diff --git a/annular_triangular_hilbert_v2.txt b/annular_triangular_hilbert_v2.txt
index 90cbb51..da573ad 100644
--- a/annular_triangular_hilbert_v2.txt
+++ b/annular_triangular_hilbert_v2.txt
@@ -202,7 +202,7 @@ form a permissible list inside S_{2^k}, so
 
 The ranks 2^k <= l < 2^{k+1} form a block of size at most 2^k, and
 each member is at most d_(2^k). The l^r norm of that block is at
-most 2^{k/r} d_(2^k). Summing over k gives the pointwise inequality
+most 2^{k/r} d_(2^k). Summing over k gives the pointwise ineuality
 
     V_r(F,G)(x,y)
         <=  sum_{k >= 0} 2^{k(1/r - 1/2)}
@@ -210,7 +210,7 @@ most 2^{k/r} d_(2^k). Summing over k gives the pointwise inequality
 
 The exponent 1/r - 1/2 is strictly negative for r > 2, so the
 geometric factor decays. Taking L^{3/2} norms, applying Minkowski's
-inequality, and invoking the count theorem,
+ineuality, and invoking the count theorem,
 
     || V_r(F,G) ||_{3/2}
         <=  C ||F||_3 ||G||_3
@@ -241,10 +241,10 @@ degree-three scaling as the trilinear form.
 
 Second, add a Gaussian heat flow to the Gram. Differentiating the
 trace in the scale parameter produces positive Hessian costs in
-the insertion directions, and a mixed trace inequality bounds the
+the insertion directions, and a mixed trace ineuality bounds the
 cyclic product by three quadratic costs, one at each vertex. A
 heat identity turns those costs into derivatives of a positive
-energy, and integrating in scale gives an inequality between the
+energy, and integrating in scale gives an ineuality between the
 accumulated costs and the initial and terminal energies. The
 energies are bounded by cubed L^3 norms. The dissipation is
 strict when the three variance rates are comparable.
@@ -278,14 +278,14 @@ product against 1/t. When the truncated integral is evaluated at
 any fixed (x, y), the resulting absolutely convergent
 two-variable sum exists outside a single null set of the plane.
 
-To see this, fix F, G in L^3(R^2) and fix t. Holder's inequality
+To see this, fix F, G in L^3(R^2) and fix t. Holder's ineuality
 in (x, y) and translation invariance give
 
     || F(. + t, .) G(., . + t) ||_{3/2}
         <= ||F||_3 ||G||_3.
 
 Integrating in t over eps < |t| < R and using Minkowski's integral
-inequality,
+ineuality,
 
     || int_{eps < |t| < R} |F(x + t, y) G(x, y + t)| dt / |t| ||_{3/2}
         <= 2 log(R / eps) ||F||_3 ||G||_3.
@@ -370,7 +370,7 @@ geometric series with sum bounded by a constant.
 
 The finite-dimensional engine has four facts: a Hessian cost on the
 energy tr(P^{3/2}), an order comparison, an insertion bound, and a
-cyclic mixed-trace inequality. Every statement below is exact, with
+cyclic mixed-trace ineuality. Every statement below is exact, with
 explicit constants.
 
 Let H_0, H_1, H_2 be finite-dimensional complex Hilbert spaces,
@@ -494,7 +494,7 @@ positive definite selfadjoint operator M on a real Hilbert space is
     <K, M^{-1} K>  =  sup_X { 2 <K, X> - <X, M X> }.
 
 Applying with M = L_{sqrt T} + R_{sqrt T} on the real Hilbert space
-of Hermitian matrices, using the averaging inequality just proved
+of Hermitian matrices, using the averaging ineuality just proved
 inside the supremum, and bounding the supremum of an integral by the
 integral of pointwise suprema, gives the claim. Uniform
 invertibility in w is not assumed; every inverse is taken on E,
@@ -531,7 +531,7 @@ the present statement is a special case.
 
     ./pdf/lewis-1995-jca02012-convex-analysis-unitarily-invariant.pdf
 
-The cyclic mixed-trace inequality is the whole point of collecting
+The cyclic mixed-trace ineuality is the whole point of collecting
 the two incident blocks into one row matrix. For Hermitian
 D_v : H_v -> H_v, put D_* = max_v ||D_v||_{op}. Then
 
@@ -555,7 +555,7 @@ projection of H onto H_v, and set A_v = M P_v M, K_v = M P_v D M.
 After permuting blocks and extending by zero, A_v = T_v and
 K_v = R_v^* D_v R_v. Because P_v commutes with D, K_v is
 Hermitian. Also ker A_v = ker(P_v M) is contained in ker K_v and
-0 <= A_v <= M^2. Lemma-order together with the inequality
+0 <= A_v <= M^2. Lemma-order together with the ineuality
 sum_v b(X_v) >= (1/3) b(sum_v X_v) for a positive quadratic form
 gives
 
@@ -607,7 +607,7 @@ yields
 
     6 |tau|  <=  5 D_* Q  <=  10 D_* E.
 
-This proves the inequality. The case tau = 0 is immediate.
+This proves the ineuality. The case tau = 0 is immediate.
 
 Complex insertions. When D_v is complex diagonal, split
 D_v = D_v^{(0)} + i D_v^{(1)} with both parts diagonal and Hermitian
@@ -734,7 +734,7 @@ under the integral is verified. Lemma-insertion gives
     Y_{v,w}   <= (3/4) ||N_w||_{op}^2 e_v,
     2 |Z_v|   <= Y_{v,v+1} + Y_{v,v-1} .
 
-The second inequality pads N_w by zero on the other column block.
+The second ineuality pads N_w by zero on the other column block.
 For any Hermitian B on the row space,
 
     | tr( sqrt(T_v) R_v^* B R_v ) |
@@ -791,7 +791,7 @@ Proof. The neighbor comparison is
 
     Y_{w+1, w} + Y_{w-1, w}  <=  sqrt(2) V_w .
 
-Write A = W_w, B = W_{w-1}^*, N = N_w. The elementary inequality
+Write A = W_w, B = W_{w-1}^*, N = N_w. The elementary ineuality
 ||A z + B z'||^2 <= 2||A z||^2 + 2||B z'||^2 implies T_w <= 2Q
 with Q = diag(A^* A, B^* B). Lemma-order gives
 
@@ -810,7 +810,7 @@ completes the neighbour comparison. In the plane identity, replace
 The two column costs carrying the score at vertex w have common
 coefficient Sigma_alpha/2 - alpha_w. Where this coefficient is
 negative, discard the terms; where positive, apply the neighbour
-comparison. This proves the first inequality. For the specific
+comparison. This proves the first ineuality. For the specific
 rates, Sigma_alpha = 16/5, and the two possible coefficients are
 -1 + (3/5) sqrt(2) and -11/10 + (1/2) sqrt(2); both are negative,
 which gives the strict dissipation.
@@ -935,7 +935,7 @@ The sum runs through the list chosen at the particular index pair
 
 where the sum runs over the distinct switch scales in the chosen
 list, and one-sided values are used at each switch. Both
-inequalities hold for complex coefficients, including simultaneous
+ineualities hold for complex coefficients, including simultaneous
 openings and closings. The first is evident. The second is a
 statement about the total variation of the mask, which is bounded
 by 2n because at each scale at most n intervals can be open and
@@ -1017,7 +1017,7 @@ using the elementary l^3 triangle bound |dA|_3 <= 2 |A_0|_3 gives
 
 The strict dissipation between switches and the initial energy
 bound at the first switch combine with the switch cost to produce
-the stated inequality. The factor n_0^2 (n_1 + n_2) is the
+the stated ineuality. The factor n_0^2 (n_1 + n_2) is the
 extreme case where the changed edge is the third input of the
 star at vertex 0; the factor n_0^3 is absorbed into the initial
 energy; the factors n_1^3 and n_2^3 are the initial energy at the
@@ -1060,7 +1060,7 @@ and trilinearity gives the desired bound
 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,
+right side of the switch-dissipation ineuality,
 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 unnormalized
@@ -1093,7 +1093,7 @@ coefficients mu_{ij,k} : Xi_D -> C satisfy
     |mu_{ij,k}(xi)|  <=  C_1 ,
     sum_k in I int_{Xi_D} |mu_{ij,k}(xi)|^2 d gamma(xi)  <=  C_1^2 ,
 
-for every output index pair (i,j), the first inequality holding
+for every output index pair (i,j), the first ineuality holding
 almost everywhere. If
 
     h^2  <=  q min_k L_k^2 / (16 D^2) ,
@@ -1243,7 +1243,7 @@ real and imaginary parts of the three window insertions. Integrate
 over base centers on Pi_0. Shifting the row center maps Pi_0 to
 Pi_d by a map of Jacobian one. The cost at vertex v is at most
 C(q) theta s V~_v; integrating over each slab with d s / s and
-over Xi_D with d gamma gives a bounded total, and the inequality
+over Xi_D with d gamma gives a bounded total, and the ineuality
 just proved bounds that total.
 
 On the other side, multiplying the three weighted windows in the
@@ -1325,8 +1325,8 @@ Gaussian envelope lemma. For F, G in L^3(R^2) put
     M_{F,G}(x,y)  =  ( M_x (|F|^{5/2})(x,y) )^{2/5}
                      ( M_y (|G|^{5/2})(x,y) )^{2/5} .
 
-The one-dimensional maximal inequality on L^{6/5}, followed by
-Holder's inequality on R^2, gives
+The one-dimensional maximal ineuality on L^{6/5}, followed by
+Holder's ineuality on R^2, gives
 
     || M_{F,G} ||_{3/2}  <=  C ||F||_3 ||G||_3 .
 
@@ -1432,7 +1432,7 @@ gives
 
     sup_{xi in Xi_D} |D^3 f_hat(xi)|  <=  C m / D .
 
-For the second bound, the translation inequality for functions of
+For the second bound, the translation ineuality for functions of
 bounded variation gives ||f'''(. + D^{-1}) - f'''||_1 <= C m / D;
 applying the supremum bound above to bound its L^2 norm squared by
 C m / D, using Plancherel f_hat''' = (i xi)^3 f_hat, and using
@@ -1451,7 +1451,7 @@ A finite frequency cutoff. Choose chi in C_c^infinity(R) supported on
 Let eta be the Schwartz convolution kernel of the multiplier chi,
 normalized so that int eta = 1, and write eta_{P_*}(z) = P_* eta(P_* z).
 Then f_*^{(j)} = f^{(j)} * eta_{P_*} for 0 <= j <= 3. The
-translation inequality gives ||f^{(j)} - f_*^{(j)}||_1 <= C m / P_*.
+translation ineuality gives ||f^{(j)} - f_*^{(j)}||_1 <= C m / P_*.
 The supremum norms are uniformly bounded, so interpolation gives
 
     ||f^{(j)} - f_*^{(j)}||_5  <=  C (m / P_*)^{1/5} .
@@ -1744,7 +1744,7 @@ the uniform family of individual bilinear bounds
 
     || B_{eps,R}(F, G) ||_{3/2}  <=  C_S ||F||_3 ||G||_3 .
 
-Holder's inequality gives one direction (from trilinear to bilinear).
+Holder's ineuality gives one direction (from trilinear to bilinear).
 For the other direction, let U in L^{3/2} be nonzero and set
 
     H  =  ( conj(U) / |U| ) ( |U|^{1/2} / ||U||_{3/2}^{1/2} ) ,
@@ -1752,7 +1752,7 @@ For the other direction, let U in L^{3/2} be nonzero and set
 defined to be zero where U = 0. Then ||H||_3 = 1 and
 int H U = ||U||_{3/2}. Applying this identity to each finite
 bilinear truncation and using the trilinear bound gives the reverse
-inequality. In particular, the optimal constant in the maximal
+ineuality. In particular, the optimal constant in the maximal
 operator is at least C_S. Equality of the scalar constants does not
 assert equality of the maximal or variation constants.
 
@@ -1764,7 +1764,7 @@ The ingredients sit in a fixed order. Hardy and Littlewood (Acta
 Mathematica 54, 1930) supply the centered line maximal as a
 localising device. Lepingle (Z. Wahrsch. Verw. Gebiete 36, 1976)
 and Jones, Seeger, and Wright (Trans. Amer. Math. Soc. 360, 2008)
-provide the variational and martingale inequalities that frame
+provide the variational and martingale ineualities that frame
 the problem. Oberlin, Seeger, Tao, Thiele, and Wright (J. Eur.
 Math. Soc. 14, 2012) prove the variation-norm Carleson theorem and
 fix the shape of the dyadic rank ordering. Daletskii (Uspekhi Mat.