\section{The affine properties}

We finish the proof of Theorem~\ref{thm:main}. The finite case is Theorem~\ref{thm:finite-b}. Let \(W\) now be one of the two affine systems.

Proposition~\ref{prop:kl-filtration} gives P4 and P15: stability of the threshold ideals gives \(a(x)\geq a(y)\) when \(x\leq_{LR}y\), and independent-parameter commutation on each layer is P15. It gives P12 as well, since the leading matrices of an original finite parabolic element have support precisely at its intrinsic degree. A proper standard parabolic of the affine system is finite, and the full-system assertion is automatic. For \(w\in S_A\), the exact trace of \(C_w\) involves only families of degree \(B\geq A\). Each contribution is \(O(v^{-B})\) from the compact matrix bound and the measure normalization. Hence \(\deg_v\tau(C_w)\leq-A\), giving P1. Moreover
\begin{equation}\label{eq:trace-at-level}
 [v^{-A}]\tau(C_w)=\tau_J(t_w).
\end{equation}
The higher degrees make no contribution to this coefficient. For the families of degree \(A\) it is the leading trace integral. The identification of the leading product and cyclicity of the matrix trace give P7, since
\[
 \gamma_{x,y,z}=\tau_J(t_x t_y t_z).
\]
Here the right side and the left side vanish unless all three indices lie in the same spectral level.

Consider the directed left multiplication graph on the basis of a fixed level of \(J\). It has an arrow from \(t_y\) to \(t_z\) if some \(t_x t_y\) has nonzero \(t_z\)-coefficient. This relation is symmetric, since
\[
 \langle t_x t_y,t_z\rangle=\langle t_y,t_x^*t_z\rangle.
\]
Such an arrow gives an original left preorder comparison, by the degree-\(A\) coefficient formula. Conversely, every original left generator edge with both endpoints at degree \(A\) lies in this graph. In fact, by P15 the ordinary left action on the degree-\(A\) quotient commutes with the right leading multiplication, obtained by taking the coefficient of degree \(A\) in its independent right parameter. A map on the unital right regular module of this leading algebra which commutes with right multiplications is left multiplication by its value on the unit. A nonzero original coordinate in its image must thus occur in some leading product. By P4 an original left preorder path between equal-degree endpoints stays at that degree. Each edge can be reversed in the original preorder, giving P9. The right version gives P10, and the same argument for paths consisting of both types gives P11. This also identifies the connected components of the left multiplication graph with the original left cells.

If \(\gamma_{x,y,z}\ne0\), the three indices have the same degree and \(z^{-1}\leq_L y\), so P9 gives \(y\sim_L z^{-1}\). The other two relations follow by P7, proving P8. To obtain P14, choose a nonzero output in \(t_x t_x^*\), whose trace is one. Its index \(z\) is in the same left cell as \(x^{-1}\) and the same right cell as \(x\) by P9 and P10. Thus \(x^{-1}\sim_{LR}z\sim_{LR}x\).

Write the unit of \(J\) as \(\sum_w m_w t_w\). Orthonormality gives \(m_w=\tau_J(t_w)\). By \eqref{eq:trace-at-level}, \(m_w\ne0\) exactly for \(w\in\mathcal D\), and then \(m_w=n_w\). Proposition~\ref{du:units} gives \(n_d=\pm1\) for \(d\in\mathcal D\) and mutually orthogonal self-adjoint projections \(p_d=n_d t_d\). Since \(t_d^*=t_{d^{-1}}\), self-adjointness gives \(d^{-1}=d\), which is P6. The right multiplications by the \(p_d\) are coordinate projections summing to the identity. Each \(t_y\) therefore has a unique right unit \(p_d\) among them. For \(d\in\mathcal D\),
\[
 [t_d](t_x t_y)
 =\tau_J(t_x t_y t_d)
 =n_d\,\delta_{x,y^{-1}}\,
         \mathbf{1}_{\{t_y p_d=t_y\}},
\]
where \([t_d]\) denotes its basis coefficient. This proves P2, P3, and P5. Every left multiplication edge preserves the right unit: associativity gives \((t_xt_y)p_d=t_xt_y\) whenever \(t_yp_d=t_y\), and right multiplication by \(p_d\) is a coordinate projection. Also \(t_dp_d=t_d\) since \(p_d\) is idempotent. The nonzero coefficient of \(t_d\) in \(t_{y^{-1}}t_y\) connects \(t_y\) to its right-unit index \(d\). Thus its original left cell contains precisely that distinguished element, and the coefficient formula gives \(\gamma_{y^{-1},y,d}\ne0\) for each \(y\) in the cell. This proves P13.