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\title[On the irrationality of $\zeta_3(5)$]{On the irrationality of the $3$-adic zeta value $\zeta_3(5)$}
\author[On the irrationality of $\zeta_3(5)$]{Antti Hippel\"ainen}
\date{October 2026}

\begin{document}

\begin{abstract}
We give a proof that the $3$-adic zeta value $\zeta_3(5)$ is irrational. The proof adapts to $p=3$ a Hankel-determinant method that a recent preprint applies to $\zeta_2(7)$. A Volkenborn integral of an explicit rational function defines a linear functional with values in $\Q+\Q\,\zeta_3(5)$, and its Hankel determinants are the values at $\zeta_3(5)$ of polynomials with rational coefficients. We bound the $3$-adic valuation of these values and the size and the denominators of the coefficients. The new ingredients at $p=3$ are a triangular basis of the polynomials that are integer valued on $3^{-1}\Z_3^\times$, a local estimate at primes of the order of $n$ in which a pole of order $3$ coming from harmonic sums cancels exactly against a zero of order $3$ of the rational function, and a non-vanishing theorem at an auxiliary prime, in which Wilson's theorem makes one block of a reduced Gram matrix invertible. Here $n$ is the parameter of the rational function. Two constants of the estimates are bounded by finite computations, and with these bounds the $3$-adic valuation exceeds the size and the denominators together by more than $4.5$ bits per $n^2$. An appendix gives a second proof, independent of the first, which adapts to $\zeta_3(5)$ and the modular curve $X_0(3)$ a determinant argument that a preprint of Yi gives for $\zeta_7(3)$, and which rests on published theorems on overconvergent modular forms.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Ap\'ery proved that $\zeta(3)$ is irrational \cite{Ap79}, and Ball and Rivoal proved that infinitely many of the numbers $\zeta(3),\zeta(5),\zeta(7),\dots$ are irrational \cite{BR01}. Both results have $p$-adic counterparts. For a prime $p$ and an odd integer $s\ge3$, the $p$-adic zeta value is $\zeta_p(s)=L_p(s,\omega^{1-s})$, the value at $s$ of a Kubota--Leopoldt $p$-adic $L$-function, where $\omega$ is the Teichm\"uller character: for odd $p$ the character of $\Z_p^\times$ with $\omega(a)^{p-1}=1$ and $\omega(a)\equiv a\pmod p$, and for $p=2$ the character with $\omega(a)=\pm1\equiv a\pmod4$ (Section~\ref{sec:volkenborn} recalls the definitions). Calegari proved that $\zeta_2(3)$ and $\zeta_3(3)$ are irrational, using overconvergent $p$-adic modular forms \cite{Ca05}, and Beukers gave another proof \cite{Be08}. The irrationality of $\zeta_2(5)$ was proved by Calegari, Dimitrov and Tang \cite[Theorem~5.2]{CDT26}, with the arithmetic holonomy bounds of \cite{CDT24}, and Lai, Sprang and Zudilin gave a second proof \cite{LSZ26}.

Other results concern several values at once. Lai proved that at least one of $\zeta_2(7)$, $\zeta_2(9)$, $\zeta_2(11)$, $\zeta_2(13)$ is irrational \cite{La25}. Lai and Sprang proved that, for every prime $p$ and every $\epsilon>0$, at least $(c_p-\epsilon)\sqrt{s/\log s}$ of the values $\zeta_p(3),\zeta_p(5),\dots,\zeta_p(s)$ are irrational once the odd integer $s$ is large enough, with an explicit constant $c_p>0$ \cite[Theorem~1.1]{LS26}. Lai, Lupu and Sprang proved that for every prime $p\ge5$ some $\zeta_p(i)$ with odd $i\in[3,\,p+p/\log p+5]$ is irrational \cite{LLS25}. At $p=3$ these results give the irrationality of $\zeta_3(3)$, and of unspecified values among $\zeta_3(3),\zeta_3(5),\dots,\zeta_3(s)$ for large $s$; as far as we know, no refereed result singles out $\zeta_3(5)$. In this note we give a proof of the following statement.

\begin{theorem}\label{thm:main}
The number $\zeta_3(5)$ is irrational.
\end{theorem}

Theorem~\ref{thm:main} has been claimed before. Two manuscripts of Long, posted on GitHub in August 2026, state the irrationality of twenty-two $p$-adic zeta values, among them $\zeta_3(s)$ for $s=3,5,7,9,11$ \cite{Lo26,Lo26b}. An issue opened in the repository of \cite{Lo26} on 27 September 2026 reports a possible gap in its Proposition~5.10, and as of 19 October 2026 the issue has no reply. Our argument uses a different method and nothing from \cite{Lo26,Lo26b}. Several other preprints claiming the irrationality of individual $p$-adic zeta values appeared in August and September 2026, among them \cite{K26} for $\zeta_2(7)$, \cite{Ki26} for $\zeta_5(3)$ and \cite{Yi26} for $\zeta_7(3)$.

After the first version of this note was made public on 4 October 2026, Kilani, the author of \cite{K26}, posted a proof of Theorem~\ref{thm:main} on 7 October 2026 \cite{Kil26}. It applies the Hankel-determinant method of \cite{K26} to a rational function that has poles of order $4$ at the integers of an interval symmetric about $0$ and no zeros, and it proves non-vanishing at an auxiliary prime just above the right end of that interval. The preprint states that its proof is formalized in Lean~4, apart from the identification of the Volkenborn integral with the Kubota--Leopoldt value. Our proofs do not use \cite{Kil26}, and the second proof (Appendix~\ref{app:second}) uses a different method.

\subsection*{Use of large language models and status of the proofs}
This note was produced by large language models, instances of the Claude models of Anthropic running as autonomous agents, in September and October 2026. They adapted the method of the preprint \cite{K26} to $p=3$ and, for the second proof (Appendix~\ref{app:second}), the method that the preprint \cite{Yi26} applies to $\zeta_7(3)$, found and wrote the new steps, wrote and ran the programs of Appendices~\ref{app:comp} and~\ref{app:second}, and wrote the text. The arguments were checked only by other instances of the same models, acting as referees and recomputing the numbers with programs of their own; such checks may share the blind spots of the models that wrote the proof. No human expert has checked either proof.

The note has not been refereed. It is made public together with this account of how it was produced and checked.

\subsection*{Code availability}
The programs for the computations of Appendices~\ref{app:comp} and~\ref{app:second}, and for the exact computations of Section~\ref{sec:nonvanishing}, are available at \url{https://github.com/ahippelainen/zeta-3-5-irrational-public} under the MIT licence.

\subsection*{Conventions}
In Sections~\ref{sec:intro}--\ref{sec:nonvanishing} and Appendices~\ref{app:analytic}--\ref{app:comp} the letter $n$ always denotes an odd positive integer; Appendix~\ref{app:second} has its own notation. We write $\log$ for the natural logarithm; $\log_2$ and $\log_3$ count bits and trits, a trit being a factor $3$, that is, $\log_23$ bits. For a prime $p$, $v_p$ is the $p$-adic valuation, extended to polynomials $F=\sum_jf_jX^j\in\Q_p[X]$ as the Gauss valuation $v_p(F)=\min_jv_p(f_j)$, which satisfies $v_p(F+G)\ge\min(v_p(F),v_p(G))$ and $v_p(FG)\ge v_p(F)+v_p(G)$. We put $\|F\|_1=\sum_j|f_j|$, which satisfies $\|FG\|_1\le\|F\|_1\|G\|_1$. We write $[z^a]F$ for the coefficient of $z^a$ in $F$, $x^+=\max(x,0)$, and $[\,\cdot\,]$ for the indicator of a condition. Implied constants in $O(\cdot)$ and $o(\cdot)$ are absolute unless a dependence is indicated, and all asymptotic statements refer to $n\to\infty$ (in Appendix~\ref{app:second}, to $d\to\infty$). Limits of $3$-adic quantities are taken in the $3$-adic absolute value $|x|_3=3^{-v_3(x)}$; for instance $3^N\to0$ in $\Z_3$ as $N\to\infty$, because $|3^N|_3=3^{-N}$. For a set $E$, $\one_E$ is its indicator function; without a subscript, $\one$ is the trivial Dirichlet character.

\subsection*{The method}
Our proof adapts to $p=3$ a method that the preprint \cite{K26} applies to $\zeta_2(7)$, and which in turn transfers to $\Q_2$ the Hankel-determinant approach of Fauzan's preprint on $\zeta(5)$ \cite{Fa26}. For an odd integer $n$ let
\begin{equation}\label{eq:weightintro}
W(y)=\frac{\qty(y+\frac n2)\prod_{j=0}^{n-1}\qty(y+j+\frac13)^3\qty(y+j+\frac23)^3}{\prod_{k=-n}^{2n}(y+k)^4}\ .
\end{equation}
The numerator is a polynomial of degree $6n+1$ and the denominator one of degree $4(3n+1)$. With the Pochhammer symbol $(x)_n=x(x+1)\cdots(x+n-1)$ the numerator is $(y+\frac n2)(y+\frac13)_n^3(y+\frac23)_n^3$, and we use this shorter form below. For polynomials $P\in\Q[y]$ put
\begin{equation}\label{eq:functionalintro}
L(P)=\sum_{r=1,2}\vint(PW)'\qty(t+\frac r3)\dd t\ ,
\end{equation}
where the integral is the Volkenborn integral (Section~\ref{sec:volkenborn}). The values of $L$ lie in $\Q+\Q\,\zeta_3(5)$ (Lemma~\ref{lem:purity}). Hence, for $K\le3n+1$, the Hankel determinant $\det[L(y^{i+j})]_{i,j<K}$ is the value at $X=\zeta_3(5)$ of a polynomial $\Delta_K(X)\in\Q[X]$ of degree at most $K$ (Definition~\ref{def:hankel}).

Suppose that $\zeta_3(5)=a/b$ and that $\Delta_K(a/b)\ne0$. Clearing the denominators of $\Delta_K(a/b)$ gives a nonzero integer, and comparing its $3$-adic valuation with its absolute value gives (Lemma~\ref{lem:criterion})
\begin{equation}\label{eq:criterionintro}
v_3\qty(\Delta_K(\zeta_3(5)))\log_23\le\log_2\|\Delta_K\|_1+\sum_{p\ne3}\qty(-v_p(\Delta_K))^+\log_2p+K\log_2\max(|a|,b)\ .
\end{equation}
We call the left side the \textit{gain}; the first two terms on the right are the \textit{size} and the \textit{denominators} of $\Delta_K$. With $K=\kappa n$, and up to terms of lower order, the gain is at least $G(\kappa)n^2$ (Theorem~\ref{thm:gain}), the size is at most $(\kappa^2-6\kappa)n^2\log_2n+F^\dagger n^2$ (Theorem~\ref{thm:archmain}), and the denominators are at most $\kappa(6-\kappa)n^2\log_2n+c\,n^2$ (Theorem~\ref{thm:denmain}). The terms of order $n^2\log n$ cancel, and the last term of \eqref{eq:criterionintro} is $O(n)$. The constants $F^\dagger$ and $c$ depend on $\kappa$, and on the range $\kappa\in[2.8,3]$ that we need, $G-F^\dagger-c$ is at least $4.58$ bits (Table~\ref{tab:margins}). For $n$ large this contradicts \eqref{eq:criterionintro}.

The inequality \eqref{eq:criterionintro} needs $\Delta_K(a/b)\ne0$. For the Hankel determinants of a positive measure on $\R$ this would follow from positivity, but $L$ is a $3$-adic functional, and no such argument is available. Following \cite{K26}, we choose an auxiliary prime $p\nmid b$ with $2.1n+2\le p\le2.159n$, and we show that the constant coefficient of $\Delta_K$ has strictly smaller $p$-adic valuation than all its other coefficients (Theorem~\ref{thm:nonvanishing}). The choice of $p$ fixes $K$, and the range of $p$ confines $\kappa$ to $[2.8,3]$.

\subsection*{What is new}
The overall structure and several of the estimates are those of \cite{K26}. We give our own proofs of the statements that we take from there (Appendices~\ref{app:analytic}--\ref{app:local}), so that the argument should not depend on the correctness of \cite{K26}. Three steps do not carry over from $p=2$. The first is the change of basis in the $3$-adic estimate: at $p=2$ it contributes a positive term of order $K^2$, while at $p=3$ the natural basis of the polynomials that are integer valued on the set $3^{-1}\Z_3^\times$ of integration points (Lemma~\ref{lem:basis}) costs about $K^2/4$ trits. The second is a local estimate at the primes of the order of $n$ (Lemma~\ref{lem:refined}), which determine the constant $c$. At such a prime, the harmonic sums that the Volkenborn integral produces behave like poles of order $3$ at points of $\frac p3\Z$, and when such a point is a zero of $W$, which has order $3$ as well, the two cancel exactly. In a numerical evaluation at $\kappa=3$ with the cruder local bound of \cite{K26} in its place, the comparison no longer leads to a contradiction (Remark~\ref{rem:crude}). The third is the non-vanishing theorem (Section~\ref{sec:nonvanishing}). There, after a change of basis and a rescaling by powers of $\sqrt p$, the constant coefficient of $\Delta_K$ becomes, modulo $\sqrt p$ in the ring $\Z_p[\sqrt p]$, a product of explicit block determinants. The block that belongs to a pair of nodes (the poles $y=-k$, $-n\le k\le2n$, of $W$) congruent modulo $p$ is invertible because the leading coefficients of $W$ at the two nodes have the ratio $-8$ modulo $p$, by Wilson's theorem (Lemma~\ref{lem:wilson}); the ratio $1$ would make it singular (Remark~\ref{rem:wilsonneeded}).

The constants $F^\dagger$ and $c$ are upper bounds obtained by finite computations: $F^\dagger$ in ball arithmetic, in which every real number is enclosed in an interval given by a midpoint and a radius and every operation returns an interval that contains the exact result \cite{Jo17}, and $c$ in exact rational arithmetic, up to a final evaluation of Euler's constant and of three logarithms in ball arithmetic (Appendix~\ref{app:comp}). The computations evaluate expressions that the lemmas of Section~\ref{sec:costs} show to be upper bounds. All other steps are proved in the text, and the coefficients and determinants needed in Section~\ref{sec:nonvanishing} are small exact computations that can be checked by hand.

Section~\ref{sec:functional} writes $\zeta_3(5)$ as a Volkenborn integral and defines the functional and the Hankel polynomial. Section~\ref{sec:plan} states the criterion and the four estimates and deduces Theorem~\ref{thm:main} from them, and Sections~\ref{sec:gain}, \ref{sec:costs} and~\ref{sec:nonvanishing} prove the estimates. The longer proofs are in the appendices: Appendix~\ref{app:analytic} treats the analytic step of the $3$-adic estimate, Appendix~\ref{app:heine} the Heine expansion, the tree bound and the size, Appendix~\ref{app:local} the local bounds and the prime sum, Appendix~\ref{app:omega} the case analysis of the non-vanishing theorem, and Appendix~\ref{app:comp} the computations. Appendix~\ref{app:second} gives a second proof of Theorem~\ref{thm:main} by a different method, the determinant argument that \cite{Yi26} uses for $\zeta_7(3)$ on $X_0(7)$, carried over to the modular curve $X_0(3)$. It uses from the rest of the note only the statement of the theorem, the normalization \eqref{eq:zeta35def} and some notation, so that a reader can check the theorem along two independent routes, but it depends on published theorems on overconvergent modular forms that the main proof does not need.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The functional and its Hankel polynomial}\label{sec:functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This section fixes the normalization of $\zeta_3(5)$, writes it as a Volkenborn integral, and shows that the functional \eqref{eq:functionalintro} takes values in $\Q+\Q\,\zeta_3(5)$. The Volkenborn integral is not translation invariant, and the defect of a translation by an integer is a harmonic sum (Lemma~\ref{lem:translation}). These harmonic sums are harmless at the prime $3$, but at the other primes they produce the virtual poles of Section~\ref{sec:singlelevel}.

\subsection{The Volkenborn integral and \texorpdfstring{$\zeta_3(5)$}{zeta3(5)}}\label{sec:volkenborn}
We use the following standard result, for which we recall the notation. $\C_p$ is the completion of an algebraic closure of $\Q_p$, and we fix embeddings of the algebraic numbers into $\C$ and into $\C_p$. A Dirichlet character $\chi$ modulo $N$ is a homomorphism $(\Z/N\Z)^\times\to\C^\times$, extended to $\Z$ by $\chi(a)=0$ for $\gcd(a,N)>1$. Its \textit{conductor} is the smallest divisor $f$ of $N$ such that $\chi$ factors through $(\Z/f\Z)^\times$, and we always use the primitive character modulo $f$ that induces $\chi$; the trivial character $\one$ has conductor $1$. The Bernoulli numbers $B_j$ are defined by $t/(e^t-1)=\sum_jB_jt^j/j!$, and the \textit{generalized Bernoulli numbers} of a character $\chi$ of conductor $f_\chi$ by $\sum_{a=1}^{f_\chi}\chi(a)\,te^{at}/(e^{f_\chi t}-1)=\sum_kB_{k,\chi}t^k/k!$ \cite[Section~4.1]{Wa97}. The \textit{Teichm\"uller character} $\omega:\Z_p^\times\to\Z_p^\times$ sends $a$ to the unique $(p-1)$-st root of unity congruent to $a$ modulo $p$; it depends only on $a$ modulo $p$, so it is a Dirichlet character of conductor $p$, and at $p=3$ it is the nontrivial character modulo $3$, $\omega(a)=\pm1\equiv a\pmod3$. Finally $\langle a\rangle=a/\omega(a)\in1+p\Z_p$ for $a\in\Z_p^\times$.

\begin{theorem}[{\cite[Section~5.2]{Wa97}}]\label{thm:washington}
Let $p$ be an odd prime, $\chi$ a Dirichlet character of conductor $f_\chi$, and $F$ a common multiple of $p$ and $f_\chi$. There is a $p$-adic meromorphic function $L_p(s,\chi)$ on $\{s\in\C_p:|s|_p<p^{(p-2)/(p-1)}\}$, analytic if $\chi\ne\one$ and with only a simple pole at $s=1$ if $\chi=\one$, such that
\begin{equation}\label{eq:interpolation}
L_p(1-k,\chi)=-\qty(1-\chi\omega^{-k}(p)\,p^{k-1})\frac{B_{k,\chi\omega^{-k}}}k\qquad(k\ge1)\ ,
\end{equation}
and
\begin{equation}\label{eq:washington}
L_p(s,\chi)=\frac1F\,\frac1{s-1}\sum_{\substack{1\le a\le F\\ p\nmid a}}\chi(a)\langle a\rangle^{1-s}\sum_{j\ge0}\binom{1-s}j\qty(\frac Fa)^jB_j\ .
\end{equation}
\end{theorem}

In \eqref{eq:interpolation}, $\chi\omega^{-k}$ denotes the character of smallest conductor attached to the product, so that $\chi\omega^{-k}(p)=0$ if $p$ divides its conductor. The interpolation property \eqref{eq:interpolation} determines $L_p(s,\chi)$ and is the reason why $L_p$ is called the $p$-adic $L$-function of $\chi$; in the proof we use only the formula \eqref{eq:washington}. The $p$-adic zeta values are $\zeta_p(s)=L_p(s,\omega^{1-s})$. At $p=3$ the Teichm\"uller character has order $2$, so $\omega^{-4}=\one$ and
\begin{equation}\label{eq:zeta35def}
\zeta_3(5)=L_3(5,\one)\ .
\end{equation}

A function $f:\Z_3\to\Q_3$ is \textit{strictly differentiable} if the difference quotients $(f(x)-f(y))/(x-y)$, $x\ne y$, converge as $(x,y)\to(a,a)$, for every $a\in\Z_3$; the limit is $f'(a)$. Polynomials are strictly differentiable, and so is every power series $\sum_me_mt^m$ that converges on a disc of radius larger than $1$, because its difference quotient $\sum_me_m\sum_{i<m}x^iy^{m-1-i}$ converges uniformly on $\Z_3^2$. For such $f$ the \textit{Volkenborn integral}
\begin{equation}\label{eq:volkenborn}
\vint f(t)\,\dd t=\lim_{N\to\infty}3^{-N}\sum_{t=0}^{3^N-1}f(t)
\end{equation}
exists \cite{Vo72}, \cite[Ch.~5]{Ro00}. We need three of its properties.

\begin{lemma}\label{lem:volkenbornfacts}
Let $f:\Z_3\to\Q_3$ be strictly differentiable.
\begin{itemize}
\item[(V1)] $\vint f(t+1)\,\dd t=\vint f(t)\,\dd t+f'(0)$. Hence, for $N\ge1$, $\vint f(t+N)\,\dd t=\vint f+\sum_{i=0}^{N-1}f'(i)$ and $\vint f(t-N)\,\dd t=\vint f-\sum_{i=1}^Nf'(-i)$.
\item[(V2)] $\vint\binom tc\,\dd t=(-1)^c/(c+1)$ for $c\ge0$, and $\vint t^c\,\dd t=B_c$.
\item[(V3)] If $|f(0)|_3\le A$ and $|f(x)-f(y)|_3\le A|x-y|_3$ for all $x,y\in\Z_3$, then $|\vint f|_3\le3A$.
\end{itemize}
\end{lemma}

\begin{proof}
(V1) The $N$-th Riemann sum of $f(t+1)-f(t)$ telescopes to $3^{-N}(f(3^N)-f(0))=\frac{f(3^N)-f(0)}{3^N-0}$. This is a difference quotient of $f$ at the two points $3^N$ and $0$, and $3^N\to0$ in $\Z_3$, since $|3^N|_3=3^{-N}$. So it tends to $f'(0)$ by strict differentiability. Apply this to $f(t+i)$, $i=0,\dots,N-1$, and add. For the second formula apply the first to $g(t)=f(t-1)$, which gives $\vint f=\vint g(t+1)\,\dd t=\vint f(t-1)\,\dd t+f'(-1)$, and iterate.

(V2) By $\sum_{t<T}\binom tc=\binom T{c+1}$, the $N$-th Riemann sum is $3^{-N}\binom{3^N}{c+1}=\frac1{c+1}\binom{3^N-1}c$. As $N\to\infty$, $3^N-1\to-1$ in $\Z_3$, and $x\mapsto\binom xc$ is continuous, so the limit is $\frac1{c+1}\binom{-1}c=\frac{(-1)^c}{c+1}$. For the monomials, Faulhaber's formula gives $3^{-N}\sum_{t<3^N}t^c=\sum_{i=0}^c\binom{c+1}iB_i\,3^{N(c-i)}/(c+1)$, and only the term $i=c$ survives the limit.

(V3) Let $\Sigma_N$ be the $N$-th Riemann sum. Then
\begin{equation}\label{eq:riemanndiff}
\Sigma_{N+1}-\Sigma_N=3^{-N-1}\sum_{t<3^N}\sum_{a=0}^2\qty(f(t+a3^N)-f(t))\ ,
\end{equation}
and every bracket has absolute value at most $A3^{-N}$. By the ultrametric inequality $|\Sigma_{N+1}-\Sigma_N|_3\le3^{N+1}A3^{-N}=3A$, and $|\Sigma_0|_3=|f(0)|_3\le A$. So $|\Sigma_N|_3\le3A$ for all $N$, and also in the limit.
\end{proof}

Property (V3) is the form in which we use the integral $3$-adically: a function whose values and Lipschitz constant are $3$-adically small has a small integral, with the loss of one factor $3$. We shall apply it to power series $\sum_me_mt^m$ whose coefficients tend to $0$ fast enough for convergence on a disc of radius larger than $1$. For such a series, $|f(0)|_3$ and the Lipschitz constant on $\Z_3$ are at most $\max_m|e_m|_3$, since $|x^m-y^m|_3\le|x-y|_3$ for $x,y\in\Z_3$.

The integration points of our functional are $t+\frac r3$ with $t\in\Z_3$ and $r\in\{1,2\}$, that is, the points of
\begin{equation}\label{eq:setS}
S=3^{-1}\Z_3^\times=\{y\in\Q_3:\ v_3(y)=-1\}\ .
\end{equation}
The substitution $t\mapsto-1-t$ maps $t+\frac r3$ to $-(t+\frac{3-r}3)$, so it acts on $S$ as $y\mapsto-y$ and exchanges the two classes $\frac13+\Z_3$ and $\frac23+\Z_3$; this removes the even zeta values.

\begin{lemma}[reflection]\label{lem:reflection}
Let $f(t)=\sum_{i\ge0}e_it^i$ with $v_3(e_i)\ge i+c_0$ for a constant $c_0$. Then $\vint f(-1-t)\,\dd t=\vint f(t)\,\dd t$. Consequently, for every odd $M\ge1$,
\begin{equation}\label{eq:reflection}
\sum_{r=1,2}\vint\qty(t+\frac r3)^{-M}\dd t=0\ .
\end{equation}
\end{lemma}

\begin{proof}
For $f=t^k$, (V2) gives $\vint(t+x)^k\,\dd t=\sum_i\binom kix^{k-i}B_i=B_k(x)$, the Bernoulli polynomial, so $\vint(-1-t)^k\,\dd t=(-1)^kB_k(1)=B_k(0)=\vint t^k\,\dd t$ by $B_k(1-x)=(-1)^kB_k(x)$. By linearity the identity holds for polynomials. For the series, split $f=f_{<J}+f_{\ge J}$ at a degree $J$. The tail $f_{\ge J}(t)$ and $f_{\ge J}(-1-t)$ are power series with coefficients of valuation at least $J+c_0$ (the coefficients of $(-1-t)^i$ are integers), so by (V3) both have integrals of absolute value at most $3^{1-J-c_0}$. Letting $J\to\infty$ proves the first claim. For the second, $g_r(t)=(t+\frac r3)^{-M}=(\frac3r)^M(1+\frac{3t}r)^{-M}$ is a series of this type, and $g_r(-1-t)=(-1)^Mg_{3-r}(t)$. So the sum $\mathcal J_M=\sum_r\vint g_r$ satisfies $\mathcal J_M=(-1)^M\mathcal J_M$, and $\mathcal J_M=0$ for odd $M$.
\end{proof}

\begin{lemma}[the value]\label{lem:value}
For even $m\ge2$,
\begin{equation}\label{eq:value}
\sum_{r=1,2}\vint\qty(t+\frac r3)^{-m}\dd t=m\,3^{m+1}L_3(m+1,\one)\ .
\end{equation}
In particular $\zeta_3(5)=\frac1{972}\sum_{r=1,2}\vint(t+\frac r3)^{-4}\,\dd t$.
\end{lemma}

\begin{proof}
Take $p=F=3$, $\chi=\one$ and $s=m+1$ in \eqref{eq:washington}. For $a\in\{1,2\}$ we have $\omega(a)=\pm1$, and $m$ is even, so $\langle a\rangle^{-m}=a^{-m}$ and
\begin{equation}\label{eq:washington3}
3m\,L_3(m+1,\one)=\sum_{a=1,2}a^{-m}\sum_{c\ge0}\binom{-m}c\qty(\frac3a)^cB_c\ .
\end{equation}
On the other side, $(t+\frac a3)^{-m}=(\frac3a)^m(1+\frac{3t}a)^{-m}=(\frac3a)^m\sum_{c\ge0}\binom{-m}c(\frac3a)^ct^c$. We integrate term by term with $\vint t^c\,\dd t=B_c$. This is legitimate: the tail $\sum_{c\ge J}$ of the series has coefficients of valuation at least $m+J$, so by (V3) its integral has absolute value at most $3^{1-m-J}$, which tends to $0$. Hence $\sum_a\vint(t+\frac a3)^{-m}\,\dd t=3^m\cdot3m\,L_3(m+1,\one)$. Finally, $4\cdot3^5=972$ and $L_3(5,\one)=\zeta_3(5)$ by \eqref{eq:zeta35def}.
\end{proof}

For $k\in\Z$ and $M\ge2$ put
\begin{equation}\label{eq:hs}
\HS_3(k,M)=\sum_{\substack{0<u<3k\\3\nmid u}}\qty(\frac3u)^M\quad(k>0)\ ,\qquad\HS_3(k,M)=-\sum_{\substack{3k<u<0\\3\nmid u}}\qty(\frac3u)^M\quad(k<0)\ ,
\end{equation}
and $\HS_3(0,M)=0$. Every term has $3$-adic valuation $M$, but at a prime $p\ne3$ the terms with $p\mid u$ have $p$-adic valuation $-M$.

\begin{lemma}[translation]\label{lem:translation}
For every $k\in\Z$ and $m\ge1$,
\begin{equation}\label{eq:translation}
\sum_{r=1,2}\vint\qty(t+\frac r3+k)^{-m}\dd t=\begin{cases}m\,3^{m+1}L_3(m+1,\one)-m\,\HS_3(k,m+1)\ ,&m\text{ even}\ ,\\[0.5ex]-m\,\HS_3(k,m+1)\ ,&m\text{ odd}\ .\end{cases}
\end{equation}
\end{lemma}

\begin{proof}
Apply (V1) to $f_r(t)=(t+\frac r3)^{-m}$, whose derivative is $f_r'(i)=-m(i+\frac r3)^{-m-1}=-m\,3^{m+1}(3i+r)^{-m-1}$. For $k>0$, (V1) adds $\sum_{i<k}f_r'(i)$, and as $r\in\{1,2\}$ and $0\le i<k$ vary, $u=3i+r$ runs exactly once over the integers $0<u<3k$ prime to $3$; this gives $-m\HS_3(k,m+1)$. For $k<0$, (V1) subtracts $\sum_{i=1}^{|k|}f_r'(-i)$, and $u=-3i+r$ runs over $3k<u<0$, $3\nmid u$, which explains the sign in \eqref{eq:hs}. The value at $k=0$ is Lemma~\ref{lem:value} for even $m$ and Lemma~\ref{lem:reflection} for odd $m$.
\end{proof}

\subsection{The functional and the Hankel polynomial}\label{sec:weightfun}
Lemma~\ref{lem:translation} shows why the functional \eqref{eq:functionalintro} takes values in $\Q+\Q\,\zeta_3(5)$. Let $W$ be a rational function whose poles are integers, all of order at most $4$. If $PW$ decays fast enough at infinity, it is the sum of its principal parts, and after the derivative in \eqref{eq:functionalintro} a pole of order $i$ of $PW$ becomes a pole of order $i+1$. By Lemma~\ref{lem:translation} with $m=i+1$, such a term contributes a harmonic sum and, for odd $i$, a multiple of $L_3(i+2,\one)$. So only $\zeta_3(3)=L_3(3,\one)$, from the simple poles, and $\zeta_3(5)$, from the poles of order $3$, can occur. The coefficient of $\zeta_3(3)$ is a multiple of the sum of the residues of $PW$, which vanishes when $PW$ decays fast enough at infinity; this is the reason for the derivative. So only $\zeta_3(5)$ survives (Lemma~\ref{lem:purity}). The zeros of $W$ do not affect this, and we choose them for the estimates of Sections~\ref{sec:gain} and~\ref{sec:costs}.

The \textit{nodes} are the integers $k\in\cN:=\{-n,-n+1,\dots,2n\}$, and the \textit{weight} is the rational function $W$ of \eqref{eq:weightintro}. The node $k$ sits at $y=-k$ and is a pole of order exactly $4$; since $n$ is odd, the zero $y=-\frac n2$ is not a node. The other zeros are the points $y=-(j+\frac r3)$, $0\le j<n$, $r\in\{1,2\}$, each of order $3$. The degree of $W$, the degree of its numerator minus that of its denominator, is $1+6n-4(3n+1)=-6n-3$. The functional $L$ of \eqref{eq:functionalintro} is well defined: the poles of $t\mapsto(PW)(t+\frac r3)$ are the points $t=-k-\frac r3$, all of absolute value $3$, so its Taylor series at $t=0$ converges on $|t|_3<3$, and the same holds for its derivative.

The weight is that of \cite{K26} with the prime $2$ replaced by $3$: there the poles have order $4$ at the same nodes, and the zeros lie on the class $\frac12+\Z$. Here the zeros, of order $3$, lie on both classes $\frac13+\Z$ and $\frac23+\Z$. Then on $S$ one of the two Pochhammer symbols becomes an integer-valued polynomial times $n!$, which produces the factor $(n!)^3$ in the $3$-adic gain (Section~\ref{sec:gain}), and at the primes $p\ne3$ the zeros cancel the virtual poles produced by the harmonic sums (Section~\ref{sec:singlelevel}). We keep the factor $y+\frac n2$ of \cite{K26}, which makes $W$ antisymmetric about the centre of the nodes, $W(-n-y)=-W(y)$. Our argument does not use this symmetry, and the factor affects the estimates only through terms of lower order and through the two nodes $\frac{n\pm p}2$ of Section~\ref{sec:nonvanishing}, where $p$ is the auxiliary prime of Theorem~\ref{thm:nonvanishing}.

For a node $k$ put
\begin{equation}\label{eq:wk}
\Wt_k(z)=z^4\,W(-k+z)\ ,\qquad H_k[b]=[z^b]\,\Wt_k(z)\ ,\qquad h_k=H_k[0]=\Wt_k(0)\ne0\ ,
\end{equation}
so that $\Wt_k$ is holomorphic at $z=0$ and its Taylor coefficients give the principal part of $W$ at the node. Explicitly,
\begin{equation}\label{eq:hk}
h_k=\frac{\qty(\frac n2-k)\prod_{0\le j<n,\,r=1,2}\qty(j+\frac r3-k)^3}{\prod_{m\in\cN\setminus\{k\}}(m-k)^4}\ ,\qquad\prod_{m\in\cN\setminus\{k\}}(m-k)=(-1)^{k+n}(k+n)!\,(2n-k)!\ .
\end{equation}
For a polynomial $P$ let $P_a(k)=[z^a]P(-k+z)$ be its Taylor coefficients at the node $k$.

\begin{lemma}[purity]\label{lem:purity}
If $\deg(PW)\le-2$, then
\begin{equation}\label{eq:purity}
L(P)=\sum_{k\in\cN}\sum_{a=0}^3c_{k,a}\,P_a(k)\ ,\qquad c_{k,a}=\beta_{k,a}+\zeta_3(5)\,\alpha_{k,a}\ ,
\end{equation}
with the rational \textit{local constants}
\begin{equation}\label{eq:localconstants}
\beta_{k,a}=\sum_{i=1}^{4-a}i(i+1)\,\HS_3(k,i+2)\,H_k[4-i-a]\ ,\qquad\alpha_{k,a}=\begin{cases}-2916\,H_k[1-a]&(a\le1)\ ,\\0&(a\ge2)\ .\end{cases}
\end{equation}
In particular, $L(y^m)\in\Q+\Q\,\zeta_3(5)$ for every $m\le6n+1$.
\end{lemma}

\begin{proof}
Since $\deg(PW)<0$, the rational function $PW$ is the sum of its principal parts at the nodes:
\begin{equation}\label{eq:partialfractions}
P(y)W(y)=\sum_{k\in\cN}\sum_{i=1}^4\frac{\mathrm{res}_{k,i}}{(y+k)^i}\ ,\qquad \mathrm{res}_{k,i}=\sum_{b\le4-i}P_b(k)\,H_k[4-i-b]\ ,
\end{equation}
the formula for $\mathrm{res}_{k,i}$ being the coefficient of $z^{-i}$ in $P(-k+z)W(-k+z)=z^{-4}P(-k+z)\Wt_k(z)$. Differentiate once, put $y=t+\frac r3$, integrate, sum over $r$ and apply Lemma~\ref{lem:translation} with $m=i+1$. The term $(k,i)$ contributes $i(i+1)\,\HS_3(k,i+2)\,\mathrm{res}_{k,i}$ and, for odd $i$, also $-i(i+1)\,3^{i+2}L_3(i+2,\one)\,\mathrm{res}_{k,i}$. For $i=1$ the latter is $-54\,\zeta_3(3)\,\mathrm{res}_{k,1}$, and $\sum_k\mathrm{res}_{k,1}$ is the sum of all residues of $PW$, which is zero because $PW(y)=O(y^{-2})$ as $y\to\infty$. For $i=3$ it is $-3\cdot4\cdot3^5\zeta_3(5)\,\mathrm{res}_{k,3}=-2916\,\zeta_3(5)\,\mathrm{res}_{k,3}$, and $\mathrm{res}_{k,3}=P_0(k)H_k[1]+P_1(k)H_k[0]$. Collecting the coefficient of each $P_a(k)$ gives \eqref{eq:localconstants}. Finally, $\deg(y^mW)=m-6n-3$, which is at most $-2$ when $m\le6n+1$.
\end{proof}

Lemma~\ref{lem:purity} writes $L(P)=\beta(P)+\zeta_3(5)\,\alpha(P)$ with rational $\beta(P)=\sum_{k,a}\beta_{k,a}P_a(k)$ and $\alpha(P)=\sum_{k,a}\alpha_{k,a}P_a(k)$. If $\zeta_3(5)$ were rational, the value $L(P)$ alone would not determine $\alpha(P)$ and $\beta(P)$, and the polynomial we work with must not depend on such a choice. We therefore define it by the local constants, with an indeterminate $X$ in place of $\zeta_3(5)$.

\begin{definition}[the Hankel polynomial]\label{def:hankel}
For $k\in\cN$ put $c_{k,a}(X)=\beta_{k,a}+X\alpha_{k,a}\in\Q[X]$ for $0\le a\le3$, and $c_{k,a}=0$ for $a\ge4$. Define $\Q[X]$-linear maps on $\Q[X][y]$ by
\begin{equation}\label{eq:lambda}
\Lambda_k(P)=\sum_{a=0}^3c_{k,a}(X)\,P_a(k)\ ,\qquad\Lambda=\sum_{k\in\cN}\Lambda_k\ ,\qquad\Lambda(P)=\beta(P)+X\,\alpha(P)\ .
\end{equation}
For $K\ge1$ the \textit{Hankel polynomial} is
\begin{equation}\label{eq:hankel}
\Delta_K(X)=\det\qty[\Lambda(y^{i+j})]_{0\le i,j<K}=\det\qty[\beta(y^{i+j})+X\,\alpha(y^{i+j})]_{0\le i,j<K}\in\Q[X]\ .
\end{equation}
\end{definition}

Every entry of the matrix in \eqref{eq:hankel} is affine in $X$, so $\deg\Delta_K\le K$. If $K\le3n+1$, then $2K-2\le6n$, so by Lemma~\ref{lem:purity} the entries at $X=\zeta_3(5)$ are the moments $L(y^{i+j})$, and
\begin{equation}\label{eq:hankelvalue}
\Delta_K(\zeta_3(5))=\det\qty[L(y^{i+j})]_{0\le i,j<K}\qquad(K\le3n+1)\ .
\end{equation}
The $3$-adic estimate concerns the right side, a determinant of $3$-adic numbers; the other estimates concern the coefficients of $\Delta_K(X)$.

By the Leibniz rule for Taylor coefficients, $(y^{i+j})_m(k)=\sum_{a+b=m}(y^i)_a(k)(y^j)_b(k)$, so
\begin{equation}\label{eq:jetform}
\Lambda(y^{i+j})=\sum_{k\in\cN}\sum_{a+b\le3}c_{k,a+b}\,(y^i)_a(k)\,(y^j)_b(k)\ ,\qquad(y^i)_a(k)=\binom ia(-k)^{i-a}\ .
\end{equation}
The Hankel matrix is therefore a sum of $3n+1$ matrices of rank at most $4$, one for each node, and the node $k$ enters only through its $4\times4$ matrix of local constants $C_k=[c_{k,a+b}]_{0\le a,b\le3}$, which vanishes below the anti-diagonal. The Heine expansion of Section~\ref{sec:heine} expands $\Delta_K$ over the ways of distributing its $K$ rows among the nodes, the denominator estimates bound each node separately, and the non-vanishing theorem chooses a basis of polynomials in which the nodes decouple modulo the auxiliary prime.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The criterion and the proof of Theorem~\ref{thm:main}}\label{sec:plan}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we state the criterion, state the four estimates that the rest of the note proves, and deduce Theorem~\ref{thm:main} from them. The criterion is the product formula for the integer obtained by clearing the denominators of $\Delta_K(a/b)$.

\begin{lemma}[criterion]\label{lem:criterion}
Let $F\in\Q[X]$ be nonzero with $\deg F\le K$, and let $a,b\in\Z$ be coprime with $b>0$ and $F(a/b)\ne0$. Then
\begin{equation}\label{eq:criterion}
v_3\qty(F(a/b))\log_23\le\log_2\|F\|_1+\sum_{p\ne3}\qty(-v_p(F))^+\log_2p+K\log_2\max(|a|,b)\ ,
\end{equation}
where the sum runs over all primes $p\ne3$, including $p=2$.
\end{lemma}

\begin{proof}
Only finitely many $v_p(F)$ are nonzero. Let $c_F=\prod_pp^{v_p(F)}$ and $Q=F/c_F$. Then $v_p(Q)=0$ for every $p$, so $Q\in\Z[X]$, and $N=b^KQ(a/b)=\sum_jq_ja^jb^{K-j}$ is an integer, nonzero because $F(a/b)\ne0$. Since $v_3(b)\ge0$,
\begin{equation}\label{eq:criterionproof}
v_3(F(a/b))-v_3(F)=v_3(Q(a/b))\le v_3(N)\le\log_3|N|\le\log_3\|Q\|_1+K\log_3\max(|a|,b)\ ,
\end{equation}
and $\log_3\|Q\|_1=\log_3\|F\|_1-v_3(F)-\sum_{p\ne3}v_p(F)\log_3p$. The terms $-v_3(F)$ on the two sides cancel. Multiply by $\log_23$ and use $-v_p(F)\le(-v_p(F))^+$.
\end{proof}

We apply the lemma with $F=\Delta_K$, under the assumption $\zeta_3(5)=a/b$. Then $F(a/b)=\Delta_K(\zeta_3(5))$, and the left side of \eqref{eq:criterion} is the $3$-adic valuation of the determinant \eqref{eq:hankelvalue}. The lemma needs $\Delta_K(a/b)\ne0$, and we obtain it from the following observation at an auxiliary prime.

\begin{lemma}[dominance]\label{lem:dominance}
Let $F=\sum_jf_jX^j\in\Q[X]$, and let $p$ be a prime with $v_p(f_0)<v_p(f_j)$ for every $j\ge1$. Then $F(a/b)\ne0$ for every $a/b\in\Q$ with $p\nmid b$.
\end{lemma}

\begin{proof}
If $p\nmid b$, then $v_p(f_j(a/b)^j)\ge v_p(f_j)>v_p(f_0)$ for $j\ge1$, so $v_p(F(a/b))=v_p(f_0)<\infty$.
\end{proof}

The same idea, a coefficient that is strictly dominant at auxiliary primes, replaces the non-vanishing hypothesis in the irrationality criterion of Lai and Sprang \cite[Lemma~2.2]{LS26}, and \cite{K26} uses it in the form of Lemma~\ref{lem:dominance}. The prime $p$ can be chosen after $b$: if dominance holds at every prime of a window whose number of primes grows with $n$, then for $n$ large one of these primes does not divide $b$.

We now state the four estimates, writing $\kappa=K/n$. The first estimate is proved in Section~\ref{sec:gain}.

\begin{theorem}[$3$-adic gain]\label{thm:gain}
For $K\le3n+1$,
\begin{equation}\label{eq:gain}
v_3\qty(\Delta_K(\zeta_3(5)))\log_23\ge G(\kappa)\,n^2-O(n\log n)\ ,\qquad G(\kappa)=\qty(10.5\,\kappa-\frac{\kappa^2}4)\log_23\ .
\end{equation}
\end{theorem}

The second and the third estimate bound the size and the denominators of $\Delta_K$. Their main terms are of order $n^2\log n$, and they cancel: $(\kappa^2-6\kappa)+\kappa(6-\kappa)=0$. Roughly, both estimates measure the same products $\prod_kh_k^{s_k}\prod_{k\ne m}(k-m)^{s_ks_m}$ coming from the Heine expansion of Section~\ref{sec:heine}, where $s_k\le4$ is the number of variables of that expansion placed at the node $k$, once with the absolute value and once with the $p$-adic valuations for $p\ne3$, and by the product formula the two agree up to the $3$-adic part and up to terms of order $n^2$. What remains are two terms of order $n^2$, whose constants $F^\dagger$ and $c$ depend on $\kappa$. We bound them separately on the five intervals
\begin{equation}\label{eq:intervals}
I_i=[\kappa_{1,i},\kappa_{2,i}]=\qty[2.80+0.04(i-1),\ 2.84+0.04(i-1)]\qquad(i=1,\dots,5)\ ,
\end{equation}
which cover $[2.8,3]$, and on each interval the bounds hold uniformly in $\kappa$.

\begin{theorem}[size]\label{thm:archmain}
There are constants $F^\dagger_i$, defined in \eqref{eq:fdagger} and bounded in Table~\ref{tab:margins}, such that for $K/n\in I_i$
\begin{equation}\label{eq:archmain}
\log_2\|\Delta_K\|_1\le(\kappa^2-6\kappa)\,n^2\log_2n+F^\dagger_i\,n^2+O(n\log n)\ ,
\end{equation}
uniformly for $K/n\in I_i$, and with an implied constant that does not depend on $i$.
\end{theorem}

\begin{theorem}[denominators]\label{thm:denmain}
There are constants $c_i$, defined in \eqref{eq:cformula} and bounded in Table~\ref{tab:margins}, such that
\begin{equation}\label{eq:denmain}
\sum_{p\ne3}\qty(-v_p(\Delta_K))^+\log_2p\le\kappa(6-\kappa)\,n^2\log_2n+c_i\,n^2+o(n^2)\ ,
\end{equation}
uniformly for $K/n\in I_i$.
\end{theorem}

Theorems~\ref{thm:archmain} and~\ref{thm:denmain} are proved in Section~\ref{sec:costs}. The fourth estimate supplies the non-vanishing and fixes $K$. For a prime $p$ with $2.1n+2\le p<3n$ let
\begin{equation}\label{eq:Kstar}
K(n,p)=\begin{cases}10n-\dfrac{10(p-1)}3\ ,&p\equiv1\pmod3\ ,\\[1.5ex]10n-\dfrac{10p-14}3\ ,&p\equiv2\pmod3\ .\end{cases}
\end{equation}
It is the number of Taylor conditions that the non-vanishing argument imposes (Section~\ref{sec:basis}).

\begin{theorem}[non-vanishing]\label{thm:nonvanishing}
Let $p$ be a prime with $2.1n+2\le p<3n$, and let $K=K(n,p)$. Then $K\le3n-2$, and $\Delta_K(a/b)\ne0$ for every $a/b\in\Q$ with $p\nmid b$. If moreover $p\le2.159n$, then $K>2.8n$.
\end{theorem}

The lower limit $2.1n+2$ for $p$ is what gives $K\le3n-2$, so that \eqref{eq:hankelvalue} applies, and the upper limit $2.159n$ gives $K>2.8n$, the left end of the range covered by the intervals \eqref{eq:intervals}. The theorem is proved in Section~\ref{sec:nonvanishing}, where we show that the constant coefficient of $\Delta_K$ is strictly $p$-adically dominant and apply Lemma~\ref{lem:dominance}.

\begin{table}[ht]
\centering
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{|c|c|c|c|c|}
\hline\hline
$I_i$ & $G(\kappa_{1,i})$ & $F^\dagger_i$ & $c_i$ & $G(\kappa_{1,i})-F^\dagger_i-c_i$ \\
\hline
$[2.80,2.84]$ & $43.49$ & $8.43$ & $30.48$ & $4.58$ \\
$[2.84,2.88]$ & $44.06$ & $8.46$ & $30.79$ & $4.81$ \\
$[2.88,2.92]$ & $44.64$ & $8.52$ & $31.10$ & $5.02$ \\
$[2.92,2.96]$ & $45.21$ & $8.58$ & $31.41$ & $5.22$ \\
$[2.96,3.00]$ & $45.78$ & $8.65$ & $31.72$ & $5.41$ \\
\hline\hline
\end{tabular}
\medskip
\caption{The constants of Theorems~\ref{thm:gain}--\ref{thm:denmain}, in bits per $n^2$. The values of $G$ are rounded down, the bounds for $F^\dagger_i$ and $c_i$ of Propositions~\ref{prop:fdagger} and~\ref{prop:cvalues} are rounded up, and the last column is computed from the printed values. The function $G$ is increasing on $[2.8,3]$, so $G(\kappa)\ge G(\kappa_{1,i})$ on $I_i$. The proof of Theorem~\ref{thm:main} needs only that the last column is positive.}
\label{tab:margins}
\end{table}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Suppose, for a contradiction, that $\zeta_3(5)=a/b$ with coprime integers $a$ and $b>0$. From now on $a$ and $b$ are fixed, and only the odd integer $n$ varies.

\textit{Step 1: the auxiliary prime.} Let $N_b$ be the number of distinct prime factors of $b$. By the prime number theorem, the interval $[2.1n+2,\,2.159n]$ contains $(0.059+o(1))\,n/\log n$ primes, a number that tends to infinity with $n$. So there is $N_1$ such that for every $n\ge N_1$ the interval contains more than $N_b$ primes, and hence a prime $p\nmid b$. This is the only use of the prime number theorem in this proof, and it concerns large $n$ only: for small $n$ the interval may contain no prime at all, but small $n$ are never used.

\textit{Step 2: one exact inequality.} Fix an odd $n\ge N_1$, a prime $p\nmid b$ in the interval, and $K=K(n,p)$. By Theorem~\ref{thm:nonvanishing}, $2.8n<K\le3n-2$ and $\Delta_K(a/b)\ne0$, so $\kappa=K/n$ lies in one of the intervals $I_i$, and Lemma~\ref{lem:criterion} applies with $F=\Delta_K$. It gives the inequality \eqref{eq:criterion} for this $n$, without any error term.

\textit{Step 3: the three estimates.} Theorems~\ref{thm:gain}, \ref{thm:archmain} and~\ref{thm:denmain} bound the three quantities in \eqref{eq:criterion}. Each bound consists of a main term and an error term, and what these theorems assert about the error terms is the following: there are functions $E_1(n)$, $E_2(n)$ and $E_3(n)$, which depend neither on $a$, $b$ and $p$ nor on $K$ in the range $K/n\in I_i$, such that $E_j(n)/n^2\to0$ as $n\to\infty$, and the gain is at least $G(\kappa)n^2-E_1(n)$, the size is at most $(\kappa^2-6\kappa)n^2\log_2n+F^\dagger_in^2+E_2(n)$, and the denominators are at most $\kappa(6-\kappa)n^2\log_2n+c_in^2+E_3(n)$. (There are five intervals, so we may take the same functions for all $i$.) Inserting these bounds into \eqref{eq:criterion}, the terms of order $n^2\log n$ cancel, because $(\kappa^2-6\kappa)+\kappa(6-\kappa)=0$ for every $\kappa$. Using $G(\kappa)\ge G(\kappa_{1,i})$ and $K\le3n$, we obtain
\begin{equation}\label{eq:contradiction}
\qty(G(\kappa_{1,i})-F^\dagger_i-c_i)\,n^2\le E_1(n)+E_2(n)+E_3(n)+3n\log_2\max(|a|,b)\ .
\end{equation}

\textit{Step 4: the contradiction.} By Table~\ref{tab:margins}, the bracket on the left is at least $4.58$ for every $i$. Dividing \eqref{eq:contradiction} by $n^2$ gives
\[
4.58\le\frac{E_1(n)+E_2(n)+E_3(n)}{n^2}+\frac{3\log_2\max(|a|,b)}n\ .
\]
The right side tends to $0$ as $n\to\infty$, because $a$ and $b$ are fixed, so there is $N_2$ such that it is smaller than $4.58$ for all $n\ge N_2$. For an odd $n\ge\max(N_1,N_2)$, Steps 2--4 give a contradiction. Hence $\zeta_3(5)$ is irrational.
\end{proof}

The argument needs a single, sufficiently large $n$. Nothing is claimed about the inequalities for small $n$, where the error terms can be larger than the main terms. In particular, the prime number theorem, which also enters the proof of Theorem~\ref{thm:denmain}, is used there only in asymptotic form: it replaces sums over primes by their main terms with an error that, divided by $n^2$, tends to $0$, and no statement is made for small $n$.

\begin{remark}[the numerical constants]\label{rem:constants}
The decimal numbers that enter the proof are not approximations whose validity depends on $n$. (The numbers introduced by ``about'' in Remark~\ref{rem:crude} and Appendix~\ref{app:comp} are quoted for orientation only and are not used.) They are of two kinds. Some are choices: the window $2.1n+2\le p\le2.159n$ for the auxiliary prime, the five intervals $I_i$, the trial density $\sigma_0$ of \eqref{eq:sigma0}, the number $\epsilon=\frac1{20}$ of Section~\ref{sec:tree}, the partition and the rationals $\xi_\nu$ and the discretization parameters of Appendix~\ref{app:comp}, and the circles of Table~\ref{tab:circles}. Every statement in which they enter remains true for every admissible choice; another choice changes the numerical margin, and the argument closes only if the margin stays positive. The others are bounds for constants that are defined exactly. The coefficient $10.5$ in $G(\kappa)$ is exact, being $9+\frac32$ from the $9n+3$ trits of \eqref{eq:factorization} and from $3v_3(n!)=\frac32n+O(\log n)$. The constants $F^\dagger_i$ and $c_i$ are defined by \eqref{eq:fdagger} and \eqref{eq:cformula}, and Propositions~\ref{prop:fdagger} and~\ref{prop:cvalues} bound them from above by rigorous computations, in exact rational arithmetic and in ball arithmetic of which only the upper ends are used, with the discretization error bounded by Lemma~\ref{lem:modulus}; the printed values are rounded up once more. So the last column of Table~\ref{tab:margins} is a lower bound for the true margin, and the proof needs only that it is positive.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The \texorpdfstring{$3$}{3}-adic gain}\label{sec:gain}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we prove Theorem~\ref{thm:gain}. Following \cite{K26}, we proceed in two steps. First we write the determinant \eqref{eq:hankelvalue} in a basis of polynomials that are integer valued on the set $S$ of integration points; this changes its valuation by an amount that depends only on $S$ and $K$. Then we show that $L$ maps every polynomial that is integer valued on $S$ to a number divisible by about $3^{10.5n}$. The second step rests on a feature of the $3$-adic place that has no archimedean counterpart: by the ultrametric inequality, every node $-k$ is at distance exactly $3$ from every point of $S$, since $|t+\frac r3+k|_3=|\frac r3|_3=3$. So each pole of order $4$ multiplies the weight by $3^4$ uniformly on $S$.

Let $\Int(S)=\{B\in\Q_3[y]:\ B(S)\subseteq\Z_3\}$. For the general theory of integer-valued polynomials on subsets we refer to Bhargava \cite{Bh97}; we need only the explicit polynomials of the following lemma.

\begin{lemma}\label{lem:basis}
For $m\ge0$ put
\begin{equation}\label{eq:basis}
F_{2m}(y)=3^m\,m!\binom{y-\frac13}m\binom{y-\frac23}m\ ,\qquad F_{2m+1}(y)=3y\,F_{2m}(y)\ .
\end{equation}
Then $F_i\in\Int(S)$, $\deg F_i=i$, and the leading coefficient of $F_i$ has $3$-adic valuation $-\nu_i$, where $\nu_i=v_3(m!)-m-e$ for $i=2m+e$, $e\in\{0,1\}$. Consequently, for every $\Q_3$-linear functional $\Lambda'$ on $\Q_3[y]$,
\begin{equation}\label{eq:basisbound}
v_3\det\qty[\Lambda'(y^{i+j})]_{i,j<K}\ge2\sum_{i<K}\nu_i+K\min_{i,j<K}v_3\,\Lambda'(F_iF_j)\ .
\end{equation}
\end{lemma}

\begin{proof}
Let $y=\frac13+t$ with $t\in\Z_3$. Then $\binom{y-1/3}m=\binom tm\in\Z_3$, and $3^mm!\binom{y-2/3}m=\prod_{i<m}(3y-2-3i)=\prod_{i<m}(3t-1-3i)$ is a product of $3$-adic units. So $F_{2m}(y)\in\Z_3$. On $\frac23+\Z_3$ the two binomial coefficients exchange their roles, and on $S$ the factor $3y$ is a unit, so also $F_{2m+1}(S)\subseteq\Z_3$. The leading coefficient of $F_{2m}$ is $3^mm!/(m!)^2=3^m/m!$, of valuation $m-v_3(m!)$, and that of $F_{2m+1}$ is $3^{m+1}/m!$. For \eqref{eq:basisbound}, write $F_i=\sum_{j\le i}f_{ij}y^j$. The matrix $T=[f_{ij}]$ is lower triangular with diagonal entries $f_{ii}$, and $[\Lambda'(F_iF_j)]=T\,[\Lambda'(y^{i+j})]\,T^{\mathsf T}$. Hence $\det[\Lambda'(y^{i+j})]=\prod_{i<K}f_{ii}^{-2}\det[\Lambda'(F_iF_j)]$, and every term of the Leibniz expansion of the last determinant is a product of $K$ entries.
\end{proof}

The first term of \eqref{eq:basisbound} is negative at $p=3$. By Legendre's formula $v_3(m!)=(m-s_3(m))/2$, where $s_3(m)\le2\log_3(3m)$ is the sum of the ternary digits of $m$, so
\begin{equation}\label{eq:basisterm}
2\sum_{i<K}\nu_i=-2\sum_{m<K/2}m+O(K\log K)=-\frac{K^2}4+O(K\log K)\ .
\end{equation}
At $p=2$, where the integration set of \cite{K26} is a translate of $\Z_2$ and the binomial basis has leading coefficients $1/i!$, the corresponding term is positive and of size about $K^2$ bits. At $p=3$ the change of basis costs rather than gains, and the whole gain has to come from the weight.

We now turn to the weight on $S$. Fix $r\in\{1,2\}$ and put $y=t+\frac r3$ with $t\in\Z_3$. The factors of $W$ behave as follows.
\begin{itemize}
\item Of the two Pochhammer symbols, the one with shift $\frac{3-r}3$ becomes $(t+1)_n=n!\binom{t+n}n$, an integer-valued polynomial times $n!$.
\item The other becomes $\prod_{i<n}(t+\frac{2r}3+i)=3^{-n}\prod_{i<n}(3t+2r+3i)$, and every factor $3t+2r+3i$ is a unit.
\item $y+\frac n2=\frac16(6t+2r+3n)$, where $6t+2r+3n$ is a unit, so this factor has valuation $-1$.
\item Every node contributes $(y+k)^{-4}=3^4(3t+r+3k)^{-4}$, where $3t+r+3k$ is a unit.
\end{itemize}
The powers of $3$ add up to $-3n-1+4(3n+1)=9n+3$, and therefore
\begin{equation}\label{eq:factorization}
W\qty(t+\frac r3)=3^{9n+3}(n!)^3\binom{t+n}n^3Q_r(t)\ ,
\end{equation}
where $Q_r$ is a unit constant times a product of factors $(a+3t)^{\pm1}$ and $a+6t$ with integers $a$ prime to $3$. Each such factor is $a^{\pm1}(1+3t/a)^{\pm1}$ or $a(1+6t/a)$, a power series whose coefficient of $t^i$ has valuation at least $i$. Hence $Q_r(t)=\sum_iq_it^i$ with $v_3(q_i)\ge i$.

Thus the poles contribute $4(3n+1)$ trits, the non-integral Pochhammer symbol and the linear factor cost $3n+1$ trits, and the integral Pochhammer symbol adds $3v_3(n!)$, about $1.5n$ trits. It remains to see how much is lost when the derivative and the integral are taken.

The following lemma is proved in Appendix~\ref{app:analytic}.

\begin{lemma}[after {\cite[Lemmas~4.2 and~4.3]{K26}}]\label{lem:analyticstep}
Let $d\ge1$, let $P$ be a polynomial of degree at most $d$ with $P(\Z_3)\subseteq\Z_3$, and let $Q(t)=\sum_iq_it^i$ with $v_3(q_i)\ge i$. Then $PQ$ is strictly differentiable on $\Z_3$, and
\begin{equation}\label{eq:analyticstep}
v_3\vint(PQ)'(t)\,\dd t\ge-\log_3d-\log_3(d+1)\ .
\end{equation}
\end{lemma}

The two losses come from the binomial basis. Writing $PQ$ in the basis $\binom tc$, the derivative of $\binom tc$ is $\sum_{i=1}^c\frac{(-1)^{i-1}}i\binom t{c-i}$, which costs at most $\log_3d$, and the integral of $\binom tc$ is $(-1)^c/(c+1)$ by (V2), which costs at most $\log_3(d+1)$. The factor $Q$ costs nothing, because its coefficients in the binomial basis decay fast enough to compensate for the shift of degrees.

\begin{proof}[Proof of Theorem~\ref{thm:gain}]
Let $A_3=9n+3+3v_3(n!)$ and $d_K=2K-2+3n$. By \eqref{eq:hankelvalue} and Lemma~\ref{lem:basis} with $\Lambda'=L$, it suffices to show that
\begin{equation}\label{eq:entrybound}
v_3L(B)\ge A_3-\log_3d_K-\log_3(d_K+1)
\end{equation}
for every $B\in\Int(S)$ of degree at most $2K-2$, since each product $F_iF_j$ with $i,j<K$ is such a $B$. Fix $r\in\{1,2\}$. By \eqref{eq:factorization}, $(BW)(t+\frac r3)=3^{9n+3}(n!)^3P(t)Q_r(t)$ with $P(t)=B(t+\frac r3)\binom{t+n}n^3$. Both factors of $P$ map $\Z_3$ into $\Z_3$, and $\deg P\le d_K$. Since $(BW)'(t+\frac r3)=\frac{\dd}{\dd t}(BW)(t+\frac r3)$, Lemma~\ref{lem:analyticstep} gives \eqref{eq:entrybound} for the $r$-th term of $L(B)$, and the sum over $r$ follows from the ultrametric inequality. Hence
\begin{equation}\label{eq:gainexplicit}
v_3\,\Delta_K(\zeta_3(5))\ge2\sum_{i<K}\nu_i+K\qty(A_3-\log_3d_K-\log_3(d_K+1))\ .
\end{equation}
By Legendre's formula $3v_3(n!)=1.5n-O(\log n)$, so $KA_3=10.5\,\kappa n^2-O(n\log n)$; by \eqref{eq:basisterm} the first term is $-\kappa^2n^2/4+O(n\log n)$; and $K\log_3(d_K(d_K+1))=O(n\log n)$. Multiplying by $\log_23$ gives \eqref{eq:gain}.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The size and the denominators}\label{sec:costs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we prove Theorems~\ref{thm:archmain} and~\ref{thm:denmain}. Both concern the coefficients of $\Delta_K(X)$, and both start from an expansion of the determinant over the nodes, which we state first.

\subsection{The Heine expansion and the size}\label{sec:heine}
Heine's formula writes the Hankel determinant of the moments of a positive measure on $\R$ as $\frac1{K!}$ times the integral of the squared Vandermonde $\prod_{i<i'}(y_{i'}-y_i)^2$ against the $K$-fold product of the measure \cite[Ch.~II]{Sz75}. The identity is algebraic and holds for every linear functional (Lemma~\ref{lem:andreief}). Our functional $\Lambda$ is a sum of the local functionals $\Lambda_k$, and $\Lambda_k$ sees only the Taylor coefficients of order at most $3$ at the node $k$. So the $K$ variables $y_1,\dots,y_K$ are distributed among the nodes, and a node that receives five or more variables contributes nothing (Step~6 of the proof in Appendix~\ref{app:heine}); the determinant becomes a finite sum over these distributions. For a node $k$ and a subset $R\subseteq\{0,1,2,3\}$ with elements $R_0<\dots<R_{|R|-1}$, write $C_k[R]=[c_{k,R_b+b'}]_{0\le b,b'<|R|}$ for the square submatrix of $C_k$ with row set $R$ and the first $|R|$ columns.

\begin{proposition}[Heine expansion; after {\cite[Proposition~5.1]{K26}}]\label{prop:heine}
Let $K\ge1$. Then
\begin{equation}\label{eq:heine}
\Delta_K(X)=\sum_s\sum_\lambda\mathrm{coef}_\lambda(s)\prod_{k\in\cN}M_{k,\lambda_k}(X)\ ,
\end{equation}
where:
\begin{itemize}
\item[(a)] $s=(s_k)_{k\in\cN}$ runs over the \textit{profiles}, the integer vectors with $0\le s_k\le4$ and $\sum_ks_k=K$, and $\lambda=(\lambda_k)_{k\in\cN}$ over the tuples of partitions with at most $s_k$ parts and $|\lambda_k|\le s_k(4-s_k)$;
\item[(b)] for each profile fix a map $\iota:\{1,\dots,K\}\to\cN$ with $|\iota^{-1}(k)|=s_k$, and put
\begin{equation}\label{eq:cross}
Q_s(w)=\prod_{\substack{i<i'\\\iota(i)\ne\iota(i')}}\qty(\iota(i')-\iota(i)+w_i-w_{i'})^2\in\Z[w_1,\dots,w_K]\ ;
\end{equation}
then $\mathrm{coef}_\lambda(s)\in\Z$ is the coefficient in $Q_s$ of any monomial whose exponents at the variables $w_i$ with $\iota(i)=k$ are, for every $k$, a rearrangement of the parts of $\lambda_k$, padded with zeros;
\item[(c)] $M_{k,\emptyset}=1$, and for $s_k\ge1$, $M_{k,\lambda_k}$ is an integer combination of the minors $\det C_k[R]$ with $|R|=s_k$ and $\sum R-\binom{s_k}2=|\lambda_k|$, whose coefficients depend only on $s_k$, $\lambda_k$ and $R$.
\end{itemize}
\end{proposition}

The proof is in Appendix~\ref{app:heine}. In a term of \eqref{eq:heine}, the local factors $M_{k,\lambda_k}$ carry the local constants, hence the numbers $h_k^{s_k}$, and the cross factor $\mathrm{coef}_\lambda(s)$ carries the node differences $(k-m)^{2s_ks_m}$, up to the corrections that come from the variables $w$. The size estimate applies the triangle inequality to \eqref{eq:heine}, and the denominator estimate applies the ultrametric inequality.

For the size, in the rescaled variable $x=k/n$, in which the node $k$, sitting at $y=-k$, has the position $x\in[-1,2]$, a profile $s$ becomes in the limit a density $\sigma$ on $[-1,2]$ with $0\le\sigma\le4$ and total mass $\kappa$. For integrable $f$ on $[-1,2]$ let
\begin{equation}\label{eq:potential}
\cL f(x)=\int_{-1}^2\log|x-y|\,f(y)\,\dd y\ ,\qquad\langle f,g\rangle=\int_{-1}^2f(x)g(x)\,\dd x\ .
\end{equation}
In the same variable, the zeros $y=-(j+\frac r3)$ of $W$ sit at $x=(j+\frac r3)/n\in(0,1)$, with density $3+3=6$ on $I_{\mathrm{in}}=(0,1)$, and its poles have density $4$ on $[-1,2]$. Accordingly, the \textit{external field} is the potential of the zeros minus that of the poles,
\begin{equation}\label{eq:field}
\cV=6\,\cL\one_{I_{\mathrm{in}}}-4\,\cL\one_{[-1,2]}=2\,\cL\one_{I_{\mathrm{in}}}-4\,\cL\one_{I_{\mathrm{out}}}\ ,\qquad I_{\mathrm{out}}=[-1,0]\cup[1,2]\ .
\end{equation}
A measurable function $\sigma$ on $[-1,2]$ is \textit{feasible} for $\kappa$ if $0\le\sigma\le4$ and $\int\sigma=\kappa$. Put
\begin{equation}\label{eq:obstacle}
\mathcal E[\sigma]=\langle\sigma,\cV\rangle+\langle\sigma,\cL\sigma\rangle\ ,\qquad P_\sigma=\cV+2\cL\sigma\ ,\qquad\gap(\sigma)=\sup_{\sigma'\ \text{feasible}}\langle P_\sigma,\sigma'-\sigma\rangle\ge0\ .
\end{equation}
The functional $\mathcal E$ is the energy of the density in the field $\cV$ plus its logarithmic self-energy, and $P_\sigma$ is its first variation. The kernel $\log|x-y|$ is conditionally negative definite, that is, $\iint\log|x-y|\,\dd\nu(x)\,\dd\nu(y)\le0$ for every signed measure $\nu$ of total mass $0$ for which the integral exists (Lemma~\ref{lem:cnd} proves the discrete form that we need), so $\mathcal E$ is concave on the feasible densities, and $\mathcal E[\sigma]+\gap(\sigma)$ is an upper bound for $\sup\mathcal E$ obtained from a single trial density $\sigma$. The following theorem is the discrete form of this statement.

\begin{theorem}[after {\cite[Theorem~6.1]{K26}}]\label{thm:arch}
Let $J\ge1$. For $n\le K\le3n+1$ and every feasible $\sigma$ that is piecewise constant with at most $J$ pieces,
\begin{equation}\label{eq:arch}
\log\|\Delta_K\|_1\le(\kappa^2-6\kappa)\,n^2\log n+n^2\qty(\mathcal E[\sigma]+\gap(\sigma))+O_J(n\log n)\ ,
\end{equation}
where the implied constant depends only on $J$.
\end{theorem}

The proof is in Appendix~\ref{app:heine}. The term $-6\kappa n^2\log n$ comes from $\sum_ks_k\log|h_k|$, as $|h_k|$ is about $n^{-6n}$ by \eqref{eq:hk}, and the term $\kappa^2n^2\log n$ comes from the cross factors $|k-m|^{2s_ks_m}$. The details of the weight that are not visible in the densities, namely the zero $y=-\frac n2$, the distance $\frac13$ of the closest zeros to a node, the harmonic sums and the weights $i(i+1)$ and $2916$ in \eqref{eq:localconstants}, affect only the error term. Since the zeros and the poles of $W$ have the same densities as in \cite{K26}, $\mathcal E$ is the functional of \cite{K26}.

To prove Theorem~\ref{thm:archmain} we use the trial density of \cite[Table~2]{K26}. Let $\sigma_0$ be the step function on the intervals $[\frac j{10},\frac{j+1}{10})$, $-10\le j\le19$, with
\begin{equation}\label{eq:sigma0}
64\,\sigma_0=95,\ 112,\ 113,\ 113,\ 112,\ 109,\ 104,\ 95,\ 70,\ 3\quad\text{on}\quad[-1,-0.9),\ \dots,\ [-0.1,0)\ ,
\end{equation}
$\sigma_0=0$ on $[0,1)$, and $\sigma_0(x)=\sigma_0(1-x)$ on $[1,2]$. Its mass is $m_0=\int\sigma_0=\frac{463}{160}$. For $\kappa\in[2.8,3]$ put $\sigma_\kappa=(\kappa/m_0)\,\sigma_0$. Since $\max\sigma_\kappa<1.84$, $\sigma_\kappa$ is feasible for $\kappa$, and it has $30$ pieces. The density vanishes on the interval $I_{\mathrm{in}}$, where the zeros of $W$ repel the variables, and puts its mass on $[-1,0]\cup[1,2]$. Define
\begin{equation}\label{eq:fdagger}
F^\dagger_i=\frac1{\log2}\sup_{\kappa\in I_i}\qty(\mathcal E[\sigma_\kappa]+\gap(\sigma_\kappa))\qquad(i=1,\dots,5)\ .
\end{equation}

\begin{proposition}\label{prop:fdagger}
The constants $F^\dagger_1,\dots,F^\dagger_5$ are at most $8.425174$, $8.458313$, $8.510836$, $8.576728$ and $8.647895$, respectively.
\end{proposition}

The proof is a finite computation in ball arithmetic, described in Appendix~\ref{app:comp}. It uses that $\mathcal E[\sigma_\kappa]$ is a quadratic polynomial in $\kappa$ with explicit coefficients, and that $\gap(\sigma_\kappa)$ is bounded by the value of a linear optimization problem whose data are values of the explicit function $P_{\sigma_\kappa}$.

\begin{proof}[Proof of Theorem~\ref{thm:archmain}]
For $K/n=\kappa\in I_i$ apply Theorem~\ref{thm:arch} with $\sigma=\sigma_\kappa$ and $J=30$, divide by $\log2$, and use $\mathcal E[\sigma_\kappa]+\gap(\sigma_\kappa)\le F^\dagger_i\log2$. The implied constant depends only on $J=30$, hence not on $i$ or $\kappa$.
\end{proof}

\subsection{The tree bound and the small primes}\label{sec:tree}
We now turn to the denominators. We bound $-v_p(\Delta_K)$ for each prime $p\ne3$ separately and then sum over $p$. The bound for a single prime is a combinatorial maximum over the profiles of Proposition~\ref{prop:heine}. Fix $\epsilon=\frac1{20}$. At the primes $p\le\epsilon n$ a crude evaluation of this maximum suffices, and its sum over $p$ produces the main term $\kappa(6-\kappa)n^2\log n$ through Mertens' theorem. The primes $p>\epsilon n$ give the positive part of the constant $c_i$, and there we need the refined local lemma of Section~\ref{sec:singlelevel} and the circle model of Section~\ref{sec:circle}. Until the end of this section $p\ne3$ is a prime, $p=2$ included, and $v=v_p$ is the Gauss valuation on $\Q_p[X]$.

For a node $k$ put $\eta_p(k)=\max_{m\in\cN\setminus\{k\}}v_p(k-m)$. For $0\le s\le4$ and $\ell\ge0$ put
\begin{equation}\label{eq:ep}
\mathsf e_p(k,s,\ell)=\max\qty{-v\qty(\det C_k[R]):\ R\subseteq\{0,1,2,3\},\ |R|=s,\ \textstyle\sum R-\binom s2=\ell}\ ,
\end{equation}
with $\mathsf e_p=-\infty$ if there is no such $R$. Let $f_p(k,0)=0$, and for $s\ge1$ let $f_p(k,s)$ be the maximum of $\mathsf e_p(k,s,\ell)+\ell\,\eta_p(k)$ over $0\le\ell\le s(4-s)$.

\begin{theorem}[tree bound; after {\cite[Theorem~7.1]{K26}}]\label{thm:tree}
For every prime $p$,
\begin{equation}\label{eq:tree}
-v_p(\Delta_K)\le T_p:=\max_s\qty[\sum_{k\in\cN}f_p(k,s_k)-\sum_{j\ge1}\ \sum_{\mathfrak c\bmod p^j}\qty(S_{\mathfrak c}^2-\sum_{k\in\mathfrak c}s_k^2)]\ ,
\end{equation}
where $s$ runs over the profiles, $\mathfrak c$ over the residue classes of the nodes modulo $p^j$, and $S_{\mathfrak c}=\sum_{k\in\mathfrak c}s_k$.
\end{theorem}

The proof is in Appendix~\ref{app:heine}. The first sum in \eqref{eq:tree} is the loss from the local factors of \eqref{eq:heine}. The second is what the cross factors save: variables placed at two nodes that are congruent modulo $p^j$ produce node differences divisible by $p^j$. The residue classes modulo $p,p^2,p^3,\dots$ form a tree, whose leaves are the single nodes, and the maximum is an optimization over this tree.

The nonzero roots and poles of $\Wt_k$ are $d=k-j-\frac r3=\frac{3k-3j-r}3$ for $0\le j<n$, $r\in\{1,2\}$ (zeros of order $3$), $d=k-\frac n2$ (a zero of order $1$), and $d=k-m$ for $m\in\cN\setminus\{k\}$ (poles of order $4$). Write $e_d$ for the order, positive for zeros and negative for poles, so that $\Wt_k(z)=h_k\prod_d(1-z/d)^{e_d}$. Let
\begin{equation}\label{eq:Dk}
D_k=-v_p(h_k)=4\sum_{m\ne k}v_p(k-m)-3\sum_{0\le j<n,\,r=1,2}v_p(3k-3j-r)-v_p\qty(\frac{2k-n}2)\ ,
\end{equation}
where the last term equals $+1$ for $p=2$, since $2k-n$ is odd, and $-v_p(2k-n)$ for odd $p$. Since the coefficient of $z^b$ in $(1-z/d)^e$ is an integer times $d^{-b}$, we have the following.

\begin{lemma}\label{lem:locval}
Let $\mu_k=\max_dv_p(d)$ over the nonzero roots and poles $d$ of $\Wt_k$. Then $v_p(H_k[b])\ge-D_k-b\mu_k$ for $b\ge0$, and $v_p(\HS_3(k,M))\ge-M\max_uv_p(u)$, the maximum over the $u$ in \eqref{eq:hs}.
\end{lemma}

Let $\lL_p=\lceil\log_p(6n)\rceil$. All integers whose valuations enter the local constants have absolute value less than $6n$, so these valuations are at most $\lL_p$. The following three lemmas are the analogues in our setting of \cite[Lemmas~8.2, 8.3 and Proposition~8.5]{K26}, and their proofs are in Appendix~\ref{app:local}. The first bounds the local loss at a single node, a leaf of the tree.

\begin{lemma}[leaf lemma]\label{lem:leaf}
For every node $k$ and $1\le s\le4$, $f_p(k,s)\le sD_k+9s\lL_p$.
\end{lemma}

\begin{lemma}[small primes]\label{lem:small}
For every prime $p\ne3$, $T_p\le K(6n+4-K)/(p-1)+9K\lL_p+150n+K$.
\end{lemma}

\begin{lemma}[middle primes]\label{lem:middle}
If $p\ge5$, $p^2>6n$ and $p\le0.4n$, then $T_p\le K(6n+4-K)/p+3K+25p$.
\end{lemma}

The leading terms $K(6n+4-K)/(p-1)$ and $K(6n+4-K)/p$, summed with weight $\log p$ over $p\le\epsilon n$, give $\kappa(6-\kappa)n^2\log n$ plus a term of order $n^2$, by Mertens' theorem. For $p>\epsilon n$ these bounds are too weak, and we treat these primes more precisely.

\subsection{Single-level primes}\label{sec:singlelevel}
A prime $p\ge5$ with $p^2>6n$ is called \textit{single-level}. Every integer that occurs in the local constants, namely $k-m$, $3k-3j-r$, the $u$ of \eqref{eq:hs} and $2k-n$, has absolute value less than $6n<p^2$, so its $p$-adic valuation is $0$ or $1$. In particular the classes modulo $p^j$ with $j\ge2$ are singletons, and the tree \eqref{eq:tree} has only one level.

At such a prime, a nonzero root or pole $d$ of $\Wt_k$ is \textit{$p$-special} if $p\mid d$. There are three kinds of $p$-special points. The poles $d=k-m$ with $m\equiv k\pmod p$ come from the \textit{partners} $m$ of $k$, the other nodes congruent to $k$. The zero $d=k-\frac n2$ is $p$-special if $p\mid2k-n$. And the zero $d=\frac{3k-3j-r}3$ is $p$-special exactly when $3k-3j-r=pt$ for an integer $t$, necessarily prime to $3$; as $3j+r$ runs over the integers in $[1,3n-1]$ prime to $3$, $pt$ runs over the interval $(3k-3n,3k)$. Besides, the terms of $\HS_3(k,M)$ with $p\mid u$ are, writing $u=pt$, the numbers $(3/(pt))^M=\rho_t^{-M}$ with $\rho_t=pt/3$. The following definition records these data.

\begin{definition}[node data]\label{def:nodedata}
Let $p$ be single-level and $k$ a node. Let $o_k$ be the number of partners of $k$, let $g_k=1$ if $p\mid2k-n$ and $g_k=0$ otherwise, and let
\begin{equation}\label{eq:vpz}
\begin{split}
&\VP_k=\{t\in\Z\setminus3\Z:\ 0<pt<3k\}\quad(k>0)\ ,\qquad\VP_k=\{t\in\Z\setminus3\Z:\ 3k<pt<0\}\quad(k<0)\ ,\\
&\VP_0=\emptyset\ ,\qquad\cZ_k=\{t\in\Z\setminus3\Z:\ 3k-3n<pt<3k\}\ .
\end{split}
\end{equation}
For $t\in\VP_k$ the point $\rho_t=pt/3$ is a \textit{virtual pole} of the node, and it is \textit{covered} if $t\in\cZ_k$. Put $z_k=3|\cZ_k|$. For $t\in\VP_k$ let $\mu_{k,t}=0$ if $t$ is covered and $o_k+z_k+g_k=3$, and $\mu_{k,t}=1$ otherwise.
\end{definition}

Counted with multiplicity, the $p$-special points of $\Wt_k$ are $4o_k$ poles and $z_k+g_k$ zeros, so $D_k=4o_k-z_k-g_k$ by \eqref{eq:Dk}, and the $\mu_k$ of Lemma~\ref{lem:locval} equals $1$ if $o_k+z_k+g_k>0$ and $0$ otherwise. The condition $\mu_{k,t}=0$ means that $t$ is covered and that the covering zero is the only $p$-special point of the node: $o_k=0$, $g_k=0$ and $\cZ_k=\{t\}$.

Split $\beta_{k,a}=\beta^\HS_{k,a}+\beta^{\mathrm{rest}}_{k,a}$, where $\beta^\HS_{k,a}$ collects the terms of the harmonic sums with $p\mid u$. The next lemma says that these terms act like additional poles of order $3$ at the points $z=\rho_t$, which are not poles of $W$ at all.

\begin{lemma}[virtual poles]\label{lem:vp}
For every node $k$,
\begin{equation}\label{eq:vp}
\beta^\HS_{k,a}=2\sgn(k)\sum_{t\in\VP_k}[z^{3-a}]\,\Wt_k(z)\,(\rho_t-z)^{-3}\ .
\end{equation}
\end{lemma}

\begin{proof}
For $k>0$ the part of $\HS_3(k,M)$ with $p\mid u$ is $\sum_{t\in\VP_k}\rho_t^{-M}$, and for $k<0$ it is $-\sum_{t\in\VP_k}\rho_t^{-M}$. From $\sum_{i\ge1}i(i+1)x^{i-1}=2(1-x)^{-3}$ we get $2(\rho-z)^{-3}=\sum_{i\ge1}i(i+1)\rho^{-(i+2)}z^{i-1}$. So the coefficient of $z^{3-a}$ in $\Wt_k(z)\cdot2(\rho_t-z)^{-3}$ is $\sum_{i\ge1}i(i+1)\rho_t^{-(i+2)}H_k[4-a-i]$, which is the contribution of $\rho_t$ to $\beta_{k,a}$ in \eqref{eq:localconstants}; the sum stops at $i=4-a$ because $H_k[b]=0$ for $b<0$.
\end{proof}

A virtual pole costs $p$-adic valuation: $\rho_t$ has $v_p(\rho_t)=1$, so the factor $(\rho_t-z)^{-3}$ contributes $p^{-3-c}$ to the coefficient of $z^c$. The bound of \cite[Section~8.3]{K26} charges every virtual pole in this way. The next lemma shows that a covered virtual pole costs less, because the zero of $W$ that covers it has the same order $3$ as the virtual pole.

\begin{lemma}[refined local lemma]\label{lem:refined}
Let $p$ be single-level. For every node $k$ and $0\le m\le3$,
\begin{equation}\label{eq:vhat}
v_p(c_{k,m})\ge\hat v_m(k):=\min\qty(-D_k-(3-m)\mu_k\ ,\ \min_{t\in\VP_k}\qty[-D_k-3-(3-m)\mu_{k,t}]\ ,\ \hat a_m(k))\ ,
\end{equation}
where $\hat a_m(k)=-D_k-(1-m)\mu_k$ for $m\le1$, $\hat a_m(k)=+\infty$ for $m\ge2$, and the minimum over $t$ is $+\infty$ if $\VP_k=\emptyset$. The three terms bound $\beta^{\mathrm{rest}}_{k,m}$, $\beta^\HS_{k,m}$ and $\alpha_{k,m}$, respectively.
\end{lemma}

\begin{proof}
Since all valuations are $0$ or $1$, $v_p(h_k)=-D_k$, and every $d$ has $v_p(d)\le\mu_k$, so $v(H_k[b])\ge-D_k-b\mu_k$ by Lemma~\ref{lem:locval}.

\textit{The part $\beta^{\mathrm{rest}}$.} The weights $i(i+1)$ are integers and the terms of the harmonic sums with $p\nmid u$ are $p$-integral, so $v(\beta^{\mathrm{rest}}_{k,m})\ge\min_{i\ge1}v(H_k[4-i-m])\ge-D_k-(3-m)\mu_k$.

\textit{The part $\alpha$.} Here $\alpha_{k,m}=-2916\,H_k[1-m]$, and $2916=2^2\cdot3^6$ is prime to $p$, which gives the third term.

\textit{The part $\beta^\HS$.} We use \eqref{eq:vp}, one virtual pole at a time. If $t$ is not covered, then $(\rho_t-z)^{-3}=\rho_t^{-3}\sum_{c\ge0}\binom{c+2}2(z/\rho_t)^c$, so the coefficient of $z^{3-m}$ in $\Wt_k(z)(\rho_t-z)^{-3}$ has valuation at least $-D_k-3-(3-m)$, and this is $-D_k-3-(3-m)\mu_{k,t}$ because $\mu_{k,t}=1$. If $t$ is covered, $\Wt_k$ contains the factor $(1-z/\rho_t)^3$ with exponent exactly $3$. Indeed, the zeros $k-j-\frac r3$ are pairwise distinct, $\rho_t$ is not a pole because the poles are integers, and $\rho_t$ is not the zero $k-\frac n2$, because $v_3(\rho_t)=-1$ while $v_3(k-\frac n2)\ge0$. Since $1-z/\rho_t=(\rho_t-z)/\rho_t$,
\begin{equation}\label{eq:cancel}
\qty(1-\frac z{\rho_t})^3(\rho_t-z)^{-3}=\rho_t^{-3}\ ,
\end{equation}
and therefore $\Wt_k(z)(\rho_t-z)^{-3}=h_k\rho_t^{-3}\prod_{d\ne\rho_t}(1-z/d)^{e_d}$. The virtual pole has disappeared, and only the constant $\rho_t^{-3}$ remains. If $\mu_{k,t}=0$, every remaining $d$ is a $p$-adic unit, and if $\mu_{k,t}=1$, every remaining $d$ has valuation at most $1$. So the coefficient of $z^b$ of the remaining product has valuation at least $-b\mu_{k,t}$, and the coefficient of $z^{3-m}$ of $\Wt_k(z)(\rho_t-z)^{-3}$ has valuation at least $-D_k-3-(3-m)\mu_{k,t}$.
\end{proof}

Stated otherwise, a covered virtual pole that is the only $p$-special point of its node contributes $v_p\ge-D_k-3$ in every Taylor order $m$, whereas the bound of \cite[Section~8.3]{K26} charges it $-D_k-6+m$, with $D_k$ already lowered by the covering zero. The cancellation \eqref{eq:cancel} rests on the coincidence of the order $3$ of the zeros of $W$ with the order $3$ of the virtual poles, which comes from the single derivative in \eqref{eq:functionalintro}. At $p=2$ in \cite{K26} the zeros have order $6$ and the virtual poles order $5$.

\begin{remark}\label{rem:crude}
Without the refined lemma our argument does not close, at least at $\kappa=3$. For primes $p>3n$ no node has a partner, and every virtual pole is covered and alone at its node, so the lemma gives $\hat v\ge0$ and these primes cost nothing (Lemma~\ref{lem:cells}). The cruder bound, by contrast, gives a positive value of the relaxed tree bound $T_p^+$ of Lemma~\ref{lem:relaxation} at every single-level prime up to $6n$. Evaluating the model of Section~\ref{sec:circle} numerically at $\kappa=3$, with the cruder bound in place of $\hat v$, gives a denominator constant of about $44$ bits per $n^2$, against about $31.5$ with $\hat v$ (the bound $c_5\le31.72$ of Table~\ref{tab:margins} is larger because it holds on all of $I_5$). Since $G(3)-F^\dagger_5$ is about $37.7$, the comparison would then give no contradiction. This evaluation is not part of the proof; we quote it only to show that the cancellation \eqref{eq:cancel} is needed in our argument.
\end{remark}

\subsection{The circle model and the prime sum}\label{sec:circle}
For the single-level primes we now evaluate the maximum $T_p$ asymptotically. The plan is the following. First, $T_p$ is bounded by a sum of independent maxima, one for each residue class modulo $p$ (Lemma~\ref{lem:relaxation}). For $p=\varrho n$ with $\varrho$ fixed, a residue class modulo $p$ corresponds to a point of the circle $\R/\varrho\Z$, and the data of its nodes depend only on that point (Lemma~\ref{lem:cells}). So $T_p$ is about $n$ times an integral over the circle, which is piecewise affine in $\varrho$ (Lemma~\ref{lem:counting}). Summing over $p$ with the prime number theorem turns it into an integral over $\varrho$ (Proposition~\ref{prop:primesum}).

\begin{lemma}[relaxation]\label{lem:relaxation}
Let $p$ be single-level and let $\check v_m(k)$, $0\le m\le3$, be integers with $v(c_{k,m})\ge\check v_m(k)$ for all $k$ and $m$, for instance $\hat v_m(k)$. Put $f^+(k,0)=0$ and, for $1\le s\le4$,
\begin{equation}\label{eq:fplus}
f^+(k,s)=\max\qty(0,\ \max_{R,\pi}\qty[\sum_{b<s}-\check v_{R_b+\pi(b)}(k)+\qty(\sum R-\binom s2)[o_k>0]])\ ,
\end{equation}
the maximum over $R=\{R_0<\dots<R_{s-1}\}\subseteq\{0,1,2,3\}$ and permutations $\pi$ of $\{0,\dots,s-1\}$ with $R_b+\pi(b)\le3$ for all $b$. For a residue class $\mathfrak c$ modulo $p$ let
\begin{equation}\label{eq:Vc}
\begin{split}
&V_{\mathfrak c}(S)=\max\qty{\sum_{k\in\mathfrak c}\qty(f^+(k,s_k)+s_k^2):\ 0\le s_k\le4,\ \sum_{k\in\mathfrak c}s_k=S}-S^2\ ,\\
&\psi_{\mathfrak c}(\xi)=\max_{0\le S\le4|\mathfrak c|}\qty(V_{\mathfrak c}(S)-\xi S)\ .
\end{split}
\end{equation}
Then $T_p\le T_p^+:=\max\{\sum_{\mathfrak c}V_{\mathfrak c}(S_{\mathfrak c}):\sum_{\mathfrak c}S_{\mathfrak c}=K\}$, and $T_p^+\le\xi K+\sum_{\mathfrak c}\psi_{\mathfrak c}(\xi)$ for every real $\xi$.
\end{lemma}

The multiplier $\xi$ prices the constraint $\sum_{\mathfrak c}S_{\mathfrak c}=K$, so that each class is optimized separately. With $\varrho=p/n$ and $x=k/n$, a class modulo $p$ consists of the nodes $k\equiv k_0\pmod p$, and it corresponds to the point $x_0=k_0/n$ of the circle $\R/\varrho\Z$. All node data of Definition~\ref{def:nodedata}, except $g_k$, are functions of the positions $(x,\varrho)$: $pt<3k$ if and only if $t\varrho<3x$, $3k-3n<pt$ if and only if $3x-3<t\varrho$, and the node range is $x\in[-1,2]$. For $\varrho>0$ and $x\in\R/\varrho\Z$ the nodes of $x$ are the points $y\in(x+\varrho\Z)\cap[-1,2]$, and the \textit{type} of $x$ is the multiset of the pairs $(\hat v(y),[o>0])$ over these nodes. Here $\hat v(y)=(\hat v_0,\dots,\hat v_3)$ is given by \eqref{eq:vhat}, with the data computed from $(y,\varrho)$ and with $g=0$, and $o+1$ is the number of nodes of $x$. For a type $\tau$ let $\psi_\tau$ be the function $\psi_{\mathfrak c}$ of \eqref{eq:Vc} for a class with these data.

\begin{lemma}[cells]\label{lem:cells}
Let $\cB=\{3j/N:\ j\in\{1,2,3\},\ N\ge1\}$. On each open interval of $(0,\infty)\setminus\cB$, which we call a \textit{cell}, the cyclic order of the seven points
\begin{equation}\label{eq:sevenpoints}
-1,\quad2,\quad0,\quad\frac\varrho3,\quad\frac{2\varrho}3,\quad1+\frac\varrho3,\quad1+\frac{2\varrho}3
\end{equation}
of $\R/\varrho\Z$ and the type of each open arc between consecutive points are constant. If each of the seven points is represented by a real number $\alpha+\beta\varrho+w\varrho$, with an integer $w$ that is kept fixed on the whole cell, the lengths of the arcs are differences of such numbers, hence affine in $\varrho$. Hence the measure $\meas_\tau(\varrho)$ of the set of points of type $\tau$ is affine in $\varrho$ on each cell; for a cell $(a,b)$ we write $\meas^{[a,b]}_\tau$ for its affine extension to the closed cell $[a,b]$. Moreover, for $\varrho>3$ every node has $\hat v\ge0$, and $T^+_p=0$ for every single-level prime $p>3n$.
\end{lemma}

The seven points are where a node enters or leaves the node range (the points $-1$ and $2$), where the range of the virtual poles changes sign ($0$), and where a virtual pole or an end of the zero range appears ($3y\in\varrho\Z$ or $3y-3\in\varrho\Z$, with the quotient prime to $3$). The proof is in Appendix~\ref{app:local}. With these measures put
\begin{equation}\label{eq:g}
\mathcal G_{[a,b]}(\xi,\varrho,\kappa)=\xi\kappa+\sum_\tau\meas^{[a,b]}_\tau(\varrho)\,\psi_\tau(\xi)\qquad(\varrho\in[a,b])\ .
\end{equation}
For fixed $\xi$ it is affine in $\varrho$ and in $\kappa$.

\begin{lemma}[counting; after {\cite[Lemma~8.7]{K26}}]\label{lem:counting}
Fix a finite set $\cA\subset\R$. There are $C$ and $n_0$ such that for all $n\ge n_0$, all primes $p\in[\epsilon n,3n]$, every closed cell $[a,b]\ni p/n$ and every $\xi\in\cA$,
\begin{equation}\label{eq:counting}
T_p\le n\,\mathcal G_{[a,b]}(\xi,p/n,K/n)+C\ .
\end{equation}
\end{lemma}

The proof is in Appendix~\ref{app:local}. It rests on the fact that a residue class modulo $p$ is a point of the grid $\frac1n\Z$ on the circle $\R/\varrho\Z$, so that, up to $O(1)$ exceptional classes, the number of classes of type $\tau$ is $n\meas_\tau(\varrho)+O(1)$. Summing \eqref{eq:counting} over the primes in $(\epsilon n,3n]$ with the prime number theorem, and the bounds of Lemmas~\ref{lem:small} and~\ref{lem:middle} over the smaller primes with Mertens' theorem, gives the following.

\begin{proposition}[the prime sum]\label{prop:primesum}
Let $\epsilon=a_0<a_1<\dots<a_M=3$ be a partition of $[\epsilon,3]$ that refines $\cB\cap[\epsilon,3]$, let $\xi_1,\dots,\xi_M$ be rational numbers, and let $[\kappa_1,\kappa_2]\subseteq[2.8,3]$. For $1\le\nu\le M$ put
\begin{equation}\label{eq:hi}
\bar{\mathcal G}_\nu=\max\qty{\mathcal G_{[a,b]}(\xi_\nu,\varrho,\kappa):\ \varrho\in\{a_{\nu-1},a_\nu\},\ \kappa\in\{\kappa_1,\kappa_2\}}\ ,
\end{equation}
where $[a,b]$ is the closed cell that contains $[a_{\nu-1},a_\nu]$. Then, uniformly for $K/n\in[\kappa_1,\kappa_2]$,
\begin{equation}\label{eq:primesum}
\begin{split}
\sum_{p\ne3}\max(T_p,0)\log p\le\ &\kappa(6-\kappa)\,n^2\log n-\kappa_1(6-\kappa_1)\qty(\gamma+\frac{\log3}2+\log\frac1\epsilon)n^2\\
&+\qty(\sum_{\nu=1}^M\max(\bar{\mathcal G}_\nu,0)(a_\nu-a_{\nu-1})+3\kappa_2\epsilon+12.5\,\epsilon^2)n^2+o(n^2)\ ,
\end{split}
\end{equation}
where $\gamma$ is Euler's constant.
\end{proposition}

The proof is in Appendix~\ref{app:local}. The first two terms come from the primes $p\le\epsilon n$, the sum over $\nu$ from the primes in $(\epsilon n,3n]$, and the primes $p>3n$ contribute nothing by Lemma~\ref{lem:cells}. For $i=1,\dots,5$ let $c_i$ be the value of
\begin{equation}\label{eq:cformula}
\frac1{\log2}\qty[-\kappa_{1,i}(6-\kappa_{1,i})\qty(\gamma+\frac{\log3}2+\log20)+\sum_{\nu=1}^M\max(\bar{\mathcal G}_\nu,0)(a_\nu-a_{\nu-1})+\frac{3\kappa_{2,i}}{20}+\frac{12.5}{400}]
\end{equation}
with $[\kappa_1,\kappa_2]=I_i$ in \eqref{eq:hi}, for the partition and the rationals $\xi_\nu$ described in Appendix~\ref{app:comp}. By Proposition~\ref{prop:primesum}, any other admissible choice would also give a valid bound.

\begin{proposition}\label{prop:cvalues}
The constants $c_1,\dots,c_5$ are at most $30.471829$, $30.783002$, $31.093465$, $31.403655$ and $31.713600$, respectively.
\end{proposition}

The sum over $\nu$ in \eqref{eq:cformula} is an exact rational number; its computation and the evaluation of the remaining terms are described in Appendix~\ref{app:comp}.

\begin{proof}[Proof of Theorem~\ref{thm:denmain}]
By Theorem~\ref{thm:tree}, $(-v_p(\Delta_K))^+\le\max(T_p,0)$ for every $p\ne3$. Apply Proposition~\ref{prop:primesum} with $[\kappa_1,\kappa_2]=I_i$ and with the partition and the $\xi_\nu$ that define $c_i$, and divide by $\log2$.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Non-vanishing at an auxiliary prime}\label{sec:nonvanishing}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we prove Theorem~\ref{thm:nonvanishing}. Throughout, $n$ is odd and $p$ is a prime with $2.1n+2\le p<3n$. This forces $n\ge5$ and $p\ge13$ (for $n=3$ the interval $[8.3,9)$ contains no prime), and $p$ is single-level. By Lemma~\ref{lem:dominance} it suffices to show that the constant coefficient of $\Delta_K$ is strictly $p$-adically dominant, and we shall compute its $p$-adic valuation exactly. The structure of the argument is that of \cite[Theorem~9.1]{K26}: we choose a basis of polynomials adapted to the nodes that are $p$-adically special, scale its elements by suitable powers of $p$, and reduce the Gram matrix of the functional modulo $\sqrt p$ (Section~\ref{sec:basis}), where it becomes block diagonal with explicit blocks.

\subsection{The nodes and their local constants}\label{sec:nodes}
Two distinct nodes are congruent modulo $p$ exactly when they differ by $p$, because $|k-m|\le3n<2p$. So the nodes $k\in[p-n,2n]$, which we call \textit{R nodes}, have the partner $k-p$; the nodes $k\in[-n,2n-p]$, the \textit{L nodes}, are these partners; and the nodes in $(2n-p,p-n)$, the \textit{U nodes}, have no partner. The last interval is nonempty because $2p>3n$. In Definition~\ref{def:nodedata} only $t\in\{\pm1,\pm2\}$ occur, since $|pt|<6n<3p$.

\begin{lemma}\label{lem:nodetypes}
The nodes fall into the types of Table~\ref{tab:nodes}. The partner of an Ra (Rb, Rb$_g$, Rc) node is an La (Lb, Lb$_g$, Lc) node, and the two nodes $\frac{n\pm p}2$ are partners.
\end{lemma}

\begin{table}[ht]
\centering
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|l|l|l|c|}
\hline\hline
type & nodes $k$ & data & $D_k$ \\
\hline
Ra & $p-n\le k<\frac{2p}3$ & $\VP_k=\{1\}$, covered; $\cZ_k=\{1\}$ & $1$ \\
Rb & $\frac{2p}3<k<n+\frac p3$, $k\ne\frac{n+p}2$ & $\VP_k=\cZ_k=\{1,2\}$, both covered & $-2$ \\
Rb$_g$ & $k=\frac{n+p}2$ & as Rb, and $g_k=1$ & $-3$ \\
Rc & $n+\frac p3<k\le2n$ & $\VP_k=\{1,2\}$, $\cZ_k=\{2\}$ & $1$ \\
La & $-n\le k<-\frac p3$ & $\VP_k=\{-1\}$, not covered; $\cZ_k=\{-2\}$ & $1$ \\
Lb & $-\frac p3<k<n-\frac{2p}3$, $k\ne\frac{n-p}2$ & $\VP_k=\emptyset$, $\cZ_k=\{-1,-2\}$ & $-2$ \\
Lb$_g$ & $k=\frac{n-p}2$ & as Lb, and $g_k=1$ & $-3$ \\
Lc & $n-\frac{2p}3<k\le2n-p$ & $\VP_k=\emptyset$, $\cZ_k=\{-1\}$ & $1$ \\
U & $2n-p<k<p-n$ & no partner; three cases, see the proof & $-3$, $-3$, $0$ \\
\hline\hline
\end{tabular}
\medskip
\caption{The nodes at a prime $p$ with $2.1n+2\le p<3n$, in the notation of Definition~\ref{def:nodedata}. All R and L nodes have exactly one partner, and $g_k=0$ except at the two nodes $\frac{n\pm p}2$.}
\label{tab:nodes}
\end{table}

\begin{proof}
By \eqref{eq:vpz}, $t=1$ lies in $\VP_k$ if and only if $k>\frac p3$, $t=2$ if and only if $k>\frac{2p}3$, $t=-1$ if and only if $k<-\frac p3$, and $t=-2$ never, since $k\ge-n>-\frac{2p}3$. Similarly $t\in\cZ_k$ if and only if $\frac{pt}3<k<n+\frac{pt}3$. Finally $g_k=1$ if and only if $2k-n=\pm p$, because $2k-n$ is odd and $|2k-n|\le3n<2p$. Since $2.1n<p<3n$, the relevant thresholds are ordered as
\begin{equation}\label{eq:thresholds}
-n<-\frac p3<n-\frac{2p}3<2n-p<p-n<\frac{2p}3<n+\frac p3<2n\ ,
\end{equation}
and moreover $n-\frac p3<p-n$ and $2n-p<\frac p3$. Each of these inequalities is equivalent to $p<3n$ or to $2p>3n$. Reading off the conditions in each interval of \eqref{eq:thresholds} gives the R and L rows of the table. For instance, an Ra node has $k\ge p-n>\frac p3$, so $1\in\VP_k$, and $k<\frac{2p}3$, so $2\notin\VP_k$; it has $\frac p3<k<n+\frac p3$, so $1\in\cZ_k$, while $2\notin\cZ_k$ because $k<\frac{2p}3$, and $-1,-2\notin\cZ_k$ because $k>n-\frac p3$. Then $D_k=4o_k-z_k-g_k=4-3-0=1$. The other R and L rows are obtained in the same way. At a U node, $o_k=0$, and there are three cases: if $\frac p3<k<p-n$, then $\VP_k=\cZ_k=\{1\}$ and $D_k=-3$; if $2n-p<k<n-\frac p3$, then $\VP_k=\emptyset$, $\cZ_k=\{-1\}$ and $D_k=-3$; otherwise $\VP_k=\cZ_k=\emptyset$ and $D_k=0$. The first two ranges are disjoint, as $n-\frac p3<\frac p3$. Finally, $\frac{n+p}2$ lies in the Rb range and $\frac{n-p}2$ in the Lb range, both being integers since $n$ and $p$ are odd, and the map $k\mapsto k-p$ sends each R range onto the L range of the same letter.
\end{proof}

Let $\Na$ and $\Nc$ be the numbers of Ra and Rc nodes, and $\Nb$ the number of Rb and Rb$_g$ nodes together. Counting the integers in the ranges of Table~\ref{tab:nodes},
\begin{equation}\label{eq:counts}
\begin{split}
&\Na=\Nb=\Nc=n-\frac{p-1}3\qquad(p\equiv1\bmod 3)\ ,\\
&\Na=\Nc=n-\frac{p-2}3\ ,\quad\Nb=n-\frac{p+1}3\qquad(p\equiv2\bmod 3)\ ,
\end{split}
\end{equation}
and in both cases the number $K(n,p)$ of \eqref{eq:Kstar} is
\begin{equation}\label{eq:Kcount}
K(n,p)=4(\Na+\Nc)+2\Nb\ .
\end{equation}
From \eqref{eq:Kstar} and $2.1n+2\le p$ we get $K\le10n-\frac{10(2.1n+2)-14}3=3n-2$, and if $p\le2.159n$, then $K\ge10n-\frac{10(2.159n-1)}3>2.8n$. This proves the first and the last statement of Theorem~\ref{thm:nonvanishing}.

At the R nodes and at the La nodes, the leading term of $\beta^\HS_{k,m}$ can be computed exactly. For such a node let $\Phi_k(w)=\sgn(k)\prod(1-w/\delta)^{e}\sum_{t\in\VP_k}(\frac t3-w)^{-3}$, where the product runs over the $p$-special points $d=p\delta$ of $\Wt_k$, with their orders $e$. From Table~\ref{tab:nodes},
\begin{equation}\label{eq:Phi}
\begin{split}
&\Phi_{\mathrm{Ra}}(w)=(1-w)^{-4}(1-3w)^3\qty(\tfrac13-w)^{-3}=27(1-w)^{-4}\ ,\\
&\Phi_{\mathrm{Rb}}(w)=(1-w)^{-4}(1-3w)^3\qty(1-\tfrac32w)^3\Sigma(w)\ ,\qquad\Phi_{\mathrm{Rb}_g}(w)=(1-2w)\,\Phi_{\mathrm{Rb}}(w)\ ,\\
&\Phi_{\mathrm{Rc}}(w)=(1-w)^{-4}\qty(1-\tfrac32w)^3\Sigma(w)\ ,\qquad\Phi_{\mathrm{La}}(w)=(1+w)^{-4}\qty(1+\tfrac32w)^3\qty(\tfrac13+w)^{-3}\ ,
\end{split}
\end{equation}
with $\Sigma(w)=(\frac13-w)^{-3}+(\frac23-w)^{-3}$. For example, at an Ra node the $p$-special points of $\Wt_k$ are the partner pole $d=k-(k-p)=p$, of order $-4$, and the covering zero $d=\frac p3$, of order $3$, and $\VP_k=\{1\}$. At an La node they are the partner pole $d=-p$ and the zero $d=-\frac{2p}3$, the virtual pole is $t=-1$, and $\sgn(k)=-1$. Let $\phi_{k,b}=[w^b]\Phi_k(w)$. These coefficients depend only on the type. Expanding \eqref{eq:Phi}, the ones that we need are
\begin{equation}\label{eq:phivalues}
\begin{split}
&\text{Ra: }27,\ 108,\ 270,\ 540\ ,\qquad\text{La: }27,\ -\tfrac{459}2,\ \tfrac{5211}4,\ -\tfrac{49761}8\qquad(b=0,1,2,3)\ ,\\
&\text{Rb: }-\tfrac{243}8,\ -\tfrac{243}8,\ 0\ ,\qquad\text{Rb}_g\text{: }-\tfrac{729}8,\ \tfrac{243}8,\ \tfrac{243}4\qquad(b=1,2,3)\ ,\qquad\text{Rc: }\phi_{\mathrm{Rc},0}=\tfrac{243}8\ .
\end{split}
\end{equation}
All coefficients of the $\Phi_k$ are $p$-integral, their denominators being powers of $2$.

\begin{lemma}\label{lem:scaled}
Let $\varepsilon_k=2h_kp^{D_k}$, a $p$-adic unit. For the nodes of types Ra, Rb, Rb$_g$, Rc and La,
\begin{equation}\label{eq:scaled}
\beta^\HS_{k,m}=\varepsilon_k\,p^{-D_k+m-6}\qty(\phi_{k,3-m}+O(p))\qquad(0\le m\le3)\ ,
\end{equation}
where $O(p)$ denotes an element of $p\Z_p$. The other local constants satisfy the following bounds, which are those of Lemma~\ref{lem:refined} with the data of Table~\ref{tab:nodes}: $v_p(\beta^{\mathrm{rest}}_{k,m})\ge-4+m$ and $v_p(\alpha_{k,m})\ge-2+m$ at the nodes of types Ra, Rc, La and Lc; $v_p(\beta^{\mathrm{rest}}_{k,m})\ge-1+m$ and $v_p(\alpha_{k,m})\ge1+m$ at Rb and Lb; and $v_p(\beta^{\mathrm{rest}}_{k,m})\ge m$ and $v_p(\alpha_{k,m})\ge2+m$ at Rb$_g$ and Lb$_g$. At the U nodes, all local constants $\beta^\HS_{k,m}$, $\beta^{\mathrm{rest}}_{k,m}$ and $\alpha_{k,m}$ have $v_p\ge0$.
\end{lemma}

\begin{proof}
Substitute $z=pw$ in \eqref{eq:vp}. Then $[z^{3-m}]f(z)=p^{-(3-m)}[w^{3-m}]f(pw)$ for any power series $f$, and
\begin{equation}\label{eq:wkpw}
\begin{split}
&\Wt_k(pw)=h_k\prod_{p\mid d}\qty(1-\frac w\delta)^{e_d}\,\Upsilon_k(pw)\ ,\qquad \Upsilon_k(z)=\prod_{p\nmid d}\qty(1-\frac zd)^{e_d}\ ,\\
&(\rho_t-pw)^{-3}=p^{-3}\qty(\frac t3-w)^{-3}\ .
\end{split}
\end{equation}
The coefficient of $z^b$ in $\Upsilon_k(z)$ is $p$-integral, so the coefficient of $w^b$ in $\Upsilon_k(pw)-1$ lies in $p\Z_p$ for every $b$. Hence $\beta^\HS_{k,m}=2h_kp^{-(3-m)-3}[w^{3-m}]\qty(\Phi_k(w)\Upsilon_k(pw))=\varepsilon_kp^{-D_k+m-6}(\phi_{k,3-m}+O(p))$, using that $\Phi_k$ has $p$-integral coefficients. The bounds for $\beta^{\mathrm{rest}}$ and $\alpha$ are the first and the third term of \eqref{eq:vhat} with $\mu_k=1$ and the $D_k$ of the table. At a U node with $\VP_k=\cZ_k=\{1\}$ we have $D_k=-3$, $\mu_k=1$ and $\mu_{k,1}=0$, so the three terms of \eqref{eq:vhat} are $m$, $0$ and $2+m$; at a U node with $\cZ_k=\{-1\}$ and no virtual pole they are $m$ and $2+m$; and at the remaining U nodes $D_k=0$ and $\mu_k=0$.
\end{proof}

The next lemma relates the units $\varepsilon_k$ at the two members of an Ra/La pair; this is where Wilson's theorem enters.

\begin{lemma}[partner congruence]\label{lem:wilson}
If $k$ is an Ra node, then $\varepsilon_{k-p}\equiv-8\,\varepsilon_k\pmod p$.
\end{lemma}

\begin{proof}
Both nodes have $D=1$, so $\varepsilon_{k-p}/\varepsilon_k=h_{k-p}/h_k$. Write $W=N_W/\prod_m(y+m)^4$ with $N_W(y)=(y+\frac n2)(y+\frac13)_n^3(y+\frac23)_n^3$, and $\Pi(k)=\prod_{m\ne k}(m-k)=(-1)^{k+n}(k+n)!\,(2n-k)!$ as in \eqref{eq:hk}. Then $h_k=N_W(-k)/\Pi(k)^4$, and
\begin{equation}\label{eq:Pratio}
\frac{\Pi(k)}{\Pi(k-p)}=(-1)^p\,\frac{\prod_{i=1}^p(n+k-p+i)}{\prod_{i=1}^p(2n-k+i)}\ .
\end{equation}
Both products run over $p$ consecutive integers in $[1,3n]$. Since $2p>3n$, the only multiple of $p$ among them is $p$ itself, and by Wilson's theorem the product of the other $p-1$ factors is $\equiv(p-1)!\equiv-1\pmod p$ in each. Hence $\Pi(k)/\Pi(k-p)\equiv-1$, and its fourth power is $\equiv1$. For each linear factor $y+c$ of $N_W$, the ratio of its values at $y=-k+p$ and $y=-k$ is $(c-k+p)/(c-k)$, which is $\equiv1\pmod p$ unless $p\mid c-k$. At an Ra node the only such factor is the covering zero, with $c-k=-\frac p3$ and multiplicity $3$, which contributes $((-\frac13+1)/(-\frac13))^3=(-2)^3=-8$.
\end{proof}

\subsection{An adapted basis and the reduction}\label{sec:basis}
The choice of the basis comes from the tree bound of Section~\ref{sec:tree} at the prime $p$. Lemma~\ref{lem:relaxation} bounds $-v_p(\Delta_K)$ by a sum over the residue classes modulo $p$, and here a class is either a single U node or a pair consisting of an R node and its partner. If a node with a partner has $\check v_m=m-D^*$ for $m\le3$, then \eqref{eq:fplus} gives $f^+(k,s)=\max(0,D^*s-s(s-1))$, because $\sum_b(R_b+\pi(b))=\sum R+\binom s2$ for every row set $R$ in \eqref{eq:fplus}. By Lemma~\ref{lem:scaled}, the bound \eqref{eq:vhat} is $-7+m$ at the Ra, Rc and La nodes, $-4+m$ at Rb and Lc, $-3+m$ at Rb$_g$, $-1+m$ at Lb, and $m$ at Lb$_g$. A short computation with \eqref{eq:Vc} then gives the class values. For a pair of type a or c, $V_{\mathfrak c}(S)=8S-S^2$ for $S\le4$, so the first four rows placed in the class have the marginal values $7$, $5$, $3$, $1$, and further rows have negative marginal values. For a pair of type b the first two rows have the marginal values $4$, $2$, and for the pair of type b$_g$ the values $3$, $1$; further rows have marginal values at most $0$. A U node accepts no row at a positive marginal value, since $\hat v\ge0$ there. So the relaxed tree bound attains its largest value, $16(\Na+\Nc)+6\Nb-2$, when $K$ is the number \eqref{eq:Kcount}. We shall show that the constant coefficient of $\Delta_K$ attains this value exactly.

Accordingly, on polynomials of degree less than $K=K(n,p)$ we prescribe the Taylor coefficients of orders $0,1,2,3$ at the Ra and Rc nodes, and those of orders $0,1$ at the Rb and Rb$_g$ nodes. We call these nodes the \textit{condition nodes}; there are $K$ conditions by \eqref{eq:Kcount}. The conditions sit at the R nodes and not at their partners, because two partners are congruent modulo $p$, and the next lemma needs condition nodes that are pairwise incongruent. We index the conditions by $x=(R_x,j_x)$, where $R_x$ is a condition node and $j_x$ the order.

\begin{lemma}\label{lem:hermite}
There are polynomials $E_x\in\Z_p[y]$ of degree less than $K$ with $(E_x)_{j_{x'}}(R_{x'})=\delta_{x,x'}$ for all conditions $x$ and $x'$, and the transition matrix from the monomials $y^i$, $i<K$, to the $E_x$ has determinant in $\Z_p^\times$.
\end{lemma}

\begin{proof}
Let $\Xi$ be the $K\times K$ matrix of the conditions $P\mapsto P_j(R)$ on the monomials $y^i$, $i<K$. Its entries $\binom ij(-R)^{i-j}$ are integers. We claim that $\Xi$ is invertible modulo $p$. The condition nodes lie in $[p-n,2n]$, an interval of length $3n-p<p$, so they are pairwise incongruent modulo $p$. Over the field $\F_p$, a polynomial of degree less than $K$ is determined by its residues modulo the coprime polynomials $(y+R)^{d_R}$, where $d_R\in\{2,4\}$ is the number of conditions at $R$ and $\sum_Rd_R=K$ (Chinese remainder theorem), and the residue modulo $(y+R)^{d_R}$ is determined by the Taylor coefficients of orders less than $d_R$ at $-R$. So the reduction of $\Xi$ is injective, hence invertible, and $\det\Xi\in\Z_p^\times$. The polynomials $E_x$ are the dual basis, whose coefficients are the entries of $\Xi^{-1}$.
\end{proof}

Let $T$ be the matrix of coefficients of the $E_x$, so that $[\Lambda(E_xE_{x'})]_{x,x'}=T\,[\Lambda(y^{i+j})]_{i,j<K}\,T^{\mathsf T}$ and $\det T\in\Z_p^\times$. Hence $\det[\Lambda(E_xE_{x'})]=(\det T)^2\Delta_K(X)$, and the coefficients of $\det[\Lambda(E_xE_{x'})]$ and of $\Delta_K(X)$ have the same $p$-adic valuations. Write
\begin{equation}\label{eq:Gamma}
\begin{split}
&\Lambda(E_xE_{x'})=\Gamma^\beta_{xx'}+X\,\Gamma^\alpha_{xx'}\ ,\qquad\Gamma^\beta=\Gamma^\HS+\Gamma^{\mathrm{rest}}\ ,\\
&\Gamma^\bullet_{xx'}=\sum_{k\in\cN}\sum_{a+b\le3}c^\bullet_{k,a+b}\,(E_x)_a(k)\,(E_{x'})_b(k)\ ,
\end{split}
\end{equation}
where $c^\HS=\beta^\HS$, $c^{\mathrm{rest}}=\beta^{\mathrm{rest}}$ and $c^\alpha=\alpha$, by \eqref{eq:jetform}. Then $[X^0]\det[\Lambda(E_xE_{x'})]=\det\Gamma^\beta$.

To evaluate the terms of \eqref{eq:Gamma} we need the Taylor coefficients of $E_x$ at every node. Since $E_x\in\Z_p[y]$, all of them lie in $\Z_p$. At a condition node $R$ the prescribed ones are given by Lemma~\ref{lem:hermite}. At the partner $\bar R=R-p$ of a condition node $R$, which sits at $y=-R+p$, Taylor's formula gives
\begin{equation}\label{eq:shift}
(E_x)_b(\bar R)=\sum_{j\ge b}\binom jb\,(E_x)_j(R)\,p^{j-b}\ .
\end{equation}
So if $R=R_x$ and $b\le j_x$, the term $j=j_x$ gives $\binom{j_x}b p^{j_x-b}$, and $(E_x)_b(\bar R)$ has valuation at least $j_x-b$. In all other cases only orders $j$ that are not prescribed contribute, and $(E_x)_b(\bar R)$ has valuation at least $4-b$ if $R$ is of type Ra or Rc, and at least $\max(0,2-b)$ if $R$ is of type Rb or Rb$_g$. Accordingly, at the partner $\bar R$ of an Ra or Rc node $R$ we write
\begin{equation}\label{eq:leadrem}
(E_x)_b(\bar R)=\xi_{x,b}+\xi'_{x,b}\ ,\qquad\xi_{x,b}=\binom{j_x}b\,p^{j_x-b}\quad(R_x=R,\ b\le j_x)\ ,\qquad\xi_{x,b}=0\quad\text{otherwise}\ ,
\end{equation}
so that $v_p(\xi'_{x,b})\ge4-b$ in all cases. At such a partner we split each term of \eqref{eq:Gamma} into the four products of these parts, and we count them as separate contributions.

To each condition $x$ we attach the weight $\theta_x$, half of the marginal value of its row in the tree bound above: $\theta_x=\frac72-j_x$ if $R_x$ is of type Ra or Rc, $\theta_x=2-j_x$ if it is of type Rb, and $\theta_x=\frac32-j_x$ if it is of type Rb$_g$. A \textit{contribution} to $\Gamma^\bullet_{xx'}$ is one term $c^\bullet_{k,a+b}(E_x)_a(k)(E_{x'})_b(k)$ of \eqref{eq:Gamma}, and its \textit{weighted valuation} is
\begin{equation}\label{eq:omega}
\varpi=v_p\qty(c^\bullet_{k,a+b})+v_p\qty((E_x)_a(k))+v_p\qty((E_{x'})_b(k))+\theta_x+\theta_{x'}\in\tfrac12\Z\ .
\end{equation}
With these weights the leading contributions at a condition node have $\varpi=0$. For instance, at an Ra or Rc node $R$, by \eqref{eq:scaled} with $D_R=1$, the contribution of $\beta^\HS_{R,j_x+j_{x'}}$ to $\Gamma^\HS_{xx'}$ with $R_x=R_{x'}=R$ has valuation $j_x+j_{x'}-7$, and $\theta_x+\theta_{x'}=7-j_x-j_{x'}$.

\begin{lemma}[weighted valuations]\label{lem:omega}
Every contribution to $\Gamma^\beta$ has $\varpi\ge0$, and every contribution to $\Gamma^\alpha$ has $\varpi\ge1$. If $R_x\ne R_{x'}$, every contribution to $\Gamma^\beta_{xx'}$ has $\varpi\ge\frac12$. If $R_x=R_{x'}=R$, the contributions to $\Gamma^\beta_{xx'}$ with $\varpi=0$ are the terms of $\beta^\HS$ at $R$ with $(a,b)=(j_x,j_{x'})$ and, if $R$ is of type Ra, the products $\beta^\HS_{\bar R,a+b}\,\xi_{x,a}\,\xi_{x',b}$ at its partner $\bar R$.
\end{lemma}

The proof is a case analysis over the types of nodes, given in Appendix~\ref{app:omega}. Since the lemma allows contributions with $\varpi=\frac12$, at the Rb nodes, the reduction below is made modulo $\sqrt p$ and not modulo $p$.

Let $\Theta$ be the diagonal matrix with entries $p^{\theta_x}$, viewed in the field $\Q_p(\sqrt p)$. The polynomial $Y^2-p$ is an Eisenstein polynomial, so this is a totally ramified quadratic extension of $\Q_p$, with valuation ring $\Z_p[\sqrt p]$ and residue field $\Z_p[\sqrt p]/(\sqrt p)=\F_p$, and the valuation $v_p$ extends with $v_p(\sqrt p)=\frac12$. By Lemma~\ref{lem:omega}, the scaled matrix $\widetilde\Gamma^\beta=\Theta\,\Gamma^\beta\,\Theta$ has entries in $\Z_p[\sqrt p]$, its entries with $R_x\ne R_{x'}$ vanish modulo $\sqrt p$, and $\widetilde\Gamma^\alpha=\Theta\,\Gamma^\alpha\,\Theta$ has entries in $p\Z_p[\sqrt p]$. So modulo $\sqrt p$ the matrix $\widetilde\Gamma^\beta$ is block diagonal, with one block for each condition node, and Lemma~\ref{lem:omega}, \eqref{eq:scaled} and \eqref{eq:shift} give the blocks.

\begin{lemma}\label{lem:blocks}
Modulo $\sqrt p$, the block of $\widetilde\Gamma^\beta$ at a condition node $R$ is
\begin{equation}\label{eq:blocks}
\begin{split}
&\varepsilon_R\,[\phi_{\mathrm{Rc},3-a-b}]_{0\le a,b\le3}\qquad(R\text{ of type Rc})\ ,\\
&\varepsilon_R\begin{pmatrix}\phi_{\mathrm{Rb},3}&\phi_{\mathrm{Rb},2}\\\phi_{\mathrm{Rb},2}&\phi_{\mathrm{Rb},1}\end{pmatrix}\qquad(R\text{ of type Rb})\ ,\qquad\varepsilon_R\begin{pmatrix}\phi_{\mathrm{Rb}_g,3}&\phi_{\mathrm{Rb}_g,2}\\\phi_{\mathrm{Rb}_g,2}&\phi_{\mathrm{Rb}_g,1}\end{pmatrix}\qquad(R\text{ of type Rb}_g)\ ,\\
&\varepsilon_R\qty(A_{\mathrm{Ra}}-8\,\mathrm{Bin}^{\mathsf T}A_{\mathrm{La}}\mathrm{Bin})\qquad(R\text{ of type Ra})\ ,
\end{split}
\end{equation}
where $\phi_b=0$ for $b<0$, $A_\bullet=[\phi_{\bullet,3-a-b}]_{0\le a,b\le3}$, and $\mathrm{Bin}=[\binom ja]_{0\le a,j\le3}$.
\end{lemma}

\begin{proof}
At an Rc, Rb or Rb$_g$ node $R$ the only contributions with $\varpi=0$ are those of $\beta^\HS_{R,j_x+j_{x'}}$ with $(E_x)_{j_x}(R)=(E_{x'})_{j_{x'}}(R)=1$. By \eqref{eq:scaled}, multiplied by $p^{\theta_x+\theta_{x'}}=p^{D_R+6-j_x-j_{x'}}$, each becomes $\varepsilon_R(\phi_{R,3-j_x-j_{x'}}+O(p))$. For Rb and Rb$_g$, $j_x,j_{x'}\in\{0,1\}$, which gives the $2\times2$ blocks. At an Ra node $R$ there is, in addition, the partner $\bar R$, of type La. By \eqref{eq:leadrem} and Lemma~\ref{lem:omega}, the contributions of $\bar R$ with $\varpi=0$ are the products with $\xi_{x,a}\xi_{x',b}$, and they sum, after scaling, to
\begin{equation}\label{eq:Lcontribution}
\varepsilon_{\bar R}\sum_{a\le j_x,\,b\le j_{x'}}\binom{j_x}a\binom{j_{x'}}b\,\phi_{\mathrm{La},3-a-b}=\varepsilon_{\bar R}\qty(\mathrm{Bin}^{\mathsf T}A_{\mathrm{La}}\mathrm{Bin})_{j_xj_{x'}}
\end{equation}
modulo $p$, and $\varepsilon_{\bar R}\equiv-8\,\varepsilon_R$ by Lemma~\ref{lem:wilson}.
\end{proof}

The determinants of the blocks are explicit. The Rc block vanishes below its anti-diagonal and has the anti-diagonal entries $\varepsilon_R\phi_{\mathrm{Rc},0}=\varepsilon_R\cdot\frac{3^5}{2^3}$, so its determinant is $\varepsilon_R^4\,3^{20}/2^{12}$. By \eqref{eq:phivalues} the Rb block is $\varepsilon_R\cdot\frac{243}8\qty(\begin{smallmatrix}0&-1\\-1&-1\end{smallmatrix})$, of determinant $-\varepsilon_R^2\,3^{10}/2^6$, and the Rb$_g$ block is $\varepsilon_R\cdot\frac{243}8\qty(\begin{smallmatrix}2&1\\1&-3\end{smallmatrix})$, of determinant $-\varepsilon_R^2\,7\cdot3^{10}/2^6$. For the Ra block, the matrix $\mathrm{Bin}^{\mathsf T}A_{\mathrm{La}}\mathrm{Bin}$ has the entries $\sum_{m}\binom{a+b}m\phi_{\mathrm{La},3-m}$, which depend only on $a+b$, and a direct computation from \eqref{eq:phivalues} gives
\begin{equation}\label{eq:rablock}
A_{\mathrm{Ra}}-8\,\mathrm{Bin}^{\mathsf T}A_{\mathrm{La}}\mathrm{Bin}=3^5\begin{pmatrix}207&163&127&98\\163&127&98&75\\127&98&75&57\\98&75&57&43\end{pmatrix}\ ,
\end{equation}
which is $3^5$ times an integer matrix of determinant $1$, so its determinant is $3^{20}$. All four determinants are products of units with powers of $2$, $3$ and $7$, so they are $p$-adic units, as $p\ge13$.

\begin{remark}\label{rem:wilsonneeded}
For a general value $\vartheta$ in place of $-8$ one finds $\det(A_{\mathrm{Ra}}+\vartheta\,\mathrm{Bin}^{\mathsf T}A_{\mathrm{La}}\mathrm{Bin})=3^{12}(\vartheta-1)^4$. So the Ra block is a unit precisely when $\vartheta\not\equiv1\pmod p$, and for $\vartheta=-8$ this holds since $p\nmid9$. The number $-8=(-2)^3$ comes from the covering zero of the Ra node: its linear factor takes the values $-\frac p3$ and $\frac{2p}3$ at the two members of the pair.
\end{remark}

\begin{proof}[Proof of Theorem~\ref{thm:nonvanishing}]
The bounds on $K$ were proved after \eqref{eq:Kcount}. By Lemma~\ref{lem:blocks} and the computation of the four block determinants, $\det\widetilde\Gamma^\beta$ is a unit of $\Z_p[\sqrt p]$. Since $\det\widetilde\Gamma^\beta=p^{2\sum_x\theta_x}\det\Gamma^\beta$, we get $v_p(\det\Gamma^\beta)=-2\sum_x\theta_x$. The sum of the weights is $\frac72+\frac52+\frac32+\frac12=8$ at each Ra and Rc node, $2+1=3$ at each Rb node and $\frac32+\frac12=2$ at the single Rb$_g$ node, so
\begin{equation}\label{eq:x0valuation}
v_p\qty([X^0]\Delta_K)=v_p\qty(\det\Gamma^\beta)=-\qty(16(\Na+\Nc)+6\Nb-2)\ .
\end{equation}
The coefficient of $X^j$ in $\det(\widetilde\Gamma^\beta+X\widetilde\Gamma^\alpha)$ is a sum of determinants of matrices with $j$ columns taken from $\widetilde\Gamma^\alpha$ and the others from $\widetilde\Gamma^\beta$, so it lies in $p^j\Z_p[\sqrt p]$. Hence $v_p([X^j]\Delta_K)\ge v_p([X^0]\Delta_K)+j$ for every $j\ge1$, and the constant coefficient of $\Delta_K$ is strictly dominant at $p$. By Lemma~\ref{lem:dominance}, $\Delta_K(a/b)\ne0$ for every $a/b\in\Q$ with $p\nmid b$.
\end{proof}

\appendix

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Lemma~\ref{lem:analyticstep}}\label{app:analytic}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We expand only the polynomial part of $PQ$ in the binomial basis and estimate the remainder directly, so that no infinite series is differentiated term by term.

A polynomial $P$ of degree at most $d$ with $P(\Z_3)\subseteq\Z_3$ can be written as $P=\sum_{c\le d}\hat P_c\binom tc$ with
\begin{equation}\label{eq:mahler}
\hat P_c=\sum_{i=0}^c(-1)^{c-i}\binom ciP(i)\in\Z_3\ ,
\end{equation}
because the $\binom tc$, $c\le d$, are a basis, and applying $c$ times the difference operator $f\mapsto f(t+1)-f(t)$, which maps $\binom tc$ to $\binom t{c-1}$, and evaluating at $0$ gives $\hat P_c$. This is the polynomial case of Mahler's expansion \cite[Ch.~4]{Ro00}. The monomial coefficients of $\binom tc$ are integers divided by $c!$, so the monomial coefficients of $P$ have valuation at least $-v_3(d!)$.

Let $\varsigma(m)=m+v_3(m!)$. It is increasing and superadditive, $\varsigma(a+b)\ge\varsigma(a)+\varsigma(b)$, since $(a+b)!/(a!\,b!)$ is an integer. The series $Q$ converges on $|t|_3<3$, so $PQ$ is strictly differentiable. Fix $M>d$ and split $Q=Q_{<M}+R_M$ with $R_M=\sum_{i\ge M}q_it^i$.

\textit{The polynomial part.} Using $t^i=\sum_{m\le i}\stirling im\,m!\binom tm$, with the Stirling numbers $\stirling im$ of the second kind, which are integers, we get $Q_{<M}=\sum_m\hat q_m\binom tm$ with $\hat q_m=m!\sum_{m\le i<M}\stirling imq_i$, and so $v_3(\hat q_m)\ge v_3(m!)+m=\varsigma(m)$, uniformly in $M$. The identity
\begin{equation}\label{eq:binomproduct}
\binom ta\binom tb=\sum_{c=\max(a,b)}^{a+b}\binom ca\binom a{c-b}\binom tc\ ,
\end{equation}
which follows by counting pairs of subsets of a $t$-element set of sizes $a$ and $b$ according to their union, has integer coefficients. So $f=PQ_{<M}=\sum_cf_c\binom tc$, where only terms with $a\le d$ and $b\ge c-a\ge c-d$ contribute to $f_c$, and hence $v_3(f_c)\ge\varsigma((c-d)^+)$, $\varsigma$ being increasing.

\textit{One derivative.} Differentiating $\binom{t+x}c=\sum_i\binom xi\binom t{c-i}$ with respect to $x$ at $x=0$, and using $\frac{\dd}{\dd x}\binom xi|_{x=0}=(-1)^{i-1}/i$ for $i\ge1$, we get $\frac{\dd}{\dd t}\binom tc=\sum_{i=1}^c\frac{(-1)^{i-1}}i\binom t{c-i}$. Hence $f'=\sum_{c'}f'_{c'}\binom t{c'}$ with $f'_{c'}=\sum_{i\ge1}\frac{(-1)^{i-1}}if_{c'+i}$. We claim that
\begin{equation}\label{eq:derivbound}
v_3(f'_{c'})\ge\varsigma((c'-d)^+)-\log_3d\ .
\end{equation}
The term $i$ of $f'_{c'}$ has valuation at least $\varsigma((c'+i-d)^+)-v_3(i)$, and there are three cases. If $c'\ge d$, then $\varsigma(c'-d+i)-v_3(i)\ge\varsigma(c'-d)+\varsigma(i)-v_3(i)\ge\varsigma(c'-d)$, because $\varsigma(i)\ge i\ge v_3(i)$. If $c'+i\le d$, then $v_3(i)\le\log_3i\le\log_3d$. If $c'<d<c'+i$, put $e=c'+i-d\ge1$; then $v_3(i)\le\log_3(d+e)\le\log_3d+\log_3(1+e)\le\log_3d+\varsigma(e)$. In all three cases the term is bounded as claimed.

\textit{The integral.} By (V2), $\vint f'=\sum_{c'}f'_{c'}(-1)^{c'}/(c'+1)$, a finite sum. In the same way as above, $v_3(c'+1)\le\log_3(d+1)+\varsigma((c'-d)^+)$: for $c'\le d$ this is clear, and for $c'=d+e$ we have $\log_3(d+1+e)\le\log_3(d+1)+\log_3(1+e)$. With \eqref{eq:derivbound}, every term has valuation at least $-\log_3d-\log_3(d+1)$, and so has $\vint f'$.

\textit{The remainder.} The monomial coefficients of $P$ and $P'$ have valuation at least $-v_3(d!)$, and those of $R_M$ and $R_M'$ at least $\max(M,m)$ at $t^m$. So $(PR_M)'=P'R_M+PR_M'=\sum_me_mt^m$ with $v_3(e_m)\ge M-v_3(d!)$ and $v_3(e_m)\ge m-d-v_3(d!)$ for all $m$. This series converges on $|t|_3<3$, its value at $0$ and its Lipschitz constant on $\Z_3$ are at most $\max_m|e_m|_3$, and (V3) gives $v_3\vint(PR_M)'\ge M-v_3(d!)-1$.

By linearity, $\vint(PQ)'=\vint f'+\vint(PR_M)'$. The left side does not depend on $M$, and letting $M\to\infty$ proves \eqref{eq:analyticstep}. \qed

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Heine expansion, the tree bound and the size}\label{app:heine}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This appendix proves Proposition~\ref{prop:heine}, Theorem~\ref{thm:tree} and Theorem~\ref{thm:arch}.
For a $\Q[X]$-linear map $\Lambda':\Q[X][w]\to\Q[X]$ and $s\ge1$ let $\Lambda'^{\otimes s}:\Q[X][w_1,\dots,w_s]\to\Q[X]$ be the $\Q[X]$-linear map with $w_1^{e_1}\cdots w_s^{e_s}\mapsto\prod_i\Lambda'(w^{e_i})$.

\begin{lemma}[Andr\'eief]\label{lem:andreief}
For polynomials $f_0,\dots,f_{s-1},g_0,\dots,g_{s-1}\in\Q[X][w]$,
\begin{equation}\label{eq:andreief}
\Lambda'^{\otimes s}\qty(\det[f_b(w_i)]_{i,b}\,\det[g_{b'}(w_i)]_{i,b'})=s!\,\det\qty[\Lambda'(f_bg_{b'})]_{b,b'<s}\ .
\end{equation}
In particular $\det[\Lambda'(w^{b+b'})]_{b,b'<s}=\frac1{s!}\Lambda'^{\otimes s}(V(w)^2)$ with $V(w)=\prod_{i<i'}(w_{i'}-w_i)$.
\end{lemma}

\begin{proof}
The product of the two determinants is $\sum_{\pi,\pi'}\sgn(\pi)\sgn(\pi')\prod_if_{\pi(i)}(w_i)g_{\pi'(i)}(w_i)$, the sum over bijections $\pi,\pi'$ from $\{1,\dots,s\}$ to $\{0,\dots,s-1\}$. The map $\Lambda'^{\otimes s}$ sends the product over $i$ to $\prod_i\Lambda'(f_{\pi(i)}g_{\pi'(i)})=\prod_b\Lambda'(f_bg_{\pi'\pi^{-1}(b)})$. For fixed $\pi$, the sum over $\pi'$ gives $\sgn(\pi)^2\det[\Lambda'(f_bg_{b'})]$, and the $s!$ choices of $\pi$ contribute equally.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:heine}]
\textit{Step 1.} By Lemma~\ref{lem:andreief}, $\Delta_K=\frac1{K!}\Lambda^{\otimes K}(V(y_1,\dots,y_K)^2)$, where $y_1,\dots,y_K$ are independent variables.

\textit{Step 2.} Since $\Lambda=\sum_k\Lambda_k$, we have $\Lambda^{\otimes K}=\sum_\iota\bigotimes_i\Lambda_{\iota(i)}$, the sum over all maps $\iota:\{1,\dots,K\}\to\cN$. The polynomial $V^2$ is symmetric, so $\bigotimes_i\Lambda_{\iota(i)}(V^2)$ depends only on the occupancy $s=(s_k)$, $s_k=|\iota^{-1}(k)|$, an integer vector with $s_k\ge0$ and $\sum_ks_k=K$, and there are $K!/\prod_ks_k!$ maps with a given occupancy. Hence $\Delta_K=\sum_s(\prod_ks_k!)^{-1}\bigotimes_i\Lambda_{\iota(i)}(V^2)$, the sum over all occupancies, with $\iota$ fixed for each occupancy $s$ as in the proposition. Since $\Lambda_k(F)$ depends only on the Taylor coefficients of $F$ of order at most $3$ at $-k$, only occupancies with $s_k\le4$, that is, the profiles, will contribute (Step~6).

\textit{Step 3.} Let $\Lambda_k^{\mathrm{loc}}:\Q[X][w]\to\Q[X]$ be the map $w^j\mapsto c_{k,j}$, so that $\Lambda_k(F)=\Lambda_k^{\mathrm{loc}}(F(-k+w))$. Substituting $y_i=-\iota(i)+w_i$, the squared Vandermonde factors into the pairs inside one node, which give $(w_{i'}-w_i)^2$, and the cross pairs, which give the factors $(\iota(i')-\iota(i)+w_i-w_{i'})^2$ of \eqref{eq:cross}. So $V^2=\prod_kV(w^{(k)})^2\cdot Q_s(w)$, where $w^{(k)}$ is the set of variables $w_i$ with $\iota(i)=k$.

\textit{Step 4.} Write $Q_s=\sum_\alpha q_\alpha w^\alpha$ with $q_\alpha\in\Z$. Permuting the variables inside one node permutes the cross pairs, and each factor of \eqref{eq:cross} is a square, so $q_\alpha$ is constant on the orbits of $\prod_k\mathrm{Sym}(w^{(k)})$. These orbits are labeled by the tuples $\lambda=(\lambda_k)$ of partitions, $\lambda_k$ being the multiset of the exponents at $w^{(k)}$. So $Q_s=\sum_\lambda\mathrm{coef}_\lambda(s)\prod_km_{\lambda_k}(w^{(k)})$, where $m_\mu$ is the monomial symmetric polynomial and $m_\emptyset=1$.

\textit{Step 5.} The map $\bigotimes_i\Lambda^{\mathrm{loc}}_{\iota(i)}$ is multiplicative on products of polynomials in disjoint groups of variables. Hence each $\lambda$ contributes $\mathrm{coef}_\lambda(s)\prod_k(\Lambda^{\mathrm{loc}}_k)^{\otimes s_k}\qty(V(w^{(k)})^2m_{\lambda_k}(w^{(k)}))$.

\textit{Step 6.} For $s\ge1$ and a partition $\mu$ with at most $s$ parts, $V(w)m_\mu(w)$ is an antisymmetric polynomial in $w_1,\dots,w_s$ with integer coefficients, homogeneous of degree $\binom s2+|\mu|$. It is therefore an integer combination $\sum_\nu n_{\mu,\nu}\det[w_i^{\nu_b}]$ of alternants with $\nu_0>\nu_1>\dots>\nu_{s-1}\ge0$ and $\sum_b\nu_b=\binom s2+|\mu|$, because an antisymmetric polynomial is determined by its coefficients at the monomials with strictly decreasing exponents. By Lemma~\ref{lem:andreief} with $f_b=w^b$ and $g_{b'}=w^{\nu_{b'}}$,
\begin{equation}\label{eq:localandreief}
(\Lambda^{\mathrm{loc}}_k)^{\otimes s}\qty(V^2m_\mu)=s!\sum_\nu n_{\mu,\nu}\det\qty[c_{k,b+\nu_{b'}}]_{b,b'<s}\ .
\end{equation}
If some $\nu_{b'}\ge4$, the corresponding column vanishes, as $c_{k,m}=0$ for $m\ge4$. So only $\nu\subseteq\{0,1,2,3\}$ contribute, and these are the row sets $R$ of the minors $\det C_k[R]$, up to transposition and sign. For $s\ge5$ there is no such $\nu$, and in general $|\mu|=\sum\nu-\binom s2\le(3s-\binom s2)-\binom s2=s(4-s)$. The factors $s_k!$ cancel those of Step~2, and the signs are absorbed in the integers $n_{\mu,\nu}$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:tree}]
By Proposition~\ref{prop:heine} and the ultrametric inequality,
\begin{equation}\label{eq:treestart}
v(\Delta_K)\ge\min_{s,\lambda}\qty[v(\mathrm{coef}_\lambda(s))+\sum_kv(M_{k,\lambda_k})]\ .
\end{equation}

\textit{Local factors.} Each $M_{k,\lambda}$ is an integer combination of minors $\det C_k[R]$ with $\sum R-\binom{s_k}2=|\lambda|$, so $v(M_{k,\lambda})\ge-\mathsf e_p(k,s_k,|\lambda|)$.

\textit{Cross factors.} Expand each factor of \eqref{eq:cross} as $\sum_{e=0}^2\binom2ed^{2-e}(w_i-w_{i'})^e$ with $d=\iota(i')-\iota(i)\ne0$, and $(w_i-w_{i'})^e=\sum_{\alpha\le e}\binom e\alpha(-1)^{e-\alpha}w_i^\alpha w_{i'}^{e-\alpha}$. A monomial of $Q_s$ arises from a choice of $(e,\alpha)$ for every cross pair, with a coefficient that is an integer times $\prod d^{2-e}$, whose valuation is $2\sum_{\mathrm{pairs}}v_p(d)-\sum_{\mathrm{pairs}}e\,v_p(d)$. Of the $e$ units of degree at a pair, $\alpha$ go to $w_i$ and $e-\alpha$ to $w_{i'}$, and a unit that goes to a variable of the node $k$ costs $v_p(d)\le\eta_p(k)$. The node $k$ receives $|\lambda_k|$ units in all, so
\begin{equation}\label{eq:crossval}
v(\mathrm{coef}_\lambda(s))\ge2\sum_{k<m}s_ks_mv_p(k-m)-\sum_k|\lambda_k|\,\eta_p(k)\ .
\end{equation}

\textit{Combination.} With $\ell_k=|\lambda_k|\le s_k(4-s_k)$, every term of \eqref{eq:heine} has valuation at least $-\sum_k\qty(\mathsf e_p(k,s_k,\ell_k)+\ell_k\eta_p(k))+2\sum_{k<m}s_ks_mv_p(k-m)\ge-\sum_kf_p(k,s_k)+2\sum_{k<m}s_ks_mv_p(k-m)$. Finally, $v_p(x)=\#\{j\ge1:\ p^j\mid x\}$ gives
\begin{equation}\label{eq:treeidentity}
2\sum_{k<m}s_ks_mv_p(k-m)=\sum_{j\ge1}\sum_{\mathfrak c\bmod p^j}\ \sum_{\substack{k,m\in\mathfrak c\\k\ne m}}s_ks_m=\sum_{j\ge1}\sum_{\mathfrak c\bmod p^j}\qty(S_{\mathfrak c}^2-\sum_{k\in\mathfrak c}s_k^2)\ ,
\end{equation}
a finite sum. Nothing in the argument uses that $p$ is odd.
\end{proof}

For Theorem~\ref{thm:arch} we need four lemmas. Put $B(d)=\frac12\log(1+d^2)$, so that $B(0)=0$ and $0\le B(d)-\log|d|\le\frac1{2d^2}$ for $d\ne0$, and for a real vector $s=(s_k)_{k\in\cN}$ let
\begin{equation}\label{eq:energy}
E_B(s)=\sum_ks_k\log|h_k|+\sum_{k,m}s_ks_mB(k-m)\ .
\end{equation}
The function $B$ replaces $\log|k-m|$ because it is defined at $k=m$ and is conditionally negative definite (Lemma~\ref{lem:cnd}).

\begin{lemma}[discrete energy]\label{lem:discrete}
$\log\|\Delta_K\|_1\le\max_sE_B(s)+O(n\log n)$, the maximum over the profiles $s$ of Proposition~\ref{prop:heine}, uniformly in $K\le3n+1$.
\end{lemma}

\begin{proof}
We apply $\|f+g\|_1\le\|f\|_1+\|g\|_1$ and $\|fg\|_1\le\|f\|_1\|g\|_1$ to \eqref{eq:heine}.

(1) There are at most $5^{3n+1}$ profiles, and for each node at most $12$ partitions $\lambda_k$, since $|\lambda_k|\le4$. So there are $e^{O(n)}$ pairs $(s,\lambda)$.

(2) The cross factor $\mathrm{coef}_\lambda(s)$ is a coefficient of $Q_s$ at a monomial of total degree $\sum_k|\lambda_k|\le4(3n+1)$. By Cauchy's estimate on the polydisc $|w_i|=\frac1{10}$, and since $|d+w_i-w_{i'}|\le|d|+\frac15$ there,
\begin{equation}\label{eq:cauchy}
|\mathrm{coef}_\lambda(s)|\le10^{4(3n+1)}\prod_{k\ne m}\qty(|k-m|+\tfrac15)^{s_ks_m}\ .
\end{equation}

(3) Write $\Wt_k(z)/h_k=\exp(-\sum_{j\ge1}\mathfrak s_jz^j/j)$ with the power sums $\mathfrak s_j=\sum_de_dd^{-j}$ over the nonzero roots and poles $d$ of $\Wt_k$. All of them have $|d|\ge\frac13$: the zeros $k-j-\frac r3$ lie in $\frac13\Z\setminus\Z$, $|k-\frac n2|\ge\frac12$, and $|k-m|\ge1$. Moreover, the $3d$ are distinct integers for the zeros of order $3$, and the $d$ are distinct integers for the poles. Summing $|e_d|/|d|$ gives $|\mathfrak s_1|=O(\log n)$, and $|\mathfrak s_j|\le C3^j$ for $j\ge2$ with an absolute constant $C$. The coefficients of $\exp(\sum_ja_jz^j)$ are polynomials in the $a_j$ with nonnegative coefficients, so $|H_k[b]|\le|h_k|\cdot O(\log n)^b$ for $b\le3$.

(4) $|\HS_3(k,M)|\le3^M\sum_{u\ge1}u^{-M}\le2\cdot3^M$ for $M\ge2$, so by \eqref{eq:localconstants} $\|c_{k,a}\|_1\le C_1\max_{b\le3}|H_k[b]|$ with an absolute constant $C_1$. The local factor $M_{k,\lambda_k}$ is a combination of at most $6$ minors with integer coefficients bounded by an absolute constant, since $s_k\le4$ and $|\lambda_k|\le4$, and each minor is a sum of at most $24$ products of $s_k$ local constants. Hence $\log\|M_{k,\lambda_k}\|_1\le s_k\log|h_k|+O(s_k\log\log n)+O(1)$.

(5) For integers $|d|\ge1$ we have $\log(|d|+\frac15)=B(d)+\mathrm{err}(d)$ with $|\mathrm{err}(d)|\le\frac1{5|d|}+\frac1{2d^2}\le\frac1{|d|}$, so $\sum_{k\ne m}s_ks_m\mathrm{err}(k-m)\le\sum_k4s_k\sum_{m\ne k}\frac1{|k-m|}=O(K\log n)$.

Collecting (1)--(5), every term of \eqref{eq:heine} has $\log\|\cdot\|_1\le E_B(s)+O(n\log n)$, and there are $e^{O(n)}$ terms.
\end{proof}

\begin{lemma}[conditional negative definiteness]\label{lem:cnd}
For real $x_i$ and real $\tau_i$ with $\sum_i\tau_i=0$, $\sum_{i,j}\tau_i\tau_jB(x_i-x_j)\le0$. Consequently, for real vectors $s$ and $\tilde s$ with $\sum_ks_k=\sum_k\tilde s_k$,
\begin{equation}\label{eq:linearize}
E_B(s)\le E_B(\tilde s)+\langle\nabla,s-\tilde s\rangle\ ,\qquad\nabla_k=\log|h_k|+2\sum_m\tilde s_mB(k-m)\ .
\end{equation}
\end{lemma}

\begin{proof}
Put $\xi_x=(\frac12-ix)/(\frac32+ix)$, so that $|\xi_x|<1$. A direct computation gives $(\frac32+ix)(\frac32-iy)-(\frac12-ix)(\frac12+iy)=2(1+i(x-y))$, hence $1+(x-y)^2=\frac14|1-\xi_x\bar\xi_y|^2|\frac32+ix|^2|\frac32+iy|^2$ and
\begin{equation}\label{eq:cayley}
B(x-y)=\log|1-\xi_x\bar\xi_y|+a(x)+a(y)-\log2\ ,\qquad a(x)=\log\abs{\tfrac32+ix}\ .
\end{equation}
The terms $a(x_i)+a(x_j)-\log2$ drop out against $\sum_i\tau_i=0$, and with $\log|1-\zeta|=-\mathrm{Re}\sum_{m\ge1}\zeta^m/m$ for $|\zeta|<1$,
\begin{equation}\label{eq:cndsum}
\sum_{i,j}\tau_i\tau_j\log|1-\xi_{x_i}\bar\xi_{x_j}|=-\sum_{m\ge1}\frac1m\abs{\sum_i\tau_i\xi_{x_i}^m}^2\le0\ .
\end{equation}
For the consequence, expand the quadratic form $E_B$ around $\tilde s$ with $\tau=s-\tilde s$: the linear term is $\langle\nabla,\tau\rangle$, and the quadratic term is $\sum_{k,m}\tau_k\tau_mB(k-m)\le0$.
\end{proof}

\begin{lemma}[modulus of continuity]\label{lem:modulus}
For $x<x'$ in $[-1,2]$ and $\delta=x'-x$,
\begin{equation}\label{eq:modulus}
\int_{-1}^2\abs{\log|x-y|-\log|x'-y|}\dd y\le w(\delta):=2\delta\qty(1+\log\frac{6+\delta}\delta)\ .
\end{equation}
Hence $|\cL f(x)-\cL f(x')|\le\|f\|_\infty\,w(|x-x'|)$.
\end{lemma}

\begin{proof}
Let $\bar x=\frac{x+x'}2$, so that $[-1,2]\subseteq[\bar x-3,\bar x+3]$. The integrand has a constant sign on each side of $\bar x$, and the reflection $y\mapsto2\bar x-y$ exchanges $x$ and $x'$. So the integral is at most $2G$, with $G=\int_{-3}^0(\log|y-\frac\delta2|-\log|y+\frac\delta2|)\,\dd y=-2\mathcal Q_1(\frac\delta2)+\mathcal Q_1(3+\frac\delta2)-\mathcal Q_1(3-\frac\delta2)$ and $\mathcal Q_1(y)=y\log|y|-y$. Here $-2\mathcal Q_1(\frac\delta2)=\delta(1+\log\frac2\delta)$, and $\mathcal Q_1(3+\frac\delta2)-\mathcal Q_1(3-\frac\delta2)=\int_{3-\delta/2}^{3+\delta/2}\log y\,\dd y\le\delta\log(3+\frac\delta2)$.
\end{proof}

\begin{lemma}[continuum limit]\label{lem:continuum}
Let $1\le\kappa\le3+\frac1n$, let $\sigma$ be feasible for $\kappa$ and piecewise constant with at most $J$ pieces, and let $\tilde s_k=c_n\sigma(k/n)$ with $c_n$ chosen so that $\sum_k\tilde s_k=K$; then $c_n=1+O_J(1/n)$. Uniformly in $k\in\cN$:
\begin{itemize}
\item[(a)] $\log|h_k|=-6n\log n+n\,\cV(k/n)+O(\log n)$;
\item[(b)] $2\sum_m\tilde s_mB(k-m)=2\kappa n\log n+2n\,\cL\sigma(k/n)+O_J(\log n)$;
\item[(c)] $E_B(\tilde s)=(\kappa^2-6\kappa)n^2\log n+n^2\mathcal E[\sigma]+O_J(n\log n)$.
\end{itemize}
\end{lemma}

\begin{proof}
We use one elementary estimate. For $\theta\in[\frac13,1]$ and integers $0\le M_1\le M_2$,
\begin{equation}\label{eq:logsum}
\sum_{M_1\le i<M_2}\log(i+\theta)=\int_{M_1}^{M_2}\log y\,\dd y+O(\log(M_2+2))\ ,
\end{equation}
because $\int_{i-1+\theta}^{i+\theta}\log\le\log(i+\theta)\le\int_{i+\theta}^{i+1+\theta}\log$ for $i\ge1$, the term $i=0$ is $O(1)$, and shifting the limits of integration by at most $2$ costs $O(\log(M_2+2))$, $\log$ being integrable at $0$. Consequently, for real $a$ with $|a|\le3n$ at distance at least $\frac13$ from $\Z$, or for $a\in\Z$ with the term $j=a$ omitted, and integers $A_1<A_2$ with $|A_1|,|A_2|\le3n$, we have $\sum_{A_1\le j<A_2}\log|a-j|=\int_{A_1}^{A_2}\log|a-y|\,\dd y+O(\log n)$; split the sum at $a$.

(a) By \eqref{eq:hk}, $\log|h_k|=\log|\frac n2-k|+3\sum_{r=1,2}\sum_{j<n}\log|k-\frac r3-j|-4\sum_{m\ne k}\log|m-k|$. The first term is $O(\log n)$. By the elementary estimate, $\sum_{j<n}\log|k-\frac r3-j|=\int_0^n\log|k-\frac r3-y|\,\dd y+O(\log n)$, and by Lemma~\ref{lem:modulus}, after the substitution $y=ny'$, the shift $\frac r3$ changes the integral by $O(\log n)$. Rescaling, $\int_0^n\log|k-y|\,\dd y=n\log n+n\,\cL\one_{I_{\mathrm{in}}}(x)$ with $x=k/n$. Likewise $\sum_{m\ne k}\log|m-k|=3n\log n+n\,\cL\one_{[-1,2]}(x)+O(\log n)$. The coefficients of $n\log n$ add up to $3\cdot2-4\cdot3=-6$, and the rest is $n(6\cL\one_{I_{\mathrm{in}}}-4\cL\one_{[-1,2]})(x)=n\cV(x)$.

(b) We have $\sum_{m\ne k}|B(k-m)-\log|k-m||\le\frac{\pi^2}6$. On each piece $[\alpha,\beta)$ of $\sigma$, the elementary estimate gives $\sum_{m\in[\alpha n,\beta n),\,m\ne k}\log|k-m|=n(\beta-\alpha)\log n+n\int_\alpha^\beta\log|x-y|\,\dd y+O(\log n)$. Multiply by the value of $\sigma$ on the piece, which is at most $4$, sum over the $J$ pieces, and use $c_n=1+O(1/n)$ and $\int\sigma=\kappa$.

(c) By Lemma~\ref{lem:modulus}, $\cV$ and $\cL\sigma$ have modulus of continuity $O(w(\delta))$, and $w(\frac1n)=O(\frac{\log n}n)$. Hence $\sum_k\sigma(k/n)\phi(k/n)=n\langle\sigma,\phi\rangle+O_J(\log n)$ for $\phi\in\{\cV,\cL\sigma\}$: the $O(n)$ intervals $[\frac kn,\frac{k+1}n)$ on which $\sigma$ is constant contribute $O(w(\frac1n))$ each, and the at most $J$ others $O(1)$ each. Sum (a) and (b) against $\tilde s_k$, and use $\sum_k\tilde s_k=\kappa n$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:arch}]
By Lemmas~\ref{lem:discrete} and~\ref{lem:cnd}, $\log\|\Delta_K\|_1\le E_B(\tilde s)+\max_s\langle\nabla,s-\tilde s\rangle+O(n\log n)$. By Lemma~\ref{lem:continuum}(a),(b), $\nabla_k=\mathrm{const}_n+n\,P_\sigma(k/n)+\delta_k$ with $\max_k|\delta_k|=O_J(\log n)$. The constant drops out because $\sum_k(s_k-\tilde s_k)=0$, and $|\langle\delta,s-\tilde s\rangle|\le2K\max_k|\delta_k|=O_J(n\log n)$. For a profile $s$ let $\bar s(x)=s_k$ on $[\frac kn,\frac{k+1}n)$ for $-n\le k<2n$, and $\sigma'=\bar s+\frac{s_{2n}}{n|E|}\one_E$ with $E=\{\bar s\le3\}$. Since $|\{\bar s>3\}|\le\kappa/4<1$, we have $|E|\ge2$, and $\sigma'$ is feasible for $n\ge3$. By Lemma~\ref{lem:modulus}, $\sum_ks_kP_\sigma(k/n)=n\langle P_\sigma,\sigma'\rangle+O(\log n)$, and by the Riemann-sum estimate of Lemma~\ref{lem:continuum}(c), $\sum_k\tilde s_kP_\sigma(k/n)=n\langle P_\sigma,\sigma\rangle+O_J(\log n)$. Hence
\begin{equation}\label{eq:archfinal}
n\sum_kP_\sigma(k/n)(s_k-\tilde s_k)\le n^2\langle P_\sigma,\sigma'-\sigma\rangle+O_J(n\log n)\le n^2\gap(\sigma)+O_J(n\log n)\ .
\end{equation}
Now insert Lemma~\ref{lem:continuum}(c). All constants depend only on $\|\sigma\|_\infty\le4$ and on $J$.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Local bounds, the circle model and the prime sum}\label{app:local}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This appendix proves Lemmas~\ref{lem:leaf}, \ref{lem:small}, \ref{lem:middle}, \ref{lem:relaxation}, \ref{lem:cells} and~\ref{lem:counting} and Proposition~\ref{prop:primesum}.

\begin{proof}[Proof of Lemma~\ref{lem:leaf}]
All integers whose valuations enter are nonzero and of absolute value less than $6n\le p^{\lL_p}$: $|k-m|\le3n$; $3k-3j-r\in[-6n+1,6n-1]$, as $3j+r\in[1,3n-1]$; the $u$ of \eqref{eq:hs} satisfy $|u|<3|k|\le6n$; and $|n-2k|\le3n$. Hence $\mu_k\le\lL_p$ (at $p=2$ the root $\frac{2k-n}2$ has $v_2=-1$), $\eta_p(k)\le\lL_p$, and $v(\HS_3(k,M))\ge-M\lL_p$. By Lemma~\ref{lem:locval} and \eqref{eq:localconstants}, whose weights $i(i+1)$ and $2916$ are integers, $v(c_{k,a})\ge-D_k-(6-a)\lL_p$ for $a\le3$. A minor $\det C_k[R]$ is a signed sum of products $\prod_bc_{k,m_b}$ of $s$ local constants with $\sum_b(6-m_b)\le6s$, so $\mathsf e_p(k,s,\ell)\le sD_k+6s\lL_p$, and $\ell\eta_p(k)\le s(4-s)\lL_p\le3s\lL_p$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:small}]
For $j\ge1$ put $o^{(j)}_k=\#\{m\ne k:\ m\equiv k\ (p^j)\}$, $z^{(j)}_k=3\#\{(i,r):\ 0\le i<n,\ r\in\{1,2\},\ 3i+r\equiv3k\ (p^j)\}$, and $g^{(j)}_k=1$ if $p^j\mid n-2k$ and $0$ otherwise, for odd $p$, and $g^{(j)}_k=0$ for $p=2$. By \eqref{eq:Dk} and $v_p(x)=\#\{j\ge1:p^j\mid x\}$, $D_k=\sum_{j\ge1}(4o^{(j)}_k-z^{(j)}_k-g^{(j)}_k)+[p=2]$. The first two quantities depend only on the class $\mathfrak c$ of $k$ modulo $q=p^j$, as $3$ is invertible modulo $q$; write $N_{\mathfrak c}$ for the number of nodes in $\mathfrak c$ and $z_{\mathfrak c}$ for $z^{(j)}_k$. By Lemma~\ref{lem:leaf}, $T_p\le9K\lL_p+[p=2]K+\max_s\sum_{j\ge1}\Xi_j(s)$ with
\begin{equation}\label{eq:phij}
\Xi_j(s)=\sum_{\mathfrak c\bmod q}\qty[S_{\mathfrak c}\qty(4(N_{\mathfrak c}-1)-z_{\mathfrak c})-\sum_{k\in\mathfrak c}s_kg^{(j)}_k-S_{\mathfrak c}^2+\sum_{k\in\mathfrak c}s_k^2]\ .
\end{equation}
If $q>3n$, every class has $N_{\mathfrak c}\le1$, and $\Xi_j(s)\le0$. If $q\le3n$, we use $\sum_{k\in\mathfrak c}s_k^2\le4S_{\mathfrak c}$ and $g\ge0$ to get $\Xi_j(s)\le\sum_{\mathfrak c}(S_{\mathfrak c}A_{\mathfrak c}-S_{\mathfrak c}^2)$ with $A_{\mathfrak c}=4N_{\mathfrak c}-z_{\mathfrak c}$. The $3n+1$ nodes are consecutive integers, so $N_{\mathfrak c}\in\{\fl{\frac{3n+1}q},\lceil\frac{3n+1}q\rceil\}$, and since $i\mapsto3i+r$ is a bijection of $\Z/q$, each of the two counts in $z_{\mathfrak c}/3$ is the number of $i\in[0,n)$ in one residue class, hence lies in $\{\fl{\frac nq},\lceil\frac nq\rceil\}$. So all $A_{\mathfrak c}$ lie in an interval of length $4+6=10$, which contains their mean $\bar A=(4(3n+1)-6n)/q=(6n+4)/q$. For real $\xi$ and $S_{\mathfrak c}$ we have $S_{\mathfrak c}A_{\mathfrak c}-S_{\mathfrak c}^2\le\xi S_{\mathfrak c}+\frac14(A_{\mathfrak c}-\xi)^2$; with $\xi=\bar A-2K/q$ and $\sum_{\mathfrak c}(A_{\mathfrak c}-\bar A)=0$,
\begin{equation}\label{eq:weakduality}
\sum_{\mathfrak c}(S_{\mathfrak c}A_{\mathfrak c}-S_{\mathfrak c}^2)\le K\bar A-\frac{K^2}q+\frac14\sum_{\mathfrak c}(A_{\mathfrak c}-\bar A)^2\le\frac{K(6n+4-K)}q+25q\ .
\end{equation}
Summing over $j$ with $p^j\le3n$, $\sum_jp^{-j}\le\frac1{p-1}$ and $\sum_{p^j\le3n}25p^j\le25\cdot3n\frac p{p-1}\le150n$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:middle}]
Here $p$ is single-level, so all valuations are $0$ or $1$ and only the classes modulo $p$ matter. By Lemma~\ref{lem:locval} with $\mu_k\le1$ and $v(\HS_3(k,M))\ge-M$, we have $v(\beta_{k,m})\ge\min_{1\le i\le4-m}(-D_k-(4-i-m)-(i+2))=-D_k-6+m$ and $v(\alpha_{k,m})\ge-D_k-1+m$. Take $\check v_m(k)=-D_k-6+m$ in Lemma~\ref{lem:relaxation}. For admissible $(R,\pi)$, $\sum_b(D_k+6-R_b-\pi(b))+(\sum R-\binom s2)[o_k>0]\le sD_k+6s-2\binom s2$, because $\sum_b\pi(b)=\binom s2$ and $\sum R\ge\binom s2$. Every class has $N_{\mathfrak c}\ge\fl{\frac{3n+1}p}\ge7$ nodes, and with $z_{\mathfrak c}\le6\lceil\frac np\rceil$ and $g_k\le1$ we get $D_k=4(N_{\mathfrak c}-1)-z_{\mathfrak c}-g_k\ge4(\frac{3n}p-2)-6(\frac np+1)-1=\frac{6n}p-15\ge0$. Hence $sD_k+6s-s(s-1)\ge0$, the clipping in \eqref{eq:fplus} is inactive, and $f^+(k,s)+s^2\le s(D'_{\mathfrak c}+7)$ with $D'_{\mathfrak c}=4(N_{\mathfrak c}-1)-z_{\mathfrak c}$. So $V_{\mathfrak c}(S)\le S(D'_{\mathfrak c}+7)-S^2$. The mean of $D'_{\mathfrak c}+7$ over the $p$ classes is $\frac{6n+4-4p}p+7$, and the values lie in an interval of length $10$, as in the proof of Lemma~\ref{lem:small}. The same weak-duality step gives $T_p\le T_p^+\le K(\frac{6n+4-4p}p+7)-\frac{K^2}p+25p=\frac{K(6n+4-K)}p+3K+25p$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:relaxation}]
A minor is $\det C_k[R]=\sum_\pi\pm\prod_bc_{k,R_b+\pi(b)}$, and the terms with some $R_b+\pi(b)\ge4$ vanish. So $-v(\det C_k[R])\le\max_\pi\sum_b-\check v_{R_b+\pi(b)}(k)$. At a single-level prime $\eta_p(k)=[o_k>0]$, so $f_p(k,s)\le f^+(k,s)$; the clipping at $0$ only increases the right side. The tree \eqref{eq:tree} has one level, and grouping its maximum by classes gives
\begin{equation}\label{eq:grouping}
T_p\le\max_s\sum_{\mathfrak c\bmod p}\qty[\sum_{k\in\mathfrak c}\qty(f^+(k,s_k)+s_k^2)-S_{\mathfrak c}^2]\le\max\qty{\sum_{\mathfrak c}V_{\mathfrak c}(S_{\mathfrak c}):\ \sum_{\mathfrak c}S_{\mathfrak c}=K}=T_p^+\ .
\end{equation}
Finally $\sum_{\mathfrak c}V_{\mathfrak c}(S_{\mathfrak c})=\xi\sum_{\mathfrak c}S_{\mathfrak c}+\sum_{\mathfrak c}(V_{\mathfrak c}(S_{\mathfrak c})-\xi S_{\mathfrak c})\le\xi K+\sum_{\mathfrak c}\psi_{\mathfrak c}(\xi)$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:cells}]
The data of a node $y$ of the class of $x$ change only when $y$ crosses $-1$ or $2$, where a node enters or leaves the node range and the number of partners changes; $0$, where the range of the virtual poles changes sign; a point of $\frac\varrho3(\Z\setminus3\Z)$, where a virtual pole or the upper end of the zero range appears ($t\varrho=3y$); or a point of $1+\frac\varrho3(\Z\setminus3\Z)$, the lower end of the zero range ($t\varrho=3y-3$). Modulo $\varrho$ these are the seven points \eqref{eq:sevenpoints}, each of the form $\alpha+\beta\varrho$ with $\alpha\in\{-1,0,1,2\}$ and $\beta\in\{0,\frac13,\frac23\}$. Two of them coincide modulo $\varrho$ if and only if $\alpha_1-\alpha_2=(N'-\beta_1+\beta_2)\varrho$ for an integer $N'$. If $\alpha_1=\alpha_2$, this is impossible for distinct points, since $\beta_1-\beta_2\notin\Z$. Otherwise $\varrho=3j/N$ with $j=|\alpha_1-\alpha_2|\in\{1,2,3\}$ and $N=|3N'-3\beta_1+3\beta_2|$, a positive integer. So on a cell the seven points are distinct, and their cyclic order is constant. If $x$ is not one of the seven points modulo $\varrho$, no node of $x$ is at a critical point, and so the type of $(x,\varrho)$ is locally constant on the set of such pairs. For each arc, the set of pairs $(x,\varrho)$ with $\varrho$ in the cell and $x$ in the arc is connected, so the type is constant on it. The arc lengths are differences of consecutive points, each represented with an integer $w$ that is fixed on the cell, hence affine in $\varrho$.

Now let $\varrho>3$. A class contains at most one node, since $[-1,2]$ has length $3<\varrho$, so $o=0$. For $y>0$, $t\in\VP$ means $0<t\varrho<3y\le6$, so $t=1$ and $y>\frac\varrho3>1$; then $3y-3\le3<\varrho$, so the virtual pole is covered, and it is the only special point of the node, because $2\varrho>6\ge3y$ excludes $t=2$ from $\cZ$ and $y>1>1-\frac\varrho3$ excludes $t=-1$. For $y<0$ there is no virtual pole, since $-\varrho<-3\le3y$. So at every node either $\VP=\cZ=\{1\}$ with $D=-3$ and $\mu_{k,1}=0$, or $\VP=\emptyset$ and $\cZ\subseteq\{-1\}$, and in all cases the three terms of \eqref{eq:vhat} are nonnegative. Then $f^+=0$ at every node, $V_{\mathfrak c}(S)=0$ for every class, and $T^+_p=0$. For a prime $p>3n$ the discrete data agree with these continuum data, because $|2k-n|\le3n<p$ gives $g_k=0$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:counting}]
For $n\ge n_0$, $p\ge\epsilon n$ implies $p\ge5$ and $p^2>6n$, so Lemma~\ref{lem:relaxation} applies with $\check v=\hat v$. The map $k_0\mapsto k_0/n$ identifies $\Z/p$ with the grid $\frac1n\Z/\varrho\Z$ of spacing $\frac1n$ on the circle $\R/\varrho\Z$, $\varrho=p/n$, and the node data of Definition~\ref{def:nodedata} at $k=k_0+mp$ are the continuum data at $y=x_0+m\varrho$, except for $g_k$: the conditions $pt<3k$, $3k-3n<pt$, $-n\le k\le2n$ and $k\gtrless0$ are the continuum conditions multiplied by $n$. Call a class \textit{exceptional} if it contains a node with $g_k=1$ or one of the nodes $-n$, $0$, $2n$; there are at most four exceptional classes, because the nodes with $g_k=1$, those with $2k\equiv n\pmod p$, form a single class. A node $k$ of a non-exceptional class is not at a critical point, because $3k=pt$ and $3k-3n=pt$ with $3\nmid t$ are impossible, as $3\nmid p$. So the type of the class is the type of the arc that contains its point, which is constant on the cell; at an endpoint $\varrho\in\{a,b\}$ the arcs of positive length carry the types of the cell, by the local constancy of the type. An open arc of length $\ell$ contains between $n\ell-1$ and $n\ell+1$ points of the grid, and there are at most seven arcs. So the number of non-exceptional classes of type $\tau$ is $n\meas^{[a,b]}_\tau(\varrho)+O(1)$. Finally $V_{\mathfrak c}(0)=0$, so $\psi_{\mathfrak c}\ge0$, and a class has at most $3/\epsilon+2$ nodes, so $0\le\psi_{\mathfrak c}(\xi)\le C_\epsilon(1+|\xi|)$ for all classes, exceptional or not. Insert all this into Lemma~\ref{lem:relaxation} with $K=\kappa n$.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:primesum}]
We use three standard facts about primes. The first is the prime number theorem \cite{Ha96,dlVP96} in the form $\vartheta(x)=\sum_{p\le x}\log p=x+o(x)$, and its consequence $\sum_{p\le x}p\log p=\frac{x^2}2+o(x^2)$, by partial summation. The second is Mertens' theorem with the value of its constant, $\sum_{p\le x}\frac{\log p}p=\log x+E+o(1)$ with $E=-\gamma-\sum_p\sum_{j\ge2}\frac{\log p}{p^j}$ \cite[(2.5), (2.8)]{RS62}. Since $\sum_{j\ge2}p^{-j}=\frac1{p(p-1)}$ and $\frac{\log p}{p-1}=\frac{\log p}p+\frac{\log p}{p(p-1)}$, it gives
\begin{equation}\label{eq:mertens}
\sum_{p\le x}\frac{\log p}{p-1}=\log x-\gamma+o(1)\ .
\end{equation}
The third is the consequence $\sum_{p\le x,\,p\ne3}\frac{\log p}{p-1}=\log x-\gamma-\frac{\log3}2+o(1)$. We split the primes $p\ne3$ into four ranges, with $Y=n^{0.9}$; any exponent in $(\frac12,1)$ would do.

(i) $p\le Y$. By Lemma~\ref{lem:small} and \eqref{eq:mertens}, the leading terms give $K(6n+4-K)(\log Y-\gamma-\frac{\log3}2)+o(n^2)$, and the other terms contribute at most $(9K\lL_2+154n)\vartheta(Y)=O(n^{1.9}\log n)$.

(ii) $Y<p\le\epsilon n$. For $n$ large these primes satisfy $p^2>6n$ and $p\le0.4n$, so Lemma~\ref{lem:middle} applies. Since $\sum_{Y<p\le\epsilon n}\frac{\log p}p=\log\frac{\epsilon n}Y+o(1)$ by Mertens' theorem, and by the prime number theorem, the sum over these primes is at most $K(6n+4-K)\log\frac{\epsilon n}Y+3K\vartheta(\epsilon n)+25\sum_{p\le\epsilon n}p\log p+o(n^2)=K(6n+4-K)\log\frac{\epsilon n}Y+(3\kappa\epsilon+12.5\epsilon^2)n^2+o(n^2)$.

(iii) $\epsilon n<p\le3n$. By Lemma~\ref{lem:counting}, $T_p\le n\,\mathcal G_{[a,b]}(\xi_\nu,p/n,K/n)+C$ for $p/n\in[a_{\nu-1},a_\nu]$. For fixed $\xi_\nu$ the function $\mathcal G_{[a,b]}$ is affine in $\varrho$ and in $\kappa$, so its maximum over $[a_{\nu-1},a_\nu]\times[\kappa_1,\kappa_2]$ is attained at a corner, and $T_p\le n\bar{\mathcal G}_\nu+C$. By the prime number theorem, $\sum\max(T_p,0)\log p\le\sum_\nu(n\max(\bar{\mathcal G}_\nu,0)+C)\qty(\vartheta(a_\nu n)-\vartheta(a_{\nu-1}n))=n^2\sum_\nu\max(\bar{\mathcal G}_\nu,0)(a_\nu-a_{\nu-1})+o(n^2)$.

(iv) $p>3n$. Then $T_p\le T_p^+=0$ by Lemma~\ref{lem:cells}.

Adding (i)--(iv), $\log Y+\log\frac{\epsilon n}Y=\log n-\log\frac1\epsilon$, and $K(6n+4-K)=\kappa(6-\kappa)n^2+O(n)$. The function $\kappa(6-\kappa)$ is increasing on $[2.8,3]$, so the negative term $-\kappa(6-\kappa)(\gamma+\frac{\log3}2+\log\frac1\epsilon)n^2$ is largest at $\kappa=\kappa_1$, and $3\kappa\epsilon$ is largest at $\kappa=\kappa_2$.

The prime number theorem and Mertens' theorem enter only in the passage from sums over primes to their main terms, and the error of this passage is contained in the term $o(n^2)$. For a fixed small $n$ the individual primes matter and this term may be large; the proposition makes no statement about such $n$, and the proof of Theorem~\ref{thm:main} uses it only for $n$ large.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Lemma~\ref{lem:omega}}\label{app:omega}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We go through the nodes $k$ by type and bound the weighted valuation \eqref{eq:omega} of every contribution, using the bounds of Lemma~\ref{lem:scaled} for the local constants and the bounds for the Taylor coefficients of the $E_x$ given after \eqref{eq:shift}. We write $m=a+b\le3$. Note that $\theta_x\ge\frac12$ for every condition $x$.

(1) \textit{U nodes.} All three parts of the local constants have $v_p\ge0$, and the Taylor coefficients of $E_x$ and $E_{x'}$ are integral, so $\varpi\ge\theta_x+\theta_{x'}\ge1$.

(2) \textit{A node $R$ of type Ra or Rc.} If $R_x\ne R$, all the Taylor coefficients $(E_x)_a(R)$, $a\le3$, vanish, and there is no contribution. So let $R_x=R_{x'}=R$. Then only $a=j_x$ and $b=j_{x'}$ contribute. The part $\beta^\HS$ has $v_p\ge m-7$ by \eqref{eq:scaled}, so $\varpi\ge(j_x+j_{x'}-7)+(\frac72-j_x)+(\frac72-j_{x'})=0$. The part $\beta^{\mathrm{rest}}$ has $v_p\ge m-4$, so $\varpi\ge3$, and $\alpha$ has $v_p\ge m-2$, so $\varpi\ge5$.

(3) \textit{A node $R$ of type Rb.} Here $\theta_x=2-j_x$ for the conditions at $R$, and $\beta^\HS$ has $v_p\ge m-4$. The orders $0,1$ are prescribed at $R$ and the orders $2,3$ are not, and since $m\le3$, at most one of $a,b$ is at least $2$.
\begin{itemize}
\item[(3i)] $R_x=R_{x'}=R$, $a=j_x$, $b=j_{x'}$: $\varpi\ge(a+b-4)+(2-j_x)+(2-j_{x'})=0$.
\item[(3ii)] $R_x=R_{x'}=R$, $a\in\{2,3\}$, $b=j_{x'}$: $\varpi\ge(a+b-4)+(2-j_x)+(2-b)=a-j_x\ge1$; likewise with the roles of $a$ and $b$ exchanged.
\item[(3iii)] $R_x=R\ne R_{x'}$: then $(E_{x'})_b(R)=0$ for $b\le1$, so $b\in\{2,3\}$, hence $a\le1$ and $a=j_x$. So $\varpi\ge(j_x+b-4)+(2-j_x)+\theta_{x'}=b-2+\theta_{x'}\ge\frac12$; likewise with $x$ and $x'$ exchanged.
\item[(3iv)] $R_x\ne R$ and $R_{x'}\ne R$: then $a,b\ge2$, so $m\ge4$, and there is no contribution.
\end{itemize}
The parts $\beta^{\mathrm{rest}}$ and $\alpha$ have $v_p\ge m-1$ and $v_p\ge m+1$, which exceed the bound $m-4$ for $\beta^\HS$ by $3$ and $5$, so their contributions have $\varpi\ge3$ and $\varpi\ge5$.

(3$'$) \textit{A node $R$ of type Rb$_g$.} The same computation with the bound $m-3$ for $\beta^\HS$ and $\theta_x=\frac32-j_x$ gives $\varpi\ge0$ in (3i), $\varpi\ge a-j_x\ge1$ in (3ii), and $\varpi\ge(j_x+b-3)+(\frac32-j_x)+\theta_{x'}=b-\frac32+\theta_{x'}\ge1$ in (3iii). The parts $\beta^{\mathrm{rest}}$ ($v_p\ge m$) and $\alpha$ ($v_p\ge m+2$) are again better by $3$ and $5$.

(4) \textit{A node $\bar R$ of type La, the partner of a node $R$ of type Ra.} Here $\beta^\HS$ has $v_p\ge m-7$. We use the decomposition \eqref{eq:leadrem}. If $R_x=R_{x'}=R$, the product with $\xi_{x,a}\xi_{x',b}$, which is nonzero only for $a\le j_x$ and $b\le j_{x'}$, has $\varpi\ge(m-7)+(j_x-a)+(j_{x'}-b)+(7-j_x-j_{x'})=0$. A product that contains $\xi'_{x,a}$ instead has a factor of valuation at least $4-a$ instead of $j_x-a$, which raises $\varpi$ by at least $1$, and likewise for $\xi'_{x',b}$. If $R_x=R\ne R_{x'}$, then $v_p((E_{x'})_b(\bar R))\ge4-b$ and $\varpi\ge(a+b-7)+(j_x-a)+(4-b)+(\frac72-j_x)+\theta_{x'}=\frac12+\theta_{x'}\ge1$. If $R_x\ne R$ and $R_{x'}\ne R$, then $\varpi\ge(m-7)+(4-a)+(4-b)+\theta_x+\theta_{x'}\ge2$. The parts $\beta^{\mathrm{rest}}$ ($v_p\ge m-4$) and $\alpha$ ($v_p\ge m-2$) are better by $3$ and $5$ in every case.

(4$'$) \textit{A node $\bar R$ of type Lc, the partner of a node $R$ of type Rc.} There is no virtual pole, so only $\beta^{\mathrm{rest}}$ ($v_p\ge m-4$) and $\alpha$ ($v_p\ge m-2$) occur. These are the bounds of (4) improved by $3$ and $5$, and the Taylor coefficients satisfy the same bounds as in (4), so $\varpi\ge3$ and $\varpi\ge5$.

(5) \textit{A node $\bar R$ of type Lb or Lb$_g$, the partner of a node $R$ of type Rb or Rb$_g$.} There is no virtual pole. At Lb, $\beta^{\mathrm{rest}}$ has $v_p\ge m-1$ and $\alpha$ has $v_p\ge m+1$; at Lb$_g$ the bounds are $m$ and $m+2$. Put $w_x(a)=a+v_p((E_x)_a(\bar R))$. By the bounds after \eqref{eq:shift}, if $R_x=R$, then $w_x(a)\ge j_x$ for $a\le j_x$ and $w_x(a)\ge\max(a,2)\ge2$ for $a>j_x$, so $w_x(a)+\theta_x\ge2$ if $R$ is of type Rb ($\theta_x=2-j_x$) and $w_x(a)+\theta_x\ge\frac32$ if $R$ is of type Rb$_g$ ($\theta_x=\frac32-j_x$). If $R_x\ne R$, then $w_x(a)\ge\max(a,2)\ge2$ and $w_x(a)+\theta_x\ge\frac52$. Hence the part $\beta^{\mathrm{rest}}$ has $\varpi\ge-1+(w_x(a)+\theta_x)+(w_{x'}(b)+\theta_{x'})\ge-1+2+2=3$ at Lb and $\varpi\ge\frac32+\frac32=3$ at Lb$_g$, and the part $\alpha$ has $\varpi\ge5$ at both.

At a condition node $R'$ different from $R_x$ and $R_{x'}$ there is no contribution, by (2), (3iv) and (3$'$); the partners of such nodes are covered by (4), (4$'$) and (5). Collecting the cases, every contribution to $\Gamma^\beta$ has $\varpi\ge0$ and every contribution to $\Gamma^\alpha$ has $\varpi\ge1$. The contributions to $\Gamma^\beta_{xx'}$ with $R_x\ne R_{x'}$ are those of (1), (3iii), (4) with $R_x=R\ne R_{x'}$ or with neither condition at $R$, (4$'$) and (5), all with $\varpi\ge\frac12$. The contributions with $\varpi=0$ are those of (2), (3i) and (3$'$) for $\beta^\HS$, and the products $\beta^\HS_{\bar R,a+b}\xi_{x,a}\xi_{x',b}$ of (4). \qed

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The computations}\label{app:comp}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This appendix describes how the upper bounds of Propositions~\ref{prop:fdagger} and~\ref{prop:cvalues} are obtained. In both cases a computer evaluates a finite expression that the lemmas of the main text show to be an upper bound for the constant in question. Real numbers are handled either as exact rationals or in ball arithmetic, in which every number is represented by a ball that contains it and every operation returns a ball that contains the exact result \cite{Jo17}. Only the upper ends of the final balls are used.

\textit{The bound for $F^\dagger_i$.} Put $t=\kappa/m_0$, $A_0=\langle\sigma_0,\cV\rangle$ and $B_0=\langle\sigma_0,\cL\sigma_0\rangle$. Since $\mathcal E$ is quadratic, $\mathcal E[\sigma_\kappa]=tA_0+t^2B_0$, $P_{\sigma_\kappa}=\cV+2t\cL\sigma_0=:P_t$ and $\langle P_t,\sigma_\kappa\rangle=tA_0+2t^2B_0$, so that
\begin{equation}\label{eq:fdaggerreduction}
\mathcal E[\sigma_\kappa]+\gap(\sigma_\kappa)=M_\kappa(P_t)-t^2B_0\ ,\qquad M_\kappa(P)=\sup\{\langle P,\sigma'\rangle:\ \sigma'\ \text{feasible for}\ \kappa\}\ .
\end{equation}
With $\mathcal Q_1(w)=w\log|w|-w$ and $\mathcal Q_2(w)=\frac{w^2}2\log|w|-\frac34w^2$, which satisfy $\mathcal Q_2'=\mathcal Q_1$ and $\mathcal Q_1'=\log|\cdot|$, we have $\int_\alpha^\beta\log|x-y|\,\dd y=\mathcal Q_1(\beta-x)-\mathcal Q_1(\alpha-x)$ and
\begin{equation}\label{eq:doubleint}
\int_{a_1}^{b_1}\int_{a_2}^{b_2}\log|x-y|\,\dd y\,\dd x=\mathcal Q_2(b_2-a_1)-\mathcal Q_2(b_2-b_1)-\mathcal Q_2(a_2-a_1)+\mathcal Q_2(a_2-b_1)\ .
\end{equation}
So $A_0$, $B_0$ and the values of $P_t$ are finite sums of values of $\mathcal Q_1$ and $\mathcal Q_2$. One finds $B_0<0$ (about $-3.78$), so $-t^2B_0$ is increasing in $t$.

To bound $M_\kappa$, split $[-1,2]$ into $N_{\mathrm{cell}}$ intervals $Q_m$ of length $\delta=3/N_{\mathrm{cell}}$, and let $\pi_m\ge\sup\{P_t(x):\ x\in Q_m,\ t\in[t_a,t_b]\}$, where $t_{a,b}=\kappa_{a,b}/m_0$ for a subinterval $[\kappa_a,\kappa_b]$ of $I_i$. For feasible $\sigma'$ put $\bar\sigma_m=\int_{Q_m}\sigma'\in[0,4\delta]$. For every real $c_*$,
\begin{equation}\label{eq:lpdual}
\langle P_t,\sigma'\rangle\le\sum_m\pi_m\bar\sigma_m=c_*\kappa+\sum_m(\pi_m-c_*)\bar\sigma_m\le c_*\kappa+4\delta\sum_m(\pi_m-c_*)^+\ .
\end{equation}
If $c_*\ge\min_m\pi_m$, it follows that $M_\kappa(P_t)\le M^+:=c_*\kappa_b+(\kappa_b-\kappa_a)\max(0,-\min_m\pi_m)+4\delta\sum_m(\pi_m-c_*)^+$ for all $\kappa\in[\kappa_a,\kappa_b]$. We take for $c_*$ the threshold of the filling with density $4$ at $\kappa_b$, that is, the value $\pi_m$ such that the intervals $Q_m$ with larger values have total length about $\kappa_b/4$; any other $c_*\ge\min_m\pi_m$ would also give a valid bound. The function $P_t(x)$ is affine in $t$, so it suffices to bound it at $t\in\{t_a,t_b\}$, and by Lemma~\ref{lem:modulus}, $\sup_{Q_m}P_t\le P_t(x_m)+b_\infty\,w(\delta/2)$, where $x_m$ is the midpoint of $Q_m$ and $b_\infty=4+2t_b\max\sigma_0$ bounds $\|2\cdot\one_{I_{\mathrm{in}}}-4\cdot\one_{I_{\mathrm{out}}}+2t\sigma_0\|_\infty$, the density whose potential is $P_t$. This is the procedure of \cite{K26}. We use $N_{\mathrm{cell}}=36000$ and cut each $I_i$ into $20$ subintervals $[\kappa_a,\kappa_b]$. All values of $\mathcal Q_1$ and $\mathcal Q_2$ are computed as balls at a precision of $96$ bits, $\pi_m$ is the upper end of the enclosure, and the bound for $F^\dagger_i$ is the largest, over the $20$ subintervals, of the upper ends of the balls for $(M^+-t_b^2B_0)/\log2$.

\textit{The bound for $c_i$.} The bound is obtained by evaluating \eqref{eq:cformula} for one explicit partition and explicit rationals $\xi_\nu$, and all steps up to the last one are done in exact rational arithmetic. The points of $\cB$ in $(\frac1{20},3)$, together with $\frac1{20}$ and $3$, divide $[\frac1{20},3]$ into cells. For each cell we fix an interior point $\varrho_*$, order the seven points \eqref{eq:sevenpoints} on $\R/\varrho_*\Z$, and record for each arc its length, as an affine function of $\varrho$, and its type, computed at the midpoint of the arc; by Lemma~\ref{lem:cells} the type does not depend on these choices. The type determines $\hat v$ at each node by \eqref{eq:vhat}, then $f^+$ by \eqref{eq:fplus}, the function $V_{\mathfrak c}$ by a dynamic program over the nodes of the class, and $\psi_\tau(\xi)=\max_S(V_{\mathfrak c}(S)-\xi S)$ as the maximum over the vertices of the upper concave hull of $V_{\mathfrak c}$. Each cell is cut into $32$ subcells of equal length; these form the partition $a_0<\dots<a_M$. On each subcell, for each interval $I_i$, a rational $\xi_\nu$ of denominator at most $10^6$ is chosen by a numerical minimization of the corner maximum \eqref{eq:hi}; the choice affects only the quality of the bound, not its validity. Then $\bar{\mathcal G}_\nu$ is computed exactly, and the sum $\sum_\nu\max(\bar{\mathcal G}_\nu,0)(a_\nu-a_{\nu-1})$ is an exact rational number. For the five intervals $I_1,\dots,I_5$ these sums are $57.5996087$, $57.8686577$, $58.1240232$, $58.3660091$ and $58.5946334$, rounded to seven decimals; we quote them so that the reader can evaluate \eqref{eq:cformula} from them. Finally, \eqref{eq:cformula} is evaluated in ball arithmetic with Euler's constant and the logarithms $\log3$, $\log20$ and $\log2$, and the upper ends of the balls give the bounds of Proposition~\ref{prop:cvalues}.

\textit{Table~\ref{tab:margins}.} The values $G(\kappa_{1,i})=(10.5\,\kappa_{1,i}-\kappa_{1,i}^2/4)\log_23$ are computed in ball arithmetic and rounded down to two decimals, the bounds of Propositions~\ref{prop:fdagger} and~\ref{prop:cvalues} are rounded up to two decimals, and the last column is computed from these printed values. The coefficients \eqref{eq:phivalues} and the determinants of the blocks in Section~\ref{sec:nonvanishing}, including \eqref{eq:rablock}, are exact rational computations that are small enough to be checked by hand.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A second proof, on the modular curve \texorpdfstring{$X_0(3)$}{X0(3)}}\label{app:second}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This appendix gives a second proof of Theorem~\ref{thm:main}, by a different method. It shares with the main proof only the statement, the normalization \eqref{eq:zeta35def} of $\zeta_3(5)$ and standard facts; it uses neither the Volkenborn functional nor any estimate of Sections~\ref{sec:plan}--\ref{sec:nonvanishing} and Appendices~\ref{app:analytic}--\ref{app:comp}. Instead, it rests on published theorems of Coleman, of Coleman and Mazur, and of Buzzard on overconvergent modular forms (Theorems~\ref{thm:coleman} and~\ref{thm:buzzard}), whose statements and hypotheses we quote but whose proofs we have not checked, and on four standard facts used in the proof of Proposition~\ref{prop:radius} and listed after it. The method is the determinant argument of Yi for $\zeta_7(3)$ on $X_0(7)$ \cite{Yi26}: a determinant of Taylor coefficients of power series built from modular forms, a bound for its denominators at the primes $\ell\ne3$ through the Eisenstein series of weight $\ell-1$, and an energy bound at the real place. We write out for level $3$ every step that we take from \cite{Yi26}, so that this proof, too, should not depend on the correctness of \cite{Yi26}. New relative to \cite{Yi26} are the $\mathrm{Sym}^4$ form of his congruence (Proposition~\ref{prop:sym4}) and the layer theorem (Theorem~\ref{thm:layers}). The notation of this appendix is independent of the main text, except for $\zeta_3(5)$, $L_3$, the trivial character $\one$, the Teichm\"uller character $\omega$ and $\langle a\rangle$ of Section~\ref{sec:volkenborn}, $\Q$, $\Z$, $v_p$, the coefficient notation $[x^a]F$ and $t^+=\max(t,0)$. In $O(\cdot)$ and $o(\cdot)$, $d\to\infty$ while $\eta_5$, $\epsilon$ and the measure $\mu$ of Section~\ref{sec:secarch} stay fixed; no other parameter depends on $d$.

\textit{Notation.} Let $\tau$ lie in the upper half plane $\mathbb H$, $q=e^{2\pi i\tau}$ and $\theta=q\,\dd/\dd q=(2\pi i)^{-1}\dd/\dd\tau$, and for $k\in\Z$ put $\sigma_k(n)=\sum_{a\mid n}a^k$, with $\sigma_k(n/3)=0$ if $3\nmid n$, and $\sigma^{(3)}_k(n)=\sum_{a\mid n,\,3\nmid a}a^k$. Let
\begin{equation}\label{eq:xdef}
x(\tau)=\qty(\frac{\eta(3\tau)}{\eta(\tau)})^{12}=q\prod_{n\ge1}\frac{(1-q^{3n})^{12}}{(1-q^n)^{12}}\in q+q^2\Z[[q]]\ ,
\end{equation}
where $\eta$ is Dedekind's function, let $q(x)\in x+x^2\Z[[x]]$ be the compositional inverse of $x(q)$, let $D=x\,\dd/\dd x$, and
\begin{equation}\label{eq:bg}
B=\theta\log x=1+12\sum_{n\ge1}\sigma^{(3)}_1(n)q^n\ ,\qquad G_5(q)=\sum_{3\nmid a}\frac{q^a}{a^5(1-q^a)}=\sum_{n\ge1}\sigma^{(3)}_{-5}(n)q^n\ .
\end{equation}
We write $B$ also for $B(q(x))\in1+x\Z[[x]]$, so that $\theta=BD$ on series in $x$, and put $u=DB/B\in x\Z[[x]]$, $P=1+27x$ and $x_e=-1/27$, the value of $x$ at the elliptic point (Lemma~\ref{lem:hauptmodul}). For a constant $c$ let
\begin{equation}\label{eq:hdef}
H_c=B^2\qty(G_5(q(x))+c)\ ,\qquad g_0=1\ ,\qquad g_i=D^{i-1}H_c\quad(1\le i\le5)\ .
\end{equation}
We call $g_0,\dots,g_5$ the germs. Finally put $\eta_5=\zeta_3(5)/2\in\Q_3$. The series $G_5$ is an Eichler integral: $\theta^5G_5$ is a modular form of weight $6$ (proof of Lemma~\ref{lem:vr}). Moreover $G_5+\eta_5$ is the $q$-expansion of an overconvergent $3$-adic modular form of weight $-4$ \eqref{eq:estar}, which the factor $B^2$, of weight $4$, turns into a function of $x$. Informally, an overconvergent $p$-adic modular form of weight $k$ is a rigid analytic section of the $k$-th power of the sheaf of invariant differentials over a region of the $p$-adic modular curve that contains the ordinary component of the cusp $\infty$ and a neighbourhood of it reaching into the supersingular locus \cite{Ka73,Co97,Bu03}; the precise statements that we use are Theorems~\ref{thm:coleman} and~\ref{thm:buzzard}. This is why, for $c=\eta_5$, the germs converge on the $3$-adic disc $|x|_3<3^6$ (Proposition~\ref{prop:radius}).

\textit{The plan.} Suppose that $\zeta_3(5)$, and hence $\eta_5$, is rational. For every $d\ge1$ we construct a nonzero rational number $\cD_d=\cD_d(\eta_5)$, the determinant of a $5d\times5d$ matrix of Taylor coefficients of the germs $x^kg_i$ with $1\le i\le5$ and $0\le k<d$ (Section~\ref{sec:secdet}). By the product formula, $\log|\cD_d|=v_3(\cD_d)\log3+\sum_{\ell\ne3}v_\ell(\cD_d)\log\ell$. We shall prove three estimates.
\begin{itemize}
\item[(i)] For every $0<\epsilon<6$, $v_3(\cD_d)\ge15(6-\epsilon)d^2-O(d)$ (Proposition~\ref{prop:secondgain}). It holds because the germs converge on the $3$-adic disc $|x|_3<3^6$.
\item[(ii)] $\sum_{\ell\ne3}v_\ell(\cD_d)\log\ell\ge-\frac{13289}{168}d^2-o(d^2)$ (Proposition~\ref{prop:cost}). It comes from Proposition~\ref{prop:sym4}, which writes the germs modulo powers of $\ell$ through rational functions with few poles.
\item[(iii)] $\log|\cD_d|\le U\,d^2+o(d^2)$ with an explicit constant $U\le16.6040183$ (Proposition~\ref{prop:c4} and \eqref{eq:ucert}). It comes from Jensen's formula on circles in the complex $q$-disc and an energy bound for Gram determinants.
\end{itemize}

\begin{proof}[Second proof of Theorem~\ref{thm:main}]
Suppose that $\eta_5\in\Q$. By the product formula and (i)--(iii), with $\epsilon=\frac1{10}$, so that $15(6-\epsilon)=\frac{177}2$,
\begin{equation}\label{eq:secondfinal}
\qty(\frac{177}2\log3-\frac{13289}{168})d^2-o(d^2)\le v_3(\cD_d)\log3+\sum_{\ell\ne3}v_\ell(\cD_d)\log\ell=\log|\cD_d|\le U\,d^2+o(d^2)\ .
\end{equation}
Since $\frac{177}2\log3-\frac{13289}{168}-U>1.52$, this is impossible for large $d$.
\end{proof}

The rest of this appendix proves (i)--(iii). Section~\ref{sec:secmod} collects the modular identities, Section~\ref{sec:secdet} constructs $\cD_d$, and Sections~\ref{sec:secgain}, \ref{sec:secprimes} and~\ref{sec:secarch} prove (i), (ii) and (iii).

\subsection{Modular identities on \texorpdfstring{$X_0(3)$}{X0(3)}}\label{sec:secmod}
This section shows that $H_c$ satisfies the inhomogeneous linear differential equation $L_5H_c=R_5$ (Lemma~\ref{lem:l5}). Here $L_5$ is an explicit operator of order $5$ whose solutions are the functions $B^2\tau^a$, $0\le a\le4$, and $R_5$ is an explicit rational function. Sections~\ref{sec:secdet} and~\ref{sec:secprimes} rest on this equation.

\begin{lemma}\label{lem:hauptmodul}
(a) The group $\Gamma_0(3)$ has genus $0$, two cusps $\infty$ and $0$, and one elliptic point, of order $3$, the orbit of $\tau_e=\frac12+\frac{i}{2\sqrt3}$ \cite{DS05}. The function $x$ is $\Gamma_0(3)$-invariant, holomorphic and nonvanishing on $\mathbb H$, has a simple zero at the cusp $\infty$ and a simple pole at the cusp $0$, and identifies $X_0(3)$ with $\mathbb P^1$ \cite[Section~3]{Ca05}. (b) $x(-1/(3\tau))=3^{-6}/x(\tau)$. (c) $j=(1+27x)(1+243x)^3/x$, and $x(\tau_e)=x_e$.
\end{lemma}

\begin{proof}
(a) The function $x$ is the Hauptmodul $(\eta(3\tau)^{24}/\eta(\tau)^{24})^{1/2}$ of $\Gamma_0(3)$ used in \cite[Section~3]{Ca05}. It is holomorphic and nonvanishing on $\mathbb H$ because $\eta$ is, its simple zero at the cusp $\infty$ can be read off from \eqref{eq:xdef}, and its simple pole at the cusp $0$ follows from (b). (b) From $\eta(-1/\tau)=(\tau/i)^{1/2}\eta(\tau)$ \cite[Theorem~3.1]{Ap90} we get
\begin{equation*}
x(-1/(3\tau))=\qty(\frac{\eta(-1/\tau)}{\eta(-1/(3\tau))})^{12}=\qty(\frac{(\tau/i)^{1/2}\eta(\tau)}{(3\tau/i)^{1/2}\eta(3\tau)})^{12}=\frac{3^{-6}}{x(\tau)}\ .
\end{equation*}
 (c) The function $j$ is $\Gamma_0(3)$-invariant and holomorphic on $\mathbb H$, hence rational in $x$ with poles only at the cusps. At $\infty$, $j=q^{-1}+O(1)$ and $x=q+O(q^2)$, a simple pole in $x$. At the cusp $0$, $j(-1/(3\tau))=j(3\tau)=q^{-3}+O(1)$, while $x(-1/(3\tau))=3^{-6}q^{-1}+O(1)$, a pole of order $3$ in $x$. So $xj$ is a polynomial in $x$ of degree at most $4$, determined by its first five Taylor coefficients at $x=0$, which one computes from the $q$-expansions. The fiber of $x\mapsto j$ over $j=0$ consists of $x=-1/243$, with ramification index $3$, and $x=-1/27$, with index $1$. Every point of $\mathbb H$ over $j=0$ is $\mathrm{SL}_2(\Z)$-elliptic of order $3$, so the unramified one is the $\Gamma_0(3)$-elliptic point, and $x(\tau_e)=-1/27$.
\end{proof}

\begin{lemma}\label{lem:bform}
The series $B$ is a holomorphic modular form of weight $2$ on $\Gamma_0(3)$ with $B(-1/(3\tau))=-3\tau^2B(\tau)$. It does not vanish at the two cusps, and its zeros in $\mathbb H$ are exactly the orbit of $\tau_e$, each of order $2$ in $\tau-\tau_e$.
\end{lemma}

\begin{proof}
Since $B=(2\pi i)^{-1}x'(\tau)/x(\tau)$ is the logarithmic derivative of an invariant function, it has weight $2$, and differentiating $\log x(-1/(3\tau))=-\log x(\tau)-6\log3$ gives $B(-1/(3\tau))\cdot(3\tau^2)^{-1}=-B(\tau)$. We have $B=1+O(q)$ at $\infty$, and $B(-1/(3\tau))/(-3\tau^2)=B(\tau)\to1$ at the cusp $0$. On $\mathbb H$, $x$ is a local coordinate except at the elliptic points, where $x-x_e$ vanishes to order $3$ in $\tau-\tau_e$; so $x'$ vanishes exactly on the orbit of $\tau_e$, to order $2$, and $x\ne0$ on $\mathbb H$.
\end{proof}

\begin{lemma}\label{lem:vr}
$V:=2Du+u^2=\dfrac{24x}{P^2}$ and $R_5:=B^{-3}\theta^5G_5=\dfrac{x(1+39x)}{P^2}$.
\end{lemma}

\begin{proof}
\textit{$V$ is invariant.} For $\gamma=\left(\begin{smallmatrix}a_\gamma&b_\gamma\\c_\gamma&d_\gamma\end{smallmatrix}\right)\in\Gamma_0(3)$ we have $B(\gamma\tau)=(c_\gamma\tau+d_\gamma)^2B(\tau)$, the operator $D$ commutes with $\tau\mapsto\gamma\tau$, and $D\tau=(2\pi iB)^{-1}$. Hence $u(\gamma\tau)=u+\delta$ with $\delta=2c_\gamma/(2\pi iB(c_\gamma\tau+d_\gamma))$, and $D\delta=-u\delta-\delta^2/2$. So $V(\gamma\tau)=2Du+2D\delta+u^2+2u\delta+\delta^2=V(\tau)$. For the Fricke involution $\tau\mapsto-1/(3\tau)$, the relation $x(-1/(3\tau))=3^{-6}/x(\tau)$ turns $D$ into $-D$, and Lemma~\ref{lem:bform} gives $u(-1/(3\tau))=-u-\delta'$ with $\delta'=2/(2\pi i\tau B)$ and $D\delta'=-u\delta'-\delta'^2/2$; the same computation shows that $V$ is invariant under it.

\textit{The poles of $V$.} So $V$ is a rational function of $x$, holomorphic wherever $B\ne0$. Near $\tau_e$, $u=(2\pi i)^{-1}B'/B^2$ has a pole of order $3$ in $\tau$ and $Du$ one of order $6$, so $V$ has order at least $-6$ in $\tau-\tau_e$, that is, at least $-2$ in $x-x_e$. At $x=0$ we have $V\in\Z[[x]]$ and $V(0)=0$, since $u=12x+O(x^2)$, and by the Fricke symmetry also $V(\infty)=0$. Hence $P^2V$ is a polynomial of degree at most $2$ vanishing at $0$, its $x^2$-coefficient is $729\,V(\infty)=0$, and $P^2V=24x$ because $V=24x+O(x^2)$.

\textit{$R_5$.} Grouping the divisors, $\theta^5G_5=\sum_n(\sigma_5(n)-\sigma_5(n/3))q^n=(E_6(3\tau)-E_6(\tau))/504$ with the Eisenstein series $E_6=1-504\sum_n\sigma_5(n)q^n$, a holomorphic form of weight $6$ on $\Gamma_0(3)$. So $R_5$ is invariant and rational in $x$, with a pole of order at most $2$ at $x_e$, since $B^3$ vanishes to order $6$ in $\tau-\tau_e$, and $R_5=x+O(x^2)$. By $E_6(-1/\tau)=\tau^6E_6(\tau)$ and Lemma~\ref{lem:bform}, $R_5(-1/(3\tau))=(729E_6(3\tau)-E_6(\tau))/(13608\,B(\tau)^3)$, which tends to $728/13608=13/243$ at $\infty$; so $R_5=13/243$ at $x=\infty$. Hence $P^2R_5=x+39x^2$, as $729\cdot13/243=39$.
\end{proof}

Since $\theta$ acts simply on $q$-series, we express $D^i(B^2Y)$, for a holomorphic function $Y$ of $\tau$ and with $D=B^{-1}\theta$, through the functions $B^{2-m}\theta^mY$, $0\le m\le i$. The coefficients are polynomials in an indeterminate $u$, to be evaluated at $u=DB/B$. Let $\mathfrak R=\Q(x)[u]$, with $u$ an indeterminate, and with the derivation $\tilde D$ that extends $D$ by $\tilde Du=(V-u^2)/2$, the Riccati form of Lemma~\ref{lem:vr}, and let $\partial=\partial/\partial u$. Define $N_{i,m}\in\mathfrak R$ by $N_{0,0}=1$ and
\begin{equation}\label{eq:nrec}
N_{i+1,m}=(2-m)\,u\,N_{i,m}+\tilde DN_{i,m}+N_{i,m-1}\ ,\qquad N_{i,m}=0\ \text{unless}\ 0\le m\le i\ .
\end{equation}

\begin{lemma}\label{lem:sl2}
(a) For every holomorphic function $Y$ of $\tau$, $D^i(B^2Y)=B^2\sum_{m=0}^iN_{i,m}(u)\,B^{-m}\theta^mY$. (b) If $V$ is kept as a symbol on which $\tilde D$ acts as $D$, then $N_{i,m}\in\Z[\frac12][u,V,DV,D^2V,\dots]$ is isobaric of weight $i-m$, where $u$ has weight $1$ and $D^jV$ weight $2+j$, and $N_{i,i}=1$. (c) $\partial N_{i,m}=\mathfrak c_m\,N_{i,m+1}$ with $\mathfrak c_m=(m+1)(4-m)/2$, that is, $(\mathfrak c_0,\dots,\mathfrak c_4)=(2,3,3,2,0)$.
\end{lemma}

\begin{proof}
(a) By induction on $i$, using $D(B^{2-m}\theta^mY\,N)=B^{2-m}\theta^mY\,((2-m)uN+DN)+B^{1-m}\theta^{m+1}Y\,N$, which follows from $D=B^{-1}\theta$ and $DB=uB$; evaluation at $u=DB/B$ is a homomorphism of differential rings by the Riccati equation. (b) Each step of \eqref{eq:nrec} raises the weight by one, since $\tilde Du$ has weight $2$. (c) On $\mathfrak R$ we have $\partial\tilde D=\tilde D\partial-u\partial$. If (c) holds at level $i$, then $\partial N_{i+1,m}=(2-m)N_{i,m}+\mathfrak c_m((1-m)uN_{i,m+1}+\tilde DN_{i,m+1})+\mathfrak c_{m-1}N_{i,m}$ and $\mathfrak c_mN_{i+1,m+1}=\mathfrak c_m((1-m)uN_{i,m+1}+\tilde DN_{i,m+1}+N_{i,m})$, and the two agree because $\mathfrak c_m=\mathfrak c_{m-1}+2-m$ and $\mathfrak c_{-1}=0$. The case $i=0$ is trivial.
\end{proof}

For example, $N_{1,0}=2u$, $N_{2,0}=3u^2+V$, $N_{2,1}=3u$, $N_{3,0}=3u^3+5Vu+DV$, $N_{3,1}=\frac12(9u^2+5V)$, $N_{3,2}=3u$, $N_{4,0}=\frac12(3u^4+24Vu^2+14\,DV\,u+5V^2+2D^2V)$, $N_{4,1}=\frac12(6u^3+24Vu+7DV)$, $N_{4,2}=3u^2+4V$ and $N_{4,3}=2u$.

\begin{lemma}\label{lem:l5}
Let $L_5=D^5+\mathfrak a_3D^3+\mathfrak a_2D^2+\mathfrak a_1D+\mathfrak a_0$ with
\begin{equation}\label{eq:l5}
\begin{split}
&\mathfrak a_3=-\frac{120x}{P^2}\ ,\qquad\mathfrak a_2=\frac{180x(27x-1)}{P^3}\ ,\qquad\mathfrak a_1=-\frac{36x(2187x^2-388x+3)}{P^4}\ ,\\
&\mathfrak a_0=\frac{24x(27x-1)(729x^2-366x+1)}{P^5}\ .
\end{split}
\end{equation}
Then $L_5(B^2Y)=B^{-3}\theta^5Y$ for every holomorphic function $Y$ of $\tau$, in particular for every $q$-series. Consequently $L_5H_c=R_5$ for every $c$, and $L_5(B^2\tau^a)=0$ for $0\le a\le4$.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:sl2}(a), $L_5(B^2Y)=B^2\sum_m(\sum_j\mathfrak a_jN_{j,m}(u))B^{-m}\theta^mY$ with $\mathfrak a_5=1$ and $\mathfrak a_4=0$. With $V=24x/P^2$, a direct computation of the inner sums, which are polynomials in $u$ over $\Q(x)$, gives $0$ for $m\le4$ and $1$ for $m=5$. Then $L_5H_c=B^{-3}\theta^5(G_5+c)=R_5$, and $\theta^5\tau^a=0$ for $a\le4$.
\end{proof}

\subsection{The determinant}\label{sec:secdet}
We show that the six germs are independent over $\C(x)$ for every $c$, that a nonzero combination of them with polynomial coefficients of degree less than $d$ vanishes at $x=0$ to order at most $6d+69$, and we construct $\cD_d$. This is Yi's argument, written for level $3$ and $\mathrm{Sym}^4$.

\begin{lemma}\label{lem:n1}
There is no $v\in\C(x)$ with $L_5v=R_5$.
\end{lemma}

\begin{proof}
At a point $x_0\notin\{0,x_e,\infty\}$, a pole of $v$ of order $k_0$ gives a pole of $L_5v$ of order $k_0+5$, while $R_5$ is regular there. At $0$ the indicial polynomial of $L_5$ is $\rho^5$, and at $x_e$ its roots are $\frac43,\frac53,2,\frac73,\frac83$, the exponents of the local solutions $B^2\tau^a$. So a pole of order $k_0\ge1$ at $0$ or at $x_e$ gives a pole of $L_5v$ of order $k_0$, respectively $k_0+5\ge6$, while $R_5$ is regular at $0$ and has a pole of order at most $2$ at $x_e$. Hence $v$ is a polynomial. All $\mathfrak a_j$ are $O(1/x)$ at $\infty$ and $D^5x^m=m^5x^m$, so if $\deg v=m>0$, the leading term of $L_5v$ is $m^5$ times that of $v$, while $R_5$ is bounded at $\infty$; so $v$ is constant. Then $L_5v=\mathfrak a_0v$, which is impossible, because $P^5\mathfrak a_0$ is a polynomial of degree $4$ while $P^5R_5=x(1+39x)P^3$ has degree $5$.
\end{proof}

A \textit{differential module} over $(\C(x),D)$ is a finite-dimensional $\C(x)$-vector space $\mathcal M$ with an additive map $D:\mathcal M\to\mathcal M$ such that $D(fm)=(Df)m+f\,Dm$ for $f\in\C(x)$ and $m\in\mathcal M$; a submodule is a $D$-stable subspace, and $\mathcal M$ is irreducible if $0$ and $\mathcal M$ are its only submodules.

\begin{lemma}\label{lem:n2}
The differential module $\mathcal M_5=\C(x)[D]/\C(x)[D]L_5$ is irreducible.
\end{lemma}

\begin{proof}
Near a point $x_0\notin\{0,x_e,\infty\}$ with a preimage $\tau_0$, the solutions of $L_5f=0$ are the span of $B^2\tau^a$, $0\le a\le4$, by Lemma~\ref{lem:l5}. A loop in $\mathbb P^1\setminus\{0,x_e,\infty\}$ lifts to a path from $\tau_0$ to $\gamma\tau_0$, and every $\gamma\in\Gamma_0(3)$ arises, since $x$ makes $\mathbb H$ minus the elliptic orbit a covering of $\mathbb P^1\setminus\{0,x_e,\infty\}$ with deck group $\Gamma_0(3)/\{\pm1\}$. Continuation sends $B^2\tau^a$ to $B^2(\gamma\tau)(\gamma\tau)^a=B^2(\tau)(a_\gamma\tau+b_\gamma)^a(c_\gamma\tau+d_\gamma)^{4-a}$, so the monodromy contains $\mathrm{Sym}^4$ of $\Gamma_0(3)$ acting on binary quartics in $(X,Y)=(\tau,1)$. The elements $\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}1&0\\3&1\end{smallmatrix}\right)$ are unipotent, with logarithms $Y\partial_X$ and $3X\partial_Y$. A subspace invariant under a unipotent element is invariant under its logarithm, a polynomial in the element minus the identity, hence under the commutator $3(Y\partial_Y-X\partial_X)$, whose eigenvalues $12,6,0,-6,-12$ on the monomials $X^aY^{4-a}$ are distinct. So a nonzero invariant subspace contains some $X^aY^{4-a}$, and then, by the two logarithms, all of them. If $\mathcal M_5$ had a proper nonzero submodule, the quotient would be cyclic, generated by the image of $1$, with annihilator a monic operator of order between $1$ and $4$ dividing $L_5$ on the right, and its local solutions would form a monodromy-invariant subspace of that dimension, a contradiction.
\end{proof}

\begin{proposition}\label{prop:n3}
Let $\mathcal W$ be the differential module with basis $e_0,\dots,e_5$, $De_0=0$, $De_i=e_{i+1}$ for $1\le i\le4$, and $De_5=R_5e_0-\mathfrak a_0e_1-\mathfrak a_1e_2-\mathfrak a_2e_3-\mathfrak a_3e_4$. Its only proper nonzero differential submodule is $\C(x)e_0$. For every $c\in\C$, the map $e_0\mapsto1$, $e_i\mapsto D^{i-1}H_c$ is an injective morphism of differential modules into $\C((x))$; in particular $1,H_c,\dots,D^4H_c$ are linearly independent over $\C(x)$.
\end{proposition}

\begin{proof}
We have $\mathcal W/\C(x)e_0\cong\mathcal M_5$ via $\bar e_1\mapsto1$, since $L_5\bar e_1=0$. Let $S$ be a proper nonzero submodule. If its image in $\mathcal M_5$ is zero, then $S=\C(x)e_0$. Otherwise the image is $\mathcal M_5$ by Lemma~\ref{lem:n2}. Since $e_0\in S$ would give $S=\mathcal W$, we have $e_0\notin S$, so $S\cap\C(x)e_0=0$, and $S$ is a complement of $\C(x)e_0$ containing $s_1=e_1-ve_0$ for some $v\in\C(x)$. Then $D^ks_1=e_{k+1}-(D^kv)e_0$ for $k\le4$, and $D^5s_1+\sum_j\mathfrak a_jD^js_1=(R_5-L_5v)e_0\in S\cap\C(x)e_0=0$, so $L_5v=R_5$, which Lemma~\ref{lem:n1} excludes. The map is a morphism because $D(D^4H_c)=R_5-\sum_j\mathfrak a_jD^jH_c$ by Lemma~\ref{lem:l5}, and its kernel is a submodule not containing $e_0$, hence zero.
\end{proof}

\begin{proposition}[zero estimate]\label{prop:n4}
Let $\mathcal F=\sum_{j=0}^5a_jg_j\ne0$ with polynomials $a_j$ of degree less than $d$, and $c\in\C$. Then $\ord_0\mathcal F\le6d+69$.
\end{proposition}

\begin{proof}
If $a_1=\dots=a_5=0$, then $\ord_0\mathcal F\le d-1$. Otherwise, identifying $\mathcal W$ with its image (Proposition~\ref{prop:n3}), the $\C(x)$-span of $\mathcal F,D\mathcal F,D^2\mathcal F,\dots$ is a differential submodule not contained in $\C(x)e_0$, hence equal to $\mathcal W$. So $\mathcal F,D\mathcal F,\dots,D^5\mathcal F$ are linearly independent over $\C(x)$, since a dependence at step $k<6$ would make the span $D$-stable of dimension $k$. Write $De=e\,\mathcal C$ for the frame $e=(e_0,\dots,e_5)$; the entries of $P^5\mathcal C$ are $0$, $P^5$, $P^5R_5$ and $-P^5\mathfrak a_j$, polynomials of degree at most $5$. Put $P^{5k}D^k\mathcal F=e\,\mathbf a^{(k)}$ with $\mathbf a^{(0)}=(a_0,\dots,a_5)^{\mathsf T}$. Then
\begin{equation}\label{eq:precur}
\mathbf a^{(k+1)}=P^5D\mathbf a^{(k)}-5k\,P^4(DP)\,\mathbf a^{(k)}+(P^5\mathcal C)\,\mathbf a^{(k)}\ ,
\end{equation}
so $\deg \mathbf a^{(k)}\le d-1+5k$, and $\mathrm{Wr}=\det(\mathbf a^{(0)},\dots,\mathbf a^{(5)})$ is a nonzero polynomial of degree at most $\sum_{k=0}^5(d-1+5k)=6d+69$. Adding to the row of $e_0$ the other rows multiplied by the germs $g_j$ does not change $\mathrm{Wr}$, and it turns that row into $(P^{5k}D^k\mathcal F)_{0\le k\le5}$, whose entries have order at least $\ord_0\mathcal F$ at $0$, as $D$ does not lower the order; the other rows are polynomials. Expanding along that row, $\ord_0\mathcal F\le\ord_0\mathrm{Wr}\le6d+69$.
\end{proof}

\textit{The determinant.} For $d\ge1$ the map $\Phi_d:\C[x]^6_{<d}\to\C[[x]]$, $a\mapsto\sum_ja_jg_j$, is injective by Proposition~\ref{prop:n3}. The orders $\ord_0\Phi_d(a)$ take exactly $6d$ distinct values $\lambda_0<\dots<\lambda_{6d-1}$, the \textit{jumps}, since each quotient $\{\ord_0\Phi_d\ge n\}/\{\ord_0\Phi_d\ge n+1\}$ has dimension at most $1$; all jumps are at most $6d+69$ by Proposition~\ref{prop:n4}. The block of $a_0$, the coefficient of $g_0=1$, realizes the orders $0,\dots,d-1$, so $\lambda_j=j$ for $j<d$, and since the jumps are distinct,
\begin{equation}\label{eq:jumps}
j\le\lambda_j\le(6d+69)-(6d-1-j)=j+70\ .
\end{equation}
Put $I_d=\{\lambda_d,\dots,\lambda_{6d-1}\}\subset[d,N_d)$ with $N_d=6d+70$, and let $M$ be the $5d\times5d$ matrix
\begin{equation}\label{eq:mdef}
M_{\lambda,(i,k)}=[x^{\lambda-k}]\,g_i\qquad(\lambda\in I_d,\ 1\le i\le5,\ 0\le k<d)\ ,\qquad\cD_d(c)=\det M\ .
\end{equation}
The sign of $\cD_d(c)$ depends on the order of rows and columns and plays no role. Let $\tilde M$ be the $6d\times6d$ matrix with entries $[x^\lambda](x^kg_i)$, rows $\lambda\in\{\lambda_0,\dots,\lambda_{6d-1}\}$ and columns $(i,k)$, $0\le i\le5$, $0\le k<d$. It is nonsingular, because in a basis of the source adapted to the order filtration it becomes triangular with nonzero diagonal. Its columns with $i=0$ are the unit vectors of the rows $\lambda_j=j<d$, so $\det\tilde M=\pm\det M$ and $\cD_d(c)\ne0$. For $c\in\Q$ the jumps are defined over $\Q$ and do not change under $\Q\subset\C$ or $\Q\subset\Q_3$, so $\cD_d(c)$ is one nonzero rational number, which we estimate at every place.

\subsection{The \texorpdfstring{$3$}{3}-adic estimate}\label{sec:secgain}
The $3$-adic estimate rests on Proposition~\ref{prop:radius}: for $c=\eta_5$ the germs converge on the open disc $|x|_3<3^6$. Our rigid analytic spaces are defined over a finite extension of the completion of $\Q_3^{\mathrm{ur}}$, as in \cite[Section~3]{Bu03}; since $H_{\eta_5}\in\Q_3[[x]]$, its radius of convergence does not depend on this choice.

We write $X_0(3)^{\mathrm{an}}$ for the rigid analytic space attached to $X_0(3)$ \cite{Con06}; through $x$, its points over $\C_3$ are the points of $\mathbb P^1(\C_3)$. A point is \textit{supersingular} if the elliptic curve that it parametrizes has potentially good supersingular reduction, which here means $|j|_3<1$, and \textit{ordinary} otherwise; this depends only on its $j$-invariant.

\begin{lemma}\label{lem:w0}
In the coordinate $x$, the ordinary locus of $X_0(3)^{\mathrm{an}}$ is $\{|x|_3\le1\}\sqcup\{|x|_3\ge3^6\}$, the two components containing the cusps $\infty$ ($x=0$) and $0$ ($x=\infty$), and the supersingular locus is the annulus $1<|x|_3<3^6$. Hence $W_0(3)$, the complement of the ordinary component containing the cusp $0$, is the open disc $\{|x|_3<3^6\}$. The elliptic point $x_e$, with $|x_e|_3=27$, lies in the supersingular annulus.
\end{lemma}

\begin{proof}
In characteristic $3$ the only supersingular $j$-invariant is $0$ \cite[Theorem~V.4.1(c) and its proof]{Si09}, so a point is supersingular if and only if $|j|_3<1$, and whether it is ordinary depends only on $j$. Put $v=v_3(x)\in\R$. By Lemma~\ref{lem:hauptmodul}(c) and the Newton polygons of $1+27x$ and $1+243x$, $v_3(j)=-v$ for $v>-3$, $v_3(j)=3$ for $-5<v<-3$, $v_3(j)=18+3v$ for $v<-5$, and $v_3(j)\ge3$ for $v=-3,-5$. So $v_3(j)\le0$ exactly for $v\ge0$ or $v\le-6$. The two ordinary pieces are closed discs, one around $x=0$ and one around $x=\infty$, so they are connected, and they contain the two cusps.
\end{proof}

We now quote the two published inputs, in our notation. For $p>2$ let $\omega$ be the Teichm\"uller character, $\langle a\rangle=a/\omega(a)$ for $a\in\Z_p^\times$ as in Section~\ref{sec:volkenborn}. For a continuous character $\kappa\ne\one$ of $\Z_p^\times$ put $\sigma^*_\kappa(n)=\sum_{a\mid n,\,p\nmid a}\kappa(a)a^{-1}$ and $G^*_\kappa(q)=\zeta^*(\kappa)/2+\sum_{n\ge1}\sigma^*_\kappa(n)q^n$, where $\zeta^*(\kappa)$ is Coleman's $p$-adic zeta function of $\kappa$; if $\kappa(a)=\langle a\rangle^s\chi(a)$ with $s\in\C_p$, $|s|<p^{(p-2)/(p-1)}$, and $\chi$ of finite order, then $\zeta^*(\kappa)=L_p(1-s,\chi)$ \cite[Section~B1]{Co97}.

\begin{theorem}[Coleman {\cite[Section~B1]{Co97}}; Coleman and Mazur {\cite[Corollary~2.2.6]{CM98}}]\label{thm:coleman}
Let $p>2$. If $\kappa(a)=\langle a\rangle^k\chi(a)$, where $k$ is an integer and $\chi$ is a character of finite order on $\Z_p^\times$ with $\chi(-1)=1$, then $G^*_\kappa(q)$ is the $q$-expansion of an overconvergent modular form of weight $k$ on $\Gamma_1(\mathrm{lcm}(p,f_\chi))$ with character $\chi\omega^{-k}$, where $f_\chi$ is the conductor of $\chi$.
\end{theorem}

Coleman states this with a sketch of proof, and Coleman and Mazur give a complete proof for the characters in the connected component $\mathcal B$ of the identity of weight space \cite[Section~1.4 and Corollary~2.2.6]{CM98}. Weight space is the rigid analytic space whose $\C_p$-points are the continuous characters $\Z_p^\times\to\C_p^\times$, and the characters in $\mathcal B$ are, by restriction, identified with the continuous characters of $1+p\Z_p$; these are the characters trivial on $\mu(\Z_p)$. In their normalization, the Eisenstein series $E_\kappa$ has constant term $1$, so that $\zeta^*(\kappa)E_\kappa=2G^*_\kappa$; the passage to $G^*_\kappa$ uses $\zeta^*(\kappa)\ne0$, which holds for these characters by \cite[Section~B1]{Co97}. At $p=3$ the Teichm\"uller character has order $2$, so for $\kappa(a)=a^{-4}=\langle a\rangle^{-4}$ we have $k=s=-4$ and $\chi=\one$, with $|s|_3=1<3^{1/2}$; the level is $\Gamma_1(3)$ with character $\omega^4=\one$, that is, $\Gamma_0(3)$. Then $\sigma^*_\kappa=\sigma^{(3)}_{-5}$ and $\zeta^*(\kappa)=L_3(5,\one)=\zeta_3(5)$, so
\begin{equation}\label{eq:estar}
G^*_{-4}:=G^*_{a^{-4}}=\eta_5+\sum_{n\ge1}\sigma^{(3)}_{-5}(n)q^n=\eta_5+G_5(q)
\end{equation}
is an overconvergent form of weight $-4$ and tame level $1$ (the tame level is the part of the level prime to $p$). The character $a\mapsto a^{-4}$ is trivial on $\mu(\Z_3)=\{\pm1\}$, so it lies in $\mathcal B$ and the complete proof of \cite{CM98} applies.

\begin{lemma}\label{lem:u3}
Atkin's operator $U_3$, which acts on $q$-expansions by $\sum_na_nq^n\mapsto\sum_na_{3n}q^n$, fixes $G^*_{-4}$.
\end{lemma}

\begin{proof}
We have $\sigma^{(3)}_{-5}(3n)=\sigma^{(3)}_{-5}(n)$ because $3n$ and $n$ have the same divisors prime to $3$; the constant term is unchanged.
\end{proof}

\begin{theorem}[Buzzard {\cite[Theorem~5.2(2)]{Bu03}}]\label{thm:buzzard}
Let $p$ be a prime, $N\ge5$ an integer prime to $p$, and $k$ an integer. Let $X_1(N;p)$ be the modular curve of level $\Gamma_1(N)\cap\Gamma_0(p)$, and let $W_0(p)\subset X_1(N;p)^{\mathrm{an}}$ be the set of points whose reduction lies in the component of the special fiber that contains the reduction of the cusp $\infty$, that is, the complement of the component of the ordinary locus which contains the cusp $0$. If $f$ is an overconvergent modular form of level $N$ and weight $k$ which is an eigenvector of $U_p$ with nonzero eigenvalue, then $f$ extends to a section of $(\underline\omega^{\mathrm{an}})^{\otimes k}$ on $W_0(p)$.
\end{theorem}

Here $\underline\omega$ is the sheaf of invariant differentials of the universal elliptic curve, the superscript $\mathrm{an}$ denotes the associated rigid analytic space, and $U_p$ acts on $q$-expansions by $\sum_na_nq^n\mapsto\sum_na_{pn}q^n$. Buzzard's setting includes $p=3$ \cite[Section~3]{Bu03}, and for small tame level he suggests adding an auxiliary level structure, working at higher level and then taking invariants \cite[Section~3]{Bu03}. We carry out this reduction for $p=3$ with the tame level $N=5$ and a trace, and we turn the resulting section of $\underline\omega^{\otimes-4}$ into a function by multiplying it with the form $B^2$ of weight $4$.

\begin{proposition}\label{prop:radius}
The series $H_{\eta_5}=B^2(G_5(q(x))+\eta_5)\in\Q_3[[x]]$ and all $D^iH_{\eta_5}$ converge on $|x|_3<3^6$. Hence for every $\epsilon>0$ there is $\Theta_\epsilon$ with
\begin{equation}\label{eq:gauss}
v_3\qty([x^n]\,g_i)\ge(6-\epsilon)n-\Theta_\epsilon\qquad(n\ge0,\ 1\le i\le5,\ c=\eta_5)\ .
\end{equation}
\end{proposition}

\begin{proof}
Away from the cusps, the points of $X_1(5;3)$ correspond to triples $(E,\iota,C)$ of an elliptic curve $E$, a point $\iota$ of exact order $5$ on $E$ and a subgroup $C\subset E[3]$ of order $3$, and those of $X_0(3)$ to pairs $(E,C)$. Let $\pi_5:X_1(5;3)\to X_0(3)$ be the morphism $(E,\iota,C)\mapsto(E,C)$ that forgets $\iota$, a finite flat morphism of smooth projective curves of some degree $\delta\ge1$. Whether a point lies in $W_0(3)$ depends only on $(E,C)$, namely on whether $E$ has supersingular reduction or ordinary reduction with $C$ the canonical subgroup \cite[Ch.~3]{Ka73}, \cite[Sections~3--4]{Bu03}; if $E$ has ordinary reduction, its canonical subgroup is the subgroup of $E[3]$ of order $3$ whose points reduce to the origin, the $3$-torsion of the formal group of $E$. On the ordinary locus of $X_0(3)$ the condition that $C$ is the canonical subgroup is open and closed; it holds at the cusp $\infty$, where $C=\mu_3$ on the Tate curve $\mathbb G_m/q^\Z$, the elliptic curve over $\Z((q))$ that describes a neighbourhood of the cusp $\infty$, and fails at the cusp $0$. Since the two ordinary components of Lemma~\ref{lem:w0} are connected, the disc $W_0(3)=\{|x|_3<3^6\}$ is exactly the set of points $(E,C)$ with $E$ supersingular or $C$ canonical. So $W_5:=\pi_5^{-1}(W_0(3))$ is Buzzard's $W_0(3)\subset X_1(5;3)^{\mathrm{an}}$, and $\pi_5:W_5\to W_0(3)$ is finite flat of degree $\delta$, with a trace $\mathrm{Tr}$ such that $\mathrm{Tr}\circ\pi_5^*$ is multiplication by $\delta$. By Theorem~\ref{thm:coleman}, $G^*_{-4}$ is overconvergent of weight $-4$ and tame level $1$. Its pullback $\pi_5^*G^*_{-4}$ is an overconvergent form of weight $-4$ and level $5$ in the sense of \cite{Bu03}, with $q$-expansion $G^*_{-4}(q)$, hence, by Lemma~\ref{lem:u3} and the $q$-expansion principle (a form on a connected region containing the cusp $\infty$ is determined by its $q$-expansion), a $U_3$-eigenvector with eigenvalue $1$. Theorem~\ref{thm:buzzard}, with $p=3$, $N=5$ and $k=-4$, extends it to a section $\phi$ of $\underline\omega^{\otimes-4}$ on $W_5$. The form $B^2$ is a classical form of weight $4$ on $\Gamma_0(3)$ with integral $q$-expansion \eqref{eq:bg} (Lemma~\ref{lem:bform}), so $\pi_5^*(B^2)$ is a global section of $\underline\omega^{\otimes4}$, and $\tilde{\mathcal H}:=\pi_5^*(B^2)\,\phi$ is a rigid analytic function on $W_5$. Put $\mathcal H=\delta^{-1}\mathrm{Tr}(\tilde{\mathcal H})$, a rigid analytic function on the open disc $W_0(3)=\{|x|_3<3^6\}$ (Lemma~\ref{lem:w0}), hence a power series in $x$ converging on that disc. On a neighbourhood of the cusp $\infty$ on which $G^*_{-4}$ is defined, $\phi=\pi_5^*G^*_{-4}$, so there $\tilde{\mathcal H}=\pi_5^*(B^2G^*_{-4})$ and $\mathcal H=B^2G^*_{-4}$. Near the cusp $\infty$, by the compatibility of the algebraic and rigid Tate curves \cite[Section~3.1]{Con06}, the function $B^2G^*_{-4}$ is $B^2(q)(G_5(q)+\eta_5)$ in the Tate parameter $q$, with $x=x(q)$, that is, $H_{\eta_5}(x)$. Hence $\mathcal H=H_{\eta_5}$ near $x=0$, and by the uniqueness of Taylor expansions $H_{\eta_5}$ is the expansion of $\mathcal H$ on the whole disc. The elliptic point $x_e$ inside the disc causes no difficulty, since $\mathcal H$ is a function on the coarse curve. Finally $D$ preserves convergence on an open disc, and \eqref{eq:gauss} follows from convergence on the closed disc of radius $3^{6-\epsilon}$.
\end{proof}

Besides Theorems~\ref{thm:coleman} and~\ref{thm:buzzard}, this proof uses four standard facts that we have not checked: that membership in $W_0(3)$ depends only on $(E,C)$ \cite[Ch.~3]{Ka73}, \cite[Sections~3--4]{Bu03}; that the algebraic and the rigid Tate curves agree \cite[Section~3.1]{Con06}; that Coleman's overconvergence at tame level $1$ pulls back to Buzzard's notion at level $5$; and the $q$-expansion principle on the ordinary component of the cusp $\infty$. Calegari uses the same description of the ordinary locus of $X_0(3)$ in his proof that $\zeta_3(3)$ is irrational \cite[Section~3]{Ca05}.

\begin{proposition}\label{prop:secondgain}
Suppose that $\eta_5\in\Q$. For every $0<\epsilon<6$, $v_3(\cD_d(\eta_5))\ge15(6-\epsilon)d^2-5\Theta_\epsilon d$, with $\Theta_\epsilon$ from \eqref{eq:gauss}.
\end{proposition}

\begin{proof}
The entries of $M$ are Taylor coefficients of the $3$-adic series of Proposition~\ref{prop:radius}, of index $\lambda-k\ge1$ for $\lambda\in I_d$ and $k<d$. By \eqref{eq:gauss}, every term of the Leibniz expansion of $\det M$ has valuation at least $(6-\epsilon)(\sum_{\lambda\in I_d}\lambda-\sum_{(i,k)}k)-5d\Theta_\epsilon$. Since $I_d$ consists of $5d$ distinct integers $\ge d$, $\sum_{\lambda\in I_d}\lambda\ge\sum_{n=d}^{6d-1}n=(35d^2-5d)/2$, and $\sum_{(i,k)}k=5d(d-1)/2$; the difference is at least $15d^2$.
\end{proof}

\subsection{The primes \texorpdfstring{$\ell\ne3$}{l not 3}}\label{sec:secprimes}
We now prove $\sum_{\ell\ne3}\nu_\ell\log\ell\le\frac{13289}{168}d^2+o(d^2)$ with $\nu_\ell=-v_\ell(\cD_d)$. The denominators of the germs at $\ell$ come only from the terms of $G_5$ with $\ell\mid a$. Modulo $\ell$, the series $u=DB/B$ agrees with the logarithmic derivative $r$ of a rational function of degree about $\ell/3$, built from the Eisenstein series of weight $\ell-1$ (Lemma~\ref{lem:hasse}). Hence, modulo $\ell^j$, the combinations $\ell^5\sum_ia_iD^{i-1}H_c$ are governed by rational functions $F_0(a),\dots,F_{j-1}(a)$ with few poles (Proposition~\ref{prop:sym4} and \eqref{eq:fn}), and counting their dimensions bounds the number $h_j$ of elementary divisors of $\ell^5M$ of valuation below $j$ (Proposition~\ref{prop:smith} and Theorem~\ref{thm:layers}). For the remaining primes a crude common denominator suffices. Fix a prime $\ell\ge5$, put $s=\fl{(\ell-1)/3}$, and let $E_{\ell-1}=1-\frac{2(\ell-1)}{\beta_{\ell-1}}\sum_n\sigma_{\ell-2}(n)q^n$ be the Eisenstein series of weight $\ell-1$, where $\beta_{\ell-1}$ is the Bernoulli number.

\begin{lemma}\label{lem:hasse}
One has $E_{\ell-1}(q(x))/B^{(\ell-1)/2}=Q_\ell(x)/P^s$ with $Q_\ell\in\Z_{(\ell)}[x]$, $Q_\ell(0)=1$, $\deg Q_\ell=s$, and leading coefficient $27^s(-3)^{(\ell-1)/2}$, an $\ell$-adic unit. The rational function
\begin{equation}\label{eq:rdef}
r:=-\frac2{\ell-1}\,D\log(Q_\ell/P^s)
\end{equation}
has only simple poles, at the roots of $Q_\ell$ and at $x_e$, is bounded at $\infty$, and lies in $\Z_{(\ell)}[[x]]$, and $\xi:=(u-r)/\ell\in\Z_{(\ell)}[[x]]$.
\end{lemma}

\begin{proof}
By the von Staudt--Clausen theorem $v_\ell(\beta_{\ell-1})=-1$, so $E_{\ell-1}\in1+\ell q\Z_{(\ell)}[[q]]$ and $E_{\ell-1}(q(x))\in1+\ell x\Z_{(\ell)}[[x]]$. The quotient $E_{\ell-1}/B^{(\ell-1)/2}$ is $\Gamma_0(3)$-invariant of weight $0$, with poles only at the zeros of $B$, of order at most $\ell-1$ in $\tau-\tau_e$ (Lemma~\ref{lem:bform}), that is, at most $s$ in $x-x_e$. It equals $1$ at $x=0$, and at $x=\infty$, by $E_{\ell-1}(-1/(3\tau))=(3\tau)^{\ell-1}E_{\ell-1}(3\tau)$ and Lemma~\ref{lem:bform}, it tends to $3^{\ell-1}(-3)^{-(\ell-1)/2}=(-3)^{(\ell-1)/2}$. So $Q_\ell=P^sE_{\ell-1}/B^{(\ell-1)/2}$ is a polynomial of degree $s$ with leading coefficient $27^s(-3)^{(\ell-1)/2}$, and its Taylor series lies in $\Z_{(\ell)}[[x]]$. Finally $D\log(Q_\ell/P^s)=D\log E_{\ell-1}(q(x))-\frac{\ell-1}2u$ gives $u-r=\frac2{\ell-1}D\log E_{\ell-1}(q(x))\in\ell\Z_{(\ell)}[[x]]$; the statements about the poles hold for the logarithmic derivative of any rational function.
\end{proof}

\begin{lemma}\label{lem:sqf}
The reduction $\bar Q_\ell\in\F_\ell[x]$ has degree $s$, is squarefree, and is prime to $\bar P$. Hence $Q_\ell$ has $s$ distinct roots in $\bar\Q_\ell$, all $\ell$-integral and different from $0$ and $x_e$.
\end{lemma}

\begin{proof}
Reducing $E_{\ell-1}=B^{(\ell-1)/2}Q_\ell/P^s$ modulo $\ell$ gives $\bar B^{(\ell-1)/2}=\bar P^s/\bar Q_\ell$, hence, as $\frac2{\ell-1}\equiv-2$,
\begin{equation}\label{eq:ubar}
\bar u=\frac2{\ell-1}D\log\frac{\bar P^s}{\bar Q_\ell}=D\log\frac{\bar Q_\ell^2}{\bar P^{2s}}\ ,\qquad2D\bar u+\bar u^2=\bar V=\frac{24x}{\bar P^2}\ .
\end{equation}
Let $\alpha$ be a root of $\bar Q_\ell$ of multiplicity $\mu$, $1\le\mu\le s<\ell$, and $t=x-\alpha$; $\alpha\ne0$ since $Q_\ell(0)=1$. If $\bar P(\alpha)\ne0$, then $\bar u=2\mu\alpha/t+O(1)$, and the coefficient of $t^{-2}$ in $2D\bar u+\bar u^2$ is $4\mu(\mu-1)\alpha^2$, which vanishes because $\bar V$ is regular at $\alpha$; so $\mu=1$. If $\alpha=x_e$, then $\bar u=\rho x_e/t+O(1)$ with $\rho=2\mu-2s$, and comparison with $\bar V=-\frac89x_e^2t^{-2}+O(t^{-1})$ gives $\rho^2-2\rho=-\frac89$, that is, $(3\rho-2)(3\rho-4)=0$ in $\F_\ell$. Since $\mu=s+\rho/2$, and $3s\equiv-1$ or $3s\equiv-2\pmod\ell$ according as $\ell=3s+1$ or $\ell=3s+2$, this gives $\mu\equiv0$ or $\mu\equiv2s+1\pmod\ell$ in both cases, and neither lies in $[1,s]$. The degree is that of Lemma~\ref{lem:hasse}, the leading coefficient being a unit, and the roots of $Q_\ell$ are integral and reduce to the distinct roots of $\bar Q_\ell$, none equal to $\bar x_e$.
\end{proof}

Now fix $\ell\ge5$ and $c\in\Z_{(\ell)}$. Write $G_5=G_5^{[\ell]}+\ell^{-5}G_5(q^\ell)$, where $G_5^{[\ell]}$ omits the terms with $\ell\mid a$ and is $\ell$-integral, and put $T_m=(\theta^mG_5)(q(x)^\ell)\in\Q[[x]]$. The coefficient of $Y^n$ in $\theta^mG_5(Y)$ is $n^m\sigma^{(3)}_{-5}(n)$, which is $\ell$-integral for $n<\ell$, and $q(x)^\ell\in x^\ell\Z[[x]]$, so $T_m$ is $\ell$-integral below $x^{\ell^2}$. Moreover $DT_m=\ell B^{-1}T_{m+1}$, since $D\log q(x)=B^{-1}$.

\begin{proposition}[the $\mathrm{Sym}^4$ form of Yi's congruence]\label{prop:sym4}
Put
\begin{equation}\label{eq:bn}
C_{m,j}=\frac1{j!}\prod_{t=m}^{m+j-1}\mathfrak c_t=\binom{m+j}{m}\frac{(4-m)!}{(4-m-j)!\,2^j}\in\Z\qty[\tfrac12]\ ,
\end{equation}
and $b_n=\sum_{m+j=n}C_{m,j}\,\xi^j\,B^{2-m}\,T_m$ for $0\le n\le4$.
Then, for $0\le i\le4$, one has the exact identity of formal series
\begin{equation}\label{eq:sym4}
\ell^5D^iH_c=\sum_{n=0}^i\ell^n\,N_{i,n}(r)\,b_n+\ell^5\mathcal E_i\ ,\qquad\mathcal E_i=D^i\qty(B^2(G_5^{[\ell]}+c))\in\Z_{(\ell)}[[x]]\ ,
\end{equation}
and every $b_n$ is $\ell$-integral below $x^{\ell^2}$.
\end{proposition}

\begin{proof}
We have $\ell^5H_c=B^2T_0+\ell^5B^2(G^{[\ell]}_5+c)$. Lemma~\ref{lem:sl2}(a) with $\theta^mT_0=\ell^mT_m$ gives $D^i(B^2T_0)=\sum_m\ell^mB^{2-m}T_mN_{i,m}(u)$. Insert $u=r+\ell\xi$ and expand the polynomial $N_{i,m}$ in $u$ by Taylor's formula; by Lemma~\ref{lem:sl2}(c), $\partial^jN_{i,m}/j!=C_{m,j}N_{i,m+j}$, and collecting the powers $\ell^{m+j}$ gives \eqref{eq:sym4}. The closed form of $C_{m,j}$ is a rearrangement of the product. The integrality follows from $\xi\in\Z_{(\ell)}[[x]]$ (Lemma~\ref{lem:hasse}), $B^{\pm1}\in\Z[[x]]$, $C_{m,j}\in\Z[\frac12]$, and the integrality of $T_m$ below $x^{\ell^2}$.
\end{proof}

At level $7$, Yi proves the analogous congruence modulo $\ell^3$ for $H=B(G+c)$, with a form $G$ of weight $-2$, and the three germs $H,DH,D^2H$ \cite[(27)]{Yi26}; Proposition~\ref{prop:sym4} is an exact identity of the same kind for weight $-4$ and five germs. From now on let $\ell\ge5$ with $\ell^2>N_d$, and $c\in\Z_{(\ell)}$; we call these primes the \textit{large primes}. Let $A=\ell^5M$, which acts on $a=(a_1,\dots,a_5)\in\Z_{(\ell)}[x]^5_{<d}\cong\Z_{(\ell)}^{5d}$ by sending $a$ to the coefficients of $\ell^5\sum_ia_iD^{i-1}H_c$ at the rows $I_d$; $A$ is $\ell$-integral by \eqref{eq:sym4}, since $I_d\subset[d,N_d)$ and $N_d<\ell^2$. Let $\mathfrak e_1,\dots,\mathfrak e_{5d}$ be the $\ell$-adic valuations of its elementary divisors over $\Z_{(\ell)}$, and $h_j=\#\{i:\mathfrak e_i<j\}$. Since $\det A=\ell^{25d}\cD_d$,
\begin{equation}\label{eq:nu}
\nu_\ell=-v_\ell(\cD_d)=25d-\sum_i\mathfrak e_i\le\sum_i\max(0,5-\mathfrak e_i)=\sum_{j=1}^5h_j\ .
\end{equation}
For $a\in\Q[x]^5_{<d}$ put $F_n(a)=\sum_{i=1}^5a_i\,N_{i-1,n}(r)$, $0\le n\le4$. By \eqref{eq:sym4},
\begin{equation}\label{eq:fn}
\ell^5\sum_{i=1}^5a_iD^{i-1}H_c=\sum_{n=0}^4\ell^nb_nF_n(a)+\ell^5\sum_{i=1}^5a_i\mathcal E_{i-1}\ ,
\end{equation}
so modulo $\ell^j$ only $F_0(a),\dots,F_{j-1}(a)$ matter. Since $N_{i,i}=1$, $F_n(a)=a_{n+1}+\sum_{i>n+1}a_iN_{i-1,n}(r)$.

\begin{proposition}\label{prop:smith}
For $1\le j\le5$,
\begin{equation}\label{eq:smith}
h_j\le\min\qty(5d,\ \#\{\lambda\in I_d:\lambda\ge\ell\},\ \sum_{n<j}\dim F_n(K_n))\ ,
\end{equation}
where $K_n=\{a\in\Q[x]^5_{<d}:\ F_0(a)=\dots=F_{n-1}(a)=0\}$.
\end{proposition}

\begin{proof}
Let $S_j=\{a\in\Z_{(\ell)}^{5d}:\ Aa\equiv0\bmod\ell^j\}$ and $\bar S_j$ its image in $\F_\ell^{5d}$. In a Smith basis, $a\in S_j$ if and only if its $i$-th coordinate lies in $\ell^{\max(0,j-\mathfrak e_i)}\Z_{(\ell)}$, so $\dim\bar S_j=\#\{i:\mathfrak e_i\ge j\}$ and $h_j=5d-\dim\bar S_j$.

\textit{The kernel bound.} $K_j$ is a $\Q$-subspace of dimension $5d-\mathrm{rank}_\Q(F_0,\dots,F_{j-1})$, and $K_j\cap\Z_{(\ell)}^{5d}$ is saturated, so it reduces injectively to a subspace of $\F_\ell^{5d}$ of that dimension. For $a$ in it, \eqref{eq:fn} reduces to $\ell^5\sum_ia_iD^{i-1}H_c=\sum_{n\ge j}\ell^nb_nF_n(a)+\ell^5\sum_ia_i\mathcal E_{i-1}$, and $F_n(a)\in\Z_{(\ell)}[[x]]$ because $r$, $V$ and the derivatives of $V$ are $\ell$-integral series and the $N_{i,m}$ have coefficients in $\Z[\frac12]$. With $b_n$ integral below $x^{\ell^2}$ this gives $a\in S_j$. So $h_j\le\mathrm{rank}_\Q(F_0,\dots,F_{j-1})=\sum_{n<j}(\dim K_n-\dim K_{n+1})=\sum_{n<j}\dim F_n(K_n)$.

\textit{The row bound.} For $\lambda<\ell$ the coefficients $[x^n]H_c$, $n\le\lambda$, are $\ell$-integral, since they involve $\sigma^{(3)}_{-5}(m)$ only for $m<\ell$; so the rows $\lambda<\ell$ of $A$ vanish modulo $\ell^5$. Modulo $\ell^j$ ($j\le5$), $A$ therefore has at most $\#\{\lambda\in I_d:\lambda\ge\ell\}$ nonzero rows, hence at most that many elementary divisors that are nonzero modulo $\ell^j$, and $h_j$ is the number of these.
\end{proof}

\begin{lemma}[weight count]\label{lem:weight}
$F_n(\Q[x]^5_{<d})\subseteq(PQ_\ell)^{-(4-n)}\Q[x]_{\le d-1+(4-n)(s+1)}$, a space of dimension $d+(4-n)(s+1)$.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:sl2}(b), $N_{i-1,n}(r)$ is a sum of monomials in $r$ and the $D^jV$ of weight $i-1-n$. The function $r$ has simple poles at the roots of $PQ_\ell$ (Lemma~\ref{lem:hasse}), $D^jV$ has a pole of order $2+j$ at $x_e$ only, and all are bounded at $\infty$. So $N_{i-1,n}(r)(PQ_\ell)^{i-1-n}$ is a polynomial of degree at most $(i-1-n)(s+1)$, and $a_iN_{i-1,n}(r)(PQ_\ell)^{4-n}$ has degree at most $d-1+(4-n)(s+1)$.
\end{proof}

With Proposition~\ref{prop:smith}, Lemma~\ref{lem:weight} already gives a bound for the denominators with which the argument closes (Remark~\ref{rem:pess}). The following local lemma improves the bounds for $n\ge1$; it more than doubles the final margin.

\begin{lemma}[local lemma]\label{lem:local}
Let $\alpha$ be a root of $Q_\ell$, $t=x-\alpha$, and $a\in K_j$ with $0\le j\le4$. Then $\ord_\alpha F_j(a)\ge2j-4$.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:sqf}, $\alpha\ne0,x_e$, so near $\alpha$ the function $V$ and its derivatives are regular, and, $\alpha$ being a simple root, $r=w+\tilde r$ with $w=\rho_\alpha/t$, $\rho_\alpha=-2\alpha/(\ell-1)\ne0$, and $\tilde r$ regular. Taylor's formula and Lemma~\ref{lem:sl2}(c) give $F_m(a)=\sum_{n\ge m}C_{m,n-m}w^{n-m}\tilde F_n$ with $\tilde F_n=\sum_ia_iN_{i-1,n}(\tilde r)$ regular at $\alpha$. With $f_m=F_m/((4-m)!\,2^m)$ and $\tilde f_n=\tilde F_n/((4-n)!\,2^n)$, the closed form of $C_{m,j}$ turns this into $f_m=\sum_n\binom nmw^{n-m}\tilde f_n$, that is, $f(Z)=\tilde f(Z+w)$ for the quartics $f(Z)=\sum_mf_mZ^m$ and $\tilde f(Y)=\sum_n\tilde f_nY^n$. Let $\hat f(y)=t^4\tilde f(y/t)=\sum_n\tilde f_nt^{4-n}y^n\in\C[[t]][y]$; then $f_m=t^{m-4}\,\partial_y^m\hat f(\rho_\alpha)/m!$. The hypothesis $f_0=\dots=f_{j-1}=0$ says that $(y-\rho_\alpha)^j$ divides $\hat f$ in $\C((t))[y]$, hence in $\C[[t]][y]$, by division by a monic polynomial with constant coefficients. Write $\hat f=\sum_{b\ge0}t^bp_b(y)$. The coefficient of $t^by^n$ is the coefficient of $t^{b-4+n}$ in $\tilde f_n$, so $y^{4-b}$ divides $p_b$ for $b\le4$, and $(y-\rho_\alpha)^j$ divides every $p_b$. For $b<j$, $\deg p_b\le4<(4-b)+j$ forces $p_b=0$. Hence $\partial_y^j\hat f(\rho_\alpha)=O(t^j)$, and $f_j=O(t^{2j-4})$.
\end{proof}

\begin{theorem}[layer theorem]\label{thm:layers}
For the large primes,
\begin{equation}\label{eq:layers}
\dim F_n(K_n)\le\mathcal L_n:=\begin{cases}d+(4-n)+(4-2n)s\ ,&n=0,1,2\ ,\\\max(0,\,d+1-2s)\ ,&n=3\ ,\\\max(0,\,d-4s)\ ,&n=4\ ,\end{cases}
\end{equation}
and therefore $h_j\le\min(5d,\ \#\{\lambda\in I_d:\lambda\ge\ell\},\ \mathcal L_0+\dots+\mathcal L_{j-1})$.
\end{theorem}

\begin{proof}
For $a\in K_n$, the function $F_n(a)$ has at $x_e$ a pole of order at most $4-n$ (Lemma~\ref{lem:weight}), at each of the $s$ distinct roots of $Q_\ell$ (Lemma~\ref{lem:sqf}) a pole of order at most $\max(0,4-2n)$ and, for $n\ge3$, a zero of order at least $2n-4$ (Lemma~\ref{lem:local}), no other finite pole, and it is $O(x^{d-1})$ at $\infty$. For $n\le2$, $P^{4-n}Q_\ell^{4-2n}F_n(a)$ is a polynomial of degree at most $d-1+(4-n)+(4-2n)s$. For $n=3$, $PF_3(a)$ is a polynomial of degree at most $d$ divisible by $Q_\ell^2$. For $n=4$, $F_4(a)=a_5$ has degree less than $d$ and is divisible by $Q_\ell^4$. Counting dimensions and applying Proposition~\ref{prop:smith} proves the claim.
\end{proof}

\begin{proposition}\label{prop:cost}
Suppose that $\eta_5\in\Q$. Then $\sum_{\ell\ne3}\nu_\ell\log\ell\le\qty(\int_0^6\Psi(z)\,\dd z)d^2+o(d^2)$, where
\begin{equation}\label{eq:fdef}
\Psi(z)=\sum_{j=1}^5\min\qty(\mathrm{rc}(z),\ \sum_{n<j}\Lambda_n(z))\ ,\qquad\mathrm{rc}(z)=\min\qty(5,(6-z)^+)\ ,
\end{equation}
with the normalized row count $\mathrm{rc}$ and $(\Lambda_0,\dots,\Lambda_4)=(1+\frac43z,\ 1+\frac23z,\ 1,\ (1-\frac23z)^+,\ (1-\frac43z)^+)$, the normalized $\mathcal L_n/d$ of \eqref{eq:layers} with $z=\ell/d$. Explicitly,
\begin{equation}\label{eq:fexplicit}
\Psi(z)=\begin{cases}15+\frac{20}3z\ ,&0\le z\le\frac34\ ,\\16+\frac{16}3z\ ,&\frac34\le z\le1\ ,\\21+\frac13z\ ,&1\le z\le\frac43\ ,\\25-\frac83z\ ,&\frac43\le z\le\frac{15}7\ ,\\30-5z\ ,&\frac{15}7\le z\le6\ ,\end{cases}\qquad\int_0^6\Psi=\frac{13289}{168}\ .
\end{equation}
\end{proposition}

\begin{proof}
Let $b_c\ge1$ be the denominator of $\eta_5$.

\textit{A common denominator.} The coefficient $[x^n]G_5(q(x))$ involves $\sigma^{(3)}_{-5}(m)$ only for $m\le n$, and $q(x)$ and $B$ have integral coefficients, so $b_c\,\mathrm{lcm}(1,\dots,n)^5[x^n]H_{\eta_5}\in\Z$; and $D$ multiplies coefficients by integers. Hence every entry of $M$ has a denominator dividing $b_c\,\mathrm{lcm}(1,\dots,N_d)^5$, and for every prime $\ell$
\begin{equation}\label{eq:crude}
\nu_\ell\le5d\qty(v_\ell(b_c)+5\fl{\log N_d/\log\ell})\ .
\end{equation}
In particular $\nu_\ell\le0$ for $\ell>N_d$ with $\ell\nmid b_c$.

\textit{The small primes.} The primes $\ell\ne3$ with $\ell=2$, $\ell\mid b_c$, or $5\le\ell\le\sqrt{N_d}$ contribute by \eqref{eq:crude} at most $5d\qty(\log b_c+5\log N_d\,(2+\log_2b_c+\sqrt{N_d}))=O(d^{3/2}\log d)$, since $b_c$ has at most $\log_2b_c$ prime factors.

\textit{The large primes.} For the others, \eqref{eq:nu} and Theorem~\ref{thm:layers} give
\begin{equation*}
\nu_\ell\le\sum_{j=1}^5\min\qty(5d,\ \#\{\lambda\in I_d:\lambda\ge\ell\},\ \mathcal L_0+\dots+\mathcal L_{j-1})\ .
\end{equation*}
We have $\#\{\lambda\in I_d:\lambda\ge\ell\}\le(6d-\ell)^++70$ by \eqref{eq:jumps}; and $|s-\ell/3|\le1$ gives $\mathcal L_n\le d\Lambda_n(\ell/d)+4$ for each $n$. So in each of the five minima the second argument exceeds $d(6-\ell/d)^+$ by at most $70$ and the third exceeds $d\sum_{n<j}\Lambda_n(\ell/d)$ by at most $20$, and $\nu_\ell\le d\,\Psi(\ell/d)+350$.

\textit{The summation.} With $\vartheta(X)=\sum_{\ell\le X}\log\ell=X+o(X)$ by the prime number theorem, and since $\Psi$ is continuous, piecewise linear and zero on $[6,\infty)$, partial summation gives $\sum_{\ell<N_d}d\,\Psi(\ell/d)\log\ell=d^2\int_0^6\Psi+o(d^2)$ and $\sum_{\ell<N_d}350\log\ell=O(d)$. The formula \eqref{eq:fexplicit} and the integral follow from \eqref{eq:fdef} by exact piecewise linear arithmetic.
\end{proof}

\begin{remark}\label{rem:pess}
The local lemma and the layer theorem are not needed for the conclusion. If Theorem~\ref{thm:layers} is replaced by the elementary bound $\dim F_n(K_n)\le d+(4-n)(s+1)$ of Lemma~\ref{lem:weight}, the same computation applies with $\Lambda_n=1+\frac{4-n}3z$, the integral becomes $\frac{33569}{420}$, and $\frac{177}2\log3-\frac{33569}{420}-U$ is still positive, about $0.70$.
\end{remark}

\subsection{The archimedean estimate}\label{sec:secarch}
We prove $\limsup_dd^{-2}\log|\cD_d|\le U(\mu)$ for every finite convex combination $\mu$ of circle measures, and we bound $U(\mu)$ for one such $\mu$. The leading Taylor coefficients of combinations of the germs are bounded by Jensen's formula on circles $|q|=r_i$ in the complex $q$-disc (Lemma~\ref{lem:c2}). The polynomial coefficients are measured in a weighted $L^2$ norm, whose Gram determinant is controlled by the logarithmic energy of the image measure (Lemma~\ref{lem:c3}). Throughout, $c\in\Q$ is fixed and embedded in $\C$.

\textit{Pullbacks and the potential.} The product \eqref{eq:xdef} and the series \eqref{eq:bg} converge locally uniformly on $|q|<1$, and $x(q)/q$ is holomorphic and nonvanishing there with value $1$ at $0$. So $H_c(x(q))=B(q)^2(G_5(q)+c)$ is holomorphic on $|q|<1$, and since $D=B^{-1}\theta$, each $g_i(x(q))$ is meromorphic with poles only at the zeros of $B(q)$. Here and below, $g_i(x(q))$ denotes the function $(B^{-1}\theta)^{i-1}(B^2(G_5+c))$ of $q$, and similarly for combinations of the germs. Near $q=0$ it is the composition of the series $g_i$ with $x(q)$, but the series $g_i$ in $x$ need not converge on the image curves used below. Choose radii $r_i=e^{-2\pi y_i}<1$ and weights $\varpi_i>0$ with $\sum_i\varpi_i=1$, and fix $\rho_0$ with $\max_ir_i<\rho_0<1$. The functions $g_i(x(q))$ have finitely many poles in $|q|\le\rho_0$, and a polynomial $\Pi(q)$ with $\Pi(0)=1$, independent of $d$, clears them for all six germs. Let $\mu_i$ be the pushforward under $x(q)$ of the normalized arc length on $|q|=r_i$, let $\mu=\sum_i\varpi_i\mu_i$, and let $\mathcal P_\mu(z)=\int\log|z-w|\,\dd\mu(w)$.

\begin{lemma}\label{lem:c1}
$\mathcal P_\mu$ is continuous on $\C$, and $\mu$ has finite logarithmic energy $I=\iint\log|z-w|\,\dd\mu(z)\,\dd\mu(w)$ (we use this sign convention, the opposite of that of \cite{ST97}).
\end{lemma}

\begin{proof}
Near a point $q_0$ of one of the circles, $x(q)=x(q_0)+\mathfrak u(q)^{m_0}$ with a local biholomorphic coordinate $\mathfrak u$ and $m_0\ge1$. For $z\in\C$ put $z'=z-x(q_0)$. If $|\mathfrak u^{m_0}-z'|\le\varrho$, then one of the $m_0$ factors $\mathfrak u-\mathfrak u_k$, $\mathfrak u_k^{m_0}=z'$, has modulus at most $\varrho^{1/m_0}$; on a short arc the inverse coordinate is Lipschitz, so the arc length of the preimage of the disc $\mathbb D(z,\varrho)$ is at most $C\varrho^{1/m_0}$, uniformly in $z$. A finite cover of the circles gives $\mu(\mathbb D(z,\varrho))\le C\varrho^{1/m_1}$ for $0<\varrho\le1$, with a fixed $C$ and $m_1$ the largest of the $m_0$. Integrating in $\varrho$, the contribution of $|z-w|<e^{-T}$ to $\mathcal P_\mu(z)$ tends to $0$ uniformly in $z$ as $T\to\infty$, so the continuous truncated potentials converge uniformly to $\mathcal P_\mu$. In particular $\mathcal P_\mu$ is bounded on the support of $\mu$, and $I=\int\mathcal P_\mu\,\dd\mu$ is finite.
\end{proof}

Put $W_i=\int\mathcal P_\mu\,\dd\mu_i$. Fix a closed disc $\Omega$ containing the image curves in its interior, and $h>0$ such that the discs of radius $h$ around all points of these curves lie in $\Omega$. Give the polynomials of degree less than $d$ the norm $\|a\|_d^2=\int_\Omega|a|^2e^{-2d\mathcal P_\mu}\dd A$, and the source $\C[x]^6_{<d}$ of $\Phi_d$ the orthogonal sum norm. Let $m_\mu(h)$ be a modulus of continuity of $\mathcal P_\mu$ on $\Omega$.

\begin{lemma}\label{lem:c2}
If a source vector of norm one has the image $\mathcal F=\sum_ja_jg_j=\mathfrak c_nx^n+O(x^{n+1})$, then $\log|\mathfrak c_n|\le dW_i+2\pi ny_i+d\,m_\mu(h)+C_h$ for every $i$, with $C_h$ independent of $d$, $n$ and the vector.
\end{lemma}

\begin{proof}
Let $z$ be a point of the image curves. For $|\zeta-z|\le h$ we have $e^{-2d\mathcal P_\mu(\zeta)}\ge e^{-2d(\mathcal P_\mu(z)+m_\mu(h))}$, so the submean inequality for $|a_j|^2$ on the disc $\mathbb D(z,h)\subset\Omega$ gives
\begin{equation*}
|a_j(z)|\le(\pi h^2)^{-1/2}e^{d(\mathcal P_\mu(z)+m_\mu(h))}\|a_j\|_d\ .
\end{equation*}
On $|q|=r_i$, $|\Pi(q)\mathcal F(x(q))|\le\sum_j|a_j(x(q))|\,|\Pi g_j(x(q))|$, and the Cauchy--Schwarz inequality over the six blocks, with fixed bounds for the $\Pi g_j$ on $|q|\le\max_ir_i$, gives $\log|\Pi\mathcal F(x(q))|\le d\mathcal P_\mu(x(q))+d\,m_\mu(h)+C_h$. The function $\psi=\Pi(q)\mathcal F(x(q))/q^n$ is holomorphic on $|q|\le r_i$, and $\psi(0)=\mathfrak c_n$ because near $q=0$ the function $\mathcal F(x(q))$ is the composition of series and $x(q)=q+O(q^2)$. Jensen's inequality $\log|\psi(0)|\le\int\log|\psi|\,\dd\vartheta/2\pi$ on $|q|=r_i$ gives the claim, since $\int\mathcal P_\mu(x(q))\,\dd\vartheta/2\pi=W_i$ and $\log r_i=-2\pi y_i$.
\end{proof}

Let $\mathcal G_d$ be the Gram matrix of $1,z,\dots,z^{d-1}$ for $\|\cdot\|_d$. Expanding the Vandermonde determinant $\det[z_k^b]_{k,b}$ and its complex conjugate and integrating term by term, which is Andr\'eief's identity, gives
\begin{equation}\label{eq:gramheine}
\det \mathcal G_d=\frac1{d!}\int_{\Omega^d}\prod_{i<j}|z_i-z_j|^2e^{-2d\sum_i\mathcal P_\mu(z_i)}\prod_i\dd A(z_i)\ .
\end{equation}

\begin{lemma}[energy inequality]\label{lem:energy}
For every Borel probability measure $\nu$ on $\Omega$, $I(\nu)-2\int\mathcal P_\mu\,\dd\nu\le-I$, where $I(\nu)=\iint\log|z-w|\,\dd\nu\,\dd\nu\in[-\infty,\infty)$.
\end{lemma}

\begin{proof}
This is \cite[Lemma~I.1.8]{ST97}, in our sign convention; we include a proof. If $I(\nu)=-\infty$ there is nothing to prove, $\mathcal P_\mu$ being bounded on $\Omega$. Otherwise $\nu$ has no atoms, and $\log|z-w|$ is absolutely integrable against $\nu\otimes\nu$, $\nu\otimes\mu$ and $\mu\otimes\mu$, because it is bounded above on $\Omega-\Omega$ and the three integrals are $I(\nu)$, $\int\mathcal P_\mu\dd\nu$ and $I$. Put $\sigma=\nu-\mu$, of total mass $0$; the claim is $I(\sigma)\le0$. Let $\sigma_\varepsilon=\sigma*\chi_\varepsilon$ with a smooth radial probability density $\chi_\varepsilon$ supported in $|z|\le\varepsilon$, and let $\mathcal P_{\sigma_\varepsilon}=\log|\cdot|*\sigma_\varepsilon$ be its potential. By the zero total mass, $\mathcal P_{\sigma_\varepsilon}=O(|z|^{-1})$ and $\nabla \mathcal P_{\sigma_\varepsilon}=O(|z|^{-2})$ at infinity, and $\Delta \mathcal P_{\sigma_\varepsilon}=2\pi\sigma_\varepsilon$, so integration by parts gives $I(\sigma_\varepsilon)=\int \mathcal P_{\sigma_\varepsilon}\,\dd\sigma_\varepsilon=-\frac1{2\pi}\int_\C|\nabla \mathcal P_{\sigma_\varepsilon}|^2\dd A\le0$. Now $I(\sigma_\varepsilon)=\iint k_\varepsilon(z-w)\,\dd\sigma\,\dd\sigma$ with $k_\varepsilon=\log|\cdot|*\chi_\varepsilon*\chi_\varepsilon$. By the submean property of the subharmonic function $\log|\cdot|$ and the radiality of $\chi_\varepsilon$, $\log|\cdot|\le k_\varepsilon$; moreover $k_\varepsilon$ is bounded above on compact sets, and $k_\varepsilon\to\log|\cdot|$ off $0$. Dominated convergence in the three pairings separately gives $I(\sigma)=\lim_{\varepsilon\to0}I(\sigma_\varepsilon)\le0$.
\end{proof}

\begin{lemma}\label{lem:c3}
$\limsup_{d\to\infty}d^{-2}\log\det \mathcal G_d\le-I$.
\end{lemma}

\begin{proof}
Let $k_T=\max(\log|z-w|,-T)$, continuous and bounded on $\Omega\times\Omega$. For $z_1,\dots,z_d\in\Omega$ with empirical measure $\nu_d=d^{-1}\sum_i\delta_{z_i}$ we have $2\sum_{i<j}\log|z_i-z_j|\le d^2\iint k_T\,\dd\nu_d\,\dd\nu_d+dT$, since the diagonal contributes $-dT$. Choose configurations $z^{(d)}$ at which the integrand in \eqref{eq:gramheine} is within a factor $2$ of its supremum. Then $\log\det \mathcal G_d\le d^2\qty(\iint k_T\dd\nu_d\dd\nu_d-2\int\mathcal P_\mu\dd\nu_d)+dT+O(d)$ for every $T$, with $\nu_d$ independent of $T$. Along a subsequence realizing the $\limsup$, $\nu_d\to\nu$ weakly, and for fixed $T$ the bracket converges to $\iint k_T\dd\nu\dd\nu-2\int\mathcal P_\mu\dd\nu$. Letting $T\to\infty$, by monotone convergence, gives at most $I(\nu)-2\int\mathcal P_\mu\dd\nu$, which is at most $-I$ by Lemma~\ref{lem:energy}.
\end{proof}

\begin{proposition}\label{prop:c4}
$\limsup_{d\to\infty}d^{-2}\log|\cD_d(c)|\le U(\mu):=\int_0^6\min_i\qty(W_i+2\pi y_ik)\,\dd k-3I$.
\end{proposition}

\begin{proof}
Choose an orthonormal basis $v_0,\dots,v_{6d-1}$ of the source adapted to the order filtration of Section~\ref{sec:secdet}: $v_j$ is a unit vector in $\{\ord_0\Phi_d\ge\lambda_j\}$ orthogonal to $\{\ord_0\Phi_d>\lambda_j\}$, so that $\Phi_d(v_j)=\mathfrak c_jx^{\lambda_j}+O(x^{\lambda_j+1})$ with $\mathfrak c_j\ne0$. The matrix $([x^{\lambda_{j'}}]\Phi_d(v_j))_{j',j}$ is triangular with diagonal $(\mathfrak c_j)$, and by Lemma~\ref{lem:c2}, $\log|\mathfrak c_j|\le\min_i(dW_i+2\pi y_i\lambda_j)+d\,m_\mu(h)+C_h$. The matrix of the monomials on the same rows has determinant $\pm\cD_d(c)$, and the change of basis from the monomials to $(v_j)$ has absolute determinant $(\det \mathcal G_d)^{-3}$, there being six orthogonal blocks. Hence
\begin{equation}\label{eq:c4}
\log|\cD_d(c)|\le3\log\det \mathcal G_d+\sum_{j=0}^{6d-1}\min_i\qty(dW_i+2\pi y_i\lambda_j)+6d^2m_\mu(h)+O_h(d)\ .
\end{equation}
Replacing $\lambda_j$ by $j$ costs $O(d)$ by \eqref{eq:jumps}, and the sum is then a Riemann sum equal to $d^2\int_0^6\min_i(W_i+2\pi y_ik)\,\dd k+O(d)$. Take $\limsup_d$ with Lemma~\ref{lem:c3}, and then let $h\to0$.
\end{proof}

To evaluate $U(\mu)$ we need the mutual energies $\mathcal I(y,y')=\iint\log|z-w|\,\dd\mu_y(z)\,\dd\mu_{y'}(w)$ of the circle measures with radii $e^{-2\pi y}$ and $e^{-2\pi y'}$.

\begin{lemma}\label{lem:kernel}
With Euler's function $\varphi$,
\begin{equation}\label{eq:kernel}
\mathcal I(y,y')=-2\pi\min(y,y')+\sum_{\substack{j\ge1\\9j^2yy'<1}}\frac{4\pi\varphi(3j)}{9j^2}\qty(\arccos\qty(3j\sqrt{yy'})-3j\sqrt{yy'}\sqrt{1-9j^2yy'})\ .
\end{equation}
\end{lemma}

\begin{proof}
Fix $\tau=u_0+iy$ outside the elliptic orbit and $w=x(e^{2\pi i\tau})$. Since $x$ identifies $X_0(3)$ with $\mathbb P^1$, the zeros of $x(q)-w$ in $|q|<1$ are the points $e^{2\pi i\gamma\tau}$ with $\gamma\in\Gamma_\infty\backslash\Gamma_0(3)$, where $\Gamma_\infty$ is the stabilizer of $\infty$ in $\Gamma_0(3)$; they are all simple. The cosets are the identity and the cosets with a primitive bottom row $(3j,m)$, $j\ge1$, and $\mathrm{Im}\,\gamma\tau=y/((3ju_0+m)^2+9j^2y^2)$. Jensen's formula on $|q|=e^{-2\pi y'}$ gives $\int\log|x(q)-w|\,\dd\vartheta/2\pi=\log|w|+\sum_\gamma2\pi(\mathrm{Im}\,\gamma\tau-y')^+$. We average over $u_0\in[0,1)$. The mean of $\log|w|$ is $-2\pi y$, because $\log|x(q)/q|$ is harmonic with value $0$ at $q=0$, and together with the identity coset this gives $-2\pi\min(y,y')$. For fixed $j$, the substitution $v=3ju_0+m$, and the fact that almost every $v\in\R$ lies in exactly $\varphi(3j)$ of the intervals $[m,m+3j]$ with $\gcd(3j,m)=1$, give the contribution
\begin{equation}\label{eq:kernelstep}
\frac{\varphi(3j)}{3j}\int_\R\qty(\frac{2\pi y}{v^2+9j^2y^2}-2\pi y')^+\dd v\ .
\end{equation}
The integrand is supported on $|v|\le v_*=\sqrt{y/y'-9j^2y^2}$, which is empty unless $9j^2yy'<1$. With $a=3j\sqrt{yy'}$, $\int_{-v_*}^{v_*}\frac{2\pi y\,\dd v}{v^2+9j^2y^2}=\frac{4\pi}{3j}\arctan\frac{\sqrt{1-a^2}}a=\frac{4\pi}{3j}\arccos a$ and $4\pi y'v_*=\frac{4\pi}{3j}a\sqrt{1-a^2}$. Only finitely many $j$ contribute, and the elliptic orbit has measure zero.
\end{proof}

\textit{The bound for $U$.} We take the $34$ circles and weights of Table~\ref{tab:circles}, with rational $y_i>0$ and $\sum_i\varpi_i=1$. Then $W_i=\sum_j\varpi_j\mathcal I(y_i,y_j)$ and $I=\sum_i\varpi_iW_i$. Assign to the circle $i$ the segment $[\alpha_i,\beta_i]$ of Table~\ref{tab:circles}. These segments tile $[0,6]$, and on each of them $\min_{i'}(W_{i'}+2\pi y_{i'}k)\le W_i+2\pi y_ik$, so
\begin{equation}\label{eq:jdef}
U(\mu)\le J-3I\ ,\qquad J=\sum_i\qty((\beta_i-\alpha_i)W_i+\pi y_i\qty(\beta_i^2-\alpha_i^2))\ .
\end{equation}
We evaluate $I$ and $J$ from \eqref{eq:kernel} in ball arithmetic at a precision of $400$ bits \cite{Jo17}, deciding every condition $9j^2y_iy_j<1$ in exact rational arithmetic. This gives $I=2.3300529587\ldots$, $J=23.5941770943\ldots$, and
\begin{equation}\label{eq:ucert}
U:=J-3I\le16.6040183\ ;
\end{equation}
we quote $I$ and $J$ so that a reader who evaluates \eqref{eq:kernel} on Table~\ref{tab:circles} can compare. With \eqref{eq:ucert}, $\frac{177}2\log3-\frac{13289}{168}-U>1.52$, as used in the proof at the beginning of this appendix.

\begin{table}[ht]
\centering\footnotesize
\renewcommand{\arraystretch}{1.05}
\begin{tabular}{|r|l|l|l|}
\hline\hline
$i$ & $y_i$ & $\varpi_i$ & segment $[\alpha_i,\beta_i]$ \\
\hline
1 & $4811/100000$ & $15863/166667$ & $[1357267/250000,\,6]$ \\
2 & $2707/50000$ & $34665/333334$ & $[4804939/1000000,\,1357267/250000]$ \\
3 & $6093/100000$ & $46900/500001$ & $[2121097/500000,\,4804939/1000000]$ \\
4 & $857/12500$ & $82975/1000002$ & $[468043/125000,\,2121097/500000]$ \\
5 & $1929/25000$ & $12424/166667$ & $[824241/250000,\,468043/125000]$ \\
6 & $8683/100000$ & $66269/1000002$ & $[2899411/1000000,\,824241/250000]$ \\
7 & $9771/100000$ & $28582/500001$ & $[127821/50000,\,2899411/1000000]$ \\
8 & $2749/25000$ & $51181/1000002$ & $[2249323/1000000,\,127821/50000]$ \\
9 & $6187/50000$ & $46379/1000002$ & $[1971069/1000000,\,2249323/1000000]$ \\
10 & $557/4000$ & $20644/500001$ & $[215419/125000,\,1971069/1000000]$ \\
11 & $1567/10000$ & $17935/500001$ & $[301631/200000,\,215419/125000]$ \\
12 & $8817/50000$ & $10775/333334$ & $[657103/500000,\,301631/200000]$ \\
13 & $4961/25000$ & $4847/166667$ & $[1139719/1000000,\,657103/500000]$ \\
14 & $22331/100000$ & $4253/166667$ & $[197323/200000,\,1139719/1000000]$ \\
15 & $2513/10000$ & $21353/1000002$ & $[85849/100000,\,197323/200000]$ \\
16 & $707/2500$ & $8519/500001$ & $[756249/1000000,\,85849/100000]$ \\
17 & $1273/4000$ & $15553/1000002$ & $[26517/40000,\,756249/1000000]$ \\
18 & $35813/100000$ & $2381/166667$ & $[577227/1000000,\,26517/40000]$ \\
19 & $20151/50000$ & $13103/1000002$ & $[498597/1000000,\,577227/1000000]$ \\
20 & $45353/100000$ & $1989/166667$ & $[427003/1000000,\,498597/1000000]$ \\
21 & $25519/50000$ & $3529/333334$ & $[14539/40000,\,427003/1000000]$ \\
22 & $28717/50000$ & $4502/500001$ & $[154727/500000,\,14539/40000]$ \\
23 & $64633/100000$ & $1247/166667$ & $[3307/12500,\,154727/500000]$ \\
24 & $36367/50000$ & $6887/1000002$ & $[111617/500000,\,3307/12500]$ \\
25 & $1637/2000$ & $6319/1000002$ & $[185321/1000000,\,111617/500000]$ \\
26 & $92109/100000$ & $2872/500001$ & $[75431/500000,\,185321/1000000]$ \\
27 & $103653/100000$ & $1729/333334$ & $[59869/500000,\,75431/500000]$ \\
28 & $23329/20000$ & $4669/1000002$ & $[91727/1000000,\,59869/500000]$ \\
29 & $4102/3125$ & $4141/1000002$ & $[66877/1000000,\,91727/1000000]$ \\
30 & $36929/25000$ & $3583/1000002$ & $[9077/200000,\,66877/1000000]$ \\
31 & $166231/100000$ & $501/166667$ & $[13673/500000,\,9077/200000]$ \\
32 & $37413/20000$ & $1189/500001$ & $[13079/1000000,\,13673/500000]$ \\
33 & $210511/100000$ & $267/166667$ & $[3471/1000000,\,13079/1000000]$ \\
34 & $7403/3125$ & $289/500001$ & $[0,\,3471/1000000]$ \\
\hline\hline
\end{tabular}
\medskip
\caption{The $34$-circle measure of \eqref{eq:ucert}. Circle $i$ has radius $r_i=e^{-2\pi y_i}$ in the $q$-disc, weight $\varpi_i$, and the assigned segment $[\alpha_i,\beta_i]\subset[0,6]$ in \eqref{eq:jdef}; the segments tile $[0,6]$, and $\sum_i\varpi_i=1$.}
\label{tab:circles}
\end{table}

\clearpage
\begin{thebibliography}{99}

\bibitem{Ap79} R.~Ap\'ery, Irrationalit\'e de $\zeta(2)$ et $\zeta(3)$, Ast\'erisque 61 (1979), 11--13, \url{https://www.numdam.org/item/AST_1979__61__11_0/}.

\bibitem{BR01} K.~Ball and T.~Rivoal, Irrationalit\'e d'une infinit\'e de valeurs de la fonction z\^eta aux entiers impairs, Invent.\ Math.\ 146 (2001), 193--207, \url{https://doi.org/10.1007/s002220100168}.

\bibitem{Ca05} F.~Calegari, Irrationality of certain $p$-adic periods for small $p$, Int.\ Math.\ Res.\ Not.\ 2005, no.~20, 1235--1249, \url{https://doi.org/10.1155/IMRN.2005.1235}; arXiv:\href{https://arxiv.org/abs/math/0408214}{math/0408214}.

\bibitem{Be08} F.~Beukers, Irrationality of some $p$-adic $L$-values, Acta Math.\ Sin.\ (Engl.\ Ser.) 24 (2008), 663--686, \url{https://doi.org/10.1007/s10114-007-1029-2}; arXiv:\href{https://arxiv.org/abs/math/0603277}{math/0603277}.

\bibitem{CDT26} F.~Calegari, V.~Dimitrov and Y.~Tang, Arithmetic holonomy bounds and effective Diophantine approximation, in: Proceedings of the International Congress of Mathematicians 2026, Vol.~3, SIAM (2026), 356--376, \url{https://doi.org/10.1137/25M1806776}; arXiv:\href{https://arxiv.org/abs/2510.04156}{2510.04156}.

\bibitem{CDT24} F.~Calegari, V.~Dimitrov and Y.~Tang, The linear independence of $1$, $\zeta(2)$, and $L(2,\chi_{-3})$, preprint (2024), arXiv:\href{https://arxiv.org/abs/2408.15403}{2408.15403}.

\bibitem{LSZ26} L.~Lai, J.~Sprang and W.~Zudilin, A note on the irrationality of $\zeta_2(5)$, Int.\ Math.\ Res.\ Not.\ IMRN 2026, no.~16, rnag180, \url{https://doi.org/10.1093/imrn/rnag180}; arXiv:\href{https://arxiv.org/abs/2505.05005}{2505.05005}.

\bibitem{La25} L.~Lai, On the irrationality of certain $2$-adic zeta values, Int.\ J.\ Number Theory 21 (2025), 207--235, \url{https://doi.org/10.1142/S1793042125500113}; arXiv:\href{https://arxiv.org/abs/2304.00816}{2304.00816}.

\bibitem{LS26} L.~Lai and J.~Sprang, Many $p$-adic odd zeta values are irrational, Michigan Math.\ J., advance publication (2026), \url{https://doi.org/10.1307/mmj/20236490}; arXiv:\href{https://arxiv.org/abs/2306.10393}{2306.10393}.

\bibitem{LLS25} L.~Lai, C.~Lupu and J.~Sprang, On the irrationality of certain $p$-adic zeta values, Res.\ Math.\ Sci.\ 12 (2025), no.~4, Paper No.~77, \url{https://doi.org/10.1007/s40687-025-00559-x}; arXiv:\href{https://arxiv.org/abs/2505.23088}{2505.23088}.

\bibitem{Lo26} C.~D.~Long, Hybrid arithmetic holonomy and twenty-two individual $p$-adic zeta values, preprint dated 24 August 2026, \url{https://github.com/octonion/p-adic-zeta-irrationality}, accessed 19 October 2026.

\bibitem{Lo26b} C.~D.~Long, Multiweight arithmetic holonomy and density-one irrationality of $p$-adic zeta values, preprint dated 26 August 2026, \url{https://github.com/octonion/p-adic-zeta-density}, accessed 19 October 2026.

\bibitem{K26} I.~Kilani, $\zeta_2(7)$ is irrational via Hankel determinants, preprint dated 25 September 2026 and signed I.~K., Zenodo, version~2 of 30 September 2026, \url{https://doi.org/10.5281/zenodo.23058762} (all versions: \url{https://doi.org/10.5281/zenodo.22961248}).

\bibitem{Ki26} S.~Kintali, Irrationality of the $5$-adic zeta value $\zeta_5(3)$, preprint dated 15 August 2026, \url{https://shivakintali.github.io/papers/Zeta-5-3.pdf}, accessed 4 October 2026.

\bibitem{Yi26} T.~Yi, A determinant argument for the irrationality of $\zeta_7(3)$. Proposed unconditional proof, preprint dated 15 September 2026, \url{https://github.com/TaifengYi/zeta7-irrationality}, accessed 4 October 2026.

\bibitem{Kil26} I.~Kilani, On the irrationality of $\zeta_3(5)$, preprint dated October 2026, Zenodo, 7 October 2026, \url{https://doi.org/10.5281/zenodo.23203596}; Lean~4 formalization at \url{https://github.com/Ijtihed/zeta3-5-lean}, accessed 19 October 2026.

\bibitem{Fa26} A.~Fauzan, $\zeta(5)$ is irrational, preprint dated 17 September 2026, Zenodo, \url{https://doi.org/10.5281/zenodo.22826419}.

\bibitem{Jo17} F.~Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans.\ Comput.\ 66 (2017), 1281--1292, \url{https://doi.org/10.1109/TC.2017.2690633}; arXiv:\href{https://arxiv.org/abs/1611.02831}{1611.02831}.

\bibitem{Wa97} L.~C.~Washington, Introduction to Cyclotomic Fields, 2nd ed., Grad.\ Texts in Math.\ 83, Springer, New York, 1997, \url{https://doi.org/10.1007/978-1-4612-1934-7}.

\bibitem{Vo72} A.~Volkenborn, Ein $p$-adisches Integral und seine Anwendungen~I, Manuscripta Math.\ 7 (1972), 341--373, \url{https://doi.org/10.1007/BF01644073}.

\bibitem{Ro00} A.~M.~Robert, A Course in $p$-adic Analysis, Grad.\ Texts in Math.\ 198, Springer, New York, 2000, \url{https://doi.org/10.1007/978-1-4757-3254-2}.

\bibitem{Bh97} M.~Bhargava, $P$-orderings and polynomial functions on arbitrary subsets of Dedekind rings, J.\ Reine Angew.\ Math.\ 490 (1997), 101--127, \url{https://doi.org/10.1515/crll.1997.490.101}.

\bibitem{Sz75} G.~Szeg\H{o}, Orthogonal Polynomials, 4th ed., Amer.\ Math.\ Soc.\ Colloq.\ Publ.\ 23, Amer.\ Math.\ Soc., Providence, RI, 1975, \url{https://doi.org/10.1090/coll/023}.

\bibitem{Ha96} J.~Hadamard, Sur la distribution des z\'eros de la fonction $\zeta(s)$ et ses cons\'equences arithm\'etiques, Bull.\ Soc.\ Math.\ France 24 (1896), 199--220, \url{https://doi.org/10.24033/bsmf.545}.

\bibitem{dlVP96} Ch.-J.~de la Vall\'ee Poussin, Recherches analytiques sur la th\'eorie des nombres premiers, Ann.\ Soc.\ Sci.\ Bruxelles 20 (1896), part~2, 183--256.

\bibitem{RS62} J.~B.~Rosser and L.~Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J.\ Math.\ 6 (1962), 64--94, \url{https://doi.org/10.1215/ijm/1255631807}.

\bibitem{Ka73} N.~M.~Katz, $p$-adic properties of modular schemes and modular forms, in: Modular Functions of One Variable III (Antwerp, 1972), Lecture Notes in Math.\ 350, Springer, Berlin, 1973, 69--190, \url{https://doi.org/10.1007/978-3-540-37802-0_3}.

\bibitem{Co97} R.~F.~Coleman, $p$-adic Banach spaces and families of modular forms, Invent.\ Math.\ 127 (1997), 417--479, \url{https://doi.org/10.1007/s002220050127}.

\bibitem{Bu03} K.~Buzzard, Analytic continuation of overconvergent eigenforms, J.\ Amer.\ Math.\ Soc.\ 16 (2003), 29--55, \url{https://doi.org/10.1090/S0894-0347-02-00405-8}.

\bibitem{DS05} F.~Diamond and J.~Shurman, A First Course in Modular Forms, Grad.\ Texts in Math.\ 228, Springer, New York, 2005, \url{https://doi.org/10.1007/978-0-387-27226-9}.

\bibitem{Ap90} T.~M.~Apostol, Modular Functions and Dirichlet Series in Number Theory, 2nd ed., Grad.\ Texts in Math.\ 41, Springer, New York, 1990, \url{https://doi.org/10.1007/978-1-4612-0999-7}.

\bibitem{Con06} B.~Conrad, Modular curves and rigid-analytic spaces, Pure Appl.\ Math.\ Q.\ 2 (2006), 29--110, \url{https://doi.org/10.4310/PAMQ.2006.v2.n1.a3}.

\bibitem{Si09} J.~H.~Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Grad.\ Texts in Math.\ 106, Springer, New York, 2009, \url{https://doi.org/10.1007/978-0-387-09494-6}.

\bibitem{CM98} R.~Coleman and B.~Mazur, The eigencurve, in: Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), London Math.\ Soc.\ Lecture Note Ser.\ 254, Cambridge Univ.\ Press, Cambridge, 1998, 1--114, \url{https://doi.org/10.1017/CBO9780511662010.003}.

\bibitem{ST97} E.~B.~Saff and V.~Totik, Logarithmic Potentials with External Fields, Grundlehren Math.\ Wiss.\ 316, Springer, Berlin, 1997, \url{https://doi.org/10.1007/978-3-662-03329-6}.

\end{thebibliography}

\end{document}