diff --git a/annular_triangular_hilbert_v2.txt b/annular_triangular_hilbert_v2.txt
index 9cfaac7..688eb79 100644
--- a/annular_triangular_hilbert_v2.txt
+++ b/annular_triangular_hilbert_v2.txt
@@ -1046,13 +1046,16 @@ and trilinearity gives the desired bound
     | Lambda_K(G_0, G_1, G_2) |
         <= C sqrt(n) ||G_0||_3 ||G_1||_3 ||G_2||_3 .
 
-The factor sqrt(n) arises as follows: the additive switch cost n
-in the dissipation is converted to a single-factor sqrt(n) after
-the A_0 rescaling, because the square root of n times the
-remaining factors n_1 + n_2 appears as (n_0^2 (n_1 + n_2))/n_0 =
-n_0 (n_1 + n_2); if all four norms are comparable, this is n^2
-divided by n_0 = n, which after rescaling gives the sqrt(n) loss
-in the final trilinear estimate.
+The sqrt(n) factor comes from a normalization. Apply the bound to
+A_0' = A_0 n^{-1/2} / ||A_0||, A_1' = A_1 / ||A_1||,
+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
+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.
 
 Passing to the continuous annulus kernel. The grid Riemann sums
 converge to the continuous scale integral I_{eps,R}. For fixed
@@ -1591,43 +1594,52 @@ two endpoints, and no annulus is charged twice.
 
 This closes the endpoint-error estimate.
 
-The hard-to-smooth comparison. Fix the third derivative of the
-Gaussian psi = -g_3''' and its primitive
+Hard-smooth comparison. Fix psi = -g_3''' and its primitive
 
-    P(v)  =  2 int_0^v psi(u) d u ,    c_0  =  P(infinite) .
+    P(v)  =  2 int_0^v psi(u) d u (v >= 0),   c_0 = P(infinite).
 
-The function psi is odd, smooth, and integrable; P is even, bounded,
-constant outside a fixed range, and P(v) - c_0 has Gaussian decay
-at infinity. The comparison between the hard annular indicator and
-the smooth kernel I_{eps,R} is the identity (hard-smooth):
+The function psi is odd, smooth, and Gaussian-tailed. The window
+k_1(t) = t g_1(t) equals -g_1'(t); convolving three Gaussian
+derivatives adds variances, so k_1 * k_1 * k_1 = -g_3''' = psi. The
+primitive P is even, smooth, bounded, equal to c_0 outside a fixed
+range, and vanishes to second order at v = 0. The identity
+(hard-smooth) is the equality of three exact forms:
 
-    beta 1_{eps < |t| < R} / t
-        =  c_0 beta I_{eps,R}(t)
-           - beta E_eps(t) - beta E_R(t) + smooth error ,
+    I_{eps,R}(t)  =  ( P(|t|/eps) - P(|t|/R) ) / t
+                  =  ( c_0 / t ) 1_{ eps < |t| < R }
+                     + E_eps(t) - E_R(t) ,
 
 where
 
-    E_rho(t)  =  ( P(|t|/rho) - c_0 1_{|t| > rho} ) / t .
+    E_rho(t)  =  ( P(|t|/rho) - c_0 1_{ |t| > rho } ) / t .
 
-Each E_rho is an odd function of t, smooth in (t, rho) away from
-t = 0 and vanishing only at t = 0 in the sense of a removable
-singularity; every derivative has Gaussian decay in t uniformly for
-rho in [eps, R]. The residual smooth error absorbs the contribution
-of the derivative of I_{eps,R} away from the two endpoints, and has
-size controlled by ||F||_3 ||G||_3 against the trilinear form.
+The middle expression is the direct consequence of the substitution
+z = |t| / sqrt(s) in the scale integral I_{eps,R}. The rightmost
+expression is the rearrangement
+
+    (c_0/t)( 1_{|t| > eps} - 1_{|t| > R} )
+      + ( P(|t|/eps) - c_0 1_{|t| > eps}
+          - P(|t|/R) + c_0 1_{|t| > R} ) / t
+    = ( P(|t|/eps) - P(|t|/R) ) / t ,
+
+in which each boundary indicator c_0 1_{|t| > rho} appears twice
+with opposite signs. Each E_rho is odd, smooth away from t = 0,
+vanishes linearly at t = 0, and every derivative has Gaussian decay
+in t uniformly for rho in any compact positive range.
 
 Multiplying (hard-smooth) by the step choice and integrating,
 
-    iiint G_0(a,b) G_1(b,c) G_2(c,a)
+    c_0 iiint G_0(a,b) G_1(b,c) G_2(c,a)
         sum_j beta_j(a,b)
         1_{eps_j < |a+b+c| < R_j} / (a+b+c) d a d b d c
-    =  c_0 Lambda_{K_smooth}(G_0, G_1, G_2)
-       - Lambda_{K-endpoint}(G_0, G_1, G_2) .
+    =  Lambda_{K_smooth}(G_0, G_1, G_2)
+       - Lambda_{E}(G_0, G_1, G_2) .
 
 The smooth term is bounded by the smooth-switches proposition. The
-endpoint term is a sum of kernels of the form E_rho, one per annular
-endpoint. Its size is bounded by the endpoint-error estimate. Since
-c_0 != 0, dividing by c_0 gives the hard-linearized bound
+endpoint term Lambda_{E}, in which E denotes the sum
+sum_j beta_j (E_{eps_j} - E_{R_j}), is controlled by the endpoint-
+error estimate. Since c_0 != 0, dividing by c_0 gives the
+hard-linearized bound
 
     | iiint G_0(a,b) G_1(b,c) G_2(c,a)
         sum_j beta_j(a,b)