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\newcommand{\dd}{,d} \newcommand{\ii}{\mathrm{i}} \newcommand{\RR}{\mathbb{R}} \newcommand{\CC}{\mathbb{C}} \newcommand{\HH}{\mathcal{H}} \newcommand{\JJ}{\mathcal{J}} \newcommand{\MM}{\mathcal{M}} \newcommand{\CCal}{\mathcal{C}} \newcommand{\KK}{\mathcal{K}} \newcommand{\eps}{\varepsilon}

\title{A Projected Wavelet Prime-Pair Criterion Equivalent to the Riemann Hypothesis}

\author{Jongmin Choi\ Independent Researcher, Seoul, Korea\ \texttt{24ping@naver.com}\ ORCID iD: 0009-0008-7448-514X}

\date{\today}

\begin{document}

\maketitle

\begin{abstract} We introduce a projected high-pass wavelet criterion for the Riemann Hypothesis.  The construction begins with compactly supported mean-zero wavelets on the logarithmic scale.  For each $0<a\leq1$, an $a$-dependent projection is imposed so that the continuous background term in the prime explicit formula is annihilated exactly.  This produces a log-weighted prime-pair energy whose diagonal contribution is uniformly bounded and whose remaining obstruction is a signed off-diagonal covariance of the von Mangoldt function.

The main result is an equivalence criterion.  For every admissible projected detector bundle, the Riemann Hypothesis is equivalent to the uniform boundedness of the associated Abel-regularized projected prime-pair energy:

\mathrm{RH}
  \quad\Longleftrightarrow\quad
  \sup_{0<a\leq1}\JJ(a)<\infty.

This paper does not claim an unconditional proof of RH.  Instead, it isolates the remaining arithmetic obstruction in the sharp form of a projected non-diagonal prime-pair covariance estimate.  The continuous background, the polar background, and the diagonal atom energy are removed or controlled by construction; the remaining problem is a second-order covariance estimate for $\Lambda(n)$ under a sign-changing logarithmic wavelet kernel. \end{abstract}

\tableofcontents

\section{Introduction}

The Riemann Hypothesis asserts that every nontrivial zero of the Riemann zeta-function has real part $1/2$.  It has many equivalent formulations.  Weil's positivity criterion expresses RH through positivity in the explicit formula \cite{Weil1952}.  Li's criterion gives a sequence of coefficients whose non-negativity is equivalent to RH \cite{Li1997}.  The Balazard--Saias--Yor criterion describes RH through an integral involving $\log|\zeta(s)|$ on the critical strip boundary \cite{BalazardSaiasYor1999}.  Connes' adelic interpretation relates the explicit formula to a trace formula on the ad`ele class space \cite{Connes1999}.  Classical references for the zeta-function and its explicit formula include \cite{Edwards1974,Titchmarsh1986,Ivic1985}.

The present paper develops a different type of equivalence.  Rather than formulating RH through a linear positivity criterion or through a boundary logarithmic norm, we isolate a second-order arithmetic obstruction: a projected, high-pass, log-weighted prime-pair covariance.

Let \begin{equation}\label{eq:H0-def} H_0(s):=-\frac{\zeta'}{\zeta}(s)-\frac{1}{s-1}. \end{equation} The pole at $s=1$ is removed.  The remaining poles of $H_0$ are the nontrivial zeros of $\zeta$.

We introduce compactly supported logarithmic wavelets

W_a(h)=a e^{ah}g_a(ah-\tfrac12)\mathbf 1_{0\le h\le1/a}.

\int_0^{1/a}W_a(h)e^{-h/2}\dd h=0.

The resulting prime-side object is

S_a(x)=
  \sum_{xe^{-1/a}\le n\le x}
  \frac{\Lambda(n)}{\sqrt n}
  W_a\!\left(\log\frac{x}{n}\right).

a\int x^{-2a}|S_a(x)|^2\frac{\dd x}{x}.

\subsection{Main theorem}

The paper constructs an admissible family of projected wavelet detectors indexed by $k\ge0$:

W_{k,a}(h)=a e^{ah}g_{k,a}(ah-\tfrac12)\mathbf 1_{0\le h\le1/a}.

\begin{equation}\label{eq:intro-detector-positivity} \MM_a(\tau):= \sum_{k=0}^{\infty}c_k|\widehat g_{k,a}(\tau)|^2>0 \qquad(\tau\ne0). \end{equation} This ensures that no off-critical zero can be hidden from all detectors.

The main theorem states that for any admissible detector bundle,

\mathrm{RH}
  \quad\Longleftrightarrow\quad
  \sup_{0<a\le1}\JJ(a)<\infty,

\subsection{What this paper does and does not prove}

This paper proves an equivalence criterion.  It does not prove RH unconditionally.  The final unresolved arithmetic task is the uniform estimate

a^2
  \sum_{k=0}^{\infty}c_k
  \sum_{\substack{m<n\le me^{1/a}}}
  \frac{\Lambda(m)\Lambda(n)}{\sqrt{mn}}n^{-2a}
  \CCal_{g_{k,a}}\!\left(a\log\frac nm\right)
  =O(1),

\section{Projected wavelet geometry}\label{sec:projection}

Let

\HH_0:=
  \left\{
  f\in C_c^\infty\!\left(-\frac12,\frac12\right):
  \int_{\RR}f(s)\dd s=0
  \right\}.

\beta(a):=1-\frac{1}{2a}.

\begin{equation}\label{eq:L-functional} L_a(f):= \int_0^1 e^{\beta(a)y} f!\left(y-\frac12\right)\dd y. \end{equation} The condition $L_a(f)=0$ is precisely the condition that the corresponding wavelet annihilates the continuous background.

There is a removable singularity at $a=1/2$ if one divides directly by $L_a(q)$, because $L_{1/2}(f)=0$ for every $f\in\HH_0$.  To avoid this, we use primitives.

For $f\in\HH_0$, define

F(s):=\int_{-1/2}^{s}f(u)\dd u.

\begin{equation}\label{eq:A-functional} A_a(f):= \int_0^1 e^{\beta(a)y} F!\left(y-\frac12\right)\dd y. \end{equation} Integration by parts gives \begin{equation}\label{eq:L-A-relation} L_a(f)=-\beta(a)A_a(f). \end{equation} Thus $A_a$ is the desingularized functional.

\begin{definition}[Projection]\label{def:projection} Choose $Q\in C_c^\infty((-1/2,1/2))$ with $Q\ge0$ and $Q\not\equiv0$, and set

q:=Q'.

A_a(q)=\int_0^1 e^{\beta(a)y}Q(y-\tfrac12)\dd y>0

\begin{equation}\label{eq:projection-def} P_af:=f-\frac{A_a(f)}{A_a(q)}q. \end{equation} \end{definition}

\begin{lemma}[Projected functions annihilate the background]\label{lem:projection-background} For every $f\in\HH_0$ and every $0<a\le1$,

L_a(P_af)=0.

\begin{proof} By definition,

A_a(P_af)=A_a(f)-\frac{A_a(f)}{A_a(q)}A_a(q)=0.

\end{proof}

Given $g_a=P_af$, define \begin{equation}\label{eq:W-def} W_{a}(h):= a e^{ah}g_a!\left(ah-\frac12\right)\mathbf 1_{0\le h\le1/a}. \end{equation} Then \begin{equation}\label{eq:W-background-zero} \int_0^{1/a}W_a(h)e^{-h/2}\dd h=0. \end{equation} Indeed, after the change of variables $y=ah$,

\int_0^{1/a}W_a(h)e^{-h/2}\dd h
  =
  \int_0^1e^{(1-\frac1{2a})y}g_a(y-\tfrac12)\dd y
  =
  L_a(g_a)=0.

\section{Existence of admissible projected detector bundles}\label{sec:admissible}

We now construct bundles satisfying both the background projection and spectral detection.

Let $I\subset(-1/2,1/2)$ be a nonempty open interval.  Choose $Q\ge0$, $Q\not\equiv0$, supported in an interval lying strictly to the left of $I$.  Let $q=Q'$.

Let ${\Phi_k}_{k\ge0}$ be a countable dense subset of $C_c^\infty(I)$ in its usual Fr'echet topology.  Put

f_k:=\Phi_k'.

For each $k$ and $a$, define \begin{equation}\label{eq:gka-def} g_{k,a}:=P_af_k

f_k-\lambda_k(a)q, \qquad \lambda_k(a):=\frac{A_a(f_k)}{A_a(q)}. \end{equation}

\begin{lemma}[Boundedness of projection coefficients]\label{lem:lambda-bounded} For every fixed $k$,

\sup_{0<a\le1}|\lambda_k(a)|<\infty.

\end{lemma}

\begin{proof} For $a$ in any compact subinterval of $(0,1]$, the denominator $A_a(q)$ is positive and continuous, hence bounded away from zero.  The numerator is continuous, so the quotient is bounded.

It remains to examine $a\downarrow0$.  Since

A_a(f_k)=\int_0^1 e^{\beta(a)y}\Phi_k(y-\tfrac12)\dd y,

\end{proof}

Choose positive coefficients $c_k>0$ so rapidly decreasing that \begin{equation}\label{eq:ck-condition} \sup_{0<a\le1} \sum_{k=0}^{\infty}c_k|g_{k,a}|{L^2(\RR)}^2<\infty \end{equation} and, for every $N\ge0$, \begin{equation}\label{eq:Schwartz-seminorm-condition} \sup{0<a\le1} \sum_{k=0}^{\infty}c_k \sup_{\xi\in\RR}(1+|\xi|)^N |\widehat{g}_{k,a}(\xi)|^2<\infty. \end{equation} Such a choice is possible because each $\lambda_k(a)$ is bounded.

Define the detector multiplier \begin{equation}\label{eq:M-def} \MM_a(\tau):=\sum_{k=0}^{\infty}c_k|\widehat{g}_{k,a}(\tau)|^2. \end{equation}

\begin{proposition}[Detector positivity]\label{prop:detector-positivity} For every $0<a\le1$ and every $\tau\ne0$,

\MM_a(\tau)>0.

\begin{proof} Suppose, to the contrary, that for some $a$ and some $\tau\ne0$,

\widehat{g}_{k,a}(\tau)=0

\widehat{P_af}(\tau)=0.

P_af=f-\frac{A_a(f)}{A_a(q)}q.

\widehat{\Phi}(\tau)
  =
  \frac{\int_0^1 e^{\beta(a)y}\Phi(y-\tfrac12)\dd y}
  {\int_0^1 e^{\beta(a)y}Q(y-\tfrac12)\dd y}
  \widehat Q(\tau).

\Phi\mapsto
  \int_0^1 e^{\beta(a)y}\Phi(y-\tfrac12)\dd y.

e^{-\ii\tau s}=C e^{\beta(a)(s+1/2)}

\end{proof}

\begin{definition}[Admissible projected detector bundle]\label{def:admissible} A collection

\mathfrak B=(q,(f_k)_{k\ge0},(c_k)_{k\ge0})

\end{definition}

\section{Continuous background cancellation}\label{sec:background}

Let

W_{k,a}(h)=a e^{ah}g_{k,a}(ah-\tfrac12)\mathbf 1_{0\le h\le1/a}.

F_{k,a}^{\mathrm{cont}}(r):=
  \int_0^{1/a}W_{k,a}(h)e^{(r-h)/2}\dd h.

F_{k,a}^{\mathrm{cont}}(r)
  =
  e^{r/2}
  \int_0^{1/a}W_{k,a}(h)e^{-h/2}\dd h=0.

\begin{proposition}[Exact cancellation of the continuous background]\label{prop:background-cancel} For every $k$ and every $0<a\le1$,

F_{k,a}^{\mathrm{cont}}(r)=0
  \qquad
  (r\in\RR).

\section{Prime atomic energy and prime-pair expansion}\label{sec:prime-energy}

For $x\ge e^{1/a}$ define \begin{equation}\label{eq:Ska-def} S_{k,a}(x):= \sum_{xe^{-1/a}\le n\le x} \frac{\Lambda(n)}{\sqrt n} W_{k,a}!\left(\log\frac{x}{n}\right). \end{equation} The projected prime atomic energy is \begin{equation}\label{eq:J-prime-def} \JJ_{\mathrm{pr}}(a):= \sum_{k=0}^{\infty}c_k, a\int_{e^{1/a}}^\infty x^{-2a}|S_{k,a}(x)|^2\frac{\dd x}{x}. \end{equation}

Expanding the square gives

\JJ_{\mathrm{pr}}(a)
  =
  \sum_{k=0}^{\infty}c_k
  \sum_{m,n\ge2}
  \frac{\Lambda(m)\Lambda(n)}{\sqrt{mn}}
  K_{k,a}(m,n),

\begin{equation}\label{eq:Kmn-def} K_{k,a}(m,n)

a\int_{e^{1/a}}^\infty x^{-2a} W_{k,a}!\left(\log\frac{x}{m}\right) W_{k,a}!\left(\log\frac{x}{n}\right) \mathbf 1_{m\le x\le me^{1/a}} \mathbf 1_{n\le x\le ne^{1/a}} \frac{\dd x}{x}. \end{equation}

Ignoring the lower cutoff for the moment and writing $u=\log m$, $v=\log n$, $r=\log x$, one obtains the full-line kernel

K_{k,a}^{\mathrm{full}}(m,n)
  =
  a\int_{\max(u,v)}^{\min(u+1/a,v+1/a)}
  e^{-2ar}W_{k,a}(r-u)W_{k,a}(r-v)\dd r.

Define the autocorrelation \begin{equation}\label{eq:Cg-def} \CCal_g(\delta) := e^\delta \int_0^{1-\delta} g!\left(y-\frac12\right) g!\left(y+\delta-\frac12\right)\dd y, \qquad 0\le\delta\le1, \end{equation} and $\CCal_g(\delta)=0$ for $\delta>1$.  Then for $m<n\le me^{1/a}$, \begin{equation}\label{eq:K-full} K_{k,a}^{\mathrm{full}}(m,n)

a^2 n^{-2a} \CCal_{g_{k,a}}!\left(a\log\frac nm\right). \end{equation}

\subsection{Diagonal term}

For $m=n$, direct computation gives

K_{k,a}^{\mathrm{full}}(n,n)
  =
  a^2 n^{-2a}\|g_{k,a}\|_2^2.

\begin{equation}\label{eq:diag-term} D(a)

a^2 \sum_{k=0}^{\infty}c_k|g_{k,a}|2^2 \sum{n\ge2}\frac{\Lambda(n)^2}{n^{1+2a}}. \end{equation}

\begin{lemma}[Diagonal boundedness]\label{lem:diag} For an admissible detector bundle,

D(a)=O(1)

\end{lemma}

\begin{proof} It is standard that

\sum_{n\ge2}\frac{\Lambda(n)^2}{n^{1+2a}}
  \ll \frac{1}{a^2}

\end{proof}

\subsection{Off-diagonal term}

The essential off-diagonal term is \begin{equation}\label{eq:offdiag-term} O(a):= 2a^2 \sum_{k=0}^{\infty}c_k \sum_{\substack{m<n\le me^{1/a}}} \frac{\Lambda(m)\Lambda(n)}{\sqrt{mn}} n^{-2a} \CCal_{g_{k,a}}!\left(a\log\frac nm\right), \end{equation} up to the initial cutoff correction.  This is the final arithmetic obstruction.

\section{Regularized Plancherel representation}\label{sec:plancherel}

The formal identity

\int_0^\infty e^{-wu}\dd\mu(u)
  =
  H_0\!\left(\frac12+w\right)

For $\eps>0$ define

\dd\mu_\eps(u):=e^{-\eps u}\dd\mu(u).

\begin{equation}\label{eq:regularized-transform} \int_0^\infty e^{-wu}\dd\mu_\eps(u)

H_0!\left(\frac12+w+\eps\right) \end{equation} in the half-plane of absolute convergence, and by meromorphic continuation elsewhere.

For each $k,a,\eps$, define

F_{k,a,\eps}(r):=
  \int_0^\infty W_{k,a}(r-u)\dd\mu_\eps(u).

\begin{equation}\label{eq:Plancherel-eps} a\int_{\RR}e^{-2ar}|F_{k,a,\eps}(r)|^2\dd r

\frac{a}{2\pi} \int_{-\infty}^{\infty} |\widehat W_{k,a}(a+\eps+\ii t)|^2 \left| H_0!\left(\frac12+a+\eps+\ii t\right) \right|^2\dd t. \end{equation}

\begin{definition}[Regularized projected energy]\label{def:Jreg} For an admissible detector bundle define \begin{equation}\label{eq:J-eps-def} \JJ_\eps(a):= \sum_{k=0}^{\infty}c_k \frac{a}{2\pi} \int_{-\infty}^{\infty} |\widehat W_{k,a}(a+\eps+\ii t)|^2 \left| H_0!\left(\frac12+a+\eps+\ii t\right) \right|^2\dd t. \end{equation} Then put \begin{equation}\label{eq:J-def} \JJ(a):=\liminf_{\eps\downarrow0}\JJ_\eps(a). \end{equation} \end{definition}

For $w=a+\ii t$ one has \begin{equation}\label{eq:W-hat-detection} \widehat W_{k,a}(a+\ii t)

e^{-\ii t/(2a)} \widehat{g}_{k,a}!\left(\frac{t}{a}\right). \end{equation}

\section{Pole detection}\label{sec:pole}

\begin{proposition}[Off-line zeros force blow-up]\label{prop:pole} Let $\mathfrak B$ be an admissible detector bundle.  If $\zeta$ has a zero

\rho_0=\frac12+\eta+\ii\gamma

\JJ(\eta)=+\infty.

\begin{proof} The case $\gamma=0$ is excluded by the classical fact that $\zeta(s)$ has no zeros on the real interval $0<s<1$.  Hence $\gamma\ne0$.

Set $a=\eta$.  Near $t=\gamma$,

H_0\!\left(\frac12+a+\eps+\ii t\right)
  =
  \frac{m}{\eps+\ii(t-\gamma)}+O(1),

\MM_a(\gamma/a)>0.

\JJ_\eps(a)
  \gg
  \int_{|t-\gamma|<\delta}
  \frac{\dd t}{\eps^2+(t-\gamma)^2}
  \gg \frac{1}{\eps}

\end{proof}

\begin{corollary}\label{cor:bound-implies-RH} If

\sup_{0<a\le1}\JJ(a)<\infty,

\end{corollary}

\begin{proof} If RH failed, then by symmetry of the functional equation there would be a zero with real part $1/2+\eta$, $\eta>0$.  Proposition \ref{prop:pole} would give $\JJ(\eta)=+\infty$, contradiction. \end{proof}

\section{Projected kernel estimates under RH}\label{sec:kernel-estimates}

Assume RH in this section.  The zeros of the centered xi-function

\Xi(z):=\xi\!\left(\frac12+z\right)

P_{x,a}(t):=
  \frac{1}{a+\ii(t-x)}+
  \frac{1}{a+\ii(t+x)}.

\begin{equation}\label{eq:Kag-def} \KK_{a,g}(x,y):= \frac{a}{2\pi} \int_{-\infty}^{\infty} |\widehat g(t/a)|^2 P_{x,a}(t)\overline{P_{y,a}(t)}\dd t. \end{equation}

\begin{lemma}[Projected packet kernel bound]\label{lem:projected-kernel-bound} Let $g\in C_c^\infty((-1/2,1/2))$.  For every $N\ge1$, \begin{equation}\label{eq:projected-kernel-bound} |\KK_{a,g}(x,y)| \ll_{N,g} \frac{a^4}{x^2y^2} + \left(1+\frac{x+y}{a}\right)^{-N} \frac{a^2}{(x-y)^2+a^2}, \end{equation} uniformly for $0<a\le1$ and $x,y\ge1$. \end{lemma}

\begin{proof} Write $t=a\tau$.  First consider bounded $\tau$-ranges.  For $x\ge1$ and $0<a\le1$,

\begin{aligned}
  P_{x,a}(a\tau)
  &=
  \frac{1}{a(1+\ii\tau)-\ii x}
  +
  \frac{1}{a(1+\ii\tau)+\ii x}  \\
  &=
  \frac{2a(1+\ii\tau)}{a^2(1+\ii\tau)^2+x^2}.
\end{aligned}

P_{x,a}(a\tau)
  =
  \frac{2a(1+\ii\tau)}{x^2}
  +
  O\!\left(\frac{a^3(1+|\tau|)^3}{x^4}\right).

For the remaining contribution, use that $P_{x,a}(t)$ is a sum of two Cauchy kernels centered at $\pm x$, while $P_{y,a}(t)$ is centered at $\pm y$.  The overlap of the same-sign peaks contributes

\frac{a^2}{(x-y)^2+a^2}.

\ll_{N,g}\left(1+\frac{|t|}{a}\right)^{-N}.

\end{proof}

\begin{lemma}[Dyadic summability of the projected kernel]\label{lem:projected-dyadic} For $N$ sufficiently large,

\sum_{\gamma,
  \gamma'>0}
  \kappa_\gamma\kappa_{\gamma'}
  |\KK_{a,g}(\gamma,\gamma')|
  \ll_{N,g}1

\end{lemma}

\begin{proof} The first term in \eqref{eq:projected-kernel-bound} gives

a^4
  \left(\sum_{\gamma>0}\frac{\kappa_\gamma}{\gamma^2}\right)^2
  \ll 1.

\left(1+\frac{\gamma+
  \gamma'}{a}\right)^{-N}
  \ll_N
  \left(1+\frac{T}{a}\right)^{-N}.

\sum_{\gamma'\in\Gamma_T}
  \frac{a^2}{(\gamma-\gamma')^2+a^2}
  \ll \log^2(T+3),

\ll
  \left(1+\frac{T}{a}\right)^{-N}
  T\log^3(T+3).

\left(1+\frac{T}{a}\right)^{-N}
  \ll_N a^N T^{-N}
  \le T^{-N}.

For far blocks $T'<2^{-1}T$ or $T'>2T$, the Cauchy denominator is no larger than in the same-block analysis and the decay factor contains

\left(1+\frac{T+T'}{a}\right)^{-N}.

\end{proof}

\section{Boundedness under RH}\label{sec:RHbounded}

Let

\Xi(z):=\xi\!\left(\frac12+z\right).

\frac{\Xi'}{\Xi}(z)
  =
  \sum_{\gamma>0}\kappa_\gamma
  \left(\frac{1}{z-\ii\gamma}+\frac{1}{z+\ii\gamma}\right),

H_0\!\left(\frac12+z\right)
  =
  -\frac{\Xi'}{\Xi}(z)+R(z),

R(z)=\frac{1}{z+\frac12}-\frac12\log\pi
  +\frac12\psi\!\left(\frac{z}{2}+\frac14\right).

\begin{proposition}[RH boundedness]\label{prop:RH-bound} Assume RH.  For every admissible detector bundle,

\sup_{0<a\le1}\JJ(a)<\infty.

\begin{proof} It is enough to bound the zero part and the regular part of $H_0$ in \eqref{eq:J-eps-def}, uniformly as $\eps\downarrow0$.

For the zero part, Lemma \ref{lem:projected-dyadic} gives

\sum_{\gamma,
  \gamma'>0}
  \kappa_\gamma\kappa_{\gamma'}
  |\KK_{a,g_{k,a}}(\gamma,\gamma')|
  \ll C_k,

For the regular term, the digamma asymptotic gives

R(a+\ii t)=O(\log(|t|+3))

|\widehat{g}_{k,a}(t/a)|^2

The Abel shift $\eps>0$ moves the line to the right and does not worsen the above estimates.  Taking $\liminf_{\eps\downarrow0}$ gives the claim. \end{proof}

\section{The main equivalence theorem}\label{sec:main}

\begin{theorem}[Projected Wavelet Prime-Pair Criterion]\label{thm:main} Let $\mathfrak B$ be an admissible projected detector bundle.  Then

\mathrm{RH}
  \quad\Longleftrightarrow\quad
  \sup_{0<a\le1}\JJ(a)<\infty,

\end{theorem}

\begin{proof} The implication

\sup_{0<a\le1}\JJ(a)<\infty\Longrightarrow \mathrm{RH}

\mathrm{RH}\Longrightarrow \sup_{0<a\le1}\JJ(a)<\infty

\end{proof}

\section{Arithmetic form of the remaining obstruction}\label{sec:arithmetic}

The diagonal term is uniformly bounded by Lemma \ref{lem:diag}.  Therefore the remaining arithmetic obstruction is the off-diagonal covariance

O(a)=
  2a^2
  \sum_{k=0}^{\infty}c_k
  \sum_{\substack{m<n\le me^{1/a}}}
  \frac{\Lambda(m)\Lambda(n)}{\sqrt{mn}}
  n^{-2a}
  \CCal_{g_{k,a}}\!\left(a\log\frac nm\right).

The projection removes the continuous background exactly:

L_a(g_{k,a})=0.

If one writes $n=m+h$, the obstruction becomes a weighted family of shifted prime-pair correlations

\sum_m\Lambda(m)\Lambda(m+h).

a^2
  \sum_{r,s,r',s'}
  \alpha_r\beta_s\alpha'_{r'}\beta'_{s'}
  \frac{(r's')^{-2a}}{\sqrt{rs\,r's'}}
  \CCal_{g_a}\!\left(a\log\frac{r's'}{rs}\right).

r's'\approx rs.

\section{Relation to existing arithmetic methods}\label{sec:methods}

\subsection{Hardy--Littlewood prime-pair heuristics}

If one inserts the Hardy--Littlewood prediction

\sum_{m\le X}\Lambda(m)\Lambda(m+h)\sim \mathfrak S(h)X,

\subsection{Vaughan and Heath--Brown identities}

Vaughan's identity \cite{Vaughan1980} and Heath-Brown's identity \cite{HeathBrown1982} decompose $\Lambda$ into Type I and Type II sums.  In the present framework, the Type I main terms are continuous and are killed by projection.  The essential difficulty is the Type II--Type II interaction.  This interaction is a four-variable multiplicative correlation, not a one-dimensional short-interval average.

\subsection{Goldston--Montgomery and prime variance}

Goldston and Montgomery related variance of primes in short intervals to pair correlation of zeros \cite{GoldstonMontgomery1987}.  The present criterion is compatible with this philosophy, but differs in the use of projected wavelets and in the precise pole-detection mechanism.  The resulting variance is logarithmic, high-pass, and background-free.

\subsection{Matom"aki--Radziwi\l\l and pretentious methods}

The Matom"aki--Radziwi\l\l theorem gives powerful cancellation for multiplicative functions in almost all short intervals \cite{MatomakiRadziwill2016}.  Hal'asz-type and pretentious methods classify large averages of multiplicative functions through approximation by characters and Archimedean twists \cite{GranvilleSoundararajan2007}.  These tools suggest a possible route to the Type II barrier, but the present obstruction is a four-variable product correlation $r's'\approx rs$, not merely a one-variable short-interval sum.

\subsection{Kuznetsov, spectral large sieve, and higher rank trace formulae}

Kuznetsov trace formulae and spectral large sieve inequalities are powerful tools for shifted convolution problems \cite{DeshouillersIwaniec1982,IwaniecKowalski2004}.  A future proof of the projected Type II bound might require a hybrid of dispersion, trace formulae, and multiplicative energy estimates.  However, no currently standard form of these tools gives the endpoint $O(1)$ estimate required here.

\section{Roadmap and final obstruction}\label{sec:roadmap}

The criterion isolates the remaining problem:

\boxed{
  \text{Prove the projected non-diagonal prime-pair covariance estimate unconditionally.}
  }

2a^2
  \sum_{k=0}^{\infty}c_k
  \sum_{\substack{m<n\le me^{1/a}}}
  \frac{\Lambda(m)\Lambda(n)}{\sqrt{mn}}
  n^{-2a}
  \CCal_{g_{k,a}}\!\left(a\log\frac nm\right)
  =O(1)

The structure of the problem is now as follows. \begin{enumerate}[label=(\roman*)] \item The continuous background is removed exactly by $L_a(g_{k,a})=0$. \item The diagonal atom variance is bounded. \item The Hardy--Littlewood continuous pair main term is annihilated formally by the same projection. \item The remaining term is a signed off-diagonal covariance of the von Mangoldt function. \item After Type II decomposition, the core barrier becomes a four-variable multiplicative geometry problem $r's'\approx rs$. \end{enumerate} Thus RH is converted into a precise second-order arithmetic cancellation problem.

\section{Concluding remarks}

The projected wavelet construction does not prove the Riemann Hypothesis.  It gives a sharp equivalence.  Its value is that it removes several layers of irrelevant structure: the continuous background, the pole at $s=1$, low-frequency drift, and the diagonal atom energy.  The only remaining obstruction is the off-diagonal signed covariance of prime pairs under a compact logarithmic high-pass kernel.

In this sense, the criterion provides a concrete target for future work:

\boxed{
  \text{Establish a uniform projected Type II prime-pair covariance bound.}
  }

\appendix

\section{A finite-cutoff Plancherel identity}

For completeness we record a fully elementary finite version of the Plancherel identity.  Let

\dd\mu_R(u):=
  \sum_{n\le e^R}
  \frac{\Lambda(n)}{\sqrt n}\delta_{\log n}(u)
  -
  e^{u/2}\mathbf 1_{0\le u\le R}\dd u.

F_{k,a,R}(r)=\int W_{k,a}(r-u)\dd\mu_R(u).

a\int_{\RR}e^{-2ar}|F_{k,a,R}(r)|^2\dd r
  =
  \frac{a}{2\pi}
  \int_{-\infty}^{\infty}
  |\widehat W_{k,a}(a+\ii t)|^2
  \left|
  \int e^{-(a+\ii t)u}\dd\mu_R(u)
  \right|^2\dd t.

\section{Autocorrelation identities}

Let $g\in C_c^\infty((-1/2,1/2))$ and set

R_g(\delta):=\int_{\RR}g(s)g(s+\delta)\dd s.

\int_{-1}^{1}R_g(\delta)\dd\delta=0.

\int_0^1 \delta R_g(\delta)\dd\delta
  =
  -\int_{\RR}\Phi(s)^2\dd s.

\widehat{R_g}(\xi)=|\widehat g(\xi)|^2.

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\end{document}