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\title[The Riemann hypothesis as a margin]{The Riemann hypothesis as a margin:\\ a two-sided criterion verified to $2\cdot10^{29}$,\\ the fourth-moment anatomy of M\"obius cancellation,\\ and one constant seen from every side}
\author{Pedro Caceres}
\address{Universidad Europea de Valencia, Valencia, Spain}
\email{pedrojesus.caceres@universidadeuropea.es}
\subjclass[2020]{Primary 11M26; Secondary 11N05, 11N37, 11M06, 11K65}
\keywords{Riemann hypothesis, Ces\`aro means, explicit formula, M\"obius function, Franel--Landau, variance, Nyman--Beurling, B\'aez-Duarte, Vasyunin sums, Weil positivity}
\date{October 2026}

\begin{document}

\begin{abstract}
This paper collects, in one place and with uniform notation, a month of new results around a single way of looking at the Riemann hypothesis: as the statement that certain explicitly computable \emph{margins} stay on one side of explicit constants for ever. Part~I studies the Ces\`aro margin $M(N)$ between a smooth curve $8\sqrt N+O(\log N)$ and the weighted prime sum $2\sum_{q<N}\Lambda(q)q^{-1/2}\log(N/q)$: RH is equivalent to $M(N)\ge0$ for all large $N$ (Suzuki), and we prove, with correctly rounded and ball arithmetic, that $0<M(N)<0.18484$ for every real $1<N\le2.15\cdot10^{29}$, with explicit unconditional two-sided bounds beyond. The margin is analyzed as a random walk against a curve, as an energy, and as a one-sided mean: RH holds if and only if a weighted mean of the prime-counting error stays below $-\widehat B$, $\widehat B=1+\frac\gamma2-\frac12\log4\pi$, for ever. Part~II dissects the M\"obius side of the same coin: the long-run variance of the truncated sawtooth sum $\sum_{d\le D}\mu(d)(\{x/d\}-\frac12)$ obeys an exact positive-definite identity, its distribution is empirically Gaussian, and the Gaussian defect of the fourth moment is taken apart exactly --- the conductor-$1$ orbit decouples, the cross term moves only at composite squarefree conductors, two of its four sectors are $O(D\log^4D)$ unconditionally, a parity sieve annihilates every sector term using an odd number of frequency slots of even conductor, a third sector is proved $O(D^{3/2+o(1)})$, and an empirical jump law, first refuted, is proved exactly in its sector; the four sectors are then computed to $D=800$, where the remaining conjectured object never exceeds $0.44$ against the conjectured $O(D)$. Part~III shows that one constant crosses everything: $2\widehat B$ is the B\'aez-Duarte constant of the Nyman--Beurling criterion, and we recompute the Nyman--Beurling distances from scratch through Vasyunin's cotangent formula, measuring $d_N^2\log N\approx0.046$ against $2\widehat B=0.04619$ with no zero of $\zeta$ used anywhere. Part~IV distils the author's twenty-two--part study of the cross-energy $C_2(n,s)$ of consecutive Dirichlet partial sums, largely consolidated with complete proofs in a companion monograph (the window studies of Part~V are proved in their own installments): the trichotomy that singles out the critical line as the trace of the pole, the energy law --- the mean energy of the prime-counting error is $\ge2\widehat B$ unconditionally, with equality if and only if RH holds with simple zeros: a fourth mirror of the same constant --- the energy slope $\partial_\sigma\log|\xi|$ as a truncated Weil functional exact in the primes, the functional reflection $C_2^{\mathrm{FR}}$ that reproduces Hardy's function from the integers below $N$ and sees the real part of every zero, and the Davenport--Heilbronn and Beurling control methodology. Part~V states precisely what none of this can do: a no-go theorem shows that weights indistinguishable from the primes' to $q^{\delta-1/2}$ satisfy every averaged statement here and still carry an off-line zero, so the exact identity $\Lambda=\mu*\log$ must enter beyond its first moments. We do not prove the Riemann hypothesis, and no statement in this paper should be read as progress towards it beyond what it literally says.
\end{abstract}

\maketitle
\tableofcontents

\section*{Introduction: the programme and the new results}
Everything in this paper is organized around one picture. The Riemann hypothesis, in each of its classical disguises, asserts that an error term is as small as square-root cancellation allows; in each of the three languages of the convolution identity $\Lambda=\mu*\log$ --- primes, M\"obius, zeros --- that assertion can be renormalized into a \emph{margin}: a computable quantity that must stay on one side of an explicit constant, for ever. The programme of this paper is to build those margins exactly, verify them as far as computation reaches, dissect their anatomy with theorems where theorems are available, and state with precision where the theorems stop. All results below are new (the paper is self-contained: it absorbs two unpublished preprints and a research note of the author, cross-checked across six rounds of independent review, and distils the consolidated cross-energy monograph, whose complete proofs it cites rather than repeats); classical inputs are cited where they enter.

\medskip\noindent\textbf{Original contributions at a glance.} For the reader who wants the value of the paper in one list, these are the results we believe are new to this work (the detailed statements, with their proved / measured / conditional status, follow by part):
\begin{enumerate}
\item the first verification of the two-sided margin criterion, $0<M(N)<0.18484$ for all real $1<N\le2.15\cdot10^{29}$, by correctly rounded arithmetic on the primes and ball-certified zeros, with explicit unconditional envelopes $E(N)$ beyond (Section~\ref{ce:sec:margin});
\item the exact description of the local minima of the margin and their recurrence, through the slope $\alpha(x)$ (Section~\ref{ce:sec:minima});
\item the reading of the margin as the variogram of the slope under RH, and the equivalence $\mathrm{RH}\iff\int_1^X\alpha^2\,dx/x=X^{o(1)}$ (Section~\ref{ce:sec:minima});
\item the unconditional energy identity for the slope, and the mean-energy lower bound attained exactly under RH with simple zeros (Section~\ref{ce:sec:energy});
\item the planted-zero no-go theorem: positive weights within $q^{\delta-1/2}$ of the primes' satisfy every averaged ingredient of the programme and still carry an off-line zero (Section~\ref{ce:sec:limits});
\item the M\"obius representation of the slope, the seed of Part~II (Section~\ref{ce:sec:limits});
\item the mean identity $M(N)=-2\int_1^NE^*K_N\,dt-2\widehat B\log N$ and the one-sided mean form $\mathrm{RH}\iff\mu_K(N)\le-\widehat B$, verified with gap $0.0171/\log N$ on the whole range (Section~\ref{ce:sec:mean});
\item the exact, positive-definite variance identity for $\widetilde\Phi_D$, diagonalized by Jordan's totient, identifying every constant in the data and the $46\%$ gcd cancellation (Section~\ref{ms:sec:variance});
\item the primitive-sawtooth decomposition and its orthogonality (Proposition~\ref{ms:prop:primitive});
\item the unit-interval lemma and the complete decoupling of the conductor-$1$ orbit from the fourth moment (Theorem~\ref{ms:thm:decouple});
\item $X(p)=X(p-1)$ at every prime, by a real-space proof with no regularization (Theorem~\ref{ms:thm:primearrival});
\item the sector decomposition of the cross term, with the $(3,1)$ and $(2,2)$ sectors proved $O(D\log^4D)$ unconditionally (Theorem~\ref{ms:thm:tame});
\item the closed forms on the $\{2,p,2p\}$ clusters --- $\operatorname{Cov}(\sigma_p^2,\sigma_{2p}^2)=-(p^2-1)/(48p^4)$ among them --- and the double-ladder orthogonality (Proposition~\ref{ms:prop:ladder}, Lemma~\ref{ms:lem:dladder});
\item the exact $2p$-jump law $-(p^2-1)/(32p^4)$, proved at every prime in its sector, with its empirical failure at $p=11$ explained term by exact term (Theorem~\ref{ms:thm:2pjump});
\item the parity sieve: every sector term using an odd number of frequency slots of even conductor vanishes identically, because a resonance must solve a congruence modulo $2$ with only odd units available (Lemma~\ref{ms:lem:parity}, Corollary~\ref{ms:cor:parity});
\item the pair-coefficient cotangent bound and the first unconditional estimate on a sector with three distinct conductors, $X_{211}\ll D^{3/2+o(1)}$, together with the computation of all four sectors to $D=800$, which locates the sign cancellation that the remaining conjecture needs (Lemma~\ref{ms:lem:paircoef}, Proposition~\ref{ms:prop:x211}, Section~\ref{ms:sec:scale});
\item the halving: the B\'aez-Duarte constant of the Nyman--Beurling criterion is $2\widehat B$, and the third mirror is an equivalence --- each off-line quadruple adds the excess $2(2\beta-1)^2/(|\rho|^2|1-\rho|^2)$ (Section~\ref{bh:sec:zeros});
\item the from-scratch recomputation of the Nyman--Beurling distances through Vasyunin's formula, $d_N^2\log N\approx0.046$ against $2\widehat B=0.04619$, with no zero of $\zeta$ used anywhere (Section~\ref{bh:sec:comp});
\item in the cross-energy series (by the companion's own accounting \cite{CEn}): the energy-law formulation and its local Gaussian-window version, the equality characterization at $C_B=2\widehat B$ --- the fourth mirror --- and the second-order constant $K=2.5125\ldots$ (Section~\ref{c2:sec:energy});
\item the exceptional-set bound and the Poisson-kernel identification and truncation of the energy slope (Theorem~\ref{c2:thm:CDE});
\item the reading of Landau's formula on the circles of the Euler factors, twisted circles included (Section~\ref{c2:thm:A} and around);
\item the functional reflection $C_2^{\mathrm{FR}}$, the restricted-Weil equivalence on its own test functions, and the Hankel criterion in the power sums of $1/(\rho(1-\rho))$;
\item the systematic control methodology: every criterion tested against the Davenport--Heilbronn function, its Euler-product twin $L(s,\chi_5)$, and a zero-sharing Beurling system;
\item the refutation of the Gaussian-mixture route for the Laguerre kernel $W$ of $\xi$, and the log-concavity conjecture left in its place (Appendix~\ref{sec:wkernel}).
\end{enumerate}

\medskip\noindent\textbf{What is due to others.} The one-variable criterion of Part~I --- the function $C(N)$, the equivalence $\mathrm{RH}\iff M(N)\ge0$ eventually, and the window $4(\log\xi)''(\frac12)$ --- is a theorem of Suzuki \cite{Su1,Su2}, rooted in Kre\u{\i}n's screw functions; our proof was written before we found his papers, and priority is his. Weil's explicit formula and criterion, the Franel--Montgomery--Vaughan kernel, the Nyman--Beurling--B\'aez-Duarte framework with Vasyunin's formula \cite{Vas}, Burnol's lower bound \cite{Burn} and the Bettin--Conrey--Farmer theorem \cite{BCF}, the zero-density and verification inputs \cite{KLN,PT}, and the classical layers of the cross-energy series (Cram\'er's theorem, Turing's method, Lagarias's pairing, Landau's formula, the hybrid products), itemized in \cite[\S1]{CEn}, are all prior art and cited where they enter. Everything in the list above is built on top of that floor.

\medskip
In more detail, by part:

\medskip\noindent\textbf{Part I (the Ces\`aro margin).}
\begin{enumerate}
\item The two-sided verification: $0<M(N)<0.18484=4(\log\xi)''(\tfrac12)$ for all real $1<N\le2.15\cdot10^{29}$ --- from the primes to $10^{10}$ with correctly rounded arithmetic (extended to $10^{12}$ in the ancillary runs), and beyond from the first $4520$ zeros certified by ball arithmetic, the verification of RH to $3\cdot10^{12}$ \cite{PT}, and an explicit zero-density bound \cite{KLN}; unconditionally, $-E(N)\le M(N)\le0.18484+E(N)$ with $E(N)$ explicit and $E(10^{20})<3\cdot10^{-5}$ (Section~\ref{ce:sec:margin}).
\item The anatomy: the local minima of $M$ described exactly by the slope $\alpha(x)=\sum_{n\le x}\Lambda(n)n^{-1/2}-2\sqrt x+\frac{c_0}2$; the margin as the variogram of $\alpha$ under RH; an unconditional energy identity, with a mean-energy lower bound attained exactly under RH with simple zeros (Sections~\ref{ce:sec:minima}--\ref{ce:sec:energy}); the walk of the primes against the curve, step by step (Section~\ref{ce:sec:walk}); a probabilistic projection of the margin to large $N$ (Section~\ref{ce:sec:projection}).
\item The mean identity: $M(N)=-2\int_1^NE^*K_N\,dt-2\widehat B\log N$, so RH holds if and only if the $K_N$-weighted mean $\mu_K(N)$ of the prime-counting error satisfies $\mu_K(N)\le-\widehat B$ for all large $N$; verified with margin $0.0171/\log N$ throughout $10^3\le N\le2.15\cdot10^{29}$ (Section~\ref{ce:sec:mean}).
\end{enumerate}

\medskip\noindent\textbf{Part II (the M\"obius sawtooths).}
\begin{enumerate}\setcounter{enumi}{3}
\item The exact variance identity for $\widetilde\Phi_D=\sum_{d\le D}\mu(d)(\{x/d\}-\frac12)$, diagonalized by Jordan's totient into a positive combination of squared restricted Mertens sums; it identifies every constant observed in the data and shows the gcd structure of $\mu$ cancels $46\%$ of the diagonal variance, unconditionally (Section~\ref{ms:sec:variance}); the decomposition into orthogonal primitive sawtooths that becomes the main tool (Proposition~\ref{ms:prop:primitive}).
\item The fourth-moment anatomy (Section~\ref{ms:sec:r4}): the conductor-$1$ orbit decouples completely by a one-line unit-interval lemma (Theorem~\ref{ms:thm:decouple}); $X(p)=X(p-1)$ at every prime, by a real-space proof with no regularization (Theorem~\ref{ms:thm:primearrival}); the four sectors of the cross term, with the $(3,1)$ and $(2,2)$ sectors $O(D\log^4D)$ unconditionally (Theorem~\ref{ms:thm:tame}); closed forms on the $\{2,p,2p\}$ clusters, including $\operatorname{Cov}(\sigma_p^2,\sigma_{2p}^2)=-(p^2-1)/(48p^4)$, and the double-ladder orthogonality (Proposition~\ref{ms:prop:ladder}, Lemma~\ref{ms:lem:dladder}); the exact $2p$-jump law $-(p^2-1)/(32p^4)$, proved at every prime in its sector, with its empirical failure at $p=11$ explained term by exact term (Theorem~\ref{ms:thm:2pjump}); the parity sieve, which annihilates half of each remaining sector (Lemma~\ref{ms:lem:parity}); the bound $X_{211}\ll D^{3/2+o(1)}$ (Proposition~\ref{ms:prop:x211}); and the computation of all four sectors to $D=800$ (Section~\ref{ms:sec:scale}). What remains open is one counting problem, Conjecture~\ref{ms:conj:r4}, whose remaining quadruple sector the data say needs sign cancellation, not absolute values.
\end{enumerate}

\medskip\noindent\textbf{Part III (one constant, three mirrors).}
\begin{enumerate}\setcounter{enumi}{5}
\item The halving $2\widehat B=\sum_\rho1/(\rho(1-\rho))=2+\gamma-\log4\pi$: the B\'aez-Duarte constant of the Nyman--Beurling criterion is exactly twice the margin constant of Part~I --- a bridge we have not found stated in the literature; and a from-scratch recomputation of the Nyman--Beurling distances through Vasyunin's cotangent formula, with $d_N^2\log N$ reaching $0.0461$ at $N=500$ against $2\widehat B=0.04619$, using no zero of $\zeta$ anywhere (Sections~\ref{bh:sec:nb}--\ref{bh:sec:comp}).
\end{enumerate}

\medskip\noindent\textbf{Part IV (the cross-energy $C_2$ and its functional reflection).}
\begin{enumerate}\setcounter{enumi}{6}
\item A distillation of the consolidated cross-energy monograph \cite{CEn}: the trichotomy $\lim_nC_2(n,s)\in\{1/(\frac14+t^2),\,0,\,+\infty\}$ that characterizes the critical line; the crossing law $n^*=\gamma^2+\frac13+\cdots$, universal in $\lambda=s(1-s)$; the reflected-pair product, negative off the line on an explicit window with depth governed by $(2\beta-1)^2$; the energy law --- the mean energy of the prime-counting error equals $\sum_\rho\lim_nC_2(n,\rho)$ when all zeros are simple, is finite iff RH holds, and is $\ge2\widehat B$ unconditionally, with equality iff RH holds with simple zeros: the fourth mirror --- with the second-order constant $K=2.5125\ldots$; the energy slope $S=\partial_\sigma\log|\xi|$ with its disc, density and truncated-Weil theorems; the functional reflection $C_2^{\mathrm{FR}}$ and the Hankel-matrix criterion in the power sums of $1/(\rho(1-\rho))$; and the Davenport--Heilbronn / Dirichlet-twin / Beurling control methodology (Section~\ref{c2:sec:energy} and around).
\end{enumerate}

\medskip\noindent\textbf{Part V (the walls).}
\begin{enumerate}\setcounter{enumi}{7}
\item The no-go theorem: positive weights within $q^{\delta-1/2}$ of $\Lambda(q)q^{-1/2}$ satisfy the energy identity, the prime number theorem and Chebyshev's identity up to $x^{1/2+\delta}$, and yet carry an off-line zero; so no argument built from size estimates and fixed-order averages alone can prove RH --- the identity $\Lambda=\mu*\log$ must be used exactly (Section~\ref{ce:sec:limits}); the computational study of Weil positivity in windows (Section~\ref{ce:sec:windows}); the precise statement of the two walls --- uniformity on the moving diagonal, and the exceptional set --- on the M\"obius side (Section~\ref{ms:sec:wall}); and, as an appendix, the refutation of a natural positivity route for the Laguerre kernel of $\xi$, with a new structural conjecture in its place (Appendix~\ref{sec:wkernel}).
\end{enumerate}

\medskip
Two remarks on reading. First, the parts are mutually reinforcing but independently readable; each keeps the notation of its subject, and the one constant that recurs across parts is $\widehat B=\sum_\rho\rho^{-1}=1+\frac\gamma2-\frac12\log4\pi$. Three notational warnings: $M(N)$ always denotes the Ces\`aro margin, while the Mertens function, which appears only in Part~II, is always written $M(x)$ with argument $x$; $V(D)$ is the variance numerator of Part~II, while $V_N(s)$ is the Bettin--Conrey--Farmer polynomial of Part~III; the cross term $X(D)$ of Part~II is unrelated to the random variable $X$ of the Rice computations in Part~I; and $E(N)$ (the explicit envelope of Part~I), $E^*(t)$ (the prime-counting error of Part~I), $E(u)$ (its normalization in Part~IV) and $e_4(D)$ (the Gaussian defect of Part~II) are four different objects. Second --- and this sentence governs everything that follows --- we do not prove the Riemann hypothesis; every theorem here either averages where RH is pointwise, or verifies where RH is perpetual, and Part~V makes that distinction into mathematics.

\PartHeader{I}{The Ces\`aro margin of the primes}
\input{part_ceb.tex}

\PartHeader{II}{How random is the M\"obius function?}
\input{part_msb.tex}

\PartHeader{III}{One constant, three mirrors}
\input{part_bhb.tex}

\PartHeader{IV}{The cross-energy of Dirichlet partial sums}
\input{part_c2.tex}

\PartHeader{V}{The walls}
The four parts above share a discipline: everything proved is an identity, a verified inequality, or an average. This part collects what that discipline cannot buy; the Davenport--Heilbronn and Beurling controls of Part~IV are its experimental face. Section~\ref{ce:sec:limits} proves it on the prime side; Section~\ref{ce:sec:windows} reports the computational study of Weil positivity; Section~\ref{ms:sec:wall} states the two walls on the M\"obius side; Appendix~\ref{sec:wkernel} closes a tempting door on the $\xi$ side.

\input{part_ce_limits.tex}
\input{part_ce_windows.tex}
\input{part_ms_wall.tex}

\input{part_ce_disc.tex}

\appendix
\section{A cognate positivity problem: the Laguerre kernel of $\xi$ and the Gaussian-mixture route}\label{sec:wkernel}
Write $F(t)=\xi(\frac12+it)=\int_{\mathbb R}\Phi(x)e^{itx}\,dx$ with the classical theta kernel $\Phi>0$, and let $L_1(t)=F'(t)^2-F(t)F''(t)$ be the first Laguerre expression; RH implies $L_1\ge0$ on $\mathbb R$, and proving this unconditionally is an open problem of the Csordas circle \cite{Cs}. One has the exact representation $L_1=\widehat W$ with the nonnegative, even kernel
\[
W(v)=\tfrac12\int_{\mathbb R}(2x-v)^2\,\Phi(x)\Phi(v-x)\,dx=(\Phi_2*\Phi_0-\Phi_1*\Phi_1)(v),\qquad\Phi_k(x)=x^k\Phi(x),
\]
so $L_1\ge0$ is equivalent, by Bochner, to $W$ being positive definite. A natural sufficient condition would be a representation $W(v)=\int_0^\infty e^{-av^2}\,d\mu(a)$ with $\mu\ge0$: positive mixtures of Gaussians are positive definite. \emph{This route is closed.} If such a $\mu$ existed, $g(u)=W(\sqrt u)=\int e^{-au}d\mu(a)$ would be logarithmically convex in $u$, by Cauchy--Schwarz on $\mu$. Computing $W$ in high-precision arithmetic (Simpson quadrature of the doubly exponentially decaying integrand, step-halving agreement to $49$ digits), the second differences of $\log W(\sqrt u)$ on $u\in[0.4,\,8.8]$ (step $0.2$) are \emph{strictly negative throughout}, between $-0.034$ and $-0.093$ --- some $38$ orders of magnitude above the numerical error. No positive Gaussian mixture represents $W$. The data suggest more (Figure~\ref{fig:wgauss}): $\log W(\sqrt u)$ appears to be \emph{concave} on all of $u>0$, with monotone second differences. We record this as a structural conjecture --- $(\,\log W(\sqrt u)\,)''<0$ for $u>0$ --- which rhymes with the doubly exponential decay $\log W(v)\asymp-2\pi e^{|v|}$ inherited from $\Phi$, and leave the positive-definiteness of $W$ where we found it: open, and not approachable by termwise positivity.

\begin{figure}[htbp]\centering\includegraphics[width=\textwidth]{fig/W_gauss.png}
\caption{The Laguerre kernel of $\xi$ in the variable $u=v^2$: $\log W(\sqrt u)$ is concave, not convex, so $W$ is not a positive mixture of Gaussians.}
\label{fig:wgauss}
\end{figure}


\section*{Data and code}
Each part carries its own ancillary code, included with this paper: for Part~I, the verification, certification, walk, mean and projection scripts of the Ces\`aro margin (including \texttt{verify\_margin.py}, run by the author to $10^{12}$, and \texttt{certify\_zeros.py}, Arb ball arithmetic); for Part~II, the exact-arithmetic suite (\texttt{xexact.py}, \texttt{decouple.py}, \texttt{sectors.py}, \texttt{ladder.py}, \texttt{xsectors.py}, \texttt{parity.py}, and the sampling and figure scripts); for Part~III, \texttt{nbmain.py} and the Vasyunin--Gram data; for Part~IV, the code and data travel with the companion monograph \cite{CEn}; for the appendix, the high-precision kernel computation (\texttt{wkernel.py}, in the ancillary). All exact rational claims were reproduced independently in separate arithmetic during review.

\section*{Use of AI assistance}
The research programme and the choice of objects are the author's. Claude (Anthropic) was used throughout as a research and writing aid: to derive and check lemmas, to write and run the computations, to referee drafts in independent sessions, and to draft text and figures. The author checked all results and is responsible for the content.

{\footnotesize
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\end{document}