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\begin{document}
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\parbox[b]{4.3in}{\small
  \href{http://www.math.toronto.edu/drorbn}{DBN}:
  \href{http://drorbn.net/AP}{AP}:
  \href{http://drorbn.net/AP/Projects}{Projects}:
  \href{http://drorbn.net/AP/Projects/HigherRank}{HigherRank}:
  {\LARGE\bf The Cast}
  }
\hfill\parbox[b]{3.4in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill initiated March 27, 2024; modified \today, \ampmtime
}~%
\raisebox{0.2em}{
  \qrcode[height=1.2em,level=L,nolink]{drorbn.net/AP/Projects/HigherRank/}
}

\vskip -3mm
\rule{\textwidth}{1pt}
\vspace{-8mm}

\begin{multicols*}{2}

{\bf The Variables:}
\[
  \begin{pmatrix}
    b_1\backslash a_1 & x_{12}            & x_{13}    \\
    y_{12}            & b_2\backslash a_2 & x_{23}    \\
    y_{13}            & y_{23}            & b_3\backslash a_3
  \end{pmatrix},
  \quad \xi_-, \alpha_-, \beta_-, \eta_-.
\]
We short ``$x_1$'' for either of $x_{12},x_{23}$ and ``$x_2$'' for $x_{13}$. Weights are intuitive on $yb$ and 3-complementary on $ax$: $\wt$: $b\to 0$, $y_1\to 1$, $y_2\to 2$, $a\to 3$, $x_1\to 2$, $x_2\to 1$. Weights are 3-complementary on the dual (greek) variables.

{\bf In $m[ij\to k]$:}
\par At $\eps=0$:\hfill
  $\ldots, \xi_1\eta_1b, \xi_2\eta_2b, \xi_1'\xi_1''x_2, \eta_2\xi_1y_1, \alpha\xi_ix_i \ldots$
\par At $\eps/\eps^2$: \hfill
  $\ldots, \xi_1\eta_1a, \xi_2\eta_2a,  \ldots$

{\bf In $\Delta[i\to jk]$:}
\par At $\eps=0$:\hfill
  $\ldots, \eta_iby_i, \ldots$

{\bf The Synthesis Equation.}
\[ 
  Z_0 \sim V,
  \quad \partial_\lambda Z_\lambda
  \sim \sum Q^{-1}_{ab}(\partial_{ab}Z_\lambda + (\partial_aZ_\lambda)(\partial_bZ_\lambda))
  \eqqcolon S(Z)
\]
In the best case $S$ is nilpotent on $V$ so $Z_\lambda$ is a polynomial in $\lambda$.

{\bf The General Heisenberg Integrand.}

\def\p#1#2{{p^{#1}_{#2}}}
\def\x#1#2{{x^{#1}_{#2}}}

$L = Q + V = Q + V_0 + V_{\geq 1} = Q + \sum_d \eps^dV_d$, where
\[ Q \sim -\hspace{-3mm}\sum_{c:(s,i,j),\alpha}\hspace{-3mm}
  \x{\alpha}{i}\left(
    \p{\alpha}{i}-T^{s\alpha}\p{\alpha}{i^+}+(T^{s\alpha}-1)\p{\alpha}{j^+}
  \right)
  + \x{\alpha}{j}\left(
    \p{\alpha}{j}-\p{\alpha}{j^+}
  \right),
\]
\[ V_0 \sim \p{\alpha_3}{i}\x{\alpha_1}{(i/j)^+}\x{\alpha_2}{(i/j)^+},
  \text{ with } \alpha_1=(1\ 0), \alpha_2=(0\ 1), \alpha_3=(1\ 1).
\]

$L$ is subject to some stable conditions:

\begin{enumerate}

\item $L$ is weight-balanced, with $\wt x^\alpha=\alpha$ and $\wt p^\alpha=-\alpha$.
\begin{itemize}
\item Should follow from both Lie reasons and Burau reasons.
\end{itemize}

\item $\deg_xV-\deg_pV \geq 1-2\deg_\eps V$.
\begin{itemize}
\item For $V_0$, this is ``$V_0$ is only trees and wheels''.
\end{itemize}

\item $\deg_pV \leq 1+\deg_\eps V$.
\begin{itemize}
\item Only holds for the connected case (taking a $\log$).
\item Reflects ``a connected arrow diagram with $k$ out arrows must have at least $k-1$ cobracket vertices''.
\end{itemize}

\end{enumerate}

Some unstable conditions can be assumed too:

\begin{enumerate}

\item $\deg_x V_1 \leq \deg_p V_1$.
\begin{itemize}
\item Or else, given $V_0$, the $\eps^1$ integral vanishes. Yet terms with $\deg_x V_1 > \deg_p V_1$ may influence the $\eps^{>1}$ integrals.
\end{itemize}

\end{enumerate}

\end{multicols*}

\end{document}

\endinput
