diff --git a/metric_entropy_duality_counterexamples_v2.txt b/metric_entropy_duality_counterexamples_v2.txt
index 198467f..b8a428e 100644
--- a/metric_entropy_duality_counterexamples_v2.txt
+++ b/metric_entropy_duality_counterexamples_v2.txt
@@ -30,7 +30,7 @@ noting the operator-theoretic origin in Pietsch's 1972 work.
 
 Available at ./pdf/artstein-milman-szarek-tomczak-2004-entropy-duality-math0407238.pdf
 
-The right-hand inequality is the substantive half. The result below shows
+The right-hand ineuality is the substantive half. The result below shows
 that it fails for every proposed pair of constants, already when L is a cube.
 
    Main theorem. For every a, b >= 1 there are a positive integer n and an
@@ -38,7 +38,7 @@ that it fails for every proposed pair of constants, already when L is a cube.
        log N(K, L)  >  b log N(L^o, a^{-1} K^o).
 
 The dimension n grows with a and b; no statement concerns a fixed dimension.
-Because the polar inequality can be forced to fail with L a cube, no universal
+Because the polar ineuality can be forced to fail with L a cube, no universal
 (a, b) survives.
 
 The surrounding landscape fixes what is and is not already known. Konig and
@@ -134,7 +134,7 @@ Proof. The set absconv{R_x} is the convex hull of the finitely many vectors
 t B_infinity^Y keeps these properties and gives K nonempty interior in the full
 space. The cube L has the same properties.
 
-For the first inequality, each point 3 R_x lies in K, and distinct such
+For the first ineuality, each point 3 R_x lies in K, and distinct such
 points are at sup-distance at least 3. A translate of L = B_infinity^Y has
 sup-diameter at most 2: if a and b lie in z + L then a - b lies in L - L = 2L
 and ||a - b||_{infinity,Y} <= 2. So one translate of L meets the set
@@ -155,7 +155,7 @@ Order the M functions of the list. Assign each lambda in B_1^Y to the first
 list function within eps of f_lambda; hypothesis (ii) guarantees one exists.
 This splits B_1^Y into at most M nonempty clusters, and from each nonempty
 cluster one coefficient vector lambda^(k) in B_1^Y is selected. For lambda and
-lambda^(k) in the same cluster the triangle inequality gives
+lambda^(k) in the same cluster the triangle ineuality gives
 ||f_lambda - f_lambda^(k)||_{infinity,X} <= 2 eps, while both vectors lie in
 B_1^Y, so ||lambda - lambda^(k)||_1 <= 1 + 1 = 2. With delta = lambda -
 lambda^(k),
@@ -165,7 +165,7 @@ lambda^(k),
 For c > 0 positivity and positive homogeneity of h_K give the equivalence
 delta in c K^o if and only if h_K(delta) <= c. Hence each cluster lies in
 lambda^(k) + (6 eps + 2 t) K^o, and the at most M translates cover
-B_1^Y = L^o. This is the second inequality. Choosing the representatives
+B_1^Y = L^o. This is the second ineuality. Choosing the representatives
 inside the coefficient ball is what keeps the added cube under control even
 when the list functions themselves lie outside F.
 
@@ -257,7 +257,7 @@ first pivot whose set contains it, with slot kappa_j(y) in {1, ..., k_j}.
 Label representatives. For a pivot z = z_{j,k} keep the single label
 L_{i_0}(z) and replace z by z' = rho_{i_0}(L_{i_0}(z)). If y = (v,T) is
 assigned to this pivot then i_0 in T, d_T(v,z) <= j, and d_T(z,z') <= 1, hence
-d_T(v,z') <= j + 1. The triangle inequality at any x gives the two
+d_T(v,z') <= j + 1. The triangle ineuality at any x gives the two
 implications d_T(x,v) <= j implies d_T(x,z') <= 2j + 1, and d_T(x,z') <= 2j+1
 implies d_T(x,v) <= 3j + 2. All distances involved are finite under the
 respective hypotheses, even when the graph is disconnected. Define
@@ -320,7 +320,7 @@ most one. The compression proposition supplies A^+, A^- in Acal with
 f_lambda to within 6 theta. The differences A^+ - A^- range over at most Q^2
 functions; ordering the list and assigning each lambda in B_1^Y to the first
 approximant within 6 theta yields at most Q^2 nonempty fibers, and the
-triangle inequality gives the 12 theta diameter. The approximants need not
+triangle ineuality gives the 12 theta diameter. The approximants need not
 lie in F, while the clusters are genuine coefficient vectors in B_1^Y.
 
 The rows come from symmetric multilinear forms over a finite field. Two
@@ -477,7 +477,7 @@ order of dependence so the field size can absorb the encoding term.
           = 2 h s (D_{h-1}/D_h) + 2 C/(D_h log p)
           = 2 h^2 s/(r + h - 1) + 2 C/(D_h log p)
           < 1/(4b) + 1/(4b) = 1/(2b),
-   the two strict inequalities following from r + h - 1 > 8 b h^2 s and from
+   the two strict ineualities following from r + h - 1 > 8 b h^2 s and from
    p > exp(8 b C/D_h). With N(L^o, a^{-1} K^o) <= Q^2 and N(K, L) >= m > 1,
        b log N(L^o, a^{-1} K^o) <= b log(Q^2) < (1/2) log m < log m
          <= log N(K, L).