diff --git a/equivariant_jiang_su_stability_v2.txt b/equivariant_jiang_su_stability_v2.txt
index e785fdd..8467728 100644
--- a/equivariant_jiang_su_stability_v2.txt
+++ b/equivariant_jiang_su_stability_v2.txt
@@ -46,6 +46,19 @@ very weak comparison.
 
 ./szabo-1904.10897-equivariant-property-si.pdf
 
+Earlier characterizations of Z-absorption invoke a finiteness restriction
+on the trace simplex, which the present argument avoids: the Gaussian
+splittings are uniform over the whole trace space rather than derived from
+comparison of positive elements.
+
+./matui-sato-1111.1637-strict-comparison-Z-absorption.pdf
+
+Theorem 1.1 (Matui and Sato). Let $A$ be a unital separable simple
+infinite-dimensional nuclear C*-algebra with finitely many extremal
+traces. Then the following are equivalent: $A \otimes Z \cong A$; $A$
+has strict comparison; every completely positive map from $A$ to $A$
+can be excised in small central sequences; has property (SI).
+
 The decisive criterion reads:
 
 Corollary C (Szabo and Wouters). Let $\alpha : G \curvearrowright A$ be
@@ -340,7 +353,9 @@ Proposition. For every finite $H \subset G$, every $\eps > 0$, and every
 integer $m \ge 1$ there are $t > 0$ and a self-adjoint $Q \in C$ such that
 $V = \exp(\mathrm{i} t Q)$ satisfies
 
-$$ \sup_{\varphi \in \mathcal T(C)} \|\bar\alpha_h(V) - V\|_{2,\varphi} < \eps \quad (h \in H), \qquad \sup_{\varphi \in \mathcal T(C)} |\varphi(V^\ell)| < \eps \quad (1 \le \ell \le m). $$
+$$ \sup_{\varphi \in \mathcal T(C)} \|\bar\alpha_h(V) - V\|_{2,\varphi} < \eps \quad (h \in H), $$
+
+$$ \sup_{\varphi \in \mathcal T(C)} |\varphi(V^\ell)| < \eps \quad (1 \le \ell \le m). $$
 
 Three estimates are needed first.
 
@@ -511,7 +526,9 @@ linear combination of these. Since the limiting variables commute and the
 closed formula for $q_E$ is real-valued, the $Q_{N_j,E}$ converge jointly
 in moments under $\varphi_j$ to the $q_E$. In particular
 
-$$ \|Q_{N_j,hF} - Q_{N_j,F}\|_{2,\varphi_j} \longrightarrow \|q_{hF} - q_F\|_{L^2}, \qquad \varphi_j(\exp(\mathrm{i} s Q_{N_j,F})) \longrightarrow \mathbb E \exp(\mathrm{i} s q_F) $$
+$$ \|Q_{N_j,hF} - Q_{N_j,F}\|_{2,\varphi_j} \longrightarrow \|q_{hF} - q_F\|_{L^2}, $$
+
+$$ \varphi_j(\exp(\mathrm{i} s Q_{N_j,F})) \longrightarrow \mathbb E \exp(\mathrm{i} s q_F) $$
 
 for every real $s$. The passage to scalar spectral laws is needed because
 the operator norms of $Q_{N,E}$ may grow with $N$: the estimates
@@ -568,7 +585,9 @@ coordinate-selection fact.
 
 Lemma. If $x = [(x_j)] \in A^\omega$ has a bounded representative, then
 
-$$ \sup_{\tau \in \mathcal T_\omega(A)} \|x\|_{2,\tau} = \lim_{j \to \omega} \sup_{\sigma \in \mathcal T(A)} \|x_j\|_{2,\sigma}, \qquad \sup_{\tau \in \mathcal T_\omega(A)} |\tau(x)| = \lim_{j \to \omega} \sup_{\sigma \in \mathcal T(A)} |\sigma(x_j)|. $$
+$$ \sup_{\tau \in \mathcal T_\omega(A)} \|x\|_{2,\tau} = \lim_{j \to \omega} \sup_{\sigma \in \mathcal T(A)} \|x_j\|_{2,\sigma}, $$
+
+$$ \sup_{\tau \in \mathcal T_\omega(A)} |\tau(x)| = \lim_{j \to \omega} \sup_{\sigma \in \mathcal T(A)} |\sigma(x_j)|. $$
 
 Finitely many such tests can therefore be realized simultaneously at a
 single coordinate with any prescribed positive slack above their
@@ -625,15 +644,19 @@ Since $V^{(m)}$ commutes with $A$, the coordinate-selection lemma, the
 weighted small moments, and the invariance estimates allow a coordinate
 $j(m) \ge m$ at which $w_m = v^{(m)}_{j(m)}$ satisfies
 
-$$ \|[w_m, a_r]\|_{2,u} < 2/m \quad (1 \le r \le m), \qquad \|\alpha_h(w_m) - w_m\|_{2,u} < 2/m \quad (h \in H_m), \qquad \sup_{\sigma \in \mathcal T(A)} |\sigma(a_r w_m^\ell)| < 2/m \quad (1 \le r, \ell \le m). $$
+$$ \|[w_m, a_r]\|_{2,u} < 2/m \quad (1 \le r \le m), $$
 
-The last line applies the trace identity of the lemma to the class
+$$ \|\alpha_h(w_m) - w_m\|_{2,u} < 2/m \quad (h \in H_m), $$
+
+$$ \sup_{\sigma \in \mathcal T(A)} |\sigma(a_r w_m^\ell)| < 2/m \quad (1 \le r, \ell \le m). $$
+
+The last display applies the trace identity of the lemma to the class
 $a_r (V^{(m)})^\ell$. All coordinate tests are finite in number, and their
 suprema already range over all traces.
 
-The class $w = [(w_m)]$ is a unitary. The first two lines, density, and
+The class $w = [(w_m)]$ is a unitary. The first two displays, density, and
 $\|[w_m, a - a_r]\|_{2,u} \le 2\|a - a_r\|$ make it central relative to $A$
-and fixed by $\bar\alpha$. The last line, first for every $a_r$, then by
+and fixed by $\bar\alpha$. The last display, first for every $a_r$, then by
 norm density for positive contractions, then by linearity for $a \in A$,
 gives $\tau(a w^\ell) = 0$ for every $a \in A$, $\tau \in \mathcal T_\omega(A)$,
 and $\ell \ge 1$. The same holds for negative powers because
@@ -657,7 +680,9 @@ for instance by replacing each indicator near its two endpoints by linear
 ramps whose total lengths tend to zero. Fix $r$ and write $f_i = f_i^{(r)}$,
 $e = e_r$. Pointwise bounds and the triangle inequality give
 
-$$ \|f_i^2 - f_i\|_{L^2(\mu)} \le e, \qquad \|f_i f_j\|_{L^2(\mu)} \le 2e \ (i \ne j), \qquad \Bigl\|\sum_i f_i - 1\Bigr\|_{L^2(\mu)} \le k e, \qquad \Bigl|\int_{\mathbb T} f_i\, d\mu - \frac1k\Bigr| \le e. $$
+$$ \|f_i^2 - f_i\|_{L^2(\mu)} \le e, \qquad \|f_i f_j\|_{L^2(\mu)} \le 2e \ \ (i \ne j), $$
+
+$$ \Bigl\|\sum_i f_i - 1\Bigr\|_{L^2(\mu)} \le k e, \qquad \Bigl|\int_{\mathbb T} f_i\, d\mu - \frac1k\Bigr| \le e. $$
 
 The first uses $|f_i^2 - f_i| \le |f_i - \chi_i|$ almost everywhere; the
 second expands
@@ -681,19 +706,23 @@ coordinate-selection lemma gives a coordinate $j(r) \ge r$ such that the
 positive contractions $y_{i,r} = f_i^{(r)}(w_{j(r)})$ satisfy, for all
 indicated indices,
 
-$$ \|y_{i,r}^2 - y_{i,r}\|_{2,u} < e_r + 1/r, \qquad \|y_{i,r} y_{j,r}\|_{2,u} < 2 e_r + 1/r \ (i \ne j), \qquad \Bigl\|\sum_i y_{i,r} - 1\Bigr\|_{2,u} < k e_r + 1/r, $$
+$$ \|y_{i,r}^2 - y_{i,r}\|_{2,u} < e_r + 1/r, \qquad \|y_{i,r} y_{j,r}\|_{2,u} < 2 e_r + 1/r \ \ (i \ne j), $$
+
+$$ \Bigl\|\sum_i y_{i,r} - 1\Bigr\|_{2,u} < k e_r + 1/r, $$
+
+$$ \|[y_{i,r}, a_s]\|_{2,u} < 1/r \quad (s \le r), \qquad \|\alpha_h(y_{i,r}) - y_{i,r}\|_{2,u} < 1/r \quad (h \in H_r), $$
 
-$$ \|[y_{i,r}, a_s]\|_{2,u} < 1/r \ (s \le r), \qquad \|\alpha_h(y_{i,r}) - y_{i,r}\|_{2,u} < 1/r \ (h \in H_r), \qquad \sup_{\sigma \in \mathcal T(A)} \Bigl|\sigma(a_s y_{i,r}) - \frac{\sigma(a_s)}{k}\Bigr| < e_r + 1/r \ (s \le r). $$
+$$ \sup_{\sigma \in \mathcal T(A)} \Bigl|\sigma(a_s y_{i,r}) - \frac{\sigma(a_s)}{k}\Bigr| < e_r + 1/r \quad (s \le r). $$
 
 This is a coordinate choice made after $f_i^{(r)}$ is fixed, so it needs no
 uniform continuity modulus, as $r$ varies, for these increasingly sharp arc
 approximations.
 
 Let $p_i = [(y_{i,r})_r] \in A^\omega$. The representatives are uniformly
-bounded positive contractions, and the first three lines make the $p_i$
-orthogonal projections summing to $1$; the next two make them central and
-invariant by density. For every sequence $(\sigma_r)$ in $\mathcal T(A)$ the
-last line gives
+bounded positive contractions. Small square defect, orthogonality, and unit
+sum make the $p_i$ orthogonal projections summing to $1$. Small commutators
+and approximate invariance make them central and invariant by density. The
+trace estimate gives, for every sequence $(\sigma_r)$ in $\mathcal T(A)$,
 
 $$ \lim_{r \to \omega} \sigma_r(a_s y_{i,r}) = \frac1k \lim_{r \to \omega} \sigma_r(a_s), $$