\section{Discussion}\label{ce:sec:discussion}
Every statement in Part~I that is equivalent to RH is equivalent through a Landau-type argument, and the same is true of the criteria of Parts~II--IV, and each reduces to the holomorphy of $\frac{\zeta'}\zeta$, or of $\frac1\zeta$, to the right of the critical line. The unconditional results are of two kinds. Some are exact identities between the primes and the margin: the Fej\'er formula, the description of the minima, the energy identity and the M\"obius representation. The others are statements of what cannot work: no uniform positive lower bound for $M$ under RH (Proposition~\ref{ce:prop:nomargin}), and no argument that survives planted weights (Theorem~\ref{ce:thm:nogo}). The second kind gives a test that any proposed argument can be run against: if it also works for the weights \eqref{ce:eq:planted}, for a Beurling system \cite{DMV}, or for the Davenport--Heilbronn function of the controls of Part~IV, it cannot prove RH. What remains is an estimate that uses $N(x)=\lfloor x\rfloor$ exactly, through the sawtooth of Proposition~\ref{ce:prop:mobiusrep}. The best unconditional estimates of that kind save $\exp(-c(\log x)^{3/5}(\log\log x)^{-1/5})$, the Vinogradov--Korobov region. A saving of $x^{-1/2+\varepsilon}$ would be RH. We do not prove the Riemann hypothesis.



\begin{remark}[Weil positivity for small support, and where this family sits]\label{ce:rem:cc}
The margin evaluates the Weil functional on the one-parameter Fej\'er family: the autocorrelation underlying $M(N)$ has multiplicative support $[N^{-1},N]$, so the primes enter only through the powers $q<N$. The regime in which Weil positivity is \emph{proved} for all test functions is the archimedean one: Connes and Consani \cite{CC} establish, through the prolate and Sonin operators, a positivity inequality for every $g$ supported in $[2^{-1/2},2^{1/2}]$ (autocorrelation in $[\frac12,2]$), precisely the range in which no prime occurs; the semilocal programme \cite{CCM} is to add the places $2,3,\dots,p$ and the support $(p^{-1},p)$ one prime at a time. Specialized to our family, the proved range is $N\le2$, where $M=R$ and positivity is the elementary estimate of Proposition~\ref{ce:prop:verified}(1); their theorem is of course much stronger there, being an operator bound valid for all $g$. The map between the two languages is Section~\ref{ce:sec:walk}: adding a place is admitting the next batch of prime powers into the walk, and the question both programmes face is the same uniformity -- positivity must survive as the support, and with it every prime, enters. Theorem~\ref{ce:thm:nogo} constrains the operator route too: any semilocal positivity argument robust under the planted weights \eqref{ce:eq:planted} cannot close RH, so the arithmetic identity $\Lambda=\mu*\log$ must enter the operator estimates at some stage. We do not see how to do this, and record the translation in the hope that the one-variable family, where every object is computable, is a useful test bench for that programme.
\end{remark}