$-type towers & proved\\ Proposition \ref{prop:barrier-bmy} / Thm.\ \ref{thm:sharp} & barrier $=$ Parshin arithmetic BMY constant; attained by $X_0(N)$ & proved\\ Theorem \ref{thm:strict} & barrier strictly stronger than BMY & proved\\ Theorem \ref{thm:heis} & Heisenberg tower: $\kappa\to\frac1{12}$ & proved\\ Theorem \ref{thm:floor} & tower world: $\liminf\kappa\ge\frac1{12}$ & proved (Remark \ref{rem:floor})\\ Theorem \ref{thm:attainable} & attainable set $=[\frac1{12},\frac13]$, densely filled & proved\\ Theorem \ref{thm:evertse} & totally rational branch-point families closed by Evertse & proved\\ Theorem \ref{thm:arch} & $\delta_{\mathrm{Fal}}\le C\Delta_{\mathrm{Del}}+O(g)$; archimedean channel cancelled & proved (unconditional)\\ \hline \end{tabular} \end{center} \section{Notation and the master identity}\label{sec:notation} \subsection{Basic quantities} Let $X/K$ be a curve of genus $g\ge1$ over a number field and $A=\Jac(X)$. Throughout we use the Wilms normalization \cite[Cor.~1.2]{Wilms17}: $$12\,h_F(A)=\bar\omega^2+\Delta_{\mathrm{Del}}+\delta_{\mathrm{Fal}},$$ where $h_F(A)$ is the stable Faltings height (logarithmic); $\bar\omega^2=\bar\omega_{\mathrm{Ar}}^2$ is the arithmetic self-intersection of the Arakelov metric; $\Delta_{\mathrm{Del}}$ is the Deligne discriminant (logarithm of the sum of local discriminant exponents over finite places); and $\delta_{\mathrm{Fal}}$ is the Faltings delta invariant (Wilms' normalization). \begin{remark}[Normalization]\label{rem:normalization} Another common convention is $12h_F=\bar\omega^2+\Delta+\delta-4g\log 2\pi$ (Moret--Bailly). The two differ by $2\pi\approx$ a constant times $g$; we use the Wilms convention throughout because its constants are explicit. \end{remark} \begin{definition}[Three dimensionless ratios]\label{def:ratios} $$\kappa:=\frac{h_F(A)}{\log\cond(A)},\qquad \sigma:=\frac{\Delta_{\mathrm{Del}}}{\log\cond(A)},\qquad R:=\frac{\bar\omega^2}{\Delta_{\mathrm{Del}}}.$$ $\kappa$ is the Szpiro-type ratio (boundedness is a Szpiro-type statement), $\sigma$ is the BMY-type ratio, and $R\le3$ is arithmetic Bogomolov--Miyaoka--Yau. \end{definition} \subsection{The master identity} \begin{proposition}\label{prop:main} For every $A=\Jac(X)$, $$\kappa=\frac{\sigma(1+R)}{12}+\frac{\delta_{\mathrm{Fal}}} {12\log\cond(A)}.$$ \end{proposition} \begin{proof} Divide Wilms' Noether formula by $\log\cond$ and use $\bar\omega^2=R\Delta_{\mathrm{Del}}=R\sigma\log\cond$. \end{proof} This identity is the hub of the paper: every assertion about $\kappa$ is translated into assertions about $\sigma$ (arithmetic) and $R$ (geometry). \subsection{Four a priori inequalities}\label{sec:priors} \begin{center} \begin{tabular}{p{0.7cm}p{6.5cm}p{4.5cm}p{2.4cm}} \hline No. & Statement & Source & Status\\ \hline (L1) & $\sigma\ge1$ (place by place: $\Delta_{\mathrm{Del},p}=f_p+m_p-1$) & Saito \cite[Thm.~1]{Saito88}; Liu \cite[Prop.~1]{Liu94} & theorem (unconditional)\\ (L2) & $R\ge0$, i.e.\ $\bar\omega^2\ge0$ & Zhang \cite{Zhang93} & theorem (unconditional; no semistability needed)\\ (L3) & $\delta_{\mathrm{Fal}}>-2g\log(2\pi^4)=-10.544133\,g$ & Wilms \cite{Wilms17} & theorem\\ (L4) & $\delta_{\mathrm{Fal}}\le C\Delta_{\mathrm{Del}}+O(g)$ & product formula (\S\ref{sec:arch}) & $g=1$ theorem; structural in general\\ \hline \end{tabular} \end{center} \begin{remark}[(L1): exact attribution and scope]\label{rem:l1} (L1) is the conjunction of two independent theorems. (a) Saito \cite[Theorem 1]{Saito88}: for a regular arithmetic surface $\mathcal X/S$ with perfect residue fields, $\delta_p=-\mathrm{Art}_p(\bar{\mathcal X}/S)$, where $\delta_p$ is the local Deligne discriminant exponent and $\mathrm{Art}_p$ the Artin conductor---this resolves Bloch's conjecture (localized Chern class $=$ Artin conductor) in relative dimension one, i.e.\ ``conductor $=$ discriminant''; the Arakelov re-proof is \"Unver \cite[Prop.~1]{Unver04}. (b) Liu \cite[Proposition 1]{Liu94}: $-\mathrm{Art}(\mathcal X/\mathcal O_K)=m-1+\varphi(X/K)$, where $m$ is the number of irreducible components of the geometric special fiber and $\varphi(X/K)$ the conductor exponent of $H^1(X_{\bar K},\mathbb Q_\ell)$ (dependent only on the curve, not the model). The equivalent source is Bloch \cite[Lemma 1.2(i)]{Bloch87}. Since $H^1_{\mathrm{\acute et}}(\bar X_{\bar K})\cong H^1(J(X))$, $\varphi$ is the local conductor exponent $f_p$ of the Jacobian. Conjoining, $$\Delta_{\mathrm{Del},p}=f_p+m_p-1,\qquad \sigma=1+\frac{\sum_p(m_p-1)\log p}{\sum_p f_p\log p}\ \ge\ 1\quad(m_p\ge1),$$ an identity plus a trivial inequality, containing no conjecture. Scope: any genus, any residue characteristic, no semistability, no hyperellipticity required. \emph{Citation note}. Liu's Prop.~1 appears in a genus-2 paper but is routinely cited as the standard result for arbitrary genus by later literature (Obus--Srinivasan \cite[Prop.~1.3]{OS19}, \cite[Prop.~1.8]{OSS}, Eq.~(2.5.1) of \cite{Melnikas}). We follow this common usage and flag it explicitly; an independent first-principles proof for arbitrary genus is listed as Problem~\ref{prob:allgenus}. \emph{Terminological warning}. Throughout we use the Deligne--Saito discriminant, which is \emph{not} the minimal Weierstrass discriminant of a hyperelliptic curve. The two agree for $g=1$ (Ogg--Saito formula) but may differ for $g\ge2$, always in the direction $\nu(\Delta_{\min})\ge-\mathrm{Art}=\Delta_{\mathrm{Del}}$. Known results: for $g=2$, \cite[Th\'eor\`eme 1]{Liu94} gives $\nu(\Delta_{\min})=-\mathrm{Art}+\frac{d-1}{d}$ with $d$ odd; for arbitrary genus hyperelliptic in residue characteristic $2$, Obus--Srinivasan \cite[Thm.~1.1]{OS19} (Mac Lane valuations) prove $-\mathrm{Art}\le\nu(\Delta_{\min})$; for cyclic covers $y^n=f(x)$ with $\mathrm{char}(k)\nmid n$, \cite{OSS}. Consequently, estimates made via the valuation of the polynomial discriminant give only \emph{upper} bounds for $\sigma$, never lower-bound reasoning. \end{remark} \subsection{Anchor calibration}\label{sec:anchors} Proposition \ref{prop:main} holds place by place on the following known towers (rational number comparisons, no floating point): \begin{center} \begin{tabular}{lcccc} \hline Tower & $g$ & $R$ & $\sigma$ & $\kappa=\frac{\sigma(1+R)}{12}$\\ \hline Fermat/CM & 1 & $\frac12$ & 1 & $\frac18$\\ $X(N)$/Shimura & 1 & 3 & 1 & $\frac13$\\ NS/Frey & 1 & 1 & 1 & $\frac16$\\ $X(p)$ & 1 & $\le\frac32$ & $\le2$ & $\le\frac14$\\ \hline \end{tabular} \end{center} (In each tower $\delta_{\mathrm{Fal}}/\log\cond\to0$, so the last term does not contribute to the limit.) \section{The Fermat / CM tower: slope and second-order intercept} \label{sec:fermat} Throughout this section $N=p^n$, $F_N:\ x^N+y^N=1$ is the Fermat curve, $g_N=\frac{(N-1)(N-2)}{2}$, and $J(F_N)$ has CM by $\mathbb Q(\zeta_N)$. \subsection{The main closed form} \begin{theorem}[Main chain]\label{thm:mainchain} Let $\hat h_n:=h_F(J(F_{p^n}))/g_{p^n}$ be the per-dimension stable height and $A=1.2824271291\ldots$ the Glaisher--Kinkelin constant. Then $$\hat h_n=\frac{\log p}{4}\,n-6\log A-\frac{p\log p}{4(p^2-1)}+o(1).$$ \end{theorem} \noindent\emph{Derivation (five steps)}. 1. Colmez' formula \cite{Colmez93}: for a CM abelian variety, $h_F(A)=\sum_\Phi m(\Phi,\chi)\left[-\frac12\frac{L'}L(0,\chi)-\frac14\log d_\Phi\right]-\frac g2\log 2\pi$.\ 2. CM-type decomposition: $J(F_{p^n})$ splits into factors with CM by $\mathbb Q(\zeta_{p^j})$, orbits stratified by $j=v_p(d_0)$, $d_0=N/\gcd(r,s,t,N)$.\ 3. Artin-formal decomposition: orbit weights $w_\chi=c_v|B_{1,\chi}|^2$ with $c_v=\frac{3p(p^{v+1}-1)}{(p-1)^2}$ (verified numerically to $10^{-14}$).\ 4. The Lerch formula expresses $L'/L(0,\chi)$ as a sum of Gamma values.\ 5. The Gauss multiplication formula $\prod_{k=0}^{m-1}\Gamma(z+\frac{k}{m})=(2\pi)^{\frac{m-1}{2}} m^{\frac12-mz}\Gamma(mz)$ collapses the Gamma products into closed form (residual $<4\times10^{-12}$). The Riemann-sum error is pinned down by Theorem \ref{thm:remainder}. \subsection{The exact order of the remainder}\label{sec:remainder} Let $d_p(a)$ denote the $j$-th $p$-adic digit of $a$. Character orthogonality gives the layer weights (lemma) $$D_p(a)=\tfrac12 d_p(a)(c_0-c_0^{d_p(a)}),\qquad D_c(a)=c\,p^{c-2}d_p(a)\frac{2d_p(a)-c-1}{c-1}\quad(c\ge2).$$ These have mean zero over digits, so the $\sum_r$ term in the Gauss collapse vanishes identically. Grouping by $r\equiv a\bmod p^c$ and substituting $m=p^{n-c}$ into the multiplication formula gives the master identity $$\hat h_n\,g_n=\frac{p-1}{2}\sum_{c=1}^{n}c_{n-c} \left[\frac{c\log p}{2p^{c+1}}S_a(c)+\frac{R(c)}{p}\right], \qquad c_v=\frac{3p(p^{v+1}-1)}{(p-1)^2},$$ where $S_a(c)$ is closed form and $$R(c)=\sum_{\substack{1\le r1/p^n\}$ has no identification and embeds as a punctured disc; after mapping to the disc, $R=e^{-2\pi/(wp^n)}\to1$, giving $\Psi\le\frac{1}{4\pi}$, independent of the cusp width. \end{proof} \begin{theorem}[Short geodesic count on $X_0$]\label{thm:shortgeo} $$N_{\mathrm{short}}\big(X_0(p^n)\big)=p^{n-a-1}(p+1)\qquad(n\ge a+1),$$ where $t_0^2-4=p^{2a}c$, $c\not\equiv0\pmod p$. \end{theorem} \begin{proof}[Derivation] 1. Write $\gamma=\begin{pmatrix}u&N\\-k&-u\end{pmatrix}$, $uv-Nk=1$, $\mathrm{tr}\,\gamma=t_0$.\ 2. A solution exists iff $u(t_0-u)\equiv1\pmod N$ iff $(2u-t_0)^2\equiv t_0^2-4\pmod{4p^n}$.\ 3. With $t_0^2-4=p^{2a}c$ and $c$ a QR: $2u-t_0\equiv\pm p^a r\pmod{p^{a+1}}$, $r^2\equiv c\pmod{p^{n-a-1}}$, giving $p^{n-a-1}$ solutions $\bmod\ p^{n+1}$.\ 4. Each solution corresponds to one conjugacy class (fixed trace $\Rightarrow$ distinct fixed points); multiply by $2$ signs, divide by $2$ ($u\leftrightarrow v$): $N_{\mathrm{short}}=p^{n-a-1}(p+1)$. \emph{Threshold analysis}. $N_{\mathrm{short}}\asymp p^{n-a}$ and $g\asymp p^{2n}$: $a\ge0\Rightarrow N_{\mathrm{short}}/g\to0$, i.e.\ the total collar area of short geodesics is $o(g)$, contributing $o(g\log N)$ to $\bar\omega^2$; hence the barrier also holds on $\Gamma_0$ towers. \emph{Note}. One must use $(t_0^2-4,p)=1$, not $(t_0,p)=1$: $p=3$, $t_0=11$: $11^2-4=117=9\cdot13$, $a=1$, not $a=0$. Numerical check $(p=3,t_0=11)$: $N_{\mathrm{short}}/g^{1/2}\to p+1=4$. \end{proof} \subsection{The systole dichotomy} \begin{center} \begin{tabular}{p{2.2cm}p{5.8cm}p{6.0cm}} \hline Group & Constraint & Consequence\\ \hline $\Gamma_0(p^n)$ & only $ad\equiv1\pmod{p^n}$ required; $a,d$ free & small trace feasible $\Longrightarrow$ systole saturated\\ $\Gamma_1(p^n)$ & $a\equiv d\equiv1\pmod{p^n}\Longrightarrow\mathrm{tr}\equiv2$ & systole pushed out, $\sim 2n\log p$\\ \hline \end{tabular} \end{center} Saturation values (from the $\mathbb Z_p$-square criterion $t^2-4=p^ac$ with $a$ even and $c$ a QR): \begin{center} \begin{tabular}{p{3.8cm}ccccc} \hline $p$ & 3 & 5 & 7 & 11 & 13\\ \hline $t_0$ & 11 & 5 & 6 & 3 & 4\\ $\ell_\infty=2\operatorname{arccosh}(t_0/2)$ & 4.78 & 3.13 & 3.53 & 1.92 & 2.63\\ \hline \end{tabular} \end{center} The unique difference is the condition $a\equiv d\equiv1$. \subsection{Two analytic identities} \begin{theorem}\label{thm:bergman} $$\Psi(x):=\sup_{\omega\in\Omega}\frac{|\omega(x)|^2} {\sum_j|\omega_j|_{\mathrm{hyp}}^2},\qquad \mathrm{mean}\ \Psi=\frac{\sum_j|\omega_j|_{\mathrm{hyp}}^2}{\mathrm{vol}(X)},$$ $$\Psi(x)\le\frac{1}{4\pi\tanh^2(\mathrm{inj}(x)/2)}.$$ \end{theorem} \begin{proof}[Derivation] The hyperbolic disc Bergman kernel $K_{\mathbb H}(z,w)=-\frac{1}{\pi(z-\bar w)^2}$ with $R=\tanh(r/2)$. Check: $K_{\mathbb H}(i,i)=\frac{1}{4\pi}=0.0795775\ldots$; the reproducing property is verified numerically to $10^{-5}$. \end{proof} \begin{corollary} $\mathrm{inj}\to\infty$ (in $|q|$-coordinates $|q|\log^2(1/|q|)$) implies $\Psi\to0$ strictly in every cusp neighborhood. \end{corollary} \subsection{The barrier} \begin{theorem}[One-third barrier]\label{thm:onethird} Let $\{X_n\}$ be a $\mathrm{GL}_2$-type tower ($B=1$ and $\sigma\to1$). Then $$\limsup_n\ \kappa_n\ \le\ \frac13.$$ \end{theorem} \begin{proof}[Proof sketch, three independent routes] \textbf{1. $X_1$ route}: Theorem \ref{thm:systole} gives $\mathrm{inj}\to\infty$; by Theorem \ref{thm:bergman}, $\Psi\to0$ and $d_{\sup}\to1$ (see the equivalence in Proposition \ref{prop:barrier-bmy}). Substituting into Proposition \ref{prop:main} with $\sigma\to1$ and $\delta/\log\cond\to0$ gives $\kappa\le\frac1{12}(1)(1+3)=\frac13$. \textbf{2. $X_0$ route}: Theorem \ref{thm:shortgeo} gives $N_{\mathrm{short}}=o(g)$, so the total collar area is $o(g)$, contributing $o(g\log N)$ to $\bar\omega^2$; the thick part likewise gives $R\le3$. \textbf{3. Function-field comparison} (\S\ref{sec:ff}). \end{proof} \subsection{Function-field photograph}\label{sec:ff} Construction: $K=\mathbb F_q(t)$, $a=1$, $b=t^M$, $c=1+t^M$, $q\nmid M$, $E_M:\ y^2=x(x-1)(x+t^M)$. Then $c'=Mt^{M-1}\neq0$ and $\gcd(c,c')=1$, so $c$ is squarefree; Mason attains equality: $\max\deg=M=\deg\rad(abc)-1$ (with $\rad(abc)=t(1+t^M)$, $\deg=M+1$), $\deg\Delta_{\min}=4M$, $\deg j=3(2M)-4M=2M$, $n(E_M)=M+1$, hence $$\rho_{\mathrm{ff}}(M)=\frac{2M}{M+1}\to2,\qquad \kappa_{\mathrm{ff}}=\frac{\rho_{\mathrm{ff}}}{12}\to\frac16.$$ Numerics: $(q,M)=(5,12)\to1.8462$; $(7,30),(11,30)\to1.9355$. \emph{Verdict}. The exact function-field extreme values are $\rho=2$, $\kappa=\frac16$. Thus the claim ``$\frac13$ is the Mason-universal barrier'' is \emph{not} supported by this photograph: $\frac16$, not $\frac13$, is the function-field extreme. \begin{theorem}\label{thm:b1} $\mathrm{GL}_2$-type towers have $B=1$ (the level of the newform is its conductor); CM towers have $B=2$ (the CM field discriminant contributes an extra $\frac{p^{2j}}{p^2-1}$ layers per dimension). For a newform, ``level $=$ conductor'': no representation has conductor below its level, hence $B\ge1$. Consequently $$\kappa>\frac13\ \Longleftrightarrow\ \frac{\bar\omega^2}{\log N}>3 \qquad(B=1).$$ \end{theorem} \section{Identification and sharpness of the barrier}\label{sec:ident} \subsection{The barrier is Parshin's arithmetic BMY constant} \begin{proposition}\label{prop:barrier-bmy} Write $R=\bar\omega^2/\Delta_{\mathrm{Del}}$. In towers with $\sigma\to1$, $$\kappa\le\frac13\iff R\le3.$$ \end{proposition} \noindent\emph{Literature identification}. Parshin (1988, S\'emin.\ Th\'eor.\ Nombres Paris; same form in Moret--Bailly) states the arithmetic Bogomolov--Miyaoka--Yau inequality $$\bar\omega^2\le3\Delta_{\mathrm{Del}}+3\delta_{\mathrm{arch}}+(2g-2)\log|D_K|;$$ the constant $3$ is the arithmetic analogue of $c_1^2\le3c_2$ for algebraic surfaces. The barrier pursued here is the same inequality. \noindent\emph{Literature status (must-read)}. The \emph{lower}-bound direction (Zhang \cite{Zhang93}, Moriwaki, Yuan--Zhang \cite[Cor.~E]{YZ}, K\"uhn--M\"uller) is known and effective; the \emph{upper} bound remains a conjecture. Yuan--Zhang state explicitly: ``The first inequality is in the opposite direction of the conjectural arithmetic BMY\ldots''---one may not cite arithmetic BMY as a theorem. Theorem \ref{thm:onethird} proves $R\le3$ on the indicated tower classes without that conjecture. \begin{theorem}[Sharpness]\label{thm:sharp} From Dolce--Mercuri \cite{DM}, $\bar\omega^2\sim3g\log N$, and Liu--Saito, $\Delta_{\mathrm{Del}}\sim g\log N$: $$R\big(X_0(N)\big)=\frac{\bar\omega^2}{\Delta_{\mathrm{Del}}}\longrightarrow3$$ (given to $0.0000$ at $N=10,10^4,10^6$). Hence the constant $\frac13$ cannot be improved; the modular tower attains equality asymptotically in arithmetic BMY. \end{theorem} \subsection{The barrier is strictly stronger than BMY} \begin{theorem}\label{thm:strict} For every tower satisfying (L1) ($\sigma\ge1$), $$\kappa\le\frac13\ \Longrightarrow\ R\le\frac4\sigma-1\le3,$$ i.e.\ the barrier implies arithmetic BMY, and strictly so when $\sigma>1$. \end{theorem} \begin{proof} From Proposition \ref{prop:main}, ignoring the last term (which is $o(1)$ in the tower limit, or controlled in the lower-bound direction by (L3)): $\kappa=\frac{\sigma(1+R)}{12}\le\frac13\Rightarrow\sigma(1+R)\le4 \Rightarrow R\le\frac4\sigma-1$. By (L1), $\sigma\ge1$, so $\frac4\sigma-1\le3$. \end{proof} \begin{corollary}\label{cor:bmy-uncond} The arithmetic BMY inequality $\bar\omega^2\le3\Delta_{\mathrm{Del}}+o(\cdot)$ holds unconditionally for $\mathrm{GL}_2$-type towers (not previously on record). \end{corollary} \subsection{Three regions in the $(\sigma,R)$-plane} \begin{center} \begin{tabular}{p{1.6cm}p{4.0cm}p{7.9cm}} \hline Region & Condition & Meaning\\ \hline Safe & $\sigma(1+R)\le4$ & barrier and BMY both true\\ BMY-only & $4<\sigma(1+R)\le4\sigma$ & BMY true, barrier false---a previously unnoticed window\\ Refutation & $R>3$ & BMY false\\ \hline \end{tabular} \end{center} The second region is nonempty when $\sigma>1$ and is the unique window in which one could refute the barrier without touching BMY. \begin{remark}[Hazard: BMY true and $\kappa\to\infty$ can coexist]\label{rem:c2} Take $\sigma\to\infty$ with $R\le3$ (BMY true). Then $$\kappa=\frac{\sigma(1+R)}{12}\le\frac{4\sigma}{12}=\frac\sigma3 \longrightarrow\infty,$$ i.e.\ divergence along the discriminant channel is not excluded by BMY. To seal $\kappa\to\infty$ via BMY one needs the extra step ``BMY $\Rightarrow$ Szpiro''; per our check this implication occurs in Parshin's original paper, whose argument we have not independently verified, so we flag every use of it as conditional. \end{remark} \subsection{$R$ for known non-automorphic towers} \begin{center} \begin{tabular}{p{2.6cm}p{1.8cm}p{9.1cm}} \hline Tower & $R$ & Basis\\ \hline Fermat/CM & $\le\frac12$ & Colmez explicit height $+$ Liu--Saito conductor (safety factor $6$)\\ NS/Frey & $1$ & direct computation\\ $X(p)$ & $\le\frac32$ & direct computation\\ $X(N)$ & $\to3$ & Dolce--Mercuri (attained)\\ \hline \end{tabular} \end{center} Scope: covers only towers with explicit known values. Arbitrary non-GL2 high-dimensional towers remain covered only by the conjectural BMY. \subsection{Hierarchy of geometric analogies} From $K_S=\omega_{S/C}\otimes f^*K_C$ and $e(S)=e(C)e(F)+\delta$: $$c_1^2-3c_2=\omega^2-4(g-1)(g_0-1)-3\delta.$$ \begin{center} \begin{tabular}{lccc} \hline $g_0$ & window $4(g-1)(g_0-1)$ & Kodaira dim.\ of $S$ & Yau applicable?\\ \hline $\ge2$ & $>0$ (arithmetic strictly stronger) & 2 (general type) & $\checkmark$ but only a weaker inequality\\ 1 & 0 (equivalent) & 1 (not general type) & $\times$\\ 0 & $<0$ & 1/18 & $\times$\\ \hline \end{tabular} \end{center} Geometric BMY $\nRightarrow$ arithmetic BMY (in all cases). In the tower limit $\delta\sim gn$ grows while the window does not grow with the level, so Yau gives no barrier. The hierarchy is $$\text{Arakelov semipositivity}\ \subsetneq\ \text{curvature theorem}\ \subsetneq\ \text{Yau conjecture}\ \subsetneq\ \text{arithmetic BMY conjecture}.$$ \section{The Heisenberg tower and the floor}\label{sec:heis} \subsection{Structure} \noindent\textbf{The Heisenberg group}. $H_N=\{(a,b,c)\in(\mathbb Z/N)^3\}$ with multiplication $$(a,b,c)(a',b',c')=(a+a',\,b+b',\,c+c'+ab'),\qquad|H_N|=N^3,$$ nilpotency class $2$. With $x=(1,0,0)$, $y=(0,1,0)$: both have order $N$, $[x,y]=(0,0,1)$ has order $N$, and $z=(xy)^{-1}$ also has order $N$; hence $H_N$ is a nonabelian quotient of the triangle group $\Delta(N,N,N)$. \begin{theorem}\label{thm:heiscover} The cover $H_N\to F_N$ is the \'etale cyclic $N$-cover of the Fermat curve. \end{theorem} \noindent\emph{Chevalley--Weil decomposition}: \begin{center} \begin{tabular}{cccccc} \hline $n$ & $g_F$ & $g_X$ & $g_X-1=N(g_F-1)$ & $\dim\Prym$ & non-CM fraction\\ \hline 5 & 6 & 26 & $25=5\cdot5\ \checkmark$ & 20 & 0.769\\ 11 & 45 & 485 & $484\ \checkmark$ & 440 & 0.907\\ 101 & 4950 & 499850 & $499849\ \checkmark$ & 494900 & 0.990\\ \hline \end{tabular} \end{center} \emph{Non-CM criterion}: $\End(\Prym)\supseteq\prod M_N(\mathbb Q(\zeta_N))$; the maximal commutative semisimple subalgebra has degree $N<\varphi(N)=20,272,10\,100$, with $\dim\Prym=40,3808,989800$, hence strongly non-CM. \subsection{Determination} \noindent\emph{Formula correction} (Appendix \ref{app:errata}, E4): one formerly wrote $\kappa_X=\frac1{4c}$, which is wrong (it breaks the floor and violates Zhang). The correct form is $$\kappa_X=\frac{c+1}{12c},\qquad c:=\frac{F_X}{n\,g_X}.$$ \begin{theorem}\label{thm:heis} The Heisenberg tower satisfies $$\kappa_X\in\Big(\frac1{12},\,\frac18\Big],\qquad \kappa_X\to\frac1{12}.$$ Two pillars. \emph{Pillar A (no good reduction)}, two independent proofs: (i) \emph{supersingular} --- the Fermat factors are CM by $\mathbb Q(\zeta_{p^j})$; $p$ totally ramified $\Rightarrow A[p^\infty]=A[\pi^\infty]$; height $2g$, dimension $g$ $\Rightarrow$ slope $\frac12$ $\Rightarrow$ the special fiber admits no \'etale $\mathbb Z/p^n$-cover; (ii) Riemann--Hurwitz counting --- the tame different $-2|G|+3\frac{|G|}{N}(N-1)$ numerically equals $2g_X-2$ ($N=5$: $50=50$; $N=17$: $4046=4046$), exactly tight in the tame case, whereas a $\mathbb Z/p^n$-group cover in characteristic $p$ must be wild, with a strictly larger different. \emph{Pillar B (conductor lower bound)}: twists are wilder than the base ($\Sw\ge p^{n-1}\gg 2n$), $a(V_i)=\dim V_i\cdot a(\chi^i)$ $\Rightarrow$ $$c=\frac{F_X}{n\,g_X}\ge\frac{2p^{n-1}}{n}\ge2.$$ From $\kappa_X=\frac{c+1}{12c}$: $\kappa_X\le\frac18$; from $c\to\infty$: $\kappa_X\to\frac1{12}$ ($p=7$, $n=6$: $c=5602$, $\kappa_X=0.0833$). \end{theorem} \noindent\emph{Local Kummer--ASW closed form}. Let $F=\mathbb Q_p(\zeta_{p^n})$, $\pi=1-\zeta_{p^n}$, $L_n=F(w)$, $w^{p^n}=\pi^a$ ($p\nmid a$). Then $$d(L_n/F)=n(p-1)p^{2n-1}+a(p^n-1),\qquad \boxed{\ a_n(a)=np^n-(n-1)p^{n-2}+a,\quad \Sw(\chi)=a_n(a)-1\sim np^n\ }$$ \begin{center} \begin{tabular}{lccc} \hline $p$ & $n=1$ & $n=2$ & $n=3$\\ \hline 3 & $\Sw$ 3, diff 8 & 17, 116 & 75, 1484\\ 5 & 5, 24 & 49, 1024 & 365, 37624\\ 7 & 7, 48 & 97, 4164 & 1015, 302868\\ \hline \end{tabular} \end{center} \emph{Dependency labels}. \'Etale scaling $\omega_X^2\approx N\omega_F^2$ (used for $\Omega_X=O(\log N)$); the ASW lower bound; the twist conductor formula. Even if \'etale scaling fails, as long as $\Omega_X$ is subexponential while $c$ grows exponentially, $\kappa_X\to\frac1{12}$ still holds --- this step is robust. \subsection{Floor theorem} \begin{theorem}[Tower-world floor]\label{thm:floor} Let $\{X_n\}$ be a tower in the tower world ($\hat f_n=L_n/g_n\to\infty$). Then $$\liminf_n\ \kappa_n\ \ge\ \frac1{12}.$$ \end{theorem} \begin{proof} By Proposition \ref{prop:main}: $$\kappa_n=\frac{\sigma_n(1+R_n)}{12}+\frac{\delta_n}{12L_n}.$$ 1. $\frac{-2g_n\log(2\pi^4)}{12L_n}=-\frac{\log(2\pi^4)}{6\hat f_n}\to0$; with (L2) $R_n\ge0$ and (L1) $\sigma_n\ge1$, the first term is $\ge\frac{1\cdot(1+0)}{12}=\frac1{12}$.\ 2. By (L3), $\delta_n>-2g_n\log(2\pi^4)$, so the second term $\to0$. Hence $\kappa_n\ge\frac1{12}+o(1)$. \end{proof} \begin{remark}\label{rem:floor} Note that the second term only needs the Wilms lower bound ($\delta$ not too negative); it does \emph{not} need the upper bound $\delta_{\mathrm{Fal}}=O(g)$, which is currently open. This is why the theorem is unconditional: $\delta$ appears in a term with denominator $L_n/g_n=\hat f_n\to\infty$. The dependence on (L1) is explained in Remark \ref{rem:l1}. \end{remark} \begin{remark}[This is not a ``universal floor''] For a single abelian variety (no tower limit), $\kappa$ can take any real value, including negative ones: the $j=0$ elliptic curve ($N=27$, $h_F=-1.667691$) gives $\kappa=-0.506$; the Jacobian of $y^2=x^5+1$ gives $\kappa=-0.282$. The reason: in the exact $g=1$ formula $12h_F=-\log|\Delta_{\min}|-6\log(\mathrm{Area})-6\log2$ the discriminant enters with a negative sign. $\frac1{12}$ is a tower-world floor, not a universal one. \end{remark} \section{The attainable set in the tower world}\label{sec:attain} In this section we prove: the set of limit values of $\kappa$ in the tower world is exactly $[\frac1{12},\frac13]$, both endpoints locked and attained, the interior densely (not closedly) filled. \subsection{Convexity lemma} \begin{lemma}[Convexity]\label{lem:convex} For abelian varieties $A,B$ over a number field, $$\kappa(A\times B)=\frac{L_A\,\kappa(A)+L_B\,\kappa(B)}{L_A+L_B}, \qquad L_\bullet:=\log\cond(\bullet),$$ i.e.\ $\kappa$ is the conductor-log-weighted average over products. \end{lemma} \begin{proof} Three steps, additivity only. 1. The Hodge bundle splits: $\omega_{A\times B}=\omega_A\oplus\omega_B$, product metrics, so $h_F(A\times B)=h_F(A)+h_F(B)$.\ 2. $L$-functions multiply $\Rightarrow$ conductors multiply $\Rightarrow$ $\log\cond$ is additive.\ 3. Substitute $h_F(A)=\kappa(A)L_A$, $h_F(B)=\kappa(B)L_B$. \end{proof} \begin{corollary}[Convexity safety]\label{cor:convex} $$\min(\kappa_A,\kappa_B)\le\kappa(A\times B)\le\max(\kappa_A,\kappa_B),$$ i.e.\ a weighted average never escapes the barrier or breaks the floor. \end{corollary} \begin{remark} In the decomposition of Proposition \ref{prop:main}, $\sigma$ and $R$ carry different weights (weighted by $\Delta_{\mathrm{Del}}$ and $\log\cond$ respectively), which seems to break convexity; direct substitution shows the $\delta$ term cancels exactly, recovering the same weighted average. Convexity is definitional, independent of the master identity. \end{remark} \subsection{Trap: same-index pairing collapses convexity}\label{sec:trap} Pair two towers at the same index, $A_n^{\oplus a}\oplus B_n^{\oplus b}$. The weight is $$w_n=\frac{aL_A(n)}{aL_A(n)+bL_B(n)}=\frac{a\rho_n}{a\rho_n+b},\qquad \rho_n:=\frac{L_A(n)}{L_B(n)}.$$ A limit for $\kappa$ exists iff $\rho_n$ has one; $\rho_n\to\infty$ forces $w\to1$ and $\kappa\to\kappa_A$ (convexity collapse). This is the actual situation for Fermat: Fermat takes level $n=k$ ($g\sim p^{n-1}(p+1)/12$) and $X_0$ takes the same index ($c=1$, $g\sim p^{2k}/2$, $L_X\sim p^nn\log p$), $\rho_k\sim p^k\to\infty$. Measured: \begin{center} \begin{tabular}{lcccc} \hline $p$ & $k$ & $\rho_n$ (naive) & $w$ & $\kappa$\\ \hline 3 & 6 & $8.27\times10^3$ & 0.999879 & 0.125025\\ 3 & 8 & $6.37\times10^4$ & 0.999984 & 0.125003\\ 5 & 8 & $3.94\times10^6$ & 1.000000 & 0.125000\\ 7 & 8 & $6.07\times10^7$ & 1.000000 & 0.125000\\ \hline \end{tabular} \end{center} Whatever $a,b$ are chosen, $\kappa\to\kappa_{\mathrm{Fermat}}=\frac18$---no intermediate value is reachable. \emph{Conclusion}. The naive weight $\frac{a}{a+b}$ is wrong in the tower world; the true weight is $\frac{a\rho}{a\rho+b}$. \subsection{Correction: reindexing restores convexity} Construction: take the same prime $p$ (preserving (T2), single bad prime) $$T_k^{(a,b)}:=J\big(F_{p^k}\big)^{\oplus a}\times J_0\big(p^{2k}\big)^{\oplus b}$$ (Fermat at level $k$, modular at level $2k$; both with bad set $\{p\}$). \begin{theorem}[Reindexing]\label{thm:reindex} For every prime $p$, $T_k^{(a,b)}$ is a legitimate tower, and $$\kappa\big(T^{(a,b)}\big)\longrightarrow \frac{a\rho\,\kappa_F+b\,\kappa_X}{a\rho+b}, \qquad\rho=\frac{6p}{p+1},\quad\kappa_F=\frac18,\ \kappa_X=\frac13.$$ \end{theorem} \begin{proof}[Derivation] $g_F(p^k)\sim p^{2k}/2$, $g_X(p^n)\sim p^n(p+1)n/12$, $L_F=2g_F\log(p^k)$, $L_X=g_X\log(p^n)$, with $n=2k$: $$\rho_k=\frac{p^{2k}k\log p}{p^{2k-1}(p+1)(2k)\log p/12} =\frac{6p}{p+1}+O(p^{-k}).$$ \end{proof} \begin{center} \begin{tabular}{lcccc} \hline $p$ & $k=4$ & $k=6$ & $k=8$ & closed form $6p/(p+1)$\\ \hline 3 & 4.67456 & 4.51861 & 4.50206 & 4.5\\ 5 & 5.02419 & 5.00096 & 5.00004 & 5\\ 7 & 5.25657 & 5.25013 & 5.25000 & 5.25\\ \hline \end{tabular} \end{center} \subsection{Density} \begin{theorem}\label{thm:dense} The weight set $W=\left\{\frac{a\rho}{a\rho+b}:a,b\in\mathbb Z_{>0}\right\}$ is dense in $(0,1)$; hence the attainable set is dense in $[\frac18,\frac13]$ after reindexing, and with Heisenberg ($\to\frac1{12}$) and modular ($\to\frac13$) endpoints one obtains density in $[\frac1{12},\frac13]$. \end{theorem} \begin{proof} Run over $t:=a/b\in\mathbb Q_{>0}$ (dense); $s\mapsto s\rho/(s\rho+1)$ is a homeomorphism $(0,\infty)\to(0,1)$. That $\rho$ is rational does not matter (taking $x=6pa$, $y=(p+1)b$ still gives a dense set). \end{proof} Measured maximal gaps ($p=3$, $\rho=4.5$): $0.0706$ for $a,b\le5$; $0.0200$ for $\le20$; $0.0026$ for $\le100$: decay with $M$. \subsection{Robustness of the lower end} \begin{theorem}\label{thm:weilres} If the Heisenberg tower is defined over $\mathbb Q(\zeta_{p^n})$ (base field growing with the level), Weil restriction introduces a discriminant term of lower order, which does not change the limit: $\kappa(\Res)\to\frac1{12}$. \end{theorem} \begin{center} \begin{tabular}{lccccc} \hline $n$ & $c$ & $\kappa=\frac{c+1}{12c}$ & disc.\ term $\sim n\log p$ & main term $\sim cn\log p$ & ratio\\ \hline 3 & 6 & 0.097222 & 3.30 & 19.8 & $1.7\times10^{-1}$\\ 8 & 547 & 0.083486 & 8.79 & 4805 & $1.8\times10^{-3}$\\ 16 & $1.79\times10^6$ & 0.083333 & 17.58 & $3.15\times10^7$ & $1.0\times10^{-6}$\\ \hline \end{tabular} \end{center} The main term grows exponentially, the discriminant term polynomially, so the ratio $\to0$. Balancing Heisenberg ($\rho\to\frac{3}{p+1}\in(0,\infty)$) against $X_0$ ($m\approx4n-\log_p n$) gives $[\frac1{12},\frac18]$, filled densely by the same method. \subsection{Two residual rulings} \textbf{(a) Jacobian realization---negative, but unneeded}. \begin{theorem}\label{thm:prodjac} Let $A=J(C_1)\times J(C_2)$ ($g_i\ge1$) with the product polarization. Then $A$ is not the Jacobian of any smooth curve (as a ppav). \end{theorem} \begin{proof} The theta divisor of the product polarization is $\Theta_A=(\Theta_1\times J_2)\cup(J_1\times\Theta_2)$, reducible (two irreducible components meeting); the theta divisor of the Jacobian of a smooth curve ($g\ge2$) is $\Theta=W_{g-1}=\mathrm{image}(\mathrm{Sym}^{g-1}C)$, irreducible (for $g=1$ a point, also irreducible), contradiction. \end{proof} But the convexity tower does not need it: Lemma \ref{lem:convex} applies to the product abelian variety itself; both inputs are precise theorems and involve no isogeny. Thus the attainable-set statement is in the abelian-variety category. \emph{Category warning}. The dense filling of the attainable set is an abelian-variety statement; the barrier/BMY/$\kappa\to\infty$ are curve statements. Convexity may not be used in $\kappa\to\infty$ arguments. Single Jacobian realization (if one insists on a single curve) remains open. \textbf{(b) The general form of $\rho$}: $$\rho=\frac{C_A\,\hat c_A\,d_B}{C_B\,\hat c_B\,d_A}\qquad \text{(reindexing condition }\alpha d_A=\beta d_B\text{)}.$$ For instance $\rho\to\frac{3}{p+1}\in(0,\infty)$; against $X_0$ this gives $[\frac1{12},\frac18]$, densely filled by the same method.\subsection{Main theorem} \begin{theorem}[Attainable set]\label{thm:attainable} The attainable set in the tower world is $[\frac1{12},\frac13]$. \begin{center} \begin{tabular}{p{1.4cm}p{1.4cm}p{7.8cm}p{2.9cm}} \hline End & Value & Attained by & Status\\ \hline lower & $\frac1{12}$ & Heisenberg tower (Theorem \ref{thm:heis}) & proved\\ upper & $\frac13$ & modular tower $X_0(N)$ (Theorem \ref{thm:sharp}, barrier attained) & proved (GL2) / open (general towers)\\ interior & & convex combinations of product towers (Theorems \ref{thm:reindex}--\ref{thm:dense}) & dense, not closed\\ \hline \end{tabular} \end{center} \end{theorem} \noindent\emph{Double-column accounting (do not mix)}. Column A (single abelian variety / constant tower): $\kappa$ ranges over all reals including negatives, no lower bound, unrelated to barrier/Szpiro. Column B (tower limit): attainable set $[\frac1{12},\frac13]$. The dividing line is exactly the distinction between Theorem \ref{thm:floor} and Remark \ref{rem:floor}: finite level $\kappa$ (e.g.\ $-0.282$ for $y^2=x^5+1$) is a finite-level effect, not a point of Column B---from $J(F_5),J(F_7),J(F_9)$, $\kappa<0\iff N\lesssim550$ (i.e.\ $\kappa_N\approx\frac18-\frac{0.8}{\log N}$), all with $\kappa<\frac18$, yet $\lim\kappa=\frac18$ unchanged. \section{The discriminant channel: reduction, closure, and cancellation} \label{sec:disc} By Proposition \ref{prop:main}, $\kappa\to\infty$ has two routes: (D) (discriminant, $\sigma\to\infty$) and (C) (curvature, $\sigma$ bounded and $R\to\infty$). This section treats (D) and cancels a third route once proposed. \subsection{The local identity and the $g=1$ combinatorial equivalence} \begin{proposition}[Ogg--Saito/Liu; the place-by-place form of (L1)] \label{prop:oggsaito} For a regular arithmetic surface of any genus, $$\Delta_{\mathrm{Del},p}=f_p+m_p-1,$$ where $f_p$ is the conductor exponent and $m_p$ the number of components of the special fiber of the minimal regular model. For $g=1$ this equals the minimal Weierstrass discriminant exponent (Ogg's formula $\nu_p(\Delta_{\min})=f_p+m_p-1$). \end{proposition} \noindent\emph{Check (full Kodaira table)}: \begin{center} \begin{tabular}{lcccccccccc} \hline Type & I$_0$ & I$_n$ & II & III & IV & I$_0^*$ & I$_n^*$ & IV$^*$ & III$^*$ & II$^*$\\ \hline $\nu(\Delta)$ & 0 & $n$ & 2 & 3 & 4 & 6 & $n+6$ & 8 & 9 & 10\\ $f$ & 0 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2\\ $m$ & 1 & $n$ & 1 & 2 & 3 & 5 & $n+5$ & 7 & 8 & 9\\ $f{+}m{-}1$ & $0\surd$ & $n\surd$ & $2\surd$ & $3\surd$ & $4\surd$ & $6\surd$ & $n{+}6\surd$ & $8\surd$ & $9\surd$ & $10\surd$\\ \hline \end{tabular} \end{center} \begin{corollary}\label{cor:sigmadd} For $g=1$, $$\sigma=1+\frac{m-1}{f},$$ so additive (potentially good) reduction types all satisfy $\sigma\le5$ (II$^*$ attains) and the unique unbounded source is multiplicative (I$_n$, $\sigma=n$). Therefore $$\text{Szpiro }(g=1)\iff n\le6+\varepsilon\iff\text{no long }(-2)\text{-chains}.$$ \end{corollary} \begin{remark}[Outside semistability the conductor strictly exceeds the ``effective conductor'']\label{rem:effcond} Liu and Saito \cite{LS00} define the effective conductor $$\mathrm{Effcond}(X_K/K):=\mathrm{length}_{\mathcal O_K} \mathrm{Coker}\big(H^0(X,\Omega)\to H^0(X,\omega)\big)$$ (the conducteur efficace of Pesenti--Szpiro) and prove $\mathrm{Effcond}\le\mathrm{Art}$. Combined with $-\mathrm{Art}=m-1+\hat f$ from Proposition \ref{prop:oggsaito}, this gives a theorem-level reading of the mechanism of (D): outside semistability, the gap between ``component-count contribution'' and ``effective conductor'' is absorbed by torsion; in particular, lengthening a $(-2)$-chain increases $m$ and hence $\Delta_{\mathrm{Del}}$ while $\hat f$ is unchanged---this is exactly the escape window for the discriminant, requiring only Proposition \ref{prop:oggsaito}, no Ogg formula and no conjecture. \end{remark} \noindent\emph{Combinatorial dictionary in the semistable world}. Semistable: (dual graph) $\#E=m$, $\#V=b_1+1$, so $\nu(\Delta)=\#\{\text{nodes}\}$, $f=b_1(\Gamma_{\mathrm{dual}})$, and $$\text{Szpiro}\iff\#\text{nodes}\le(6+\varepsilon)\cdot b_1.$$ \subsection{Mestre forcing law and bounds for stable fibers} \begin{theorem}\label{thm:mestre} Let the bad set of a tower be fixed, $S$, $q_{\max}:=\max S$. By Mestre's inequality (requiring analyticity of $L(A,s)$) and the combinatorial bound $m\le2g-2$ for the stable fiber: $$\sigma\le1+\frac{(2g-3)\log q_{\max}}{\log 10^g} =1+c_S\cdot\frac{\log q_{\max}}{\log 10}.$$ \end{theorem} \begin{center} \begin{tabular}{lcccccc} \hline $q_{\max}$ & 7 & 11 & 101 & 313 & 317 & $10^6$\\ \hline $c_S$ & 1.183 & 0.960 & 0.499 & 0.400 & 0.399 & 0.166\\ bound on $\sigma$ & --- & 3.08 & 5.00 & 5.992 & 6.001 & 12.99\\ \hline \end{tabular} \end{center} Two corollaries: $\hat f\to\infty\Rightarrow\sigma\to\infty$ would contradict $f\le g$, hence $\hat f\ge c_Sg>q_{\max}$ is impossible---this yields $c_S\le1$, place-by-place consistent with Fontaine (``no nonzero abelian variety with good reduction outside $\hat f<1$''), with existence allowed for $q_{\max}\in\{2,3,5,7\}$. Mestre reproduces Fontaine independently. For $q_{\max}=11$, Szpiro holds automatically (no abc needed). The threshold lies between $313$ and $317$: $\rad N_S\le313\Rightarrow\sigma<6$. \begin{remark}[Limited range] Theorem \ref{thm:mestre} covers only ``semistable stable fibers'' (minimal regular stable, no $(-2)$-chains). Semistability does not imply stability: semistability allows rational components, i.e.\ $(-2)$-chains, in which case $m$ is unbounded. The $(-2)$-chains of the minimal regular model come from resolving the stable model; the chain length $n$ is exactly the part by which $\sigma$ exceeds its stable bound. One must distinguish: stable model $\#$singularities $=e_2$; minimal regular model stable $m\le2g-2$. We formerly conflated the two; corrected (Appendix \ref{app:errata}, E5). \end{remark} \subsection{Cancellation of the archimedean channel}\label{sec:arch} The third route once proposed was $d:=\delta_{\mathrm{Fal}}/\log\cond\to\infty$. \begin{theorem}\label{thm:arch} $\delta_{\mathrm{Fal}}$ has logarithmic singularities along the discriminant divisor, and by the product formula the finite discriminant is the non-archimedean compensation of the same quantity: $$\delta_{\mathrm{Fal}}=-c_g\log|\Delta_{\mathrm{disc}}|_\infty+O(g),\qquad \Delta_{\mathrm{Del}}=-\sum_p\log|\Delta_{\mathrm{disc}}|_p =-\log|\Delta_{\mathrm{disc}}|_\infty+O(1).$$ Hence $$\delta_{\mathrm{Fal}}\le C\Delta_{\mathrm{Del}}+O(g)\ \Longrightarrow\ d\le C\sigma+o(1),$$ i.e.\ the third channel is fully absorbed into the discriminant channel: $d\to\infty$ forces $\sigma\to\infty$. The archimedean channel is cancelled. \end{theorem} \noindent\emph{Sharp calibration for $g=1$}. From $\eta(iy)\sim e^{-\pi y/12}$ and $-24\log|\eta|\sim2\pi y$: $\delta_{\mathrm{Fal}}\sim2\pi\,\mathrm{Im}\,\tau=\log|j|$. The Legendre family gives $N\Longrightarrow\lambda=1/N$, $\delta_{\mathrm{Fal}}\sim2\log N$, $\Delta_{\mathrm{Del}}\sim4\log N$, $\log\cond\sim\log N$: $$\frac{\delta_{\mathrm{Fal}}}{\Delta_{\mathrm{Del}}}\to\frac12,\qquad d\to2,\qquad\sigma\to4.$$ The third channel not only fails to diverge but converges to a constant, and $2d=\sigma\cdot\frac12$: it is half of the discriminant channel. \emph{Correction} (Appendix \ref{app:errata}, E3): one formerly wrote $\delta\sim\frac13\log|j|$; the coefficient is $2\pi\,\mathrm{Im}\,\tau=\log|j|$. The earlier argument ``$S$-unit equations bound $\sigma$ for fixed $S$, hence no tower exists'' is correct at fixed genus but does not constitute an escape along the third channel. \subsection{Two closure theorems (negative results)} \begin{theorem}[Evertse closure]\label{thm:evertse} Let $y^2=f(x)$, $f\in\mathbb Q[x]$, $\deg f=2g+1$. If the bad set is fixed, $S$, then all prime factors of $\mathrm{disc}(f)=\prod_{i-2g\log(2\pi^4),\qquad \delta_{\mathrm{Fal}}=-24H+2\varphi-8g\log2\pi.$$ Derivation: solve Noether for $\bar\omega^2=12h_F-\Delta_{\mathrm{Del}}-\delta_{\mathrm{Fal}}$; use $\Delta_{\mathrm{Del}}\ge0$ for $-\Delta_{\mathrm{Del}}\le0$ and the Wilms lower bound for $-\delta_{\mathrm{Fal}}<2g\log(2\pi^4)$. Unconditional: no abc, no BMY, no conjecture. \begin{corollary}[Reduction]\label{cor:reduction} $$\text{Faltings--Shafarevich }(h_F=O_S(g))\ \Longrightarrow\ \text{arithmetic Bogomolov }(\bar\omega^2=O(g)).$$ \end{corollary} \emph{Meaning}. The task ``bound $\bar\omega^2$ above'' is replaced by ``bound $h_F$ above''---the direction changes and so does the toolbox (Faltings finiteness, moduli spaces, Belyi bounds). But see Remark \ref{rem:reverse}: the tools are still missing. \subsection{A necessary correction} A statement made in three rounds---``fixed bad set $S$ implies $\bar\omega^2\le C(S)g$''---is \textbf{false}. Counterexample: $S=\{p\}$ fixed, while Dolce--Mercuri \cite{DM} give $\bar\omega^2\sim3g\log N$ for $X_0(p^n)$: $\bar\omega^2\to\infty$, not $O(1)$. The correct statement is $$\text{$S$ fixed}\ \&\ \hat f=\frac{\log\cond}{g}=O(1)\ \Longrightarrow\ \bar\omega^2=O_S(g).$$ Since $\hat f=O(1)$ is exactly the region where $\kappa$ diverges, the correction affects no existing conclusion; the hypothesis must be added (Appendix \ref{app:errata}, E11). \subsection{A non-circular cross-check} Solving Proposition \ref{prop:main} against (L3) (write $\hat f=\log\cond/g$): $$R<\frac{12\kappa}{\sigma}-1+\frac{2\log(2\pi^4)}{\sigma\hat f}.$$ Substituting the Fermat values $\kappa=\frac18$, $\sigma=1$, $\hat f\to\infty$: $$R\longrightarrow\frac{12\cdot\frac18}{1}-1=\frac12,$$ matching the independently obtained $R\le\frac12$ from Colmez' explicit height plus the Liu--Saito conductor, with disjoint inputs: the latter uses CM theory and $L$-values, the former only Noether and the pure-analytic Wilms lower bound. The limiting $R$-values of all four towers (Fermat, $X_0$, NS, Heisenberg) align with independently known values. \emph{Difference from a withdrawn ``check''}. An earlier version solved the identity \emph{for} $\kappa=\frac18$ and read off $R$: that contains no information (circular). The direction here is opposite---the independent Wilms lower bound constrains $R$---hence non-circular. \subsection{The reverse direction is unreachable}\label{rem:reverse} From $12h_F=\bar\omega^2+\Delta_{\mathrm{Del}}+\delta_{\mathrm{Fal}}$, obtaining $\bar\omega^2=O_S(g)$ requires: \begin{center} \begin{tabular}{ll} \hline Needed & Status\\ \hline $\Delta_{\mathrm{Del}}=O_S(g)$ & $=$ Szpiro/abc, open\\ $\delta_{\mathrm{Fal}}=O(g)$ (upper bound) & open (lower bound by Wilms; upper bound only a compactness argument with $g$-dependent constants)\\ \hline \end{tabular} \end{center} And the whole chain already exists in the literature: this corollary appears in Parshin's 1988 paper. We merely named the intermediate step. In the $\hat f=O(1)$ region, arithmetic Bogomolov is equivalent to ``no $\kappa\to\infty$'' (since $\bar\omega^2=O(g)\iff R\sigma=O(1)$), so it is \emph{not} an independent seal alongside Szpiro---we formerly estimated so, and retract it (Appendix \ref{app:errata}, E12). \subsection{What survives} Arithmetic Bogomolov is strictly weaker than BMY: if $\sigma=O(1)$ then BMY gives $\bar\omega^2\le3\sigma$, i.e.\ Bogomolov; the converse fails. Refuting $\kappa\to\infty$ does not require full BMY, only the scale $R=O(1)$, not the constant. Theorem \ref{thm:reduction} is unconditional and reduces the problem to an upper bound on the height. \emph{Kept judgment}. Arithmetic Bogomolov $R=O(1)$ is more likely true than BMY---it is true in the function field (Deligne's semipositivity of the Hodge bundle) and is a strict weakening of BMY. The two can be separated; they need not share a fate. \section{Open problems and the function-field comparison}\label{sec:open} \subsection{Function-field comparison: a missing pair of inputs}\label{sec:fftower} \begin{center} \begin{tabular}{p{3.8cm}p{5.9cm}p{4.5cm}} \hline Channel & Function-field mechanism & Number-field status\\ \hline $R=O(1)$ ($\bar\omega^2/g=O(1)$) & Deligne (Hodge bundle semipositivity) & missing $=$ arithmetic Bogomolov\\ $\sigma=O(1)$ ($\Delta/g=O(1)$) & Mason & missing $=$ Szpiro/abc\\ spectral interpretation of the finite $\Delta$ & dual-graph Laplacian Green $=$ N\'eron local height (Werner \cite{Werner96}; Wei \cite{Wei18}) & available; a side confirmation, no new bound on $\sigma$\\ \hline \end{tabular} \end{center} The function field seals both channels with two independent theorems; the number field has neither. The precise form of ``the function field has Mason, the number field only the abc conjecture'' is: Mason is only one of the two. \emph{On the spectral row}. The interpretation of the N\'eron local height pairing via the Laplacian of the dual graph on semistable curves is established in the published literature (Werner, Wei) for Mumford curves; a recent preprint extends it to arbitrary semistable reduction and attempts an identification with boundary two-point functions in $p$-adic string theory. In that language $$\sigma=\#\text{nodes}/b_1\quad\Longrightarrow\quad \text{Szpiro}\iff\text{boundedness of the scale of the two-point function on the unit-torus rank}.$$ This is an independent paraphrase and confirmation of channel (D). But the equivalence is published only for the Mumford-graph case; the extension preprint gives no new bound and is not peer-reviewed. \textbf{We do not use it as input to any derivation.} \subsection{Open problems} \begin{problem}\label{prob:allgenus} General arithmetic BMY: does $\bar\omega^2<3\Delta_{\mathrm{Del}}+O(g)$ hold for all towers? $=$ the curvature channel; in towers with $\sigma\to1$ it is the same question as the barrier. \end{problem} \begin{problem} Can $\kappa_n\to\infty$ occur? (Discriminant channel $\sigma\to\infty$ or curvature channel.) All known towers have $\sigma\to1$, $R\le3$. \end{problem} \begin{problem} Search space: must have ``branch points not totally rational'' or ``non-hyperelliptic'' (Corollary \ref{cor:searchspace}). Mutually exclusive with computability. \end{problem} \begin{problem} Explicit $g=1$ barrier-breaker: threshold $|c_4|\ge10^{47}$, the known record is short by a factor $\sim10$ (lattice reduction). \end{problem} \begin{problem} The Siegel/$L$-value channel: in all $g>1$ cases it does not bind, and is ``provable but not computable''. Rated low priority. \end{problem} \begin{problem}\label{prob:mestre-hyp} Mestre's conductor bound $f\ge\log B_M$ depends on the analyticity of $L(A,s)$ (Langlands-type hypothesis). Bypassable on automorphic towers; conditional on general towers. \end{problem} \begin{problem} Tension between the two normalizations of $h_F$ (period formula vs.\ Noether): affects the threshold reading in the $g=1$ branch. \end{problem} \begin{problem} Curve realization of convexity towers (single Jacobian): the abelian-variety version is complete (Theorem \ref{thm:prodjac}). \end{problem} \begin{problem} Towers neither non-hyperelliptic nor Belyi: no attainable construction currently. \end{problem} \begin{problem}[O19: the unique corner covered by no tool]\label{prob:o19} Partially ramified orbits with $\mathbf B>0$ but $\mathbf B\ll g$ (few ramified orbits among many unramified ones). Then $\mathbf B$ is nonzero yet does not exclude $\sigma\to\infty$, and the neighboring--bad-set tension argument does not apply directly. This is currently the unique corner covered by no tool. \end{problem} \begin{problem}\label{prob:l1genus} An independent first-principles proof of (L1) for arbitrary genus: Liu 1994 Prop.~1 is commonly cited for arbitrary genus, but no explicit independent proof has been found (Remark \ref{rem:l1}). \end{problem} \subsection{The final form of the discriminant channel: thickness language and an explicit bound}\label{sec:thickness} \emph{Thickness formula}. Let $\mathcal X^{\mathrm{st}}$ be the stable model; the local equation of the special fiber at node $i$ is $xy=p^{N_i}$, and $N_i$ is the thickness of the node. The minimal regular model resolves these singularities, inserting $N_i-1$ copies of $\mathbb A^1$ at each. One checks $b_1=\#\text{nodes}$, $m=m^{\mathrm{st}}+\sum_i(N_i-1)$, $\nu_p=\sum_iN_i$, $f_p=b_1(\Gamma_p)+\Sw_p$ (check: $n$ nodes of thickness $1$ gives $\nu=n$; or a single self-node of thickness $n$ gives $\nu=n$: the two readings agree). The stable fiber has $\#V\le2g-2$, so the unique source of $\#V\gg g$ is the thickness itself: $$\text{\bfseries(D)}\ \Longleftrightarrow\ \frac{\sum_iN_i}{b_1+\Sw_p}\to\infty\ \Longleftrightarrow\ \frac{\nu}{f}\to\infty,$$ i.e.\ channel (D) \emph{is} the higher-genus Szpiro conjecture in thickness language. \emph{An extremal candidate passing Noether}. Take the stable fiber to be a $\mathbb P^1$-cactus: one rational curve with $g$ self-nodes, thicknesses $N_1,\ldots,N_g$. Then $f=b_1=g$ (torus rank, $\Sw_p=0$), $\nu=\sum_iN_i$, $\sigma=\frac{\sum_iN_i}{g}$. Its minimal regular model (one vertex with hanging self-chains of lengths $N_i$) passes Noether place by place: $\#V=1$, $B=g$, $L=2B-2=2g-2$, $F\cdot b_1=1$, $\nu=B(L+1)$, $\sigma=L+1$ arbitrary in $N_i$. On known towers $N_i=O(1)$, $\sigma\to1$---consistent with \S\ref{sec:anchors}. \emph{A new mechanism specific to $g\ge2$}. For $g=1$, $b_1=1$, so a semistable singular fiber can only be an $n$-gon, $b_1=n$: every bad prime has $f_p\ge1$ and the discriminant has no ``free lunch''. For $g\ge2$ there are tree-like fibers with $b_1=0$ (e.g.\ two elliptic curves meeting at one point): then $f_p=0$ (the Jacobian in fact has good reduction at $p$) while $\nu_p=m_p-1>0$. The discriminant can be supported at primes invisible to the conductor. \textbf{Szpiro intuition does not transplant to higher genus.} \begin{theorem}\label{thm:sigmabound} Write $\hat f=\log\cond/g$. From $12\kappa=\Omega+\sigma+d$ and three inputs--- (L2) Zhang $\Omega\ge0$; (L3) Wilms $\delta_{\mathrm{Fal}}>-Wg$, $W=2\log(2\pi^4)=10.544133$; the conductor bound $\hat f\ge\log B_M$: $$d=\frac{\delta_{\mathrm{Fal}}}{g\hat f}\ge-\frac{W}{\log B_M} \ \Longrightarrow\ \sigma\ \le\ 12\kappa+\frac{W}{\log B_M}.$$ With Mestre's $B_M=2.334374$: $\sigma\le12\kappa+4.5169$; with $B_M=10.323\log$: for GL2 towers $\sigma\le8.5169$ (and $\sigma\le4$ when $\hat f\to\infty$). \end{theorem} Channel (D) is closed for all GL2 (automorphic/CM) towers, consistent with all five anchors of \S\ref{sec:anchors}. \emph{The third input is conditional}. Mestre's lower bound requires, per four-source verification, that $L(J,s)$ be an analytic $L$-function: Tchamitchian \cite{Te25} (assumption added in v2/v3), Mestre 1986 (zbMATH abstract), Kowalski (MO \#10860), and the Persiflage blog. \emph{Version-level evidence}: the paper's v1 abstract read ``Any abelian variety over $\mathbb Q$ of dimension $g$ has conductor greater than $10^{0.323g}$'' (no hypothesis); v2/v3 add the hypothesis. Even the specialist author once omitted it---Mestre must always be cited with the hypothesis. Hence Theorem \ref{thm:sigmabound} is unconditional on automorphic towers, conditional on general towers. \subsection{The structural mismatch between tools and loopholes}\label{sec:mismatch} This is the last and methodologically most important observation of the paper. What Mestre covers (automorphic towers) is exactly the class of the five anchors of \S\ref{sec:anchors}, where Theorem \ref{thm:sigmabound} already gives $\sigma\le8.5169$; the (D) loophole requires average thickness (\S\ref{sec:thickness}), which is necessarily non-semistable and expected non-automorphic---exactly outside Mestre's range; and the unconditional conductor bound is only $\hat f\ge O(1)$ (Fontaine: no abelian varieties above $\hat f<1$; Abrashkin-type results). Thus $\hat f\to0$ cannot be excluded, and then $d=\delta_{\mathrm{Fal}}/(g\hat f)\to-\infty$ is not ruled out, so all constraint from (L2)$+$(L3) is lost. (\S\ref{sec:bk} gives an unconditional bound for hyperelliptic Jacobians, repairing this over a wide range; but it can vanish identically at genuine bad primes, so the gap is moved, not closed.) \textbf{The inability to rule out $\kappa\to\infty$ is not because the crack is wide, but because the only tool is effective on the half where the answer is already known.} This sentence explains why \S\ref{sec:disc}--\S\ref{sec:bk} keep refining the reduction everywhere without producing a new theorem. This is not a temporary technical gap but a structural one. We propose to leave (D) and Problem \ref{prob:allgenus} as open problems for follow-up, and judge accordingly: the expected return of further search along the discriminant channel is negative. \subsection{A theorem-level closure: cyclotomic towers}\label{sec:cyclo} Let $p$ be an odd prime with fixed bad set $\{2,p\}$. \emph{Cluster picture and two exact quantities}. $C_m:\ y^2=x^{p^m}+1$, $g_m=\frac{p^m-1}{2}$. The root set is $\{-\zeta_{p^m}^k\}$, so pairwise valuations depend only on $v(1-\zeta_{p^M})=\frac{1}{p^{M-1}(p-1)}$; the cluster picture is a full $v$-adic tree: clusters of size $p^s$, each in number $p^{m-s}$ ($s=1,\ldots,m$), depth $\frac{1}{p^{s-1}(p-1)}$. $p$ odd $\Rightarrow$ all cluster sizes odd $\Rightarrow$ no even clusters $\Rightarrow$ torus rank $t=0$. Hence $$f_p=2g+\Sw_p,\qquad\nu_p(\Delta)=mp^m,$$ i.e.\ the tower is potentially good, and $\Sw_p$ is the only unknown. \emph{Conductor lower bound, verified}. By the explicit form of the conductor formula for induced representations, $a(\mathrm{Ind}\,\psi)=\mathrm{val}(\mathfrak d_{L/k})\dim\psi+f_{L/k}\,a(\psi)$, for Fermat--Jacobi quotients $N_{r,s,t}=|\mathrm{disc}\,\mathbb Q(\mu_n)|\cdot N_{K/\mathbb Q}(\mathfrak f)$; Cain (2016) gives the formula, and Rohrlich \cite{Rohrlich86} proves $f\mid p^2$, so each Hecke character contributes $c_j\le2$. The value $v_p(d_{\mathbb Q(\zeta_{p^j})})=p^{j-1}(j(p-1)-1)$ differs from the lower bound by exactly $1$ or $2$, cross-confirming Rohrlich's bound. The earlier concern ``reflex fields might overestimate $\mathbf B$'' is excluded: Weil's (1952) Jacobi-sum Gr\"ossencharaktere are defined on $\mathbb Q(\zeta_{p^j})$ itself; Gallese--Goodson--Lombardo \cite{GGL24} prove each simple factor nondegenerate; Cain's formula uses exactly the discriminant of $\mathbb Q(\zeta_{p^j})$. $$f_p\ge S_m:=\sum_{j=1}^m v_p\big(d_{\mathbb Q(\zeta_{p^j})}\big) =\frac{p^m(m(p-1)-2)+2}{p-1},$$ $$S_m\le f_p\le S_m+2m\ \Longrightarrow\ \frac{mp^m}{S_m+2m}\le\sigma_p\le\frac{mp^m}{S_m},$$ both ends tending to $1$. At $q=2$ no patch is needed: the denominator takes a partial sum (safe in the discard direction), and the contribution $\nu_2=4g\sim4\log2$ is $o(mp^m)$. With the unconditional lower bound (L1), $\sigma\ge1$: $$\boxed{\sigma\to1\quad\text{(theorem level, no Mestre needed)}}$$ \emph{Meaning}: Theorem \ref{thm:sigmabound} gives only a constant upper bound on $\sigma$; this section gives the asymptotic. The two are complementary, and the route here bypasses Mestre entirely---on this tower class (D) is sealed not by ``a tool covering the half where the answer is known'' but by an independent conductor formula. This sharpens the meta-observation of \S\ref{sec:mismatch}: there is more than one rescue mechanism. \emph{Honest labels}. Cain's formula is for a single Fermat factor; the decomposition of $\mathfrak f$ comes from eigenvalue analysis, the literature gives only ``each simple factor nondegenerate'' without listing CM fields one by one; only $c_j\le2$ is known. Cain states the composite case only for a sketch at $n=2$, so taking $c_j\le2$ factor by factor is a corollary, not a direct citation. \subsection{An unconditional conductor bound, and why it still does not suffice}\label{sec:bk} \emph{The Brumer--Kramer bound}. Let $K$ be a $\mathfrak p$-adic field, $p$ an odd prime; $h(x)\in K[x]$ a separable polynomial of odd degree $\deg h=2g+1$, $B=J(y^2=h)$. Factor $h=h_1\cdots h_r$ with root fields $M_i=K(\theta_i)$. Then \cite[Lemma 6.3]{BK94} $$f_{\mathfrak p}(B/K)\ \ge\ \mathbf B:=\sum_{i=1}^r v_K\big(\mathfrak d_{M_i/K}\big)\,f_{M_i/K},$$ with equality when $h$ is irreducible and totally ramified. The key point: it uses only the Artin conductor of $B[2]$---no $L$-functions, no modularity, no analytic continuation---and it is fully computable from the factorization of $h$ over $K$ and the relative discriminants of the root fields. Three hard restrictions: $p$ odd; the shape $y^2=h$; $\deg h$ odd and separable. \emph{Comparison with \S\ref{sec:cyclo}: $\mathbf B=S_m$}. On the cyclotomic tower $y^2=x^{p^m}+1$, $$x^{p^m}+1=(x+1)\prod_{j=1}^m\Phi_{2p^j}(x),\qquad M_j=\mathbb Q_p(\zeta_{p^j}),$$ totally ramified, $v_p(\mathfrak d_{M_j/\mathbb Q_p})=p^{j-1}(j(p-1)-1)$: $$\mathbf B=\sum_j p^{j-1}\big(j(p-1)-1\big)=S_m.$$ \begin{center} \begin{tabular}{lccccc} \hline $m$ & $g$ & $\nu_p$ & $S_m$ & $\sigma\le\nu_p/S_m$ & $S_m/2g$\\ \hline 3 & 13 & 81 & 55 & 1.473 & 2.12\\ 4 & 40 & 324 & 244 & 1.328 & 3.05\\ 6 & 364 & 4374 & 3646 & 1.200 & 5.01\\ \hline \end{tabular} \end{center} Identical to \S\ref{sec:cyclo} place by place. But this route needs neither Cain, Rohrlich, nor Weil, nor reflex-field arguments---the risk drops from ``a four-section corollary chain'' to ``one standard 1994 lemma''. Thus $\sigma\to1$ is a theorem-level conclusion confirmed by two independent paths. \emph{Counterexample: $\mathbf B$ can vanish identically at a genuine bad prime} (Appendix \ref{app:errata}, E10). Take $u=2$ (a non-square in $\mathbb F_3$), $p=3$, $$h=(x^2-u)\big(x^2-2p^mx+(p^{2m}-u)\big)(x-1).$$ The two quadratic orbits have discriminants $4u=8$ and $4u=8$, both unramified, so $\mathbf B=0$. But they collide mod $3$ ($h_1\equiv h_2\equiv x^2+1$), and $p=3$ is a genuine bad prime: \begin{center} \begin{tabular}{lcccc} \hline $m$ & $\nu_3(\Res)$ & $\nu_3(\mathrm{disc}\,h)$ & $\mathbf B$ & $\nu_3/\mathbf B$\\ \hline 1 & 2 & 4 & 0 & $\infty$\\ 2 & 4 & 8 & 0 & $\infty$\\ 3 & 6 & 12 & 0 & $\infty$\\ \hline \end{tabular} \end{center} The criterion is clean: every $M_i/K$ unramified $\iff\mathbf B=0$. So this is not ``the bound is too weak'' but ``the bound can vanish outright''. And the mechanism scales with genus: take $$h=\prod_{i=1}^s(x^2-u_i)\prod_{i=1}^s\big(x^2-2p^mx+p^{2m}-u_i\big)(x-c), \qquad g=2s,\ \nu_p=2gm,\ \mathbf B=0,$$ $m\asymp g\Rightarrow\nu_p\asymp g^2$, of which Brumer--Kramer knows nothing. Measured at $p=5$, $s=2$, $g=4$: $m=1,2,3$ give $\nu_5=8,16,24$, confirmed place by place. \emph{Neighboring--bad-set tension}. Why does this not save (D)? Because $\mathrm{Res}=p^{2m}(p^{2m}-4u_i)$, and keeping unramifiedness requires $u_i$ to be a unit mod $p$, so $p^{2m}-4u_i\equiv-4u_i\not\equiv0\pmod p$: an integer of this size must acquire new prime factors beyond $p$. Measured: $p=7$, $m=3$ gives $p^{2m}-4u=117641=47\times2503$. The larger $p,m$, the more new primes, so the bad set must grow. Thus the hyperelliptic family trichotomizes: \begin{center} \begin{tabular}{p{4.0cm}p{3.6cm}p{6.6cm}} \hline Case & Criterion & Consequence\\ \hline (i) all branch points in $K$ (split completely) & $\mathbf B$ degenerate & Evertse closure (Theorem \ref{thm:evertse}); no tower\\ (ii) every orbit unramified & $\mathbf B=0$ & neighboring--bad-set tension: bad set grows with $m$\\ (iii) some orbit ramified & $\mathbf B>0$ & the bound bites\\ \hline \end{tabular} \end{center} Cases (i) and (ii) are completely different mechanisms ($S$-unit equations vs.\ conductor-bound degeneration), yet are captured by the same criterion $\mathbf B=0$---the most unexpected finding of this round. \emph{Three comparison families: BK is region-optimal, not universal}: \begin{center} \begin{tabular}{p{3.0cm}p{4.5cm}p{6.7cm}} \hline Family & What BK gives & Comparison with other tools\\ \hline wild $X_1$ & $\mathbf B=404$ vs $2g=80$ & strongest\\ twist $y^2=x+p^{2n}$ & $\sigma\le k\bmod 2$, $n\le4g+1$ & weaker than Noether (Zhang$+$Noether give $O(1)$)\\ CM $y^2=x^{p^m}+1$ & $\mathbf B=S_m$ & matches \S\ref{sec:cyclo}; confirmed by two paths\\ \hline \end{tabular} \end{center} \emph{Verdict: the gap is moved, not closed}. BK does not close the gap of \S\ref{sec:mismatch}; it only moves it from (D) into a weak version: for hyperelliptic Jacobians, case (iii) now has an unconditional bound independent of Mestre; but case (ii) shows the bound can vanish at genuine bad primes, and then the bad set must grow, so the conclusion retreats to ``a weak version with growing bad set'' rather than the original problem. The remaining exposure is Problem \ref{prob:o19}. \emph{Honest labels}. The three hard restrictions of \cite[Lemma 6.3]{BK94} ($p$ odd; $y^2=h$; $\deg h$ odd separable) are not removable; $\nu_p=2gm$ is measured only up to $s=2$ (a larger $p$ is needed, since $\mathbb F_3$ has only one non-square class, not accommodating several distinct $u_i$); $s\ge3$; the tension of \S\ref{sec:thickness} is presently a mechanism argument plus numerical confirmation, not a theorem; Brumer--Kramer's upper bound (their Theorem 6.2) we could not reproduce---the definition of $\lambda_{\mathfrak p}$ is missing from the available text, and the most natural guess (digit sum) at $p=3$, $g=40$ gives $20824$, conflicting with the lower bound. Hence that upper bound is unusable; only the lower-bound direction is used here, which is unaffected. \subsection{A meta-observation}\label{sec:meta} Three ``geometrically dangerous'' instances are all rescued by arithmetic quantities, and each rescue covers only automorphic/CM towers: \begin{center} \begin{tabular}{p{3.2cm}p{3.2cm}p{4.0cm}p{3.4cm}} \hline Dangerous instance & Geometric diagnosis & Rescue mechanism & Scope\\ \hline Fermat (thin-and-dense, $\#\Gamma_0$) & geometric routes fail & Colmez explicit height ($R\le\frac12$) & CM only\\ Heisenberg (non-CM $+$ thin-dense) & geometric routes fail & wild Swan conductor ($c>1$) & arithmetic towers only\\ long $(-2)$-chains ($\sigma\to\infty$) & combinatorial extreme allowed & Mestre $f\ge c_Sg$ & needs analyticity of $L(A,s)$\\ \hline \end{tabular} \end{center} Geometric side (Bergman/systole) has zero effect on all three instances; the arithmetic side rescues three times, but all unconditionally only on automorphic towers. \textbf{$\kappa\to\infty$ is not excluded not because the crack is wide, but because all our tools are effective only on arithmetic towers.} \subsection{A tension concerning BMY $\Rightarrow$ effective Mordell} BMY implies effective Mordell. If GRH $\Rightarrow$ BMY extended to general towers, then GRH $\Rightarrow$ effective Mordell---suspiciously strong. Our judgment: GRH $\Rightarrow$ BMY most likely holds only for CM/automorphic families (where $L$-value formulas exist), while general BMY is exactly Problem \ref{prob:allgenus}, out of range. This is a judgment, not a conclusion; we register it. \appendix \section{Falsification of an external framework: the rigidity-spectrum test} \label{app:rigid} This appendix is independent of the main line, but records a complete falsifiability test, as a methodological specimen. \subsection{Background} A recent phenomenological framework (self-described as ``not a theorem, analogous to critical phenomena and renormalization group in physics'') postulates a universal set of dimensionless constants $$\mathrm{LOW}=0.035,\qquad\mathrm{BAND}=[0.62,0.71],\qquad \mathrm{SAT}\approx0.97\sim1,$$ called the rigidity spectrum, and asserts that ``most arithmetic problems fall in this band'', with the screening criterion ``nonvanishing leading term of a function equation at the critical point''. All five anchor towers of \S\ref{sec:anchors} satisfy this screening criterion (all are standard automorphic/CM objects). Hence, if the framework is self-consistent, they should land in the band. Taking our $\kappa$ as the candidate for its ``modulus'' provides the test. \subsection{Pre-registration (written before computing)} \begin{center} \begin{tabular}{p{0.7cm}p{8.7cm}p{3.8cm}} \hline No. & Prediction & Outcome\\ \hline P1 & No raw $\kappa$ lands in the band & $\checkmark$ (0/5)\\ P2 & Pure scaling rescaling cannot send all five values into the band & $\checkmark$ (ratio spread $4>$ band ratio $1.145$)\\ P3 & The most natural normalization $3\kappa$ spreads the five towers over $[0.25,1]$ rather than clustering & $\checkmark$ (band catches 1/5)\\ P4 & With $\sigma$ as modulus the five towers degenerate to the saturation end; band still empty & $\checkmark$\\ P5 & The band contains several mutually unrelated constants, suggesting a density artifact of constant-scanning & $\checkmark$ (4/10)\\ \hline \end{tabular} \end{center} \subsection{Results} $T_0$ (raw): $\kappa\in\{0.083,0.125,0.167,0.208,0.333\}$; hits 0/5. The maximum misses the lower edge of the band by $0.29$. $S_1$ (structural exclusion---the strongest): scaling preserves the spread ratio, and $$\frac{\kappa_{\max}}{\kappa_{\min}}=\frac{1/3}{1/12}=4,\qquad \frac{0.71}{0.62}=1.1452,\qquad 4>1.145.$$ Hence \textbf{no} $c$ can send all five values into the band simultaneously. This is independent of the specific values of $\kappa$, far stronger than case-by-case comparison---consequently ``the normalization was not chosen correctly'' is not an available explanation. $S_2$ (affine transformation works but is ad hoc): $(a,b)=(0.36,0.59)$ sends the five towers to $0.62,0.635,0.65,0.665,0.71$, all in the band. But the two coefficients have no mathematical meaning in either framework; adopting them is ad hoc fitting. $T_1$--$T_4$ (four motivated transformations): the most natural normalization by the proved barrier gives $3\kappa\in\{0.25,0.375,0.5,0.625,1.0\}$: the five towers spread uniformly over $[0.25,1]$, and the band touches only one value ($0.625$, just $0.005$ above the lower edge). Alternative candidate moduli: $\sigma\to1$ (the five towers collapse to one point); $R\in\{0,\frac12,1,\frac32,3\}$. Band fill rates: 0/5 throughout. \subsection{Two side findings} \emph{Constant density}. $[0.62,0.71]$ contains $\ln2=0.6931$, $1-1/e=0.6321$, $1/\sqrt2=0.7071$, $2/3=0.6667$---four constants of completely unrelated origin. A window of width $0.09$ containing four unrelated constants is more plausibly a density artifact of constant-scanning. $1/\varphi=0.6180$ misses the band by $0.002$, showing the band edge is highly sensitive to fitting choices. \emph{The spectrum may not be independent}. The framework states $\Phi_{\mathrm{Tate}}\approx0.7\in\mathrm{BAND}$ and $\Phi_{\mathrm{cusp}}\approx0.07$, while $0.07/2=0.035=\mathrm{LOW}$. If both hold, the ``rigidity spectrum'' is really $\{\Phi_{\mathrm{cusp}}/2,\ \Phi_{\mathrm{Tate}},\ 1\}$: two amplitudes plus a trivial critical point, not three independent eigenvalues. \emph{Multiple comparisons}. $3\times4+10=22$ variables, 4 transformations, 10 constants, 4 chances: best result $1/5$. After correction this is not a signal. \subsection{Verdict and the disposition of this paper} \begin{center} \begin{tabular}{p{4.8cm}p{8.0cm}} \hline Item & Verdict\\ \hline $\kappa$ vs.\ the spectrum & falsified (0/5 $+$ structural exclusion)\\ $\sigma$, $R$ vs.\ the spectrum & falsified (band empty)\\ Whether the spectrum is independent of $\Phi$ & doubtful; pending confirmation by the framework's side\\ The true kernel of the framework & kept: the natural boundary of the prime zeta function\\ \hline \end{tabular} \end{center} We do not merge the framework: the main line is theorem-driven, the framework is data-driven, their correctness standards differ, and merging would dilute the proved part. But two independent convergence points are retained as cross-checks: the framework's ``dangerous direction $=$ bands with multiplicative parameter $(-2)$-chains'' $\longleftrightarrow$ our independent identification of the (D) loophole (additive reduction $+$ long $(-2)$-chains, \S\ref{sec:thickness}); and the framework's ``Ogg rewrite $\Rightarrow$ Szpiro form'' $\longleftrightarrow$ our independent discovery of the Ogg even -graph Euler characteristic $\nu=\#E$, $f=b_1$, $m=\#V-1$ (\S\ref{sec:disc}). Two programs, completely different methods, pointing to the same conclusion---such independent convergences are worth more than the framework's own fits. \emph{Final word (to the framework's side)}. What it lacks is not more fitting but one a priori quantitative prediction about a new object---written down before measurement. Do it once, and the nature of the framework changes. \emph{Honest labels}. The band edges are taken from a paraphrase, not verified against the original verbatim; this test assumes $\kappa$ is a legitimate candidate for the ``modulus''; if the framework defines it otherwise, the test must be rerun under its definition with the five towers' values. If the reply is ``these five towers belong to another sector'', that reply makes the spectrum unfalsifiable---the most common failure mode of phenomenological frameworks. \section{Revision history}\label{app:errata} \small This is the third version (final, 2026-10-09). The second version (2026-10-08) added: \S\ref{sec:remainder} (exact order of the remainder of the main chain; the first version's $\Theta(n\log p\cdot p^{-n})$ was wrong); the revision history itself; the thickness language for the discriminant channel. The third version adds \S\ref{sec:bk} (the Brumer--Kramer unconditional bound) and E9, E10 below. All corrections: \begin{center} \begin{tabular}{p{0.7cm}p{7.0cm}p{5.5cm}} \hline No. & Correction & Disposition\\ \hline E1 & Remainder order $\Theta(n\log p\cdot p^{-n})$ & should be $O(p^{-n})$; slope attribution in Corollary \ref{cor:slope}\\ E2 & Conductor density $c\to\infty$ & should be $c=2$ (direct Liu--Saito recomputation)\\ E3 & $\delta_{\mathrm{Fal}}\sim\frac13\log|j|$ & coefficient is $2\pi\,\mathrm{Im}\,\tau=\log|j|$\\ E4 & $\kappa_X=\frac1{4c}$ & should be $\kappa_X=\frac{c+1}{12c}$ (breaks the floor; violates Zhang)\\ E5 & stable vs.\ minimal regular model conflated & $(-2)$-chains make $m$ unbounded; must distinguish\\ E6 & convexity weight $\frac{a}{a+b}$ & true weight $\frac{a\rho}{a\rho+b}$ (\S\ref{sec:trap})\\ E7 & Heisenberg tower judged CM by density & actually $\End(\Prym)\supseteq M_N(\mathbb Q(\zeta_N))$, strongly non-CM\\ E8 & Easton--Urz\'ua and Jang constructions misclassified as arithmetic BMY counterexamples & the former has no $g\ge2$ fibration; the latter is not a counterexample\\ E9 & Mestre bound usable without hypothesis & requires analyticity of $L(A,s)$; v1 abstract omitted it, v2/v3 restore it\\ E10 & expectation ``Brumer--Kramer closes the \S\ref{sec:mismatch} gap'' withdrawn & $\mathbf B$ can vanish at genuine bad primes (counterexample in \S\ref{sec:bk})\\ E11 & fixed bad $S\Rightarrow\bar\omega^2=O(g)$ & false; hypothesis $\hat f=O(1)$ must be added\\ E12 & ``arithmetic Bogomolov an independent seal alongside Szpiro'' & retracted; equivalent in the $\hat f=O(1)$ region\\ E13 & (L1) upgraded to unconditional & Remark \ref{rem:l1} rewritten per Saito \cite{Saito88} $+$ Liu \cite{Liu94}\\ E14 & Brumer--Kramer upper bound (Thm.\ 6.2) unusable & $\lambda_{\mathfrak p}$ definition missing; only the lower bound used\\ E15 & layer--condensation identity & $10^{-14}$-level numerical; combinatorial proof still missing\\ \hline \end{tabular} \end{center} \normalsize \section{Verification and compilation note}\label{app:verify} Every conclusion in this paper has been reproduced place by place on at least one known anchor and cross-checked on a second, independent anchor; the anchors are the Fermat tower ($\kappa=\frac18$), the $X_0(N)$ tower ($\kappa=\frac13$), the elliptic curve ($j=0$, $h_F=-1.667691$), and the CM discriminant ($d=163$, $h_F=0.231116$). 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