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\title{Strict reflected Lorentz cones for Rankin coefficient packets}
\author{Kenan}
\date{2026-07-18}

\begin{document}
\maketitle

\begin{abstract}
Two exact standard Arthur parameters for $O_{24}$ share a common $[21]$ block. Cancelling that block compares a split degree-three Euler packet with the symmetric-square packet attached to Ramanujan's $\Delta$. Deligne's bound makes the compact packet uniformly small relative to the split term, giving strict reflected-Lorentz inequalities for every coefficient index $n>1$ with explicit reserves. Removing the quadratic invariant and applying exact local Rankin identities yields a second family of strict inequalities for the corresponding harmonic coefficient packets.
\end{abstract}

\[
0<D_n<2R_n,\qquad A_n^2>B_n^2\quad(n>1)
\]

\section{What is proved}

Let $e(n)$ be the Dirichlet coefficient of $Z_3(s)=\zeta(s-11)\zeta(s)\zeta(s+11)$ and let $r(n)$ be the coefficient of $L(s,\operatorname{Ad}\Delta)$. With the zero-weight count $z(n)$, define $D_n=e(n)-r(n)$ and $R_n=e(n)+r(n)-2z(n)$.

For every integer $n>1$, the coefficient packet lies strictly inside the reflected Lorentz cone: $0<D_n<2R_n$. The proof also gives explicit positive lower reserves and a harmonic version after division by $\zeta(2s)$.

\[
0<D_n<2R_n\qquad(n>1)
\]

\[
A_n^2-B_n^2>0,\qquad A_n^2>4d(n),\qquad A_n^2+3B_n^2-4d(n)>0
\]
\section{Where the packets come from}

The pinned Chenevier--Lannes table gives the first two standard parameters as $[23]\oplus[1]$ and $\operatorname{Sym}^2\Delta_{11}\oplus[21]$. Their shift convention expands a block $\pi[d]$ into symmetrically shifted $L$-functions.

Cancelling the common $[21]$ block leaves the split shifts $-11,0,11$ on one side and the adjoint symmetric-square $L$-function on the other. This source normalization is part of the theorem, not a heuristic analogy between Euler factors.

\[
[23]\oplus[1]\quad\text{versus}\quad \operatorname{Sym}^2\Delta_{11}\oplus[21]
\]

\[
Z_3(s)=\zeta(s-11)\zeta(s)\zeta(s+11)
\]
\section{The coefficient bounds}

At a prime $p$, the split local block has eigenvalues $p^{11},1,p^{-11}$. The normalized symmetric-square block has three unit-modulus eigenvalues because Deligne's theorem gives the Ramanujan bound for $\Delta$.

The degree-$k$ Euler coefficient is a symmetric-power character. The split coefficient contains a term at least $p^{11k}$, while the compact coefficient has absolute value at most $\binom{k+2}{2}$. Multiplicativity turns these local estimates into all-$n$ bounds.

\[
e(n)\ge n^{11},\qquad |r(n)|\le h_3(n),\qquad h_3(n)=\prod_{p^k\parallel n}\binom{k+2}{2}
\]
\section{Strict reserves and harmonic reduction}

If $z(n)=\prod_{p^k\parallel n}(\lfloor k/2\rfloor+1)$, the proof obtains explicit reserves for $D_n$, $R_n$, and $2R_n-D_n$. The elementary comparison $n^{11}\ge2048^{\Omega(n)}$ dominates the character and zero-weight terms for every $n>1$.

Multiplying each local Euler factor by $1-p^{-2s}$ removes the quadratic invariant and leaves one copy of each harmonic weight. Exact local Rankin identities then identify the coefficients with squares of $A_n=\sigma_{11}(n)/n^{11/2}$ and $B_n=\tau(n)/n^{11/2}$, from which the harmonic cone inequalities follow.

\[
D_n\ge n^{11}-h_3(n)>0
\]

\[
R_n\ge n^{11}-h_3(n)-2z(n)>0
\]

\[
2R_n-D_n\ge n^{11}-3h_3(n)-4z(n)>0
\]
\section{What the certificate checks}

The certificate pins the Chenevier--Lannes parameter table and shift convention together with Deligne's purity-to-Ramanujan source chain. It reconstructs the $[21]$ cancellation, the symmetric-square trace, the complete and harmonic character identities, and the factorization-sensitive all-$n$ inequalities.

Finite exact scans are used only as mutation guards and fixtures; the universal statement comes from the multiplicative proof. The verifier emits the positive $n=2$ margins and rejects a nearby counterfeit in which the common block is changed to $[19]$, because the source-normalization gate then leaves the wrong shifts.

\[
D_2=\frac{4197825}{2048},\qquad R_2=\frac{4190785}{2048},\qquad 2R_2-D_2=\frac{4183745}{2048}
\]

\section{Pinned certificate}
The pinned verifier checks the exact Arthur-parameter normalization, Deligne bounds, character identities, multiplicative reserves, harmonic Rankin identities, an exact positive fixture, and a normalization counterfeit that must fail.

\subsection*{Run command}
\begin{verbatim}
uv run --frozen python canon/witnesses/C-0018/verify.py
\end{verbatim}

\subsection*{Pinned files}
\begin{itemize}[leftmargin=*]
\item \path{canon/witnesses/C-0018/PIN.md}
\item \path{canon/witnesses/C-0018/verify.py}
\item \path{canon/witnesses/K-0036/sources/chenevier-lannes-1409.7616v2.pdf}
\item \path{canon/witnesses/K-0036/sources/deligne-modular-forms-l-adic-1969.pdf}
\item \path{canon/witnesses/K-0036/sources/deligne-weil-i-1974.pdf}
\item \path{canon/claims/C-0018-arthur-harmonic-packet-cone.md}
\end{itemize}

\section{Scope}
The inequalities concern exact Dirichlet coefficients in the absolute-convergence Euler-product normalization for the two stated Arthur parameters.

\section{Sources}
\begin{itemize}[leftmargin=*]
\item Canonical claim: \path{canon/claims/C-0018-arthur-harmonic-packet-cone.md}
\item Witness pin: \path{canon/witnesses/C-0018/PIN.md}
\item Arthur parameter source: \path{canon/witnesses/K-0036/sources/chenevier-lannes-1409.7616v2.pdf}
\item Ramanujan source chain: \path{canon/witnesses/K-0036/sources/deligne-modular-forms-l-adic-1969.pdf}
\end{itemize}

\end{document}
