diff --git a/docs/1_fixed_effects_and_block_methods.md b/docs/1_fixed_effects_and_block_methods.md
index 6c31f0c..9ef7ad9 100644
--- a/docs/1_fixed_effects_and_block_methods.md
+++ b/docs/1_fixed_effects_and_block_methods.md
@@ -74,11 +74,11 @@ The $m \times m$ Gramian inherits a natural block structure from the factor part
 
 $$
 G = \begin{pmatrix}
-{\color{royalblue}D_1} & C_{12} & C_{13} & \cdots & C_{1Q} \\
-C_{12}^\top & {\color{crimson}D_2} & C_{23} & \cdots & C_{2Q} \\
-C_{13}^\top & C_{23}^\top & {\color{forestgreen}D_3} & \cdots & C_{3Q} \\
+{\color{blue}D_1} & C_{12} & C_{13} & \cdots & C_{1Q} \\
+C_{12}^\top & {\color{red}D_2} & C_{23} & \cdots & C_{2Q} \\
+C_{13}^\top & C_{23}^\top & {\color{green}D_3} & \cdots & C_{3Q} \\
 \vdots & \vdots & \vdots & \ddots & \vdots \\
-C_{1Q}^\top & C_{2Q}^\top & C_{3Q}^\top & \cdots & {\color{goldenrod}D_Q}
+C_{1Q}^\top & C_{2Q}^\top & C_{3Q}^\top & \cdots & {\color{orange}D_Q}
 \end{pmatrix}
 $$
 
@@ -109,21 +109,22 @@ The Gramian has $Q = 3$ diagonal blocks and $\binom{3}{2} = 3$ cross-tabulation
 
 $$
 G = \begin{pmatrix}
-{\color{royalblue}D_W} & {\color{gray}C_{WF}} & {\color{gray}C_{WY}} \\
-{\color{gray}C_{WF}^\top} & {\color{crimson}D_F} & {\color{gray}C_{FY}} \\
-{\color{gray}C_{WY}^\top} & {\color{gray}C_{FY}^\top} & {\color{forestgreen}D_Y}
+{\color{blue}D_W} & {\color{gray}C_{WF}} & {\color{gray}C_{WY}} \\
+{\color{gray}C_{WF}^\top} & {\color{red}D_F} & {\color{gray}C_{FY}} \\
+{\color{gray}C_{WY}^\top} & {\color{gray}C_{FY}^\top} & {\color{green}D_Y}
+\end{pmatrix}
+$$
+
+$$
+= \begin{pmatrix}
+{\color{blue}2} & {\color{blue}0} & {\color{blue}0} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} \\
+{\color{blue}0} & {\color{blue}2} & {\color{blue}0} & {\color{gray}2} & {\color{gray}0} & {\color{gray}1} & {\color{gray}1} \\
+{\color{blue}0} & {\color{blue}0} & {\color{blue}2} & {\color{gray}0} & {\color{gray}2} & {\color{gray}1} & {\color{gray}1} \\
+{\color{gray}1} & {\color{gray}2} & {\color{gray}0} & {\color{red}3} & {\color{red}0} & {\color{gray}2} & {\color{gray}1} \\
+{\color{gray}1} & {\color{gray}0} & {\color{gray}2} & {\color{red}0} & {\color{red}3} & {\color{gray}1} & {\color{gray}2} \\
+{\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{gray}1} & {\color{green}3} & {\color{green}0} \\
+{\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{green}0} & {\color{green}3}
 \end{pmatrix}
-= \left(\begin{array}{ccc|cc|cc}
-{\color{royalblue}2} & {\color{royalblue}0} & {\color{royalblue}0} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} \\
-{\color{royalblue}0} & {\color{royalblue}2} & {\color{royalblue}0} & {\color{gray}2} & {\color{gray}0} & {\color{gray}1} & {\color{gray}1} \\
-{\color{royalblue}0} & {\color{royalblue}0} & {\color{royalblue}2} & {\color{gray}0} & {\color{gray}2} & {\color{gray}1} & {\color{gray}1} \\
-\hline
-{\color{gray}1} & {\color{gray}2} & {\color{gray}0} & {\color{crimson}3} & {\color{crimson}0} & {\color{gray}2} & {\color{gray}1} \\
-{\color{gray}1} & {\color{gray}0} & {\color{gray}2} & {\color{crimson}0} & {\color{crimson}3} & {\color{gray}1} & {\color{gray}2} \\
-\hline
-{\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{gray}1} & {\color{forestgreen}3} & {\color{forestgreen}0} \\
-{\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{forestgreen}0} & {\color{forestgreen}3}
-\end{array}\right)
 $$
 
 $D_W$ is $3 \times 3$ (one row/column per worker) with 2s on the diagonal because each worker appears in exactly 2 observations (e.g. W1 in obs 1, 2). Off-diagonals are zero because no observation belongs to two workers. $D_F$ is $2 \times 2$ with 3s on the diagonal because each firm appears in 3 observations (F1 in obs 1, 3, 4; F2 in obs 2, 5, 6). The cross-tabulation block $C_{WY}$ is $3 \times 2$ (3 workers $\times$ 2 years); entry $[j,k]$ counts observations where worker $j$ is observed in year $k$. Here every worker appears once per year, so $C_{WY}$ is all ones.
diff --git a/docs/2_solver_architecture.md b/docs/2_solver_architecture.md
index 619ebea..3b55b81 100644
--- a/docs/2_solver_architecture.md
+++ b/docs/2_solver_architecture.md
@@ -37,7 +37,7 @@ As discussed in [Part 1, Section 5.2](1_fixed_effects_and_block_methods.md#52-th
 
 ## 2. Graph Structure of the Gramian
 
-Part 1 derived the block structure of $G = D^\top W D$, with diagonal blocks $D_q$ and cross-tabulation blocks $C_{qr}$. It is convenient to write $G = \mathcal{D} + \mathcal{C}$, where $\mathcal{D} = \operatorname{block-diag}(D_1, \ldots, D_Q)$ collects the diagonal blocks and $\mathcal{C}$ collects the off-diagonal cross-tabulation blocks. This section describes the graph-theoretic properties that drive the domain decomposition.
+Part 1 derived the block structure of $G = D^\top W D$, with diagonal blocks $D_q$ and cross-tabulation blocks $C_{qr}$. It is convenient to write $G = \mathcal{D} + \mathcal{C}$, where $\mathcal{D} = \text{block-diag}(D_1, \ldots, D_Q)$ collects the diagonal blocks and $\mathcal{C}$ collects the off-diagonal cross-tabulation blocks. This section describes the graph-theoretic properties that drive the domain decomposition.
 
 ### 2.1 Factor-pair bipartite blocks