diff --git a/switch_chain_polynomial_mixing_v2.txt b/switch_chain_polynomial_mixing_v2.txt
index fda358f..0364349 100644
--- a/switch_chain_polynomial_mixing_v2.txt
+++ b/switch_chain_polynomial_mixing_v2.txt
@@ -119,13 +119,13 @@ Every vertex degree is unchanged: u lost v as a friend but gained x,
 v lost u but gained y, x lost y but gained u, y lost x but gained v.
 That is the whole point -- switches preserve the degree sequence.
 
-Two distinct perfect matchings of the complete graph on four vertices
-are exactly the six ways to pair up {u,v,x,y} as two disjoint edges:
+There are exactly three ways to pair up {u,v,x,y} into two
+disjoint edges:
 
-    {uv,xy}, {ux,vy}, {uy,vx}
+    {uv,xy}, {ux,vy}, {uy,vx}.
 
-and the "swap" reading of the same three pairings, so six ordered
-(F,F') pairs total.  The switch chain picks one of these at every step.
+The chain picks a uniformly random ordered pair (F,F') of distinct
+matchings from this list, so 3 * 2 = 6 ordered pairs per step.
 
 Precise discrete-time convention (this matters for the mixing constant):
 At each step,
@@ -805,9 +805,11 @@ Explanation of the "extra column" column:
 
   e = 1: the extra column has a single 1 at row i and a single 1 at
   row j (the endpoints of the edge), so the "extra column sum" is 2.
-  Encoding: entries (1,1,0) on (i,j,k).  Each vertex of T loses 1 to
-  the internal edge, so its internal-external degree is d_i - 1 from
-  the outside; the external column sums are exactly that.
+  Encoding: entries (1,1,0) on (i,j,k).  The two endpoints of the edge
+  each lose 1 to the internal edge, while the third vertex of T loses
+  0.  So the external degrees are d_i - 1 at the edge endpoints and
+  d_i unchanged elsewhere; adding the extra column gives augmented row
+  sums of d_i, d_j, d_k.
 
   e = 2: a path has a center (say vertex i, with two edges i-j, i-k).
   The extra column has a single 1 at the center i (sum 1).  Each