diff --git a/three_manifold_no_conjugate_v2.txt b/three_manifold_no_conjugate_v2.txt
index 8a89e67..034ebb9 100644
--- a/three_manifold_no_conjugate_v2.txt
+++ b/three_manifold_no_conjugate_v2.txt
@@ -2112,8 +2112,8 @@ with t_* > 1/4, the following hold with e = (1, 0) (in the boundary
 frame) and p the passage's conserved momentum:
 
     (a)  S(e, e)                <=  L (1 + C_e δ);
-    (b)  |S(p, e) - L p_1|      <=  C_m δ L |p|;
-    (c)  S(p, p)                <=  L  +  C_m δ L |p|^2.
+    (b)  |S(p, e) - L p_1|      <=  C_m δ L |p_1|;
+    (c)  S(p, p)                <=  L (1 + v_0^2).
 
 Proof.  We use the explicit structure (9.9) and the cometric bounds
 (Lemma 6.4).
@@ -2164,50 +2164,88 @@ by (9.12).  Therefore
 Combining: S(e,e) <= L(1+2δ) + 2C_0δL <= L(1 + C_e δ) with C_e = 2+2C_0.
                                                                           (a) ✓
 
-Proof of (b).  First term of S(p,e):
+Proof of (b).  We split S(p,e) into its two terms,
+    A := ∫_0^L e^T P(t(s)) p ds,      B := (Φ_p, Φ_e)_{D*}.
+The goal is |A + B - Lp_1| ≤ C_m δL|p_1|.
 
-    ∫_0^L p^T P(t) e ds   =       ∫_0^L p_1 · P_{11}(t) ds  +  ∫_0^L p_2 · P_{21}(t) ds.
+Estimate of A - Lp_1.  Since e^T P(0)p = p_1 (as P(0)=diag(1,b)),
 
-Here P_{21}(t) = d_2(t) H_1(t) H_2(t) = d_2(t) · (εχ(t)) · 1 (since
-H = e_2 + εχe_1, so H_2 = 1).  Therefore |P_{21}(t)| <= (1+δ)δ² <= 2δ²,
-and |P_{11}(t) - 1| <= 2δ as before.         Thus
+    A - Lp_1 = ∫_0^L (e^T P(t(s)) p - p_1) ds.
 
-    |p^T P e - p_1|  <=  |p_1|·2δ  +  |p_2|·2δ²  <=  2δ|p| + 2δ²|p|  <=  3δ|p|
+For t ∈ [0, t_*], set D(t) := P'(0) - P'(t) ∈ Sym_2.  Since P'' ≤ 0
+(Lemma 6.4(c)), P' is nonincreasing, so D(t) ⪰ 0.  Also
 
-for δ <= 1.  Integrating against ds from 0 to L:
+    ||D(t)|| ≤ ||P'(0)|| + ||P'(t)|| ≤ 2C_0δ              [by (6.4b)],
 
-    |∫_0^L (p^T P e - p_1) ds|  <=  3δ L |p|.                                  (9.14)
+and, since F'(t) = p^T P'(t)p ≥ 0 for t ∈ [0,t_*] (Lemma 8.3(C5)),
 
-For the second term (Φ_p, Φ_e)_{D*}: by Cauchy-Schwarz in D*
+    p^T D(t)p = p^T P'(0)p - p^T P'(t)p = κp_1² - F'(t) ≤ κp_1² = δa(0)p_1².
 
-    |(Φ_p, Φ_e)_{D*}|  <=  ||Φ_p||_{D*} · ||Φ_e||_{D*}.
+Cauchy-Schwarz for the positive semidefinite form D(t) gives
 
-We computed ||Φ_e||^2_{D*} <= 2C_0δL in (a).  And by (6.4d) with r = p:
+    |e^T D(t)p|² ≤ (e^T D(t)e)(p^T D(t)p) ≤ (2C_0δ)(δa(0)p_1²),
+    |e^T D(t)p| ≤ √(2C_0a(0)) · δ|p_1|.
 
-    |Φ_p(w)|  =  |∫ (p^T P' p) w ds|  <=  |p| · ∫ |P' p| · |w| ds
-              <=  |p| sqrt(C_0 δ) ∫ sqrt(-F'')|w| ds  <=  |p| sqrt(2C_0 δ L) ||w||_D.
+Since e^T P'(0)p = κp_1 = a(0)δp_1,
 
-So ||Φ_p||^2_{D*} <= 2C_0δL|p|².  Combining:
+    |e^T P'(t)p| ≤ |e^T P'(0)p| + |e^T D(t)p| ≤ (a(0)+√(2C_0a(0))) δ|p_1|.
 
-    |(Φ_p, Φ_e)_{D*}|  <=  sqrt(2C_0δL|p|²) · sqrt(2C_0δL)  =  2C_0 δ L |p|.  (9.15)
+Integrating from 0 to t (in the t variable),
 
-Total: |S(p,e) - Lp_1| <= 3δL|p| + 2C_0δL|p| = (3 + 2C_0)δL|p|, so
-C_m = 3 + 2C_0 works.
+    |e^T P(t)p - p_1| = |∫_0^t e^T P'(u)p du| ≤ (a(0)+√(2C_0a(0))) δt|p_1|.
+
+Since 0 ≤ t = t(s) ≤ t_* ≤ 1 throughout the passage,
+
+    |e^T P(t(s))p - p_1| ≤ (a(0)+√(2C_0a(0))) δ|p_1| for every s ∈ [0,L],
+
+hence integrating in s,
+
+    |A - Lp_1| ≤ (a(0)+√(2C_0a(0))) δL|p_1|.                                (9.14)
+
+Estimate of B.  Recall Φ_p(w) = ∫_0^L (p^T P'(t)p) w ds = ∫_0^L F'(t) w ds.
+Since F'(t(s)) = -2v'(s) (Eq. 7.1d) and w(0) = w(L) = 0,
+
+    Φ_p(w) = -2∫_0^L v'(s) w(s) ds = 2∫_0^L v(s) ẇ(s) ds.
+
+Thus  |Φ_p(w)| ≤ 2(∫_0^L v² ds)^{1/2} (∫_0^L ẇ² ds)^{1/2} ≤ 2(∫v²)^{1/2}||w||_D.
+By Lemma 9.1, ∫v² ds ≤ Lv_0²/3, so
+
+    ||Φ_p||_{D*} ≤ 2√(Lv_0²/3).                                         (9.15)
+
+From part (a), ||Φ_e||_{D*} ≤ √(2C_0δL).
+
+By Cauchy-Schwarz in D*:
+
+    |B| ≤ ||Φ_p||_{D*} ||Φ_e||_{D*} ≤ 2√(Lv_0²/3) √(2C_0δL) = 2√(2C_0δL²v_0²/3).
+
+By (9.1), v_0² ≤ κp_1² = δa(0)p_1², so v_0 ≤ √(δa(0)) |p_1|.  Then
+
+    |B| ≤ 2√(2C_0δ²L²a(0)/3) |p_1| = √(8C_0a(0)/3) δL|p_1|.          (9.16)
+
+Combining (9.14) and (9.16):
+
+    |S(p,e) - Lp_1| ≤ C_m δL|p_1|,
+
+with  C_m := a(0) + √(2C_0a(0)) + √(8C_0a(0)/3).
                                                                           (b) ✓
 
 Proof of (c).  First term of S(p,p):
 
-    ∫_0^L p^T P(t) p ds  =  ∫ F(t(s)) ds  =  ∫ (1 - v(s)^2) ds  =  L - ∫ v^2 ds  <=  L.
+    ∫_0^L p^T P(t) p ds = ∫_0^L F(t(s)) ds = ∫_0^L (1 - v(s)^2) ds = L - ∫_0^L v^2 ds.
 
-Second term: ||Φ_p||^2_{D*} <= 2C_0δL|p|² (already shown in (b)).
-Therefore
+Second term, from (9.15):  ||Φ_p||²_{D*} ≤ 4∫_0^L v^2 ds.
+
+Adding:
+
+    S(p,p) ≤ L - ∫_0^L v^2 ds + 4∫_0^L v^2 ds = L + 3∫_0^L v^2 ds
+           ≤ L + 3 ⋅ (L v_0^2 / 3) = L (1 + v_0^2),                        (c) ✓
 
-    S(p,p)  <=  L  +  2C_0 δ L |p|^2,
+where the last line uses Lemma 9.1.
 
-which is (c) with C_m = 2C_0 (or the max of the two constants).  ✓
+This proves all three bounds, with
 
-This proves all three bounds with
-    C_m  :=  max(3 + 2C_0, 2C_0)  =  3 + 2C_0,       C_e  :=  2 + 2C_0,
+    C_e := 2 + 2C_0,
+    C_m := a(0) + √(2C_0a(0)) + √(8C_0a(0)/3),
 provided δ_1 <= 1, which we take henceforth.  ∎
 
 9.4  The turning estimate (conclusion of Part 9)