Let $X_n(s)=\sum_{k\le n}k^{-s}$ be the partial sums of the Dirichlet series of $\zeta$. The \emph{cross-energy} of consecutive partial sums,
\begin{equation}\label{c2:eq:def}
C_2(n,s)=|X_n(s)|^2-|X_{n-1}(s)|^2-n^{-2\sigma}=2\operatorname{Re}\bigl(X_{n-1}(s)\,\overline{n^{-s}}\bigr),
\end{equation}
records how the newest term interacts with all previous ones. A twenty-two--part series of the author studies this object at the zeros of $\zeta$; the parts on the strip, the energy, the slope, the Euler product and the functional reflection are consolidated, with complete proofs, in the companion monograph \cite{CEn}, while the window studies quoted in Part~V are proved in \cite{P20,P21,P22}. This part states the main results --- those the companion's own accounting marks as new, together with the elementary structure (Theorem~\ref{c2:thm:A}) that frames them --- and the threads that tie them to Parts I--III. Throughout this part $\gamma_0$ is Euler's constant, $\rho=\beta+i\gamma$ runs over the nontrivial zeros, $\Theta=\sup\beta$, and
\[
C_B=\sum_\rho\frac1{\rho(1-\rho)}=2+\gamma_0-\log4\pi=2\widehat B ,
\]
the constant of Parts~I and~III once more. Proofs are in \cite{CEn} at the stated locations; what is classical there is credited there (Cram\'er's theorem, Turing's method, Lagarias's pairing, Weil's criterion, the density theorems), and we repeat none of those attributions here.

\subsection{The trichotomy and the crossing law}
\begin{theorem}[{the trichotomy; \cite[Thm.~A, \S2]{CEn}}]\label{c2:thm:A}
For $0<\sigma<1$,
\[
X_{n-1}(s)\,\overline{n^{-s}}=\frac{n^{1-2\sigma}}{1-s}+\zeta(s)\,n^{-\sigma}e^{it\log n}-\tfrac12n^{-2\sigma}+O\bigl(|s|\,n^{-2\sigma-1}\bigr).
\]
Consequently $\lim_nC_2(n,s)$ equals $1/(\frac14+t^2)$, $0$ or $+\infty$ according as $\sigma=\frac12$, $\sigma>\frac12$ or $\sigma<\frac12$, at every point of the strip, and the zeros of $\zeta$ appear only in the second term. The first term is the trace of the pole at $s=1$: the critical line is where it survives exactly.
\end{theorem}

On the line, the planar curve described by $X_{n-1}\overline{n^{-s}}$ is a helix whose natural orientation gives it a \emph{signed} radius, and that signed radius is Hardy's function $Z(t)$: each zero on the line is a crossing of the axis, and an off-line pair is a missing crossing (Turing's method). At a zero, $C_2(\cdot,\rho)$, extended to real $n$ by the Hurwitz tail $X_{n-1}(s)=\zeta(s)-\zeta(s,n)$, changes sign exactly once beyond $|\gamma|^{3/2}$, at
\[
n^*=\gamma^2+\tfrac13+\tfrac1{180\gamma^2}+O(\gamma^{-4})\qquad(\beta=\tfrac12),
\]
displaced by $(2\beta-1)\gamma^2/(2(1-\beta))+O(1)$ otherwise \cite[Prop.~2.2]{CEn}; in the variable $\lambda=s(1-s)$ the law is a universal series with rational coefficients,
\[
n^*=\lambda+\frac1{12}+\frac1{180\lambda}-\frac{53}{5040\lambda^2}+\cdots,
\]
and the $X$--$Y$ energies obey the same rule: at a zero on the line $|X_N^C|^2=N/\lambda-\frac1{6N}+(\frac1{90}+\frac\lambda{240})N^{-3}+\cdots$ to all orders, two zeros interact only through $\lambda_1-\lambda_2$, and the normalised energy gap at a zero tends to $-\frac16\operatorname{Re}\lambda(\rho)/|1-\rho|^2$, which equals $-B_2$ exactly when $\beta=\frac12$. At a zero on the line $C_2(n,\rho)=1/(\gamma^2+\frac14)-\frac1n-\frac1{12n^2}+O(|\gamma|^5n^{-5})$ increases monotonically to its limit, and \emph{RH is equivalent to the boundedness of $C_2(\cdot,\rho)$ at every zero} \cite[Cor.~2.1]{CEn}. Three arithmetical by-products \cite[\S2]{CEn}: a zero detector on the critical line; the reading of Landau's formula on the circles of the Euler factors --- placed on the circle $|p^{-s}|=p^{-1/2}$ of a prime $p$, the zeros lean toward the angle $\pi$, and toward $\pi+\arg\chi(p)$ for a Dirichlet $L$-function, while on the circle of $e$ they are uniformly distributed; and, for the finite sums $\sum_{k\le N}k^{-s}$, a polygon condition in $\pi(\sqrt N)$ angles governing the real parts of their zeros, with the rules that adding a prime never moves the right edge left, adding a power of $2$ moves it left for $2^k\le16$ --- and $11^2$ is an exception to the corresponding rule for prime powers.

\subsection{Reflected pairs and the functional reflection}
For a reflected pair $\rho$, $1-\bar\rho$ the weights $n^{-2\beta}$ and $n^{-2(1-\beta)}$ compensate.

\begin{proposition}[{reflected pairs; \cite[Prop.~2.2$'$, 2.2$''$]{CEn}}]\label{c2:prop:refl}
The product $C_2(n,\rho)\,C_2(n,1-\bar\rho)$ tends to a finite positive limit for every $\beta$, and is a square on the line (at every point of the line, zero or not). Off the line, beyond $|\gamma|^{3/2}$, it is negative exactly between the two crossings, an interval that contains the crossing $n^*(\frac12+i\gamma)$ of the line, with minimum asymptotically $-(2\beta-1)^2/(4\beta(1-\beta))$ times the limit; for the zeros above an effective but uncomputed height, the critical line is characterized by the nonnegativity of this product. The product equals $(\operatorname{Re}\mathcal C_2)^2-(\operatorname{Re}\mathcal W)^2$, with $\mathcal C_2(N;s,1-s)$ the two-variable cross-energy and $\mathcal W$ the Wronskian of consecutive partial sums, purely imaginary on the line. Summed over the zeros with a weight, the product is Weil's quadratic form at test functions built from the integers below $n$.
\end{proposition}

Note the family resemblance to Part~III: the off-line excess there was $2(2\beta-1)^2/(|\rho|^2|1-\rho|^2)$; the deformation parameter $(2\beta-1)^2$ is the common currency of every detector in this circle. The zeros of the truncations spiral into their limits: with the endpoint term $\frac12N^{-s}$ restored, along helices of radius $|\rho|N^{-\beta-1}/(12|\zeta'(\rho)|)$, proved with explicit constants and verified for the first $100$ zeros; for the Davenport--Heilbronn function the two zeros of an off-line pair converge at rates that differ by $2(\beta-\frac12)$ \cite[Prop.~2.3$'$]{CEn}.

\begin{figure}[htbp]\centering\includegraphics[width=0.85\textwidth]{fig/CE2_reflected.png}
\caption{The reflected-pair product, wave removed, at a synthetic off-line point $s=0.7+i\gamma_1$: negative exactly between the two crossings, with depth governed by $(2\beta-1)^2$. (Figure from \cite{CEn}.)}
\label{c2:fig:reflected}
\end{figure}

The \emph{functional reflection} of the cross-energy is the deepest tool of the series:
\begin{equation}\label{c2:eq:c2fr}
C_2^{\mathrm{FR}}(N,s)=G_N\bigl(s(1-s)\bigr)+\zeta(s)\,K_N(s),
\end{equation}
with $G_N$ explicit and $K_N$ the carrier of the functional equation, whose zeros lie on the critical line \cite[Prop.~2.2$'''$]{CEn}. On the line it equals Hardy's function times $2\cos(t\log N-\vartheta(t))$, \emph{computed from the integers below $N$}, and it sees every zero with its real part. Applied to any number system from its integers alone, it sees the off-line real zero of a Beurling system --- which drifts with $N$, because the system has no functional equation --- while the off-line zero of the Davenport--Heilbronn function is fixed \cite[Prop.~2.2$^{(6)}$]{CEn} (Figure~\ref{c2:fig:c2frB}). Weil's criterion restricted to the cross-energy's own test functions $f*\mu_n$ is still equivalent to RH \cite[Prop.~2.2$^{(4)}$]{CEn}, its positivity residing in the fluctuation of the primes in the dual range $[n,n^2]$, which neither the density of the primes, nor Chebyshev's identity, nor random or Beurling models reproduce; a variant trades the dual range for the primes up to $ne^{2a}$ and the divisor function \cite[Prop.~2.2$^{(5)}$]{CEn}. The margin of the restricted form at height $T$ is of order $T^{-2}$ to $T^{-4}$ of the Gamma term, and as $n\to\infty$ the restricted criterion becomes the positivity of the Hankel matrices of the power sums of $1/(\rho(1-\rho))$ --- the sequence whose first term is $C_B=2\widehat B$ --- computable from $\xi$ at $s=1$ alone, and verified to size $43$ \cite[Prop.~2.2$^{(8)}$]{CEn}.

\begin{figure}[htbp]\centering\includegraphics[width=0.85\textwidth]{fig/CE2_c2frB.png}
\caption{The functional reflection applied to a Beurling system from its integers alone: the off-line real zero is seen, and drifts with $N$ because the system has no functional equation. (Figure from \cite{CEn}.)}
\label{c2:fig:c2frB}
\end{figure}

\subsection{The energy law: a fourth mirror of the same constant}\label{c2:sec:energy}
\begin{theorem}[{the energy law; \cite[Thm.~B, \S3]{CEn}}]\label{c2:thm:B}
Let $E(u)$ be the normalized error term of the prime number theorem. When all zeros are simple,
\[
\limsup_{U\to\infty}\frac1U\int_0^UE(u)^2\,du=\sum_\rho\lim_{n\to\infty}C_2(n,\rho)\qquad\text{in }[0,\infty];
\]
in general both sides are finite or both are infinite, and RH holds if and only if they are finite. The exponential growth rate of the energy is $2\Theta-1$. An unconditional local version expresses the energy in a Gaussian window as an absolutely convergent double sum over zeros; under RH it stays in a band of explicit width around $C_B$.
\end{theorem}

\begin{corollary}[{\cite[Cor.~3.7]{CEn}}]\label{c2:cor:floor}
The mean energy is at least $C_B=2\widehat B$ unconditionally, with equality if and only if RH holds and all zeros are simple.
\end{corollary}

So the constant of Parts~I and~III has a fourth mirror: it is the minimal possible mean energy of the prime-counting error, attained exactly on RH with simple zeros --- the primes up to $10^8$ measure $0.0453$ against $0.0462$. Under RH, simple zeros and a spacing condition, the energy up to $X$ is
\[
C_B\log X+K+\Phi(\log X)+o(1),\qquad K=2.5125\ldots,
\]
with $K$ explicit and $\Phi$ an almost periodic interference term from pairs of zeros, which the primes up to $10^7$ follow with correlation $0.999$ \cite[Prop.~3.8]{CEn} (Figure~\ref{c2:fig:K}); the Gram matrix of the zeros has the $C_2$ limits on its diagonal and as its asymptotic spectrum, and random models of the primes and of $\mu$ (Cram\'er, Denjoy, Wintner) carry $20$ to $150$ times more energy \cite[Rem.~3.9]{CEn} --- an echo, in the cross-energy metric, of the $46\%$ gcd cancellation of Part~II: arithmetic is far quieter than chance. Each zero contributes an orbit of the prime signal, a closed ellipse if $\beta=\frac12$ and an expanding spiral if $\beta>\frac12$, whose phase also determines $\beta$.

\begin{figure}[htbp]\centering\includegraphics[width=0.85\textwidth]{fig/CE2_K.png}
\caption{The second-order energy law $C_B\log X+K+\Phi(\log X)$, $K=2.5125\ldots$: the primes up to $10^7$ follow the interference term $\Phi$ with correlation $0.999$. (Figure from \cite{CEn}.)}
\label{c2:fig:K}
\end{figure}

\subsection{The energy slope}
Let $S(\sigma,t)=\partial_\sigma\log|\xi(\sigma+it)|$, so that RH is equivalent to $S>0$ for $\sigma>\frac12$.

\begin{theorem}[{\cite[Thms.~C, D, E]{CEn}}]\label{c2:thm:CDE}
\emph{(C)} An off-line pair $\rho,1-\bar\rho$ contributes negatively to $S(\sigma,t)$ exactly inside the disc $(\sigma-\frac12)^2+(t-\gamma)^2<(\beta-\frac12)^2$.
\emph{(D)} For $\frac12<\sigma<1$, the set of $t\in[0,T]$ where $S(\sigma,t)\le0$ has measure at most $2\sqrt{\sigma(1-\sigma)}\,N(\sigma,T+1)$: zero-density estimates make it sparse.
\emph{(E)} $S(\sigma,t)$ is Weil's functional evaluated at a Poisson kernel of width $\delta=\sigma-\frac12$; truncating the kernel to a window of length $L$ gives an exact finite expression in the prime powers $n\le e^L$, which under RH lies between $(1-e^{-\delta L})^2$ and $(1+e^{-\delta L})^2$ times $2S(\sigma,t)$. Consequently, under RH, primes up to $P$ resolve $S$ at distance $\delta$ from the line once $\delta\log P\gtrsim\log(1/\varepsilon)$; for the Davenport--Heilbronn function, an off-line zero at horizontal distance $\beta-\sigma$ from the probe becomes visible at $L\approx4.5/(\beta-\sigma)$.
\end{theorem}

Two companions of the slope: the argument of de la Vall\'ee Poussin gives $S>0$ in a region of width $c/\log t$ to the left of $\sigma=1$ \cite[Prop.~4.4]{CEn}; and in the square $v=(\sigma-\frac12)^2$ of the distance to the line, \emph{RH is equivalent to the concavity of $\log|\xi|$ in $v$ at every height} \cite[Prop.~4.3]{CEn} --- the structural conjecture of Appendix~\ref{sec:wkernel}, concavity of $\log W(\sqrt u)$ for the Laguerre kernel, is a cognate statement one level up.

\subsection{The Euler product inside the strip}
The companion's Section~6 identifies each partial Euler product with the $C_2$ energy of the partial sums restricted to smooth integers, and proves that a corrected partial product converges to $\zeta$ in $\frac12<\sigma<1$ \emph{exactly when RH holds}, decaying at a zero on the line like $|\zeta'(\rho)|e^{-\gamma_0}/\log x$. On the critical line the corrected partial products, read along $u=\log x$, are a superposition of helices, one per zero, whose mean energy at $\frac12+it$ is the curvature $-(\log|\xi|)''(t)$ --- the quantity of Appendix~\ref{sec:wkernel} again; computed with care, the helix built from the primes up to $10^8$ fixes the real parts of known zeros to within about $4\cdot10^{-5}$, with symmetric off-line pairs entering only at second order (Figure~\ref{c2:fig:helix}). Sieving the wave $C_2+iS_2$ by the primes and removing the pole correction turns its free amplitude into the Euler helix, so the partial sums and the partial products describe the same object; with a smooth weight in $\log x$ the identity becomes local --- $\zeta$ is reproduced to $10^{-12}$ by the primes and the zeros within distance $20$ of $t$, and $\zeta'(\rho)$ at every simple zero is given exactly, \emph{without} the Deep Riemann Hypothesis \cite[Prop.~6.1$'$]{CEn}. On the line each prime acts on the wave by the real operator $1-p^{-1}T_p$, the same at every height; the sieved functional equation shows why this rigidity does not combine with the reflection --- it maps the half of the strip where the Euler product converges onto the half where it diverges. Three honesty results frame the section: the amplification that proves RH for curves becomes over $\mathbb Q$ a one-sided bound for $\psi(x)$, where, on short intervals near $10^8$, Selberg's sieve loses a factor close to $2$ exactly along Liouville's function --- the parity barrier; the sums over zeros that see $\beta$ are non-holomorphic and are reached from the primes only through energies, the damped energy having its first singularity at $2\Theta-1$ and detecting an off-line zero at height $T$ only with the primes up to about $T^{1/\delta}$; and \emph{no combination of cosines without a constant term can be nonnegative at every $\log p$} \cite[Prop.~6.12]{CEn}: positivity prime by prime requires the pole, and without it positivity can only be quadratic. Finally, the Jensen-moment route: mollifying by the Euler--C\'aceres factor instead of a Dirichlet polynomial removes the pole barrier $0.0386$; numerically, the resulting bound for the off-line excess decays like $1.828\,W(s_0)/\log X$ for $\zeta$ and grows with $X$ for the Davenport--Heilbronn function; an attained off-line supremum of the real parts forces it to grow like a power of $X$, under RH it tends to $0$, and RH is equivalent to its convergence to $0$ --- and, when the supremum of the real parts is attained, to the boundedness of a single mean square of $\zeta/E^*$ on the critical line \cite[Prop.~6.1$'''$--Cor.~6.1$^{(6)}$]{CEn}.

\begin{figure}[htbp]\centering\includegraphics[width=0.85\textwidth]{fig/CE2_helix.png}
\caption{The Euler helix on the critical line at $s=\frac12+20i$: corrected partial products over the primes, one helix per zero, with the running mean of the helix energy against the curvature $-(\log|\xi|)''(20)$. (Figure from \cite{CEn}.)}
\label{c2:fig:helix}
\end{figure}

\subsection{The controls}
Every criterion of the series is tested against the Davenport--Heilbronn function $f$ (same type of functional equation, zeros off the line) and its Euler-product twin $L(s,\chi_5)$ (same conductor and gamma factor): $f$ violates each criterion in the way its off-line zeros predict --- the energy grows at the rate its zeros dictate, the failure set of the pointwise inequality consists of one interval per off-line zero and fills $86$--$90\%$ of Theorem~\ref{c2:thm:CDE}(D)'s bound, the truncated Weil functional goes negative but outside the discs of (C), and a scan of Hardy's function to height $10^5$ shows no extremum that fails to cross for $\zeta$ against eight such extrema for $f$, one per off-line pair --- while $L(s,\chi_5)$ behaves like $\zeta$ in every test. The zeros of the minimiser of Weil's form \cite{Co26} approximate those of $f$ as well as those of $\zeta$ until shortly after the form loses positivity at $L^*\approx3.44$ (which high precision locates below the double-precision value $3.66$), after which all approximations degrade at once and one zero of the minimiser locks onto the ordinate of the off-line pair. Weil's form with two windows detects an isolated zero $\beta=\frac12+\delta$ at height $\gamma$ with the prime powers up to $(C\log\gamma)^{1/\delta}$; the corresponding positivity holds for $\zeta$ up to $3\cdot10^6$ but fails for a random multiplicative model. And a Beurling system of Broucke--Debruyne--R\'ev\'esz type, built to $3\cdot10^7$, shares every zero of $\zeta$, has a real zero at $0.6$, and violates Chebyshev's identity at $\rho_1$ with remainder of order $N^{-0.4}$: \emph{the regularity of the integers cannot replace the functional equation} \cite[Prop.~6.3$'$]{CEn}. These controls are the experimental counterpart of the no-go theorem of Part~V: criteria blind to $f$, or satisfied by zero-sharing Beurling systems, cannot prove RH.