\section{Computations}\label{app:computations} This appendix lists the computations on which the proofs of Parts~I--III depend, with what each computes and its output: items C1--C3 concern Part~I, item C4 Part~II, which also uses C2, and item C5 Part~III. They are finite exact computations in $\Z$, $\Z[\omega]$, $\Z[\zeta_9]$, number fields, finite fields and the rings $\Z/7^N$, except items C4 and C5, which are evaluations of integrals with certified error bounds, in interval and ball arithmetic \cite{Johansson17}. The programs are written in PARI/GP \cite{PARI} (version 2.17.3), Python 3.12 (with python-flint 0.8.0 on FLINT 3.3.0 for ball arithmetic, mpmath 1.4.1 and SymPy 1.14.0), GAP \cite{GAP} 4.15.1 and SageMath 10.9 (used only for cross-checks), with one program in C. For each of the items C1--C5 the record \cite{Companion} also contains a second, separately written program (for C1, at $\lp$ and for one $m'$ in each class), which gives the same results. On a laptop each program for items C1--C5 runs in a minute or less, most in a few seconds, except the program of item C1 that checks \cref{lem:trace7} for $k=1$ and $s=2$ by counting the fixed points of $F\circ g$ directly, which takes about three minutes. The computations for Part~IV are items~S12--S24 of the Supplement; the subsection after item~C5 summarizes the search of items~S14--S16 and the $7$-adic logarithms of items~S22 and~S24. We write $\omega$ for the image of $\omega\in K$ in the coefficients, and identify $\OO/\lp$ and $\OO/\lpb$ with $\F_7$ by $\omega\mapsto2$ and $\omega\mapsto4$. \subsection*{C1.\ Traces of \texorpdfstring{$\VV$}{V} at the primes above 7} This is the input of \cref{prop:frob7}. For $\q\in\{\lp,\lpb\}$, each $m'\in(\Z/9)^\times$, $k\in\{1,2,3\}$ and every $s\in\F_{7^k}\smallsetminus\{0,1\}$, the sums of \cref{lem:trace7} were evaluated exactly, as integer combinations of powers of a primitive ninth root of unity. The character $\vartheta$ is the power map $z\mapsto z^{38}$ into $\mu_9(\F_{343})$, identified with the ninth roots of unity of the coefficients so that the cube roots of unity correspond to the reductions of $1,\omega,\omega^2$. Every value lies in $\Z[\omega]$. The results depend only on the class of $m'$ modulo $3$: \begin{center}\small \begin{tabular}{@{}lll@{}} \toprule & class I & class II \\ \midrule traces at $s=2$ ($k=1,2,3$) & $6-3\omega,\ -99-87\omega,\ -267-486\omega$ & $9+3\omega,\ -12+87\omega,\ 219+486\omega$\\ $P(T)$ at $s=2$ & $P^+(T)$ & $P^-(T)$\\ power sums on $H^1_c$ & $1+2\omega,\ 95+94\omega,\ -285$ & $-1-2\omega,\ 1-94\omega,\ -285$\\ $\Omega(T)$ & $(T-1)(T^2-2\omega T+49\omega^2)$ & $(T-1)(T^2-2\omega^2T+49\omega)$\\ \bottomrule \end{tabular} \end{center} At $\lp$, class I is $m'\equiv2\pmod 3$; at $\lpb$, it is $m'\equiv1\pmod 3$. \subsection*{C2.\ The irreducibility test} This is Step~5 of the proof of \cref{thm:irreducible}; it is used again in the proof of \cref{prop:member-irreducible}. With $\varpi=2-\omega$, for each assignment $(P_\lp,P_\lpb)\in\{(P^+,P^-),(P^-,P^+)\}$, each type $(k,k')\in\mathcal T$ and each $(\zeta_1,\zeta_2)\in\mu_3^2$, the elements $P_\lp(\zeta_1\varpi^k\bar\varpi^{k'})$ and $P_\lpb(\zeta_2\bar\varpi^k\varpi^{k'})$ of $\OO$ are nonzero, and the ideal they generate was factored. The rational primes below its prime factors, taken together over all $(\zeta_1,\zeta_2)$, are: \begin{center} \begin{tabular}{@{}lccccc@{}} \toprule type & $(0,0)$ & $(1,1)$ & $(2,2)$ & $(0,2)$ & $(2,0)$\\ \midrule $(P^+,P^-)$ & $2,3$ & $2,3,7$ & $2,3,7$ & $2,3,13$ & $2,3,7$\\ $(P^-,P^+)$ & $2,3$ & $2,3,7$ & $2,3,7$ & $2,3,7$ & $2,3,13$\\ \bottomrule \end{tabular} \end{center} No prime $p\ge17$ occurs. \subsection*{C3.\ Ray class groups} This is used in the proof of \cref{prop:limit-trivial}. The ray class groups $\Cl_{\p^k}(K)$ for $k=1,\dots,8$ have orders $1,1,3,9,27,81,243,729$; $\Cl_{\p^4}(K)\isom(\Z/3)^2$. The classes of $\lp=(2-\omega)$ and $\lpb=(3+\omega)$ generate $\Cl_{\p^4}(K)$. Equivalently, the images of $2-\omega$ and $3+\omega$ generate $(\OO/\p^4)^\times/\mu_6$, a group of order $9$. \subsection*{C4.\ Lower bounds for root discriminants} This is the input of \cref{prop:odlyzko}. The right side of \eqref{eq:rdbound}, with $r_1=0$ and the test function $\Phi_0(x/b)/\cosh(x/2)$ of \cref{sec:odlyzko}, was evaluated with certified error bounds in the following cases. The third column gives the exponential of the value found, truncated, and the last the constant with which it is compared in the text. \begin{center}\small \begin{tabular}{@{}rlll@{}} \toprule $n$ & $b$ & bound for $\rd(L)$ & constant\\ \midrule $18$ & $23/4$ & $9.27091\ldots$ & $3^{1/2}\cdot5=8.66025\ldots$\\ $120$ & $13$ & $16.98861\ldots$ & $3^{1/2}\cdot5^{7/5}=16.48612\ldots$\\ $2002$ & $36$ & $21.41368\ldots$ & $3^{1/2}\cdot5^{3/2}=19.36491\ldots$\\ \bottomrule \end{tabular} \end{center} The values of $b$ are, to within $\frac14$, those that maximize the bounds. \subsection*{C5.\ Root discriminants} This is the input of \cref{prop:three-odlyzko}, with the constants of \cref{prop:three-rd}. The right side of \eqref{eq:rdbound}, with $r_1=0$ and the test function $\Phi_0(x/b)/\cosh(x/2)$, was evaluated in two ways, each with certified error bounds. The first uses ball arithmetic with Arb \cite{Johansson17}, at $160$ bits. With $\delta=10^{-8}$, the integral over $[\delta,b]$ is computed by Arb's rigorous numerical integration, and the integral over $[0,\delta]$ is bounded by $0\le\int_0^\delta\big(1-\Phi_0(x/b)/\cosh(x/2)\big)\,\frac{dx}{2\sinh(x/2)}\le\big(\frac{\pi+\pi^2}{2b^2}+\frac18\big)\frac{\delta^2}2$, an elementary consequence of $\Phi_0(0)=1$, $\Phi_0'(0)=0$, $0\le\Phi_0\le1$ and $|\Phi_0''|\le\pi+\pi^2$; the integral over $[b,\infty)$ is $-\log\tanh(b/4)$ in closed form, and $I_2=4b/\pi^2$. The second uses interval arithmetic with Taylor models of degree $12$ on a dyadic subdivision, at $200$ bits. The results, truncated: \begin{center}\small \begin{tabular}{@{}rllll@{}} \toprule $n$ & $b$ & bound for $\rd(L)$ & constant & margin in $\log\rd$\\ \midrule $1378$ & $63/2$ & $21.1515190\ldots$ & $3^{449/162}=21.0083345\ldots$ & $6.79\cdot10^{-3}$\\ $674$ & $25$ & $20.4476673\ldots$ & $T_3=3^{437/162}=19.3664184\ldots$ & $5.43\cdot10^{-2}$\\ $118$ & $12$ & $16.9179477\ldots$ & $T_2=3^{137/54}=16.2358224\ldots$ & $4.11\cdot10^{-2}$\\ $80$ & $10$ & $15.6281490\ldots$ & $3^{5/2}=15.5884572\ldots$ & $2.54\cdot10^{-3}$\\ $24$ & $6$ & $10.5939560\ldots$ & $3^{11/6}=7.4941485\ldots$ & $0.346$\\ $20$ & $6$ & $9.7691231\ldots$ & $T_1=3^{37/18}=9.5664156\ldots$ & $2.09\cdot10^{-2}$\\ $4$ & $2$ & $3.2422670\ldots$ & $3^{1/2}=1.7320508\ldots$ & $0.626$\\ \bottomrule \end{tabular} \end{center} The last column is a lower bound for the logarithm of the bound minus that of the constant. The smallest margin, $2.54\cdot10^{-3}$ at $n=80$, is more than $10^5$ times the width of the enclosures: both evaluations enclose each bound in an interval of width below $4\cdot10^{-9}$ in $\log\rd$. \subsection*{Part IV: the search and the \texorpdfstring{$7$}{7}-adic logarithms} \emph{Items S14--S16.} For \cref{thm:3zp-unconditional}~(b) and~(c) the program takes each prime $p$ in a range and runs through $n=2,4,6,\dots$ up to a bound, keeping the $n$ for which $q=np+1$ is prime. For such a $q$ it computes $a_q(W)$ and tests~(A) of \cref{prop:3zp-aux}; if (A) holds, it runs through the $n$ elements $\zeta$ of $9^{1-k}\mu_n(\F_q)$, tests whether $d_2=(12\zeta-1)/3$ is a square in $\F_q$, and if it is, computes $a_q(E_\zeta)$ for one square root $d$ and compares $a_q(E_\zeta)^2$ with $a_q(W)^2$ modulo $p$. It records the least $n$ for which (A) and (B$_k$) hold. The traces $a_q$ are computed by PARI's \texttt{ellap}, and $\mu_n(\F_q)$ is generated by $g^p$ for a primitive root $g$ modulo $q$. For $k=1$ the $5\,761\,453$ primes $5\le p<10^8$ were treated with $n$ up to $600$, $2000$ and $4000$ for $p\le10^5$, $10^5