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\title{Subset Sum in time $O(2^{0.4877n})$: a complete proof}
\author{}
\date{12 October 2026}
\begin{document}
\maketitle
This note proves that Subset Sum can be solved in time $O(2^{0.4877n})$, in the model and with the guarantees of
OpenAI result \#138, ``Subset Sum in Time $O(2^{0.49n})$'' (OpenAI, 4 October 2026). We call that work \emph{the paper}.
References such as ``Lemma 2.3'' or ``(4.11)'' point to the paper. Appendix B restates, without proof, every result,
definition and equation of the paper that this note uses, under the paper's numbering. We rely on the paper's proofs of
these results and do not repeat them. With Appendix B, this note can be read without the paper.

The algorithm keeps the paper's framework: isolation, a capped preliminary search, a modular checksum, and the
subtraction of aliases. We replace two of its parts. A new alias step (Theorem A) works with a smaller modulus, and a
family of 43 checksum filters (Theorem B) uses that smaller modulus. Section 1.2 lists the numerical checks.

\textbf{Contents.}

\begin{enumerate}
\item The result
\item Notation, parameters and facts from the paper
\item An averaged small-layer lemma
\item The preliminary search
\item Alias phase sums (Theorem A)
\item The checksum with a filter family (Theorem B)
\item Proof of Theorem 1
\item The certificates
\end{enumerate}

Appendices:

\begin{itemize}
\item A. The 43 filters.
\item B. Results used from the paper.
\item C. The multipliers of the 430 moment certificates.
\item D. The two verification scripts.
\end{itemize}

\medskip\noindent\rule{\linewidth}{0.4pt}\medskip

\section*{1. The result}\addcontentsline{toc}{section}{1. The result}

\textbf{Theorem 1.} There is a uniform randomized classical algorithm for Subset Sum with the following properties in the
word-RAM model of the paper (word length (1.1)). For every input with $n \ge 2$, it halts on every execution and
answers correctly with probability at least $2/3$. For every fixed positive integer $c$, there are constants $C_c > 0$
and $N_c \ge 2$ such that every input with $n \ge N_c$ and $b \le n^c$ is processed in at most $C_c\,2^{0.4877n}$
word operations, on every outcome of the random choices. The program does not depend on $c$.

Here $b$ is the largest bit length of the input integers and the target, as in the paper. The intermediate bound is
\[2^{(\tau_0 + 20\epsilon)n + O(n^{0.7})}\,\mathrm{poly}(n+b+2),\qquad \tau_0 = 0.48769,\quad \epsilon = 10^{-8}.\]
Since $\tau_0 + 20\epsilon = 0.4876902 < 0.4877$, the gap absorbs every factor $2^{O(n^{0.7})}\mathrm{poly}(n+n^c)$.

\subsection*{1.1 Changes relative to the paper}

{\small
\begin{longtable}{>{\raggedright\arraybackslash}p{0.223\dimexpr\linewidth-6\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.268\dimexpr\linewidth-6\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.509\dimexpr\linewidth-6\tabcolsep\relax}}
\hline
\textbf{Part} & \textbf{Paper} & \textbf{This note} \\ \hline
\endhead
Work exponent & $\tau = 0.48996$ & $\tau_0 = 0.48769$ \\
Bits of the large prime $P$ & $\eta n$ with $\eta = 0.51772$ & $\eta n$ with $\eta = 0.51613$ \\
Preliminary search & Proposition 2.4 & the same procedure with cap $2^{\lceil\tau_0 n\rceil}$ (Proposition P) \\
Checksum & Theorem 4.1, one filter & Theorem B: 43 filters, one chosen for each pattern \\
Alias step & Theorem 5.1, lists the aliases & Theorem A: returns the phase sum over the aliases \\
Input guard & $b \le 2^{0.00001n}$ & $10^6\lceil\log(b+1)\rceil \le n$ \\
\hline
\end{longtable}}

The new alias step has five features.

\begin{enumerate}
\item One random partition $[n] = M \cup L \cup R$ with a large block, $|M| \approx 0.473n$. It stays fixed for all trials.
\item The value maps of $L$ and $R$ (distinct values with summed phases) are built once.
\item The trials repeat about $2^{\hat F}$ times. Here $\hat F$ is a sampled estimate of $-\log\mathbb E_V 2^{-d(V)}$
over random $(|M| - 2k)$-subsets $V$ of $M$.
\item In each trial, the right masks are grouped into classes with equal reduced and unreduced weights.
\item Coverage uses Corollary D, an averaged form of the small-layer property.
\end{enumerate}

A large block gives each alias many representations, so the alias step works with a large prime of only $0.51613n$
bits. The checksum costs $2^{\tau_0 n}$ with $\tau_0 = \eta - \Delta$. At this $\eta$, the filter family certifies the
budget $\Delta = 0.02844$, so $\tau_0 = 0.48769$.

\subsection*{1.2 What is machine-checked}

Only the numerical inequalities are machine-checked:

{\small
\begin{longtable}{>{\raggedright\arraybackslash}p{0.361\dimexpr\linewidth-4\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.639\dimexpr\linewidth-4\tabcolsep\relax}}
\hline
\textbf{Statement} & \textbf{Check} \\ \hline
\endhead
Constraints (C1)--(C8) of the alias step & Arb ball arithmetic: script of Appendix D.1, on the parameters of Section 2.2 \\
The 430 moment inequalities of Proposition B & Arb ball arithmetic: script of Appendix D.2, on the data of Appendices A and C \\
\hline
\end{longtable}}

I re-ran both checks on 12 October 2026 (Section 8). Everything else is a written proof. That includes the parts
where an error is most likely: the cost accounting of Theorem A (Section 5.7), the application of Corollary D to a
fixed signing (Lemma A5), and the changes to the checksum (Section 6). The proof uses the paper's lemmas as the paper
states them. Where an argument of the paper carries over, the note says which one and what changes.

\medskip\noindent\rule{\linewidth}{0.4pt}\medskip

\section*{2. Notation, parameters and facts from the paper}\addcontentsline{toc}{section}{2. Notation, parameters and facts from the paper}

\subsection*{2.1 Notation}

\begin{itemize}
\item $\log$ means $\log_2$. The binary entropy is $h(x) = -x\log x - (1-x)\log(1-x)$.
\item $[n] = \{1, \dots, n\}$. For integer weights $w_1, \dots, w_n$ and $I \subseteq [n]$, $w(I) = \sum_{i\in I} w_i$.
\item The transformed weights and targets are $c_i = Ca_i + u_i$ and $t_s = Ct + s$ for $0 \le s \le nN_0$, as in (2.1).
They have $O(b + \log n)$ bits.
\item $\Sigma(V) = \{c(I) : I \subseteq V\}$ and $d(V) = |V| - \log|\Sigma(V)|$, as in (2.2).
\item $f(q) = \mathbb E\, d(V_q)$ for a uniform $q$-subset $V_q$ of $[n]$.
\item $e_P(x) = \exp(2\pi i x/P)$. For phases $\varphi_i \in \mathbb Z/P\mathbb Z$, $\chi(I) = e_P(\sum_{i \in I}\varphi_i)$.
\item The paper writes $a = \Delta/\Gamma$ and $v(x) = 2D(x) - A(x) - E(x)$. This note writes $a_\Gamma$ and $w(x)$,
because $a$ is the mask fraction and $v$ is a set size.
\end{itemize}

\subsection*{2.2 Parameters}

{\small
\begin{longtable}{>{\raggedright\arraybackslash}p{0.090\dimexpr\linewidth-6\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.239\dimexpr\linewidth-6\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.672\dimexpr\linewidth-6\tabcolsep\relax}}
\hline
\textbf{Symbol} & \textbf{Value} & \textbf{Meaning} \\ \hline
\endhead
$\tau_0$ & $0.48769$ & work exponent \\
$\eta$ & $0.51613$ & the large prime $P$ has about $\eta n$ bits \\
$\Delta$ & $\eta - \tau_0 = 0.02844$ & checksum budget \\
$\Gamma$ & $2\tau_0 - \eta = 0.45925$ & checksum budget \\
$a_\Gamma$ & $\Delta/\Gamma = 0.0619270\ldots$ &  \\
$\epsilon$ & $10^{-8}$ & pattern accuracy, as in the paper \\
$\zeta$ & $10^{-6}$ & slack in the preliminary threshold \\
$\mu$ & $0.473$ & the block $M$ has about $\mu n$ elements \\
$a$ & $0.0082$ & the left masks have about $a\lvert M\rvert$ elements \\
$\gamma$ & $0.06035$ & the small prime $P'$ has about $\gamma\lvert M\rvert$ bits \\
$l$ & $0.48377$ & the left part $L$ has about $ln$ elements \\
$r$ & $1 - \mu - l = 0.04323$ & the right part $R$ has about $rn$ elements \\
\hline
\end{longtable}}

\textbf{Alias constraints.} The parameters satisfy these strict inequalities. Section 8.1 checks them in Arb.

{\small
\begin{longtable}{>{\raggedright\arraybackslash}p{0.094\dimexpr\linewidth-6\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.562\dimexpr\linewidth-6\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.344\dimexpr\linewidth-6\tabcolsep\relax}}
\hline
 & \textbf{Inequality} & \textbf{Value of the left side} \\ \hline
\endhead
(C1) & $\gamma < h(2a)/2$ & $\gamma = 0.06035$, $h(2a)/2 = 0.06035987$ \\
(C2) & $l + (h(a) - \gamma)\mu < \tau_0$ & $0.48767640$ \\
(C3) & $r + (1 - \gamma)\mu < \tau_0$ & $0.48768445$ \\
(C4) & $1 - \mu + \big(h(a) + h(\tfrac12 - a) - \gamma\big)\mu - \eta < \tau_0$ & $0.48768463$ \\
(C5) & $h(\tfrac12 - a)\mu + \tfrac12 - \tau_0 < \tau_0$ & $0.48521823$ \\
(C6) & $0 < r < \tau_0$ and $l < \tau_0$ & $r = 0.04323$, $l = 0.48377$ \\
(C7) & $\mu(1 - 2a) < l$ & $\mu(1-2a) = 0.46524280$ \\
(C8) & $\tfrac{\mu}{2}\log 3 < \eta$ & $0.37484363$ \\
\hline
\end{longtable}}

Let $\varepsilon_0$ be the least slack in (C1)--(C8). Arb gives $\varepsilon_0 \ge 5.37\cdot 10^{-6}$; (C4) is the
tightest. We use $\zeta + 10^{-6} < \varepsilon_0$.

\textbf{Sizes.} For each $n$ put
\[m = 2\lfloor \mu n/2\rfloor,\qquad \sigma = m/2,\qquad k = \lfloor am\rfloor,\qquad v = m - 2k,\qquad
|L| = \lfloor ln\rfloor,\qquad |R| = n - m - |L|.\]
For all large $n$ these sizes satisfy:

\begin{itemize}
\item (S1) $4k \le m$, so $2k \le \sigma$; and $2\sigma = v + 2k$.
\item (S2) $v < |L|$, by (C7), and $v \le \lfloor n/2\rfloor$.
\item (S3) $\binom{\sigma}{k}\binom{v}{k} \le 2^{v}$. Indeed the left side is at most $2^{m(h(2a)/2 + h(a))}$, and
$h(2a)/2 + h(a) = 0.129 < 1 - 2a$.
\item (S4) $\binom{\sigma}{k} \ge 2^{\gamma m}$, by (C1).
\item (S5) $\binom{m}{k} \le 2^{h(a)m}$ and $\binom{m}{\sigma - k} \le 2^{h(1/2 - a)m + 1}$.
\end{itemize}

\textbf{Input guard.} We use the guard $10^6\lceil\log(b+1)\rceil \le n$. It implies $b + 2 \le 2^{n/10^6 + 1}$. For each
fixed $c$, inputs with $b \le n^c$ satisfy it for all large $n$.

\subsection*{2.3 Facts from the paper}

Appendix B states these results in full. We use them without proof.

\begin{itemize}
\item \textbf{(F1)} Lemma 2.2. For arbitrary integer weights, $d(\varnothing) = 0$ and $d \ge 0$. Adding an index increases $d$
by at least 0 and at most 1. An exchange of one index changes $d$ by at most 1. Hence $d(U) \le d(V)$ for
$U \subseteq V$.
\item \textbf{(F2)} Signs. Negating the weights of a set $F$ maps $\Sigma(V)$ to its translate by $-c(F \cap V)$. So $d$ does
not change.
\item \textbf{(F3)} Lemma 2.3. The function $f$ is nondecreasing. Its proof builds the exposure martingale of a uniform
$q$-subset and shows that the increments have absolute value at most 1. The same proof works for a uniform $q$-subset
of any fixed finite ground set. Azuma's inequality then gives, for $q \le n$,
\[\Pr\big[\,\lvert d(V_q) - \mathbb E\,d(V_q)\rvert > n^{0.6}\big] \le 2e^{-n^{0.2}/2}.\]
\item \textbf{(F4)} Lemma 2.5. There are absolute constants $K$ and $L_*$ with this property. Let $p$ be uniform among the
primes of $[2^{L'}, 2^{L'+1})$ with $L' \ge L_*$, and let $D \ne 0$ be an integer with $|D| < 2^B$. Then
$\Pr[p \mid D] \le KBL'2^{-L'}$.
\item \textbf{(F5)} Lemma 2.1 (isolation) for the transformation (2.1).
\item \textbf{(F6)} Section 6.2: bounded uniform draws, primes by trial division, and finite-precision complex sums. A bounded
draw fails with probability at most $2^{-n^3}$, and a bounded prime draw with probability at most $e^{-c_0n^2}$.
\item \textbf{(F7)} Section 3 and Lemma 3.2. The proof of Lemma 3.2 uses only that $r$ and $s$ are fixed trigonometric
polynomials. So its conclusions hold for every such pair, except the numerical moment bound (3.7).
\item \textbf{(F8)} Section 4: the checksum construction, Lemma 4.2, and the estimates (4.1)--(4.25).
\end{itemize}

\medskip\noindent\rule{\linewidth}{0.4pt}\medskip

\section*{3. An averaged small-layer lemma}\addcontentsline{toc}{section}{3. An averaged small-layer lemma}

Let $w$ be integer weights on a finite index set; equal weights are allowed. For a finite index set $V$ and $j \ge 0$,
put
\[\Sigma_j(V) = \{w(T) : T \subseteq V,\ |T| = j\},\qquad a_j(V) = |\Sigma_j(V)|\Big/\binom{|V|}{j},\]
and $\Sigma(V) = \bigcup_j\Sigma_j(V)$, $d(V) = |V| - \log|\Sigma(V)|$.

\textbf{Lemma 3.1 (covering).} Let $0 \le k \le t \le |V|$, and let $W$ be a uniform $(t-k)$-subset of $V$. Then
$a_t(V) \le \mathbb E_W\, a_k(V\setminus W)$.

\emph{Proof.} For each $x \in \Sigma_t(V)$, fix a $t$-set $T_x \subseteq V$ with $w(T_x) = x$. Consider the pairs $(x, W)$ with
$W \subseteq T_x$ and $|W| = t - k$. The map $(x, W) \mapsto (W, x - w(W))$ is injective, because
$x = (x - w(W)) + w(W)$. Its second coordinate is $w(T_x\setminus W) \in \Sigma_k(V\setminus W)$. There are
$\binom tk|\Sigma_t(V)|$ pairs, so
\[\binom tk\,|\Sigma_t(V)| \le \sum_{W\subseteq V,\ |W| = t-k}|\Sigma_k(V\setminus W)|.\]
Divide by $\binom{|V|}{t}\binom tk = \binom{|V|}{t-k}\binom{|V| - t + k}{k}$. $\square$

\textbf{Lemma 3.2 (sampling).} Let $k \le s \le |V|$, and let $S$ be a uniform $s$-subset of $V$. Then
$\mathbb E_S\, a_k(S) \ge a_k(V)$.

\emph{Proof.} For each $x \in \Sigma_k(V)$, fix a $k$-set $K_x$ with $w(K_x) = x$. Then $x \in \Sigma_k(S)$ whenever
$K_x \subseteq S$. This event has probability $\binom{|V|-k}{s-k}\big/\binom{|V|}{s} = \binom sk\big/\binom{|V|}{k}$.
Sum over $x$ and divide by $\binom sk$. $\square$

\textbf{Lemma 3.3 (a good layer).} Some $t$ with $2t \le |V|$ has $|\Sigma(V)| \le (|V|+1)\,|\Sigma_t(V)|$. In particular
$a_t(V) \ge 2^{-d(V)}/(|V|+1)$.

\emph{Proof.} The $|V| + 1$ layers cover $\Sigma(V)$, so some layer $j$ has $|\Sigma_j(V)| \ge |\Sigma(V)|/(|V|+1)$.
Complementation gives $\Sigma_{|V|-j}(V) = w(V) - \Sigma_j(V)$, so we may take $t = \min(j, |V| - j)$. Finally
$\binom{|V|}{t} \le 2^{|V|}$. $\square$

\textbf{Corollary D (averaged small layer).} Let $M$ be a finite index set with integer weights. Let $k, \sigma, v$ be
integers with $2k \le \sigma$, $2\sigma \le v + 2k$, $v \le |M|$ and $\binom\sigma k\binom vk \le 2^v$. Let $S$ be a
uniform $\sigma$-subset and $V$ a uniform $v$-subset of $M$. Then
\[\mathbb E_S\, a_k(S) \;\ge\; \frac{\mathbb E_V\, 2^{-d(V)}}{v+1}.\]

\emph{Proof.} The hypotheses give $k \le \sigma \le v$. Fix a $v$-set $V \subseteq M$, and let $S_V$ be a uniform
$\sigma$-subset of $V$. We show that $2^{-d(V)}/(v+1) \le \mathbb E\, a_k(S_V)$. Take $t$ from Lemma 3.3.

\emph{Case $t \ge k$.} Lemma 3.1 gives $a_t(V) \le \mathbb E_W\, a_k(V\setminus W)$, where
$|V\setminus W| = v - t + k \ge v/2 + k \ge \sigma$. Lemma 3.2 for $V \setminus W$ and $s = \sigma$ gives
$a_k(V\setminus W) \le \mathbb E\, a_k(S')$ for a uniform $\sigma$-subset $S'$ of $V\setminus W$. When $W$ is uniform,
$S'$ is a uniform $\sigma$-subset of $V$. Hence $2^{-d(V)}/(v+1) \le a_t(V) \le \mathbb E\, a_k(S_V)$.

\emph{Case $t < k$.} Then $t < k \le v/2$, so $|\Sigma_t(V)| \le \binom vt \le \binom vk$. Lemma 3.3 gives
$2^{v - d(V)} \le (v+1)\binom vk$. With the hypothesis, $2^{-d(V)}/(v+1) \le \binom vk2^{-v} \le 1/\binom\sigma k$.
Every $\sigma$-set $S$ has $a_k(S) \ge 1/\binom\sigma k$, because $\Sigma_k(S)$ is not empty.

Finally average over $V$. A uniform $\sigma$-subset of a uniform $v$-subset of $M$ is a uniform $\sigma$-subset of $M$.
$\square$

\textbf{Remark.} Signs do not change $d$ (F2), but they change $a_k$. Theorem A applies Corollary D to one fixed signing of
the block for each alias (Lemma A5, Step 2).

\medskip\noindent\rule{\linewidth}{0.4pt}\medskip

\section*{4. The preliminary search}\addcontentsline{toc}{section}{4. The preliminary search}

\textbf{Proposition P.} Run the procedure of Proposition 2.4 with the list cap $B_n = 2^{\lceil\tau_0 n\rceil}$ in place of
$2^{\lceil 0.489985n\rceil}$. It returns the correct decision or CONTINUE. On every execution it uses at most
$2^{\tau_0 n}\,\mathrm{poly}(n+b+2)$ word operations. If $\mathbb E\, d(V) > (\tfrac12 - \tau_0 + \zeta)n$ for a uniform
$\lfloor n/2\rfloor$-subset $V$, it returns CONTINUE with probability $O(1/n)$.

\emph{Proof.} Correctness and the time bound are those of Proposition 2.4. An answer comes from two complete lists, and
every merge stops after $O(B_n)$ steps. Let $U$ be either side of the random split; it has $\lfloor n/2\rfloor$ or
$\lceil n/2\rceil$ elements. Since $f$ is nondecreasing, $\mathbb E\, d(U) > (\tfrac12 - \tau_0 + \zeta)n$. Lemma 2.3 gives
$\mathrm{Var}\, d(U) \le n$, and Chebyshev's inequality gives
\[\Pr\big[d(U) < (\tfrac12 - \tau_0 + \tfrac\zeta2)n\big] \le \frac{n}{(\zeta n/2)^2} = \frac{4}{\zeta^2n}.\]
Otherwise $\log|\Sigma(U)| = |U| - d(U) \le \tfrac n2 + \tfrac12 - (\tfrac12 - \tau_0 + \tfrac\zeta2)n < \tau_0 n$ for
$n > 1/\zeta$. Then both lists fit under the cap, and the search decides. The sampler of the split fails with
probability at most $n2^{-n^2}$. $\square$

So, for fixed tie weights, either the preliminary search decides with probability $1 - O(1/n)$, or
\[\mathbb E_{V\subseteq[n],\ |V| = \lfloor n/2\rfloor}\, d(V) \;\le\; (\tfrac12 - \tau_0 + \zeta)\,n. \tag{LD}\]
Theorem A uses (LD) only to bound the number of trials and to prove coverage. The algorithm never evaluates it.

\medskip\noindent\rule{\linewidth}{0.4pt}\medskip

\section*{5. Alias phase sums (Theorem A)}\addcontentsline{toc}{section}{5. Alias phase sums (Theorem A)}

Let $P$ be a prime in $[2^{\lfloor\eta n\rfloor}, 2^{1+\lfloor\eta n\rfloor})$ and $b_i = c_i \bmod P \in \{0, \dots, P-1\}$.
The aliases of the target $t_s$ are
\[\mathcal A_s(P) = \{I \subseteq [n] : c(I) \equiv t_s \pmod P,\ c(I) \ne t_s\}.\]
An alias $I$ has $b(I) \equiv t_s \pmod P$ and $0 \le b(I) \le n(P-1)$. So it has a unique \emph{carry}
$q \in \{0, \dots, n-1\}$ with
\[b(I) = T_{s,q} := (t_s \bmod P) + qP.\]

\textbf{Idea.} In a trial, an alias $I$ is represented by every split of the set $S = (I\cap M)\triangle F$ into a left mask $X$
of size $k$ and a right mask $Y$ of size $\sigma - k$. The small prime $P'$ keeps a $1/P'$ fraction of the left
records. So a trial finds $I$ when the left masks of $S$ take about $P'$ distinct values. Corollary D supplies these
values on average over $S$, and $2^{\hat F}$ repeated trials turn the average into coverage of every alias. The
deficiencies pay for the repetitions: $d(L)$ pays on the left side, and $d(M)$ pays on the right side, because the
right masks are grouped by value.

\subsection*{5.1 Statement}

\textbf{Theorem A.} There is a randomized procedure with the following properties for all sufficiently large $n$. Its input is
the transformed weights and targets of (2.1) under the guard of Section 2.2, a prime
$P \in [2^{\lfloor\eta n\rfloor}, 2^{1+\lfloor\eta n\rfloor})$, and phases $\varphi \in (\mathbb Z/P\mathbb Z)^n$. It returns
FAIL or complex numbers $\hat A_s$ for $0 \le s \le nN_0$.

\begin{enumerate}
\item Assume (LD). Let $P$ be uniform among the primes of the interval, and let $\varphi$ be arbitrary; it may depend on $P$.
Then, with probability $1 - o(1)$, the procedure does not fail and
\[\Big|\hat A_s - \sum_{I\in\mathcal A_s(P)}\chi(I)\Big| \le \frac1{16}\qquad\text{for every } s.\]
The $o(1)$ depends only on $n$.
\item On every execution the procedure uses at most $2^{\tau_0 n + 2n^{0.7}}\,\mathrm{poly}(n+b+2)$ word operations.
\end{enumerate}

\subsection*{5.2 The procedure}

The procedure makes its random draws with the bounded samplers of Section 6.2 (F6). For $F \subseteq M$, the signed
weights on $M$ are
\[c^F_i = (-1)^{[i\in F]}c_i,\qquad b^F_i = (-1)^{[i \in F]}b_i\qquad (i \in M).\]
For every $Z \subseteq M$ they satisfy
\[c^F(Z) = c(Z\triangle F) - c(F),\qquad b^F(Z) = b(Z\triangle F) - b(F). \tag{5.1}\]
Indeed $c(Z\triangle F) - c(F) = c(Z\setminus F) + c(F\setminus Z) - c(F) = c(Z\setminus F) - c(Z\cap F)$.

The \emph{key} of a set $A$ is the pair $\kappa(A) = (c(A), b(A)) \in \mathbb Z^2$. For a key $\kappa = (x, y)$ we write
$c(\kappa) = x$ and $b(\kappa) = y$.

\begin{enumerate}
\item \textbf{Partition.} Draw a uniform $m$-subset $M \subseteq [n]$, then a uniform $|L|$-subset $L$ of $[n]\setminus M$. Put
$R = [n]\setminus(M \cup L)$. The partition stays fixed.
\item \textbf{Value maps.} For every $A \subseteq L$, compute $\kappa(A)$ and $\chi(A)$. Sort, and store each distinct key
$\kappa$ once with $\Phi_L(\kappa) = \sum_{A\subseteq L:\ \kappa(A) = \kappa}\chi(A)$. Do the same for $R$.
\item \textbf{Trial count.} Draw $n^2$ independent uniform $v$-subsets $V_1, \dots, V_{n^2}$ of $M$. Compute each $|\Sigma(V_j)|$
exactly by the merge procedure of Proposition 2.4, without a cap. Put
\[\hat S = \frac1{n^2}\sum_{j=1}^{n^2}\frac{|\Sigma(V_j)|}{2^v} = \frac1{n^2}\sum_{j=1}^{n^2}2^{-d(V_j)},\qquad
\hat F = \log\frac1{\hat S} + e_n,\qquad T = n^7\,2^{e_n}\Big\lceil\frac1{\hat S}\Big\rceil,\]
where $e_n$ is an integer with $2n^{0.6} \le e_n \le 2n^{0.6} + 1$. Then $n^72^{\hat F} \le T \le 2n^72^{\hat F}$.
\item \textbf{Trials.} For every target index $s$ and every carry $q \in \{0, \dots, n-1\}$, run $T$ independent trials.
A trial for $(s, q)$ does the following.
\begin{itemize}
\item (a) Draw a uniform $F \subseteq M$. Put $T^F = T_{s,q} - b(F)$ and $t^F = t_s - c(F)$.
\item (b) Draw a uniform prime $P' \in [2^{\lfloor\gamma m\rfloor}, 2^{1+\lfloor\gamma m\rfloor})$ and a uniform
$j' \in \mathbb Z/P'\mathbb Z$.
\item (c) \emph{Left records:} all pairs $(\kappa_L, X)$ with $\kappa_L$ a key of $L$, $X \subseteq M$, $|X| = k$, and
$b(\kappa_L) + b^F(X) \equiv j' \pmod{P'}$. To list them, sort the keys of $L$ by $b(\kappa_L) \bmod P'$; for
each $X$, find the keys with the needed residue by binary search.
\item (d) \emph{Right classes:} list all $Y \subseteq M$ with $|Y| = \sigma - k$, and group them by
$\lambda(Y) = (b^F(Y), c^F(Y))$. A \emph{class} is a value $\lambda = (b_\lambda, c_\lambda)$ that occurs, together with
the list of its members $Y$.
\item (e) \emph{Right class records:} all pairs $(\kappa_R, \lambda)$ with $\kappa_R$ a key of $R$, $\lambda$ a class, and
$b(\kappa_R) + b_\lambda \equiv T^F - j' \pmod{P'}$. List them as in (c).
\item (f) \emph{Matching:} group the right class records by the integer $b(\kappa_R) + b_\lambda$, and sort each group by the
integer $c(\kappa_R) + c_\lambda$. For each left record $(\kappa_L, X)$, find the group with
$b(\kappa_R) + b_\lambda = T^F - b(\kappa_L) - b^F(X)$. In it, find by two binary searches the interval of records
with $c(\kappa_R) + c_\lambda = t^F - c(\kappa_L) - c^F(X)$. Skip that interval, and \emph{visit} every other record of
the group.
\item (g) \emph{Expansion:} for each visited $(\kappa_R, \lambda)$ and each member $Y$ of $\lambda$, test $X \cap Y = \varnothing$.
If the test passes, put $Z = (X\cup Y)\triangle F$, and insert the key $(\kappa_L, \kappa_R, Z)$ into the
dictionary $\mathcal D_s$ unless it is there.
\end{itemize}
\item \textbf{Output.} For each $s$, return $\hat A_s$, a finite-precision value of
$\sum_{(\kappa_L, \kappa_R, Z)\in\mathcal D_s}\Phi_L(\kappa_L)\,\Phi_R(\kappa_R)\,\chi(Z)$, computed as in Section 6.2.
\item \textbf{Cap.} Count the elementary steps of Steps 1--5: subsets enumerated, merge operations, masks listed, keys sorted,
records formed, records visited, members expanded, dictionary operations, terms summed, and trial divisions. Return
FAIL as soon as the count would exceed $2^{\lceil\tau_0 n\rceil + \lceil n^{0.7}\rceil}$. Also return FAIL if a
bounded sampler fails.
\end{enumerate}

Each step costs $\mathrm{poly}(n+b)$ word operations. Keys have $O(b + \log n)$ bits, masks have $m$ bits, and the
dictionaries are balanced search trees. The finite-precision sums use $n^2$ fractional bits (Section 6.2). With the
cap, this proves part 2 of Theorem A. The rest of this section proves part 1.

\subsection*{5.3 Keys, classes and soundness}

For keys $\kappa_L$ of $L$ and $\kappa_R$ of $R$ and a set $Z \subseteq M$, the \emph{class} of $(\kappa_L, \kappa_R, Z)$ is
\[\mathcal C(\kappa_L, \kappa_R, Z) = \{A\cup Z\cup B : A \subseteq L,\ B \subseteq R,\ \kappa(A) = \kappa_L,\ \kappa(B) = \kappa_R\}.\]
Each $I \subseteq [n]$ lies in exactly one class: the class of $(\kappa(I\cap L), \kappa(I\cap R), I\cap M)$. All
members of a class have the same $c(I)$ and the same $b(I)$. So either every member is an alias of $t_s$ or none is.
In the first case we call the class an \emph{alias class} of $t_s$. Since phases add,
\[\sum_{I \in \mathcal C(\kappa_L, \kappa_R, Z)}\chi(I) = \Phi_L(\kappa_L)\,\Phi_R(\kappa_R)\,\chi(Z). \tag{5.2}\]

\textbf{Lemma A1 (soundness).} Every key inserted into $\mathcal D_s$ is an alias class of $t_s$.

\emph{Proof.} An insertion comes from a left record $(\kappa_L, X)$, a visited record $(\kappa_R, \lambda)$, and a member $Y$ of
$\lambda$ with $X \cap Y = \varnothing$; then $Z = (X\cup Y)\triangle F$. Take $A, B$ with $\kappa(A) = \kappa_L$ and
$\kappa(B) = \kappa_R$, and put $I = A\cup Z\cup B$. By (5.1), $b^F(X) + b^F(Y) = b^F(X\cup Y) = b(Z) - b(F)$, and the same
holds for $c$. The group condition gives $b(A) + b(Z) - b(F) + b(B) = T^F$. So $b(I) = T_{s,q}$, and
$c(I) \equiv b(I) \equiv t_s \pmod P$. The skipped interval gives $c(A) + c(Z) - c(F) + c(B) \ne t^F$, so $c(I) \ne t_s$.
Hence $I$ is an alias, and so is every member of its class. $\square$

The alias classes of $t_s$ partition $\mathcal A_s(P)$, and the dictionary holds each key once. So, by (5.2) and
Lemma A1: if $\mathcal D_s$ contains every alias class of $t_s$, then the exact output sum equals
$\sum_{I\in\mathcal A_s(P)}\chi(I)$.

\textbf{Lemma A2 (key and class counts).}

\begin{enumerate}
\item $L$ has at most $(n+1)\,|\Sigma(L)| = (n+1)\,2^{|L| - d(L)}$ keys, and each value $c(\kappa)$ occurs in at most $n+1$
keys. The same holds for $R$.
\item Every trial has at most $(2m+1)\,2^{m - d(M)}$ right classes.
\end{enumerate}

\emph{Proof.} (1) A key $(x, y)$ of $L$ has $y \equiv x \pmod P$ and $0 \le y \le |L|(P-1)$. So $y$ takes at most
$|L| + 1 \le n + 1$ values for each $x$. (2) By (5.1), $c^F(Y) = c(Y\triangle F) - c(F)$ takes at most
$|\Sigma(M)| = 2^{m - d(M)}$ values. For each of them, $b^F(Y) \equiv c^F(Y) \pmod P$ and $|b^F(Y)| < mP$. This leaves at
most $2m+1$ values of $b^F(Y)$. $\square$

\subsection*{5.4 The trial count}

Let $V$ be a uniform $v$-subset of $M$, and put
\[g_M = -\log\mathbb E_V\, 2^{-d(V)},\qquad e_M = \mathbb E_V\, d(V).\]
Jensen's inequality gives $g_M \le e_M$, and (F1) gives $e_M \le d(M)$.

\textbf{Lemma A3.} With probability $1 - o(1)$ over $M$, $L$ and Step 3, the following hold.

\begin{enumerate}
\item $\hat F \ge g_M$.
\item $\hat F \le e_M + 3n^{0.6} + 1 \le d(M) + 3n^{0.6} + 1$.
\item $\hat F \le d(L) + 5n^{0.6} + 1$.
\item Under (LD): $g_M \le e_M \le (\tfrac12 - \tau_0 + \zeta)n + n^{0.6}$ and $\hat F \le (\tfrac12 - \tau_0 + \zeta)n + 4n^{0.6} + 1$.
\end{enumerate}

\emph{Proof.} (1) $\hat S$ is the mean of $n^2$ independent samples of $2^{-d(V)}$, so $\mathbb E\,\hat S = 2^{-g_M}$. Markov's
inequality gives $\hat S < 2^{2n^{0.6}}2^{-g_M}$, except with probability $2^{-2n^{0.6}}$. Then
$\hat F \ge \log(1/\hat S) + 2n^{0.6} > g_M$.

(2) Apply (F3) inside $M$. For fixed $M$, every $d(V_j)$ is at most $e_M + n^{0.6}$, except with probability
$2n^2e^{-n^{0.2}/2}$. Then $\hat S \ge 2^{-e_M - n^{0.6}}$, so $\hat F \le e_M + n^{0.6} + e_n \le e_M + 3n^{0.6} + 1$.
Also $e_M \le d(M)$.

Next we show $e_M \le f(v) + n^{0.6}$ with high probability. Exchange one element $i \in M$ for $j \notin M$. Couple $V$
with $V' = V$ if $i \notin V$, and $V' = (V\setminus\{i\})\cup\{j\}$ otherwise. Then $V'$ is uniform in the new block, and
$|d(V) - d(V')| \le 1$ by (F1). So $e_M$ changes by at most 1 under an exchange. The exposure martingale of $M$ (as in
(F3)) and Azuma's inequality give $e_M \le \mathbb E_M\, e_M + n^{0.6} = f(v) + n^{0.6}$, except with probability
$e^{-n^{0.2}/2}$.

(3) $L$ is a uniform $|L|$-subset of $[n]$. By (F3), $d(L) \ge f(|L|) - n^{0.6}$, except with probability
$2e^{-n^{0.2}/2}$. Since $|L| > v$ (S2) and $f$ is nondecreasing,
$\hat F \le e_M + 3n^{0.6} + 1 \le f(v) + 4n^{0.6} + 1 \le d(L) + 5n^{0.6} + 1$.

(4) $v \le \lfloor n/2\rfloor$ (S2), so $f(v) \le f(\lfloor n/2\rfloor) \le (\tfrac12 - \tau_0 + \zeta)n$ under (LD).
Combine this with the bounds above. $\square$

Let $\mathcal E_3$ be the event of Lemma A3. It depends only on $M$, $L$ and Step 3, which are independent of $P$.

\subsection*{5.5 Reduction modulo $P$ on half blocks}

Call a $\sigma$-subset $S \subseteq M$ \emph{bad} for $P$ if some $\lambda \in \{-1, 0, 1\}^S$ has
$\sum_{i\in S}\lambda_ic_i \ne 0$ and $P \mid \sum_{i\in S}\lambda_ic_i$. Put $\delta = (\eta - \tfrac\mu2\log3)/2$.
Then $\delta > 0.0706$ by (C8).

\textbf{Lemma A4.} For every fixed $M$, with probability $1 - o(1)$ over $P$, at most a $2^{-\delta n}$ fraction of the
$\sigma$-subsets of $M$ are bad.

\emph{Proof.} Fix $S$. There are fewer than $3^\sigma$ vectors $\lambda$. Each nonzero $\sum\lambda_ic_i$ has absolute value
below $2^{b + O(\log n)}$. By (F4) with $L' = \lfloor\eta n\rfloor$, $P$ divides it with probability at most
$\mathrm{poly}(n)(b + \log n)2^{-\eta n}$. So $S$ is bad with probability at most
$\mathrm{poly}(n)(b+\log n)\,3^\sigma2^{-\eta n} \le \mathrm{poly}(n)(b + \log n)\,2^{-2\delta n}$, as $\sigma \le \mu n/2$.
The expected bad fraction has the same bound. By Markov's inequality, the fraction exceeds $2^{-\delta n}$ with
probability at most $\mathrm{poly}(n)(b+\log n)2^{-\delta n}$. This is $o(1)$ by the guard. $\square$

On a good set, reduction modulo $P$ keeps all distinctions between signed subset sums. Precisely: let $S$ be good, let
$\epsilon_i \in \{\pm1\}$ for $i \in S$, and let $X, X' \subseteq S$. Then
\[\sum_{i\in X}\epsilon_ic_i \ne \sum_{i\in X'}\epsilon_ic_i \implies \sum_{i\in X}\epsilon_ib_i \ne \sum_{i\in X'}\epsilon_ib_i. \tag{5.3}\]
The difference of the left sides is $\sum_{i\in S}\lambda_ic_i$ with $\lambda_i = \epsilon_i([i\in X] - [i\in X'])$. It is
not zero, so $P$ does not divide it. The difference of the right sides is congruent to it modulo $P$, so it is not zero.

\subsection*{5.6 Coverage}

\textbf{Lemma A5 (one trial).} Assume (LD), $\mathcal E_3$, and the event of Lemma A4. Fix $s$, an alias $I \in \mathcal A_s(P)$
with carry $q$, and one trial for $(s, q)$. After the trial, $\mathcal D_s$ contains the key of the class of $I$ with
probability at least $2^{-g_M}n^{-6}$. The threshold for $n$ does not depend on $I$, $s$, $P$ or $M$.

\emph{Proof.} Put $A = I\cap L$, $B = I\cap R$ and $S = (I\cap M)\triangle F$.

\emph{Step 1 (half of the block).} $F$ is a uniform subset of $M$, so $S$ is too. Hence $|S| = \sigma$ with probability
$\binom m\sigma2^{-m} \ge 1/(m+1)$. Conditional on this event, $S$ is a uniform $\sigma$-subset of $M$, and
$F = S\triangle(I\cap M)$.

\emph{Step 2 (the signs on $S$ are fixed).} Let $i \in S$. Either $i \in I\setminus F$, and then $c^F_i = c_i$; or
$i \in F\setminus I$, and then $c^F_i = -c_i$. So on $S$ the signed weights equal the weights
\[c^I_i = c_i\ \ (i \in I\cap M),\qquad c^I_i = -c_i\ \ (i \in M\setminus I).\]
These depend on $I$ but not on $S$. The same holds for $b^F$ and $b^I$.

\emph{Step 3 (many layer values).} Apply Corollary D to the weights $c^I$ on $M$, with $k, \sigma, v$ of Section 2.2;
(S1)--(S3) give its hypotheses. By (F2), $d$ is the same for $c^I$ and for $c$. Hence
\[\mathbb E_S\, a_k(S; c^I) \ge \frac{\mathbb E_V\,2^{-d(V)}}{v+1} = \frac{2^{-g_M}}{v+1}.\]
Let $W_S = \{b(A) + b^F(X) : X \subseteq S,\ |X| = k\}$ and $B_S = |W_S|$. If $S$ is good, then (5.3) and Step 2 give
$B_S \ge |\Sigma_k(S; c^I)| = a_k(S; c^I)\binom\sigma k$. Bad sets have probability at most $2^{-\delta n}$. By Lemma
A3(4), $2^{-\delta n} \le 2^{-g_M}/(2(v+1))$, because $\tfrac12 - \tau_0 + \zeta < 0.013 < \delta$. Since $a_k \le 1$,
\[\mathbb E_S\Big[\mathbf 1_{S\ \text{good}}\,\frac{B_S}{\binom\sigma k}\Big] \ge \frac{2^{-g_M}}{v+1} - 2^{-\delta n} \ge \frac{2^{-g_M}}{2(v+1)}.\]

\emph{Step 4 (the small prime keeps many residues).} Fix a good $S$. The set $W_S$ depends on $P$, $M$, $L$, $I$ and $S$, but
not on $P'$ or $j'$. Its elements lie in $(-nP, nP)$, so differences have at most $\eta n + \log n + 3$ bits. For
distinct $x, x' \in W_S$, (F4) with $L' = \lfloor\gamma m\rfloor$ gives $\Pr[P' \mid x - x'] \le K_0n^22^{-\gamma m}$, for
an absolute constant $K_0 \ge 1$. Let $z_y$ be the number of elements of $W_S$ in the residue class $y$ modulo $P'$. Then
\[\mathbb E_{P'}\sum_y z_y^2 \le B_S + K_0n^2B_S^2\,2^{-\gamma m}.\]
If $B_S \le 2^{\gamma m}$, the right side is at most $2K_0n^2B_S$. Otherwise it is at most $2K_0n^2B_S^22^{-\gamma m}$.
By Markov's inequality, with probability at least $1/2$ over $P'$, $\sum_yz_y^2$ is at most twice that bound. Then
the Cauchy--Schwarz inequality $(\sum_yz_y)^2 \le \#\{y : z_y > 0\}\cdot\sum_yz_y^2$ shows that at least
$\min(B_S, 2^{\gamma m})/(4K_0n^2)$ residues are occupied. Since $P' < 2^{1+\gamma m}$, a uniform $j'$ hits an occupied
residue with probability at least
\[\frac{\min(B_S2^{-\gamma m}, 1)}{8K_0n^2} \ge \frac{B_S}{8K_0n^2\binom\sigma k}.\]
The last step uses $B_S \le \binom\sigma k$ and $2^{\gamma m} \le \binom\sigma k$ (S4).

\emph{Step 5 (a hit is recovered).} Suppose $j' \equiv b(A) + b^F(X) \pmod{P'}$ for some $X \subseteq S$ with $|X| = k$. Then the
left record $(\kappa(A), X)$ is listed. Put $Y = S\setminus X$; so $|Y| = \sigma - k$ and $X\cap Y = \varnothing$. Since
$S\triangle F = I\cap M$, (5.1) gives
\[b(A) + b^F(X) + b(B) + b^F(Y) = b(A) + b(B) + b(I\cap M) - b(F) = T_{s,q} - b(F) = T^F.\]
So the class record $(\kappa(B), \lambda(Y))$ has the residue $T^F - j'$. It is listed, and it lies in the group of the
left record. The same computation with $c$ gives the unreduced sum $c(I) - c(F)$. This differs from
$t^F = t_s - c(F)$, because $I$ is an alias. So the record is not in the skipped interval, and it is visited. The
expansion reaches its member $Y$ and inserts $(\kappa(A), \kappa(B), (X\cup Y)\triangle F) = (\kappa(A), \kappa(B), I\cap M)$,
unless the key is already there.

\emph{Step 6 (product).} The draws of $F$, $P'$ and $j'$ are independent. Steps 1--5 give a success probability of at least
\[\frac1{m+1}\cdot\mathbb E_S\Big[\mathbf 1_{S\ \text{good}}\cdot\frac12\cdot\frac{B_S}{8K_0n^2\binom\sigma k}\Big]
\ge \frac{2^{-g_M}}{32K_0n^2(m+1)(v+1)} \ge 2^{-g_M}n^{-6}\]
for all large $n$. $\square$

\textbf{Corollary A6.} Assume (LD), $\mathcal E_3$, and the event of Lemma A4. Then, except with probability $o(1)$, every
alias class of every target is in its dictionary at the end of Step 4.

\emph{Proof.} Condition on $P$, $\varphi$, $M$, $L$ and Step 3. The $T$ trials for a pair $(s, q)$ are independent, and
$T \ge n^72^{\hat F} \ge n^72^{g_M}$ by Lemma A3(1). By Lemma A5, all of them miss a fixed alias with carry $q$ with
probability at most $(1 - 2^{-g_M}n^{-6})^T \le e^{-n}$. There are at most $(nN_0+1)\,2^n$ pairs of a target and a
subset, and $(nN_0 + 1)\,2^ne^{-n} = o(1)$. $\square$

\subsection*{5.7 Expected work}

\textbf{Lemma A7.} Assume (LD) and $\mathcal E_3$. Condition on $M$, $L$ and Step 3, and average over $P$ and the trials; the
phases are arbitrary. Then the uncapped procedure takes in expectation at most
\[(b+2)\,2^{(\tau_0 - \varepsilon_0 + \zeta)n + O(n^{0.6})}\]
steps.

\emph{Proof.} Recall $T \le 2n^72^{\hat F}$, and that there are $O(n^3)$ pairs $(s, q)$. Polynomial factors go into
$2^{O(n^{0.6})}$. We bound each kind of step.

\begin{enumerate}
\item \emph{Value maps and Step 3:} $2^{|L|} + 2^{|R|} + n^22^{v}$ steps. The exponents $l$, $r$ and $\mu(1-2a)$ are below
$\tau_0 - \varepsilon_0$, by (C6) and (C7).
\item \emph{Masks:} each trial lists $\binom mk + \binom m{\sigma-k} \le 2^{h(1/2-a)m + 2}$ masks and sorts them into classes.
By Lemma A3(4) and (S5), the total is $2^{(h(1/2-a)\mu + 1/2 - \tau_0 + \zeta)n + O(n^{0.6})}$. By (C5) the exponent
is below $(\tau_0 - \varepsilon_0 + \zeta)n$.
\item \emph{Small primes:} each trial tests at most $n^3$ candidates, each by at most $2^{\gamma m/2 + 1}$ divisions. The total
is $2^{(1/2 - \tau_0 + \zeta + \gamma\mu/2)n + O(n^{0.6})}$, far below the bound.
\item \emph{Sorting keys by residue:} each trial sorts at most $(n+1)(2^{|L| - d(L)} + 2^{|R|})$ keys. By Lemma A3(3) and (4),
the total is at most $2^{ln + O(n^{0.6})} + 2^{(r + 1/2 - \tau_0 + \zeta)n + O(n^{0.6})}$.
\item \emph{Left records:} for fixed choices other than $j'$, a pair $(\kappa_L, X)$ is listed with probability $1/P'$, and
$1/P' \le 2^{1-\gamma m}$. So a trial lists at most $(n+1)2^{|L| - d(L)}\binom mk2^{1-\gamma m}$ left records in
expectation. With Lemma A3(3), the total is $2^{(l + (h(a) - \gamma)\mu)n + O(n^{0.6})}$. Use (C2).
\item \emph{Right class records:} by Lemma A2, a trial handles at most $(2m+1)2^{m-d(M)}$ classes. It lists at most
$(n+1)2^{|R|}(2m+1)2^{m-d(M)}2^{1-\gamma m}$ class records in expectation. With Lemma A3(2), the total is at most
$2^{(r + (1-\gamma)\mu)n + O(n^{0.6})} + 2^{\mu n + O(n^{0.6})}$. Use (C3), and $\mu < \tau_0 - \varepsilon_0$.
\item \emph{Visits and expansions.} Each expansion step handles one tuple $(\kappa_L, X, \kappa_R, Y)$, where $Y$ is a member of a
visited class record. Call such a tuple \emph{visited}. Fix a trial's $F$ and $P'$. For values $x_L \in \Sigma(L)$ and
$x_R \in \Sigma(R)$ and masks $X, Y$ of sizes $k$ and $\sigma - k$, put
\[D = x_L + c^F(X) + x_R + c^F(Y) - t^F.\]
$D$ does not depend on $P$. A visited tuple with $c(\kappa_L) = x_L$ and $c(\kappa_R) = x_R$ needs three events.
First, $D \ne 0$, because the exact interval is skipped. Second, $P \mid D$, because the reduced weights add to
$T^F \equiv t^F \pmod P$ and $b_i \equiv c_i$. Third, both small-prime conditions hold; given everything else, this
happens for at most one residue $j'$. By (F4), $\Pr_P[P \mid D] \le \mathrm{poly}(n)(b+\log n)2^{-\eta n}$, as
$|D| < 2^{b + O(\log n)}$. By Lemma A2, at most $(n+1)^2$ key pairs share the values $x_L$ and $x_R$. Hence one
trial visits in expectation at most
\[\mathrm{poly}(n)\,(b + \log n)\,2^{|L| - d(L) + |R| - d(R)}\binom mk\binom m{\sigma-k}2^{-\eta n - \gamma m}\]
tuples. With Lemma A3(3) and $d(R) \ge 0$, the total over all trials is
\[(b + \log n)\,2^{(1 - \mu + (h(a) + h(1/2-a) - \gamma)\mu - \eta)n + O(n^{0.6})}.\]
By (C4) the exponent is below $(\tau_0 - \varepsilon_0)n$.
\item \emph{Matching, dictionaries and output:} polynomial work for each left record, each class record, each visited tuple
and each dictionary key.
\end{enumerate}

Every item is at most $(b+2)\,2^{(\tau_0 - \varepsilon_0 + \zeta)n + O(n^{0.6})}$. $\square$

The proof of item 7 averages over $P$, but it does not condition on the event of Lemma A4. Only coverage uses that event.

\subsection*{5.8 Proof of Theorem A}

Part 2 holds by Section 5.2. For part 1, consider these failure events.

\begin{itemize}
\item $\mathcal E_3$ fails: probability $o(1)$, by Lemma A3.
\item The event of Lemma A4 fails: probability $o(1)$. Here $M$ is independent of $P$.
\item Some alias class is missed, given the two events above: probability $o(1)$, by Corollary A6.
\item The cap is reached. By Markov's inequality and Lemma A7, this has probability at most
\[o(1) + \frac{(b+2)\,2^{(\tau_0 - \varepsilon_0 + \zeta)n + O(n^{0.6})}}{2^{\tau_0 n + n^{0.7}}}.\]
The guard gives $b + 2 \le 2^{n/10^6 + 1}$, and $10^{-6} + \zeta < \varepsilon_0$. So this is $o(1)$.
\item A bounded sampler fails. A capped execution makes at most $2^{O(n)}$ draws, so this has probability
$2^{O(n)}(2^{-n^3} + e^{-c_0n^2}) = o(1)$.
\end{itemize}

Outside these events, $\mathcal D_s$ contains every alias class of $t_s$ (Corollary A6) and nothing else (Lemma A1). Then
the exact output sum is $\sum_{I\in\mathcal A_s(P)}\chi(I)$. The evaluation of Section 6.2 computes each phase to error
$2^{-2n^2}$ and rounds each term to $n^2$ fractional bits. The sums $\Phi_L$ and $\Phi_R$ have at most $2^n$ terms, each
dictionary has at most $2^n$ keys, and every modulus is at most $2^n$. So the total error is $2^{-n^2 + O(n)} \le 1/16$
for large $n$. $\square$

\medskip\noindent\rule{\linewidth}{0.4pt}\medskip

\section*{6. The checksum with a filter family (Theorem B)}\addcontentsline{toc}{section}{6. The checksum with a filter family (Theorem B)}

\subsection*{6.1 The family}

The family has $J = 43$ filters $\mathsf f_v = (r_v, s_v, \kappa_v)$, $v = 0, \dots, 42$, with
\[r_v(x) = \sum_{j=1}^8 r_{v,j}\cos jx,\qquad s_v(x) = \sum_{j=1}^8 s_{v,j}\cos jx,\qquad \kappa_v \in [1.02371,\ 1.06368].\]
The coefficients are decimal rationals with at most five digits after the point (Appendix A). For each filter, Section 3 of the paper defines $F_v = 2^{r_v}$, $G_v = 2^{s_v}$, the
densities $f_v$ and $g_v$, their normalized powers $f_{*,v}$ and $g_{*,v}$ with exponent $\kappa_v$, and the functions
$D_v, A_v, E_v$ of (3.2). The paper uses one filter with $\kappa = 1.0315$; here each filter has its own $\kappa_v$. Put
\[u_v = A_v + E_v - D_v,\qquad w_v = 2D_v - A_v - E_v.\]
These functions exist on the continuous circle $\mathbb T$ and, for each $P$, on the finite circle $\mathbb Z/P\mathbb Z$
(F7).

\textbf{Regions.} Put $b_0 = 1$, $b_1 = 1.25$, $b_2 = 1.6$, $b_3 = 2$, $b_4 = 2.6$, $b_5 = 3.5$, $b_6 = 5$, $b_7 = 8$,
$b_8 = 15$ and $b_9 = \infty$. A real number $W$ lies in the region $\rho_0$ if $W \le \Gamma$, and in the region
$\rho_i$ ($1 \le i \le 9$) if $\Gamma b_{i-1} < W \le \Gamma b_i$.

\textbf{Certificate data.} For each filter $v$ and region $\rho$, Appendix C gives a vector $z = z^{(v,\rho)}$ with
nonnegative entries and a set $C = C^{(v,\rho)} \subseteq \{0, \dots, 42\}\setminus\{v\}$. They define the row vector
$G = G^{(v,\rho)}(x)$ and a condition on $z_0$:

\begin{itemize}
\item for $\rho_0$: $G = \big(\Delta - u_v,\ \Gamma - w_v,\ (u_v - u_l)_{l\in C}\big)$, and $z_0 \ge 1$;
\item for $\rho_i$ with $1 \le i \le 8$: $G = \big(a_\Gamma w_v - u_v,\ w_v - \Gamma b_{i-1},\ \Gamma b_i - w_v,\ (u_v - u_l)_{l\in C}\big)$, and $z_0b_{i-1} \ge 1$;
\item for $\rho_9$: $G = \big(a_\Gamma w_v - u_v,\ w_v - 15\Gamma,\ (u_v - u_l)_{l\in C}\big)$, and $15z_0 \ge 1$.
\end{itemize}

Every certificate has $\sum_tz_t \le 418.36$.

\textbf{Proposition B (certified moments).} For each of the $43\times10 = 430$ pairs $(v, \rho)$,
\[\int_{\mathbb T}M(x)^2\,2^{z\cdot G(x)}\,dx < 2^{\eta - 0.00025},\]
where the measure on $\mathbb T = \mathbb R/2\pi\mathbb Z$ is normalized and $M(x)^2 = 2 + 2\cos x$.

\emph{Proof.} Section 8.2. The least $\log_2$ margin is $0.00025379$, at filter 28 in region $\rho_0$. $\square$

\textbf{Lemma B1 (finite circle).} There is an absolute $P_0$ with the following property. For all $P \ge P_0$ and every
filter, the finite-circle functions satisfy the conclusions of Lemma 3.2 other than (3.7). Moreover,
\[\frac1P\sum_{x\in\mathbb Z/P\mathbb Z}M(x)^2\,2^{z\cdot G(x)} < 2^{\eta - 0.0002}\qquad\text{for all 430 pairs } (v,\rho).\]

\emph{Proof.} By (F7), the proof of Lemma 3.2 applies to each filter. It shows that the finite-circle $D_v$, $A_v$ and $E_v$
converge uniformly to the continuous ones at the grid points. It also gives the uniform bounds, the order (3.3), and the
effective approximations. Each row of $G$ is a fixed linear combination of these functions and constants. So the finite
averages converge to the integrals of Proposition B, by the same Riemann-sum estimate. There are finitely many pairs, and
each has the slack $2^{\eta - 0.0002} - 2^{\eta - 0.00025} > 0$. $\square$

\subsection*{6.2 The construction}

The checksum procedure is the one of the paper's Section 4, with $\tau = \tau_0$ and $\eta = 0.51613$, and with three
changes.

\begin{enumerate}
\item \textbf{Patterns.} For each block $B_j$ and each filter $v$, a frequency's pattern records rational approximations
$D_{v,j}, A_{v,j}, E_{v,j}$ of the block averages of $D_v, A_v, E_v$ at its coordinates. They follow the paper's
rounding rule, so each has error at most $\epsilon$, and $D_{v,j} \ge 0$, $A_{v,j} \le D_{v,j}$, $E_{v,j} \le D_{v,j}$.
The collection $\mathcal P$ of all such blockwise tuples has $Q_n = 2^{O(\sqrt n)}$ elements, since $J$ is fixed. For
$\pi \in \mathcal P$, put $(D_{v,\pi}, A_{v,\pi}, E_{v,\pi}) = \frac1n\sum_jd_j(D_{v,j}, A_{v,j}, E_{v,j})$,
$U_{v,\pi} = A_{v,\pi} + E_{v,\pi} - D_{v,\pi}$ and $W_{v,\pi} = 2D_{v,\pi} - A_{v,\pi} - E_{v,\pi}$.
\item \textbf{Filter choice.} For $\pi\in\mathcal P$, let $v^*(\pi)$ be the least index $v$ that maximizes $U_{v,\pi}$.
\item \textbf{Tables.} For a pattern $\pi$ with $v = v^*(\pi)$, run the paper's construction with filter $v$. That is, $\alpha_\pi$
and $H_\pi$ come from (4.8) with $(D_\pi, A_\pi, E_\pi, U_\pi, W_\pi) := (D_{v,\pi}, A_{v,\pi}, E_{v,\pi}, U_{v,\pi}, W_{v,\pi})$.
The table counts are $R_j = 2^{\lceil(\alpha_\pi D_{v,j} + 4\epsilon)d_j\rceil}$. The scores are $\log f_{*,v}$ and
$\log g_{*,v}$, and the thresholds are $\alpha_\pi A_{v,j} - 3\epsilon$ and $\alpha_\pi E_{v,j} - 3\epsilon$. The pattern
is summed directly if $\alpha_\pi U_{v,\pi} \ge \Delta$. Otherwise it is sampled with $N = 2^{\lceil\tau_0 n\rceil}$
draws.
\end{enumerate}

Multiplicities, caps, sampling and the finite-precision arithmetic do not change.

Lemma 4.2 holds for this construction with $U_\pi := U_{v^*(\pi),\pi}$. Its proof treats one pattern at a time, with one
filter. Part (i) uses only that the pattern of $k$ records the block averages at the $x_i(k)$ for the filter of that
pattern. Parts (ii) and (iii) use only (4.5), (4.8), (4.9), the normalization of $f_{*,v}$ and $g_{*,v}$, and the joint
moment (4.14). These hold for every filter.

\subsection*{6.3 Domination and the error bound}

Let $\mathcal G$ be the event of Lemma 4.2(iii). For a frequency $k$, put $x_i = x_i(k)$, and let $\bar u_l$ and $\bar w_l$
be the averages of $u_l$ and $w_l$ over the coordinates $x_1, \dots, x_n$. As in (4.21), the pattern $\pi$ of $k$
satisfies, for every filter $l$,
\[|U_{l,\pi} - \bar u_l| \le 3\epsilon,\qquad |W_{l,\pi} - \bar w_l| \le 4\epsilon. \tag{6.1}\]

\textbf{Lemma B2 (domination).} Let $\pi$ be a sampled pattern, $v = v^*(\pi)$, and $\rho$ the region of $W_{v,\pi}$. Let $z$ and
$G$ be the certificate data for $(v, \rho)$. Then every frequency $k$ of pattern $\pi$ satisfies
\[-\alpha_\pi U_{v,\pi} \le -\Delta + \frac1n\sum_{i=1}^nz\cdot G(x_i) + 6\epsilon\sum_tz_t.\]

\emph{Proof.} Let $\bar G = \frac1n\sum_iG(x_i)$. Let $G^\pi$ be the same rows with $\bar u_v$, $\bar w_v$ and $\bar u_l$ replaced
by $U_{v,\pi}$, $W_{v,\pi}$ and $U_{l,\pi}$. Each row is affine in at most two of these quantities, with coefficients of
absolute value at most 1 ($a_\Gamma < 1$). By (6.1), $|\bar G_t - G^\pi_t| \le 6\epsilon$ for every row $t$. In detail,
the rows have errors at most $3\epsilon$, $4\epsilon$, $6\epsilon$, $(4a_\Gamma + 3)\epsilon$, $4\epsilon$ and $4\epsilon$.

The rows $t \ge 1$ have $G^\pi_t \ge 0$. The rows $\Gamma - W$, $W - \Gamma b_{i-1}$ and $\Gamma b_i - W$ are nonnegative,
because $W_{v,\pi}$ lies in the region $\rho$. The rows $U_{v,\pi} - U_{l,\pi}$ are nonnegative, because $v$ maximizes
$U_{\cdot,\pi}$. Since $z \ge 0$,
\[z\cdot\bar G \ge z_0G^\pi_0 - 6\epsilon\textstyle\sum_tz_t.\]

\emph{Region $\rho_0$.} Here $W_{v,\pi} \le \Gamma$, so $\alpha_\pi = 1$. Sampling means $U_{v,\pi} < \Delta$. Then
$G^\pi_0 = \Delta - U_{v,\pi} > 0$ and $z_0 \ge 1$ give $-U_{v,\pi} = -\Delta + G^\pi_0 \le -\Delta + z_0G^\pi_0$.

\emph{Region $\rho_i$, $i \ge 1$.} Here $W_{v,\pi} > \Gamma b_{i-1} \ge \Gamma$. So $\alpha_\pi = \Gamma/W_{v,\pi} < 1/b_{i-1} \le z_0$,
and $\alpha_\pi a_\Gamma W_{v,\pi} = \Delta$. Sampling means $\alpha_\pi U_{v,\pi} < \Delta$, that is,
$G^\pi_0 = a_\Gamma W_{v,\pi} - U_{v,\pi} > 0$. Then $-\alpha_\pi U_{v,\pi} = -\Delta + \alpha_\pi G^\pi_0 \le -\Delta + z_0G^\pi_0$.

In both cases, combine with the display above. $\square$

\textbf{Lemma B3 (one sampled pattern).} Let $\pi$ be a sampled pattern and $\rho$ a target residue, and let $E_{\pi,\rho}$ be the
error of the ideal estimate for $\pi$. Then $\mathbb E\big[\mathbf 1_{\mathcal G}|E_{\pi,\rho}|^2\big] \le 2^{-0.00017n + 2}$
for all large $n$.

\emph{Proof.} Put $c_z = 6\sum_tz_t \le 2511$. By (4.11), (4.4) and Lemma B2, every $k < P$ satisfies
\[\mathbf 1_{\mathcal G}\mathbf 1_{\{\mathrm{pattern}(k) = \pi\}}\,L_\pi\,|J_\rho(k)|^2 \le
2^{(\eta - \Delta + (18 + c_z)\epsilon)n}\prod_{i=1}^nM(x_i(k))^2\,2^{z\cdot G(x_i(k))}.\]
The right side does not depend on the pattern or on $\mathcal G$. For fixed $k$, the $x_i(k)$ are independent and uniform
on $\mathbb Z/P\mathbb Z$ under the original phase law. So Lemma B1 bounds the expectation of the right side by
$2^{(\eta - \Delta + (18 + c_z)\epsilon)n}\,2^{(\eta - 0.0002)n}$. As in (4.25), take the expectation of (4.20) and sum over
the $P$ frequencies:
\[\mathbb E\big[\mathbf 1_{\mathcal G}|E_{\pi,\rho}|^2\big] \le \frac{2^{(2\eta - \Delta + (18 + c_z)\epsilon - 0.0002)n}}{NP}
\le 2^{((18 + c_z)\epsilon - 0.0002)n + 2}.\]
The last step uses $N \ge 2^{\tau_0 n}$, $P \ge 2^{\eta n - 2}$ and $\eta - \Delta - \tau_0 = 0$. Finally
$(18 + c_z)\epsilon < 2.6\cdot10^{-5}$. $\square$

\subsection*{6.4 Theorem B}

\textbf{Theorem B.} Fix a constant $d > 0$. There is a uniform randomized procedure with the following properties for all
sufficiently large $n$. Its inputs are an integer $P$ with $|\log P - \eta n| \le 2$, residues $b_1, \dots, b_n$ modulo
$P$, phases $\varphi_1, \dots, \varphi_n$, and a list $\mathcal T$ of at most $n^d$ target residues. It returns FAIL or
estimates $\hat V_\rho$, $\rho \in \mathcal T$, of
\[V_\rho = \sum_{I\subseteq[n]:\ b(I)\equiv\rho\ (\mathrm{mod}\ P)}\chi(I).\]
If the phases are independent and uniform, then with probability $1 - o(1)$ over the phases and the procedure's own
draws, it does not fail and $|\hat V_\rho - V_\rho| \le 1/8$ for every $\rho \in \mathcal T$. The $o(1)$ is uniform in
$P$, the residues and $\mathcal T$. On every execution the procedure uses at most
$2^{(\tau_0 + 20\epsilon)n + O(\sqrt n)}\,\mathrm{poly}(n + |\mathcal T|)$ word operations.

\emph{Proof.} This is Theorem 4.1 with new constants, a filter family, and phases given as input. The paper's proof applies
with these changes.

\begin{itemize}
\item \emph{Sections 4.1--4.3.} Use the construction of Section 6.2 above; Lemma 4.2 holds, as explained there. The paper's
procedure draws the phases itself. Its analysis uses only that the phases are uniform, independent, and independent
of the tables and the samples. This holds here.
\item \emph{Section 4.4.} Direct patterns are exact on $\mathcal G$, as before. For sampled patterns, Lemma B3 replaces
(4.24)--(4.25). Cauchy--Schwarz over the $Q_n = 2^{O(\sqrt n)}$ patterns gives
$\mathbb E\big[\mathbf 1_{\mathcal G}|\sum_\pi E_{\pi,\rho}|^2\big] \le Q_n^2\,2^{-0.00017n + 2}$. Markov's inequality and a
union bound over $\mathcal T$ show that, with probability $1 - o(1)$, $\mathcal G$ holds and every ideal estimate has
error at most $1/16$.
\item \emph{Section 4.5.} Resources and precision do not change. The pattern of a frequency needs $3Jq$ approximate block
averages; Lemma B1 computes each in polynomial time. The arithmetic error is at most $1/16$. $\square$
\end{itemize}

\medskip\noindent\rule{\linewidth}{0.4pt}\medskip

\section*{7. Proof of Theorem 1}\addcontentsline{toc}{section}{7. Proof of Theorem 1}

\textbf{The algorithm.}

\begin{enumerate}
\item If $n < n_*$ or $10^6\lceil\log(b+1)\rceil > n$, run the exact meet-in-the-middle fallback of Section 6.3, and stop.
\item Draw the tie weights, and form $c_i$ and $t_s$ by (2.1).
\item Run the preliminary search of Proposition P. If it decides, return its answer.
\item Draw the prime $P$ uniformly from $[2^{\lfloor\eta n\rfloor}, 2^{1+\lfloor\eta n\rfloor})$ (Section 6.2). Draw independent
uniform phases $\varphi_1, \dots, \varphi_n \in \mathbb Z/P\mathbb Z$, and retain them. Put $b_i = c_i \bmod P$.
\item Run the procedure of Theorem A on $(c, t, P, \varphi)$. If it fails, return NO.
\item Run the procedure of Theorem B on $(P, b, \varphi)$ with the target residues $t_s \bmod P$. If it fails, return NO.
\item For each $s$, put $\hat B_s = \hat V_{t_s\bmod P} - \hat A_s$. Return YES if $|\hat B_s| \ge 1/2$ for some $s$, and NO
otherwise. The test compares the sum of the squares of the dyadic real and imaginary parts with $1/4$.
\end{enumerate}

\textbf{Correctness.} Since $b_i \equiv c_i \pmod P$, the sum $V_{t_s\bmod P}$ runs over the subsets with
$c(I) \equiv t_s \pmod P$. So for every $s$,
\[V_{t_s\bmod P} - \sum_{I\in\mathcal A_s(P)}\chi(I) = B_s := \sum_{I:\ c(I) = t_s}\chi(I).\]
Fix the tie weights. If (LD) fails, Proposition P decides with probability $1 - O(1/n)$. If (LD) holds, Theorem A gives
$|\hat A_s - \sum_{\mathcal A_s(P)}\chi| \le 1/16$ for all $s$ with probability $1 - o(1)$. Its success event depends only
on $P$ and its own draws, not on the phases. For every fixed $P$, Theorem B gives $|\hat V_\rho - V_\rho| \le 1/8$ for all
targets, with probability $1 - o(1)$ over the phases and its own draws. As in the paper, the main procedures also draw
their randomness when the preliminary search decides; then a union bound composes the three statements. So, except with
probability $o(1)$, either the preliminary answer is returned, or $|\hat B_s - B_s| < 1/4$ for all $s$.

In the second case the decision is correct, as in Section 6.3. On a NO instance every $B_s$ is $0$, by Lemma 2.1. On a
YES instance, with probability at least $11/12$ some $t_s$ has exactly one representing subset, and then $|B_s| = 1$. All
$o(1)$ terms depend only on $n$, so a fixed cutoff $n_*$ gives success probability at least $5/6$. Below $n_*$ and when the
guard fails, the fallback is exact.

\textbf{Running time.} Take an execution with $n \ge n_*$ that passes the guard.

\begin{itemize}
\item Tie weights and the transformation: $\mathrm{poly}(n+b)$.
\item Proposition P: $2^{\tau_0 n}\,\mathrm{poly}(n+b+2)$.
\item The prime $P$: at most $n^3$ candidates, each tested by trial division in $2^{\eta n/2 + O(1)}\,\mathrm{poly}(n)$.
\item Theorem A: $2^{\tau_0 n + 2n^{0.7}}\,\mathrm{poly}(n+b+2)$.
\item Theorem B: $2^{(\tau_0 + 20\epsilon)n + O(\sqrt n)}\,\mathrm{poly}(n)$, as there are $O(n^2)$ targets.
\end{itemize}

So the total is at most $K_0\,2^{(\tau_0 + 20\epsilon)n + 2n^{0.7}}(n+b+2)^K$ for fixed constants $K_0$ and $K$. Fix $c$.
For large $n$, every $b \le n^c$ passes the guard, and
\[K_0\,2^{0.4876902n + 2n^{0.7}}(n + n^c + 2)^K = O_c(2^{0.4877n}).\]
The address-space argument of Section 6.3 applies unchanged: the logarithm of the operation count is
$0.4877n + O(\log(n+b+2))$, which is below the word length $w$. The fallback halts on every input and is exact. $\square$

\medskip\noindent\rule{\linewidth}{0.4pt}\medskip

\section*{8. The certificates}\addcontentsline{toc}{section}{8. The certificates}

\subsection*{8.1 The alias constraints}

The script of Appendix D.1 reads $\tau_0, \eta, a, \mu, \gamma, l$ (Section 2.2) as exact decimal
rationals. It evaluates (C1)--(C8) in Arb ball arithmetic at 200-bit precision, with
$h(x) = -(x\ln x + (1-x)\ln(1-x))/\ln2$. The run on 12 October 2026 printed:

\begin{lstlisting}
h(2a)/2                  [0.060359873 +/- 1.01e-10]
left list                [0.48767640 +/- 1.94e-9]
right classes            [0.48768445 +/- 3e-13]
pairs                    [0.48768463 +/- 6.24e-10]
right masks, all trials  [0.48521823 +/- 2.57e-9]
left masks, all trials   [0.044761952 +/- 6.27e-11]
(mu/2) log2 3            [0.37484363 +/- 1.43e-9]
v                        [0.46524280 +/- 3e-13]
l                        [0.48377000 +/- 3e-13]
r                        [0.043230000 +/- 1e-14]
least slack eps0 >= [5.3706e-6 +/- 2.34e-11] (lower end of the Arb balls)
ALL ALIAS CONSTRAINTS PASS (strict)
\end{lstlisting}

The ``left masks'' line is the left-mask analogue of (C5). It is implied by (C5), because $h(a) < h(\tfrac12 - a)$.

\subsection*{8.2 The 430 region integrals}

The script of Appendix D.2 follows the method of the paper's Appendix A (see B.5). It uses $N = 8192$ quadrature nodes, the inner
strip half-width $\sigma = 0.45$, the outer strip half-width $2\beta = 0.38$, and 200-bit precision. For each filter
$(r, s, \kappa)$ it does the following.

\begin{enumerate}
\item \textbf{Node values.} For $x_k = 2\pi k/N$, Arb computes $r(x_k)$, $s(x_k)$, $F(x_k) = 2^{r(x_k)}$ and $G(x_k) = 2^{s(x_k)}$.
\item \textbf{Normalizers.} $Z_f = \int F$ lies in $T_N(F) + [-e, e]$ with $e = 2B_0/(e^{N\sigma} - 1)$ and $B_0 = 2^{R}$,
$R = \sum_j|r_j|\cosh(j\sigma)$. This is the analytic trapezoid bound (A.6) of B.5 with $N$ nodes in place of 64, because
$|F| \le 2^{R}$ on the strip $|\mathrm{Im}\,z| \le \sigma$. The same holds for
$Z_g$, $Z_{f,\kappa} = \int F^\kappa$ and $Z_{g,\kappa}$.
\item \textbf{Convolutions.} The script encloses $\mathcal P(x) = \int F(x+w)G(w)\,dw$ and
$\mathcal Y(x) = \int F(x+w)G(w)\,(r(x+w) + s(w))\,dw$ at all nodes. The trapezoid sums are exact cyclic correlations,
computed by Arb polynomial products. The error bounds use $B_0 = 2^{R_r + R_s}$ and $(R_r + R_s)2^{R_r + R_s}$.
\item \textbf{Scores.} Put $L = \log\mathcal P - \log(Z_fZ_g)$, $B = \mathcal Y/\mathcal P - \log(Z_fZ_g)$ and
$H = \log Z_{f,\kappa} + \log Z_{g,\kappa} - \kappa\log(Z_fZ_g)$. The identities (A.2) give $D = B - L$,
$u = L + (\kappa-1)B - H$ and $w = (2-\kappa)B - 2L + H$. So the resulting balls contain the exact values $u(x_k)$ and
$w(x_k)$.
\item \textbf{Outer strip.} As in the paper's Appendix A.4 (see B.5), on the strip $|\mathrm{Im}\,z| \le 2\beta$: let $r_\beta$ have the coefficients
$r_j\cosh(j\beta)$, and put $A_f = \int2^{2r_\beta}/Z_f^2$; define $A_g$ in the same way. Then
$q_0 = \sqrt{(A_f-1)(A_g-1)}$ bounds $|Q - 1|$, and
$q_b = \sqrt{A_fA_g}\,\big(\sum_j|r_j|\cosh(j\beta) + \sum_j|s_j|\cosh(j\beta) + |\log(Z_fZ_g)|\big)$ bounds $|QB|$.
All 43 filters have $q_0 < 0.677$. Then $|\mathrm{Re}\,L| \le -\log(1 - q_0)$ and $|B| \le q_b/(1-q_0)$. These
give bounds $U^*$ on $|\mathrm{Re}\,u|$ and $W^*$ on $|\mathrm{Re}\,w|$.
\end{enumerate}

For each certificate $(v, \rho)$, the script averages the node values of $M^22^{z\cdot G}$. On the outer strip, the real
part of each row is at most the matching combination of $\Delta$, $\Gamma$, $b_i$, $U^*$ and $W^*$. With
$|M(z)^2| \le 2 + 2\cosh2\beta$, this bounds the integrand by some $B_0$. The outer error is at most
$2B_0/(e^{2\beta N} - 1)$, which is below $10^{-300}$ for every certificate. The script checks that the average plus the
error is below $2^\eta$. It also asserts $z \ge 0$ and the condition on $z_0$.

The trapezoid bound needs analyticity on a neighborhood of the closed strip. The inner integrands are entire. For the
outer integrand, the bounds hold with $q_0 < 1$ strictly; so one can apply the bound on the strip of half-width
$0.38 - \delta$ for a small $\delta > 0$. This changes the error term by a negligible amount.

\textbf{Result.} All 430 inequalities hold. The run on 12 October 2026 reproduced the log of 9 October 2026 line by line.
The least $\log_2$ margins, by region:

{\small
\begin{longtable}{>{\raggedright\arraybackslash}p{0.477\dimexpr\linewidth-6\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.386\dimexpr\linewidth-6\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.136\dimexpr\linewidth-6\tabcolsep\relax}}
\hline
\textbf{Region} & \textbf{Least $\log_2$ margin} & \textbf{Filter} \\ \hline
\endhead
$\rho_0$: $W \le \Gamma$ & 0.00025379 & 28 \\
$\rho_1$: $1 < W/\Gamma \le 1.25$ & 0.00035951 & 34 \\
$\rho_2$: $1.25 < W/\Gamma \le 1.6$ & 4.1558 & 3 \\
$\rho_3$: $1.6 < W/\Gamma \le 2$ & 10.605 & 3 \\
$\rho_4$ to $\rho_9$: $W/\Gamma > 2$ & at least 17.967 & 3 \\
\hline
\end{longtable}}

I also checked the certificate data directly: every filter has all ten regions, every $z$ is nonnegative, every
condition on $z_0$ holds, and no set $C^{(v,\rho)}$ contains $v$.

\subsection*{8.3 Repeating the checks}

Both scripts need Python with python-flint 0.9.0. Each takes one argument, a JSON file that holds the data printed in
this note. For the script of Appendix D.1, the file is one object with the keys \texttt{tau0}, \texttt{eta}, \texttt{a}, \texttt{mu}, \texttt{gamma} and
\texttt{l}, set to the values of Section 2.2. For the script of Appendix D.2, it is one object with the keys \texttt{tau0} and \texttt{eta},
a list \texttt{filters} of objects \texttt{\{r\_\allowbreak{}coef, s\_\allowbreak{}coef, kappa\}} from Appendix A, and a list \texttt{certs} of objects
\texttt{\{v, case, lo, hi, z, cons\}} from Appendix C. Here \texttt{case} is \texttt{"case1"} for $\rho_0$ and \texttt{"band"} otherwise; \texttt{lo} and
\texttt{hi} are $b_{i-1}$ and $b_i$ for $\rho_i$, and \texttt{hi} is \texttt{null} for $\rho_9$; \texttt{cons} is the set $C$. The second script
takes a few minutes.

\textbf{Provenance.} A cutting-plane search produced the filters, and linear programming produced the multipliers $z$.
A separate optimization produced the alias parameters. These programs are not part of the proof. The certificates are
checked independently of them.

\subsection*{8.4 Limits}

The alias constraints (C1)--(C4) are tight to about $10^{-5}$, and the checksum certificate has a margin of $0.00025$. So this
design cannot certify $\tau_0 = 0.4876$. At $\tau_0 = 0.4876$ the cutting-plane search for filters stalls at a
$\log_2$ margin of $-0.0015$. Further progress needs a new idea.

\medskip\noindent\rule{\linewidth}{0.4pt}\medskip

\section*{Appendix A. The 43 filters}\addcontentsline{toc}{section}{Appendix A. The 43 filters}

$r_v(x) = \sum_{j=1}^8r_{v,j}\cos jx$ and $s_v(x) = \sum_{j=1}^8s_{v,j}\cos jx$. Appendix C gives the multipliers $z^{(v,\rho)}$ and the sets $C^{(v,\rho)}$.

{\scriptsize
\begin{longtable}{>{\raggedright\arraybackslash}p{0.066\dimexpr\linewidth-8\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.066\dimexpr\linewidth-8\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.428\dimexpr\linewidth-8\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.440\dimexpr\linewidth-8\tabcolsep\relax}}
\hline
\textbf{$v$} & \textbf{$\kappa_v$} & \textbf{$r_{v,1},\dots,r_{v,8}$} & \textbf{$s_{v,1},\dots,s_{v,8}$} \\ \hline
\endhead
0 & 1.02371 & 0.99307, 0.47365, 0.24703, -0.26955, 0.07942, -0.1185, -0.00587, 0.06006 & 1.29217, -0.81333, 0.44354, 0.08221, 0.05022, 0.11128, -0.02479, -0.0816 \\
1 & 1.03833 & 1.11312, 0.4803, -0.14032, -0.06862, 0.07166, -0.03251, -0.0345, 0.02296 & 1.37762, -0.74295, 0.12852, -0.0223, 0.02354, 0.00279, -0.04016, -0.04383 \\
2 & 1.05826 & 1.20389, 0.34962, -0.15858, -0.02392, 0.02946, -0.00605, -0.00528, 0.0034 & 1.41719, -0.61945, 0.04713, -0.00707, 0.00439, 0.00127, -0.00975, -0.00124 \\
3 & 1.06368 & 0.6425, 0.54635, 0.73657, -0.56823, -0.21433, 0.03123, -0.00347, 0.01585 & 0.87763, -0.69266, 0.73949, 0.38951, -0.20075, 0.0364, -0.09908, 0.01145 \\
4 & 1.04768 & 0.76631, 0.61145, 0.24893, -0.41359, -0.49022, 0.10407, -0.01777, 0.05843 & 1.07613, -0.70375, 0.20271, 0.45459, -0.60319, 0.14968, -0.10179, -0.04873 \\
5 & 1.0433 & 0.83495, 0.46517, 0.5449, -0.47927, -0.07681, 0.34334, -0.11624, -0.00504 & 1.08519, -0.70466, 0.6481, 0.22751, -0.03469, -0.23636, -0.12224, 0.0097 \\
6 & 1.03556 & 0.87031, 0.62799, 0.1799, -0.35158, -0.09333, 0.32235, -0.36841, 0.01375 & 1.13882, -0.81379, 0.36137, 0.19229, -0.1388, -0.11502, -0.3531, 0.01982 \\
7 & 1.03283 & 0.91928, 0.45906, 0.37137, -0.36271, -0.39634, 0.24445, 0.0778, 0.02384 & 1.16315, -0.65175, 0.4156, 0.24669, -0.31226, -0.1391, 0.05494, -0.08916 \\
8 & 1.03767 & 0.78095, 0.64691, 0.41453, -0.41692, -0.25846, 0.07514, 0.03248, 0.12195 & 1.12144, -0.82371, 0.47376, 0.28329, -0.23722, 0.02516, -0.05553, -0.09075 \\
9 & 1.03461 & 0.86927, 0.53001, 0.41132, -0.44655, -0.2227, 0.32734, -0.04267, 0.01694 & 1.11897, -0.71831, 0.49366, 0.25437, -0.15994, -0.2237, -0.03326, -0.05703 \\
10 & 1.03567 & 0.8321, 0.52819, 0.46889, -0.20175, -0.42154, 0.04074, 0.24988, -0.0618 & 1.16305, -0.75504, 0.56489, 0.07749, -0.3697, 0.06344, 0.05195, 0.09625 \\
11 & 1.03551 & 0.84221, 0.53758, 0.37518, -0.42183, -0.31017, -0.0051, 0.31945, 0.00946 & 1.13288, -0.69997, 0.35377, 0.34385, -0.34748, 0.10405, 0.1547, -0.03961 \\
12 & 1.03316 & 0.87274, 0.48576, 0.45208, -0.38394, -0.35036, 0.1767, 0.08996, 0.03503 & 1.14932, -0.69434, 0.48876, 0.25489, -0.32142, -0.05422, 0.01517, -0.03626 \\
13 & 1.03419 & 0.85174, 0.56596, 0.38584, -0.3491, -0.16892, 0.23733, -0.2596, 0.02587 & 1.13489, -0.79903, 0.53282, 0.1833, -0.15998, -0.07574, -0.28489, -0.00352 \\
14 & 1.03247 & 0.88559, 0.53721, 0.30733, -0.25829, -0.41024, 0.37263, 0.04931, 0.01665 & 1.14677, -0.72577, 0.40769, 0.13659, -0.29855, -0.24429, 0.06347, -0.0611 \\
15 & 1.03454 & 0.84001, 0.57254, 0.38289, -0.34192, -0.38508, 0.27044, 0.04167, 0.01266 & 1.12958, -0.75004, 0.4546, 0.21648, -0.3285, -0.11539, -0.00938, -0.0199 \\
16 & 1.03481 & 0.8362, 0.53334, 0.45578, -0.40199, -0.29198, -0.0118, 0.27142, 0.01226 & 1.13334, -0.72772, 0.46413, 0.26541, -0.28542, 0.08643, 0.14305, -0.03728 \\
17 & 1.03175 & 0.89625, 0.50135, 0.36394, -0.37738, -0.32793, 0.31977, 0.05621, 0.02428 & 1.15022, -0.68644, 0.42637, 0.24349, -0.26955, -0.19193, 0.04992, -0.06989 \\
18 & 1.03342 & 0.86094, 0.53068, 0.4075, -0.35513, -0.35546, 0.27595, 0.10329, -0.00102 & 1.13535, -0.70827, 0.47222, 0.21607, -0.28977, -0.15573, 0.06196, -0.0245 \\
19 & 1.03267 & 0.88294, 0.51847, 0.38488, -0.42188, -0.29302, 0.27218, 0.0419, 0.04304 & 1.13389, -0.70929, 0.43803, 0.28072, -0.25033, -0.14536, 0.0215, -0.08702 \\
20 & 1.03545 & 0.86281, 0.45994, 0.48878, -0.32787, -0.25329, 0.29811, -0.2039, 0.01034 & 1.13221, -0.72677, 0.62432, 0.16134, -0.20135, -0.14383, -0.23787, 0.06405 \\
21 & 1.03455 & 0.88834, 0.49837, 0.38691, -0.38945, -0.28447, 0.25054, 0.31107, -0.25291 & 1.1201, -0.67256, 0.45506, 0.18534, -0.22116, -0.17539, 0.20721, 0.12574 \\
22 & 1.03499 & 0.85001, 0.54566, 0.40484, -0.39481, -0.26303, 0.19392, 0.26159, -0.26262 & 1.13131, -0.7394, 0.49309, 0.17757, -0.20576, -0.11705, 0.15214, 0.17794 \\
23 & 1.03534 & 0.85197, 0.53211, 0.41166, -0.42436, -0.31667, 0.23063, 0.24482, -0.19896 & 1.10977, -0.6779, 0.45797, 0.24574, -0.27742, -0.11887, 0.13567, 0.12986 \\
24 & 1.03477 & 0.84514, 0.54903, 0.42997, -0.37293, -0.30081, 0.27646, -0.08542, 0.02302 & 1.1227, -0.75372, 0.52559, 0.22297, -0.26354, -0.11398, -0.13015, 0.00362 \\
25 & 1.03483 & 0.8658, 0.58243, 0.25803, -0.37795, -0.25582, 0.38649, -0.26534, 0.05029 & 1.11181, -0.75235, 0.371, 0.25799, -0.26227, -0.15765, -0.25746, -0.00535 \\
26 & 1.03467 & 0.85888, 0.55394, 0.36415, -0.42775, -0.20245, 0.27706, -0.21773, 0.06224 & 1.11769, -0.74839, 0.46479, 0.29293, -0.22279, -0.08262, -0.24819, -0.00591 \\
27 & 1.03436 & 0.86478, 0.5278, 0.41777, -0.40473, -0.25776, 0.25915, 0.24767, -0.20953 & 1.12348, -0.70627, 0.50431, 0.17544, -0.18192, -0.19054, 0.17509, 0.11446 \\
28 & 1.03394 & 0.86308, 0.53287, 0.42654, -0.42302, -0.25014, 0.29894, -0.08177, 0.0007 & 1.1256, -0.73481, 0.52328, 0.24054, -0.20542, -0.15082, -0.11524, -0.00236 \\
29 & 1.03456 & 0.86313, 0.55186, 0.34889, -0.36638, -0.304, 0.36213, -0.15893, 0.03608 & 1.1167, -0.74128, 0.45205, 0.23333, -0.26878, -0.17088, -0.17039, -0.00989 \\
30 & 1.03415 & 0.85818, 0.54628, 0.38759, -0.44335, -0.29154, 0.27179, -0.01381, 0.03202 & 1.11312, -0.71038, 0.43662, 0.31628, -0.28431, -0.11752, -0.05788, -0.04319 \\
31 & 1.0341 & 0.85707, 0.54367, 0.37415, -0.43456, -0.26503, 0.33393, -0.07451, 0.00062 & 1.13404, -0.72594, 0.47385, 0.27029, -0.23227, -0.17966, -0.08723, -0.01861 \\
32 & 1.03491 & 0.85072, 0.55482, 0.38188, -0.40032, -0.2798, 0.30596, -0.15929, 0.06331 & 1.10328, -0.72304, 0.46223, 0.27734, -0.26758, -0.13156, -0.1789, -0.03912 \\
33 & 1.03398 & 0.88051, 0.50341, 0.40888, -0.45168, -0.23285, 0.23687, 0.26848, -0.21947 & 1.12135, -0.67729, 0.46221, 0.22873, -0.18142, -0.17482, 0.19505, 0.11387 \\
34 & 1.03437 & 0.86871, 0.51508, 0.42945, -0.43509, -0.25321, 0.2272, 0.23493, -0.22168 & 1.11865, -0.69451, 0.489, 0.22189, -0.21026, -0.14509, 0.13319, 0.14005 \\
35 & 1.03391 & 0.85057, 0.53343, 0.44509, -0.40952, -0.3385, 0.19825, 0.10004, -0.00846 & 1.12894, -0.70834, 0.48297, 0.26695, -0.30557, -0.06563, 0.01507, -0.00375 \\
36 & 1.03451 & 0.86692, 0.50471, 0.42791, -0.43793, -0.27782, 0.25233, 0.2257, -0.18455 & 1.12258, -0.68146, 0.48267, 0.23354, -0.22862, -0.15992, 0.13775, 0.11201 \\
37 & 1.03474 & 0.85647, 0.51996, 0.43532, -0.40925, -0.30279, 0.23825, 0.216, -0.19066 & 1.1223, -0.69736, 0.50061, 0.20757, -0.24504, -0.1422, 0.12123, 0.12733 \\
38 & 1.03474 & 0.86209, 0.51154, 0.43612, -0.44135, -0.27888, 0.22172, 0.23067, -0.19378 & 1.11689, -0.68778, 0.47777, 0.24619, -0.24013, -0.12521, 0.13192, 0.11959 \\
39 & 1.03469 & 0.864, 0.51097, 0.42096, -0.42708, -0.28147, 0.25407, 0.22187, -0.20024 & 1.11794, -0.68476, 0.47639, 0.23281, -0.23104, -0.16673, 0.13745, 0.11816 \\
40 & 1.03426 & 0.87586, 0.50613, 0.42275, -0.447, -0.24132, 0.22823, 0.26037, -0.19339 & 1.11444, -0.68726, 0.4671, 0.23682, -0.19155, -0.16238, 0.18424, 0.09582 \\
41 & 1.03452 & 0.86807, 0.50413, 0.43113, -0.42692, -0.27906, 0.24109, 0.21854, -0.17893 & 1.11829, -0.68867, 0.49081, 0.23105, -0.23279, -0.15416, 0.13344, 0.10685 \\
42 & 1.0341 & 0.85861, 0.53793, 0.39525, -0.43541, -0.24138, 0.34497, -0.06676, 0.00099 & 1.12556, -0.72213, 0.49236, 0.26456, -0.19709, -0.19857, -0.08363, -0.02398 \\
\hline
\end{longtable}}

\medskip\noindent\rule{\linewidth}{0.4pt}\medskip

\section*{Appendix B. Results used from the paper}\addcontentsline{toc}{section}{Appendix B. Results used from the paper}

The paper is: OpenAI, \emph{Subset Sum in Time $O(2^{0.49n})$}, 4 October 2026 (OpenAI result \#138). This appendix restates,
without proof, everything of the paper that this note uses. It keeps the paper's numbering, so ``Lemma 2.3'' in the body
means Lemma 2.3 below. We rely on the paper's proofs of these statements. Where a statement depends on the paper's
constants, we state it in the general form that the paper's proof gives; the note points out each such use.

\subsection*{B.1 Model, isolation and deficiency (paper, Sections 1--2)}

\textbf{Model (1.1).} An instance has positive integers $a_1, \dots, a_n$ ($n \ge 2$) and a target $t \ge 0$; equal integers
and the empty subset are allowed. Let $b = \max\{1, \lceil\log(1 + \max\{t, a_1, \dots, a_n\})\rceil\}$. The machine is a
word RAM with word length $w = \lceil 4(n + b + \log(n+2))\rceil$. Unit-cost operations act on a constant number of
words: reads and writes, indirect addressing, comparisons, addition, subtraction, multiplication, quotient and
remainder, Boolean operations and shifts. A fresh uniform random word costs one operation. Randomness is one-way: a
random word can be reused only if the program stored it. The program is one fixed program for all $n$ and $b$, with no
advice.

\textbf{Theorem 1.1 (paper).} The statement of our Theorem 1 with $0.49$ in place of $0.4877$.

\textbf{Transformation (2.1).} Let $N_0$ be the power of two with $12n \le N_0 < 24n$. Draw independent uniform
$u_i \in \{1, \dots, N_0\}$, and put
\[C = 1 + nN_0,\qquad c_i = Ca_i + u_i,\qquad t_s = Ct + s\quad (0 \le s \le nN_0).\]
These integers have $O(b + \log n)$ bits, and there are $O(n^2)$ targets.

\textbf{Lemma 2.1 (isolation).} For every choice of the $u_i$: some $I$ has $a(I) = t$ if and only if some $s$ and $I$ have
$c(I) = t_s$. If the instance has a solution, then with probability at least $11/12$ some $t_s$ has exactly one subset
$I$ with $c(I) = t_s$.

\textbf{Deficiency (2.2).} For integer weights and $V \subseteq [n]$: $\Sigma(V) = \{c(I) : I\subseteq V\}$ and
$d(V) = |V| - \log|\Sigma(V)|$.

\textbf{Lemma 2.2.} For arbitrary integer weights: $d(\varnothing) = 0$ and $d(V) \ge 0$. For $i \notin V$,
$0 \le d(V\cup\{i\}) - d(V) \le 1$. For disjoint $U, V$, $d(U\cup V) \ge d(U) + d(V)$. If $V, V'$ have equal size and
differ by the exchange of one index, then $|d(V) - d(V')| \le 1$.

\textbf{Lemma 2.3.} Let $V_q$ be a uniform $q$-subset of $[n]$. Then $\mathrm{Var}\,d(V_q) \le q$, and $\mathbb E\,d(V_q)$ is
nondecreasing in $q$. The proof exposes an ordered sample $X_1, \dots, X_q$ and shows that the Doob martingale
$M_r = \mathbb E[d(\{X_1, \dots, X_q\}) \mid X_1, \dots, X_r]$ has increments of absolute value at most 1. The same proof
applies to a uniform $q$-subset of any fixed finite ground set, and to any set function that changes by at most 1 under
the exchange of one element. (Our (F3) adds Azuma's inequality to this martingale bound.)

\textbf{Proposition 2.4 (capped preliminary search).} The procedure: choose a uniform $\lfloor n/2\rfloor$-subset $V$ by
bounded rejection sampling. For each of the two sides $V$ and $[n]\setminus V$, build the sorted list of distinct subset
sums by inserting one index at a time: merge the current list with its translate by $c_i$ and remove duplicates. Stop
with CONTINUE as soon as a list exceeds the cap $B_n$. If both lists finish, check for every $s$ whether two entries sum
to $t_s$, by a linear scan, and return YES exactly when this happens. The procedure returns the correct decision or
CONTINUE, and it runs in $O(B_n)\,\mathrm{poly}(n + b + 2)$ time on every execution. The paper uses
$B_n = 2^{\lceil 0.489985n\rceil}$ and proves: if $\mathbb E\,d(V) > 0.01004n$, then CONTINUE has probability $O(1/n)$.
Our Proposition P uses another cap and threshold, with the same proof (given in Section 4).

\textbf{Lemma 2.5 (random-prime collisions).} There are absolute constants $K$ and $L_*$ with this property. Let $L' \ge L_*$,
and let $p$ be uniform among the primes in $[2^{L'}, 2^{L'+1})$. If $D \ne 0$ is an integer with $|D| < 2^B$ and
$B \ge 1$, then $\Pr[D \equiv 0 \pmod p] \le KBL'2^{-L'}$.

\subsection*{B.2 The filter functions (paper, Section 3)}

Let $\mathbb T = \mathbb R/2\pi\mathbb Z$ with normalized measure. Let $f, g$ be strictly positive continuous probability
densities on $\mathbb T$, and $\kappa > 0$. Put
\[Q(x) = \int_{\mathbb T}f(x+w)g(w)\,dw,\qquad K_x(w) = \frac{f(x+w)g(w)}{Q(x)},\qquad
f_* = \frac{f^\kappa}{\int f^\kappa},\qquad g_* = \frac{g^\kappa}{\int g^\kappa}. \tag{3.1}\]
\[D(x) = \int K_x\log K_x,\quad A(x) = \int K_x(w)\log f_*(x+w)\,dw,\quad E(x) = \int K_x(w)\log g_*(w)\,dw,\]
\[u = A + E - D,\qquad v = 2D - A - E. \tag{3.2}\]
(This note writes $w$ for the paper's $v$.) These are relative entropies, so
\[D \ge 0,\qquad D - A \ge 0,\qquad D - E \ge 0. \tag{3.3}\]
The paper's filter takes trigonometric polynomials $r(x) = \sum_j r_j\cos jx$ and $s(x) = \sum_j s_j\cos jx$, puts
$F = 2^r$, $G = 2^s$, $f = F/\int F$ and $g = G/\int G$, and uses fixed constants $\kappa$, $r_j$, $s_j$. Its
Proposition 3.1 proves two moment bounds (3.6) for these constants; this note replaces them by Proposition B.

\textbf{Finite circle.} For an integer $P \ge 2$, identify $j \in \mathbb Z/P\mathbb Z$ with the angle $2\pi j/P$. Replace
every integral above, including all normalizers, by the average $P^{-1}\sum_{j=0}^{P-1}$.

\textbf{Lemma 3.2 (finite filtering data).} There is an integer $P_0$ such that for $P \ge P_0$ the finite-circle functions
satisfy (3.3) and the finite form (3.7) of the moment bounds. The absolute values of $D, A, E, u, v, \log f_*, \log g_*$ and
$\log K_x(w)$ are bounded independently of $P$. For each fixed rational $\rho > 0$, one fixed deterministic procedure
approximates each of these quantities to error $\rho$ in time polynomial in $\log P$, without a sum over the $P$ points.
The proof uses only that $r$ and $s$ are fixed trigonometric polynomials: the finite functions converge uniformly to the
continuous ones, and Riemann sums converge for the outer integrals.

\subsection*{B.3 The checksum (paper, Section 4)}

Fix $\tau$ and $\eta$ and put $\Delta = \eta - \tau$, $\Gamma = 2\tau - \eta$, $a_\Gamma = \Delta/\Gamma$ and
$\epsilon = 10^{-8}$. Let $M(x)^2 = 2 + 2\cos x$.

\textbf{Theorem 4.1 (modular checksum estimator).} Fix $d > 0$. There is a uniform randomized procedure with the following
guarantee for large $n$. Its input is an integer $P$ with $|\log P - \eta n| \le 2$, residues $b_1, \dots, b_n$ modulo $P$,
and at most $n^d$ target residues $\rho \in \mathcal T$. It draws and keeps uniform phases $\varphi_i$, and returns FAIL or
estimates $\hat V_\rho$ of $V_\rho = \sum_{I:\ b(I)\equiv\rho}e_P(\sum_{i\in I}\varphi_i)$. With probability $1 - o(1)$,
uniformly in $P$, the residues and $\mathcal T$, it does not fail and $|\hat V_\rho - V_\rho| \le 1/8$ for all
$\rho \in \mathcal T$. Every execution takes $2^{(\tau + 20\epsilon)n + O(\sqrt n)}\mathrm{poly}(n + |\mathcal T|)$
operations. The paper proves this for its constants; Theorem B is our version.

The construction and the facts we use are these.

\begin{itemize}
\item \textbf{Fourier form (4.3)--(4.4).} $V_\rho = P^{-1}\sum_{k=0}^{P-1}J_\rho(k)$ with
$J_\rho(k) = e_P(-k\rho)\prod_{i=1}^n(1 + e_P(x_i(k)))$ and $x_i(k) = \varphi_i + kb_i \bmod P$. So
$|J_\rho(k)|^2 = \prod_iM(x_i(k))^2$. For fixed $k$ and uniform independent phases, the $x_i(k)$ are independent and
uniform.
\item \textbf{Blocks and patterns (4.5)--(4.6).} Partition $[n]$ into $q = \Theta(\sqrt n)$ blocks $B_j$ of sizes
$d_j = \Theta(\sqrt n)$. A frequency's pattern records, for each block, rational approximations $D_j, A_j, E_j$ of the
block averages of $D, A, E$ at its coordinates $x_i(k)$, each with error at most $\epsilon$, on a fixed grid, rounded so
that $D_j \ge 0$, $A_j \le D_j$, $E_j \le D_j$ (4.5). The set $\mathcal P$ of all such grid tuples has
$Q_n = 2^{O(\sqrt n)}$ elements (4.6). For $\pi \in \mathcal P$: $(D_\pi, A_\pi, E_\pi) = \frac1n\sum_jd_j(D_j, A_j, E_j)$,
$U_\pi = A_\pi + E_\pi - D_\pi$ and $W_\pi = 2D_\pi - A_\pi - E_\pi$.
\item \textbf{Parameters (4.8)--(4.9).} $\alpha_\pi = 1$ if $W_\pi \le \Gamma$ and $\alpha_\pi = \Gamma/W_\pi$ otherwise;
$H_\pi = \tau - \alpha_\pi(D_\pi - A_\pi)$. Then $\Delta \le H_\pi \le \tau$ and
$\eta - H_\pi + \alpha_\pi(D_\pi - E_\pi) = \Delta + \alpha_\pi W_\pi \le \tau$.
\item \textbf{Digits.} $B_\pi = 2^{\lceil(\eta - H_\pi)n\rceil}$. High digits $0 \le h < \lceil P/B_\pi\rceil$ give vectors
$y_i(h) = \varphi_i + hB_\pi b_i$; low digits $0 \le \ell < B_\pi$ give $z_i(\ell) = -\ell b_i$. The pair $(h, \ell)$
represents $k = hB_\pi + \ell$, and $y_i(h) - z_i(\ell) = x_i(k)$.
\item \textbf{Tables (4.7).} Block $B_j$ gets $R_j = 2^{\lceil(\alpha_\pi D_j + 4\epsilon)d_j\rceil}$ independent tables. Each
coordinate of a table has label $*$ with probability $1 - \alpha_\pi$, and otherwise a uniform $\theta \in \mathbb Z/P\mathbb Z$.
A label $\theta$ gives the high score $\log f_*(y_i - \theta)$ and the low score $\log g_*(z_i - \theta)$; a label $*$
gives 0. Fixed approximations to error $\epsilon$ replace the scores. A vector passes a block table if its average
score is at least $\alpha_\pi A_j - 3\epsilon$ (high side) or $\alpha_\pi E_j - 3\epsilon$ (low side). A whole table is
a tuple of one block table per block; an entry is a digit with a whole table that it passes; a match is a high entry
and a low entry with the same whole table. $L_\pi$ is the number of matches.
\item \textbf{Lemma 4.2.} (i) Conditional on any phases, with probability $1 - o(1)$ every $k < P$ has a match in its own pattern.
(ii) In each pattern, both entry counts have expectation at most $2^{(\tau + 8\epsilon)n + O(\sqrt n)}$, and
$\mathbb E L_\pi \le 2^{(\eta - \alpha_\pi U_\pi + 12\epsilon)n + O(\sqrt n)}$ (4.10). (iii) With probability $1 - o(1)$,
all frequencies are covered and, in every pattern, both entry counts are at most $2^{(\tau + 14\epsilon)n}$ and
\[L_\pi \le 2^{(\eta - \alpha_\pi U_\pi + 18\epsilon)n}. \tag{4.11}\]
Call this event $\mathcal G$. The proof of (ii) uses the joint moment
$\mathbb E_{\varphi,\text{tables}}2^{S_h + S_\ell} = 1$ (4.14) for the total scores of a digit pair and a table tuple.
\item \textbf{Multiplicity (4.16).} $\mu_\pi(k)$ is the number of matches of $k$ in pattern $\pi$; it is computed exactly by
retesting the block tables. Entry lists are capped at $C_n = 2^{\lceil(\tau + 20\epsilon)n\rceil}$ (4.17).
\item \textbf{Estimation.} If $\alpha_\pi U_\pi \ge \Delta$, the procedure enumerates all matches and adds
$J_\rho(k)/(P\mu_\pi(k))$ for each match with $k < P$ and pattern $\pi$ (4.18); on $\mathcal G$ this is exact. Otherwise
it averages $N = 2^{\lceil\tau n\rceil}$ independent uniform matches, each with value $L_\pi J_\rho(k)/(P\mu_\pi(k))$ if
$k < P$ has pattern $\pi$ and 0 otherwise (4.19). Given the phases and tables on $\mathcal G$, this is unbiased, and its
mean square error is at most
\[\frac1N\,\frac{L_\pi}{P^2}\sum_{k < P:\ \mathrm{pattern}(k) = \pi}|J_\rho(k)|^2. \tag{4.20}\]
\item \textbf{Pattern errors (4.21).} For a frequency of pattern $\pi$, with $\bar u, \bar v$ the coordinate averages of $u, v$ at
the $x_i(k)$: $|U_\pi - \bar u| \le 3\epsilon$ and $|W_\pi - \bar v| \le 4\epsilon$.
\item \textbf{Error bound (4.24)--(4.25).} The paper proves
\[\mathbf 1_{\mathcal G}\mathbf 1_{\{\mathrm{pattern}(k)=\pi\}}L_\pi|J_\rho(k)|^2 \le
2^{(\eta - \Delta + O(\epsilon))n}\prod_iM(x_i(k))^2\,2^{R(x_i(k))}\]
with its two functions $R$. It takes the expectation
over the phases, and uses its moment bounds to get $\mathbb E[\mathbf 1_{\mathcal G}|E_{\pi,\rho}|^2] \le 2^{-0.0007n}$.
Then Cauchy--Schwarz over the $Q_n$ patterns, Markov's inequality and a union bound over $\mathcal T$ give total ideal
error at most $1/16$ with probability $1 - o(1)$. Our Lemmas B2 and B3 replace this step.
\item \textbf{Resources (Section 4.5).} Caps bound the work on every execution; the finite-precision arithmetic of Section 6.2
adds error at most $1/16$.
\end{itemize}

\subsection*{B.4 Implementation and the main proof (paper, Section 6)}

\begin{itemize}
\item \textbf{Bounded draws (6.2).} A uniform draw from $\{0, \dots, q-1\}$ uses rejection on $\lceil\log q\rceil$ random bits, and
fails after $n^3$ rejections (probability at most $2^{-n^3}$). A prime in $[2^h, 2^{h+1})$ is drawn by testing at most
$n^3$ uniform candidates by trial division; conditional on success it is uniform among the primes, and failure has
probability at most $e^{-c_0n^2}$. Coupled with ideal samplers, all capped executions differ from ideal ones with
probability $o(1)$.
\item \textbf{Complex sums (6.2).} Each phase is computed to error $2^{-2n^2}$ with dyadic parts, each weighted term is rounded to
$n^2$ fractional bits, and the sums are exact. With $2^{O(n)}$ terms of size $2^{O(n)}$, the total error is
$2^{-n^2 + O(n)}$.
\item \textbf{Decision and composition (6.1, 6.3).} $\hat B_s$ is the checksum estimate minus the alias phase sum; the answer is
YES if some $|\hat B_s| \ge 1/2$. Any failure of a main procedure gives NO. The main procedures draw their randomness
even when the preliminary search decides, so a union bound composes the success events.
\item \textbf{Fallback (6.3).} Below a fixed cutoff $n_*$, or when the input guard fails, the program runs exact
meet-in-the-middle in $2^{n/2}\mathrm{poly}(n+b+2)$ time.
\item \textbf{Address space (6.3).} If the operation count is $K_02^{cn}(n+b+2)^K$ with $c < 1$, then every address fits in a
word of $w$ bits for large $n$.
\end{itemize}

\subsection*{B.5 The numerical method (paper, Appendix A)}

\begin{itemize}
\item \textbf{Identities (A.2).} With $\mathcal P(x) = \int F(x+w)G(w)\,dw$, $\mathcal Y(x) = \int F(x+w)G(w)(r(x+w) + s(w))\,dw$,
$L = \log Q = \log\mathcal P - \log(Z_fZ_g)$, $B = \mathcal Y/\mathcal P - \log(Z_fZ_g)$ and
$H = \log Z_{f,\kappa} + \log Z_{g,\kappa} - \kappa\log(Z_fZ_g)$, where $Z_f = \int F$, $Z_{f,\kappa} = \int F^\kappa$ and so on:
\[D = B - L,\qquad u = L + (\kappa - 1)B - H,\qquad v = (2 - \kappa)B - 2L + H.\]
\item \textbf{Trapezoid bound (A.6).} If a $2\pi$-periodic $h$ is analytic on a neighborhood of the strip $|\mathrm{Im}\,z| \le \sigma$
and $|h| \le B_0$ there, then the mean of $h$ at $N$ equally spaced nodes differs from $\int h$ by at most
$2B_0/(e^{N\sigma} - 1)$. (The paper states it for $N = 64$; the proof is the same for every $N$.)
\item \textbf{Strip bounds (A.4).} $\int|f(w + iy)|^2\,dw$ is nondecreasing in $|y|$, and equals $Z_f^{-2}\int2^{2r_y}$, where $r_y$
has coefficients $r_j\cosh(jy)$. For $|y| \le 2\beta$, $Q(x + iy) = \int f(x + w + iy/2)\,g(w - iy/2)\,dw$, and
Cauchy--Schwarz gives $|Q - 1| \le \sqrt{(A_f - 1)(A_g - 1)}$ with $A_f = \int|f(\cdot + i\beta)|^2$ and $A_g$ likewise.
The same split gives $|QB| \le \sqrt{A_fA_g}\,\big(\sup_{|\mathrm{Im}| \le \beta}|r| + \sup_{|\mathrm{Im}|\le\beta}|s| + |\log(Z_fZ_g)|\big)$.
\end{itemize}

\medskip\noindent\rule{\linewidth}{0.4pt}\medskip

\section*{Appendix C. The multipliers of the 430 moment certificates}\addcontentsline{toc}{section}{Appendix C. The multipliers of the 430 moment certificates}

Each row gives a filter $v$, a region $\rho_i$ (Section 6.1), the set $C = C^{(v,\rho)}$, and the vector
$z = z^{(v,\rho)}$. The entries of $z$ follow the order of the rows of $G$ in Section 6.1: first $z_0$, then the region
rows, then one entry for each $l \in C$ in the listed order.

{\scriptsize
\begin{longtable}{>{\raggedright\arraybackslash}p{0.065\dimexpr\linewidth-8\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.065\dimexpr\linewidth-8\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.261\dimexpr\linewidth-8\tabcolsep\relax}>{\raggedright\arraybackslash}p{0.609\dimexpr\linewidth-8\tabcolsep\relax}}
\hline
\textbf{$v$} & \textbf{region} & \textbf{$C$} & \textbf{$z$} \\ \hline
\endhead
0 & $\rho_{0}$ & 9, 13, 17, 19, 22, 26, 33 & 9.1679, 0.0, 0.6464, 0.9449, 4.8989, 0.0037, 3.2564, 0.7714, 2.5871 \\
0 & $\rho_{1}$ & 9, 13, 14, 22, 24, 27 & 1.0001, 40.0, 0.0, 1.0117, 8.8167, 2.1195, 14.1535, 7.6113, 4.0057 \\
0 & $\rho_{2}$ & 9, 13, 14, 22, 24, 27 & 0.8001, 40.0, 0.0, 0.7875, 8.8962, 1.8576, 13.1238, 8.737, 3.4303 \\
0 & $\rho_{3}$ & 9, 13, 14, 22, 24, 27 & 0.6251, 40.0, 0.0, 0.5848, 8.9607, 1.6289, 12.2357, 9.7076, 2.9428 \\
0 & $\rho_{4}$ & 9, 13, 14, 22, 24, 27 & 0.5001, 40.0, 0.0, 0.4364, 9.004, 1.4658, 11.6087, 10.3927, 2.6034 \\
0 & $\rho_{5}$ & 5, 9, 13, 14, 22, 24, 27 & 0.3848, 40.0, 0.0, 0.009, 0.2951, 9.0455, 1.3235, 11.0516, 11.0089, 2.2703 \\
0 & $\rho_{6}$ & 5, 9, 13, 14, 22, 24, 27 & 0.2859, 40.0, 0.0, 0.1118, 0.1531, 9.1172, 1.2862, 10.753, 11.4234, 1.7128 \\
0 & $\rho_{7}$ & 5, 9, 13, 14, 22, 24, 27 & 0.2001, 40.0, 0.0, 0.1996, 0.029, 9.1779, 1.2529, 10.4943, 11.781, 1.2372 \\
0 & $\rho_{8}$ & 5, 13, 14, 22, 24, 27 & 0.1251, 40.0, 0.0, 0.2565, 9.2408, 1.2155, 10.2707, 12.0537, 0.809 \\
0 & $\rho_{9}$ & 5, 13, 14, 22, 24, 27 & 0.0668, 40.0, 0.2946, 9.2921, 1.1838, 10.0974, 12.2539, 0.4746 \\
1 & $\rho_{0}$ & 0, 6, 7, 21 & 9.9712, 0.0, 15.3809, 0.4218, 4.9982, 0.1638 \\
1 & $\rho_{1}$ & 0, 4, 6, 14 & 1.5686, 40.0, 0.0, 40.0, 1.5232, 8.2982, 0.663 \\
1 & $\rho_{2}$ & 0, 4, 6, 14 & 1.5686, 40.0, 0.0, 40.0, 1.5232, 8.2982, 0.663 \\
1 & $\rho_{3}$ & 0, 4, 6, 14 & 1.5686, 40.0, 0.0, 40.0, 1.5232, 8.2982, 0.663 \\
1 & $\rho_{4}$ & 0, 4, 6, 14 & 1.5686, 40.0, 0.0, 40.0, 1.5232, 8.2982, 0.663 \\
1 & $\rho_{5}$ & 0, 4, 6, 14 & 1.5686, 40.0, 0.0, 40.0, 1.5232, 8.2982, 0.663 \\
1 & $\rho_{6}$ & 0, 4, 6, 14 & 1.5686, 40.0, 0.0, 40.0, 1.5232, 8.2982, 0.663 \\
1 & $\rho_{7}$ & 0, 4, 6, 14 & 1.5686, 40.0, 0.0, 40.0, 1.5232, 8.2982, 0.663 \\
1 & $\rho_{8}$ & 0, 4, 6, 14 & 1.5686, 40.0, 0.0, 40.0, 1.5232, 8.2982, 0.663 \\
1 & $\rho_{9}$ & 0, 4, 6, 14 & 1.5686, 40.0, 40.0, 1.5232, 8.2982, 0.663 \\
2 & $\rho_{0}$ & 0, 1, 7, 21 & 10.2703, 0.0, 5.9733, 3.8695, 6.9658, 1.6915 \\
2 & $\rho_{1}$ & 0, 1, 7, 21 & 1.4183, 40.0, 0.0, 23.4401, 29.9709, 6.8452, 0.4075 \\
2 & $\rho_{2}$ & 0, 1, 7, 21 & 1.4183, 40.0, 0.0, 23.4401, 29.9709, 6.8452, 0.4075 \\
2 & $\rho_{3}$ & 0, 1, 7, 21 & 1.4183, 40.0, 0.0, 23.4401, 29.9709, 6.8452, 0.4075 \\
2 & $\rho_{4}$ & 0, 1, 7, 21 & 1.4183, 40.0, 0.0, 23.4401, 29.9709, 6.8452, 0.4075 \\
2 & $\rho_{5}$ & 0, 1, 7, 21 & 1.4183, 40.0, 0.0, 23.4401, 29.9709, 6.8452, 0.4075 \\
2 & $\rho_{6}$ & 0, 1, 7, 21 & 1.4183, 40.0, 0.0, 23.4401, 29.9709, 6.8452, 0.4075 \\
2 & $\rho_{7}$ & 0, 1, 7, 21 & 1.4183, 40.0, 0.0, 23.4401, 29.9709, 6.8452, 0.4075 \\
2 & $\rho_{8}$ & 0, 1, 7, 21 & 1.4183, 40.0, 0.0, 23.4401, 29.9709, 6.8452, 0.4075 \\
2 & $\rho_{9}$ & 0, 1, 7, 21 & 1.4183, 40.0, 23.4401, 29.9709, 6.8452, 0.4075 \\
3 & $\rho_{0}$ & 5, 16 & 17.211, 27.1977, 8.3302, 3.0053 \\
3 & $\rho_{1}$ & 5, 8, 16, 35 & 7.7547, 0.0, 0.0, 4.2699, 2.2635, 0.461, 3.4032 \\
3 & $\rho_{2}$ & 4, 25 & 0.8001, 40.0, 0.0, 10.1405, 0.4377 \\
3 & $\rho_{3}$ & 4, 25 & 0.6251, 40.0, 0.0, 9.9415, 0.5728 \\
3 & $\rho_{4}$ & 4, 25 & 0.5001, 40.0, 0.0, 9.8013, 0.6685 \\
3 & $\rho_{5}$ & 4, 25 & 0.3848, 40.0, 0.0, 9.6733, 0.7561 \\
3 & $\rho_{6}$ & 4, 25 & 0.2859, 40.0, 0.0, 9.5646, 0.8307 \\
3 & $\rho_{7}$ & 4, 25 & 0.2001, 40.0, 0.0, 9.4711, 0.8949 \\
3 & $\rho_{8}$ & 4, 25 & 0.1251, 40.0, 0.0, 9.3899, 0.9508 \\
3 & $\rho_{9}$ & 4, 25 & 0.0668, 40.0, 9.3271, 0.9941 \\
4 & $\rho_{0}$ & 7, 11, 15, 25 & 10.6428, 7.303, 4.0335, 1.1143, 2.334, 3.8337 \\
4 & $\rho_{1}$ & 11, 15, 30, 35 & 7.8954, 0.0, 0.0, 1.9351, 3.5653, 4.1533, 0.954 \\
4 & $\rho_{2}$ & 3 & 0.8001, 40.0, 0.0, 11.481 \\
4 & $\rho_{3}$ & 3 & 0.6251, 40.0, 0.0, 11.3729 \\
4 & $\rho_{4}$ & 3 & 0.5001, 40.0, 0.0, 11.2948 \\
4 & $\rho_{5}$ & 3 & 0.3848, 40.0, 0.0, 11.2222 \\
4 & $\rho_{6}$ & 3 & 0.2859, 40.0, 0.0, 11.1595 \\
4 & $\rho_{7}$ & 3 & 0.2001, 40.0, 0.0, 11.1048 \\
4 & $\rho_{8}$ & 3 & 0.1251, 40.0, 0.0, 11.0567 \\
4 & $\rho_{9}$ & 3 & 0.0668, 40.0, 11.0192 \\
5 & $\rho_{0}$ & 9, 28, 33, 36, 38 & 9.9165, 4.6818, 2.575, 2.3437, 1.6297, 3.3531, 0.1018 \\
5 & $\rho_{1}$ & 20, 28, 36 & 7.9993, 0.0, 0.0, 1.4983, 8.7892, 0.5363 \\
5 & $\rho_{2}$ & 3, 14, 21, 25 & 0.8001, 40.0, 0.0, 15.8577, 1.6266, 7.9554, 12.4666 \\
5 & $\rho_{3}$ & 3, 14, 21, 25 & 0.6251, 40.0, 0.0, 15.5041, 2.2225, 6.9464, 12.1888 \\
5 & $\rho_{4}$ & 3, 14, 21, 25 & 0.5001, 40.0, 0.0, 15.2517, 2.6369, 6.2395, 11.9955 \\
5 & $\rho_{5}$ & 3, 14, 21, 25 & 0.3848, 40.0, 0.0, 15.019, 3.011, 5.5974, 11.8208 \\
5 & $\rho_{6}$ & 3, 14, 21, 25 & 0.2859, 40.0, 0.0, 14.8197, 3.3254, 5.055, 11.6738 \\
5 & $\rho_{7}$ & 3, 14, 21, 25 & 0.2001, 40.0, 0.0, 14.6471, 3.5932, 4.5909, 11.5484 \\
5 & $\rho_{8}$ & 3, 14, 21, 25 & 0.1251, 40.0, 0.0, 14.4963, 3.8239, 4.1894, 11.4402 \\
5 & $\rho_{9}$ & 3, 14, 21, 25 & 0.0668, 40.0, 14.379, 4.001, 3.8801, 11.357 \\
6 & $\rho_{0}$ & 13, 25, 26, 33 & 10.3545, 4.961, 1.1834, 7.1305, 0.7517, 4.2545 \\
6 & $\rho_{1}$ & 13, 21, 22, 25, 26 & 8.4537, 0.0, 0.0, 3.9932, 0.6808, 0.8, 6.5429, 2.8242 \\
6 & $\rho_{2}$ & 3, 4, 21 & 0.8001, 40.0, 0.0, 9.0517, 5.9704, 0.0499 \\
6 & $\rho_{3}$ & 3, 4 & 0.6251, 40.0, 0.0, 8.9388, 5.8095 \\
6 & $\rho_{4}$ & 3, 4 & 0.5001, 40.0, 0.0, 8.8533, 5.6888 \\
6 & $\rho_{5}$ & 3, 4 & 0.3848, 40.0, 0.0, 8.7739, 5.5788 \\
6 & $\rho_{6}$ & 3, 4 & 0.2859, 40.0, 0.0, 8.7056, 5.4855 \\
6 & $\rho_{7}$ & 3, 4 & 0.2001, 40.0, 0.0, 8.6461, 5.4055 \\
6 & $\rho_{8}$ & 3, 4 & 0.1251, 40.0, 0.0, 8.5939, 5.336 \\
6 & $\rho_{9}$ & 3, 4 & 0.0668, 40.0, 8.5532, 5.2823 \\
7 & $\rho_{0}$ & 17, 21, 33 & 10.4052, 5.7867, 4.9545, 0.7827, 3.4876 \\
7 & $\rho_{1}$ & 12, 17, 19, 23, 36 & 8.4306, 0.0, 0.0, 3.0378, 4.2803, 1.4702, 5.61, 0.1052 \\
7 & $\rho_{2}$ & 4, 5, 20, 21 & 0.8001, 40.0, 0.0, 19.085, 18.8588, 0.8077, 13.5458 \\
7 & $\rho_{3}$ & 4, 5, 20, 21 & 0.6251, 40.0, 0.0, 18.8324, 18.5875, 1.0624, 12.8465 \\
7 & $\rho_{4}$ & 4, 5, 20, 21 & 0.5001, 40.0, 0.0, 18.6518, 18.3896, 1.2477, 12.3563 \\
7 & $\rho_{5}$ & 4, 5, 20, 21, 23 & 0.3848, 40.0, 0.0, 18.3187, 18.0314, 1.3823, 11.3486, 1.0118 \\
7 & $\rho_{6}$ & 4, 5, 20, 21, 23 & 0.2859, 40.0, 0.0, 17.9285, 17.6139, 1.4748, 10.1356, 2.5156 \\
7 & $\rho_{7}$ & 4, 5, 20, 21, 23 & 0.2001, 40.0, 0.0, 17.5905, 17.251, 1.556, 9.0877, 3.818 \\
7 & $\rho_{8}$ & 4, 5, 20, 21, 23 & 0.1251, 40.0, 0.0, 17.2948, 16.9327, 1.6279, 8.1735, 4.9568 \\
7 & $\rho_{9}$ & 4, 5, 20, 21, 23 & 0.0668, 40.0, 17.0649, 16.6846, 1.6842, 7.4643, 5.842 \\
8 & $\rho_{0}$ & 22, 24, 26, 30, 32 & 8.8616, 1.223, 1.2823, 2.7043, 2.5334, 1.7898, 2.7054 \\
8 & $\rho_{1}$ & 22, 24, 26, 30, 32 & 8.2246, 0.0, 0.0, 0.2761, 4.2858, 2.4287, 0.9726, 2.596 \\
8 & $\rho_{2}$ & 3, 4, 6 & 0.8001, 40.0, 0.0, 14.5934, 9.3125, 10.3444 \\
8 & $\rho_{3}$ & 3, 4, 6 & 0.6251, 40.0, 0.0, 14.2681, 9.0025, 9.8548 \\
8 & $\rho_{4}$ & 3, 4, 6 & 0.5001, 40.0, 0.0, 14.037, 8.7828, 9.5098 \\
8 & $\rho_{5}$ & 3, 4, 6 & 0.3848, 40.0, 0.0, 13.8246, 8.5813, 9.1948 \\
8 & $\rho_{6}$ & 3, 4, 6 & 0.2859, 40.0, 0.0, 13.6434, 8.4097, 8.9274 \\
8 & $\rho_{7}$ & 3, 4, 6 & 0.2001, 40.0, 0.0, 13.4868, 8.2616, 8.6977 \\
8 & $\rho_{8}$ & 3, 4, 6 & 0.1251, 40.0, 0.0, 13.3503, 8.1327, 8.4981 \\
8 & $\rho_{9}$ & 3, 4, 6 & 0.0668, 40.0, 13.2443, 8.0327, 8.3439 \\
9 & $\rho_{0}$ & 26, 28, 31, 34, 42 & 8.5363, 0.0, 0.4313, 4.8268, 1.0636, 3.0003, 16.624 \\
9 & $\rho_{1}$ & 26, 28, 34, 42 & 8.1097, 0.766, 0.0, 0.7322, 7.9623, 1.7086, 20.6672 \\
9 & $\rho_{2}$ & 3, 4, 5, 8, 25, 29, 31, 42 & 0.8001, 40.0, 0.0, 0.6502, 5.9345, 40.0, 11.2999, 13.8658, 17.3183, 23.2391, 40.0 \\
9 & $\rho_{3}$ & 3, 4, 5, 8, 25, 29, 31, 42 & 0.6251, 40.0, 0.0, 0.272, 6.0412, 40.0, 11.8999, 13.0328, 16.9991, 21.2834, 40.0 \\
9 & $\rho_{4}$ & 3, 4, 5, 8, 25, 29, 31, 42 & 0.5001, 40.0, 0.0, 0.0059, 6.0947, 40.0, 12.3291, 12.4248, 16.7902, 20.1103, 40.0 \\
9 & $\rho_{5}$ & 4, 5, 8, 25, 29, 31, 42 & 0.3848, 40.0, 0.0, 5.0047, 40.0, 12.8333, 11.337, 17.1158, 30.2913, 40.0 \\
9 & $\rho_{6}$ & 4, 5, 8, 25, 29, 31, 42 & 0.2859, 40.0, 0.0, 4.0465, 40.0, 13.2663, 10.3942, 17.4114, 39.2537, 40.0 \\
9 & $\rho_{7}$ & 4, 5, 8, 25, 29, 31, 42 & 0.2001, 40.0, 0.0, 3.8422, 39.7563, 13.3779, 10.0605, 17.5452, 40.0, 40.0 \\
9 & $\rho_{8}$ & 4, 5, 8, 25, 29, 31, 42 & 0.1251, 40.0, 0.0, 3.6941, 39.3384, 13.4883, 9.6, 17.8184, 40.0, 40.0 \\
9 & $\rho_{9}$ & 4, 5, 8, 25, 29, 31, 42 & 0.0668, 40.0, 3.5789, 39.0141, 13.5734, 9.2426, 18.0318, 40.0, 40.0 \\
10 & $\rho_{0}$ & 18, 23, 24, 37 & 8.745, 0.6735, 1.1431, 2.3284, 3.1588, 4.3415 \\
10 & $\rho_{1}$ & 23, 24, 37 & 8.3077, 0.0, 0.0, 2.427, 4.9925, 3.7514 \\
10 & $\rho_{2}$ & 3, 4, 21, 25 & 0.8001, 40.0, 0.0, 14.079, 5.5689, 3.3718, 13.8544 \\
10 & $\rho_{3}$ & 3, 4, 21, 25 & 0.6251, 40.0, 0.0, 13.9046, 5.3119, 2.5519, 14.1084 \\
10 & $\rho_{4}$ & 3, 4, 21, 25 & 0.5001, 40.0, 0.0, 13.7793, 5.1284, 1.9784, 14.2884 \\
10 & $\rho_{5}$ & 3, 4, 21, 25 & 0.3848, 40.0, 0.0, 13.6629, 4.9591, 1.4579, 14.4535 \\
10 & $\rho_{6}$ & 3, 4, 21, 25 & 0.2859, 40.0, 0.0, 13.5628, 4.8141, 1.0185, 14.5941 \\
10 & $\rho_{7}$ & 3, 4, 21, 25 & 0.2001, 40.0, 0.0, 13.4758, 4.6886, 0.6426, 14.7153 \\
10 & $\rho_{8}$ & 3, 4, 21, 25 & 0.1251, 40.0, 0.0, 13.3994, 4.5788, 0.3175, 14.8208 \\
10 & $\rho_{9}$ & 3, 4, 21, 25 & 0.0668, 40.0, 13.3398, 4.4935, 0.067, 14.9025 \\
11 & $\rho_{0}$ & 23, 26, 30, 32 & 8.5281, 0.0, 5.5848, 2.1883, 0.8096, 3.2527 \\
11 & $\rho_{1}$ & 16, 23, 26, 30, 32 & 8.3201, 0.0, 0.0, 0.2064, 4.8721, 1.843, 0.6515, 4.3175 \\
11 & $\rho_{2}$ & 3, 4, 6, 21, 25 & 0.8001, 40.0, 0.0, 13.7106, 20.5765, 8.9862, 17.4401, 13.3446 \\
11 & $\rho_{3}$ & 3, 4, 6, 21, 25 & 0.6251, 40.0, 0.0, 13.5737, 19.9583, 8.6882, 16.4445, 13.4554 \\
11 & $\rho_{4}$ & 3, 4, 6, 21, 25 & 0.5001, 40.0, 0.0, 13.4747, 19.522, 8.4779, 15.7457, 13.5344 \\
11 & $\rho_{5}$ & 3, 4, 6, 21, 25 & 0.3848, 40.0, 0.0, 13.3825, 19.1233, 8.2857, 15.1098, 13.6069 \\
11 & $\rho_{6}$ & 3, 4, 6, 21, 25 & 0.2859, 40.0, 0.0, 13.3027, 18.7845, 8.1226, 14.5716, 13.6688 \\
11 & $\rho_{7}$ & 3, 4, 6, 21, 25 & 0.2001, 40.0, 0.0, 13.2331, 18.4932, 7.9823, 14.1105, 13.7222 \\
11 & $\rho_{8}$ & 3, 4, 6, 21, 25 & 0.1251, 40.0, 0.0, 13.1718, 18.24, 7.8605, 13.7109, 13.7687 \\
11 & $\rho_{9}$ & 3, 4, 6, 21, 25 & 0.0668, 40.0, 13.1239, 18.0441, 7.7663, 13.4026, 13.8047 \\
12 & $\rho_{0}$ & 23, 31, 35, 36, 37 & 8.5784, 0.0, 0.5944, 1.0368, 6.3616, 4.0629, 2.5742 \\
12 & $\rho_{1}$ & 35, 36, 37 & 8.0772, 0.9811, 0.0, 10.662, 4.5469, 2.7765 \\
12 & $\rho_{2}$ & 3, 4, 5, 10, 23 & 0.8001, 40.0, 0.0, 4.63, 15.0086, 19.4035, 10.5572, 27.1491 \\
12 & $\rho_{3}$ & 3, 4, 5, 10, 23 & 0.6251, 40.0, 0.0, 4.5826, 14.7861, 19.132, 10.4615, 26.2338 \\
12 & $\rho_{4}$ & 3, 4, 5, 10, 23 & 0.5001, 40.0, 0.0, 4.549, 14.6255, 18.9362, 10.3957, 25.5935 \\
12 & $\rho_{5}$ & 3, 4, 5, 10, 23 & 0.3848, 40.0, 0.0, 4.518, 14.476, 18.7541, 10.3368, 25.0123 \\
12 & $\rho_{6}$ & 3, 4, 5, 10, 23 & 0.2859, 40.0, 0.0, 4.4916, 14.3469, 18.5971, 10.2877, 24.5216 \\
12 & $\rho_{7}$ & 3, 4, 5, 10, 23, 35 & 0.2001, 40.0, 0.0, 4.4657, 14.2224, 18.4494, 10.2417, 24.074, 0.1565 \\
12 & $\rho_{8}$ & 3, 4, 5, 10, 23, 35 & 0.1251, 40.0, 0.0, 4.4272, 14.0499, 18.2626, 10.1784, 23.5391, 1.1156 \\
12 & $\rho_{9}$ & 3, 4, 5, 10, 23, 35 & 0.0668, 40.0, 4.3973, 13.9159, 18.1175, 10.1294, 23.125, 1.8588 \\
13 & $\rho_{0}$ & 24, 26, 27, 32, 37 & 8.4877, 0.0, 3.0138, 4.3261, 3.5085, 0.4597, 0.5967 \\
13 & $\rho_{1}$ & 24, 26, 27, 37 & 7.7034, 1.7489, 0.0, 4.0823, 3.8813, 0.7355, 2.1616 \\
13 & $\rho_{2}$ & 3, 6, 10, 20, 22 & 0.8001, 40.0, 0.0, 16.3858, 38.793, 14.471, 2.8242, 14.1504 \\
13 & $\rho_{3}$ & 3, 6, 10, 20, 22 & 0.6251, 40.0, 0.0, 15.9878, 37.2826, 14.3161, 2.2365, 13.5223 \\
13 & $\rho_{4}$ & 3, 6, 10, 20, 22 & 0.5001, 40.0, 0.0, 15.7028, 36.2036, 14.2002, 1.8202, 13.0856 \\
13 & $\rho_{5}$ & 3, 6, 10, 20, 22 & 0.3848, 40.0, 0.0, 15.4393, 35.2077, 14.0896, 1.4385, 12.6911 \\
13 & $\rho_{6}$ & 3, 6, 10, 20, 22 & 0.2859, 40.0, 0.0, 15.2132, 34.3544, 13.992, 1.1134, 12.3594 \\
13 & $\rho_{7}$ & 3, 6, 10, 20, 22 & 0.2001, 40.0, 0.0, 15.017, 33.615, 13.9056, 0.833, 12.0765 \\
13 & $\rho_{8}$ & 3, 6, 10, 20, 22 & 0.1251, 40.0, 0.0, 14.8452, 32.9684, 13.8285, 0.5888, 11.8325 \\
13 & $\rho_{9}$ & 3, 6, 10, 20, 22 & 0.0668, 40.0, 14.7115, 32.4657, 13.7676, 0.3996, 11.6448 \\
14 & $\rho_{0}$ & 17, 18, 21, 29, 37 & 8.4737, 0.0, 1.4952, 4.6461, 1.039, 6.6959, 0.0326 \\
14 & $\rho_{1}$ & 18, 21, 29, 39 & 8.152, 0.61, 0.0, 4.6508, 0.2222, 7.2421, 1.7712 \\
14 & $\rho_{2}$ & 3, 4, 25 & 0.8001, 40.0, 0.0, 9.8357, 4.3953, 23.0982 \\
14 & $\rho_{3}$ & 3, 4, 25 & 0.6251, 40.0, 0.0, 9.7559, 4.1763, 22.5913 \\
14 & $\rho_{4}$ & 3, 4, 25 & 0.5001, 40.0, 0.0, 9.697, 4.0218, 22.2415 \\
14 & $\rho_{5}$ & 3, 4, 25 & 0.3848, 40.0, 0.0, 9.6414, 3.8807, 21.9274 \\
14 & $\rho_{6}$ & 3, 4, 25 & 0.2859, 40.0, 0.0, 9.5928, 3.7607, 21.6648 \\
14 & $\rho_{7}$ & 3, 4, 25 & 0.2001, 40.0, 0.0, 9.55, 3.6576, 21.442 \\
14 & $\rho_{8}$ & 3, 4, 25 & 0.1251, 40.0, 0.0, 9.512, 3.568, 21.2507 \\
14 & $\rho_{9}$ & 3, 4, 25 & 0.0668, 40.0, 9.4822, 3.4987, 21.1041 \\
15 & $\rho_{0}$ & 18, 24, 32, 35, 37 & 8.49, 0.0, 0.8847, 0.6571, 6.9361, 0.4502, 2.1524 \\
15 & $\rho_{1}$ & 23, 24, 32, 35 & 7.71, 1.6389, 0.0, 0.6785, 4.5247, 6.1184, 0.3694 \\
15 & $\rho_{2}$ & 3, 4, 8, 10, 14, 25 & 0.8001, 40.0, 0.0, 14.9333, 16.7966, 23.4241, 20.443, 40.0, 17.4896 \\
15 & $\rho_{3}$ & 3, 4, 8, 10, 14, 25 & 0.6251, 40.0, 0.0, 14.7048, 16.4281, 23.7546, 20.0366, 40.0, 16.5246 \\
15 & $\rho_{4}$ & 3, 4, 8, 10, 14, 25 & 0.5001, 40.0, 0.0, 14.5413, 16.1672, 23.9894, 19.7531, 40.0, 15.8435 \\
15 & $\rho_{5}$ & 3, 4, 8, 10, 14, 25 & 0.3848, 40.0, 0.0, 14.3902, 15.9282, 24.2054, 19.4962, 40.0, 15.221 \\
15 & $\rho_{6}$ & 3, 4, 8, 10, 14, 25 & 0.2859, 40.0, 0.0, 14.2606, 15.7245, 24.3898, 19.2797, 40.0, 14.6919 \\
15 & $\rho_{7}$ & 3, 4, 8, 10, 14, 25 & 0.2001, 40.0, 0.0, 14.1465, 15.5449, 24.5337, 19.0874, 39.9721, 14.2428 \\
15 & $\rho_{8}$ & 3, 4, 8, 10, 14, 25 & 0.1251, 40.0, 0.0, 13.9921, 15.2414, 24.1106, 18.6607, 38.9561, 14.0657 \\
15 & $\rho_{9}$ & 3, 4, 8, 10, 14, 25 & 0.0668, 40.0, 13.8719, 15.0054, 23.7797, 18.3293, 38.1634, 13.9299 \\
16 & $\rho_{0}$ & 24, 28, 32, 38 & 8.4893, 0.0, 0.9621, 0.4743, 4.7923, 4.436 \\
16 & $\rho_{1}$ & 24, 28, 32, 38 & 7.9258, 0.9111, 0.0, 2.1236, 1.0062, 4.188, 2.6765 \\
16 & $\rho_{2}$ & 3, 4, 10, 11, 13, 20, 22 & 0.8001, 40.0, 0.0, 16.0979, 19.1761, 1.8956, 38.375, 36.5682, 25.7049, 21.2476 \\
16 & $\rho_{3}$ & 3, 4, 10, 11, 13, 20, 22 & 0.6251, 40.0, 0.0, 15.6679, 19.0948, 1.3008, 35.7174, 36.0578, 25.2266, 20.3188 \\
16 & $\rho_{4}$ & 3, 4, 10, 11, 13, 20, 22 & 0.5001, 40.0, 0.0, 15.3611, 19.038, 0.8753, 33.8248, 35.6936, 24.8898, 19.6629 \\
16 & $\rho_{5}$ & 3, 4, 10, 11, 13, 20, 22 & 0.3848, 40.0, 0.0, 15.0782, 18.9865, 0.4823, 32.0821, 35.3575, 24.5823, 19.0628 \\
16 & $\rho_{6}$ & 3, 4, 10, 11, 13, 20, 22 & 0.2859, 40.0, 0.0, 14.8359, 18.9431, 0.1452, 30.5915, 35.0697, 24.3213, 18.5524 \\
16 & $\rho_{7}$ & 3, 4, 11, 13, 20, 22 & 0.2001, 40.0, 0.0, 14.6603, 18.8364, 29.5641, 34.8092, 23.9942, 18.142 \\
16 & $\rho_{8}$ & 3, 4, 11, 13, 20, 22, 25 & 0.1251, 40.0, 0.0, 14.5383, 18.6622, 28.8076, 34.4859, 23.5841, 17.7888, 0.0727 \\
16 & $\rho_{9}$ & 3, 4, 11, 13, 20, 22, 25 & 0.0668, 40.0, 14.4499, 18.4561, 27.9295, 33.9437, 23.1421, 17.4429, 0.3763 \\
17 & $\rho_{0}$ & 18, 23, 29, 31, 36, 39 & 8.623, 0.0, 1.4471, 1.1251, 0.7058, 4.1178, 0.1343, 4.9324 \\
17 & $\rho_{1}$ & 18, 23, 29, 31, 36, 39 & 8.1251, 1.1251, 0.0, 3.2212, 0.9476, 0.9886, 5.1325, 0.6111, 4.5907 \\
17 & $\rho_{2}$ & 4, 5, 14, 21, 36, 42 & 0.8001, 40.0, 0.0, 10.8709, 14.214, 30.2561, 1.5133, 40.0, 19.8607 \\
17 & $\rho_{3}$ & 4, 5, 14, 21, 36, 42 & 0.6251, 40.0, 0.0, 10.5898, 13.8601, 30.0739, 0.7062, 40.0, 20.0283 \\
17 & $\rho_{4}$ & 4, 5, 14, 21, 36, 42 & 0.5001, 40.0, 0.0, 10.3885, 13.6062, 29.9434, 0.1399, 40.0, 20.1491 \\
17 & $\rho_{5}$ & 4, 5, 14, 36, 42 & 0.3848, 40.0, 0.0, 10.2271, 13.3872, 29.6978, 39.6571, 20.3131 \\
17 & $\rho_{6}$ & 4, 5, 14, 36, 42 & 0.2859, 40.0, 0.0, 10.125, 13.2934, 29.4202, 38.897, 20.5002 \\
17 & $\rho_{7}$ & 4, 5, 14, 36, 42 & 0.2001, 40.0, 0.0, 10.0356, 13.2102, 29.1815, 38.2479, 20.6613 \\
17 & $\rho_{8}$ & 4, 5, 14, 36, 42 & 0.1251, 40.0, 0.0, 9.9568, 13.136, 28.9741, 37.6872, 20.8015 \\
17 & $\rho_{9}$ & 4, 5, 14, 36, 42 & 0.0668, 40.0, 9.8951, 13.0774, 28.8136, 37.2557, 20.9099 \\
18 & $\rho_{0}$ & 28, 31, 32, 37 & 8.528, 0.0, 0.5692, 1.4029, 4.1416, 4.7187 \\
18 & $\rho_{1}$ & 24, 32, 35, 37, 42 & 7.9285, 1.2778, 0.0, 2.637, 5.0418, 0.0354, 4.0037, 0.7809 \\
18 & $\rho_{2}$ & 4, 5, 10, 14, 15, 20, 23, 29 & 0.8001, 40.0, 0.0, 11.2099, 36.9048, 20.9929, 22.9765, 40.0, 1.0497, 24.7907, 27.6067 \\
18 & $\rho_{3}$ & 4, 5, 10, 14, 15, 20, 23, 29 & 0.6251, 40.0, 0.0, 10.8137, 36.2402, 20.6667, 21.7313, 40.0, 0.7695, 23.5879, 27.5377 \\
18 & $\rho_{4}$ & 4, 5, 10, 14, 15, 20, 23, 29 & 0.5001, 40.0, 0.0, 10.5325, 35.763, 20.4315, 20.8478, 40.0, 0.5795, 22.7348, 27.4884 \\
18 & $\rho_{5}$ & 4, 5, 10, 14, 15, 20, 23, 29 & 0.3848, 40.0, 0.0, 10.2741, 35.3209, 20.213, 20.0367, 40.0, 0.4111, 21.9517, 27.4429 \\
18 & $\rho_{6}$ & 4, 5, 10, 14, 15, 20, 23, 29 & 0.2859, 40.0, 0.0, 10.0536, 34.9405, 20.0245, 19.3447, 40.0, 0.2724, 21.2838, 27.4037 \\
18 & $\rho_{7}$ & 4, 5, 10, 14, 15, 20, 23, 29 & 0.2001, 40.0, 0.0, 9.8632, 34.61, 19.8605, 18.7474, 40.0, 0.156, 20.7074, 27.3699 \\
18 & $\rho_{8}$ & 4, 5, 10, 14, 15, 20, 23, 29 & 0.1251, 40.0, 0.0, 9.6969, 34.3203, 19.7166, 18.2264, 40.0, 0.0564, 20.2052, 27.3403 \\
18 & $\rho_{9}$ & 4, 5, 10, 14, 15, 23, 29 & 0.0668, 40.0, 9.571, 34.0869, 19.596, 17.8207, 40.0, 19.8049, 27.3192 \\
19 & $\rho_{0}$ & 23, 30, 31, 32, 37, 38 & 8.6098, 0.0, 0.3903, 0.2715, 4.5217, 3.0355, 1.3507, 3.3631 \\
19 & $\rho_{1}$ & 18, 23, 30, 31, 32, 37 & 6.9298, 6.5593, 0.0, 1.4487, 0.1027, 19.0842, 11.2171, 2.145, 10.5777 \\
19 & $\rho_{2}$ & 4, 8, 9, 15, 22, 23, 30, 31, 35, 42 & 0.8001, 40.0, 0.0, 8.0702, 2.7437, 22.851, 0.4658, 0.7696, 40.0, 40.0, 19.8155, 18.4227, 40.0 \\
19 & $\rho_{3}$ & 4, 8, 9, 15, 23, 30, 31, 35, 42 & 0.6251, 40.0, 0.0, 7.7174, 3.1069, 23.1329, 1.1565, 39.9798, 40.0, 16.4951, 17.8332, 40.0 \\
19 & $\rho_{4}$ & 4, 8, 9, 15, 23, 30, 31, 35, 42 & 0.5001, 40.0, 0.0, 7.4234, 3.1837, 21.9692, 1.4009, 38.9229, 40.0, 16.3843, 18.1183, 40.0 \\
19 & $\rho_{5}$ & 4, 8, 9, 15, 23, 30, 31, 35, 42 & 0.3848, 40.0, 0.0, 7.1497, 3.2541, 20.871, 1.6248, 37.9523, 40.0, 16.3157, 18.3825, 40.0 \\
19 & $\rho_{6}$ & 4, 8, 9, 15, 23, 30, 31, 35, 42 & 0.2859, 40.0, 0.0, 6.9135, 3.3143, 19.9116, 1.8153, 37.1242, 40.0, 16.2823, 18.6097, 40.0 \\
19 & $\rho_{7}$ & 4, 8, 9, 15, 23, 30, 31, 35, 42 & 0.2001, 40.0, 0.0, 6.7076, 3.3662, 19.0673, 1.9796, 36.4093, 40.0, 16.2714, 18.8071, 40.0 \\
19 & $\rho_{8}$ & 4, 8, 9, 15, 23, 30, 31, 35, 42 & 0.1251, 40.0, 0.0, 6.5266, 3.4116, 18.3189, 2.1225, 35.7859, 40.0, 16.2756, 18.9801, 40.0 \\
19 & $\rho_{9}$ & 4, 8, 9, 15, 23, 30, 31, 35, 42 & 0.0668, 40.0, 6.385, 3.4467, 17.7294, 2.2332, 35.3022, 40.0, 16.2892, 19.1154, 40.0 \\
20 & $\rho_{0}$ & 27, 28, 32, 37, 41 & 8.3632, 0.0, 0.7483, 4.6966, 0.1833, 3.4444, 1.3918 \\
20 & $\rho_{1}$ & 24, 26, 28, 32, 37, 38 & 7.917, 0.8422, 0.0, 0.5183, 0.0243, 3.7384, 1.734, 3.0335, 0.768 \\
20 & $\rho_{2}$ & 3, 10, 21, 25 & 0.8001, 40.0, 0.0, 16.4125, 3.418, 32.0545, 4.3774 \\
20 & $\rho_{3}$ & 3, 10, 21, 25 & 0.6251, 40.0, 0.0, 16.0301, 3.5656, 30.4958, 4.6758 \\
20 & $\rho_{4}$ & 3, 10, 21, 25 & 0.5001, 40.0, 0.0, 15.757, 3.6674, 29.394, 4.8867 \\
20 & $\rho_{5}$ & 3, 10, 21, 25 & 0.3848, 40.0, 0.0, 15.505, 3.7589, 28.3856, 5.0798 \\
20 & $\rho_{6}$ & 3, 10, 21, 25 & 0.2859, 40.0, 0.0, 15.2892, 3.8354, 27.5277, 5.2439 \\
20 & $\rho_{7}$ & 3, 10, 21, 25 & 0.2001, 40.0, 0.0, 15.1022, 3.9003, 26.7892, 5.3851 \\
20 & $\rho_{8}$ & 3, 10, 21, 25 & 0.1251, 40.0, 0.0, 14.9386, 3.9561, 26.1466, 5.508 \\
20 & $\rho_{9}$ & 3, 10, 21, 25 & 0.0668, 40.0, 14.8115, 3.9988, 25.6493, 5.603 \\
21 & $\rho_{0}$ & 27, 32, 34, 39, 40 & 8.5072, 0.0, 0.8849, 3.6145, 0.4717, 5.8188, 0.1611 \\
21 & $\rho_{1}$ & 23, 27, 32, 39, 40 & 8.2948, 0.0582, 0.0, 0.9113, 2.5345, 3.6052, 4.7568, 0.1153 \\
21 & $\rho_{2}$ & 3, 4, 25 & 0.8001, 40.0, 0.0, 11.7496, 3.6971, 17.5473 \\
21 & $\rho_{3}$ & 3, 4, 25 & 0.6251, 40.0, 0.0, 11.6497, 3.4713, 18.0065 \\
21 & $\rho_{4}$ & 3, 4, 25 & 0.5001, 40.0, 0.0, 11.5761, 3.3119, 18.3176 \\
21 & $\rho_{5}$ & 3, 4, 25 & 0.3848, 40.0, 0.0, 11.5066, 3.1662, 18.5926 \\
21 & $\rho_{6}$ & 3, 4, 25 & 0.2859, 40.0, 0.0, 11.4458, 3.0424, 18.8193 \\
21 & $\rho_{7}$ & 3, 4, 25 & 0.2001, 40.0, 0.0, 11.3924, 2.936, 19.0093 \\
21 & $\rho_{8}$ & 3, 4, 25 & 0.1251, 40.0, 0.0, 11.345, 2.8435, 19.1707 \\
21 & $\rho_{9}$ & 3, 4, 25 & 0.0668, 40.0, 11.3077, 2.772, 19.2932 \\
22 & $\rho_{0}$ & 28, 32, 34, 37 & 8.457, 0.0, 0.646, 5.0085, 3.1523, 1.22 \\
22 & $\rho_{1}$ & 24, 28, 32, 34 & 7.6971, 1.4924, 0.0, 2.8168, 0.8236, 3.8957, 1.7391 \\
22 & $\rho_{2}$ & 3, 4, 6, 25 & 0.8001, 40.0, 0.0, 12.6692, 4.1182, 0.0957, 19.1734 \\
22 & $\rho_{3}$ & 3, 4, 6, 25 & 0.6251, 40.0, 0.0, 12.5188, 4.1095, 0.6053, 18.2366 \\
22 & $\rho_{4}$ & 3, 4, 6, 25 & 0.5001, 40.0, 0.0, 12.4097, 4.0999, 0.9548, 17.5833 \\
22 & $\rho_{5}$ & 3, 4, 6, 25 & 0.3848, 40.0, 0.0, 12.3078, 4.0887, 1.2672, 16.9919 \\
22 & $\rho_{6}$ & 3, 4, 6, 25 & 0.2859, 40.0, 0.0, 12.2196, 4.0775, 1.5273, 16.4934 \\
22 & $\rho_{7}$ & 3, 4, 6, 25 & 0.2001, 40.0, 0.0, 12.1426, 4.0666, 1.7472, 16.0677 \\
22 & $\rho_{8}$ & 3, 4, 6, 25 & 0.1251, 40.0, 0.0, 12.0747, 4.0562, 1.9355, 15.6999 \\
22 & $\rho_{9}$ & 3, 4, 6, 25 & 0.0668, 40.0, 12.0217, 4.0475, 2.0794, 15.4168 \\
23 & $\rho_{0}$ & 32, 37, 38, 39 & 8.4654, 0.0, 5.1563, 1.6763, 3.4869, 1.464 \\
23 & $\rho_{1}$ & 32, 37, 38 & 7.9483, 0.8114, 0.0, 6.7412, 4.2362, 2.9363 \\
23 & $\rho_{2}$ & 3, 4, 5, 21, 22, 25 & 0.8001, 40.0, 0.0, 14.3944, 16.8813, 6.4856, 40.0, 17.3881, 14.9366 \\
23 & $\rho_{3}$ & 3, 4, 5, 21, 22, 25 & 0.6251, 40.0, 0.0, 13.8561, 17.2811, 7.6465, 40.0, 18.562, 14.3323 \\
23 & $\rho_{4}$ & 3, 4, 5, 21, 22, 25 & 0.5001, 40.0, 0.0, 13.481, 17.5523, 8.4424, 40.0, 19.3795, 13.9093 \\
23 & $\rho_{5}$ & 3, 4, 5, 21, 22, 25 & 0.3848, 40.0, 0.0, 13.1416, 17.7923, 9.1532, 40.0, 20.119, 13.525 \\
23 & $\rho_{6}$ & 3, 4, 5, 21, 22, 25 & 0.2859, 40.0, 0.0, 12.8556, 17.9903, 9.7447, 40.0, 20.7414, 13.2003 \\
23 & $\rho_{7}$ & 3, 4, 5, 20, 21, 22, 25 & 0.2001, 40.0, 0.0, 12.6169, 18.1677, 10.1784, 0.1387, 40.0, 21.231, 12.9555 \\
23 & $\rho_{8}$ & 3, 4, 5, 20, 21, 22, 25 & 0.1251, 40.0, 0.0, 12.4127, 18.3229, 10.5244, 0.3103, 40.0, 21.6383, 12.7562 \\
23 & $\rho_{9}$ & 3, 4, 5, 20, 21, 22, 25 & 0.0668, 40.0, 12.2556, 18.4408, 10.7886, 0.4414, 40.0, 21.9519, 12.6023 \\
24 & $\rho_{0}$ & 28, 32, 37 & 8.4468, 0.0, 2.8607, 4.6434, 2.7779 \\
24 & $\rho_{1}$ & 28, 32, 38 & 7.6081, 1.8327, 0.0, 1.6571, 3.3198, 1.4285 \\
24 & $\rho_{2}$ & 3, 5, 8, 10, 15, 22, 25, 29 & 0.8001, 40.0, 0.0, 15.0424, 18.0017, 21.0568, 24.7704, 33.7546, 10.2306, 9.1279, 40.0 \\
24 & $\rho_{3}$ & 3, 5, 8, 10, 15, 22, 25, 29 & 0.6251, 40.0, 0.0, 14.5812, 17.8577, 21.4365, 24.733, 32.2825, 9.2929, 8.4526, 40.0 \\
24 & $\rho_{4}$ & 3, 5, 8, 10, 15, 22, 25, 29 & 0.5001, 40.0, 0.0, 14.2548, 17.7469, 21.6992, 24.7026, 31.2279, 8.633, 7.9728, 40.0 \\
24 & $\rho_{5}$ & 3, 5, 8, 10, 15, 22, 25, 29 & 0.3848, 40.0, 0.0, 13.9557, 17.6389, 21.9355, 24.6719, 30.2522, 8.0311, 7.5316, 40.0 \\
24 & $\rho_{6}$ & 3, 4, 5, 8, 10, 15, 22, 25, 29 & 0.2859, 40.0, 0.0, 12.4219, 1.027, 21.2866, 25.1056, 26.8688, 27.5949, 6.6037, 6.7714, 40.0 \\
24 & $\rho_{7}$ & 3, 4, 5, 8, 10, 15, 22, 25, 29 & 0.2001, 40.0, 0.0, 10.5081, 2.3871, 26.1589, 29.2102, 29.7897, 24.4573, 4.9504, 5.9374, 40.0 \\
24 & $\rho_{8}$ & 3, 4, 5, 8, 10, 15, 22, 25, 29 & 0.1251, 40.0, 0.0, 8.8393, 3.5732, 30.4046, 32.7873, 32.3352, 21.7171, 3.5097, 5.2094, 40.0 \\
24 & $\rho_{9}$ & 3, 4, 5, 8, 10, 15, 22, 25, 29 & 0.0668, 40.0, 7.5466, 4.4918, 33.6913, 35.5569, 34.3058, 19.5923, 2.3944, 4.6448, 40.0 \\
25 & $\rho_{0}$ & 23, 26, 29, 33 & 8.4155, 0.0, 0.7535, 3.9191, 7.1124, 1.3003 \\
25 & $\rho_{1}$ & 21, 22, 26, 29, 32 & 8.0401, 0.6077, 0.0, 0.8529, 0.0811, 3.249, 4.5745, 4.2571 \\
25 & $\rho_{2}$ & 3, 4, 6, 21 & 0.8001, 40.0, 0.0, 9.9933, 6.8874, 1.2555, 0.3972 \\
25 & $\rho_{3}$ & 3, 4, 6, 21 & 0.6251, 40.0, 0.0, 9.9045, 6.813, 1.6848, 0.2793 \\
25 & $\rho_{4}$ & 3, 4, 6, 21 & 0.5001, 40.0, 0.0, 9.8391, 6.7588, 1.9831, 0.2078 \\
25 & $\rho_{5}$ & 3, 4, 6, 21 & 0.3848, 40.0, 0.0, 9.7773, 6.708, 2.2524, 0.1507 \\
25 & $\rho_{6}$ & 3, 4, 6, 21 & 0.2859, 40.0, 0.0, 9.7233, 6.6639, 2.4788, 0.1084 \\
25 & $\rho_{7}$ & 3, 4, 6, 21 & 0.2001, 40.0, 0.0, 9.6758, 6.6254, 2.6716, 0.0764 \\
25 & $\rho_{8}$ & 3, 4, 6, 21 & 0.1251, 40.0, 0.0, 9.6337, 6.5914, 2.8379, 0.0518 \\
25 & $\rho_{9}$ & 3, 4, 6, 21 & 0.0668, 40.0, 9.6006, 6.5648, 2.9657, 0.0349 \\
26 & $\rho_{0}$ & 28, 32, 34, 38 & 8.4664, 0.0, 1.3666, 5.3536, 1.5467, 1.4069 \\
26 & $\rho_{1}$ & 24, 28, 32, 38 & 7.9562, 0.9097, 0.0, 0.9326, 1.5826, 4.5767, 2.2312 \\
26 & $\rho_{2}$ & 3, 4, 6, 11, 22, 25 & 0.8001, 40.0, 0.0, 16.3611, 1.7703, 5.1, 12.2398, 14.558, 36.6664 \\
26 & $\rho_{3}$ & 3, 4, 6, 11, 22, 25 & 0.6251, 40.0, 0.0, 15.9516, 2.0101, 5.2357, 11.2086, 14.2607, 34.8126 \\
26 & $\rho_{4}$ & 3, 4, 6, 11, 22, 25 & 0.5001, 40.0, 0.0, 15.6603, 2.1785, 5.3306, 10.4795, 14.0565, 33.5006 \\
26 & $\rho_{5}$ & 3, 4, 6, 11, 22, 25 & 0.3848, 40.0, 0.0, 15.3923, 2.3318, 5.4167, 9.8119, 13.8738, 32.2984 \\
26 & $\rho_{6}$ & 3, 4, 6, 11, 22, 25 & 0.2859, 40.0, 0.0, 15.1634, 2.4616, 5.4894, 9.2439, 13.7216, 31.2747 \\
26 & $\rho_{7}$ & 3, 4, 6, 11, 22, 25 & 0.2001, 40.0, 0.0, 14.9655, 2.5728, 5.5516, 8.7546, 13.5929, 30.3925 \\
26 & $\rho_{8}$ & 3, 4, 6, 11, 22, 25 & 0.1251, 40.0, 0.0, 14.7927, 2.6693, 5.6054, 8.3288, 13.4827, 29.6244 \\
26 & $\rho_{9}$ & 3, 4, 6, 11, 22, 25 & 0.0668, 40.0, 14.6587, 2.7437, 5.6468, 7.9992, 13.3985, 29.0294 \\
27 & $\rho_{0}$ & 28, 32, 34, 37, 39 & 8.459, 0.0, 1.5991, 3.9299, 2.0208, 0.9315, 1.5485 \\
27 & $\rho_{1}$ & 24, 28, 32, 34, 37 & 8.0031, 0.6933, 0.0, 0.4752, 2.2035, 4.2074, 0.2738, 1.8622 \\
27 & $\rho_{2}$ & 3, 5, 6, 10, 14, 20, 21, 22, 25 & 0.8001, 40.0, 0.0, 11.83, 11.3211, 2.0953, 2.1662, 5.5224, 25.5822, 40.0, 40.0, 40.0 \\
27 & $\rho_{3}$ & 3, 5, 6, 10, 14, 20, 21, 22, 25 & 0.6251, 40.0, 0.0, 10.5936, 16.044, 2.3876, 5.6925, 3.7508, 20.1871, 40.0, 40.0, 40.0 \\
27 & $\rho_{4}$ & 3, 5, 6, 10, 14, 20, 21, 22, 25 & 0.5001, 40.0, 0.0, 9.7151, 19.3875, 2.5943, 8.1922, 2.4891, 16.3639, 40.0, 40.0, 40.0 \\
27 & $\rho_{5}$ & 3, 5, 6, 10, 14, 20, 21, 22, 25 & 0.3848, 40.0, 0.0, 8.9077, 22.4517, 2.7836, 10.4855, 1.3274, 12.8573, 40.0, 40.0, 40.0 \\
27 & $\rho_{6}$ & 3, 5, 6, 10, 14, 20, 21, 22, 25 & 0.2859, 40.0, 0.0, 8.2182, 25.0615, 2.9447, 12.4404, 0.3339, 9.8687, 40.0, 40.0, 40.0 \\
27 & $\rho_{7}$ & 3, 5, 6, 10, 20, 21, 22, 25 & 0.2001, 40.0, 0.0, 7.8204, 26.5255, 3.0974, 13.4489, 8.5447, 40.0, 40.0, 40.0 \\
27 & $\rho_{8}$ & 3, 5, 6, 10, 20, 21, 22, 25 & 0.1251, 40.0, 0.0, 7.5837, 27.3611, 3.2381, 13.9492, 8.0907, 40.0, 40.0, 40.0 \\
27 & $\rho_{9}$ & 3, 5, 6, 10, 20, 21, 22, 25 & 0.0668, 40.0, 7.4006, 28.005, 3.347, 14.3348, 7.7427, 40.0, 40.0, 40.0 \\
28 & $\rho_{0}$ & 32, 34, 37, 38, 39 & 8.46, 0.0, 3.7252, 0.0692, 0.3588, 1.3805, 1.2413 \\
28 & $\rho_{1}$ & 24, 32, 38 & 7.806, 1.319, 0.0, 1.8426, 3.4987, 1.9034 \\
28 & $\rho_{2}$ & 5, 8, 15, 20, 23, 24, 30, 32, 42 & 0.8001, 40.0, 0.0, 40.0, 19.5051, 2.2287, 10.5766, 19.9014, 40.0, 30.6988, 40.0, 21.7588 \\
28 & $\rho_{3}$ & 5, 8, 15, 20, 23, 24, 30, 32, 42 & 0.6251, 40.0, 0.0, 40.0, 18.4595, 2.6703, 7.7488, 19.9469, 40.0, 28.0138, 40.0, 18.0201 \\
28 & $\rho_{4}$ & 5, 8, 15, 20, 23, 24, 30, 32, 42 & 0.5001, 40.0, 0.0, 40.0, 17.7166, 2.9829, 5.7341, 19.9889, 40.0, 26.0938, 40.0, 15.3643 \\
28 & $\rho_{5}$ & 5, 8, 15, 20, 23, 24, 30, 32, 42 & 0.3848, 40.0, 0.0, 40.0, 17.0335, 3.2693, 3.8781, 20.0347, 40.0, 24.3196, 40.0, 12.9234 \\
28 & $\rho_{6}$ & 5, 8, 15, 20, 23, 24, 30, 32, 42 & 0.2859, 40.0, 0.0, 40.0, 16.4501, 3.5133, 2.2899, 20.0792, 40.0, 22.7974, 40.0, 10.839 \\
28 & $\rho_{7}$ & 5, 8, 15, 20, 23, 24, 30, 32, 42 & 0.2001, 40.0, 0.0, 40.0, 15.9461, 3.7236, 0.9154, 20.1217, 40.0, 21.4769, 40.0, 9.0389 \\
28 & $\rho_{8}$ & 5, 8, 15, 23, 24, 30, 32, 42 & 0.1251, 40.0, 0.0, 40.0, 15.542, 3.9177, 19.955, 40.0, 20.7281, 40.0, 7.6801 \\
28 & $\rho_{9}$ & 5, 8, 15, 23, 24, 29, 30, 32, 42 & 0.0668, 40.0, 40.0, 15.2272, 3.6259, 18.9957, 40.0, 0.8582, 21.6082, 40.0, 5.8662 \\
29 & $\rho_{0}$ & 28, 32, 37, 39 & 8.4836, 0.0, 1.1816, 5.7239, 0.5264, 2.5149 \\
29 & $\rho_{1}$ & 24, 27, 32, 37, 42 & 7.5979, 2.0507, 0.0, 1.8679, 0.2116, 9.2619, 1.3267, 0.0703 \\
29 & $\rho_{2}$ & 3, 4, 14, 21, 22, 25 & 0.8001, 40.0, 0.0, 14.843, 6.3571, 15.556, 5.5528, 5.7576, 40.0 \\
29 & $\rho_{3}$ & 3, 4, 14, 21, 22, 25 & 0.6251, 40.0, 0.0, 14.7655, 6.0631, 16.1838, 4.5794, 6.2574, 40.0 \\
29 & $\rho_{4}$ & 3, 4, 14, 21, 22, 25 & 0.5001, 40.0, 0.0, 14.7077, 5.8556, 16.6223, 3.8966, 6.6157, 40.0 \\
29 & $\rho_{5}$ & 3, 4, 14, 21, 22, 25 & 0.3848, 40.0, 0.0, 14.6526, 5.6658, 17.0199, 3.2757, 6.9473, 40.0 \\
29 & $\rho_{6}$ & 3, 4, 14, 21, 22, 25 & 0.2859, 40.0, 0.0, 14.6041, 5.5044, 17.3554, 2.7506, 7.2321, 40.0 \\
29 & $\rho_{7}$ & 3, 4, 14, 21, 22, 25 & 0.2001, 40.0, 0.0, 14.5611, 5.3656, 17.6423, 2.3007, 7.4793, 40.0 \\
29 & $\rho_{8}$ & 3, 4, 14, 21, 22, 25 & 0.1251, 40.0, 0.0, 14.5228, 5.2449, 17.8904, 1.9112, 7.6958, 40.0 \\
29 & $\rho_{9}$ & 3, 4, 14, 21, 22, 25 & 0.0668, 40.0, 14.4926, 5.1515, 18.0814, 1.6108, 7.8643, 40.0 \\
30 & $\rho_{0}$ & 28, 31, 32, 38 & 8.4753, 0.0, 1.3556, 1.1162, 4.6796, 3.0421 \\
30 & $\rho_{1}$ & 28, 32, 35, 38 & 7.9935, 0.7745, 0.0, 2.5639, 6.0856, 0.8719, 1.229 \\
30 & $\rho_{2}$ & 4, 5, 8, 22, 26, 42 & 0.8001, 40.0, 0.0, 27.6138, 32.195, 11.8912, 1.8721, 22.3933, 16.7236 \\
30 & $\rho_{3}$ & 4, 5, 8, 22, 25, 26, 42 & 0.6251, 40.0, 0.0, 26.389, 32.4625, 12.2138, 1.5335, 1.6203, 19.6765, 12.0937 \\
30 & $\rho_{4}$ & 4, 5, 8, 22, 25, 26 & 0.5001, 40.0, 0.0, 24.3495, 34.4861, 12.5199, 1.112, 6.4365, 13.4941 \\
30 & $\rho_{5}$ & 4, 5, 8, 22, 25, 26 & 0.3848, 40.0, 0.0, 23.9452, 34.0414, 12.71, 0.951, 6.2881, 13.1063 \\
30 & $\rho_{6}$ & 4, 5, 8, 22, 25, 26 & 0.2859, 40.0, 0.0, 23.5999, 33.6606, 12.8699, 0.8176, 6.1596, 12.7776 \\
30 & $\rho_{7}$ & 4, 5, 8, 22, 25, 26 & 0.2001, 40.0, 0.0, 23.3017, 33.3309, 13.0061, 0.7055, 6.047, 12.4957 \\
30 & $\rho_{8}$ & 4, 5, 8, 22, 25, 26 & 0.1251, 40.0, 0.0, 23.0416, 33.0428, 13.1235, 0.61, 5.9477, 12.251 \\
30 & $\rho_{9}$ & 4, 5, 8, 22, 25, 26 & 0.0668, 40.0, 22.8397, 32.8188, 13.2137, 0.5373, 5.87, 12.062 \\
31 & $\rho_{0}$ & 28, 32, 38, 39 & 8.4669, 0.0, 2.0055, 4.1501, 0.9035, 2.1279 \\
31 & $\rho_{1}$ & 23, 28, 32, 37, 38 & 7.7427, 1.5111, 0.0, 0.2012, 4.3219, 6.8835, 0.1089, 0.8989 \\
31 & $\rho_{2}$ & 4, 5, 8, 22, 25, 42 & 0.8001, 40.0, 0.0, 21.8583, 38.6821, 4.2934, 4.3617, 26.2854, 40.0 \\
31 & $\rho_{3}$ & 4, 5, 8, 22, 25, 42 & 0.6251, 40.0, 0.0, 21.3883, 37.7995, 4.6644, 4.1145, 25.4473, 40.0 \\
31 & $\rho_{4}$ & 4, 5, 8, 22, 25, 42 & 0.5001, 40.0, 0.0, 21.054, 37.1712, 4.924, 3.9471, 24.8502, 40.0 \\
31 & $\rho_{5}$ & 4, 5, 8, 22, 25, 42 & 0.3848, 40.0, 0.0, 20.7466, 36.5927, 5.1597, 3.7993, 24.3002, 40.0 \\
31 & $\rho_{6}$ & 4, 5, 8, 22, 25, 42 & 0.2859, 40.0, 0.0, 20.4839, 36.0981, 5.3587, 3.6775, 23.8297, 40.0 \\
31 & $\rho_{7}$ & 4, 5, 8, 22, 25, 42 & 0.2001, 40.0, 0.0, 20.2568, 35.6703, 5.529, 3.5756, 23.4227, 40.0 \\
31 & $\rho_{8}$ & 4, 5, 8, 22, 25, 42 & 0.1251, 40.0, 0.0, 20.0586, 35.2966, 5.6763, 3.4893, 23.0672, 40.0 \\
31 & $\rho_{9}$ & 4, 5, 8, 22, 25, 42 & 0.0668, 40.0, 19.9047, 35.0064, 5.7898, 3.4239, 22.791, 40.0 \\
32 & $\rho_{0}$ & 28, 34, 37, 38, 39 & 8.46, 0.0, 1.685, 0.0692, 0.3588, 1.3805, 1.2413 \\
32 & $\rho_{1}$ & 24, 28, 38 & 7.7141, 1.6056, 0.0, 2.0404, 1.2528, 2.2207 \\
32 & $\rho_{2}$ & 3, 4, 5, 20, 22, 25 & 0.8001, 40.0, 0.0, 9.355, 16.7155, 20.5904, 2.8685, 28.1601, 39.0966 \\
32 & $\rho_{3}$ & 3, 4, 5, 20, 22, 24, 25 & 0.6251, 40.0, 0.0, 9.3693, 15.9842, 19.1977, 2.972, 26.9122, 0.4184, 37.7918 \\
32 & $\rho_{4}$ & 3, 4, 5, 20, 22, 24, 25 & 0.5001, 40.0, 0.0, 9.3444, 15.5059, 18.3538, 2.9442, 26.0789, 1.0339, 36.9474 \\
32 & $\rho_{5}$ & 3, 4, 5, 20, 22, 24, 25 & 0.3848, 40.0, 0.0, 9.3209, 15.0671, 17.5798, 2.919, 25.3174, 1.5965, 36.1709 \\
32 & $\rho_{6}$ & 3, 4, 5, 20, 22, 24, 25 & 0.2859, 40.0, 0.0, 9.3003, 14.6929, 16.9199, 2.8977, 24.6702, 2.0747, 35.5077 \\
32 & $\rho_{7}$ & 3, 4, 5, 20, 22, 24, 25 & 0.2001, 40.0, 0.0, 9.282, 14.37, 16.3509, 2.8793, 24.1137, 2.4861, 34.9345 \\
32 & $\rho_{8}$ & 3, 4, 5, 20, 22, 24, 25 & 0.1251, 40.0, 0.0, 9.2659, 14.0886, 15.8548, 2.8637, 23.6296, 2.8436, 34.4343 \\
32 & $\rho_{9}$ & 3, 4, 5, 20, 22, 24, 25 & 0.0668, 40.0, 9.2533, 13.8703, 15.4701, 2.8517, 23.2551, 3.1203, 34.0462 \\
33 & $\rho_{0}$ & 31, 32, 34, 36, 40, 42 & 8.4858, 0.0, 0.6407, 3.6349, 2.9376, 2.8963, 2.7034, 0.0095 \\
33 & $\rho_{1}$ & 27, 32, 34, 39, 40, 42 & 7.8569, 1.3641, 0.0, 0.1874, 3.0897, 1.8933, 4.1996, 12.6495, 0.4099 \\
33 & $\rho_{2}$ & 3, 5, 21, 22, 23, 25 & 0.8001, 40.0, 0.0, 14.2149, 0.9381, 40.0, 0.3832, 14.2852, 29.2081 \\
33 & $\rho_{3}$ & 3, 5, 21, 22, 23, 25 & 0.6251, 40.0, 0.0, 13.868, 1.501, 40.0, 0.8903, 14.057, 28.9425 \\
33 & $\rho_{4}$ & 3, 5, 21, 22, 23, 25 & 0.5001, 40.0, 0.0, 13.6242, 1.8876, 40.0, 1.2549, 13.8824, 28.7492 \\
33 & $\rho_{5}$ & 3, 5, 21, 22, 23, 25 & 0.3848, 40.0, 0.0, 13.4021, 2.2334, 40.0, 1.5929, 13.7132, 28.5683 \\
33 & $\rho_{6}$ & 3, 5, 21, 22, 23, 25 & 0.2859, 40.0, 0.0, 13.2139, 2.5218, 40.0, 1.8836, 13.5621, 28.4114 \\
33 & $\rho_{7}$ & 3, 5, 21, 22, 23, 25 & 0.2001, 40.0, 0.0, 13.0524, 2.7657, 40.0, 2.1362, 13.427, 28.2742 \\
33 & $\rho_{8}$ & 3, 5, 21, 22, 23, 25 & 0.1251, 40.0, 0.0, 12.9123, 2.9748, 40.0, 2.3576, 13.3057, 28.1531 \\
33 & $\rho_{9}$ & 3, 5, 21, 22, 23, 25 & 0.0668, 40.0, 12.8041, 3.1347, 40.0, 2.53, 13.2093, 28.0584 \\
34 & $\rho_{0}$ & 28, 32, 37, 38, 39 & 8.46, 0.0, 1.685, 3.7252, 0.3588, 1.3805, 1.2413 \\
34 & $\rho_{1}$ & 28, 32, 37 & 8.0578, 0.5422, 0.0, 2.4521, 4.3254, 2.2871 \\
34 & $\rho_{2}$ & 3, 5, 21, 22, 23, 25, 27, 39 & 0.8001, 40.0, 0.0, 13.7208, 5.0245, 19.6336, 40.0, 40.0, 13.1679, 26.5879, 34.2393 \\
34 & $\rho_{3}$ & 3, 5, 21, 22, 23, 25, 27, 39 & 0.6251, 40.0, 0.0, 13.2916, 5.1613, 17.1356, 40.0, 40.0, 12.7892, 27.9057, 34.0223 \\
34 & $\rho_{4}$ & 3, 5, 21, 22, 23, 25, 27, 39 & 0.5001, 40.0, 0.0, 12.9866, 5.256, 15.3696, 40.0, 40.0, 12.5154, 28.849, 33.8607 \\
34 & $\rho_{5}$ & 3, 5, 21, 22, 23, 25, 27, 39 & 0.3848, 40.0, 0.0, 12.7062, 5.3411, 13.753, 40.0, 40.0, 12.2606, 29.7206, 33.7066 \\
34 & $\rho_{6}$ & 3, 5, 21, 22, 23, 25, 27, 39 & 0.2859, 40.0, 0.0, 12.4667, 5.4125, 12.3776, 40.0, 40.0, 12.0406, 30.4681, 33.5706 \\
34 & $\rho_{7}$ & 3, 5, 21, 22, 23, 25, 27, 39 & 0.2001, 40.0, 0.0, 12.2598, 5.4733, 11.1932, 40.0, 40.0, 11.8489, 31.116, 33.4496 \\
34 & $\rho_{8}$ & 3, 5, 21, 22, 23, 25, 27, 39 & 0.1251, 40.0, 0.0, 12.0792, 5.5257, 10.1625, 40.0, 40.0, 11.6803, 31.683, 33.342 \\
34 & $\rho_{9}$ & 3, 5, 21, 22, 23, 25, 27, 39 & 0.0668, 40.0, 11.9391, 5.5658, 9.3646, 40.0, 40.0, 11.5487, 32.1242, 33.2564 \\
35 & $\rho_{0}$ & 28, 32, 37, 38 & 8.4785, 0.0, 1.8096, 4.0834, 1.3459, 2.2544 \\
35 & $\rho_{1}$ & 24, 28, 32, 38 & 7.967, 0.8534, 0.0, 1.7215, 1.7706, 3.6903, 2.3521 \\
35 & $\rho_{2}$ & 3, 4, 5, 10, 23, 24 & 0.8001, 40.0, 0.0, 8.2122, 16.0399, 11.1008, 3.2521, 27.8334, 38.2439 \\
35 & $\rho_{3}$ & 3, 4, 5, 10, 23, 24 & 0.6251, 40.0, 0.0, 8.1294, 15.6306, 10.6351, 3.0732, 26.5334, 38.3422 \\
35 & $\rho_{4}$ & 3, 4, 5, 10, 23, 24 & 0.5001, 40.0, 0.0, 8.0704, 15.3392, 10.3038, 2.9468, 25.6154, 38.4109 \\
35 & $\rho_{5}$ & 3, 4, 5, 10, 16, 23, 24 & 0.3848, 40.0, 0.0, 7.997, 15.0939, 10.0609, 2.8402, 0.1118, 24.6759, 38.4192 \\
35 & $\rho_{6}$ & 3, 4, 5, 10, 16, 23, 24 & 0.2859, 40.0, 0.0, 7.8888, 14.9387, 10.0017, 2.7713, 0.4747, 23.6377, 38.2964 \\
35 & $\rho_{7}$ & 3, 4, 5, 10, 16, 23, 24 & 0.2001, 40.0, 0.0, 7.7955, 14.8042, 9.9496, 2.7119, 0.7869, 22.7439, 38.1904 \\
35 & $\rho_{8}$ & 3, 4, 5, 10, 16, 23, 24 & 0.1251, 40.0, 0.0, 7.7141, 14.6864, 9.9035, 2.6603, 1.0585, 21.9662, 38.0979 \\
35 & $\rho_{9}$ & 3, 4, 5, 10, 16, 23, 24 & 0.0668, 40.0, 7.651, 14.5948, 9.8672, 2.6203, 1.2687, 21.3642, 38.0261 \\
36 & $\rho_{0}$ & 28, 32, 38, 39, 41 & 8.4679, 0.0, 1.8976, 3.2031, 2.0689, 3.9792, 0.0146 \\
36 & $\rho_{1}$ & 28, 32, 38, 39, 41, 42 & 7.8724, 1.103, 0.0, 1.0529, 3.8946, 1.3251, 2.0595, 12.8877, 0.7308 \\
36 & $\rho_{2}$ & 3, 4, 5, 21, 22, 23, 25, 27, 33, 39 & 0.8001, 40.0, 0.0, 4.6584, 17.1524, 40.0, 40.0, 2.7476, 40.0, 6.553, 13.5827, 31.1995, 40.0 \\
36 & $\rho_{3}$ & 3, 4, 5, 21, 22, 23, 25, 27, 33, 39 & 0.6251, 40.0, 0.0, 4.4093, 16.1698, 40.0, 40.0, 5.1801, 40.0, 7.8025, 7.6136, 25.4956, 40.0 \\
36 & $\rho_{4}$ & 3, 4, 5, 21, 22, 23, 25, 27, 33, 39 & 0.5001, 40.0, 0.0, 4.2314, 15.4779, 40.0, 40.0, 6.8994, 40.0, 8.6747, 3.4137, 21.4945, 40.0 \\
36 & $\rho_{5}$ & 3, 4, 5, 21, 22, 23, 25, 33, 39 & 0.3848, 40.0, 0.0, 4.069, 14.8953, 40.0, 40.0, 8.3079, 40.0, 9.3834, 18.0789, 40.0 \\
36 & $\rho_{6}$ & 3, 4, 5, 21, 22, 23, 25, 33, 39 & 0.2859, 40.0, 0.0, 3.9424, 14.7409, 40.0, 40.0, 8.3598, 40.0, 9.4125, 16.7566, 40.0 \\
36 & $\rho_{7}$ & 3, 4, 5, 21, 22, 23, 25, 33, 39 & 0.2001, 40.0, 0.0, 3.833, 14.609, 40.0, 40.0, 8.4079, 40.0, 9.4341, 15.6363, 40.0 \\
36 & $\rho_{8}$ & 3, 4, 5, 21, 22, 23, 25, 33, 39 & 0.1251, 40.0, 0.0, 3.7374, 14.4952, 40.0, 40.0, 8.4523, 40.0, 9.4502, 14.6754, 40.0 \\
36 & $\rho_{9}$ & 3, 4, 5, 21, 22, 23, 25, 33, 39 & 0.0668, 40.0, 3.6633, 14.4076, 40.0, 40.0, 8.4883, 40.0, 9.461, 13.9404, 40.0 \\
37 & $\rho_{0}$ & 28, 32, 34, 38, 39 & 8.46, 0.0, 1.685, 3.7252, 0.0692, 1.3805, 1.2413 \\
37 & $\rho_{1}$ & 24, 28, 32, 38 & 7.9204, 0.8944, 0.0, 0.7723, 2.1886, 4.1463, 0.2466 \\
37 & $\rho_{2}$ & 3, 4, 5, 10, 21, 22, 23, 27 & 0.8001, 40.0, 0.0, 6.0626, 23.9406, 39.0678, 15.5292, 40.0, 40.0, 4.3521, 39.3821 \\
37 & $\rho_{3}$ & 3, 4, 5, 10, 21, 22, 27 & 0.6251, 40.0, 0.0, 5.905, 23.9094, 38.7627, 15.2526, 40.0, 40.0, 37.9198 \\
37 & $\rho_{4}$ & 3, 4, 5, 10, 21, 22, 27 & 0.5001, 40.0, 0.0, 5.7412, 23.7059, 38.8547, 15.5102, 40.0, 40.0, 36.6537 \\
37 & $\rho_{5}$ & 3, 4, 5, 10, 21, 22, 27 & 0.3848, 40.0, 0.0, 5.5664, 23.4987, 38.8482, 15.6429, 39.4074, 40.0, 35.909 \\
37 & $\rho_{6}$ & 3, 4, 5, 10, 21, 22, 27 & 0.2859, 40.0, 0.0, 5.384, 23.2971, 38.7325, 15.626, 38.1328, 40.0, 35.8074 \\
37 & $\rho_{7}$ & 3, 4, 5, 10, 21, 22, 27 & 0.2001, 40.0, 0.0, 5.2269, 23.1222, 38.63, 15.6104, 37.0339, 40.0, 35.7196 \\
37 & $\rho_{8}$ & 3, 4, 5, 10, 21, 22, 27 & 0.1251, 40.0, 0.0, 5.0902, 22.9691, 38.5387, 15.5959, 36.0766, 40.0, 35.6428 \\
37 & $\rho_{9}$ & 3, 4, 5, 10, 21, 22, 27 & 0.0668, 40.0, 4.9842, 22.8498, 38.4667, 15.5843, 35.3349, 40.0, 35.5832 \\
38 & $\rho_{0}$ & 28, 32, 34, 37, 39 & 8.46, 0.0, 1.685, 3.7252, 0.0692, 0.3588, 1.2413 \\
38 & $\rho_{1}$ & 24, 28, 32, 37 & 8.0134, 0.6396, 0.0, 0.3063, 2.3275, 4.2612, 1.7726 \\
38 & $\rho_{2}$ & 3, 4, 5, 21, 22, 23, 25, 33, 39 & 0.8001, 40.0, 0.0, 12.9905, 14.3216, 18.8796, 22.9238, 38.1451, 40.0, 4.1588, 24.292, 40.0 \\
38 & $\rho_{3}$ & 3, 4, 5, 21, 22, 23, 25, 33, 39 & 0.6251, 40.0, 0.0, 12.9515, 13.0823, 17.4601, 22.4132, 36.0402, 40.0, 5.4192, 18.5827, 40.0 \\
38 & $\rho_{4}$ & 3, 4, 5, 21, 22, 23, 25, 33, 39 & 0.5001, 40.0, 0.0, 12.9215, 12.2062, 16.4581, 22.0462, 34.559, 40.0, 6.306, 14.5508, 40.0 \\
38 & $\rho_{5}$ & 3, 4, 5, 21, 22, 23, 25, 33, 39 & 0.3848, 40.0, 0.0, 12.8923, 11.4041, 15.5416, 21.7058, 33.208, 40.0, 7.115, 10.8627, 40.0 \\
38 & $\rho_{6}$ & 3, 4, 5, 21, 22, 23, 25, 33, 39 & 0.2859, 40.0, 0.0, 12.8663, 10.7215, 14.7625, 21.4129, 32.0618, 40.0, 7.8012, 7.7263, 40.0 \\
38 & $\rho_{7}$ & 3, 4, 5, 21, 22, 23, 25, 33, 39 & 0.2001, 40.0, 0.0, 12.8429, 10.1336, 14.0919, 21.1581, 31.0774, 40.0, 8.3905, 5.0269, 40.0 \\
38 & $\rho_{8}$ & 3, 4, 5, 21, 22, 23, 25, 33, 39 & 0.1251, 40.0, 0.0, 12.8218, 9.6218, 13.5086, 20.9347, 30.2226, 40.0, 8.9023, 2.6783, 40.0 \\
38 & $\rho_{9}$ & 3, 4, 5, 21, 22, 23, 25, 33, 39 & 0.0668, 40.0, 12.8051, 9.2254, 13.0571, 20.7606, 29.5619, 40.0, 9.2978, 0.8606, 40.0 \\
39 & $\rho_{0}$ & 28, 32, 34, 37, 38 & 8.46, 0.0, 1.685, 3.7252, 0.0692, 0.3588, 1.3805 \\
39 & $\rho_{1}$ & 28, 32, 34, 37, 38 & 7.881, 0.9904, 0.0, 2.3541, 5.0639, 0.6916, 3.7929, 0.7231 \\
39 & $\rho_{2}$ & 3, 4, 5, 21, 22, 23, 25 & 0.8001, 40.0, 0.0, 4.4531, 16.6514, 40.0, 40.0, 34.6448, 40.0, 15.4614 \\
39 & $\rho_{3}$ & 3, 4, 5, 21, 22, 23, 25 & 0.6251, 40.0, 0.0, 4.2896, 16.5103, 40.0, 40.0, 34.7025, 40.0, 15.2344 \\
39 & $\rho_{4}$ & 3, 4, 5, 21, 22, 23, 25 & 0.5001, 40.0, 0.0, 4.1719, 16.4111, 40.0, 40.0, 34.7546, 40.0, 15.0698 \\
39 & $\rho_{5}$ & 3, 4, 5, 21, 22, 23, 25 & 0.3848, 40.0, 0.0, 4.0627, 16.3206, 40.0, 40.0, 34.8104, 40.0, 14.9162 \\
39 & $\rho_{6}$ & 3, 4, 5, 21, 22, 23, 25 & 0.2859, 40.0, 0.0, 3.9686, 16.2439, 40.0, 40.0, 34.8637, 40.0, 14.7834 \\
39 & $\rho_{7}$ & 3, 4, 5, 21, 22, 23, 25 & 0.2001, 40.0, 0.0, 3.8868, 16.1781, 40.0, 40.0, 34.9139, 40.0, 14.6675 \\
39 & $\rho_{8}$ & 3, 4, 5, 21, 22, 23, 25 & 0.1251, 40.0, 0.0, 3.815, 16.1209, 40.0, 40.0, 34.9607, 40.0, 14.5655 \\
39 & $\rho_{9}$ & 3, 4, 5, 21, 22, 23, 25 & 0.0668, 40.0, 3.759, 16.0767, 40.0, 40.0, 34.9988, 40.0, 14.4857 \\
40 & $\rho_{0}$ & 28, 32, 34, 36, 38 & 8.4641, 0.0, 1.1682, 4.4575, 2.2769, 1.9541, 0.8421 \\
40 & $\rho_{1}$ & 27, 28, 32, 34, 38 & 8.0576, 0.5755, 0.0, 1.0797, 2.0556, 4.8491, 1.2892, 2.7233 \\
40 & $\rho_{2}$ & 3, 5, 6, 21, 22, 23, 25, 33 & 0.8001, 40.0, 0.0, 16.2802, 10.0071, 0.8967, 40.0, 13.2398, 29.6258, 40.0, 40.0 \\
40 & $\rho_{3}$ & 3, 5, 6, 21, 22, 23, 25, 33 & 0.6251, 40.0, 0.0, 16.1565, 9.8913, 0.6795, 40.0, 13.4879, 27.8043, 40.0, 40.0 \\
40 & $\rho_{4}$ & 3, 5, 6, 21, 22, 23, 25, 33 & 0.5001, 40.0, 0.0, 16.0669, 9.8071, 0.5206, 40.0, 13.6789, 26.5124, 40.0, 40.0 \\
40 & $\rho_{5}$ & 3, 5, 6, 21, 22, 23, 25, 33 & 0.3848, 40.0, 0.0, 15.9587, 9.6872, 0.3073, 40.0, 13.8761, 25.1584, 40.0, 39.1112 \\
40 & $\rho_{6}$ & 3, 5, 21, 22, 23, 25, 33 & 0.2859, 40.0, 0.0, 15.7173, 9.4906, 40.0, 13.9328, 23.6801, 39.7836, 33.8486 \\
40 & $\rho_{7}$ & 3, 5, 21, 22, 23, 25, 33 & 0.2001, 40.0, 0.0, 15.4874, 9.4508, 40.0, 13.7933, 23.0911, 39.2752, 29.5496 \\
40 & $\rho_{8}$ & 3, 5, 21, 22, 23, 25, 33 & 0.1251, 40.0, 0.0, 15.2864, 9.4166, 40.0, 13.6738, 22.5771, 38.8308, 25.8031 \\
40 & $\rho_{9}$ & 3, 5, 21, 22, 23, 25, 33 & 0.0668, 40.0, 15.1301, 9.3901, 40.0, 13.5824, 22.1784, 38.4854, 22.8975 \\
41 & $\rho_{0}$ & 28, 32, 37, 38, 39 & 8.4617, 0.0, 2.048, 3.4139, 0.7549, 2.1776, 2.5777 \\
41 & $\rho_{1}$ & 28, 32, 37, 38 & 7.9215, 0.8957, 0.0, 3.274, 4.1968, 4.3058, 2.4521 \\
41 & $\rho_{2}$ & 3, 4, 5, 20, 21, 22, 23, 25, 29, 32, 36, 37, 38, 39 & 0.8001, 40.0, 0.0, 4.2353, 15.2321, 40.0, 16.3957, 40.0, 24.1538, 40.0, 5.2109, 0.2405, 27.3646, 40.0, 40.0, 40.0, 40.0 \\
41 & $\rho_{3}$ & 3, 4, 5, 20, 21, 22, 23, 25, 32, 36, 37, 38, 39 & 0.6251, 40.0, 0.0, 3.8072, 14.9576, 40.0, 16.5738, 40.0, 23.9801, 40.0, 3.3489, 32.333, 40.0, 40.0, 40.0, 40.0 \\
41 & $\rho_{4}$ & 3, 4, 5, 20, 21, 22, 23, 25, 32, 36, 37, 38, 39 & 0.5001, 40.0, 0.0, 3.5157, 14.7537, 40.0, 16.6401, 40.0, 23.8734, 40.0, 1.9975, 35.6723, 40.0, 40.0, 40.0, 40.0 \\
41 & $\rho_{5}$ & 3, 4, 5, 20, 21, 22, 23, 25, 32, 36, 37, 38, 39 & 0.3848, 40.0, 0.0, 3.2474, 14.5665, 40.0, 16.701, 40.0, 23.7793, 40.0, 0.7564, 38.7337, 40.0, 40.0, 40.0, 40.0 \\
41 & $\rho_{6}$ & 3, 4, 5, 20, 21, 22, 23, 25, 32, 36, 37, 38, 39 & 0.2859, 40.0, 0.0, 3.0923, 14.4424, 40.0, 16.6592, 40.0, 23.7056, 40.0, 0.1685, 40.0, 40.0, 40.0, 40.0, 40.0 \\
41 & $\rho_{7}$ & 3, 4, 5, 20, 21, 22, 23, 25, 32, 36, 37, 38, 39 & 0.2001, 40.0, 0.0, 2.9118, 14.3698, 40.0, 16.7854, 39.1052, 23.81, 40.0, 0.1101, 40.0, 40.0, 40.0, 40.0, 40.0 \\
41 & $\rho_{8}$ & 3, 4, 5, 20, 21, 22, 23, 25, 32, 36, 37, 38, 39 & 0.1251, 40.0, 0.0, 2.7218, 14.3081, 40.0, 16.9674, 38.0552, 23.9516, 40.0, 0.0784, 40.0, 40.0, 40.0, 40.0, 40.0 \\
41 & $\rho_{9}$ & 3, 4, 5, 20, 21, 22, 23, 25, 32, 36, 37, 38, 39 & 0.0668, 40.0, 2.5742, 14.2601, 40.0, 17.1084, 37.2414, 24.0626, 40.0, 0.0532, 40.0, 40.0, 40.0, 40.0, 40.0 \\
42 & $\rho_{0}$ & 28, 31, 32, 39 & 8.4489, 0.0, 2.9557, 0.5842, 3.6518, 2.8956 \\
42 & $\rho_{1}$ & 28, 32, 37 & 7.7615, 1.3738, 0.0, 5.2569, 5.1678, 1.1169 \\
42 & $\rho_{2}$ & 3, 4, 5, 8, 25, 29 & 0.8001, 40.0, 0.0, 2.7474, 11.5296, 40.0, 3.1096, 25.1785, 20.4246 \\
42 & $\rho_{3}$ & 3, 4, 5, 8, 25, 29 & 0.6251, 40.0, 0.0, 2.3752, 11.4155, 40.0, 3.8174, 24.0895, 20.2282 \\
42 & $\rho_{4}$ & 3, 4, 5, 8, 25, 29 & 0.5001, 40.0, 0.0, 2.1107, 11.3338, 40.0, 4.3177, 23.3171, 20.0931 \\
42 & $\rho_{5}$ & 3, 4, 5, 8, 25, 29 & 0.3848, 40.0, 0.0, 1.8676, 11.2583, 40.0, 4.7757, 22.6083, 19.9723 \\
42 & $\rho_{6}$ & 3, 4, 5, 8, 25, 29 & 0.2859, 40.0, 0.0, 1.66, 11.1935, 40.0, 5.1653, 22.0037, 19.8716 \\
42 & $\rho_{7}$ & 3, 4, 5, 8, 25, 29 & 0.2001, 40.0, 0.0, 1.4806, 11.1373, 40.0, 5.5009, 21.482, 19.7864 \\
42 & $\rho_{8}$ & 3, 4, 5, 8, 25, 29 & 0.1251, 40.0, 0.0, 1.324, 11.088, 40.0, 5.7929, 21.0271, 19.7135 \\
42 & $\rho_{9}$ & 3, 4, 5, 8, 25, 29 & 0.0668, 40.0, 1.2025, 11.0496, 40.0, 6.0191, 20.6745, 19.6578 \\
\hline
\end{longtable}}

\medskip\noindent\rule{\linewidth}{0.4pt}\medskip

\section*{Appendix D. The verification scripts}\addcontentsline{toc}{section}{Appendix D. The verification scripts}

\subsection*{D.1 The alias constraints}

\begin{lstlisting}
"""Rigorous check (Arb) of the alias-step constraints for the big-block design with value classes of right masks.
Exponents per n; worst case over all deficiency profiles with f(1/2) <= 1/2 - tau0 (else the preliminary search wins).
- 2^F repeated trials, F = -log2 E_V 2^{-d(V)} over (m-2k)-subsets V of M, so F <= d(M) and F <= 1/2 - tau0.
- Right masks Y are grouped by (b^F(Y), c^F(Y)); at most (2m+1) 2^{m - d(M)} classes, so the right list over all trials
  is at most r + (1 - gamma) mu.  The left side pays with d(L) >= F, which needs l >= mu(1-2a).
Usage: python verify_alias_bigclass.py params.json"""
from flint import arb, ctx
import json, sys
ctx.prec = 200
P = json.load(open(sys.argv[1]))
Q = lambda x: arb(str(x))
tau0, eta, a, mu, gam, l = Q(P["tau0"]), Q(P["eta"]), Q(P["a"]), Q(P["mu"]), Q(P["gamma"]), Q(P["l"])
r = 1 - mu - l
def h(x): return -(x*x.log() + (1-x)*(1-x).log())/arb(2).log()
half = Q("0.5"); v = mu*(1 - 2*a); Fmax = half - tau0
log2_3 = arb(3).log()/arb(2).log()
vals = {
  "h(2a)/2": h(2*a)/2,
  "left list": l + (h(a) - gam)*mu,
  "right classes": r + (1 - gam)*mu,
  "pairs": 1 - mu + (h(a) + h(half - a) - gam)*mu - eta,
  "right masks, all trials": h(half - a)*mu + Fmax,
  "left masks, all trials": h(a)*mu + Fmax,
  "(mu/2) log2 3": (mu/2)*log2_3, "v": v, "l": l, "r": r}
checks = {
  "coverage: gamma < h(2a)/2": gam < h(2*a)/2,
  "left list:  l + (h(a) - gamma) mu < tau0": vals["left list"] < tau0,
  "right classes: r + (1 - gamma) mu < tau0": vals["right classes"] < tau0,
  "pairs: 1 - mu + (h(a)+h(1/2-a)-gamma) mu - eta < tau0": vals["pairs"] < tau0,
  "right masks in every trial: h(1/2-a) mu + (1/2 - tau0) < tau0": vals["right masks, all trials"] < tau0,
  "left masks in every trial: h(a) mu + (1/2 - tau0) < tau0": vals["left masks, all trials"] < tau0,
  "value maps of L and R: l, r < tau0": (l < tau0) and (r < tau0) and (r > 0),
  "left side pays the trials: l > mu(1-2a)": l > v,
  "reduction mod P on half blocks: (mu/2) log2 3 < eta": vals["(mu/2) log2 3"] < eta,
  "sampled set fits in half the input: mu(1-2a) <= 1/2": v <= half,
}
for k, x in vals.items(): print(f"{k:24s} {x.str(8)}")
ok = True
for k, c in checks.items():
    print(("PASS " if c else "FAIL ") + k); ok &= bool(c)
slacks = [h(2*a)/2 - gam, tau0 - vals["left list"], tau0 - vals["right classes"], tau0 - vals["pairs"],
          tau0 - vals["right masks, all trials"], l - v, eta - vals["(mu/2) log2 3"]]
eps0 = min(x.lower() for x in slacks)
print(f"least slack eps0 >= {eps0.str(5)} (lower end of the Arb balls)")
print("ALL ALIAS CONSTRAINTS PASS (strict)" if ok else "SOME CONSTRAINT FAILS")
\end{lstlisting}

\subsection*{D.2 The 430 region integrals}

\begin{lstlisting}
"""Rigorous certificate (Arb ball arithmetic) for a FAMILY of checksum filters chosen per pattern.

For every filter v and every region of patterns assigned to v (case 1, or a band of W in case 2), the JSON lists
multipliers z >= 0 (z[0] is the Chernoff exponent c) and the rows G of the domination inequality:
  case 1:            G = [Delta - u_v, Gamma - w_v, u_v - u_l (l in cons)]
  band [lo, hi):     G = [a w_v - u_v, w_v - Gamma lo, Gamma hi - w_v (if hi), u_v - u_l (l in cons)]
We prove   int M(x)^2 2^{z . G(x)} dx < 2^eta   (normalised circle integral) for every listed certificate.

Method: the paper's Appendix A, as in verify_moments_gen.py.  Inner convolutions are evaluated exactly at N nodes
with Arb polynomial products; trapezoid errors use analytic-strip bounds.  On the outer strip |Im x| <= 2 beta,
|Re u| <= U* and |Re w| <= W* follow from the Cauchy-Schwarz bounds on Q and QB, which bounds Re(z . G)."""
from flint import arb, arb_poly, ctx
import json, sys, time
ctx.prec = 200
A = arb
P = json.load(open(sys.argv[1]))
tau, eta = A(str(P["tau0"])), A(str(P["eta"]))
Delta = eta - tau; Gamma = 2*tau - eta; aG = Delta/Gamma
N = int(P.get("N", 1024)); sig, beta = A(str(P.get("sig", "0.45"))), A(str(P.get("beta", "0.19")))
pi, ln2 = A.pi(), A(2).log()
pow2 = lambda t: (t*ln2).exp()
log2 = lambda t: t.log()/ln2
xs = [2*pi*k/N for k in range(N)]
def trap_err(B0, width): return 2*B0/((A(N)*width).exp() - 1)
def ball(x, err): return x + A(0, err.upper())
def abs_sum(coef, y): return sum(abs(coef[j])*(A(j+1)*y).cosh() for j in range(len(coef)))
mean = lambda v: sum(v)/N
COS = {}
def cosj(j):
    if j not in COS: COS[j] = [(A(j)*x).cos() for x in xs]
    return COS[j]
def trig_nodes(coef): return [sum(coef[j]*cosj(j+1)[k] for j in range(len(coef))) for k in range(N)]
def corr(Fv, Gv):
    """c[j] = (1/N) sum_i F[(i+j) mod N] G[i], via one polynomial product."""
    pa = arb_poly(Fv)
    b = [A(0)]*(N + 1)
    for i in range(N): b[N - i] = Gv[i]
    pr = pa*arb_poly(b)
    co = [pr[t] for t in range(2*N)]
    return [(co[N + j] + co[j])/N for j in range(N)]

def filter_data(fp):
    r = [A(str(x)) for x in fp["r_coef"]]; s = [A(str(x)) for x in fp["s_coef"]]; kappa = A(str(fp["kappa"]))
    rk, sk = trig_nodes(r), trig_nodes(s)
    F = [pow2(t) for t in rk]; G = [pow2(t) for t in sk]
    Rs, Ss = abs_sum(r, sig), abs_sum(s, sig)
    Zf = ball(mean(F), trap_err(pow2(Rs), sig)); Zg = ball(mean(G), trap_err(pow2(Ss), sig))
    Zfk = ball(mean([pow2(kappa*t) for t in rk]), trap_err(pow2(kappa*Rs), sig))
    Zgk = ball(mean([pow2(kappa*t) for t in sk]), trap_err(pow2(kappa*Ss), sig))
    LZ = log2(Zf*Zg); H = log2(Zfk) + log2(Zgk) - kappa*LZ
    eP = trap_err(pow2(Rs + Ss), sig); eY = trap_err(pow2(Rs + Ss)*(Rs + Ss), sig)
    Pc = corr(F, G)
    Yc1 = corr([F[k]*rk[k] for k in range(N)], G); Yc2 = corr(F, [G[k]*sk[k] for k in range(N)])
    u, w = [], []
    for j in range(N):
        Pj = ball(Pc[j], eP); Yj = ball(Yc1[j] + Yc2[j], eY)
        L = log2(Pj) - LZ; B = Yj/Pj - LZ
        u.append(L + (kappa - 1)*B - H); w.append((2 - kappa)*B - 2*L + H)
    # outer strip bounds (|Im x| <= 2 beta)
    rb = [r[j]*(A(j+1)*beta).cosh() for j in range(len(r))]; sb = [s[j]*(A(j+1)*beta).cosh() for j in range(len(s))]
    Ir = ball(mean([pow2(2*t) for t in trig_nodes(rb)]), trap_err(pow2(2*abs_sum(rb, sig)), sig))
    Is = ball(mean([pow2(2*t) for t in trig_nodes(sb)]), trap_err(pow2(2*abs_sum(sb, sig)), sig))
    Af, Ag = Ir/(Zf*Zf), Is/(Zg*Zg)
    q0 = ((Af - 1)*(Ag - 1)).sqrt()
    qb = (Af*Ag).sqrt()*(abs_sum(r, beta) + abs_sum(s, beta) + abs(LZ))
    assert q0 < 1, "q0 >= 1: outer strip too wide"
    Bmax = qb/(1 - q0); lneg = -log2(1 - q0)
    Ustar = lneg + abs(kappa - 1)*Bmax + abs(H)
    Wstar = 2*lneg + abs(2 - kappa)*Bmax + abs(H)
    return dict(u=u, w=w, Ustar=Ustar, Wstar=Wstar, q0=q0)

t0 = time.time()
FD = [filter_data(fp) for fp in P["filters"]]
print(f"{len(FD)} filters, N = {N}, prepared in {time.time() - t0:.0f} s; max q0 = "
      f"{max(float(d['q0'].upper()) for d in FD):.4f}", flush=True)
for i, d in enumerate(FD):
    print(f"  filter {i}: q0 {float(d['q0'].upper()):.4f}  U* {float(d['Ustar'].upper()):.3f}  W* {float(d['Wstar'].upper()):.3f}")
STRIP = [dict(U=float(d['Ustar'].upper()), W=float(d['Wstar'].upper())) for d in FD]
if len(sys.argv) > 2 and sys.argv[2] == "strip":
    json.dump(STRIP, open(sys.argv[1] + ".strip.json", "w")); sys.exit(0)
m2 = [2 + 2*x.cos() for x in xs]; M2s = 2 + 2*(2*beta).cosh(); target = pow2(eta)
worst = None; ok = True
for c in P["certs"]:
    v, z = c["v"], [A(str(t)) for t in c["z"]]; d = FD[v]; cons = c["cons"]
    if c["case"] == "case1":
        rows = [[Delta - d["u"][k], Gamma - d["w"][k]] for k in range(N)]
        sup = [Delta + d["Ustar"], Gamma + d["Wstar"]]
        assert z[0] >= 1
    else:
        lo, hi = A(str(c["lo"])), (A(str(c["hi"])) if c["hi"] is not None else None)
        rows = [[aG*d["w"][k] - d["u"][k], d["w"][k] - Gamma*lo] + ([Gamma*hi - d["w"][k]] if hi is not None else [])
                for k in range(N)]
        sup = [aG*d["Wstar"] + d["Ustar"], d["Wstar"] - Gamma*lo] + ([Gamma*hi + d["Wstar"]] if hi is not None else [])
        assert z[0]*lo >= 1, "Chernoff exponent must be >= alpha <= 1/lo"
    for l in cons:
        for k in range(N): rows[k].append(d["u"][k] - FD[l]["u"][k])
        sup.append(d["Ustar"] + FD[l]["Ustar"])
    assert len(z) == len(rows[0]) and all(t >= 0 for t in z)
    T = mean([m2[k]*pow2(sum(z[i]*rows[k][i] for i in range(len(z)))) for k in range(N)])
    B0exp = sum(z[i]*sup[i] for i in range(len(z)))
    err = trap_err(M2s*pow2(B0exp), 2*beta)
    I = T + A(0, err.upper())
    good = bool(I < target); ok &= good
    marg = eta - log2(I)
    if worst is None or float(marg.lower()) < float(worst[0].lower()): worst = (marg, c)
    print(f"filter {v} {c['case']}{'' if c['case'] == 'case1' else ' ' + str(c['lo'])}: I <= {I.upper().str(10)}"
          f"  log2 margin {marg.str(5)}  (outer err {err.str(3)}) {'PASS' if good else 'FAIL'}", flush=True)
print(f"worst log2 margin {worst[0].str(5)} at filter {worst[1]['v']} {worst[1]['case']}")
print("ALL CERTIFICATES PASS" if ok else "SOME CERTIFICATE FAILS")
\end{lstlisting}
\end{document}