diff --git a/annular_triangular_hilbert_v2.txt b/annular_triangular_hilbert_v2.txt
index 688eb79..0544e83 100644
--- a/annular_triangular_hilbert_v2.txt
+++ b/annular_triangular_hilbert_v2.txt
@@ -23,10 +23,14 @@ When the same kernel is applied to a bilinear product
     BH(f,g)(x) = p.v. int_{-infinity}^{infinity} f(x + t) g(x - t) dt / t,
 
 the two inputs sit on opposite sides of one corner. Boundedness of
-BH on a wide range of L^p products is Lacey and Thiele (1997). The proof decomposes the frequency plane into tilted wave packets
-and selects coherent trees. Demeter and Thiele
-later isolate a two-dimensional analogue in which the two inputs
-move along perpendicular axes:
+BH on a wide range of L^p products is due to Lacey and Thiele; the proof decomposes the
+frequency plane into tilted wave packets and selects coherent
+trees.
+
+    ./lacey-thiele-1997.pdf
+
+Demeter and Thiele later isolate a two-dimensional analogue in
+which the two inputs move along perpendicular axes.
 
     ./0803.1268.pdf
 
@@ -59,7 +63,7 @@ estimate for B at (3,3,3/2).
 
 The two-variable dyadic model with two general inputs and a third
 of specified structure is constructed by Kovac, Thiele, and
-Zorin-Kranich. That paper also states the cyclic trace formulation
+Zorin-Kranich. That work also states the cyclic trace formulation
 and the equivalence of the flat and simplex forms.
 
     ./1506.00861.pdf
@@ -76,7 +80,7 @@ Durcik, Kovac, Skreb, and Thiele obtain square-function and
 norm-variation estimates for the triangular form over finitely many
 smooth scales. Durcik, Kovac, and Thiele obtain power-type
 cancellation for hard truncations. Permuting and interpolating the
-(4,4,2) bounds of the latter paper gives a bound of order
+(4,4,2) bounds of the latter work give a bound of order
 sqrt(log(R/eps)) at the symmetric point. Both estimates retain a
 dependence on the range of scales, and neither places the supremum
 over output-dependent partitions inside the output norm.
@@ -100,7 +104,14 @@ energy method.
 
 Grafakos, Oliveira e Silva, Pramanik, Seeger, and Stovall collect
 open problems at the symmetric point. Problem 13 in Section 9 of
-that survey asks precisely for the scalar estimate proved below.
+that survey asks precisely for the scalar estimate proved below:
+
+    Problem 13. Prove the a priori estimate
+
+        |Lambda_3(f, g, h)|
+            <= C ||f||_{L^3} ||g||_{L^3} ||h||_{L^3}
+
+    with a universal constant C.
 
     ./1701.06637.pdf
 
@@ -1052,7 +1063,7 @@ A_2' = A_2 / ||A_2||, whose L^3 norms are (n^{-1/2}, 1, 1). The
 right side of the switch-dissipation inequality,
 n(n^{-3/2} + 2n^{-1}) + n^{-3/2} + 2, is then bounded by an
 absolute constant, and the initial energy sum_v n_v^3 becomes
-n^{-3/2} + 2, also bounded. By trilinearity, the original
+n^{-3/2} + 2, also bounded. By trilinearity, the unnormalized
 trilinear form is n^{1/2} ||A_0|| ||A_1|| ||A_2|| times the
 trilinear form on the normalized arrays, which proves the
 sqrt(n) factor in the final smooth-switches bound.
@@ -1669,15 +1680,20 @@ rows covers every measurable choice. Passing to the supremum over
 all such menus gives the count theorem for smooth compactly
 supported F, G, and G_0.
 
-Now let F and G general L^3. If F, G are smooth compactly supported
-and G_0 is smooth compactly supported, the bound holds with constant
-C. By density of smooth compactly supported functions in L^3(R^2),
-the bound transfers to general complex F and G: a standard
-approximation argument applies in the linearized trilinear form
-once F, G are kept separately L^3-fixed, and once the supremum over
-annuli is taken, the bound holds for arbitrary F and G.
+Now take general complex F, G in L^3(R^2). Choose smooth compactly
+supported F_m -> F and G_m -> G in L^3. For each fixed annulus
+(eps, R), bilinearity and the 2 log(R/eps) bound give
+
+    || B_{eps,R}(F_m, G_m) - B_{eps,R}(F, G) ||_{3/2}
+        <= 2 log(R/eps) ( ||F_m - F||_3 ||G_m||_3
+                          + ||F||_3 ||G_m - G||_3 ) -> 0 .
 
-This proves the count theorem.
+For a fixed finite collection of rational lists of disjoint
+annuli, the finite maximum is over finitely many sums of such
+truncation terms; it converges in L^{3/2} as m -> infinity. The
+count bound passes to the limit. Exhausting the countable family
+of rational lists by monotone convergence proves the count
+theorem for arbitrary F, G in L^3(R^2).
 
 Maximal operator corollary. Take n = 1 and a single annulus in the
 count supremum. The count theorem gives
@@ -1757,3 +1773,5 @@ cube root of a positive matrix. Lewis (J. Convex Anal. 2, 1995)
 supplies the gradient formula across matrix rank changes. Kwong
 (Proc. Amer. Math. Soc. 51, 1975) supplies the Loewner order facts
 for the square root of a positive matrix.
+
+    ./0910.1555.pdf