\section{One constant}\label{bh:sec:intro}
Write $\rho$ for the nontrivial zeros of the Riemann zeta function, and
\begin{equation}\label{bh:eq:Bhat}
\widehat B:=\sum_\rho\frac1\rho=1+\frac\gamma2-\frac12\log4\pi=0.023095708966\ldots,
\end{equation}
the sum taken with zeros paired with their conjugates, as usual; the closed form is classical (see \cite[\S12]{Dav}). Three statements, proved in three different languages, put this one number at the centre of the Riemann hypothesis:
\begin{enumerate}
\item (\emph{primes}) RH $\iff$ a specific weighted mean of the error $\psi(t)-t$ stays below $-\widehat B$ for ever (Section~\ref{bh:sec:primes});
\item (\emph{M\"obius}) RH $\iff$ the constant function $1$ can be approximated in a weighted $L^2$ by M\"obius-weighted combinations of the sawtooths $\{1/nt\}$ --- and the squared error is expected to decay exactly like $2\widehat B/\log N$ (Section~\ref{bh:sec:nb});
\item (\emph{zeros}) the doubling $2\widehat B=\sum_\rho1/(\rho(1-\rho))$ is an identity, equal to $\sum_\rho|\rho|^{-2}$ precisely on RH (Section~\ref{bh:sec:zeros}).
\end{enumerate}
The three languages are the three corners of the convolution identity $\Lambda=\mu*\log$: prime side, M\"obius side, zero side. The purpose of this part is modest and expository-plus-numerical: to state the three mirrors with matching normalizations --- the halving between $\widehat B$ and the B\'aez-Duarte constant $2+\gamma-\log4\pi$ seems not to be pointed out explicitly in the literature --- to recompute the Nyman--Beurling distances from scratch, and to record how early the constant becomes visible in the data. We claim no progress towards RH, and the last section of this part says precisely why none of the three mirrors, verified however far, could supply any.

Throughout, $\gamma$ is Euler's constant, $\psi(t)=\sum_{q\le t}\Lambda(q)$, and all numerical work is reproducible from the ancillary code.

\section{The mirror of the zeros}\label{bh:sec:zeros}
Pairing $\rho$ with $1-\rho$ (the functional equation preserves the zero set),
\begin{equation}\label{bh:eq:double}
\sum_\rho\frac1{\rho(1-\rho)}=\sum_\rho\Bigl(\frac1\rho+\frac1{1-\rho}\Bigr)=2\sum_\rho\frac1\rho=2\widehat B=2+\gamma-\log4\pi=0.046191417\ldots,
\end{equation}
an unconditional, absolutely convergent identity. On RH one has $\bar\rho=1-\rho$, so $|\rho|^2=\rho\bar\rho=\rho(1-\rho)$ and
\begin{equation}\label{bh:eq:rhform}
\sum_\rho\frac1{|\rho|^2}=2\widehat B\qquad\text{(on RH)}.
\end{equation}
Moreover \eqref{bh:eq:rhform} is itself \emph{equivalent} to RH: each off-line quadruple $\{\rho,1-\rho,\bar\rho,1-\bar\rho\}$, $\rho=\beta+i\gamma$ with $\beta\ne\frac12$, contributes the strictly positive excess $2(2\beta-1)^2/(|\rho|^2|1-\rho|^2)$ to $\sum_\rho|\rho|^{-2}-\sum_\rho1/(\rho(1-\rho))$, so $\sum_\rho|\rho|^{-2}\ge2\widehat B$ always, with equality if and only if every zero lies on the critical line. The third mirror is thus an equivalence, exactly like the other two. The quantity $\sum_\rho|\rho|^{-2}$ is exactly the constant of the Nyman--Beurling asymptotics below: the inverse-square energy of the zeros.

\section{The mirror of the primes: the Ces\`aro margin}\label{bh:sec:primes}
This mirror is Part~I: RH is equivalent to $M(N)\ge0$ for all large $N$ (a theorem of Suzuki \cite{Su1}), and by the mean identity of Section~\ref{ce:sec:mean}, with $\mu_K(N)$ the $K_N$-weighted mean of the prime-counting error $E^*$,
\begin{equation}\label{bh:eq:onesided}
M(N)=-2\log N\,\bigl(\mu_K(N)+\widehat B\bigr),\qquad\text{RH}\iff\mu_K(N)\le-\widehat B\ \text{for all large }N,
\end{equation}
and the verification of Part~I gives $\mu_K(N)\le-\widehat B-0.0171/\log N$ throughout $10^3\le N\le2.15\cdot10^{29}$. The mean of the prime-counting error must hug $-\widehat B$ from below, for ever.

\section{The mirror of the M\"obius sawtooths: Nyman--Beurling}\label{bh:sec:nb}
The Nyman--Beurling criterion \cite{Ny,Be}, in B\'aez-Duarte's strengthening to integer dilations \cite{BD}, states that RH holds if and only if the constant function $1$ lies in the closed span of the dilated sawtooths $t\mapsto\{1/nt\}$, $n\ge1$, in the natural weighted $L^2$. In the equivalent Dirichlet-polynomial normalization used by \cite{BCF},
\begin{equation}\label{bh:eq:dN}
d_N^2=\inf_{A_N}\frac1{2\pi}\int_{-\infty}^{\infty}\bigl|1-\zeta A_N\bigl(\tfrac12+it\bigr)\bigr|^2\,\frac{dt}{\frac14+t^2},
\qquad A_N(s)=\sum_{n\le N}\frac{a_n}{n^s},
\end{equation}
and RH $\iff d_N\to0$. Burnol \cite{Burn} proved the unconditional lower bound
\begin{equation}\label{bh:eq:burnol}
\liminf_{N\to\infty}\,d_N^2\log N\;\ge\;\sum_{\Re\rho=1/2}\frac{m(\rho)^2}{|\rho|^2},
\end{equation}
$m(\rho)$ the multiplicity, sharpening \cite{BDBLS}; on RH with simple zeros the right side is $\sum_\rho|\rho|^{-2}=2\widehat B$ by \eqref{bh:eq:rhform}, and this value is expected to be the true asymptotic:
\begin{equation}\label{bh:eq:bdasym}
d_N^2\;\sim\;\frac{2\widehat B}{\log N}=\frac{2+\gamma-\log4\pi}{\log N}.
\end{equation}
Bettin, Conrey and Farmer \cite{BCF} proved that, under RH and a mild growth hypothesis on $\sum_\rho|\zeta'(\rho)|^{-2}$, the asymptotic \eqref{bh:eq:bdasym} is already attained by the explicit polynomial
\begin{equation}\label{bh:eq:VN}
V_N(s)=\sum_{n\le N}\Bigl(1-\frac{\log n}{\log N}\Bigr)\mu(n)\,n^{-s}:
\end{equation}
a \emph{Ces\`aro-smoothed M\"obius polynomial} --- the smoothing of Part~I and the M\"obius weights of Part~II in a single object. Thus the B\'aez-Duarte constant \emph{is} $2\widehat B$: the margin constant of Section~\ref{bh:sec:primes}, doubled by the pairing $\rho\leftrightarrow1-\rho$. We have not found this halving stated explicitly, trivial as it is; it is the bridge between the one-sided prime criterion and the two-sided $L^2$ criterion.

\section{The computation}\label{bh:sec:comp}
We recomputed the Nyman--Beurling quantities from scratch. Expanding \eqref{bh:eq:dN}, with $\langle f,g\rangle=\frac1{2\pi}\int f\bar g\,dt/(\frac14+t^2)$ on the critical line, one needs $\|1\|^2=1$, the linear functionals $c_n=\langle1,\zeta n^{-s}\rangle$, and the Gram matrix $G_{mn}=\langle\zeta m^{-s},\zeta n^{-s}\rangle$.

\emph{The linear term.} Writing $\frac1{\frac14+t^2}=\frac1{s(1-s)}$ with $s=\frac12+it$,
\[
c_n=\frac1{2\pi i}\int_{(1/2)}\frac{\zeta(s)\,n^{-s}}{s(1-s)}\,ds
=-\operatorname*{Res}_{s=1}\frac{\zeta(s)\,n^{-s}}{s(1-s)}
=\frac{\gamma-1-\log n}{n},
\]
by closing the contour to the right across the double pole at $s=1$ (expand $\zeta(s)=\frac1{s-1}+\gamma+O(s-1)$ and $n^{-s}=n^{-1}(1-(s-1)\log n+\cdots)$).

\emph{The Gram matrix.} By Mellin--Plancherel, $G_{mn}$ is the autocorrelation $\int_0^\infty\{1/mt\}\{1/nt\}\,dt$, for which Vasyunin \cite{Vas} (see also \cite[p.~141]{BDBLS}) gave the exact closed form: for coprime $m,n$,
\begin{equation}\label{bh:eq:vasyunin}
mn\int_0^\infty\Bigl\{\frac1{nt}\Bigr\}\Bigl\{\frac1{mt}\Bigr\}dt
=\frac{\log2\pi-\gamma}2(m+n)+\frac{m-n}2\log\frac nm-\frac\pi2\bigl(S(m,n)+S(n,m)\bigr),
\end{equation}
\[
S(m,n)=\sum_{k=1}^{n-1}\Bigl\{\frac{mk}n\Bigr\}\cot\frac{k\pi}n ,
\]
and $G_{mn}=g^{-1}G_{m'n'}$ for $g=(m,n)$, $m=gm'$, $n=gn'$. We validated \eqref{bh:eq:vasyunin} against direct numerical integration of $|\zeta(\frac12+it)|^2(n/m)^{it}/(\frac14+t^2)$ out to $t=6000$ at six pairs, with agreement to the size of the truncation tail ($\sim10^{-3}$), and checked that the resulting Gram matrices are positive definite. Then $d_N^2=1-c^{\mathsf T}G^{-1}c$ over all $n\le N$, solved by spectral decomposition with monitored conditioning (relative residuals below $10^{-13}$ throughout the table), and the energy of the Bettin--Conrey--Farmer polynomial $V_N$ is evaluated directly, with no inversion, over the squarefree indices.

\begin{center}
\begin{tabular}{lcccccccc}
\toprule
$N$ & $20$ & $40$ & $80$ & $160$ & $320$ & $500$ & $640$ & $800$\\
\midrule
$d_N^2\log N$ & $0.0494$ & $0.0469$ & $0.0481$ & $0.0474$ & $0.0459$ & $0.0461$ & $0.0453$ & $0.0449$\\
\bottomrule
\end{tabular}
\medskip

\begin{tabular}{lcccccc}
\toprule
$N$ & $100$ & $400$ & $700$ & $1000$ & $1500$ & $2000$\\
\midrule
$\|1-\zeta V_N\|^2\log N$ & $0.822$ & $0.640$ & $0.587$ & $0.562$ & $0.533$ & $0.512$\\
\bottomrule
\end{tabular}
\end{center}

Two readings (Figure~\ref{bh:fig:nb}). The optimal distance locks onto the constant almost immediately: $d_N^2\log N$ oscillates between $0.0448$ and $0.0495$ from $N=20$ on, against $2\widehat B=0.04619$ --- and the computation inputs \emph{no zero of zeta}: only fractional parts, cotangent sums and two residues. The inverse-square energy of the zeros emerges from sawtooth geometry alone. One point of candour: the last two entries ($0.0453$, $0.0449$) lie \emph{below} the floor $2\widehat B$. Burnol's bound \eqref{bh:eq:burnol} constrains only the liminf, so a finite-$N$ undershoot is logically harmless --- the sequence must come back up --- but it is a concrete reminder that finite data cannot certify even the floor, which is the moral of Section~\ref{bh:sec:wall} in miniature. The natural arithmetic approximant $V_N$, by contrast, is still a factor $\sim11$ above the constant at $N=2000$ and descending slowly; its asymptotic optimality \cite{BCF} is invisible at accessible $N$ --- a known phenomenon (compare the graphs cited in \cite{BCF}), and a reminder of how slowly the M\"obius function surrenders its cancellation.

\begin{figure}[htbp]\centering\includegraphics[width=\textwidth]{fig/NB_en.png}
\caption{Left: $d_N^2\log N$ against $2\widehat B=0.04619$. Right: the same, with the Ces\`aro--M\"obius polynomial $V_N$ of \eqref{bh:eq:VN}. No zeros of $\zeta$ enter the computation.}
\label{bh:fig:nb}
\end{figure}

\section{The triangle, and what it cannot do}\label{bh:sec:wall}
The convolution identity $\Lambda=\mu*\log$ has three corners, and $\widehat B$ is pinned at all three. From the zero corner it is a definition, \eqref{bh:eq:Bhat}. From the prime corner it is the exact level that the weighted mean of $\psi(t)-t$ must respect one-sidedly, \eqref{bh:eq:onesided}. From the M\"obius corner it is (doubled) the exact price of building the constant function out of sawtooths, \eqref{bh:eq:bdasym}. The three statements are not three theorems about three constants; they are one constant crossing the explicit formula in three directions.

None of this, verified however far, approaches a proof of RH, and the reasons are the same two walls described in Parts~I and~II: the prime mirror is a one-sided inequality to be held \emph{for ever}, and no finite verification or mean-value bound excludes a late, measure-zero failure; the M\"obius mirror is a limit statement whose rate carries RH only in the limit, and Burnol's bound \eqref{bh:eq:burnol} shows the constant $2\widehat B$ is a \emph{floor}, not a certificate --- reaching it numerically is consistent with RH, and proves nothing. What the triangle does provide is calibration: any proposed approach to RH through means, margins or $L^2$ distances must, at minimum, reproduce $\widehat B$ exactly at all three corners. It is a sharp sanity check, and we offer it as such.