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\title[The energy of the Chebyshev race]{Cross-Energy of Dirichlet Partial Sums.\\ Part 38: The energy of the Chebyshev race:\\ an explicit unconditional floor for the prime race between the lanes $6k\pm1$}
\author{Pedro Caceres}
\address{Universidad Europea de Valencia, Valencia, Spain}
\email{pedrojesus.caceres@universidadeuropea.es}
\thanks{ORCID: 0000-0002-6540-8993}
\subjclass[2020]{Primary 11N13; Secondary 11M20, 11M26}
\keywords{Chebyshev bias, prime races, Dirichlet $L$-functions, explicit constants, energy of arithmetic signals}
\date{November 2026 (revised 1 and 10 November 2026; formerly a standalone note). Part 38 of the book-series \emph{Cross-Energy of Dirichlet Partial Sums} (forty-seven parts, Parts 0--46; block VII: M\"obius cancellation and prime races). Part~0 is the guide. In book version~1 (2 November 2026) this was Part~35; the present numbering is that of book version~2 (10 November 2026).}

\begin{document}

\begin{abstract}
Let $R(x)=\pi(x;3,2)-\pi(x;3,1)$ be the Chebyshev race between the two prime lanes $6k\mp1$ (the prime $2$ counted, by the congruence $2\equiv-1$ mod $3$, in the $6k-1$ lane). We study the race as an \emph{energy} and establish three things (the energy law uses two estimates imported from the $\zeta$ case; see Proposition~\ref{prop:cramer}). First, an elementary ledger identity: the entire bias of the race is the bookkeeping of prime squares, which all land in the lane $6k+1$; subtracting the ledger $\tfrac12\pi(\sqrt x)$ turns a walk with \emph{zero} sign changes up to $10^8$ into one with $15{,}329$ ($21{,}329$ if every passage between $D>0$ and $D\le0$ at a prime is counted). Second, the mean energy of the normalized race signal $e(u)=e^{-u/2}\psi(e^u,\chi_3)$ admits an explicit unconditional lower bound for its upper limit, which we call the floor,
\[
\limsup_{U\to\infty}\frac1U\int_0^U e(u)^2\,du\;\ge\;\kappa_3
:=\sum_{\rho}\frac{1}{\rho(1-\rho)}
=\log\frac3\pi-\gamma_E+2\,\frac{L'}{L}(1,\chi_3)
=0.113229969857\ldots,
\]
the sum running over the nontrivial zeros of $L(s,\chi_3)$ with multiplicity, and the limit exists and attains the floor if and only if the Riemann hypothesis holds for $L(s,\chi_3)$ with all zeros simple. The constant is known: under GRH it is the variance of the limiting distribution of Rubinstein and Sarnak for $(\log x/\sqrt x)R(x)$ (four times smaller than in their normalization $E(x;3,2)-E(x;3,1)$), and its closed form is that of Fiorilli and Martin (there under GRH); it is also the variance $V_\chi$ of Part~37, on the $6k\pm1$ race, where the prime-square mechanism of the bias is identified; what is added here is the unconditional floor with its equality case and exponential-growth law, the ledger correction of the counting race, and the exact measurement. The constant $\kappa_3$ is the exact analogue, for the prime race, of the B\'aez-Duarte constant $2+\gamma_E-\log4\pi$ of the Nyman--Beurling criterion. Third, we measure: from all prime powers up to $10^8$, by exact piecewise integration, the cumulative mean energy is $0.1201$ at $x=10^8$ and lies $4$ to $12\%$ above $\kappa_3$ for $10^4\le x\le10^8$. This is consistent with the theorem and does not test it: the floor is a statement about an upper limit, obtained from the dichotomy between GRH and its failure, and means over single decades fall below $\kappa_3$ ($0.102$ on $[10^4,10^5]$, $0.113$ on $[10^7,10^8]$). The second-order term is $(\text{mean}-\kappa_3)\cdot U\approx0.11$ (between $0.09$ and $0.13$), the race analogue of the second-order constant $K=2.5124797\ldots$ of the prime energy. The proofs of the energy law transplant the $C_2$ energy law for $\zeta$ to $L(s,\chi_3)$, where the absence of a pole simplifies every step.
\end{abstract}

\maketitle

\section{Introduction: the race, the ledger, and the constant}\label{sec:intro}

Among the primes other than $3$, every one lies in one of the two residue lanes mod $3$ --- for $p>3$ these are the familiar lanes $6k-1$ and $6k+1$, and the prime $2$ joins the first by $2\equiv-1$ (mod $3$). The two lanes are not symmetric in the data: writing
\[
R(x)=\pi(x;3,2)-\pi(x;3,1)=-\sum_{p\le x}\chi_3(p),
\]
where $\chi_3$ is the nontrivial character mod $3$, the lane $6k-1$ leads almost always. This is Chebyshev's bias \cite{Ch} in its modulus-three form --- exactly the race of Bays and Hudson \cite{BH}. The lead is overwhelming in any computable range: in our data $R(x)\ge1$ throughout $2\le x\le10^8$ --- zero sign changes --- and the first reversal is known to occur only at $x=608{,}981{,}813{,}029$ \cite{BH}. Yet the lead changes sign infinitely often \cite{Li14}, and under the Riemann hypothesis for $L(s,\chi_3)$ together with linear independence of the zeros, the logarithmic density of the set $\{R(x)>0\}$ is $0.9990\ldots$ \cite{RS}.

This note looks at the race through two elementary moves and one constant.

\emph{The ledger.} The bias is not a mystery about primes; it is bookkeeping about their powers. The square of every prime lands in the lane $6k+1$, because $\chi_3(p^2)=\chi_3(p)^2=1$. In the Chebyshev--von Mangoldt accounting, where the natural object is $\psi(x,\chi_3)=\sum_{n\le x}\chi_3(n)\Lambda(n)$, the lane $6k+1$ therefore carries the full weight of the prime squares, and the \emph{prime} race inherits the debt with the opposite sign. Subtracting the ledger --- the explicit term $\tfrac12\pi(\sqrt x)$ --- removes the entire first-order bias: the corrected race
\[
D(x)=R(x)-\tfrac12\pi(\sqrt x)
\]
changes sign $15{,}329$ times up to $10^8$ (Section~\ref{sec:ledger}, Figure~\ref{fig:race}, left). The operation ``subtract the primes, add their powers'' is exactly the operation that makes the race fair.

\emph{The energy.} Once the drift is understood, the fluctuation remains, and the right way to measure a fluctuation driven by zeros is its mean energy. Let
\[
e(u)=e^{-u/2}\,\psi(e^u,\chi_3),\qquad u=\log x .
\]
Because $L(s,\chi_3)$ has no pole, $e(u)$ is pure oscillation: a superposition of the modes $e^{(\rho-1/2)u}/\rho$, one per nontrivial zero. Its mean energy obeys an exact law with an explicit, unconditional lower bound for its upper limit.

\emph{The constant.} Define
\[
\kappa_3\;=\;\sum_{\rho}\frac{1}{\rho(1-\rho)},
\]
the sum over the nontrivial zeros of $L(s,\chi_3)$, counted with multiplicity; the series converges absolutely, with no hypothesis. In Section~\ref{sec:kappa} we evaluate it in closed form,
\begin{equation}\label{eq:kappa}
\kappa_3=\log\frac3\pi-\gamma_E+2\,\frac{L'}{L}(1,\chi_3)
=0.113229969857\ldots,
\end{equation}
using $L(1,\chi_3)=\pi/(3\sqrt3)$ and $L'(1,\chi_3)=0.2226629870\ldots$ The constant $\kappa_3$ plays, for the race, precisely the role that the B\'aez-Duarte constant $2+\gamma_E-\log4\pi=\sum_\rho1/(\rho(1-\rho))$ (zeros of $\zeta$) plays in the Nyman--Beurling criterion and in Cram\'er's mean square of the prime-counting error \cite{BD,BDBLS,Cr}: it is an unconditional lower bound for the upper limit of a mean energy, attained as a limit exactly at ``RH and simple zeros''. The constant is not new: under GRH it is the variance $\sum_\gamma(\frac14+\gamma^2)^{-1}$ of the limiting distribution of the race \cite{RS}, and the closed form \eqref{eq:kappa} is the case $q=3$ of the formula of Fiorilli and Martin for that variance \cite[Sect.~3]{FM}.

\begin{theorem}[race energy law; proved in Section~\ref{sec:law}]\label{thm:main}
With $e(u)$ as above and $F(U)=\int_0^U e(u)^2\,du$:
\begin{enumerate}
\item[(i)] Unconditionally, $\displaystyle\limsup_{U\to\infty}\frac{F(U)}U\;\ge\;\kappa_3$.
\item[(ii)] The limit $\lim_{U\to\infty}F(U)/U$ exists and equals $\kappa_3$ if and only if all nontrivial zeros of $L(s,\chi_3)$ lie on the critical line and are simple.
\item[(iii)] If $\Theta_3=\sup_\rho\operatorname{Re}\rho>\tfrac12$, then $\limsup_U\frac1U\log F(U)=2\Theta_3-1$: a single off-line zero makes the energy grow exponentially in $u$.
\end{enumerate}
\end{theorem}

\begin{remark}[what the floor is]\label{rem:floor}
Part (i) is proved by a dichotomy. If GRH holds for $L(s,\chi_3)$, the limit exists and is at least $\kappa_3$ (Proposition~\ref{prop:cramer}); if it fails, the energy grows exponentially (Proposition~\ref{prop:uncond}). So (i) is a statement about the upper limit only. It gives no inequality $F(U)/U\ge\kappa_3$ at any finite $U$, and no finite computation can confirm or refute it. Under GRH with simple zeros the mean tends to $\kappa_3$, and nothing keeps means over windows above $\kappa_3$; in the data they fall below it (Section~\ref{sec:numerics}).
\end{remark}

The proof transplants the $C_2$ energy law of the author's cross-energy series --- the growth-rate theorem and the energy law for $\zeta$, \cite[Theorem~2.2, Theorem~B and Corollary~2.3]{CE5}, consolidated in \cite[\S3]{CP} --- from $\zeta$ to $L(s,\chi_3)$. The transplant is a simplification, not a generalization: the pole of $\zeta$, which there had to be subtracted, is absent here, and every step shortens.

Finally we measure. From all prime powers up to $10^8$, with the integral $F(U)$ computed \emph{exactly} between consecutive prime-power jumps, the mean energy is
\[
0.1255,\quad 0.1208,\quad 0.1209,\quad 0.1212,\quad 0.1201
\qquad\text{at }x=10^4,10^5,10^6,10^7,10^8,
\]
against $\kappa_3=0.11323$: above it by $6$--$11\%$ (nothing forces this at finite $U$; see Remark~\ref{rem:floor}), and with the excess times $U$ hovering between $0.09$ and $0.13$ --- the race analogue of the second-order constant $K=2.512479790006\ldots$ determined for $\zeta$ in \cite[Theorem~5 and Corollary~9]{CE6} (closed form without zeros, certified under RH; see also \cite[\S3]{CP}) (Section~\ref{sec:numerics}).

\emph{Relation to the author's earlier work.} This part continues two earlier parts of the series, and repeats part of each. Part~37, on the $6k\pm1$ race \cite{CE37}, already identifies the prime squares as the cause of the bias (in the normalized $\theta$-race they shift the $6k+1$ side by exactly $1$, \cite[Prop.~4.2]{CE37}), evaluates the constant of this part in closed form --- there it is the variance $V_\chi=\sum_\rho1/(\rho(1-\rho))=0.113230$, \cite[Prop.~4.3]{CE37}, with the same Hadamard proof as our Proposition~\ref{prop:kappa} --- states that under GRH with simple zeros it is the mean square of $\psi(x,\chi_3)/\sqrt x$, and measures that mean square on $[10^4,2^{27}]$. The energy law that we transplant, with its unconditional growth-rate part, is the author's law for $\zeta$ \cite[Theorem~2.2, Theorem~B, Corollary~2.3]{CE5}, and the second-order constant we compare with is that of \cite[Theorem~5, Corollary~9]{CE6}. The $\zeta$-row of the same instrument is the energy gauge, now a section of Part~6 \cite{CE6}.

\emph{What is new, and what is not.} The bias mechanism (the prime squares) is classical, implicit in every treatment since Chebyshev and explicit in the framework of Rubinstein and Sarnak \cite{RS}; the first-order debiasing is folklore, and the mechanism is already in Part~37 \cite{CE37}. The distribution theory of races under GRH and linear independence is \cite{RS}; we assume neither. The constant $\kappa_3$ and its closed form \eqref{eq:kappa} are known \cite{RS,FM}; they are also \cite[Prop.~4.3]{CE37}. What we believe is new here is: the \emph{unconditional} floor with the equality characterisation and the exponential-growth law (Theorem~\ref{thm:main}), which in Part~37 \cite{CE37} appear only under GRH and for a different signal; the ledger correction of the \emph{counting} race, $D(x)=R(x)-\tfrac12\pi(\sqrt x)$, with its sign-change count; and the exact piecewise measurement of the cumulative race energy from $u=0$ against the floor, with the measured second-order constant. Unconditional lower bounds of comparable strength for related mean values go back to the comparative theory of Knapowski and Tur\'an and to Pintz's mean-value theorems; we have not found the floor-with-equality characterization stated outside the author's series. Theorem~\ref{thm:main} is a reformulation that locates GRH for $L(s,\chi_3)$ as a statement about one number attached to one signal.

\emph{Place in the book.} This is Part~38 of the series, in block~VII (M\"obius cancellation and prime races). It builds on Part~5 (the growth-rate theorem and the energy law, transplanted here from $\zeta$ to $L(s,\chi_3)$), Part~6 (the second-order constant $K$ and the energy gauge, which is the $\zeta$-row of the same instrument) and Part~37 (the prime squares as the mechanism of the bias, and the constant $\kappa_3$). It is used by Part~6, whose energy-gauge section is its $\zeta$-analogue.

\section{The ledger identity and the fair race}\label{sec:ledger}

Everything in this section is elementary and unconditional. Write $\theta(x,\chi)=\sum_{p\le x}\chi(p)\log p$, with all characters mod $3$, so that $\theta(x)=\theta(x,\chi_0)+\log3$ for $x\ge3$.

\begin{lemma}[the ledger]\label{lem:ledger}
For $x\ge2$,
\[
\theta(x,\chi_3)\;=\;\psi(x,\chi_3)\;-\;\theta(\sqrt x,\chi_0)\;-\;\theta(x^{1/3},\chi_3)\;-\;\theta(x^{1/4},\chi_0)\;-\;\cdots,\qquad \theta(y,\chi_0)=\theta(y)-\log3\ \ (y\ge3),
\]
the sum having at most $\log_2x$ terms, the even-index terms carrying the principal-character weight ($\chi_3(p^{2k})=1$ for $p\ne3$) and the odd-index terms the character weight. Consequently
\[
\theta(x,\chi_3)=\psi(x,\chi_3)-\theta(\sqrt x)+O\!\big(x^{1/3}\big),
\]
and, by partial summation, with $\Pi_\chi(x)=\sum_{p^k\le x}\chi_3(p^k)/k$,
\[
-R(x)=\sum_{p\le x}\chi_3(p)=\Pi_\chi(x)-\tfrac12\pi(\sqrt x)+O\!\big(x^{1/3}\big),
\qquad
\Pi_\chi(x)=\frac{\psi(x,\chi_3)}{\log x}+\int_2^x\frac{\psi(t,\chi_3)}{t\log^2t}\,dt .
\]
\end{lemma}

\begin{proof}
$\psi(x,\chi)=\sum_{k\ge1}\theta(x^{1/k},\chi^k)$ by grouping prime powers, and $\chi_3^k$ is principal for even $k$, equal to $\chi_3$ for odd $k$; the principal-character $\theta$ differs from $\theta$ by the single prime $3$. The second display keeps the $k=2$ term and absorbs $k\ge3$ into the error. The third is partial summation against $d\psi$, each prime power $p^k$ receiving the weight $1/\log(p^k)=1/(k\log p)$, which is the definition of $\Pi_\chi$; replacing $\Pi_\chi-\sum_{p\le x}\chi_3(p)$ by its $k=2$ term costs $O(\pi(x^{1/3}))$.
\end{proof}

The content is the sign pattern: $\psi(x,\chi_3)$ is driftless (it is the pure zero oscillation; see \eqref{eq:explicit} below), while the race $R$ carries the drift $+\tfrac12\pi(\sqrt x)$ inherited from the squares. The data agree with the accounting to the hilt:

\begin{center}
\begin{tabular}{lrr}
\toprule
 up to $x=10^8$ & sign changes & size at $10^8$\\
\midrule
raw race $R(x)$ & $0$ & $+420$\\
ledger $\tfrac12\pi(\sqrt x)$ & --- & $614.5$\\
fair race $D(x)=R(x)-\tfrac12\pi(\sqrt x)$ & $15{,}329$ & $-194.5$\\
\bottomrule
\end{tabular}
\end{center}

\noindent The raw race never trails in a hundred million integers --- consistent with the first known reversal at $6.09\cdot10^{11}$ \cite{BH} --- and the corrected race changes sign fifteen thousand times (Figure~\ref{fig:race}, left). The count depends on the convention. $D$ jumps at the primes and at the squares of primes and takes the value $0$ often (at $10{,}820$ primes). Counting the changes of sign of $D$ over all its jump points, with the zeros skipped, gives $15{,}329$. Counting every passage between $D>0$ and $D\le0$ at the primes, so that a touch of zero counts twice, gives $21{,}329$, the number printed in the first version of this part. One subtraction, dictated by Lemma~\ref{lem:ledger}, is the difference between a loaded coin and a fair one.

\begin{figure}[htbp]\centering\includegraphics[width=\textwidth]{fig/R1_race.png}
\caption{Left: the race $R(x)$, the ledger $\tfrac12\pi(\sqrt x)$, and the fair race $D(x)$, to $x=10^8$. Centre: the normalized signal $e(u)=\psi(x,\chi_3)/\sqrt x$, the superposition of the modes of the zeros of $L(s,\chi_3)$. Right: the running mean energy $F(U)/U$ against the floor $\kappa_3$ of Theorem~\ref{thm:main}.}
\label{fig:race}
\end{figure}

\section{The constant $\kappa_3$ in closed form}\label{sec:kappa}

\begin{proposition}\label{prop:kappa}
Let $\kappa_3=\sum_\rho1/(\rho(1-\rho))$, over the nontrivial zeros of $L(s,\chi_3)$ with multiplicity. The series converges absolutely, and
\[
\kappa_3\;=\;2\,\frac{\xi'}{\xi}(1,\chi_3)\;=\;\log\frac3\pi-\gamma_E+2\,\frac{L'}{L}(1,\chi_3)\;=\;0.113229969857\ldots
\]
where $\xi(s,\chi_3)=(3/\pi)^{(s+1)/2}\,\Gamma\!\big(\tfrac{s+1}2\big)\,L(s,\chi_3)$ is the completed $L$-function.
\end{proposition}

This is \cite[Prop.~4.3]{CE37}; we repeat the proof, adding the absolute convergence and the absence of real zeros in $(0,1]$.

\begin{proof}
Absolute convergence: $|\rho(1-\rho)|^{-1}\asymp\gamma^{-2}$ and the zero-counting function of $L(s,\chi_3)$ grows like $\tfrac T\pi\log\tfrac T{2\pi e/3}$, so the sum over $|\gamma|\ge1$ converges; there are no zeros with $\gamma=0$ since $L(s,\chi_3)>0$ on $(0,1]$ (indeed $L(\sigma,\chi_3)=\sum_{k\ge0}\big((3k+1)^{-\sigma}-(3k+2)^{-\sigma}\big)$ is a sum of positive terms).

$\chi_3$ is odd and primitive, so $\xi(s,\chi_3)$ is entire of order one, its zeros are exactly the nontrivial zeros of $L$, and Hadamard's factorization gives
\[
\frac{\xi'}{\xi}(s)=B+\sum_\rho\Big(\frac1{s-\rho}+\frac1\rho\Big),
\]
the sum absolutely and locally uniformly convergent in the paired sense. At $s=0$ the summands vanish termwise, so $\xi'/\xi(0)=B$. At $s=1$,
\[
\frac{\xi'}{\xi}(1)=B+\sum_\rho\Big(\frac1{1-\rho}+\frac1\rho\Big)=B+\sum_\rho\frac1{\rho(1-\rho)}=B+\kappa_3 .
\]
The functional equation $\xi(1-s,\overline{\chi_3})=\epsilon\,\xi(s,\chi_3)$ with $|\epsilon|=1$ and $\overline{\chi_3}=\chi_3$ gives, on taking logarithmic derivatives at $s=1$, $\xi'/\xi(0)=-\xi'/\xi(1)$. Hence $B=-\xi'/\xi(1)$ and $\kappa_3=2\,\xi'/\xi(1)$. Finally
\[
\frac{\xi'}{\xi}(1)=\tfrac12\log\tfrac3\pi+\tfrac12\psi\!\big(1\big)+\frac{L'}{L}(1,\chi_3)
=\tfrac12\log\tfrac3\pi-\tfrac{\gamma_E}2+\frac{L'}{L}(1,\chi_3),
\]
since $\psi(1)=-\gamma_E$, which is the stated formula.

For the numerical value: $L(1,\chi_3)=\pi/(3\sqrt3)=0.604599788078\ldots$ is classical (Dirichlet). We computed $L'(1,\chi_3)=-\sum_{n\ge1}\chi_3(n)\log n/n$ by pairing consecutive terms $n=3k+1,3k+2$ --- after pairing the series converges absolutely, the general term being $O(\log k/k^2)$ --- summing $10^7$ pairs and correcting with the integral tail; the same scheme reproduces $L(1,\chi_3)$ to $16$ digits, and gives
\[
L'(1,\chi_3)=0.22266298697\ldots,\qquad
2\,\frac{L'}{L}(1,\chi_3)=0.73656323194\ldots,
\]
whence $\kappa_3=-0.04611759718-0.57721566490+0.73656323194=0.11322996986$, reliable to the printed digits (paired tails of first order at $10^7$ pairs).
\end{proof}

\begin{remark}
A closed form for $L'(1,\chi_3)$ in terms of $\log\Gamma(1/3)$ follows from the Lerch--Kronecker limit formulas; we do not need it. We record one curiosity with a warning: $\log(3/\pi)=-0.0461176\ldots$ while the B\'aez-Duarte constant for $\zeta$ is $2+\gamma_E-\log4\pi=+0.0461914\ldots$; the near-coincidence of digits is numerical accident, not identity --- the two differ in the fourth decimal.
\end{remark}

\section{The race energy law}\label{sec:law}

We now prove Theorem~\ref{thm:main}, following the $C_2$ energy law of \cite[\S3]{CP} step by step; we keep the proofs short where the transplant is verbatim and spell out what changes. Throughout, $\rho=\beta+i\gamma$ runs over the nontrivial zeros of $L(s,\chi_3)$ with multiplicity $m_\rho\ll\log(|\gamma|+2)$, and $\Theta_3=\sup_\rho\beta$.

The explicit formula for $\psi(x,\chi_3)$ has no main term, because $L(s,\chi_3)$ has no pole \cite[\S19--20]{Dav}: for $x\ge2$,
\begin{equation}\label{eq:explicit}
\psi(x,\chi_3)=-\sum_{|\gamma|\le T}\frac{x^\rho}{\rho}
+O\!\Big(\frac{x\log^2(xT)}{T}+\log x\Big),
\end{equation}
and in the limit $e(u)=e^{-u/2}\psi(e^u,\chi_3)=-\sum_\rho e^{(\rho-1/2)u}/\rho+O(ue^{-u/2})$ in the symmetric sense. On the critical line each zero contributes a unimodular mode $e^{i\gamma u}/\rho$ of energy $1/|\rho|^2=1/(\rho(1-\rho))$.

\begin{proposition}\label{prop:cramer}
If all nontrivial zeros of $L(s,\chi_3)$ lie on the critical line, then
\[
\lim_{U\to\infty}\frac1U\int_0^Ue(u)^2\,du
=\sum_{\rho\ \mathrm{distinct}}\frac{m_\rho^2}{|\rho|^2}\;\in\;[\kappa_3,\infty),
\]
and the value equals $\kappa_3$ if and only if all zeros are simple.
\end{proposition}

\begin{proof}
This is Cram\'er's argument \cite{Cr} in the form of \cite[Prop.~2.1]{CE5} and \cite[Prop.~3.1(a)]{CP} (for $L(s,\chi_3)$ and simple zeros the statement is in \cite[Prop.~4.3]{CE37}), with one simplification: no pole term has to be removed from $e(u)$. Group the zeros by height: $e(u)=-\sum_{\gamma}c_\gamma e^{i\gamma u}+o(1)$ in mean square, where for each distinct ordinate the coefficient collects the multiplicity, $|c_\gamma|=m_\rho/|\rho|$. The system $\{e^{i\gamma u}\}$ is orthogonal in mean on growing intervals and $\sum|c_\gamma|^2\,$ converges or diverges with $\sum m_\rho^2/|\rho|^2$; the standard almost-periodic mean-value theorem gives the limit $\sum_{\mathrm{distinct}}m_\rho^2/|\rho|^2$. Two quantitative inputs are imported from \cite[\S3]{CP}, where they are established for $\zeta$ using only $N(T)\asymp\frac T\pi\log T$, $m_\rho\ll\log|\gamma|$ and $\sum|\rho|^{-2}<\infty$ --- all three available for $L(s,\chi_3)$: the mean-square truncation of the explicit formula, and the off-diagonal bound of Montgomery--Vaughan \cite{MV}. The sum is finite, because $m_\rho\ll\log(|\gamma|+2)$ and $\sum_\rho\log(|\gamma|+2)/|\rho|^2<\infty$. On the line, $|\rho|^2=\rho(1-\rho)$, so
\[
\sum_{\mathrm{distinct}}\frac{m_\rho^2}{|\rho|^2}\ \ge\ \sum_{\mathrm{distinct}}\frac{m_\rho}{|\rho|^2}=\sum_\rho\frac1{\rho(1-\rho)}=\kappa_3,
\]
by $m^2\ge m$, with equality precisely when every $m_\rho=1$. \emph{Status.} The proof is by import: the two quantitative inputs are proved in \cite[\S3]{CP} for $\zeta$, and we assert, without rewriting those proofs, that they use only the three properties listed. A reader who wants a self-contained proof must check that transfer.
\end{proof}

\begin{proposition}\label{prop:uncond}
Unconditionally, with $F(U)=\int_0^Ue(u)^2du$ and $I(\delta)=\int_0^\infty e(u)^2e^{-2\delta u}du$:
\begin{enumerate}
\item[(a)] If $I(\delta)<\infty$ for every $\delta>0$, then $L(s,\chi_3)$ has no zeros in $\operatorname{Re}s>\tfrac12$.
\item[(b)] If $\Theta_3>\tfrac12$, then $\limsup_U\frac1U\log F(U)=2\Theta_3-1$; in particular $\limsup_U F(U)/U=\infty$.
\end{enumerate}
\end{proposition}

\begin{proof}
(a) For $\operatorname{Re}s>1$,
\[
\int_1^\infty\psi(x,\chi_3)\,x^{-s-1}\,dx=-\frac1s\,\frac{L'}{L}(s,\chi_3),
\]
with \emph{no} pole term on the right --- this is the transplant's one structural simplification. Writing $x=e^u$, the left side is the Laplace transform $\int_0^\infty e(u)\,e^{-(s-\frac12)u}\,du$. If $I(\delta)<\infty$, Cauchy--Schwarz bounds the transform for $\operatorname{Re}s>\tfrac12+\delta$ by $\big(I(\delta)\big)^{1/2}\big(2(\operatorname{Re}s-\tfrac12-\delta)\big)^{-1/2}$, locally uniformly, so the left side is analytic there; hence $L'/L(s,\chi_3)$ continues analytically to $\operatorname{Re}s>\tfrac12+\delta$ and $L$ has no zeros there. Letting $\delta\to0$ gives (a).

(b) is the Laplace--Stieltjes abscissa argument of \cite[Theorem~2.2]{CE5} (\cite[Thm.~3.2]{CP}), unchanged: the abscissa $\delta_c$ of $I$ satisfies $\limsup\frac1U\log F=2\delta_c$; the bound $\psi(x,\chi_3)\ll x^{\Theta_3}\log^2x$ gives $\delta_c\le\Theta_3-\tfrac12$, and $\delta_c<\Theta_3-\tfrac12$ would continue $L'/L$ past a zero by the argument of (a).
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
(iii) is Proposition~\ref{prop:uncond}(b). For (i) and (ii): if $\Theta_3>\tfrac12$, the energy grows exponentially, so the $\limsup$ in (i) is $+\infty>\kappa_3$ and the limit in (ii) does not exist finitely --- and on the right side of (ii) a zero off the line exists, so both sides of the equivalence fail. If $\Theta_3=\tfrac12$, all zeros are on the line (the zeros are symmetric under $\rho\mapsto1-\bar\rho$), and Proposition~\ref{prop:cramer} applies: the limit exists, equals $\sum m_\rho^2/|\rho|^2\ge\kappa_3$, with equality iff all zeros are simple. This proves (ii), and (i) in the remaining case.
\end{proof}

\begin{remark}
As with the $C_2$ energy law for $\zeta$, the theorem is a \emph{reformulation}: proving that the race energy is bounded is proving the Riemann hypothesis for $L(s,\chi_3)$. Its value is the same as there --- it converts a sign problem into a size problem with an explicit constant, and the constant can be measured against data (Section~\ref{sec:numerics}).
\end{remark}

\begin{remark}[general modulus]\label{rem:general}
Nothing is special to $q=3$ except the tidiness of the lanes. For a primitive character $\chi$ mod $q$ with parity $a\in\{0,1\}$ ($\chi(-1)=(-1)^a$), the same Hadamard argument gives, for the energy of $|e_\chi(u)|^2$, $e_\chi(u)=e^{-u/2}\psi(e^u,\chi)$, the floor
\[
\kappa_\chi=\sum_\rho\frac1{\rho(1-\rho)}
=\log\frac q\pi+\psi\!\Big(\frac{1+a}2\Big)+2\,\operatorname{Re}\frac{L'}{L}(1,\chi),
\]
with $\psi(\tfrac12)=-\gamma_E-2\log2$ in the even case, and Theorem~\ref{thm:main} holds verbatim with $\kappa_\chi$ and GRH for $L(s,\chi)$. The ledger generalizes too: $\chi(p^2)$ is the square character, so the biased lanes are the squares modulo $q$, which is Chebyshev's bias in the generality of \cite{RS}. For $\zeta$ itself, with the pole subtracted, the same instrument is the energy gauge of Part~6 \cite{CE6}, whose slope constant is $2\widehat B=2+\gamma_E-\log4\pi$.
\end{remark}

\section{The measurement}\label{sec:numerics}

We computed every prime power $p^k\le10^8$ with its weight $\chi_3(p^k)\log p$ ($5{,}762{,}859$ events), the running $\psi(x,\chi_3)$, and the energy integral \emph{exactly} between events: on a maximal interval $[x_i,x_{i+1})$ where $\psi(\cdot,\chi_3)=C_i$ is constant,
\[
\int_{\log x_i}^{\log x_{i+1}}e(u)^2\,du=C_i^2\Big(\frac1{x_i}-\frac1{x_{i+1}}\Big),
\]
so $F(U)$ is a finite exact sum (in double precision) with no quadrature error; the integral starts at $u=0$, and $\psi(x,\chi_3)=0$ on $[1,2)$. The results:

\begin{center}
\begin{tabular}{lccccc}
\toprule
$x$ & $10^4$ & $10^5$ & $10^6$ & $10^7$ & $10^8$\\
\midrule
mean energy $F(U)/U$ & $0.12553$ & $0.12080$ & $0.12086$ & $0.12121$ & $0.12015$\\
excess $\times\,U$:\ $(F(U)/U-\kappa_3)\,U$ & $0.113$ & $0.087$ & $0.105$ & $0.129$ & $0.127$\\
\bottomrule
\end{tabular}
\end{center}

Three remarks, in the spirit of strict bookkeeping. First, the cumulative mean $F(U)/U$ lies above $\kappa_3=0.113230$ for every $10^4\le x\le10^8$, between $0.1181$ and $0.1273$. This is consistent with Theorem~\ref{thm:main} and is not a test of it: part (i) concerns the upper limit and holds whatever the data to $10^8$ are (Remark~\ref{rem:floor}). Means over windows are not bounded below by $\kappa_3$: $(F(U_2)-F(U_1))/(U_2-U_1)$ is $0.1019$, $0.1211$, $0.1233$, $0.1127$ on the four decades from $10^4$ to $10^8$, and $0.1148$ on $[10^4,10^8]$. What the data can test is the closed form \eqref{eq:kappa} in the conditional sense: under GRH with simple zeros the mean tends to $\kappa_3$, and the window $[10^4,10^8]$ is within $1.4\%$ of it. (The table uses $U=\log x$ and includes the last interval up to $x$; the first version stopped at the last prime power below $x$, which gave $0.12544$ at $10^4$.) Second, the excess times $U$ is roughly stable, between $0.09$ and $0.13$ over four decades: this is the signature of a finite-$U$ expansion $F(U)=\kappa_3U+K_3+\Phi_3(U)+o(1)$ with a second-order constant $K_3\approx0.11$ and an almost-periodic interference term $\Phi_3$, exactly parallel to the expansion $\int_1^X(\psi(x)-x)^2x^{-2}dx=C_B\log X+K+\Phi(\log X)+o(1)$, $K=2.512479790006\ldots$, established for $\zeta$ under RH, simple zeros and a spacing condition in \cite[Theorem~5]{CE6} (closed form without zeros and certified value in \cite[Prop.~8, Cor.~9]{CE6}; see also \cite[\S3]{CP}); we record $K_3$ as \emph{measured}, with the conditional derivation deferred. Third, $K_3$ is much smaller than $K$; we have no explanation for this.

For comparison, \cite[\S4]{CE37} measures the variance of the normalized $\theta$-race, sampled uniformly in $\log x$ on $[10^4,2^{27}]$, as $0.1124$; the present measurement is a different functional (the cumulative energy from $u=0$, computed exactly), and both place the race energy within about $10\%$ of $\kappa_3$.

The sign-change counts of Section~\ref{sec:ledger} ($0$ raw, $15{,}329$ debiased, to $10^8$) and Figure~\ref{fig:race} complete the picture: drift explained by the ledger, fluctuation measured by the energy, energy measured against $\kappa_3$.

\section{What this does and does not do}\label{sec:honesty}

The theorem converts ``GRH for $L(s,\chi_3)$ with simple zeros'' into the statement that one number attached to the prime race equals one computable constant. It does not make the number easier to bound: as in the $\zeta$ case \cite[\S3]{CP}, proving $F(U)=O(U)$ \emph{is} the Riemann hypothesis for this $L$-function, and nothing in our measurement to $10^8$ --- four decades within $12\%$ of $\kappa_3$ --- constitutes evidence that can distinguish GRH from a zero of real part $0.500001$. The distributional theory of races (densities, comparisons between moduli) remains that of Rubinstein--Sarnak \cite{RS} and its descendants \cite{GM}, which assume GRH and linear independence; our floor assumes nothing, and characterises rather than estimates. We do not prove the Riemann hypothesis for $L(s,\chi_3)$.

\section*{Data and code}
All computations are reproducible from the ancillary scripts: \texttt{race.py} (sieve to $10^8$, the race, the ledger, the sign-change counts in both conventions, the exact piecewise energy and the window means; revised 10 November 2026), \texttt{kappa3.py} (the paired series for $L(1,\chi_3)$, $L'(1,\chi_3)$ and $\kappa_3$), \texttt{racefig.py} (Figure~\ref{fig:race}); \texttt{recheck\_10nov.py} (the sign-change counts in the two conventions, the energy with exact end points, the window means). Sums are accumulated in double precision over exact integer event positions; the series for $L$ and $L'$ are validated against the closed form $L(1,\chi_3)=\pi/(3\sqrt3)$ to $16$ digits.

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

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