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\title{All-order positive kernels for curves over finite fields}
\author{Kenan}
\date{2026-07-18}

\begin{document}
\maketitle

\begin{abstract}
For a smooth projective geometrically connected curve $C/\mathbf F_q$, the numerator of its zeta function can be centered into an even entire function $X_C$. The Rosati involution on the Jacobian forces every Frobenius root to have modulus $\sqrt q$. This puts the zeros of $X_C$ on explicit imaginary frequency towers, and the logarithmic derivative becomes a Stieltjes sum whose negative divided difference is a Gram kernel. A positive diagonal congruence then proves that the descended kernel $Q_C$ is positive semidefinite at every matrix order and every positive node.
\end{abstract}

\[
Q_C(\lambda_i,\lambda_j)\succeq 0\quad(\lambda_i>0,\ \text{all matrix orders})
\]

\section{What is proved}

Let $P_C$ be the numerator of the zeta function of $C/\mathbf F_q$, and center it by $X_C(z)=e^{g(\log q)z}P_C(q^{-1/2}e^{-(\log q)z})$. Write $L_C=X_C'/X_C$.

For positive nodes, the theorem concerns the confluent kernel $Q_C(\lambda,\mu)=4[\mu L_C(\lambda)-\lambda L_C(\mu)]/(\mu^2-\lambda^2)$. Its diagonal value is defined by continuity. Every finite matrix formed from this kernel is positive semidefinite.

\[
Q_C(\lambda_i,\lambda_j)\succeq0\qquad(\lambda_i>0)
\]
\section{The polarization input}

Let $J=\operatorname{Jac}(C)$ and let $\pi$ be its $q$-power Frobenius endomorphism. A polarization gives the positive Rosati involution $\dagger$, and the pinned source identity is $\pi^\dagger\pi=[q]_J$.

Rosati positivity also shows directly that $\mathbf Q[\pi]$ is a product of fields stable under $\dagger$. On each complex factor, the involution acts as complex conjugation. Therefore every characteristic root $\alpha$ of Frobenius satisfies $|\alpha|^2=q$.

\[
\pi^\dagger\pi=[q]_J,\qquad |\alpha|=\sqrt q
\]
\section{From point counts to an even function}

The numerator identity is derived from the point-count formula over every finite extension, rather than imported from a source passage with a polynomial-convention mismatch. Formal logarithmic summation identifies $P_C(u)=\det(1-u\pi\mid V_\ell J)$.

Integrality and purity pair each root with $q/\alpha$, including multiplicity. The resulting functional equation makes $X_C$ even and periodic. The fixed normalized roots $+1$ and $-1$ have even multiplicity, so the exceptional zero frequencies can be grouped without losing their multiplicities.

\[
N_n=1+q^n-\sum_j\alpha_j^n
\]

\[
P_C(u)=q^g u^{2g}P_C\!\left(\frac1{qu}\right)
\]
\section{The Gram-kernel route}

The paired frequency product has no hidden exponential term because $X_C$ is even. If $m_0$ is the multiplicity at zero and $m_\gamma$ is the multiplicity at $i\gamma$, logarithmic differentiation gives a Stieltjes expansion.

With $H_C(x)=X_C'(\sqrt x)/(\sqrt x\,X_C(\sqrt x))$, its negative divided difference is a sum of rank-one positive kernels. Setting $x_i=\lambda_i^2$ then gives $Q=DMD$ with a positive diagonal matrix $D$.

\[
H_C(x)=\frac{m_0}{x}+2\sum_{\gamma>0}\frac{m_\gamma}{x+\gamma^2}
\]

\[
M_C(x,y)=\frac{m_0}{xy}+2\sum_{\gamma>0}\frac{m_\gamma}{(x+\gamma^2)(y+\gamma^2)}
\]

\[
Q=DMD,\qquad D=\operatorname{diag}(2\lambda_i)
\]
\section{What the certificate checks}

The pinned certificate checks the Milne source hashes and theorem anchors, reconstructs the power-sum derivation of the zeta numerator, and verifies the reciprocal-pair and exceptional-multiplicity bookkeeping. It also checks the confluent diagonal and the exact diagonal congruence.

As exact fixtures, it reconstructs five curve numerators from finite-field point counts. The curve $y^2=x^5-x$ over $\mathbf F_5$ has numerator $(1-5u^2)^2$ and exercises a zero at the origin, both self-reciprocal angles, and repeated frequencies. Arb enclosures certify all principal minors of direct $3\times3$ test matrices at positive nodes including $1/4$.

\[
P_C(u)=(1-5u^2)^2,\qquad X_C(z)=4\sinh^2((\log5)z)
\]

\section{Pinned certificate}
The certificate pins the polarization and point-count sources, checks the exact Frobenius-to-canonical-product derivation, verifies the Gram decomposition and confluent kernel identity, and exercises the exceptional multiplicities on exact finite-field curves.

\subsection*{Run command}
\begin{verbatim}
uv run --frozen python canon/witnesses/C-0016/verify.py
\end{verbatim}

\subsection*{Pinned files}
\begin{itemize}[leftmargin=*]
\item \path{canon/witnesses/C-0016/PIN.md}
\item \path{canon/witnesses/C-0016/verify.py}
\item \path{scratch/proof-system--c0016-rosati-purity-loewner-audit/AUDIT.md}
\item \path{canon/claims/C-0016-function-field-polarization-positive-loewner-kernel.md}
\end{itemize}

\section{Scope}
The all-order kernel theorem applies to smooth projective geometrically connected curves over finite fields. The spectral Gram statement holds at every positive node, and the closed-point Euler series is used for $\operatorname{Re}z>1/2$.

\section{Sources}
\begin{itemize}[leftmargin=*]
\item Canonical claim: \path{canon/claims/C-0016-function-field-polarization-positive-loewner-kernel.md}
\item Witness pin: \path{canon/witnesses/C-0016/PIN.md}
\item Source and proof audit: \path{scratch/proof-system--c0016-rosati-purity-loewner-audit/AUDIT.md}
\end{itemize}

\end{document}
