diff --git a/three_manifold_no_conjugate_v2.txt b/three_manifold_no_conjugate_v2.txt
index 39717b3..fc5ae59 100644
--- a/three_manifold_no_conjugate_v2.txt
+++ b/three_manifold_no_conjugate_v2.txt
@@ -1802,7 +1802,7 @@ The right-hand side depends only on the "end-to-end displacement"
 V.  Because the field V^gauge has infinitely many degrees of freedom,
 we cannot try to make (8.43) hold by minimization over shape:  we
 must SHOW it whenever V satisfies our gauge.  That is the content
-of Part 9: the inequality (8.43) is proved for EVERY smooth field V
+of Part 9: the ineuality (8.43) is proved for EVERY smooth field V
 along the passage [0, L], not merely the minimizer.  The condition
 "q_V affine" only enters via (8.41), which converts ∫|q_V'|²
 into the "cheap" term (p^T Δz)^2/L.
@@ -1813,7 +1813,7 @@ PART 9.  THE RETURNING PASSAGE ESTIMATE
 ================================================================
 
 By Lemma 8.5, all passages except case (H) of 8.6 are already handled.
-This part treats case (H).  Recall from 8.6 the required inequality:
+This part treats case (H).  Recall from 8.6 the required ineuality:
 
     E_[0,L](V) + (κ v_0/2)(z_1(0)^2 + z_1(L)^2)  >=  (p^T Δz)^2 / L,   (9.0)
 
@@ -1908,8 +1908,8 @@ requires a more explicit value than β < 1; only the numeric quantity
 
 9.2  The bilinear form S and the Hilbert-space duality
 
-In this subsection we turn the inequality (9.0) into a 2x2 matrix
-inequality.  The mechanism is Hilbert-space Cauchy-Schwarz.
+In this subsection we turn the ineuality (9.0) into a 2x2 matrix
+ineuality.  The mechanism is Hilbert-space Cauchy-Schwarz.
 
 Setup.  Fix a returning passage [0, L] as in 9.1.  Its geodesic
 gamma = (t, x) is fixed; the field V = (w, z) varies.  Recall the
@@ -2032,7 +2032,7 @@ frame) and p the passage's conserved momentum:
 Proof.  We use the explicit structure (9.9) and the cometric bounds
 (Lemma 6.4).
 
-Preliminary: D-norm inequality.  For every w ∈ D and every
+Preliminary: D-norm ineuality.  For every w ∈ D and every
 -∞ <= s_1 < s_2 <= +∞,
 
     ∫_{s_1}^{s_2} sqrt(-F''(t(s))) |w(s)| ds  <=  sqrt(2(s_2 - s_1)) ||w||_D.       (9.12)
@@ -2222,7 +2222,7 @@ This is (ii).  ∎
 LEMMA 9.7 (Turning estimate).  There exists δ_2 ∈ (0, δ_1] such that,
 for every 0 < δ <= δ_2, every returning passage [0, L] with t_* > 1/4
 and every smooth field V = (w, z) with w(0) = w(L) = 0 satisfies the
-key inequality (9.0):
+key ineuality (9.0):
 
     E_[0,L](V)  +  (κ v_0/2)(z_1(0)² + z_1(L)²)         >=  (p^T Δz)²/L.         (9.0) again
 
@@ -2234,7 +2234,7 @@ ranging over R²:
     (c^T X)²  <=  S(c,c)  ·  E_[0,L](V)     for every c ∈ R².
 
 This is equivalent (standard 2x2 Cauchy-Schwarz) to the matrix
-inequality
+ineuality
 
     X^T S^{-1} X  <=  E_[0,L](V).                              (9.21)
 
@@ -2300,8 +2300,8 @@ The specific threshold r_0 and constant δ_2 are chosen once and for
 all, depending only on (C_m, C_e, β, a(0)), and independent of δ, of
 B_1, B_2, and of the particular passage.
 
-CONCLUSION.  With δ <= δ_2, the inequality (target) holds, hence by
-Lemma 9.6 the matrix inequality X^T M X <= L E (equivalently,
+CONCLUSION.  With δ <= δ_2, the ineuality (target) holds, hence by
+Lemma 9.6 the matrix ineuality X^T M X <= L E (equivalently,
 (p^T Δz)² <= L E + Lμ (e^T Δz)²) holds.  Since
 (κv_0/2)(z_1(0)² + z_1(L)²) >= (κv_0/4)(Δz_1)² = μ(e^T Δz)² (using
 2(z_1(0)² + z_1(L)²) >= (Δz_1)²), this gives (9.0).  ∎
@@ -2342,7 +2342,7 @@ of Lemma 8.3.
 
     by Lemma 9.7.  And  I_J(V) = I_J^⊥(V) + ∫_J |q_V'|² >= I_J^⊥(V) >= 0.
 
-So both inequalities hold in every case.  ∎
+So both ineualities hold in every case.  ∎
 
 
 ================================================================