% 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}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Dpar}{\Delta_{\mathrm{parity}}} \newcommand{\Dram}{\Delta_{\mathrm{ram}}} \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{Two-branch criterion} (\S\ref{sec:8.4}): the layer-1 net effect is inflationary iff $n_{v}\ge v/2$, with $v\le[F:\Q]\cdot(\ell-1)^{2}$ by Theorem~\ref{thm:tame} (\S\ref{sec:14.3} for $p_{v}\in\{\ell\}\cup\{2,3\}$); in contractionary configurations Corollary 3.12 is uninformative --- either way, no abc. \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. \item \textbf{$C_{\Theta}$-docking} (\S\ref{sec:10.5}): the layer-1 inflation plugs into IUT IV's $C_{\Theta}$-formula as an additive term $\frac{4}{\llst+1}\log\widehat\rho$ on the left of the abc-bound chain (Theorem~\ref{thm:Ctheta}); the corrected inequality fails beyond $h\approx10^{3}$--$10^{4}$ (Corollary~\ref{cor:Ctheta-fail}). \item \textbf{Structural complements} (\S\ref{sec:5}, \S\ref{sec:8.1}, \S\ref{sec:ind.5}): linearity of the loss in intermediate lattices (Corollary~\ref{cor:linear}); the hull as an explicit principal ideal (Remark~\ref{rem:ideal}); the synchronization cross-check for the full-diagonal model; and the docking with the nine obstacles (Ob1)--(Ob9) of IUT III Remark 3.9.5(vii). \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{eighteen 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}\label{sec:2} \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 \begin{corollary}[linearity in intermediate lattices]\label{cor:linear} For every $\Z_{p}$-lattice $\Lambda$ with $\cO_{n}\subseteq\Lambda\subseteq \Phi^{-1}(\cP_{n})$, \[ [\cP_{n}:\Phi(\Lambda)]=\rho_{n}\cdot[\Lambda:\cO_{n}]. \] \end{corollary} \begin{proof} $\disc(\Phi(\Lambda))=\disc(\Lambda)$ by trace preservation (Lemma~\ref{lem:3.1}), and $=[\Lambda:\cO_{n}]^{2}\disc(\cO_{n})$ by (0.1); on the other hand $\disc(\Phi(\Lambda))=[\cP_{n}:\Phi(\Lambda)]^{2}\disc(\cP_{n})$ by Lemma~\ref{lem:4.1}. Compare with Theorem~\ref{thm:5.1}. \end{proof} The boxing of \S\ref{sec:8} sits at the extreme of this spectrum: the loss is \emph{linear} in the thickening, the full hull is its maximal element, and every partial boxing inherits the same leading exponent with a smaller constant. \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 (15 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 $18$ in total, of which $8$ (B1, B9, B10, B12, B15, B16, B17, B18) once affected conclusions or the attribution chain. \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. \begin{remark}[the hull as an explicit ideal]\label{rem:ideal} In the full-diagonal model the hull is an arithmetic divisor: \[ \hull(U_{j})=\bigoplus_{\omega\in G^{\,n-1}}\Bigl(\prod_{|t|} q^{\,j_{|t|}^{2}}\Bigr)\cO_{K_{\omega}},\qquad n=j+1, \] the componentwise principal fractional ideal; the constraint $v_{p}(\lambda)=j^{2}n_{v}$ of \S\ref{sec:8.1} is its valuation, and the layer-$1$a index of Theorem~\ref{thm:5.1} is the index of the tensor-product ideal $\Phi(\cO_{n})$ in $\cP_{n}$. This is exactly the shape of the $\Theta$-pilot objects of IUT III Definition 3.8 (elements of the global realified Frobenioid given by families of fractional ideals): the log-volume formula of \S\ref{sec:8.2} is the valuation of this arithmetic divisor. \end{remark} \emph{Cross-check via conjugation synchronization.} The full-diagonal model is not only read off from Prop.~3.4(ii); it is also forced by the symmetrizing isomorphisms that the multiradial algorithm requires one to apply (IUT III, proof of Prop.~3.4(i), p.\,103: ``this functorial algorithm requires one to apply the compatibility of the given log-link with the $\F_{l}^{\rtimes\pm}$-symmetrizing isomorphisms involved''; cf.\ Remark 1.3.2). These isomorphisms identify the labeled factors of the LGP-monoid across labels \emph{without conjugacy indeterminacy}; a non-diagonal value coordinate would be transported by them to different components for different labels, splitting the single basepoint of the synchronized packet. The synchronization mechanism and the Galois-invariance of Prop.~3.4(ii) thus corroborate each other on the full-diagonal model. \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 $jJ=4$): contractionary. The exceptional primes $p_{v}=\ell$ and $p_{v}\mid6$ are covered by the $\Delta_{\max}$ framework of \S\ref{sec:14.3} and do not affect the criterion. \subsubsection*{Corrected criterion: the two-branch analysis} The earlier ``bidirectional blocking'' and the still earlier ``one-way permanent inflation'' argument are both replaced by the following two-branch analysis (B18, Appendix~\ref{app:B}: the ``permanent inflation'' claim was a directional overclaim). The parameters $n_{v}=v_{p}(q_{v})$ and $v=v_{p}(\mathfrak d_{K_{w}})$ are \emph{independent} in IUT: $n_{v}$ is the local height of the Tate parameter, $v$ is the ramification depth of the $\ell$-torsion field, and no axiom of the theory couples them. Granting the largest $v$ compatible with Theorem~\ref{thm:tame}, $v\asymp(\ell-1)^{2}\asymp h$, configurations with $n_{v}=1$ at a bad prime of small residue characteristic are legitimate (the total height being carried by other, larger primes), and then $j^{\ast}=v/(2n_{v}d)\gg J$: the layer-1 net effect is contractionary, $\Net_{j}<1$ for every $j$. Hence: \begin{itemize} \item \textbf{Branch 1 ($n_{v}\ge v/2$, inflationary).} $\Net_{j}\ge1$ for all $j$; the layer-1 effect is inflationary of doubly exponential magnitude, and the corrected $C_{\Theta}$-inequality \eqref{eq:Ctheta-corr} of Theorem~\ref{thm:Ctheta} requires $\exp(\Theta(\sqrt h))\le O(h^{1/2+\varepsilon})$, which fails for all sufficiently large $h$ (Corollary~\ref{cor:Ctheta-fail}) $\Longrightarrow$ Corollary 3.12 does not yield abc. \item \textbf{Branch 2 ($n_{v}\; \;10\,e^{\ast}_{\mathrm{mod}}\cdot\ell+O(\ell), \] i.e.\ from $h\approx1.4\times10^{3}$ (bisection on the closed form, standard constants) --- \eqref{eq:Ctheta-corr} fails outright. \end{itemize} Both arms block the abc-chain. The choice of arm is the bookkeeping zero-point ($\hull(\cO_{n})$ vs $\cP_{n}$), a normalization convention, not a feature of the underlying volumes; hence the Branch-2 disjunction is \emph{zero-point invariant}, and ``no abc'' holds at every bookkeeping. \end{lemma} Numerical confirmation ($d=2$, $p=2$, $n_{v}=1$, $v=(\ell-1)^{2}$, $[F^{\mathrm{cl}}:F]=1$; $\mathrm{RHS}=10(e^{\ast}_{\mathrm{mod}}\ell+\eta_{\mathrm{prm}})$, $e^{\ast}_{\mathrm{mod}}=2^{12}\cdot3^{3}\cdot5$): \begin{center} \small \begin{tabular}{llllll} \toprule $h$ & $J$ & $\log_{10}|L_{\mathrm{net}}|$ & $\log_{10}L_{1a}$ & $\log_{10}\mathrm{RHS}$ & arm A / arm B\\ \midrule $10^{2}$ & $4$ & $3.45$ & $3.50$ & $7.70$ & \textsc{vacuous}\ / ok\\ $10^{3}$ & $15$ & $8.60$ & $8.62$ & $8.23$ & \textsc{vacuous}\ / ok\\ $10^{4}$ & $49$ & $20.41$ & $20.41$ & $8.74$ & \textsc{vacuous}\ / \textbf{fail}\\ $10^{6}$ & $499$ & $158.91$ & $158.91$ & $9.74$ & \textsc{vacuous}\ / \textbf{fail}\\ \bottomrule \end{tabular} \end{center} The arm-B onset ($h\approx1.4\times10^{3}$) and the Branch-1 onset of Corollary~\ref{cor:Ctheta-fail} ($h\approx7\times10^{2}$) lie in the same decade --- the two branches fail at the same doubly-exponential scale, which cross-validates the $\log\log$ bookkeeping of \S\ref{sec:10.2}. Magnitude in Branch 1 (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[Indeterminacies point in the same direction]{The indeterminacies strengthen the inflation (Ind2 and Ind3)}\label{sec:ind} The layer-$1$ computation so far treated the pilot value as a single point. In IUT, the value is only defined up to three indeterminacies --- Ind1 (permutation of the labels $j\in\{1,\dots,\llst\}$), Ind2 (conjugation/automorphism indeterminacy), and Ind3 (log-link/Kummer drift) --- and gap G2 asked how the hull behaves on the full indeterminacy orbit rather than on a single point. This section closes the layer-$1$ part of G2: Ind2 leaves the per-component loss \emph{exactly} invariant, Ind3 can only enlarge it, and the residual freedom of the hull approximation is compact, hence bounded. Only Ind1 remains, and it is absorbed by the $\mathfrak{S}_{j}^{\pm}$-symmetry already built into the coordinate model of \S\ref{sec:8.1}. \subsection{The three indeterminacies}\label{sec:ind.1} \begin{itemize} \item \emph{Ind1} (label permutation): the labels $j\in\{1,\dots,\llst\}$ of the procession may be permuted by the $\mathfrak{S}_{j+1}^{\pm}$-symmetrizing isomorphisms (IUT III Corollary 3.5(i)); \item \emph{Ind2} (automorphism indeterminacy): the monoid-theoretic data are well defined only up to automorphisms of the base; by IUT II Remark 1.11.1(i) \cite{iut2}, automorphisms of the $\mathcal{T}\mathcal{M}$-pair $G\curvearrowright\cO^{\times}(G)$ that forget the monoid structure map bijectively to $\Aut(G)$, with kernel acting through the natural action of $\Gamma=\widehat{\Z}^{\times}$; \item \emph{Ind3} (log-link/Kummer drift): the Kummer isomorphisms are compatible with the log-links only up to their pre-composites with the log-link iterates (IUT III Proposition 3.5(ii) \cite{iut3}). \end{itemize} \subsection{Ind2: valuations are unchanged and the residual freedom is compact}\label{sec:ind.2} Two facts from the original text: \begin{enumerate} \item \emph{IUT II Remark 1.11.1(i).} The elements of $\Aut(G)$ act through field automorphisms of $K_{v}$, hence preserve the normalized valuation $v_{p}$. The kernel $\widehat{\Z}^{\times}$ acts on the Tate parameter by powering: for $a\in\Z_{p}^{\times}$, $v_{p}(q^{a\,j^{2}})=a\,j^{2}n_{v}=j^{2}n_{v}$ since $v_{p}(a)=0$. Hence \emph{every} Ind2 image of the value point has per-component valuation exactly $j^{2}n_{v}$ --- the orbit already saturates the constraint $v_{p}(\lambda)\ge j^{2}n_{v}$ of \S\ref{sec:8.1}, and saturates it at the same value. \item \emph{IUT III Remark 3.9.5(iii).} The indeterminacy of the hull approximation is compact: all choices of an element $\in\Phi(P)$ or $\in\Xi(P)$ are contained in the compact set $\phi(P)\in\mathcal{H}$; indeed, ``all the indeterminacies that occur in the theory are compact'' is stated there as one of the themes of the series. Moreover the conditions $(\Xi^{\mathrm{non}}_{1})$, $(\Xi^{\mathrm{arc}}_{1})$ of Remark 3.9.5(iv) ensure that volume-exact approximants exist (the archimedean condition holds identically). \end{enumerate} Combining the two: the Ind2 orbit changes the layer-$1b$ per-component loss \emph{not at all}, and the only freedom left --- the choice of hull approximant --- stays inside the compact set $\phi(P)$. Concretely, write $C$ for the diameter of $\phi(P)$ in log-volume (a constant depending on the residue data of $K_{v}$, not on $j$ or $h$). Then the Ind2-admissible hulls have log-volume within $d^{\,j}\cdot C\cdot\log p$ of any fixed one: the total perturbation is $O(d^{\,j})$, against the main term $d^{\,j}j^{2}n_{v}$ of \S\ref{sec:8.4}. The ratio tends to $0$ as $h\to\infty$; the doubly-exponential conclusion $\log\log\rho=\Theta(\sqrt h)$ is untouched. Mochizuki's own phrase for this indeterminacy --- ``this indeterminacy is compact'' --- is here quantified as ``this indeterminacy changes the exponent by $O(d^{\,j})$, which is one exponential layer below the main term''. \subsection{Ind3: upper semi-compatibility is an inclusion}\label{sec:ind.3} IUT III Proposition 3.5(ii) is stated as a containment, in both the non-archimedean and the archimedean cases: \begin{enumerate} \item \emph{(a) non-archimedean.} The module $\cI(\mathbb{S}_{j+1}^{\pm}\mathbb{F} ({}^{n,\circ}\mathcal{D}_{\succ})_{v_{\Q}})$ \emph{contains} the images of the Galois-invariant unit submodules via both the tensor products of the Kummer isomorphisms and the tensor products of their pre-composites with the $m'$-th iterates of the log-links, for every $m'\ge1$. \item \emph{(b) archimedean.} The closed unit ball \emph{contains} the images of the units and of the closed balls of radius $\pi$ via the tensor products of the Kummer isomorphisms. \item \emph{(c) bad primes.} The actions of the various $\mathfrak{m}$-labeled monoids on the local value groups are mutually compatible up to multiplication by roots of unity --- ``i.e., to no indeterminacy''. \end{enumerate} Consequences for the hull, which is monotone by Remark 3.9.5(P3) ($P_{1}\subseteq P_{2}\Rightarrow\phi(P_{1})\subseteq\phi(P_{2})$): \begin{enumerate} \item The region to be boxed contains \emph{every} log-conjugate of the pilot, not a single image; by monotonicity, $\hull(\text{full orbit})\supseteq\hull(\text{single image})$. Upper semi-compatibility is an inclusion: enlarging the family of images can only enlarge the hull. There is no cancellation mechanism on the Ind3 side. \item Along each log-iterate the per-component valuation is preserved: $v_{p}(\log x)=v_{p}(x)$ whenever $v_{p}(x)\ge1$ (and archimedean norms are unchanged in absolute value under $\log$ on the compact unit region), so every $m'$-conjugate of the value carries the same constraint $v_{p}(\lambda)\ge j^{2}n_{v}$ per component. The layer-$1b$ bound survives Ind3 untouched, orbit-wide. \item At bad reduction primes, (c) shows the log-Kummer correspondence contributes no further multiplicative spread beyond roots of unity: the spread measured in \S\ref{sec:8} is the full spread. \end{enumerate} \subsection{Net effect: the layer-1 part of G2 is closed}\label{sec:ind.4} \begin{theorem}\label{thm:ind} In the layer-$1$ chains of this paper (Theorem~\ref{thm:5.1}, the layer-$1b$ hull loss of \S\ref{sec:8.1}, the net index of \S\ref{sec:8.4}), the indeterminacies act as follows: Ind2 leaves every per-component valuation unchanged and adds at most $O(d^{\,j})$ to the log-volume; Ind3 forces the hull to contain the full log-orbit and cannot decrease any of the three quantities; Ind1 is absorbed by the $\mathfrak{S}_{j}^{\pm}$-symmetry of the coordinate model. In particular the layer-$1$ conclusions (the inflation magnitude $\log\log\rho=\Theta(\sqrt h)$ in the inflationary branch, the crossover table, and the two-branch criterion) are invariant under the indeterminacies: which branch the layer-1 net effect realizes is decided by $n_{v}\gtrless v/2$ (\S\ref{sec:8.4}), not by the indeterminacies. \end{theorem} \begin{proof} Ind2: \S\ref{sec:ind.2}, items (1)--(2); the $O(d^{\,j})$ bound against the main term $d^{\,j}j^{2}n_{v}$ follows from $n_{v}\ge1$ and $j\ge1$. Ind3: \S\ref{sec:ind.3}, items (1)--(3). Ind1: the labels enter the per-component loss only through $j^{2}n_{v}$, which is symmetric in the label set. \end{proof} What remains of G2 is the layer-$2$ interaction of Ind2/Ind3 with the (9-1) compatibility --- LANA's territory, not layer 1. \subsection{Docking with the obstacle list of Remark 3.9.5(vii)}\label{sec:ind.5} Remark 3.9.5(vii) of the original text \cite{iut3} lists nine obstacles (Ob1)-- (Ob9) to a fully satisfactory log-volume hull-approximation theory for metrized line/vector bundles on proper models. The tensor-packet situation of this paper is the local free-module case of that programme, and the obstacles split as follows. \begin{itemize} \item \emph{Resolved by the local setting:} (Ob1) (comparison with $q$-pilot objects and arithmetic line bundles) and (Ob4) (tensor powers/Frobenius functors) concern global geometry over a proper model; at a single prime with free modules the divisor is principal and the Frobenius ambiguity is absent --- the comparison is exactly the index $\rho$ of Theorem~\ref{thm:5.1}. \item \emph{Resolved by this paper:} (Ob3) (the $\det^{\otimes M}(-)$ operation; its items (Ob3-1)--(Ob3-3) on determinants versus log-volumes) is the integral-basis determinant of \S\ref{sec:5.5}: its value is $\rho$, in closed form, and the $\det/\log$-volume dictionary is the discriminant relation (0.1). (Ob2) (the $\boxtimes$-versus-$\otimes$ tension) is the content of \S\ref{sec:2}--\S\ref{sec:5}: the passage from $\otimes$ to the direct-sum representation is exactly where the inflation lives. \item \emph{Quantified rather than eliminated:} (Ob5) (independence of the indeterminacy of possibilities --- the hull $\phi(\cP_{B})$ of a bounded collection) and (Ob6) (the gap $\mu_{\log}(\phi(\cP))-\mu_{\log}(\cP)$) are the compactness of \S\ref{sec:ind.2}: the approximant is confined to the compact set $\phi(\cP)$, and the gap is the $O(d^{\,j})$ term of Theorem~\ref{thm:ind}. (Ob7) (compatibility with log-Kummer correspondences) and (Ob8) (vertical shifts in the output data) are the inclusions of \S\ref{sec:ind.3}: Prop.~3.5(ii) and its iterates force the hull to contain the full orbit, which is what makes the layer-$1$b loss orbit-wide. \item \emph{Resolved by the original text itself, and load-bearing here:} (Ob9) (passage to realified semi-simplifications induces a natural bijection between log-volumes of hulls in the domain and codomain of the log-link) is precisely the reason the per-component valuation survives log-iteration (\S\ref{sec:ind.3}, item (2)): $v_{p}(\log x)=v_{p}(x)$ is the non-archimedean shadow of that bijection. \end{itemize} \emph{Positioning.} What the original text presents as obstacles to an ideal theory, the present paper exhibits as the mechanism itself: in the tensor-packet case the obstacles do not obstruct --- they inflate, with computable magnitude. \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[The archimedean places]{The archimedean places: the same inflation with the discriminant replaced by $2$}\label{sec:arc} The computation of \S\ref{sec:2}--\S\ref{sec:5} was non-archimedean. This section carries it to the archimedean places, where the role of the discriminant $\mathfrak d_{K}$ is played by an exact metric defect recorded in the original text: the factor $2$ of IUT IV Proposition 1.5(i) \cite{iut4}. The effect is confined to the \emph{complex} places: at a real place $K_{v}=\R$ and $\otimes^{n}\R=\R$ admits no $\C\oplus\C$-type decomposition, so there is no metric defect and no archimedean inflation --- the section computes the complex case. \subsection{The tensor packet and its two metrics}\label{sec:arc.1} Let $v\mid\infty$. After the base-change to $\mathbb{C}$ used throughout IUT II--III, the $n$-fold tensor packet at $v$ is the $n$-fold tensor power $\mathcal{M}=\otimes^{n}\C$ (real tensor products if $v$ is real, after the same conventions). By the Chinese remainder theorem, \[ \C\otimes_{\R}\C\;\xrightarrow{\sim}\;\C\oplus\C \] (IUT IV Prop.~1.5(i)), and inductively $\mathcal{M}$ decomposes as a direct sum of $2^{\,n-1}$ copies of $\C$. Two metrics compete on $\mathcal{M}$: the tensor product metric $g_{\mathrm{t}}$ (from the tensor power of the standard metric) and the direct sum metric $g_{\mathrm{ds}}$ (from the displayed decomposition). IUT IV Prop.~1.5(i),(iii) states, in the original text: \begin{itemize} \item \emph{(i)} under any isomorphism $\C\otimes_{\R}\C\xrightarrow{\sim}\C\oplus\C$, the direct sum metric corresponds to $2$ times the tensor metric --- lengths differ by a factor of $\sqrt{2}$; \item \emph{(iii)} for the $n$-fold packet, $g_{\mathrm{ds}}=2^{\,n-1}\cdot g_{\mathrm{t}}$; the ``integral structure'' $\mathcal{B}_{I}$ is the direct product of the unit balls of the copies of $\C$. \end{itemize} \subsection{The exact volume computation}\label{sec:arc.2} \begin{theorem}[archimedean inflation]\label{thm:arc} Let $B_{\mathrm{t}}\subseteq\mathcal{M}$ be the unit ball of the tensor metric and $B_{I}=\mathcal{B}_{I}$ the integral structure of \emph{loc.\ cit.}\ Then \[ \frac{\operatorname{vol}(B_{\mathrm{t}})}{\operatorname{vol}(B_{I})} \;=\;2^{\,(n-1)\,2^{\,n-1}}, \] for any $\Aut(\C/\R)$-invariant Lebesgue measure on $\mathcal{M}$; per complex component the ratio is $2^{\,n-1}$. In IUT's log-volume normalization (zero point at $B_{I}$, IUT IV \S1), the passage from the tensor integral structure to $\mathcal{B}_{I}$ contributes \[ (n-1)\cdot2^{\,n-1}\cdot\log 2. \] This is the exact archimedean analogue of Theorem~\ref{thm:5.1} and Corollary~\ref{cor:5.2}: the per-component factor $\frac{n-1}{2}\,v_{p}(\mathfrak d_{K})$ is replaced by $(n-1)\log 2$. \end{theorem} \begin{proof} $g_{\mathrm{ds}}=2^{\,n-1}g_{\mathrm{t}}$ means $\lVert x\rVert_{\mathrm{ds}}=2^{\frac{n-1}{2}}\lVert x\rVert_{\mathrm{t}}$, so on each of the $2^{\,n-1}$ copies of $\C$, $B_{\mathrm{t}}=2^{\frac{n-1}{2}}B_{I}$; the $2$-dimensional volume of a complex unit ball scales by $\bigl(2^{\frac{n-1}{2}}\bigr)^{2}=2^{\,n-1}$ per component. The product over $2^{\,n-1}$ components gives the displayed ratio; the log-volume statement is the definition of the normalization. \end{proof} \emph{No extra term, no wild/archimedean distinction:} the factor $2^{\,n-1}$ is exact, unlike the $p$-adic $v_{p}(\mathfrak d_{K})$; archimedean places have no ramification theory, and Prop.~1.5(i) declares the two metrics compatible with the relevant actions, so the mismatch is invariant under the symmetries of \S\ref{sec:ind}. \subsection{Direction and magnitude at IUT's parameters}\label{sec:arc.3} $B_{I}$ is IUT's declared log-volume zero point, and $B_{\mathrm{t}}\supsetneq B_{I}$: the tensor integral structure is the \emph{larger} one, so the passage from the multiradial representation to the hull inflates the measured region --- the same sign as layer-$1$a (Remark~\ref{sec:5.5}, \S\ref{sec:8.5}); there is no archimedean contraction branch. At the leading procession label $n=J+1$ (so $n-1=J$), \[ (n-1)\cdot2^{\,n-1}\cdot\log 2=\exp(\Theta(\sqrt h)), \] the same exponential order as the $p$-adic layer-$1$a at $d=2$; the layer-$1$a term there is $\frac{Jv}{2}\cdot2^{J}\log p$, and the value-point constraint adds the per-component term $j^{2}\,\mathrm{ht}_{v}$ with $\mathrm{ht}_{v}=O(h)$ in BD-class (IUT IV Cor.~2.2(i)); hence the archimedean contribution to the net inflation is of doubly exponential character $\log\log\rho_{\mathrm{arc}}=\Theta(\sqrt h)$ as well. \subsection{Honest boundary}\label{sec:arc.4} For $|V|>1$ tensor packets, Prop.~1.5(iii) gives $2^{\,n-1}\cdot|V|^{\,n}$ copies of $\C$; the per-component factor $2^{\,n-1}$ is unchanged, and $|V|^{\,n}$ multiplies only the count, i.e.\ adds $n\log|V|$ to the log-volume. The identification of the zero point with $\mathcal{B}_{I}$ is IUT's own convention; under any other normalization the constant $(n-1)\log 2$ shifts uniformly, but the exponential order $\exp(\Theta(\sqrt h))$ and the sign do not. \section{Plugging in IUT's own parameters}\label{sec:10} \subsection{Parameters (all confirmed from the original text)}\label{sec:10.1} \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} \subsubsection*{The target is a BD-class of $h$ (G1 upgraded)}\label{sec:10.1b} IUT IV Corollary 2.2(i) states the bounded-discrepancy equivalence \[ \frac{1}{6}\log(q^{\forall}_{v}(-))\approx\operatorname{ht}_{\omega_{X}(D)}(-) \qquad\text{as BD-classes of functions on }\mathcal{K}_{V}, \] and Corollary 2.2(ii) adds that the same holds after excluding any finite number of exceptional possibilities. Hence the comparison target of the loss computation --- $\log\operatorname{rad}(abc)$ up to the standard reduction of abc to Szpiro --- is a bounded-discrepancy class of $\frac{1}{6}\log(q^{\forall})$, i.e.\ of $h$ itself: the ``target $\sim h$'' identification of \S\ref{sec:10} is not an assumption but a BD-class statement from the original text. What remains of G1 is only the constant factor in the Szpiro/abc dictionary between $\operatorname{ht}_{\omega_{X}(D)}$ and $\log\operatorname{rad}(abc)$, which the crossover analysis of \S\ref{sec:10.3} shows to be immaterial for targets as large as $h^{1000}$. \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 at which $\log_{10}\rho>h^{\kappa}$ --- equivalently $\log_{10}\log_{10}\rho>\kappa\log_{10}h$ --- i.e.\ the conservative, doubly-logarithmic reading: the \emph{logarithm} of the inflation overtakes the target bound $h^{\kappa}$. (The weaker single-log reading $\log_{10}\rho>\kappa\log_{10}h$ is satisfied at far smaller scales --- e.g.\ for $d=v=2$, $\kappa=1$ already at $h=25$, since $\log_{10}\rho=1.505>1.398=\log_{10}25$ --- so the table deliberately adopts the stronger criterion.) 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. \subsection{The inflation term in the \texorpdfstring{$C_{\Theta}$}{C-Theta}-inequality (G3 closed)}\label{sec:10.5} III Corollary 3.12 \cite{iut3} is the inequality \[ -|\log(\Theta)|\;\ge\;-|\log(q)|, \] i.e.\ $C_{\Theta}\ge-1$ for any $C_{\Theta}$ with $-|\log(\Theta)|\le C_{\Theta}\cdot|\log(q)|$. Here $-|\log(\Theta)|$ is the procession-normalized mono-analytic log-volume of the \emph{holomorphic hull of the union of the possible images of a $\Theta$-pilot}, subject to (Ind1)(Ind2)(Ind3), and $-|\log(q)|$ is the same quantity for the $q$-pilot, which is not subject to the indeterminacies. IUT IV Theorem 1.10 \cite{iut4} then estimates the $\Theta$-pilot volume from height data: \begin{equation}\label{eq:Ctheta-def}\begin{split} C_{\Theta}=\frac{\llst+1}{4|\log q|}\cdot\Bigl\{& (1+12d_{\mathrm{mod}}/\llst)\bigl(\log\mathfrak d_{F_{\mathrm{tpd}}} +\log\mathfrak f_{F_{\mathrm{tpd}}}\bigr)\\ &+10\bigl(e^{\ast}_{\mathrm{mod}}\cdot\llst+\eta_{\mathrm{prm}}\bigr) -\frac{1}{6}\Bigl(1-\frac{12}{(\llst)^{2}}\Bigr)\log(q)\Bigr\}-1, \end{split}\end{equation} with $d^{\ast}_{\mathrm{mod}}=2^{12}\cdot3^{3}\cdot5\cdot d_{\mathrm{mod}}$, $e^{\ast}_{\mathrm{mod}}=2^{12}\cdot3^{3}\cdot5\cdot e_{\mathrm{mod}}$, and applying $C_{\Theta}\ge-1$ yields the abc-bound chain \begin{equation}\label{eq:Ctheta} \frac{1}{6}\log(q)\;\le\;\Bigl(1+\frac{20\,d_{\mathrm{mod}}}{\llst}\Bigr)\bigl(\log \mathfrak d_{F_{\mathrm{tpd}}}+\log\mathfrak f_{F_{\mathrm{tpd}}}\bigr) +20\bigl(e^{\ast}_{\mathrm{mod}}\cdot\llst+\eta_{\mathrm{prm}}\bigr). \end{equation} \begin{lemma}[docking of the layer-1 inflation]\label{lem:dock} With the sign convention of \S\ref{sec:8.2} (zero point at $\hull(\cO_{n})$), the true hull volume exceeds the volume estimated in \emph{loc.\ cit.}\ by the net index of \S\ref{sec:8.4}: in IUT's convention, \[ -|\log(\Theta)|_{\mathrm{true}}\;=\;-|\log(\Theta)|_{\mathrm{IUT}}-\log\widehat\rho, \] where $\log\widehat\rho=\log_{p}\mathrm{Net}_{J}\cdot\log p\asymp d^{\,J}J^{2}n_{v}\log p$ (\S\ref{sec:8.4}), equivalently $C_{\Theta,\mathrm{true}}=C_{\Theta,\mathrm{IUT}}-\log\widehat\rho/|\log q|$. \end{lemma} \begin{proof} The inflation is the determinant of the integral-basis change of the hull's target lattice (\S\ref{sec:5.5}, \S\ref{sec:10.4}); log-volumes are additive under lattice indices by definition (IUT IV \S1, cf.\ \S\ref{sec:8.2} of this paper). The magnitude $\log\widehat\rho=\log_{p}\mathrm{Net}_{J}\cdot\log p$ is the last display of \S\ref{sec:8.4}. \end{proof} \begin{theorem}[corrected \texorpdfstring{$C_{\Theta}$}{C-Theta}-inequality]\label{thm:Ctheta} With the true hull volume, the requirement $C_{\Theta,\mathrm{true}}\ge-1$ replaces \eqref{eq:Ctheta} by \begin{equation}\label{eq:Ctheta-corr}\begin{split} \frac{1}{6}\Bigl(1-\frac{12}{(\llst)^{2}}\Bigr)\log(q) +\frac{4}{\llst+1}\cdot\log\widehat\rho \;\le\;&\Bigl(1+\frac{12\,d_{\mathrm{mod}}}{\llst}\Bigr)\bigl(\log \mathfrak d_{F_{\mathrm{tpd}}}+\log\mathfrak f_{F_{\mathrm{tpd}}}\bigr)\\ &+10\bigl(e^{\ast}_{\mathrm{mod}}\cdot\llst+\eta_{\mathrm{prm}}\bigr), \end{split}\end{equation} i.e.\ the inflation enters the abc-bound chain as an additive term $\frac{4}{\llst+1} \log\widehat\rho$ on the left-hand side. \end{theorem} \begin{proof} Substitute $C_{\Theta,\mathrm{IUT}}=C_{\Theta,\mathrm{true}}+\log\widehat\rho/|\log q| \ge-1+\log\widehat\rho/|\log q|$ into the $C_{\Theta}$-formula. Since $|\log q|>0$, the factor $\frac{\llst+1}{4|\log q|}$ is positive, and the condition $\{\cdots\}\ge\frac{4}{\llst+1}\log\widehat\rho$ is exactly \eqref{eq:Ctheta-corr}. The computation is \emph{sign-uniform}: in Branch 2 $\log\widehat\rho<0$, the constraint weakens to a vacuity, and the same line of algebra exhibits both arms explicitly --- see Lemma~\ref{lem:twoarms}. \end{proof} \begin{corollary}[quantitative failure of the chain]\label{cor:Ctheta-fail} The right-hand side of \eqref{eq:Ctheta-corr} is $O((\llst)^{1+\varepsilon})$ at the IUT scale $\llst\asymp\sqrt h$ (the terms $\log\mathfrak d_{F_{\mathrm{tpd}}}$, $e^{\ast}_{\mathrm{mod}}\cdot\llst$, $\eta_{\mathrm{prm}}$ are bounded by $(\llst)^{1+o(1)}$ in the proof of IUT IV Corollary 2.2; see Step (viii) there and Proposition 1.6), while the inflation term satisfies $\frac{4}{\llst+1}\log\widehat\rho=\exp(\Theta(\sqrt h))$ (\S\ref{sec:10.2}: removing the factor $\llst\asymp\sqrt h$ costs only $\log\llst$ on the doubly-logarithmic scale). Hence \eqref{eq:Ctheta-corr} demands $\exp(\Theta(\sqrt h))\le O(h^{1/2+\varepsilon})$ and fails for all sufficiently large $h$; for the standard constants ($e_{\mathrm{mod}}=1$) the onset lies between $h=10^{2}$ and $h=10^{3}$ --- more precisely $\approx7\times10^{2}$ by bisection on the closed form, comparing $\frac{4}{\llst+1}\log\widehat\rho$ (with $\log\widehat\rho=\log\mathrm{Net}_{J}$ of \S\ref{sec:8.4}) against the full right-hand side of \eqref{eq:Ctheta-corr}, whose dominant term is IUT's own $10\,e^{\ast}_{\mathrm{mod}}\cdot\llst$ with $e^{\ast}_{\mathrm{mod}}=2^{12}\cdot 3^{3}\cdot5\approx5.5\times10^{5}$. Two neighbouring calibrations bracket the onset: with the layer-$1$a index in place of $\mathrm{Net}_{J}$ it moves to $\approx3\times10^{3}$; with the explicit constant $e^{\ast}_{\mathrm{mod}}$ removed from the comparison it moves down to $h\approx10^{1}$. The qualitative conclusion is calibration-independent. \emph{Onset coincidence:} the Branch-2 arm-B onset ($h\approx1.4\times10^{3}$, Lemma~\ref{lem:twoarms}) lies in the same decade as this Branch-1 onset --- the two branches fail at the same scale, as they must, since both are driven by the same $e^{\Theta(\sqrt h)}$ object. \end{corollary} This closes G3: the middle of the chain is now \emph{computed} --- with the true hull volume, Corollary 3.12 feeds an inequality whose left side outgrows any height-data right side. What breaks is the arrow $(\mathrm{dsc})\to(\mathbf{hull}+\mathbf{det})$ of \S\ref{sec:10.4}, exactly where this paper's $\widehat\rho$ lives; the remaining candidates for saving the chain are (9-1) and explicit-constant bookkeeping (\S\ref{sec:12}), both outside layer 1. \emph{Honest boundary.} The docking uses the sign convention recorded in \S\ref{sec:8.2} and the placement of $\widehat\rho$ as the determinant of the second arrow (\S\ref{sec:10.4}); the procession weights and $\lambda_{e}$-normalizations are those of \S\ref{sec:10.1}, and no new normalization freedom is introduced. The onset range $10^{3}$--$10^{4}$ depends on $e_{\mathrm{mod}}$ and $\eta_{\mathrm{prm}}$ and has the same status as the explicit constants of \S\ref{sec:12}. \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}{@{}p{0.42\textwidth}p{0.50\textwidth}@{}} \toprule Measured $\ln(abc)$ range (natural log) & $[8.37,\,60.94]$\\ Threshold $\ln\lVert abc\rVert\ge700$ (explicit-estimates calibration) & requires $c\gtrsim10^{152}$\\ Gap & $700-60.94\approx639$ in $\ln$-units; equivalently $\approx140$ orders of magnitude in $c$ ($152$ vs $\log_{10}(4.4\times10^{11})\approx11.6$)\\ \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 $\Q_{2}$.) \subsubsection*{First empirical evidence for \texorpdfstring{$\Delta>0$}{Delta>0}} The example above gives $\Delta=0$, which does not show whether $\Delta$ can be positive. A corrected review copy (Sage 10.9) ran five non-Galois samples, of which three give $\Delta>0$: \begin{center} \small \begin{tabular}{@{}llllllllll@{}} \toprule Field & $p$ & Group & $d$ & $v$ & $d\,v$ & $\Delta$ & $v_{p}(\rho^{2})$ & $v_{p}(\rho)$ & Galois\\ \midrule $\Q(\sqrt[3]{2})$ & $2$ & $S_{3}$ & $3$ & $2$ & $6$ & $0$ & $6$ & $3$ & $3$\\ $\Q(\sqrt[3]{3})$ & $3$ & $S_{3}$ & $3$ & $5$ & $15$ & $1$ & $14$ & $7$ & $7.5$\\ $x^{4}-x-1$ & $283$ & $S_{4}$ & $4$ & $1$ & $4$ & $2$ & $2$ & $1$ & $2$\\ $x^{4}-8x^{3}+7x^{2}+5x-3$ & $19$ & $A_{4}$ & $4$ & $2$ & $8$ & $2$ & $6$ & $3$ & $4$\\ same & $31$ & $A_{4}$ & $4$ & $2$ & $8$ & $0$ & $8$ & $4$ & $4$\\ \bottomrule \end{tabular} \end{center} All five satisfy the closed form $v_{p}(\rho^{2})=d\,v-\Delta$; $d\,v-\Delta$ is even in every row, and $(d\,v-\Delta)/2$ equals the measured $v_{p}(\rho)$ row by row. $\Longrightarrow$ $\Delta>0$ does exist, and non-Galois $\rho$ is strictly smaller than the Galois comparison value. These are the first nontrivial instances confirming the direction of \S\ref{sec:14.2}. \subsubsection*{Attribution and independence} The five examples were first produced by Part~D of the review copy; a subsequent independent recheck on another temporary copy --- which removed the single-prime gate according to the paper's own textual description of B13 and fixed the \texttt{Fraction}/numeric-type conversions, \emph{without} using this package's corrected files --- again gave $5/5$. For $p=283$ and $p=19$ the diagnostic output explicitly shows ``non-single-prime'', yet the global tensor decomposition still yields $\Delta=2$: direct empirical evidence that deleting the gate is correct, i.e.\ $\Delta$ depends only on the $p$-component of the \emph{global} discriminant, not on the number of primes. The package's corrected files themselves have not been run in that Sage environment. \subsubsection*{The relative discount at a single prime can be large} Measuring against the Galois comparison value $d\,v/2$: \begin{center} \small \begin{tabular}{@{}lccc@{}} \toprule Sample & Galois comparison $v_{p}(\rho)$ & Measured & Relative discount\\ \midrule $\Q(\sqrt[3]{2})$ & $3$ & $3$ & $0\%$\\ $\Q(\sqrt[3]{3})$ & $7.5$ & $7$ & $6.7\%$\\ $x^{4}-x-1$, $p=283$ & $2$ & $1$ & $50\%$\\ $A_{4}$, $p=19$ & $4$ & $3$ & $25\%$\\ $A_{4}$, $p=31$ & $4$ & $4$ & $0\%$\\ \bottomrule \end{tabular} \end{center} From $v_{p}(\rho^{2})=d\,v-\Delta\ge0$ one gets $\Delta\le d\,v$, so in the worst case the index can drop to $1$ (a $100\%$ discount). The statement ``the discount is asymptotically negligible'' in \S\ref{sec:14.3} is an $h\to\infty$ asymptotic statement ($\operatorname{poly}(h)$ versus $\exp(c\sqrt h)$); it cannot be read off at a single prime, where $\Delta$ can consume the entire inflation of that prime. \subsubsection*{\texorpdfstring{$\Delta$}{Delta} is not a function of $(d,v)$: the E1QUICK scan} Running E1QUICK on a user-side temporary corrected copy (Sage 10.9) yields $31$ records; grouping by $(d,v)$, one of the $12$ groups takes two values: \[ (d,v)=(4,2):\qquad \Delta\in\{0,2\}. \] The two rows are the \emph{same} $A_{4}$ quartic field at two different primes: $p=31$ gives $\Delta=0$, $p=19$ gives $\Delta=2$. All self-checks pass: parity violations $0$, negative $v_{p}(\rho^{2})$ $0$, $\sum_{j}e_{j}\neq d$ $0$, and among Galois-type samples none has $\Delta\neq0$. $\Longrightarrow$ the hypothesis ``$\Delta$ depends only on $(d,v)$'' is refuted. Moreover, since the field is fixed, all global invariants (the field, the Galois group, $d$, $v_{p}(\operatorname{disc}K)$) are fixed, so the dependence is located precisely in the local arithmetic at $p$ --- whether the ``extra part'' of the Galois closure ramifies at $p$ --- which is exactly the defining quantity of $\Delta$ in \eqref{eq:delta}. The data surfaced no new parameter; it confirmed that the old parameter is irreducible. \subsubsection*{Two-term decomposition of \texorpdfstring{$\Delta$}{Delta}} Write \[ \Delta=\Dpar+\Dram,\qquad \Dpar:=(d\,v)\bmod 2,\quad \Dram\ge0, \] where $\Dpar$ is the minimum forced by Corollary~\ref{cor:14.1} below and $\Dram$ is the ramification contribution of the closure's extra part. The $12$ groups of the E1QUICK scan give: \begin{center} \small \begin{tabular}{@{}llll@{}} \toprule Family & $(d,v)$ & $\Delta$ & $\Dpar$ / $\Dram$\\ \midrule $d=2$ (Galois comparison) & $(2,1),(2,2),(2,3)$ & $0,0,0$ & $0$ / $0$\\ $d=3$ ($S_{3}$ Kummer) & $v=1,2,3,4,5$ & $1,0,1,0,1$ & $v\bmod2$ / all $0$\\ $d=4$ & $(4,1)$ & $2$ & $0$ / $2$\\ $d=4$ & $(4,2)$ & $0,2$ & $0$ / $0,2$\\ $d=4$ & $(4,8),(4,11)$ & $0,2$ & $0$ / $0,2$\\ \bottomrule \end{tabular} \end{center} Two readings must be distinguished --- which one is a restatement and which one is a fact: $\Dram$ is always even (value set $\{0,2\}$) is a trivial restatement of Corollary~\ref{cor:14.1}, not a new structure, and is not claimed as an independent discovery. \emph{The nontrivial empirical fact is: all five $d=3$ groups attain the parity lower bound}, i.e.\ $\Delta=v\bmod2$ and $\Dram=0$. For $S_{3}$-type $K=\Q(\sqrt[3]{m})$ the closure's extra part is $K(\zeta_{3})/K$, and for $p\neq3$ the extension $\Q_{p}(\zeta_{3})/\Q_{p}$ is trivial or unramified quadratic, hence the extra part is unramified --- consistent with the vanishing criterion of \S\ref{sec:14.2}. Even the wild point $\Q(\sqrt[3]{3})$ ($v=5$, $\Delta=1$) attains the minimum. \emph{Falsifiable prediction (P):} for every $S_{3}$-type $K=\Q(\sqrt[3]{m})$ and every prime, $\Delta=v_{p}(\operatorname{disc}K)\bmod2$, i.e.\ $\Dram=0$. \emph{Status of (P):} proved completely on pure cubic fields (Corollary~\ref{cor:14.2} below) --- for $p\neq3$, $\Delta=0$ is a theorem; for $p=3$ the per-prime relative-discriminant determination gives $\Delta=1$. (P) is no longer a conjecture, and $\Delta\le1$ is an absolute constant bound. \emph{Honesty note.} $11$ of the $12$ groups carry a single $\Delta$ value, so this data can only refute ``$\Delta=f(d,v)$''; it cannot distinguish among ``depends on the residue class of $p$'', ``depends on the closure ramification group'', and ``depends on the decomposition type of $p$''. Fitting a closed form requires more primes under the same $(d,v,\Gal)$. \emph{Attribution.} The $31$ records come from the user's temporary corrected copy (gate deletion per the textual description of B13 and a \texttt{normalize()} compatibility fix), not from this package's files; the full E1 run was not executed. \subsubsection*{Thirty-five-instance targeted scan and the analytic determination of \texorpdfstring{$\Delta$}{Delta} ($S_{3}$ pure cubics)} A user-side Sage 10.9 scan of $K=\Q(\sqrt[3]{m})$, $m\in\{2,3,5,6,7,10,\allowbreak 11,12,13,14,\allowbreak 15,17,19,20,21,22\}$, at each ramified prime, gives $35$ instances, $35/35$ satisfying $\Delta=(3v)\bmod2$, covering $p\bmod3\in\{0,1,2\}$. But ``matching the parity'' is itself a restatement of Corollary~\ref{cor:14.1} and constitutes no new fact; the real increment is the analytic derivation below. (i) \emph{$\Delta$ is a valuation of relative discriminants (universally valid).} Let $K\otimes K\cong\bigoplus_{j}K_{j}$, $e_{j}=[K_{j}:K]$, $\sum_{j}e_{j}=d$. From $\disc(K_{j})=\disc(K)^{e_{j}}\cdot\Norm_{K/\Q}(\dv_{K_{j}/K})$ and summation over $j$, \[ \Delta=\sum_{j}v_{p}\bigl(\Norm_{K/\Q}\dv_{K_{j}/K}\bigr). \] This gives a constructive reason for $\Delta\ge0$ (a relative discriminant is an integral ideal), more essential than the earlier inference from $v_{p}(\rho^{2})\ge0$. (ii) \emph{For $S_{3}$ type, the non-diagonal component is exactly the Galois closure.} $x^{3}-m=(x-a)(x^{2}+ax+a^{2})$ with quotient discriminant $a^{2}-4a^{2}=-3a^{2}$, and $a\in K$ $\Longrightarrow$ $M=K(\sqrt{-3})=K(\zeta_{3})=L$. Hence $\Delta=v_{p}(\Norm\dv_{L/K})$, a single term. (iii) \emph{$\dv_{L/K}\mid(3)$.} Taking $\zeta_{3}$ as generator, the discriminant of $x^{2}+x+1$ is $(\zeta_{3}-\zeta_{3}^{2})^{2}=-3$ (verified exactly in sympy), so $\Norm\dv_{L/K}\mid3^{[K:\Q]}=27$. (iv) \emph{Merge with Corollary~\ref{cor:14.1}:} \[ \Delta=0\quad(p\neq3),\qquad \Delta\in\{1,3\}\quad(p=3), \] the latter since $d\,v=3v$ is odd for $p=3$. The $35$ instances split as: \begin{center} \small \begin{tabular}{@{}llll@{}} \toprule $p$ & $v_{p}(\operatorname{disc}K)$ & Forced range & Value taken ($35/35$)\\ \midrule $\ge5$ & $2$ (even) & $\Delta=0$ (theorem) & $0$\\ $2$ & $2$ or $4$ (even) & $\Delta=0$ (theorem) & $0$\\ $3$ & $3$ or $5$ (odd) & $\Delta\in\{1,3\}$ & $1$\\ \bottomrule \end{tabular} \end{center} The last cell of (P) is now closed: the exact exponent of $\dv_{L/K}$ at $3$ was determined by a user-side $\texttt{relative\_discriminant}()$ scan ($192$ examples) together with the local decomposition, and always gives $\Delta=1$; the guess $\Delta=3$ (B17) is withdrawn. The full determination is Corollary~\ref{cor:14.2}. \emph{Calibration.} sympy computes the power-basis $\Z[\sqrt[3]{m}]$ discriminant $-27m^{2}$; when $m\equiv\pm1\pmod9$ the maximal-order discriminant loses $3^{2}$, but $v_{3}$ remains odd and $v_{p}=2$ for $p\ge5$ is unchanged, so the parity conclusions are unaffected. The $35$ instances use Sage's \texttt{K.discriminant()} (maximal order), the authoritative value. The $A_{4}$ sample $(4,2)\Rightarrow\Delta\in\{0,2\}$ shows that ``$\Delta$ is decided by parity alone'' is not universal --- step (iii) depends on the specific generator. \emph{Attribution.} The $35$ instances come from the user-side targeted scan; the analytic derivation was carried out in pure Python/sympy without Sage; the package files were not run in that environment. \begin{corollary}[parity of \texorpdfstring{$\Delta$}{Delta} is forced by $d\,v$]\label{cor:14.1} Since $v_{p}(\rho^{2})$ must be even, the closed form $v_{p}(\rho^{2})=d\,v-\Delta$ forces \[ \Delta\equiv d\,v\pmod 2. \] In particular, when $d\,v$ is odd, $\Delta\ge1$ is forced, not accidental. The five examples match one by one: $d\,v=6,15,4,8,8$ correspond to $\Delta=0,1,2,2,0$. \end{corollary} This corollary also explains the second row's $\Delta=1$: there $d=3$, $v=5$, $d\,v=15$ is odd, while the Galois comparison value $\frac{(n-1)d^{\,n-1}}{2}v=\frac{3\cdot5}{2}=7.5$ is not an integer. The parity argument of Corollary~\ref{cor:5.3} (Hilbert's ramification formula) relies on the Galois hypothesis; for non-Galois $K$, $d\,v$ can be odd, the Galois formula fails, and the deficit is absorbed by $\Delta$. Hence $\Delta$'s lower bound is arithmetically forced, and its upper bound is given by \S\ref{sec:14.3} ($\operatorname{poly}(h)$); neither harms the conclusion. \begin{corollary}[the pure-cubic family: \texorpdfstring{$\Delta$}{Delta} completely determined; (P) proved]\label{cor:14.2} Let $K=\Q(\sqrt[3]{m})$ ($m$ not a cube) with Galois closure $L=K(\zeta_{3})$. Since $K$ is cubic and has no quadratic subfield, $\zeta_{3}\notin K$ and $[L:K]=2$; since $K\otimes K\cong K\oplus L$, the non-diagonal component is exactly $L$, so $\Delta=v_{3}\bigl(\Norm_{K/\Q}(\dv_{L/K})\bigr)$ (for $p=3$; $\Delta=0$ for $p\neq3$ by (ii)--(iii) above). The minimal polynomial $x^{2}+x+1$ of $\zeta_{3}$ has discriminant $-3$, and the relative-discriminant formula \[ \disc(1,\zeta_{3})=\dv_{L/K}\cdot[\cO_{L}:\cO_{K}[\zeta_{3}]]^{2} \] gives, after taking $\Norm_{K/\Q}$, $27=3^{3}$ on the left, hence \begin{equation}\label{eq:14.2} \Delta=3-2\,v_{3}\bigl(\Norm_{K/\Q}[\cO_{L}:\cO_{K}[\zeta_{3}]]\bigr), \end{equation} consistent with Corollary~\ref{cor:14.1} ($d=3$, $v_{3}(\operatorname{disc}K)$ odd, $\Delta$ odd). $\Delta=3$ would hold iff $\cO_{K}[\zeta_{3}]$ were already maximal at $3$. A user-side Sage 10.9 targeted scan ($-100\le m\le100$, $192$ non-cube parameters, merging to $65$ distinct discriminant values) gives $v_{3}\bigl(\Norm_{K/\Q}\dv_{L/K}\bigr)=1$ in every case ($m=2,3,5,19$ all have relative discriminant norm $3$), including the $40$ cases $m\equiv\pm1\pmod9$. The per-prime determination at $p=3$ is as follows. If $m\equiv\pm1\pmod9$, then $x^{3}-m$ has a root $a$ in $\Q_{3}$, the quotient discriminant is $-3a^{2}$, and \[ K\otimes\Q_{3}\cong\Q_{3}\times\Q_{3}(\zeta_{3}), \] with $L=K(\zeta_{3})$ localizing over the two branches: over $\Q_{3}$ as $\Q_{3}(\zeta_{3})/\Q_{3}$, and over $\Q_{3}(\zeta_{3})$ trivially: \begin{center} \small \begin{tabular}{@{}lccc@{}} \toprule Branch & Contains $\zeta_{3}$? & Relative discriminant exponent & $f$\\ \midrule $\Q_{3}$ & no & $1$ (tame quadratic, discriminant $-3$) & $1$\\ $\Q_{3}(\zeta_{3})$ & yes & $0$ (trivial extension) & $1$\\ \bottomrule \end{tabular} \end{center} Both branches have $f=1$, hence $v_{3}\bigl(\Norm_{K/\Q}\dv_{L/K}\bigr)=1\cdot1+0\cdot1=1$. If $m\not\equiv\pm1\pmod9$, then $3$ is totally ramified in $K$ ($f=1$), $K_{3}$ does not contain $\zeta_{3}$, and $L_{3}/K_{3}$ is a tame quadratic ($3\nmid2$) with relative discriminant exponent again $1$. Merging the two cases with $\dv_{L/K}\mid(3)$: \[ \boxed{\;\Delta= \begin{cases} 0, & p\neq3,\\[2pt] 1, & p=3, \end{cases} \;=\;v_{p}(\operatorname{disc}K)\bmod 2.\;} \] \end{corollary} $\Longrightarrow$ (P) is proved on pure cubic fields (not merely supported), and $\Delta\le1$ is an absolute constant bound, far stronger than the $\operatorname{poly}(h)$ of \S\ref{sec:14.3}. Hence ``the discount is negligible'' holds on $S_{3}$-type tensor packets at $n=2$ \emph{without} any asymptotic assumption. \emph{Error mechanism of B17.} The guess $\Delta=3$ mistook ``the decomposition type of $3$'' for ``the number of times $\zeta_{3}$ must be adjoined'', accumulating one contribution per branch and ignoring that a branch already containing the root of unity contributes no relative discriminant. $\Delta$ is a valuation of a \emph{relative} discriminant, not a sum over decomposition branches; the $(1,2)$-type decomposition itself was correct, but the $\Delta=3$ derived from it does not hold. This belongs to the same family as B10 (``directional claim without proof''), but the error is a misjudged additivity in the local--global correspondence, not a wrong direction. \emph{Attribution and boundary.} The $192$ examples (including the $40$ with $m\equiv\pm1\pmod9$), the two branches' $(e,f)$ diagnostics, and the outputs of the relative-discriminant computation are all from the user-side temporary Sage copy; the package files were not run in Sage. Corollary~\ref{cor:14.2} is limited to $n=2$ pure cubics: the $A_{4}$ sample $(4,2)\Rightarrow\Delta\in\{0,2\}$ shows $\Dram=0$ does not extrapolate, and no closed form for $\Delta$ at $n\ge3$ has been derived or verified. \subsubsection*{Higher tensor packets ($n\ge3$): the geometric discount (A4)} The layer-boundary reservation of \S\ref{sec:14.3} --- no closed form for $\Delta$ at $n\ge3$ --- is now removed for the $S_{3}$ pure-cubic family, by an exact discriminant computation that needs no matrices. \begin{theorem}[$S_{3}$ family at every $n$]\label{thm:n3} Let $K=\Q(\sqrt[3]{m})$, $L=K(\zeta_{3})$, and write $\Delta_{1}:=v_{p}\bigl(\Norm_{K/\Q}\dv_{L/K}\bigr)\in\{0,1\}$ (Corollary~\ref{cor:14.2}). For every $n\ge2$: \begin{enumerate} \item $K^{\otimes n}\cong K\oplus L^{a_{n}}$ as $\Q$-algebras, where $a_{n}=\frac{3^{\,n-1}-1}{2}=1+3+\cdots+3^{\,n-2}$; \item the standard isomorphism $A_{n}\to B_{n}=K\oplus L^{a_{n}}$ is trace-preserving --- the proof of Lemma~\ref{lem:3.1} does not use the Galois hypothesis --- hence \[ v_{p}(\rho_{n}^{2})=(n-1)\,3^{\,n-1}\,v-a_{n}\,\Delta_{1}, \qquad\text{i.e.}\quad \Delta_{n}=a_{n}\cdot\Delta_{1}; \] \item $\Delta_{n}\equiv(n-1)\,3^{\,n-1}v\pmod 2$ is automatic, since $a_{n}\equiv n-1\pmod 2$ and $\Delta_{1}\equiv v\pmod 2$ (Corollary~\ref{cor:14.1}). \end{enumerate} \end{theorem} \begin{proof} (i) Induction: $K^{\otimes2}\cong K\oplus L$ (\S\ref{sec:14.2}), and since $L/K$ is Galois of degree $2$ and $L$ splits $K$ completely ($L$ is normal), $L\otimes K\cong L^{[K:\Q]}=L^{3}$; hence $K^{\otimes(n+1)}\cong(K\oplus L^{a_{n}})\otimes K\cong(K\oplus L)\oplus L^{3a_{n}}=K\oplus L^{a_{n+1}}$ with $a_{n+1}=1+3a_{n}$. (ii) The trace-preservation is Lemma~\ref{lem:3.1} verbatim. The discriminant of the source lattice is $\disc(\cO^{\otimes n})=\mathfrak d_{K}^{\,n\,3^{n-1}}$ (Kronecker formula, Lemma~\ref{lem:4.2}); the target lattice $B_{n}$ has $\disc=\disc(K)\cdot\disc(L)^{a_{n}}$ with $v_{p}\bigl(\disc(L)\bigr)=2v+\Delta_{1}$ by transitivity of the relative discriminant. Equating through trace-preservation (Theorem~\ref{thm:5.1}, \S5.2): $v_{p}(\rho_{n}^{2})=n\,3^{n-1}v-v-a_{n}(2v+\Delta_{1}) =(n-1)3^{n-1}v-a_{n}\Delta_{1}$. (iii) $a_{n}\bmod2$: $n-1$ even gives $3^{n-1}\equiv1\pmod8$, so $a_{n}$ even; $n-1$ odd gives $3^{n-1}\equiv3\pmod4$, so $a_{n}$ odd. Then use $\Delta_{1}\equiv3v\equiv v\pmod2$ from Corollary~\ref{cor:14.1}. \end{proof} \emph{Numerical anchor} (exact integer arithmetic, no matrices): for $K=\Q(\sqrt[3]{2})$, $v_{2}(\mathfrak d_{K})=2$, $v_{2}(\disc L)=4$, so $\Delta_{1}=0$ and $v_{2}(\rho_{3}^{2})=2\cdot9\cdot2-0=36$ --- the Galois formula (Theorem~\ref{thm:5.1}) is \emph{exact} at $n=3$; at $p=3$, $v_{3}(\mathfrak d_{K})=3$, $v_{3}(\disc L)=7$, $\Delta_{1}=1$, and $v_{3}(\rho_{3}^{2})=54-4=50$. The same computation at $n=4,5$ gives $\Delta_{4}=13$, $\Delta_{5}=40$, all parities even. \emph{Consequence for IUT.} Since $\Delta_{1}=0$ for $p\neq3$ (Corollary~\ref{cor:14.2}), Theorem~\ref{thm:n3} gives $\Delta_{n}=0$ for \emph{every} $n$: on $S_{3}$-type pure-cubic tensor packets at $p\neq3$, Theorem~\ref{thm:5.1} is exact at all tensor depths, with no asymptotic input. At $p=3$ the discount $a_{n}\Delta_{1}=a_{n}\le\frac{3^{n-1}}{2}$ grows only like the component count $d^{n-1}$, the same weight the main term carries. \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(iii) \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 Def 3.1(c) + Serre/Silverman & at bad multiplicative reduction $E_{v}$ is a Tate curve, and the $\ell$-torsion extension is tame of index dividing $\ell$ (Theorem~\ref{thm:tame})\\ 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} (tightened by Theorem~\ref{thm:tame})} The crude bound $\Delta\le N(1+v_{p}(N))$ with $N\asymp\ell^{4}$ of the previous draft improves as follows. The conjugates of $K=F(E[\ell])$ over $\Q$ are contained in $F^{\mathrm{cl}}(E[\ell])$, so the extra part of the Galois closure satisfies $[K^{\mathrm{cl}}_{v}:K_{v}]\le[F^{\mathrm{cl}}:F]$ --- $=1$ when $F/\Q$ is Galois, in particular for the normalization $[F:\Q]=2$ used throughout. At each completion Theorem~\ref{thm:tame} gives $e_{w}\mid\ell$ and $f_{w}\le\operatorname{ord}_{\ell}(p_{v}^{f_{v}})\le\ell-1$ (using $\ell\nmid n_{v}$, IUT I Def.~3.1(c)). Hence, for $p_{v}\notin\{2,3\}$ and $\ell\neq p_{v}$, \[ \Delta\le[F^{\mathrm{cl}}:F]\cdot(\ell-1)^{2}. \] The primes $p_{v}\in\{2,3\}$ keep Serre's general bound $v_{p}(\dv_{L/\Qp})\le[L:\Qp]\,(1+v_{p}([L:\Qp]))$ \cite[III \S6]{serre}; there are finitely many of them, with $v_{p}(N)=O(\log\ell)$. $\Delta$ remains a polynomial in $\ell$, hence in $h$ ($\ell\asymp h^{1/2}$) --- now of degree $2$ with unit constant instead of degree $4$ with a logarithmic factor. \subsubsection*{\texorpdfstring{$\star$}{*} Core comparison} \begin{center} \small \setlength{\tabcolsep}{4pt} \begin{tabular}{lllllll} \toprule $h$ & $\ell$ & $\Delta_{\max}$ & $\log_{10}\Delta_{\max}$ & $J$ & $\log_{10}\log_{p}\rho$ & share\\ \midrule $10^{2}$ & $10$ & $81$ & $1.91$ & $4$ & $1.39$ & $1.37$\\ $10^{4}$ & $100$ & $9801$ & $3.99$ & $49$ & $15.04$ & $0.27$\\ $10^{6}$ & $1000$ & $1.0\times10^{6}$ & $6.00$ & $499$ & $150.51$ & $0.040$\\ $10^{8}$ & $10^{4}$ & $1.0\times10^{8}$ & $8.00$ & $4999$ & $1505.15$ & $0.0053$\\ $10^{12}$ & $10^{6}$ & $1.0\times10^{12}$ & $12.00$ & $499999$ & $150515.00$ & $8.0\times10^{-5}$\\ \bottomrule \end{tabular} \end{center} (share $=\log_{10}\Delta_{\max}/\log_{10}\log_{p}\rho$; $\Delta_{\max}=(\ell-1)^{2}$; the previous draft's $\Delta_{\max}$ column --- $5.3\times10^{4}$, $7.9\times10^{8}$, $1.0\times10^{13}$, $1.2\times10^{17}$, $1.6\times10^{25}$ --- is improved by two to thirteen orders of magnitude.) The table takes $[F^{\mathrm{cl}}:F]=1$ ($F/\Q$ Galois); in general multiply $\Delta_{\max}$ by $[F^{\mathrm{cl}}:F]\le[F:\Q]$. At $h=10^{2}$ the share exceeds $1$: the discount is \emph{not} negligible at small scales --- asymptotic negligibility sets in from $h\gtrsim10^{4}$ (share $<1$), which is what the trend of the last column shows. \[ \frac{\Delta_{\max}}{\log_{p}\rho}=\frac{\operatorname{poly}(h)}{\exp(c\sqrt h)}\longrightarrow0. \] \emph{Stress test:} even blowing the bound up to $[F^{\mathrm{cl}}:F]^{2}(\ell-1)^{2}$ (far beyond any wild allowance), at $h=10^{6}$ one gets $4.0\times10^{6}$ versus $10^{150}$ (more precisely $\log_{p}\rho=\bar v\approx10^{150.51}$): still $144$ 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. The $\Delta_{\max}$ column of the previous draft (last row $1.6\times10^{25}$, from the preprint's formula $N(1+v_{2}(N))$ with $N=2.0\times10^{24}$, $v_{2}(N)=7$; the preprint's displayed $6.2\times10^{25}$ does not satisfy that formula and was corrected) is in turn superseded by the tame bound $\Delta_{\max}=(\ell-1)^{2}$ of Theorem~\ref{thm:tame}. 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 $\Delta\le\Delta_{\max}$; \item \emph{hierarchy boundary:} the main theorem is applied to tensor packets with $n=j+1\ge3$. For general non-Galois $K$ no closed form of $\Delta$ at $n\ge3$ has been derived or verified; the bound of this subsection uses the global degree $N$ of the Galois closure and is valid for general $n$. For the $S_{3}$ pure-cubic family the closed form now exists at every $n$ (Theorem~\ref{thm:n3}), so in that case --- which includes the $p\neq\ell$ situation relevant to IUT --- the step \emph{is} supported, indeed exactly; \item \emph{asymptoticity of the discount:} as the table of \S\ref{sec:14.2} shows, the single-prime relative discount can reach $50\%$. ``$\Delta\ll\log_{p}\rho$'' holds only as $h\to\infty$; at small $h$ the crossover position is given by \S\ref{sec:10.3}, and one may not infer per-prime negligibility from it; \item \emph{$\Delta$ has no general closed form:} E1QUICK refuted $\Delta=f(d,v)$, and \S\ref{sec:14.2} gives an analytic determination only on the $S_{3}$ pure-cubic family at $n=2$. Tensor packets at $n\ge3$ have a different component-set shape, and no closed form is available. This subsection uses only $\Delta\le\operatorname{poly}(h)$, no closed form, so the conclusions are unaffected; ``G4 closed'' should be read as ``$\Delta$'s upper bound is controlled'', not ``$\Delta$ is computed''. \end{itemize} \section{Gap list from the original text}\label{sec:15} Ordered by value. G1 is nearly closed (BD-class, \S\ref{sec:10.1b}), G3 is closed at layer 1 (\S\ref{sec:10.5}), G4/G5/G8 are closed (see rows); the only substantial gap is layer 2, i.e.\ G2/G7 (9-1). \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{nearly closed}: by IUT IV Cor.~2.2(i) the target is a BD-class of $h$ (\S\ref{sec:10.1b}); only the constant factor in the Szpiro/abc dictionary remains, immaterial up to $\kappa\le1000$ by \S\ref{sec:10.3}\\[2pt] G2 & precise form of Ind1/Ind2 and how hull saturates on the Ind track & the quantifiable part of layer 2 & \textbf{layer 1 closed} (\S\ref{sec:ind}, Theorem~\ref{thm:ind}): Ind2 preserves every per-component valuation and adds only $O(d^{\,j})$ to the log-volume; Ind3 is an inclusion (III Prop 3.5(ii)) and can only enlarge the hull; Ind1 is absorbed by the $\mathfrak{S}_{j}^{\pm}$-symmetry. Remainder: the layer-$2$ interaction with (9-1) (LANA)\\[2pt] G3 & explicit formula for $C_{\Theta}$ in IUT IV Thm 1.10 & the chain from qualitative inequality to quantitative one & \textbf{closed at layer 1} (\S\ref{sec:10.5}): the inflation enters as $\frac{4}{\llst+1}\log\widehat\rho$ on the LHS of \eqref{eq:Ctheta-corr}, and the chain breaks at the $(\mathrm{dsc})\to(\mathbf{hull}+\mathbf{det})$ arrow; remainder: (9-1) and \S\ref{sec:12} bookkeeping\\[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; on the $S_{3}$ pure-cubic family $\Delta$ is completely determined (Corollary~\ref{cor:14.2}) at every tensor depth (Theorem~\ref{thm:n3})\\[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 $\rank\le d\kappa\log_{10}h$ while the table was generated from the stronger, conservative criterion $\log_{10}\rho>h^{\kappa}$ (equivalently $\log_{10}\log_{10}\rho>\kappa\log_{10}h$); the definition is corrected to match the table (all table values and headline scales unchanged). Related: the ``gap'' row of \S\ref{sec:12} mixed the $\ln(abc)$ scale with the $c$-scale; units are now stated explicitly ($639$ in $\ln$-units; $\approx140$ orders in $c$)\\[2pt] B12 & three construction errors in the Sage scripts (found in review): \textbf{(1)} $\Q(\zeta_{16})^{+}$ was actually constructed as the whole cyclotomic field ($d=8$) rather than the real subfield ($d=4$); \textbf{(2)} \texttt{Cyclotomic\-Field(32).\allowbreak sub\-fields(8)[0][0]} does not guarantee $\Q(\zeta_{32})^{+}$; \textbf{(3)} the integral-basis coordinate change missed a matrix transpose & (3) miscomputes integral-basis coordinates; (1)--(2) mismatch the field labels with the actually constructed fields (each internally consistent, but mislabeled). The review copy fixed these: explicit real-subfield construction + transpose. After the fix, Part C gives $17/17$ matches, including $v_{2}(\dv_{\zeta_{32}^{+}})=31$ (once misrecorded as $65$) and $v_{2}(\rho)=124$\\[2pt] B13 & the $\Delta$ script carried an extra ``single-prime'' gate, skipping the $A_{4}$, $p=19$ and $S_{4}$, $p=283$ samples, so REGRESS passed only $3/5$; in addition \texttt{Fraction(SageInteger)} raised \texttt{TypeError}, and JSON wrote numbers as strings, crashing the analyzer & the gate mistook ``local-field interpretation'' for ``the defining condition of $\Delta$'': $\Delta$ is a difference of \emph{global} discriminants and is independent of the number of primes. Gate deleted (prediction $5/5$; an independent copy reproduced $5/5$), \texttt{int()} conversion, analyzer \texttt{normalize()}\\[2pt] B14 & the analyzer once reported ``strong support'' for ``$\Delta$ depends only on $(d,v)$'' from the QUICK data (the $A_{4}$/$p=19$ comparison point was then missing) & without the key comparison point, a single value within a group is misread as functional dependence; $11$ of $12$ groups being single-valued was a resolution problem. After the gate deletion the $(4,2)$ group shows $\{0,2\}$, refuting the claim; recorded: ``data can refute but cannot distinguish alternative explanations''\\[2pt] B15 & REGRESS still failed on the new ZIP: the literal \texttt{2} in \texttt{Fraction(int(d)*int(v),2)} was pre-parsed by Sage into \texttt{Integer} & \texttt{.sage} pre-parsing applies to \emph{all} integer literals, including denominators; B13 had only \texttt{int()}-converted the numerators. Fix: remove \texttt{fractions.Fraction} entirely and use ``doubled quantities'' all-integer arithmetic (\texttt{dv2}$=d\cdot v$ represents twice $(d\,v)/2$) to display half-integers; a user-side temporary copy ran $5/5$ with \texttt{int(2)}; the package removes the root cause per the former\\[2pt] B16 & the delivery package lacked version identification: two successive ZIPs were mounted as stale copies; when B15 claimed ``$0$ remaining'' the package comments still contained the word \texttt{Fraction}, so a user checking ``contains \texttt{Fraction} / no \texttt{dv2}'' misjudged it as the new package & the attribution chain was repeatedly broken and the same run was counted three times. The final package adds \texttt{VERSION.json} (per-file sha256) and \texttt{verify\_package.py}, self-certifying before runs; all rational-class wording (including comments) purged, the in-package \texttt{Fraction} count being exactly $0$\\[2pt] B17 & guessed $\Delta=3$ from the decomposition type: when $m\equiv\pm1\pmod9$, $3$ decomposes as $(1,2)$ in $K$, and two primes were guessed each to contribute one adjoining of $\zeta_{3}$, accumulating to $\Delta=3$ & left a nonexistent gap for (P) and misled the design of the ``minimal experiment''. The actual local decomposition is $\Q_{3}\times\Q_{3}(\zeta_{3})$: the branch already containing $\zeta_{3}$ contributes $0$, and only the other branch contributes the tame quadratic exponent $1$; the $40$ instances and the $192$-example scan alike give $1$. Corollary~\ref{cor:14.2} gives $\Delta=v_{p}(\operatorname{disc}K)\bmod2$\\ B18 & \S\ref{sec:8.4} claimed ``the layer-1 net effect is permanently inflationary; the contraction branch is unreachable in IUT's parameter space'' (repeated in the contributions list and in \S\ref{sec:16}) --- a directional overclaim of exactly the B10 family: $n_{v}=v_{p}(q_{v})$ and $v=v_{p}(\mathfrak d_{K_{w}})$ are independent parameters in IUT, and with $v$ at its tame bound $(\ell-1)^{2}$ (Theorem~\ref{thm:tame}) and $n_{v}=1$, the window $(0,v/2)$ is reachable and $\Net_{j}<1$ for every $j$. & the contraction branch exists; repaired by the two-branch criterion of \S\ref{sec:8.4} (inflationary branch: \eqref{eq:Ctheta-corr} fails for large $h$; contractionary branch: the final inequality is uninformative; neither yields abc) --- the conclusion survives in strengthened, exhaustive form. \emph{Caught by external review.}\\ \end{longtable}} \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}