% hull_inflation_paper.tex % English synthesis of the revised Chinese preprint % "An Integral-Structure Inflation in the Hull Operation of IUT Corollary 3.12" % All numerical values in this file were independently recomputed and verified. \documentclass[11pt,a4paper]{amsart} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage{booktabs,array,longtable} \usepackage{enumitem} \usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} %------------------------- macros ------------------------- \newcommand{\Q}{\mathbf{Q}} \newcommand{\Qp}{\mathbf{Q}_{p}} \newcommand{\Qv}{\mathbf{Q}_{v_{\Q}}} \newcommand{\Zp}{\mathbf{Z}_{p}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\N}{\mathbf{N}} \newcommand{\dv}{\mathfrak{d}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\disc}{\operatorname{disc}} \newcommand{\Norm}{\operatorname{N}} \newcommand{\hull}{\operatorname{hull}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\Net}{\mathrm{Net}} \newcommand{\Gal}{\operatorname{Gal}} \newcommand{\F}{\mathbf{F}} \newcommand{\cO}{\mathcal{O}} \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cI}{\mathcal{I}} \newcommand{\llst}{\ell^{\ast}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \begin{document} \title[An integral-structure inflation in the hull operation]{An Integral-Structure Inflation in the Hull Operation\\ of IUT Corollary 3.12} \author{Chunping Zeng} \date{October 2026} \begin{abstract} This paper identifies and rigorously quantifies a third-class gap in the passage from Theorem~3.11 to Corollary~3.12 in S.~Mochizuki's inter-universal Teichm\"uller theory (IUT). It occupies the same position as, but is of a different type from, and neither implies nor is implied by, the Scholze--Stix gap (non-commutativity of the $j^{2}$-scaling) and the compatibility gap (9-1) recorded in the LANA project. The holomorphic hull of IUT~III, Remark~3.9.5, is the minimum over the constrained family $\{\lambda\cdot\cO_{\cI^{\Q}}\}$: each direct summand of $\cI^{\Q}\cong\bigoplus_{\alpha}F_{\alpha}$ is scaled independently. This operation ``boxes'' the image of the tensor-product integral structure and introduces an index $\rho$. The main mathematical result of this paper is the exact closed form of this index, \begin{equation*} \rho \;=\; \lvert\dv_{K}\rvert^{(n-1)d^{\,n-1}/2},\qquad d=[K:\Qp], \end{equation*} proved as an identity of \emph{positive integers} (not merely a $p$-adic valuation), where $\dv_{K}$ is the discriminant of $K/\Qp$. Its $p$-adic corollary is $v_{p}(\rho)=\frac{(n-1)d^{\,n-1}}{2}\,v_{p}(\dv_{K})$: the loss equals $\frac{n-1}{2}\times$ (number of components) $\times$ (the $p$-adic valuation of the local discriminant). The formula is uniform for tame and wild ramification, with no extra term; wild ramification only replaces the tame value $(d-1)$ by the actual $v_{p}(\dv_{K})$. Substituting IUT's own parameters ($n=j+1$, $\llst=(\ell-1)/2\asymp h^{1/2}$, $v\in V^{\mathrm{bad}}$), the inflation is doubly exponential: $\log\log\rho=\Theta(\sqrt{h})$, i.e.\ $\rho=\exp(\exp(\Theta(\sqrt{h})))$. Measured by the crossover scale $h^{\ast}(\kappa)$, the conclusion is robust: even if the target bound were as generous as $h^{1000}$, the $d=2$ inflation overtakes it at $h\approx4\times10^{9}$ and the $d=4$ inflation at $h\approx9\times10^{8}$. This paper is \emph{not} a verdict on IUT. What it proves is the narrower statement: even if (9-1) holds, layer-1 inflation alone already suffices to prevent Corollary~3.12 from yielding abc. \end{abstract} \maketitle \keywords{inter-universal Teichm\"uller theory; abc conjecture; holomorphic hull; local tensor products; discriminant; integral-structure mismatch} \subjclass[2020]{11G99; 11R04; 14H30; 11D75} \section*{0.\ Notation and conventions}\phantomsection \addcontentsline{toc}{section}{0. Notation and conventions} \begin{itemize}[leftmargin=1.8em,itemsep=2pt] \item $p$ is a fixed prime; $K/\Qp$ is a finite \emph{Galois} extension of degree $d=[K:\Qp]$, $G=\Gal(K/\Qp)$ (so $|G|=d$). \item $\cO=\cO_{K}$; $\dv_{K}$ is the (field) discriminant, $|\dv_{K}|$ its absolute value; $v_{p}$ is the normalized valuation. \item For $n\ge1$ write $A_{n}=\bigotimes_{\Qp}^{n}K$ (the $n$-fold tensor product as a $\Qp$-algebra) and $\cO_{n}=\bigotimes_{\Zp}^{n}\cO$. \item Write $B_{n}=\bigoplus_{g\in G^{\,n-1}}K$ with integral lattice $\cP_{n}=\bigoplus_{g\in G^{\,n-1}}\cO$. \item Elements of $G^{\,n-1}$ are written $g=(g_{2},\dots,g_{n})$, with $g_{1}=\mathrm{id}$ fixed once and for all (normalization). \item $\Tr_{K}$ is the field trace; the traces on $A_{n}$ and $B_{n}$ are $\Tr_{A_{n}}$, $\Tr_{B_{n}}$. \item For a $\Zp$-lattice $\Lambda$, $\disc(\Lambda):=\det\bigl(\Tr_{A}(e_{i}e_{j})\bigr)$ is its discriminant (in the ambient algebra $A$). \end{itemize} A basic fact, used repeatedly: if $\Lambda'\subseteq\Lambda$ are $\Zp$-lattices of the same rank, then \begin{equation}\label{eq:0.1} \disc(\Lambda')=[\Lambda:\Lambda']^{2}\,\disc(\Lambda). \tag{0.1} \end{equation} \section{Introduction} \subsection{Background} The abc conjecture (Oesterl\'e--Masser, 1985) asserts that for every $\varepsilon>0$ there are only finitely many primitive triples $(a,b,c)$ with $a+b=c$, $\gcd(a,b,c)=1$ and $c>\operatorname{rad}(abc)^{1+\varepsilon}$. It is equivalent to the Szpiro conjecture and implies Fermat's Last Theorem together with most other famous Diophantine problems. In 2012 Shinichi Mochizuki released the four papers of inter-universal Teichm\"uller theory (IUT) \cite{iut1,iut2,iut3,iut4}, claiming a proof of the abc conjecture. For over a decade the verification of the proof has been in a rare state: on the one hand no accepted refutation has been given; on the other hand, several teams of experts have reported steps that they could not close. \subsection{The gaps of the three camps} In the entire IUT argument, the passage \emph{from Theorem 3.11 to Corollary 3.12} is the unique load-bearing point: the hundreds of pages of construction before it (Hodge theater, $\Theta$-link, log-link, multiradial algorithms) are all ``identity machines''; every piece of structural work is cashed out in this final step as a single inequality. Three independent teams have identified a gap at the same position: \begin{center} \footnotesize \begin{tabular}{@{}p{2.05cm}p{4.05cm}p{2.95cm}l@{}} \toprule \textbf{Camp} & \textbf{Location of the gap} & \textbf{Type} & \textbf{Status}\\ \midrule Scholze--Stix \cite{ss} & $j^{2}$-scaling non-commutativity & diagram not exchanged / $O(\ell^{2})$ & disputed\\ LANA (2026-07) \cite{lana} & q-pilot double-construction compatibility (9-1) & ``not manifestly false, but no proof'' & undecided\\ \textbf{this paper} & hull's boxing inflation & \textbf{magnitude} (exactly computable) & \textbf{quantified}\\ \bottomrule \end{tabular} \end{center} LANA \S 10 explicitly excludes the Scholze--Stix hexagon diagram (``\emph{our analysis does not give rise to this particular diagram}''). The three gaps occupy the same position (all in $3.11\!\to\!3.12$), are of different types, and are mutually independent — a fact of decisive importance for the present paper's orientation: this paper neither depends on nor disproves either of the other two. \subsection{Contributions of this paper} \begin{enumerate}[leftmargin=2em,itemsep=2pt] \item \textbf{Exact closed form} (Theorem~\ref{thm:5.1}): $\rho_{n}=|\dv_{K}|^{(n-1)d^{n-1}/2}$, an identity of positive integers, with complete proof. \item \textbf{Trace-preservation lemma} (Lemma~\ref{lem:3.1}): $\Tr_{B_{n}}(\Phi(z))=\prod_{i}\Tr_{K}(z_{i})$ — the pivot of the entire proof, and an identity valid for \emph{every} Galois $K$ and \emph{every} $n$. \item \textbf{Integrality/parity corollary} (Corollary~\ref{cor:5.3}): $(n-1)d^{\,n-1}v_{p}(\dv_{K})$ is always even, proved by the Hilbert ramification-group formula; an internal consistency check on the closed form. \item \textbf{Sparsity of the value set} (\S\ref{sec:8}): $\rank\langle\Theta\rangle\le\min\bigl(\llst,[\Qp(q):\Qp]\bigr)\le d$ (the bound is saturated and attainable, not an identity; see B8), while the hull lives in dimension $d^{\,n}$; and $\hull(\Theta)=\hull(\cO_{n})$ up to the normalization zero point — hence $\rho$ is a strict lower bound. \item \textbf{Robustness} (\S\ref{sec:10.3}): crossover $h^{\ast}(\kappa)$ quantifies extreme insensitivity to assumptions on the target magnitude. \item \textbf{Direction lock} (\S\ref{sec:8.4}): $n_{v}=h_{v}\in\Z_{\ge1}$ with $n_{v}\asymp h$ (IUT~I Definition 3.1(c) + IUT~IV Corollary 2.2) $\Longrightarrow$ the layer-1 net effect is \emph{permanently} inflationary; the contraction branch is unreachable in IUT's parameter space. \item \textbf{Non-Galois treatment} (\S\ref{sec:14}): compositum closed form $\rho_{\mathrm{NG}}^{2}=\rho_{\mathrm{G}}^{2}/p^{\Delta}$, with $\Delta\le\operatorname{poly}(h)$ so the discount is asymptotically negligible. \end{enumerate} \subsection{Methodological statement} All conclusions of this paper carry reproducible numerical verification. \S\ref{sec:6.4} and Appendix~\ref{app:B} record in full the \emph{ten errors} caught by the project's self-checks and their corrections — these errors once produced wrong conclusions (including a negative exponent and a directional misjudgment), and they are recorded as found. Two of them (B1, B10) carry methodological lessons, see Appendix~\ref{app:B}. In particular, \S\ref{sec:9} records the unique conceivable falsifier of this paper (procession normalization accidentally cancelling the low-rank degeneration) and gives the original-text check: the mechanism does not exist. \section{Standard decomposition of local tensor products} \subsection{The standard isomorphism \texorpdfstring{$\Phi$}{Phi}} Define the $\Qp$-linear map \begin{align*} \Phi\colon A_{n}=\otimes^{n}K &\longrightarrow B_{n}=\bigoplus_{g\in G^{\,n-1}}K\\ \Phi(x_{1}\otimes\dots\otimes x_{n}) &:=\Bigl(x_{1}\cdot\sigma_{g_{2}}(x_{2})\cdots\sigma_{g_{n}}(x_{n})\Bigr)_{g=(g_{2},\dots,g_{n})} \end{align*} Dimension check: $\dim A_{n}=d^{\,n}$; $\dim B_{n}=|G|^{\,n-1}\cdot d=d^{\,n-1}\cdot d=d^{\,n}$. \checkmark \subsection[Lemma 2.1]{Lemma \texorpdfstring{$\ref{lem:2.1}$}{2.1}: \texorpdfstring{$\Phi$}{Phi} is a \texorpdfstring{$\Qp$}{Qp}-algebra isomorphism}\label{sec:2.2} \begin{lemma}\label{lem:2.1} $\Phi$ is an isomorphism of $\Qp$-algebras. \end{lemma} \begin{proof} \emph{Multiplicativity.} For pure tensors, \begin{align*} \Phi\bigl((x_{1}\otimes\dots\otimes x_{n})(y_{1}\otimes\dots\otimes y_{n})\bigr) &=\Phi\bigl(x_{1}y_{1}\otimes\dots\otimes x_{n}y_{n}\bigr)\\ =\Bigl(\prod_{i}\sigma_{g_{i}}(x_{i}y_{i})\Bigr)_{g} &=\bigl(\prod_{i}\sigma_{g_{i}}(x_{i})\bigr)_{g}\,\bigl(\prod_{i}\sigma_{g_{i}}(y_{i})\bigr)_{g} =\Phi(x)\,\Phi(y) \end{align*} (with $\sigma_{g_{1}}=\mathrm{id}$). By bilinearity and linear extension, $\Phi$ is an algebra homomorphism. \emph{Surjectivity (standard Galois descent, by induction).} For $n=1$, $A_{1}=K=B_{1}$ and $\Phi=\mathrm{id}$, which is clear. For $n=2$: since $K/\Qp$ is Galois, $K\otimes K$ splits completely over its primitive idempotents into $\bigoplus_{G}K$. Concretely, the map \[ x\otimes y\longmapsto\bigl(x\,\sigma_{g}(y)\bigr)_{g\in G} \] is the classical isomorphism $K\otimes K\cong\operatorname{Map}(G,K)$ (equivalently, $K\otimes_{\Qp}K\cong\prod_{G}K$; see \cite[III]{sga1}). We recall the argument in detail, since the induction step reuses it. For each $g\in G$ define \[ e_{g}:=\sum_{\tau\in G}a_{\tau,g}\otimes b_{\tau,g}\in K\otimes K \] to be the idempotent projecting onto the $\sigma=g$-coordinate: equivalently, choose a normal basis $K=\bigoplus_{\tau}\Qp\cdot\tau(\theta)$ ($\theta\in K$ with $\{\tau(\theta)\}_{\tau}$ a $\Qp$-basis, possible by the normal basis theorem) and set \[ e_{g}=\sum_{\tau\in G}\tau(\theta)\otimes g^{-1}(\tau(\theta)^{\vee}), \] where $(\cdot)^{\vee}$ is the dual basis with respect to the trace pairing $\Tr_{K/\Qp}(xy)$, which is nondegenerate. The $e_g$ are pairwise orthogonal idempotents ($e_{g}e_{h}=\delta_{gh}e_{g}$): orthogonality follows from the character relations \[ \sum_{\tau}\tau(\theta)\,\tau\bigl((g^{-1}h)(\theta)^{\vee}\bigr) =\Tr\bigl(\theta\,(g^{-1}h)(\theta)^{\vee}\bigr)=\delta_{g,h}, \] summed against the dual basis. They satisfy $\sum_{g}e_{g}=1$, and on the $g$-th component the multiplication $x\otimes y\mapsto x\,\sigma_g(y)$ identifies $K\otimes K\cdot e_{g}\cong K$ as a $\Qp$-algebra. Hence $K\otimes K\cong\bigoplus_{g\in G}K$, and under this identification $x\otimes y$ goes to $\bigl(x\,\sigma_g(y)\bigr)_g$. So $\Phi$ is surjective for $n=2$ (in fact an isomorphism onto $\bigoplus_G K$). For $n>2$: write $\otimes^{n}K=K\otimes\bigl(\otimes^{n-1}K\bigr)$. By the induction hypothesis the right factor admits the analogous splitting $\otimes^{n-1}K\cong\bigoplus_{G^{n-2}}K$; applying the $n=2$ step to each summand against the left factor splits each component into $|G|$ copies of $K$, giving $|G|\cdot|G|^{n-2}=|G|^{n-1}$ copies of $K$, i.e.\ all of $B_n$. The induction closes. \emph{Isomorphism.} $\Phi$ is surjective and $\dim A_{n}=\dim B_{n}=d^{\,n}$, hence $\Phi$ is a $\Qp$-linear isomorphism (thus has zero kernel). \end{proof} \begin{remark}[Fix of the previous draft]\label{rem:2.1-fix} The previous draft argued injectivity from ``$\Phi(1)=1\neq0$ and equal dimension, hence it suffices to prove surjectivity'' — insufficient: a non-zero kernel is not excluded by that argument (equal dimension only turns surjectivity into injectivity; $\Phi(1)\neq0$ does not directly give injectivity). The present text proves surjectivity \emph{first} and obtains the isomorphism from equal dimension. The logic is now closed. \end{remark} \subsection[Lemma 2.2]{Lemma \texorpdfstring{$\ref{lem:2.2}$}{2.2}: \texorpdfstring{$\Phi(\cO_{n})\subseteq\cP_{n}$}{Phi(O\_n) subset P\_n}} \begin{lemma}\label{lem:2.2} $\Phi(\cO_{n})\subseteq\cP_{n}$. \end{lemma} \begin{proof} Each $\sigma_{g}$ preserves $\cO$, and $\cO$ is closed under multiplication. Hence for a pure tensor $\otimes_{i}x_{i}$ with $x_{i}\in\cO$, every component of its image is a product of elements of $\cO$, so it lies in $\cO$. \end{proof} \begin{definition}[Inflation index]\label{def:2.3} \[ \rho_{n}:=[\cP_{n}:\Phi(\cO_{n})] \] (finite, as both lattices have rank $d^{\,n}$). \end{definition} The goal of this paper is to compute $\rho_{n}$. \section{The trace-preservation lemma (pivot of the proof)}\label{sec:3} \subsection{Statement} \begin{lemma}[Trace preservation]\label{lem:3.1} For any $z=x_{1}\otimes\dots\otimes x_{n}\in A_{n}$: \[ \Tr_{B_{n}}(\Phi(z))=\prod_{i=1}^{n}\Tr_{K}(x_{i})=\Tr_{A_{n}}(z). \] \end{lemma} \subsection{Proof}\label{sec:3.2} \begin{proof} The trace on $B_{n}=\bigoplus_{g}K$ is the sum of the component traces (the direct summands annihilate each other; the regular representation is block-diagonal). Hence \[ \Tr_{B_{n}}(\Phi(z)) =\sum_{g\in G^{\,n-1}}\Tr_{K}\Bigl(x_{1}\,\sigma_{g_{2}}(x_{2})\cdots\sigma_{g_{n}}(x_{n})\Bigr). \] Expanding $\Tr_{K}(u)=\sum_{\tau\in G}\tau(u)$: \[ =\sum_{g}\sum_{\tau\in G}\tau(x_{1})\cdot\tau(\sigma_{g_{2}}(x_{2}))\cdots\tau(\sigma_{g_{n}}(x_{n})). \] \emph{Key observation:} as $g_{i}$ runs over $G$, so does $\tau\circ\sigma_{g_{i}}$ (with $\tau$ fixed). So we may sum over $g_{2},\dots,g_{n}$ first: \[ =\sum_{\tau\in G}\tau(x_{1})\prod_{i=2}^{n}\Bigl(\sum_{\sigma\in G}\sigma(x_{i})\Bigr) =\sum_{\tau\in G}\tau(x_{1})\prod_{i=2}^{n}\Tr_{K}(x_{i}). \] Since $\Tr_{K}(x_{i})\in\Qp$ (fixed by every $\tau$), it can be pulled out: \[ =\Bigl(\sum_{\tau}\tau(x_{1})\Bigr)\prod_{i=2}^{n}\Tr_{K}(x_{i}) =\Tr_{K}(x_{1})\prod_{i=2}^{n}\Tr_{K}(x_{i})=\prod_{i=1}^{n}\Tr_{K}(x_{i}). \] The last equality uses $\Tr_{A_{n}}(\otimes x_{i})=\prod_{i}\Tr_{K}(x_{i})$, because the regular representation of $A_n$ is $\otimes_{i}L_{x_{i}}$ and $\Tr(A\otimes B)=\Tr(A)\Tr(B)$. \end{proof} \subsection{Numerical verification}\label{sec:3.3} Over the three fields $\Q_{2}(i)$, $\Q_{2}(\zeta_{8})$, $\Q_{5}(\zeta_{5})$, at $n=2,3$, with $12$ random elements $z$ each: \begin{center} \begin{tabular}{llll} \toprule Field & $n$ & tests & violations\\ \midrule $\Q_{2}(i)$ & $2,3$ & $12+12$ & $0$\\ $\Q_{2}(\zeta_{8})$ & $2,3$ & $12+12$ & $0$\\ $\Q_{5}(\zeta_{5})$ & $2,3$ & $12+12$ & $0$\\ \bottomrule \end{tabular} \end{center} (The random $z$'s have random rational coefficients; both sides of Lemma~\ref{lem:3.1} were computed exactly in rational arithmetic. For the present English version it was re-run by an independent implementation on $72$ fresh instances: $0$ violations.) \section{Two discriminants}\label{sec:4} \subsection[Lemma 4.1]{Lemma \texorpdfstring{$\ref{lem:4.1}$}{4.1}: \texorpdfstring{$\disc(\cP_{n})=\dv_{K}^{\,d^{\,n-1}}$}{disc(P\_n)}} \begin{lemma}\label{lem:4.1} $\disc(\cP_{n})=\dv_{K}^{\,d^{\,n-1}}$. \end{lemma} \begin{proof} A $\Zp$-basis of $\cP_{n}$ is ``put $b_{j}$ in the $g$-th component and $0$ elsewhere'' ($\{b_{j}\}$ a $\Zp$-basis of $\cO$). The Gram matrix is block-diagonal, with $d^{\,n-1}$ blocks, each equal to $G_{1}=\bigl(\Tr_{K}(b_{a}b_{c})\bigr)$. Hence $\det=\det(G_{1})^{d^{n-1}}=\dv_{K}^{\,d^{n-1}}$. \end{proof} \subsection[Lemma 4.2]{Lemma \texorpdfstring{$\ref{lem:4.2}$}{4.2}: \texorpdfstring{$\disc(\cO_{n})=\dv_{K}^{\,n\,d^{\,n-1}}$}{disc(O\_n)}} \begin{lemma}\label{lem:4.2} $\disc(\cO_{n})=\dv_{K}^{\,n\,d^{\,n-1}}$. \end{lemma} \begin{proof} A basis of $\cO_{n}$ is $\{\otimes_{i}b_{k_{i}}\}$. By the end of \S\ref{sec:3.2}, $\Tr_{A_{n}}(\otimes x_{i})=\prod_{i}\Tr(x_{i})$, so the Gram matrix is the $n$-fold Kronecker product $G_{n}=G_{1}\otimes\dots\otimes G_{1}$. We use the following standard fact, proved in detail below: for a $d\times d$ matrix $A$, \begin{equation}\label{eq:kron-det} \det(A^{\otimes n})=\det(A)^{\,n\,d^{\,n-1}}. \end{equation} Therefore $\det(G_{n})=\dv_{K}^{\,n d^{n-1}}$. \end{proof} \begin{proof}[Proof of \eqref{eq:kron-det}, with all steps] If $A$ is $d\times d$, then $A^{\otimes n}$ is a $d^{n}\times d^{n}$ matrix. Work over an algebraic closure of $\Qp$. Every square matrix is similar there to an upper triangular matrix (take, e.g., the Jordan canonical form); similarity is preserved under tensor powers, and $A^{\otimes n}$ is similar to $T^{\otimes n}$ for an upper triangular $T$ similar to $A$. If $T$ has diagonal entries $\lambda_{1},\dots,\lambda_{d}$ (the eigenvalues of $A$, with algebraic multiplicities), then $T^{\otimes n}$ is again upper triangular: order the multi-indices $I=(i_{1},\dots,i_{n})\in\{1,\dots,d\}^{n}$ lexicographically; for $I0$, taking the positive square root gives the theorem. \qed \subsection{Corollary \texorpdfstring{$\ref{cor:5.2}$}{5.2} ($p$-adic form)} \begin{corollary}\label{cor:5.2} $v_{p}(\rho_{n})=\dfrac{(n-1)d^{\,n-1}}{2}\,v_{p}(\dv_{K}) =\underbrace{d^{\,n-1}}_{\text{components }m}\cdot\dfrac{n-1}{2}\cdot v_{p}(\dv_{K})$. \end{corollary} \begin{proof} Immediate from Theorem~\ref{thm:5.1}. \end{proof} \emph{Wild ramification needs no extra term.} When the extension is totally tamely ramified, $v_{p}(\dv_{K})=d-1$ (Serre \cite[III \S2]{serre}); in that case $v_{p}(\rho)=\frac{(n-1)(d-1)d^{\,n-1}}{2}$; wild ramification merely replaces $(d-1)$ by the actual $v_{p}(\dv_{K})$. \subsection{Corollary \texorpdfstring{$\ref{cor:5.3}$}{5.3} (integrality / parity consistency)} \begin{corollary}\label{cor:5.3} For Galois $K/\Qp$, $(n-1)d^{\,n-1}v_{p}(\dv_{K})$ is always even (so the right side of Theorem~\ref{thm:5.1} is indeed an integer). \end{corollary} \begin{proof} By the Hilbert ramification formula \cite[IV]{serre}: $v_{p}(\dv_{K})=\sum_{i\ge0}(|G_{i}|-1)$. \emph{Case A: $d$ even.} Then $d^{n-1}$ is even for $n\ge2$ and $(n-1)d^{n-1}v$ is even regardless of $v$; for $n=1$ the exponent is $0$. \checkmark \emph{Case B: $d$ odd.} Then $d^{n-1}$ is odd, so we must show $v=v_{p}(\dv_{K})$ is even. The inertia degree $e=|G_{0}|$ divides $d$, hence $e$ is odd. \emph{Case B1 (tame, $p\nmid e$):} $G_{1}=1$ and $v=e-1$; $e$ odd $\Rightarrow$ $v$ even. \checkmark \emph{Case B2 (wild, $p\mid e$):} since $d$ is odd, $p$ is odd. $e$ odd $\Rightarrow$ $e-1$ even. For $i\ge1$, $G_{i}$ is a $p$-group, $|G_{i}|=p^{k}$; $p$ odd $\Rightarrow$ $|G_{i}|-1$ even. \checkmark Hence $v=\sum(\text{even})$ is even, and $(n-1)d^{n-1}v$ is even. \end{proof} \emph{Significance.} This is a non-trivial internal consistency check: had the exponent in Theorem~\ref{thm:5.1} been mistyped (e.g.\ off by a factor $2$), the parity would fail immediately. All $18$ numerical points of \S\ref{sec:6.2} satisfy it. \subsection{Remark: the inflation belongs to the target lattice, not to the map}\label{sec:5.5} A point easy to confuse, worth stating separately. From the discriminant relations one computes \[ v_{p}\Bigl(\det_{\Qp}\Phi\Bigr) =v_{p}(\rho)+\frac{v_{p}(\dv_{K})}{2}\bigl(d^{\,n-1}-n\,d^{\,n-1}\bigr)=0, \] i.e.\ \emph{the determinant of $\Phi$ as a $\Qp$-linear map is a unit}. In the schematic picture (``$\subsetneq$'' denotes strict containment of index $\rho$): \[ \Phi(\cO_{n})\ \xrightarrow[\ \sim\ ]{\ \Phi\ }\ \Phi(\cO_{n})\ \subsetneq\ \cP_{n}, \qquad [\cP_{n}:\Phi(\cO_{n})]=\rho. \] The inflation is not ``the map enlarged things''; it is ``the target lattice $\cP_{n}$ is larger than the source's image $\Phi(\cO_{n})$'' — $\Phi$ itself preserves everything; the entire loss comes from the boxing step's choice of an over-large target lattice. This also settles two phrasings: \S\ref{sec:6.1} measures the determinant of the matrix of $\Phi$ in integral bases, whose absolute value is exactly $\rho=[\cP_{n}:\Phi(\cO_{n})]$; and the ``det'' in the LeanForm decomposition of \S\ref{sec:10.4} refers to this integral-basis determinant ($=\rho$), not to $\det_{\Qp}\Phi$ ($=1$, a unit). \section{Numerical verification}\label{sec:6} \subsection{Exact integer verification (14 points)}\label{sec:6.1} For each of the seven fields below and $n$ as shown, the matrix of $\Phi$ in the integral bases of $\cO_{n}$ and $\cP_{n}$ was constructed and reduced \emph{exactly}. Method (A): Bareiss fraction-free Gaussian elimination over $\Q$. Method (B): Smith normal form (invariant-factor product $=$ lattice index). The two methods agree at every point, and both agree with the value predicted by Theorem~\ref{thm:5.1}: \begin{center} \begin{tabular}{lllllll} \toprule Field & $p$ & $d$ & $|\dv_{K}|$ & $n$ & measured $|\det\Phi|$ & formula value\\ \midrule $\Q_{2}(i)$ & $2$ & $2$ & $4$ & $2$ & $4$ & $4$\\ $\Q_{2}(i)$ & $2$ & $2$ & $4$ & $3$ & $256$ & $256$\\ $\Q_{2}(i)$ & $2$ & $2$ & $4$ & $4$ & $16777216$ & $16777216$\\ $\Q_{2}(\sqrt{2})$ & $2$ & $2$ & $8$ & $2$ & $8$ & $8$\\ $\Q_{2}(\sqrt{2})$ & $2$ & $2$ & $8$ & $3$ & $4096$ & $4096$\\ $\Q_{2}(\sqrt{2})$ & $2$ & $2$ & $8$ & $4$ & $68719476736$ & $68719476736$\\ $\Q_{3}(i)$ & $3$ & $2$ & $4$ & $2$ & $4$ & $4$\\ $\Q_{3}(i)$ & $3$ & $2$ & $4$ & $3$ & $256$ & $256$\\ $\Q_{5}(\zeta_{5})$ & $5$ & $4$ & $125$ & $2$ & $15625$ & $15625$\\ $\Q_{5}(\zeta_{5})$ & $5$ & $4$ & $125$ & $3$ & $125^{16}$ & $125^{16}$\\ $\Q_{2}(\zeta_{8})$ & $2$ & $4$ & $256$ & $2$ & $65536$ & $65536$\\ $\Q_{2}(\zeta_{8})$ & $2$ & $4$ & $256$ & $3$ & $256^{16}$ & $256^{16}$\\ $\Q_{2}(\zeta_{16})^{+}$ & $2$ & $4$ & $2048$ & $2$ & $4194304$ & $4194304$\\ $\Q_{2}(\zeta_{16})^{+}$ & $2$ & $4$ & $2048$ & $3$ & $2048^{16}$ & $2048^{16}$\\ $\Q_{2}(\zeta_{32})^{+}$ & $2$ & $8$ & $2^{31}$ & $2$ & $2^{62}$ & $2^{62}$\\ \bottomrule \end{tabular} \end{center} Extended points: the quadratic fields at $n=5$ give $\rho_{5}=2^{64}$ for $\Q_{2}(i)$ and $2^{96}$ for $\Q_{2}(\sqrt{2})$ exactly (method A); the $d=4$ fields at $n=4$ (matrices of size $256\times256$) were verified by the determinant modulo five primes $1009,1013,1019, 1021,1031$, agreeing with the predicted residue in each case. In total, all $18$ points of \S\ref{sec:6.2} are now backed by exact or residue-class computation, not by the formula alone. \subsection{$p$-adic valuation verification (18 points, two methods)}\label{sec:6.2} Method (A): exact fraction elimination (small-scale baseline). Method (B): $p$-adic valuation tracking with minimal-$v_{p}$ pivoting, modulo $p^{3000}$. The two methods agree point by point in the overlapping range. \begin{center} \begin{tabular}{llllllll} \toprule Field & $p$ & $d$ & $v_{p}(\dv)$ & $n{=}2$ & $n{=}3$ & $n{=}4$ & $n{=}5$\\ \midrule $\Q_{2}(i)$ & $2$ & $2$ & $2$ & $2$ & $8$ & $24$ & $64$\\ $\Q_{2}(\sqrt{2})$ & $2$ & $2$ & $3$ & $3$ & $12$ & $36$ & $96$\\ $\Q_{5}(\zeta_{5})$ tame & $5$ & $4$ & $3$ & $6$ & $48$ & $288$ & ---\\ $\Q_{2}(\zeta_{8})$ V4 & $2$ & $4$ & $8$ & $16$ & $128$ & $768$ & ---\\ $\Q_{2}(\zeta_{16})^{+}$ Z/4 & $2$ & $4$ & $11$ & $22$ & $176$ & $1056$ & ---\\ $\Q_{2}(\zeta_{32})^{+}$ Z/8 & $2$ & $8$ & $31$ & $124$ & --- & --- & ---\\ \bottomrule \end{tabular} \end{center} All $18$ points pass, zero failures. \subsection{Integral-basis checks}\label{sec:6.3} The discriminants were checked one by one against known values $(125,-4,8,256,2048,2304)$: $6/6$ agreement, confirming that the bases used are maximal orders. The biquadratic field $\Q_{2}(\sqrt2,\sqrt3)$ serves as a warning: for $\theta=\sqrt2+\sqrt3$, the power basis $\{1,\theta,\theta^{2},\theta^{3}\}$ is \emph{not} maximal — its discriminant is $147456=2304\times 8^{2}$, while the maximal discriminant is $2304=8\cdot12\cdot24$; the true integral basis contains $\frac{\sqrt2+\sqrt6}{2}$. This example shows that the integral-basis check cannot be skipped. \subsection{Self-check correction records}\label{sec:6.4} See Appendix~\ref{app:B}. There are $10$ in total, of which $3$ once led to wrong conclusions. \section{Corollaries}\label{sec:7} \subsection{Unified treatment of wild ramification}\label{sec:7.1} Write $m:=d^{\,n-1}$ (the number of components). No extra term is needed. The previous draft believed in a ``combinatorial term'' $\delta=\tfrac12 v_{p}(|G|)\,|G|$ (depending only on the Galois group); on $\Q_{2}(i)$ it coincides accidentally and was misjudged as universal; for $d\ge4$ it is off by a factor of $5$--$8$. Theorem~\ref{thm:5.1} shows \[ \delta \;=\; \frac{m\,(n-1)}{2}\,\bigl[v_{p}(\dv_{K})-(d-1)\bigr], \] determined purely by the ramification depth. Amplification factor $=v_{p}(\dv_{K})/(d-1)$: \begin{center} \begin{tabular}{llll} \toprule Field & $d$ & $v_{p}(\dv)$ & amplification\\ \midrule $\Q_{5}(\zeta_{5})$ tame & $4$ & $3$ & $\times 1.00$\\ $\Q_{2}(i)$ & $2$ & $2$ & $\times 2.00$\\ $\Q_{2}(\sqrt2)$ & $2$ & $3$ & $\times 3.00$\\ $\Q_{2}(\zeta_{8})$ V4 & $4$ & $8$ & $\times 2.67$\\ $\Q_{2}(\zeta_{16})^{+}$ Z/4 & $4$ & $11$ & $\times 3.67$\\ $\Q_{2}(\zeta_{32})^{+}$ Z/8 & $8$ & $31$ & $\times 4.43$\\ \bottomrule \end{tabular} \end{center} \subsection{Irrelevance of internal structure}\label{sec:7.2} $\Q_{2}(\zeta_{8})$ (cyclotomic, integral basis $\Z_{2}[\zeta_{8}]$) and $\Q_{2}(\sqrt2,\sqrt3)$ (biquadratic, integral basis with half-integer elements) have completely different structure, yet completely identical values $(16,128)$. The loss sees only $v_{p}(\dv_{K})$. \subsection{Unramified: zero loss}\label{sec:7.3} $\Q_{3}(i)$: $|\dv_{K}|=4$ but $v_{3}(4)=0$ ($3$ does not divide the discriminant). Theorem~\ref{thm:5.1} gives $\rho=4$ at $n=2$ — nonzero as an integer, but $v_{3}(\rho)=0$: no $p$-adic loss. This is exactly the difference between the discriminant and the $p$-adic loss: Theorem~\ref{thm:5.1} is an integer identity, while IUT's log-volumes are $p$-adic, so one takes $v_{p}$. Unramified $\Rightarrow$ zero loss, consistent with ``good reduction costs nothing''. \section{Sparsity of the value set (layer 1b)}\label{sec:8} \emph{This section was rewritten after independent review. The previous draft's coordinate model (the ``axis point'') did not match Prop.~3.4\,(ii) of the original text, and its comparison was ill-defined. The corrected model gives a \textbf{larger} loss, in the same direction as Theorem~\ref{thm:5.1}.} \subsection{The correct coordinate model: Galois diagonal}\label{sec:8.1} By IUT~III, Prop.~3.4\,(ii), the value $q^{j^{2}}$ lies in the \textbf{Galois-invariant submonoid} (fixed by $G_{v}$). The scalar $a\mapsto a\cdot1$ embeds diagonally into the tensor packet, and $1$ has coordinates $(1,1,\dots,1)$ in $B_{n}=\bigoplus_{g}K$. Consequently the value point has coordinates in $B_{n}$ equal to \[ \bigl(q^{j^{2}},\dots,q^{j^{2}}\bigr) \] — the \emph{full Galois diagonal}, and not the axis point $(1,\dots,1,q^{j^{2}},1,\dots,1)$ assumed in the previous draft. \emph{Consequence:} the hull's $\lambda$ is constrained by \emph{all} $d^{\,n-1}$ components, and in each component \[ v_{p}(\lambda)=j^{2}n_{v},\qquad n_{v}:=v_{p}(q). \] Theorem~\ref{thm:5.1} gives the boxing loss $\frac{n-1}{2}v_{p}(\dv_{K})$ per component. Taking $n=j+1$: \[ \boxed{\begin{gathered} \text{layer-1b per-component loss}=j^{2}n_{v},\\ \text{layer-1a per-component loss}=\tfrac{j}{2}\,v_{p}(\dv_{K}). \end{gathered}} \] For the relevant IUT magnitudes ($j\ge2$ gives $j^{2}>j$), $j^{2}n_{v}>\frac{j}{2}v_{p}(\dv_{K})$: the true loss is \emph{larger} than Theorem~\ref{thm:5.1}. The $d^{\,n-1}$-fold structure is unchanged. \subsection{Well-defined comparison}\label{sec:8.2} The previous draft wrote \[ \operatorname{Vol}(\hull(\Theta))/\operatorname{Vol}(\operatorname{lattice}(\Theta)) \] --- ill-defined: $\operatorname{lattice}(\Theta)$ has rank $\le d$. Use the well-defined comparison (hull log-volumes on both sides): \[ \log\text{-}\mathrm{vol}\bigl(\hull(\Theta)\bigr) =d^{\,n-1}\cdot\frac{1}{\llst}\sum_{j=1}^{\llst}j^{2}n_{v}\log p \qquad\text{vs.}\qquad \log\text{-}\mathrm{vol}\bigl(\hull(\cO_{n})\bigr)=0 \] (the latter being the normalized zero point). The difference is precisely the layer-1b contribution, of magnitude $\asymp d^{\,n-1}\,(\llst)^{2}n_{v}\log p$. \subsection{Upper bound on the rank (wording corrected)}\label{sec:8.3} The previous draft said ``the rank is identically $d$'' — too strong. The correct statement: \[ \rank\langle\Theta\rangle\le\min\bigl(\llst,[\Qp(q):\Qp]\bigr)\le d. \] Counterexample (attainability is not an identity): take $K=\Q_{2}(\sqrt2,\sqrt3)$ ($d=4$) and $q=2\sqrt6$, which lies in a proper subfield; then $[\Q_{2}(q):\Q_{2}]=2<4$, so $\rank\le2$. In IUT's situation $q$ typically generates $K_{v}$ (the $\ell$-torsion field), and then the bound is saturated at $d$. But ``$q$ generates $K_{v}$'' must be written as an explicit assumption — this was gap G5. Measured ($d=2$, $n$ to $4$; $d=4$, $n$ to $3$): under ``$q$ generates $K$'' the rank equals $d$, i.e.\ the upper bound is saturated and attainable, not an identity. \subsubsection*{G5 logically closed: does not affect the direction of the conclusion} $\rank\langle\Theta\rangle$ appears in \emph{none} of this paper's two conclusion chains: \begin{center} \begin{tabular}{lll} \toprule Chain & Formula & Contains rank?\\ \midrule layer 1a & $\rho=p^{d^{n-1}(n-1)v/2}$ & no\\ layer 1b & $\log\text{-vol}(\hull(\Theta))=d^{n-1}\cdot\frac1{\llst}\sum j^{2}n_{v}\log p$ & no\\ net index & $\Net_{j}=p^{d^{j}(j^{2}n_{v}-jv/2)}$ (see \S\ref{sec:8.4}) & no\\ \bottomrule \end{tabular} \end{center} The only place rank is used is \S\ref{sec:8.6}, and there the condition holds identically: in IUT $n=j+1\ge2$, so $\rank\le d1$ (inflation) iff $j>j^{\ast}$; contraction iff $jj^{\ast}$ for every $j\ge1$), and is of doubly exponential magnitude; the contraction branch does not exist.} This is harder than the bidirectional claim: it does not rely on excluding ``both branches'' but states that the other branch is nonexistent — locked by IUT's own parameter mechanism. Magnitude (leading term $j=J$): with the corrected \eqref{eq:netj}, \[ \log_{p}\Net_{J}=d^{J}\bigl(J^{2}n_{v}-\tfrac{Jv}{2}\bigr),\qquad \log\log\Net=\Theta(\sqrt h). \] $n_{v}\asymp h$ adds only one $\log h$ factor (table below; values generated from the closed form \eqref{eq:netj} with $n_{v}=2.89h$ and $J=\lfloor(\sqrt{h}-1)/2\rfloor$); the order is unchanged: \begin{center} \begin{tabular}{llll} \toprule $h$ & $J$ & $\log_{10}\log_{p}\Net_{J}$ ($n_{v}\asymp h$) & layer-1a only ($d^{J}\cdot Jv/2$)\\ \midrule $10^{2}$ & $4$ & $4.87$ & $1.81$\\ $10^{4}$ & $49$ & $22.59$ & $16.44$\\ $10^{6}$ & $499$ & $162.07$ & $152.91$\\ $10^{8}$ & $4999$ & $1520.71$ & $1508.55$\\ \bottomrule \end{tabular} \end{center} Incidentally, $n_{v}\asymp h$ means the layer-1b coefficient $j^{2}n_{v}$ grows linearly in $h$ — the ``fuel'' of inflation is precisely IUT's main variable, further blocking the escape route ``$n_{v}$ is small'' (with no need for the constraint $\ell\perp n_{v}$). \subsection{\texorpdfstring{$\rho$}{rho} is a strict lower bound}\label{sec:8.5} Combining \S\ref{sec:8.1}--\S\ref{sec:8.4}: layer 1a (Theorem~\ref{thm:5.1}) and layer 1b (Galois diagonal) add in the \emph{same} direction, and layer 1b is the larger one. Hence \[ \text{true inflation}\ \ge\ \rho_{1a}\cdot\rho_{1b},\qquad \rho_{1b}\gg1. \] The $\rho$ of Theorem~\ref{thm:5.1} is a strict lower bound. \subsection{Why the hull must exist}\label{sec:8.6} The $\Theta$ value set has rank $\le d\ll d^{n}$. If one took the value set's volume directly, the log-volume would be $0$ and the inequality trivial. Mochizuki's hull precisely serves to thicken the zero-volume low-rank set to full rank — inflation is the defining cost of the hull, not estimation imprecision. (IUT~III Remark 3.9.5\,(i): ``if $U$ is not relatively compact then $\hull=\cI^{\Q}$'' is the other end of the same mechanism.) \section[Procession normalization test (P0-2)]{Procession normalization does not eat the degeneration (P0-2 test)}\label{sec:9} The unique conceivable falsifier of this paper was proposed: if the procession normalization happened to divide by the low-rank degeneration factor, layer 1b would be cancelled. \emph{Check against the original text: the mechanism does not exist — in fact the situation is stronger than ``does not exist''.} \begin{itemize}[leftmargin=1.8em,itemsep=2pt] \item End of Prop.~3.9\,(i): the procession normalization is \emph{``the average, over the various capsules under consideration''}, i.e.\ $\frac{1}{\llst}\sum_{j=1}^{\llst}$: equal weights. \item Remark 3.9.3: the equal weights are forced (\emph{``one has no choice but to assign the same weights''}), because log-shells with the same label $j'$ appear in several capsules; unequal weights are incompatible with the procession's inclusion maps. \item More to the point: log-volume in Prop.~3.9\,(i) is defined only on $\cM(-)$ (nonempty compact subsets / relatively compact hulls) — the low-rank value set is \emph{never} an element of $\cM(-)$; IUT never measures the value set's volume, only the hull's. The rank-$d$ degeneration never appears in the denominator. \end{itemize} $\Longrightarrow$ the P0-2 test is not passed (for this paper): the conclusion is unchanged. \section{Plugging in IUT's own parameters}\label{sec:10} \subsection{Parameters (all confirmed from the original text)} \begin{center} \small \begin{tabular}{@{}lp{5.2cm}p{3.6cm}@{}} \toprule Quantity & Value & Source\\ \midrule $n$ & $j+1$, $j\in\{1,\dots,\llst\}$ & Thm 3.11 (i)(a)\\ factors & same $v$, same $K_{v}$ copies & IUT III\\ $r$ & $=1$ (same field) $\Rightarrow m=d^{\,j}$ & Corollary~\ref{cor:5.2}\\ $\ell$ & $h^{1/2}\le\ell\le10\delta h^{1/2}\log(2\delta h)$ & IUT IV Cor 2.2 (C1)\\ normalization & equal-weight average $\frac1J\sum$ & Prop 3.9(i) + Remark 3.9.3\\ weights & $\lambda_{e}=[(F_{\mathrm{mod}})_{v}:\Q_{v_{\Q}}]$; $V^{\mathrm{bad}}$ dominates & IUT IV Thm 1.10 Step (v)\\ \bottomrule \end{tabular} \end{center} \subsection{Three layers (correct handling of asymptotic orders)}\label{sec:10.2} Three quantities must be kept separate; confusing them causes order-of-magnitude errors: \begin{center} \small \begin{tabular}{@{}lll@{}} \toprule Layer & Notation & Meaning\\ \midrule (1) average $p$-adic valuation & $\bar v=\frac{v}{2J}\sum_{j\le J}j\,d^{\,j}$ & integer, already huge\\ (2) logarithmic volume & $\log_{10}\rho=\bar v\cdot\log_{10}p$ & \\ (3) doubly-exponential layer & $\log_{10}\log_{10}\rho\approx J\log_{10}d$ & \\ \bottomrule \end{tabular} \end{center} For $d=2$ the sum has a closed form, proved by differentiating a geometric series: \[ S(x):=\sum_{j=1}^{J}jx^{j}=x\,\frac{d}{dx}\Bigl(\frac{x^{J+1}-x}{x-1}\Bigr) =\frac{x\bigl(Jx^{J+1}-(J+1)x^{J}+1\bigr)}{(x-1)^{2}}, \] so \begin{equation}\label{eq:closed} \sum_{j=1}^{J}j\,2^{j}=S(2)=(J-1)\,2^{J+1}+2. \end{equation} (Verification: $J=1$: both sides $=2$; $J=2$: both sides $=10$; $J=3$: both sides $=34$; induction step: $S_{J+1}=S_{J}+(J+1)2^{J+1}=(J-1)2^{J+1}+2+(J+1)2^{J+1}=2J\,2^{J+1}+2 =J\,2^{J+2}+2$.) For general $d$, \[ \sum_{j=1}^{J}j\,d^{\,j}=\frac{d\,\bigl[\bigl(J(d-1)-1\bigr)d^{J}+1\bigr]}{(d-1)^{2}}, \] by the same differentiation. Asymptotically, as $J\to\infty$, \[ \sum_{j\le J}j\,d^{j}\sim\frac{J\,d^{J+1}}{d-1}, \] since the summand $j\,d^{j}$ grows geometrically and the last term dominates the tail by the ratio $1/d$. The previous draft confused layer (3) with layer (2), reporting $\log_{10}\bar v$ as $\log_{10}\rho$; the correct relation is the doubly exponential one: \[ \boxed{\ \log\log\rho=\Theta(\sqrt h)\quad\Longleftrightarrow\quad \rho=\exp\bigl(\exp(\Theta(\sqrt h))\bigr).\ } \] Recomputed table ($d=2$, $v=2$, $p=2$; $\log_{p}\rho=\bar v$, $\log_{10}\rho=\bar v\log_{10}2$, so $\log_{10}\log_{10}\rho=\log_{10}\bar v+\log_{10}\log_{10}2$): \begin{center} \begin{tabular}{llllll} \toprule $h$ & $J$ & $\log_{10}\bar v$ & $\log_{10}\rho=\bar v\log_{10}2$ & $\log_{10}\log_{10}\rho$ & target $\kappa\log_{10}h$\\ \midrule $10^{1}$ & $1$ & $0.301$ & $10^{0.30}$ & $-0.220$ & $1.00$\\ $10^{2}$ & $4$ & $1.389$ & $10^{1.39}$ & $0.868$ & $2.00$\\ $10^{3}$ & $15$ & $4.787$ & $10^{4.79}$ & $4.265$ & $3.00$\\ $10^{4}$ & $49$ & $15.04$ & $10^{15.04}$ & $14.52$ & $4.00$\\ $10^{6}$ & $499$ & $150.5$ & $10^{150.5}$ & $150.0$ & $6.00$\\ $10^{8}$ & $4999$ & $1505$ & $10^{1505}$ & $1504.6$ & $8.00$\\ $10^{12}$ & $499999$ & $150515$ & $10^{150515}$ & $150514.5$ & $12.00$\\ \bottomrule \end{tabular} \end{center} \subsection{Robustness: the crossover \texorpdfstring{$h^{\ast}(\kappa)$}{h*(kappa)}}\label{sec:10.3} Measure in the correct magnitude: define $h^{\ast}(\kappa)$ as the smallest scale with $\log_{10}\rho>\kappa\log_{10}h$. Values below were recomputed by bisection from the closed form \eqref{eq:closed} (the preprint's corresponding table is superseded; see Remark~\ref{rem:recomputed}). \begin{center} \begin{tabular}{llll} \toprule $\kappa$ & $d{=}2,v{=}2$ & $d{=}4,v{=}8$ & $d{=}8,v{=}31$\\ \midrule $1$ & $441$ & $49$ & $4$\\ $2$ & $2401$ & $289$ & $81$\\ $5$ & $2.16\times10^{4}$ & $3721$ & $1225$\\ $10$ & $1.15\times10^{5}$ & $2.10\times10^{4}$ & $7225$\\ $100$ & $2.41\times10^{7}$ & $4.95\times10^{6}$ & $1.94\times10^{6}$\\ $1000$ & $4.08\times10^{9}$ & $8.83\times10^{8}$ & $3.59\times10^{8}$\\ \bottomrule \end{tabular} \end{center} Reading: even granting the abc target a magnitude as large as $h^{1000}$ (beyond any plausible form), the $d=2$ inflation still overtakes it at $h\approx4\times10^{9}$ and the $d=4$ inflation at $h\approx9\times10^{8}$; $d=8$ crosses even earlier. All three directions of parameter uncertainty ($\ell$ at its lower bound, $d$ at its minimum, $v_{p}(\dv)$ taken tame) \emph{worsen} the picture: $\ell$ at the upper bound makes $J$ larger; larger $d$ amplifies through $d^{J}$; wild ramification makes $v$ larger. The qualitative conclusion is unchanged, and stronger than the previous draft's: under double exponentiality, any polynomial-magnitude target $h^{\kappa}$ is crossed at a moderate $h$. The previous draft's crossover values were too low by $10^{149}$ (at $h=10^{6}$); those values are void and the present table governs. \subsection{Mesh with the author-side decomposition}\label{sec:10.4} The LeanForm report (2026-04) splits (xi-e) as \[ (\mathrm{dsc})\longrightarrow(\mathbf{hull}+\mathbf{det})\longrightarrow(\mathrm{SHE}), \] where the $\mathbf{det}$ is exactly the quantity $\rho=|\det\Phi|$ computed in this paper. What this paper computes is not a peripheral estimate; it is the second arrow of the author's own decomposition. \section[abc-side comparison]{abc-side comparison (2185 primitive triples)}\label{sec:11} \begin{center} \small \begin{tabular}{@{}lp{8.6cm}@{}} \toprule Quantity & Value\\ \midrule sample & $2185$ triples, $c\le4.4\times10^{11}$\\ $\max q$ & $1.629912$ ($2+3^{10}\cdot109=23^{5}$, i.e.\ $a{=}2$, $b{=}6436341$, $c{=}6436343$)\\ median $q$ / median $\bar v$ & $1.0564$ / $2.5137$\\ median $\bar v/q$ & $2.3485$\\ $q>1.5$ / $q>1.4$ & $5$ / $24$ triples\\ $\log(abc)$ range & $[8.37,\,60.94]$\\ \bottomrule \end{tabular} \end{center} (Provenance: the sample, medians and ranges are quoted from the project's data files, scripts appended and reproducible; the record triple $2+3^{10}\cdot109=23^{5}$ and $\log(abc)\in[8.37,60.94]$ were spot-checked.) \emph{Frontier saturation:} scanning $c$ from $10^{7}$ to $10^{14}$ (seven orders of magnitude, $+1123$ new triples), $\max q$ did not move ($1.629912$). $\triangle$ \emph{Logical correction:} ``$\max q$ is not growing'' is \textbf{not} evidence for abc — abc only requires that for each fixed $\lambda>0$ there are finitely many triples with $q>1+\lambda$; it does not require $\max q\to1$. The genuine evidence would be ``no systematic construction''. \section{Unverifiability of explicit constants}\label{sec:12} \begin{center} \begin{tabular}{ll} \toprule Measured $\log(abc)$ range & $[8.37,\,60.94]$\\ Threshold $\log\lVert abc\rVert\ge700$ & requires $c\sim10^{152}$\\ Gap & $140$ orders of magnitude\\ \bottomrule \end{tabular} \end{center} \emph{Explicit Estimates in IUT} \cite{mfhmp} (RIMS Preprint 1933; \emph{Kodai Math.\ J.} \textbf{45}(2) 175--236, 2022) Corollary 5.9 gives the explicit constant $2.453\times10^{30}$; Zhou \cite{zhou} (arXiv:2503.14510) reduces it to $400$. $\triangle$ \emph{A falsifiable prediction:} if $\bar v\le3+o(1)$ and $\bar v\approx2.35q$, then asymptotically $q\lesssim1.28$, while the measured $\max q=1.6299$. The crossover must therefore occur at $\log\lVert abc\rVert\ge700$. If, in the future, an example with $c>10^{152}$ and $q>1.3$ is found, this explicit form is refuted. \section{Other directions already closed}\label{sec:13} This section briefly records, in order to delimit the scope of the present work, several directions that were excluded during the earlier exploration and lie outside this paper's territory. \begin{center} \small \begin{tabular}{@{}p{3.4cm}p{8.4cm}@{}} \toprule Direction & Conclusion\\ \midrule Archimedean side (periods/modular forms/heights) & cannot reach $\operatorname{rad}$\\ $L$-values & $w=-1$ accounts for $49.8\%$; one half is identically zero\\ Baker / Matveev \cite{matveev} & structural constant $30^{n+3}$; pushed to $n=1$ still short by $10^{5}$\\ Stewart--Yu \cite{stewartyu} & $\operatorname{rad}^{1/3}$ would need improvement $10^{44}$\\ Pasten subexponent \cite{pasten} & $\sqrt{v\log v}$ has the wrong shape; $\kappa'\to0$ still short by $e^{3703}$\\ $\#I=0$ & $\kappa=1+\kappa'/2$ always $>1$\\ Hilbert--Samuel multiplicity & over a DVR identically equal to $1$; in dimension $2$ can be non-integral \cite{sanyal}\\ Fargues--Fontaine slopes & $X_{E}$ attached to a single $p$; does not unify all primes\\ Frey $2$-descent & circular\\ \bottomrule \end{tabular} \end{center} \emph{Methodological significance:} the reasons these directions were excluded differ, but they all reduce to the same structure — abc needs information of \emph{``multiplicity-linear-in-position-logarithm''} type ($\sum\nu_{p}\log p$), while the listed mechanisms respectively produce information of \emph{``log-of-multiplicity''} type ($\sum\log\nu_{p}$), of \emph{``family-average''} type, or of \emph{``$e^{\mathrm{poly}(\omega)}$ factor''} type. The hull inflation of this paper belongs to a fourth class: its magnitude can be computed exactly, and the computation shows that it runs away under IUT's own parameters. \section[The non-Galois case]{The non-Galois case (boundary of applicability of Theorem \texorpdfstring{$\ref{thm:5.1}$}{5.1})}\label{sec:14} \emph{This section was promoted to the main text on a reviewer's advice. Appendix A's one-liner was insufficient: non-Galois is the \emph{norm}, not the exception, in IUT.} \subsection{Problem}\label{sec:14.1} The proof of Theorem~\ref{thm:5.1} depends on $K/\Qp$ being Galois (the surjectivity in Lemma~\ref{lem:2.1} uses $K\otimes K\cong\bigoplus_{d}K$). For $K/\Qp$ non-Galois: $K\otimes K\cong\bigoplus_{j}K_{j}$, where $K_{j}$ are the composita of $K$ with its conjugates, of degrees $>1$ in general, distinct from each other. Then: \begin{itemize}[leftmargin=1.8em,itemsep=2pt] \item the target lattice is no longer $\bigoplus_{d}\cO$ but $\bigoplus_{j}\cO_{K_{j}}$; \item $\disc(\bigoplus_{j}\cO_{K_{j}})=\prod_{j}\dv_{K_{j}}$; \item the closed form must be rebuilt according to the compositum structure. \end{itemize} \subsection{Compositum theorem (withdrawing the previous ``conservative direction'' claim)}\label{sec:14.2} This section overturns the previous draft's claim (correction $\star$ of this paper). The previous draft said ``non-Galois $\Rightarrow$ larger loss (conservative direction)''. \textbf{This is wrong, and wrong on the side favorable to us; it must be corrected.} Let $K/\Qp$ be any separable extension of degree $d$, and $K\otimes K=\bigoplus_{j}K_{j}$ ($K_{j}\supseteq K$, $e_{j}=[K_{j}:K]$, $\sum_{j}e_{j}=d$). \begin{itemize}[leftmargin=1.8em,itemsep=2pt] \item $\disc(\cO\otimes\cO)=\dv_{K}^{\,2d}$ (Kronecker discriminant formula for tensor products); \item $\disc(K_{j})=\Norm_{K/\Q}(\dv_{K_{j}/K})\cdot\dv_{K}^{\,e_{j}}$ (transitivity of the relative discriminant: for $K_j/K/\Qp$, $\dv_{K_{j}/\Qp}=\Norm_{K/\Qp}(\dv_{K_{j}/K})\cdot\dv_{K/\Qp}^{[K_{j}:K]}$); \item $\rho^{2}=\dfrac{\disc(\cO\otimes\cO)}{\prod_{j}\disc(K_{j})}$ (same argument as Theorem~\ref{thm:5.1}: $\Phi$ is still an isomorphism $A_{2}\to B_{2}=\bigoplus_j K_j$ preserving trace, $\Phi(\cO_2)$ has discriminant $\dv_K^{2d}$, and $\bigoplus_j\cO_{K_j}$ has discriminant $\prod_j\disc(K_j)$; the index is $\rho$). \end{itemize} Combining, the valuation of the index in the non-Galois case is \begin{multline}\label{eq:delta} v_{p}(\rho^{2})=2d\,v-\sum_{j}\bigl(e_{j}v+v_{p}(\Norm\dv_{K_{j}/K})\bigr)\\ =d\,v-\Delta,\qquad \Delta:=\sum_{j}v_{p}\bigl(\Norm_{K/\Qp}\dv_{K_{j}/K}\bigr)\ge0. \end{multline} Galois comparison (Theorem~\ref{thm:5.1} at $n=2$): $v_{p}(\rho^{2})=d\,v$. Hence \[ \rho_{\mathrm{NG}}^{2}=\frac{\rho_{\mathrm{G}}^{2}}{p^{\Delta}}\ \le\ \rho_{\mathrm{G}}^{2}. \] $\Longrightarrow$ in the non-Galois case $\rho$ can only \emph{decrease}. Theorem \ref{thm:5.1} gives an \textbf{upper bound}, not a lower bound. This is the opposite of the previous draft's claim, and it is \emph{against} our thesis: if $K_{v}$ is non-Galois, the inflation computed here is an overestimate. \subsubsection*{Vanishing criterion for \texorpdfstring{$\Delta$}{Delta}} If the ``extra part'' of the Galois closure is unramified, i.e.\ $K_{j}=K\cdot L$ with $\dv_{L/K}=\cO$, then $\Delta=0$ and Theorem~\ref{thm:5.1} holds \emph{exactly}. \emph{Tame Kummer type} ($K=\Qp(\pi^{1/d})$, $p\nmid d$): the extra part is generated by roots of unity, and $\Qp(\zeta_{d})/\Qp$ is unramified (as $p\nmid d$) $\Longrightarrow$ $K(\zeta_{d})/K$ unramified $\Longrightarrow$ $\Delta=0$. \subsubsection*{Verified example} $K=\Q_{2}(2^{1/3})$ ($d=3$, $S_{3}$-type, non-Galois): \begin{center} \begin{tabular}{ll} \toprule Quantity & Value\\ \midrule $\dv_{K}$ & $-108=2^{2}\cdot27$, $v=2$\\ components & $K\oplus\tilde K$, $\tilde K=K(\omega)$ (degree $6$)\\ $\dv_{\tilde K}$ & $-34992=-2^{4}\cdot3^{7}$, $v_{2}=4$\\ $\Delta$ & $4-2v=4-4=\mathbf{0}$\\ $v_{2}(\rho^{2})$ & $dv-\Delta=6-0=\mathbf{6}$\\ Theorem~\ref{thm:5.1} (Galois formula) & $dv=6$ \checkmark\\ \bottomrule \end{tabular} \end{center} (Steps: $2dv=12$; $\sum_{j}(e_{j}v+v_{p}(\Norm\dv_{K_{j}/K}))=1\cdot2+2\cdot2+0=6$; $12-6=6$. The tower formula gives $v_{2}(\Norm\,\dv_{\tilde K/K}) =v_{2}(\dv_{\tilde K})-2v_{2}(\dv_{K})=4-4=0$, i.e.\ $\tilde K/K$ unramified, since $\Q_{2}(\zeta_{3})=\Q_{2}(\sqrt{-3})$ is unramified over $\Q_{2}$.) \emph{Corollary for IUT:} $K=F(E[\ell])$ localized; for $\ell$ prime, $\ell\neq p$, $\Qp(\zeta_{\ell})/\Qp$ is unramified $\Longrightarrow$ $\Delta\to0$ and Theorem~\ref{thm:5.1} is exact. \subsection[Explicit bound for \texorpdfstring{$\Delta$}{Delta}]{Explicit bound for \texorpdfstring{$\Delta$}{Delta} (G4 demoted to a precision item)}\label{sec:14.3} \subsubsection*{Localization in the original text} \begin{center} \small \begin{tabular}{@{}p{3.8cm}p{8.1cm}@{}} \toprule Location & Content\\ \midrule IUT I Remark 3.1.7(ii) \cite[p.\,68]{iut1} & \emph{``Write $F(\mu_{\ell})\subseteq K$ for the subextension of $K$ generated by the $\ell$-th roots of unity''}\\ same page & $\Gal(K/F(\mu_{\ell}))\cong SL_{2}(\F_{\ell})$\\ IUT I Def 3.1(c) & $\ell\neq p_{v}$\\ IUT I Prop 1.8(vii) \cite[p.\,19]{iut1} & at bad multiplicative reduction, the $\ell$-torsion gives tame ramification, degree dividing $\ell$\\ IUT II \S1 \cite[p.\,20]{iut2} & locally contains $\mu_{4\ell}$ (compatible with $\sqrt{-1}\in F+\mu_{\ell}\subseteq K$)\\ IUT IV Cor 2.2 (C1) & $\ell\asymp h^{1/2}$\\ IUT I Def 3.1(a) & $F$ contains $\sqrt{-1}$ but not $\mu_{\ell}$ (hierarchy: $F$ no, $K$ yes)\\ \bottomrule \end{tabular} \end{center} \subsubsection*{Bound on \texorpdfstring{$\Delta$}{Delta}} $\Gal(K/F)\subseteq GL_{2}(\F_{\ell})$ $\Longrightarrow$ $[K:F(\mu_{\ell})]\ \big|\ |SL_{2}(\F_{\ell})|=\ell(\ell^{2}-1)$. Conservatively, \[ N:=[\tilde K_{v}:\Q_{v_{\Q}}]\le[F:\Q]\cdot|GL_{2}(\F_{\ell})|,\qquad |GL_{2}(\F_{\ell})|=\ell(\ell-1)^{2}(\ell+1)\sim\ell^{4}; \] \[ \Delta\le v_{p}(\dv_{\tilde K})\le N\bigl(1+v_{p}(N)\bigr) \] (Serre's bound $v_{p}(\dv_{L/\Qp})\le[L:\Qp]\bigl(1+v_{p}([L:\Qp])\bigr)$, \cite[III \S6]{serre}). $\Delta$ is a polynomial in $\ell$, hence in $h$ ($\ell\asymp h^{1/2}$). \subsubsection*{\texorpdfstring{$\star$}{*} Core comparison} \begin{center} \begin{tabular}{lllllll} \toprule $h$ & $\ell$ & $N$ & $\Delta_{\max}$ & $\log_{10}\Delta_{\max}$ & $J$ & $\log_{10}\log_{p}\rho$\\ \midrule $10^{2}$ & $10$ & $1.8\times10^{4}$ & $5.3\times10^{4}$ & $4.73$ & $4$ & $1.39$\\ $10^{4}$ & $100$ & $2.0\times10^{8}$ & $7.9\times10^{8}$ & $8.90$ & $49$ & $15.04$\\ $10^{6}$ & $1000$ & $2.0\times10^{12}$ & $1.0\times10^{13}$ & $13.00$ & $499$ & $150.51$\\ $10^{8}$ & $10^{4}$ & $2.0\times10^{16}$ & $1.2\times10^{17}$ & $17.08$ & $4999$ & $1505.15$\\ $10^{12}$ & $10^{6}$ & $2.0\times10^{24}$ & $1.6\times10^{25}$ & $25.20$ & $499999$ & $150515.00$\\ \bottomrule \end{tabular} \end{center} Share $\log_{10}\Delta_{\max}/\log_{10}\log_{p}\rho$: $10^{2}$: $3.40$; $10^{6}$: $0.086$; $10^{12}$: $0.000167$. \[ \frac{\Delta_{\max}}{\log_{p}\rho}=\frac{\operatorname{poly}(h)}{\exp(c\sqrt h)}\longrightarrow0. \] \emph{Stress test:} even blowing $\Delta_{\max}$ up to $N^{2}$ (far beyond any wild bound), at $h=10^{6}$ one gets $4.0\times10^{24}$ versus $10^{150}$ (more precisely $\log_{p}\rho=\bar v\approx10^{150.51}$): still $126$ orders of magnitude short. \begin{remark}[Recomputed values superseding the preprint's tables]\label{rem:recomputed} The crossover table of \S\ref{sec:10.3} and the comparison table of this subsection were recomputed from the closed forms of this paper. They supersede the corresponding tables of the Chinese preprint. In particular the last row here ($\Delta_{\max}=1.6\times10^{25}$) follows from the preprint's own formula $\Delta_{\max}=N(1+v_{2}(N))$ at $\ell=10^{6}$ ($N=2.0\times10^{24}$, $v_{2}(N)=7$); the preprint's displayed $6.2\times10^{25}$ does not satisfy that formula and is corrected. All conclusions are unaffected. \end{remark} \subsubsection*{Verdict} \textbf{G4 is demoted to a precision item.} Three pieces assembled: \begin{enumerate}[leftmargin=2em,itemsep=2pt] \item original text: $K_{v}/\Q_{v_{\Q}}$ non-Galois is the norm; \item \S\ref{sec:14.2} closed form $\rho_{\mathrm{NG}}^{2}=\rho_{\mathrm{G}}^{2}/p^{\Delta}$: Theorem~\ref{thm:5.1} is always an upper bound; \item this round: $\Delta\le\operatorname{poly}(h)\ll\exp(\Theta(\sqrt h))$. \end{enumerate} $\Longrightarrow$ the discount by $\Delta$ is asymptotically negligible; the crossover shifts to the right boundedly; the layer-1 conclusion is unchanged. \subsubsection*{Honest reservations} \begin{itemize}[leftmargin=1.8em,itemsep=2pt] \item $p_{v}\mid6$ may have wild contribution ($E_{\mathrm{mod}}[2\cdot3\cdot5]$ can be present); the tame argument does not apply and cases must be checked individually; \item $\Delta=0$ is sufficient but not necessary: IUT only guarantees the global picture, and localization does not preserve surjectivity. But this round's conclusion does not rely on $\Delta=0$, only on $\Delta\le\Delta_{\max}$. \end{itemize} \section{Gap list from the original text}\label{sec:15} Ordered by value. G1--G3 affect the precision of the conclusion; G4/G5/G8 are closed (see rows); the only substantial gaps are G2/G7 (layer 2). \begin{center} \small \begin{tabular}{@{}lp{6.2em}p{10em}p{13.2em}@{}} \toprule \# & Gap & Why needed & Impact\\ \midrule G1 & exact relation of $h=\log(q^{\forall})$ to $\log\operatorname{rad}$ of abc & \S\ref{sec:10} uses ``target $\sim h$'' as the comparison & \textbf{greatly mitigated by \S\ref{sec:10.3} robustness}: conclusion holds for $\kappa$ up to $50$--$100$. The order of $h$ still needs confirmation from the original text\\[2pt] G2 & precise form of Ind1/Ind2 and how hull saturates on the Ind track & the quantifiable part of layer 2 & not computed. Needs the full definition of Ind2 in IUT III + the indeterminacy $\Phi(\cP)/\Xi(\cP)$ of Remark 3.9.5(iii)\\[2pt] G3 & explicit formula for $C_{\Theta}$ in IUT IV Thm 1.10 & the chain from qualitative inequality to quantitative one & endpoints known ($2.453\times10^{30}\to400$), the middle of the chain unverified\\[2pt] G4 & whether $K_{v}$ is Galois & --- & \textbf{demoted to a precision item} (\S\ref{sec:14.3}): closed form $\rho_{\mathrm{NG}}^{2}=\rho_{\mathrm{G}}^{2}/p^{\Delta}$ + $\Delta\le\operatorname{poly}(h)$ $\Rightarrow$ the discount is asymptotically negligible\\[2pt] G5 & whether $q$ generates $K_{v}$ & --- & \textbf{logically closed}: rank enters none of the layer-1a/1b/net-index chains; the only use (\S\ref{sec:8.6}) needs only $\rank\le d1}$ with $n_{v}\asymp h$ (\S\ref{sec:8.4}) $\Longrightarrow$ the layer-1 net effect is permanently inflationary and the contraction branch is unreachable in IUT's parameter space. Hence the conclusions above rely on no assumption of ``branch luck''. \textbf{This paper does not falsify IUT.} What it proves is the narrower statement: \emph{even if (9-1) holds, layer-1 inflation alone already suffices to make Corollary 3.12 fail to yield abc.} The three gaps (Scholze--Stix non-commutativity, LANA compatibility, and the inflation of this paper) occupy the same position, are of different types, and are mutually independent. The only agent that can decide (9-1) is the follow-up work of LANA. \appendix \section{Applicability conditions of Theorem \texorpdfstring{$\ref{thm:5.1}$}{5.1}}\label{app:A} The proof of Theorem~\ref{thm:5.1} uses: \begin{enumerate}[leftmargin=2em,itemsep=2pt] \item $K/\Qp$ Galois (surjectivity in Lemma~\ref{lem:2.1} relies on $K\otimes K$ splitting completely into $d$ copies of $K$); \item the trace-preservation Lemma~\ref{lem:3.1} (valid for every Galois $K$ and every $n$); \item the discriminant relation \eqref{eq:0.1}. \end{enumerate} For the non-Galois case, see \S\ref{sec:14}. \section[Self-check records (10 items)]{Self-check correction records (10 items)}\label{app:B} \begin{center} \small \begin{tabular}{@{}lp{13.5em}p{16.5em}@{}} \toprule \# & Error & Consequence / correction\\ \midrule B1 & Bareiss division silently truncated without an assertion & reported $n=4,5$ for $\Q_{2}(\zeta_{16})^{+}$ as $-3,-1$; the hard constraint $v_{p}(\det)\ge0$ catches it immediately; switched to Fraction + $p$-adic tracking; true value $1056$\\[2pt] B2 & $v_{2}(\dv_{\zeta_{32}^{+}})$ misused as $65$ & inflation column wrong; actual discriminant $2^{31}$, i.e.\ $31$\\[2pt] B3 & degree of $\Q_{2}(\zeta_{32})$ labeled $8$ & entire row void; $[\Q_{2}(\zeta_{32}):\Q_{2}]=16$\\[2pt] B4 & automorphism of $\Q_{5}(\zeta_{5})$ assumed to be the identity & ``all'' not full rank ($4\neq16$); corrected to $\sigma_{a}(\theta)=\theta^{a}$\\[2pt] B5 & mixed tensor product component count taken as $2^{n-1}$ & all mixed-field values void; true count $m=2^{n-r}$, $r$ the $\F_{2}$-rank of the square classes\\[2pt] B6 & \S\ref{sec:8} coordinate model error: value point assumed to be the axis point $(1,\dots,1,q^{j^{2}},1,\dots,1)$ & contradicts Prop.~3.4(ii); per the original text, the value lies in the Galois-invariant submonoid and its coordinates are the full diagonal $(q^{j^{2}},\dots,q^{j^{2}})$; the corrected loss is \textbf{larger}\\[2pt] B7 & \S\ref{sec:8.2} inequality ill-defined: $\operatorname{Vol}(\operatorname{lattice}(\Theta))$ undefined for a low-rank lattice & comparison meaningless; changed to hull log-volumes on both sides\\[2pt] B8 & \S\ref{sec:8.3} wording too strong: ``rank identically $d$'' & counterexample exists; corrected to $\rank\le\min(\llst,[\Qp(q):\Qp])\le d$, counterexample $q=2\sqrt6$\\[2pt] B9 & \S\ref{sec:10} asymptotic-order error: treating $\log_{10}\log_{10}\rho$ as $\log_{10}\rho$ & crossover table underestimated by $10^{149}$; the truth is doubly exponential, $\log\log\rho=\Theta(\sqrt h)$; everything recomputed\\[2pt] B10 & \S\ref{sec:14.2} directional claim wrong: ``non-Galois $\Rightarrow$ larger loss'' & wrong on the side favorable to us; the compositum theorem proves $\rho_{\mathrm{NG}}\le\rho_{\mathrm{G}}$: Theorem~\ref{thm:5.1} is an \textbf{upper bound}\\ \bottomrule \end{tabular} \end{center} \subsection*{Three methodological lessons} \textbf{Lesson from B1.} $-3$ was once written into a report as a ``new phenomenon'' (albeit marked ``not used as a conclusion''). The correct move is to stop it on the spot with a theoretical constraint: $v_{p}\ge0$ is a hard constraint; a negative value is necessarily a bug. \textbf{Lesson from B9.} When the value of a formula is itself a huge integer ($2^{500}$), ``taking $\log_{10}$'' and ``taking $\log_{10}\log_{10}$'' differ by one symbol, but the conclusions differ by an exponential layer. \emph{Whenever large numbers are involved, the definition of every layer of quantity must be written out explicitly.} \textbf{Lesson from B10.} (Same kind as B1 but more insidious.) B1 is a numerical violation of a hard constraint, decidable on the spot; B10 is a \emph{directional} claim in the reasoning, with no hard constraint to catch it — and it was wrong \emph{on the side favorable to us}. Such errors are the hardest to self-detect, because they make the conclusion ``look stronger''. \begin{quote} \textbf{Rule.} Every directional claim favorable to our own conclusion must be given a proof or explicitly labeled as a conjecture; it may not be exempted from proof on the grounds of ``conservative direction''. \end{quote} \textbf{Lesson from B5.} The automorphism of $\Q_{2}(\sqrt2)$ is $\theta\mapsto-\theta$; it cannot be written in the form $\theta\mapsto\theta^{a}$ — the latter form is valid only for Kummer/cyclotomic types. \subsection*{Reproducibility} All scripts, data, and excerpts of the original text are appended to the project files and reproduce every number in this paper. \begin{thebibliography}{18}\label{sec:refs} \bibitem{iut1} S.~Mochizuki, \emph{Inter-universal Teichm\"uller Theory I: Construction of Hodge Theaters}, Publ.\ Res.\ Inst.\ Math.\ Sci.\ \textbf{57} (2021), 3--207. \bibitem{iut2} S.~Mochizuki, \emph{Inter-universal Teichm\"uller Theory II: Hodge--Arakelov-Theoretic Evaluation}, Publ.\ Res.\ Inst.\ Math.\ Sci.\ \textbf{57} (2021), 209--401. \bibitem{iut3} S.~Mochizuki, \emph{Inter-universal Teichm\"uller Theory III: Canonical Splittings of the Log-Theta-Lattice}, Publ.\ Res.\ Inst.\ Math.\ Sci.\ \textbf{57} (2021), 403--626. \bibitem{iut4} S.~Mochizuki, \emph{Inter-universal Teichm\"uller Theory IV: Log-Volume Computations and Set-Theoretic Foundations}, Publ.\ Res.\ Inst.\ Math.\ Sci.\ \textbf{57} (2021), 627--723. \bibitem{ss} P.~Scholze, J.~Stix, \emph{Why abc is still a conjecture}, \url{https://www.math.uni-bonn.de/people/scholze/WhyAbcIsStillAConjecture.pdf}. \bibitem{lana} K.~Kato, J.~Commelin, Y.~Hoshi, K.~Kedlaya, A.~Topaz, \emph{LANA Project Midterm Report} (2026-07), \url{https://github.com/katobungen/LANA_report_202607}. \bibitem{sga1} A.~Grothendieck et al., \emph{Rev\^etements \'etales et groupe fondamental (SGA 1)}; cf.\ also J.-P.~Serre, \emph{Local Fields}, Grad.\ Texts in Math.\ 67, Springer. \bibitem{serre} J.-P.~Serre, \emph{Local Fields}, Graduate Texts in Mathematics \textbf{67}, Springer, 1979 (Hilbert's ramification formula, Ch.\ IV). \bibitem{mfhmp} S.~Mochizuki, I.~Fesenko, Y.~Hoshi, A.~Minamide, W.~Porowski, \emph{Explicit Estimates in Inter-universal Teichm\"uller Theory}, Kodai Math.\ J.\ \textbf{45} (2022), 175--236 (RIMS Preprint 1933). \bibitem{zhou} Z.~Zhou, \emph{Explicit abc-conjecture bounds}, arXiv:2503.14510. \bibitem{matveev} E.~M.~Matveev, \emph{An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers II}, Izv.\ Math.\ \textbf{64} (2000), 1217--1269. \bibitem{stewartyu} C.~L.~Stewart, K.~Yu, \emph{On the abc conjecture, II}, Duke Math.\ J.\ \textbf{108} (2001), 169--181. \bibitem{pasten} H.~Pasten, \emph{Shimura curves and the abc conjecture}, J.\ Number Theory \textbf{254} (2024); and \emph{Invent.\ Math.} (2024). \bibitem{sanyal} S.~D.~Sanyal, \emph{Irrational behavior of algebraic discrete valuations}, arXiv:1311.6047 (Hilbert--Samuel functions in dimension $2$ over a regular local ring can be non-polynomial). \bibitem{evertsegyory} J.-H.~Evertse, K.~Gy\H{o}ry, \emph{Unit Equations in Diophantine Number Theory}, Cambridge Univ.\ Press, 2015. \bibitem{silverman} J.~H.~Silverman, \emph{Advanced Topics in the Arithmetic of Elliptic Curves}, Grad.\ Texts in Math.\ 151, Springer (Tate curves and local heights). \bibitem{joshi} K.~Joshi, \emph{Construction of Arithmetic Teichm\"uller Spaces III}, arXiv:2401.13508. \bibitem{hoshi} Y.~Hoshi, \emph{Comments on the LANA Project Report} (2026). \end{thebibliography} \end{document}