diff --git a/qma_coulomb_hardness_v2.txt b/qma_coulomb_hardness_v2.txt
index bc1ca73..153b5af 100644
--- a/qma_coulomb_hardness_v2.txt
+++ b/qma_coulomb_hardness_v2.txt
@@ -840,7 +840,7 @@ polynomial size).
 Proposition 1 (continuous one-particle control).
   With the density and separated centers constructed above, once the
   polynomial scale is sufficiently large, there are fixed constants
-  c > 0 and C < infinity such that, as quadratic-form inequalities,
+  c > 0 and C < infinity such that, as quadratic-form ineualities,
 
     (I-P) L_c (I-P)  >=  c (I-P)          on the complement,
     L_c              >=  - Delta_y/2 - C  (kinetic lower bound),
@@ -858,7 +858,7 @@ bounded everywhere, L_c is at least the separable operator
             - e_* ,                                                  (41)
 
 minus an error O(S^3/H) + O(p(n,D)/lambda) that is uniform in all states
-(as a form inequality). Call the first bracket (planar) A_pl and the second
+(as a form ineuality). Call the first bracket (planar) A_pl and the second
 (vertical) A_v, each shifted so its ground energy is near zero:
 A_pl has ground energy -1/2 (mode phi at each site) and A_v has ground
 energy omega/2 (mode g).
@@ -901,7 +901,7 @@ localization costs O(S^{-2}) in the form, and the rank-one mode is P_g =
     A_v  >=  omega/2 + c_2 (I - P_g) - o(1) I .                      (44)
 
 Step 4: tensor the gaps. On the spatial space P = P_r (tensor) P_g. The
-operator inequality
+operator ineuality
 
     (I - P_r) (tensor) I + I (tensor) (I - P_g)  >=  I - P           (45)
 
@@ -914,7 +914,7 @@ through (45), the separable operator A_sep has, on I - P, a fixed gap
 
 since e_* = -1/2 + omega/2 makes the constant term zero. Choosing the
 polynomial scale so the total error O(S^3/H) + O(p(n,D)/lambda) + o(1) is
-at most c/2 gives the first inequality in (40). The slowly small errors
+at most c/2 gives the first ineuality in (40). The slowly small errors
 o(d^{-2}), o(s^{-2}) are only needed to establish a fixed gap, not a per-
 particle energy accuracy; no budget is blown.
 
@@ -938,7 +938,7 @@ shifted by e_*). So the residual L_c psi_j is (i) the other wells W_i,
 i != j, acting on the small planar tails of psi_j, and (ii) the term
 (b_j/lambda) W_j on psi_j, and (iii) the errors already bounded in (47).
 All three are of size at most p(n,D) lambda^{-0.9} on unit mode vectors,
-giving the last inequality in (40). QED
+giving the last ineuality in (40). QED
 
 A fixed complementary gap is not enough: one needs the matrix on the modes to
 a sharper scale, because the continuum energy will be compared to H_d after
@@ -1137,7 +1137,7 @@ over a cube of volume h^3 with eight points.) Apply this (a) to the
 singular Coulomb kernel away from its singularity, then (b) to the smooth
 fields K_ij coming from the modes (Step 3).
 
-Step 2: the general form estimate (first inequality in (60)). For fixed
+Step 2: the general form estimate (first ineuality in (60)). For fixed
 evaluation point y, the charge integral is the kernel |y - G(b)|^{-1} against
 uniform density in b. Split cubes Q into close ones (whose images touch a
 ball of radius a fixed large multiple C' h about y) and far ones.
@@ -1149,7 +1149,7 @@ ball of radius a fixed large multiple C' h about y) and far ones.
   |y - b_alpha| <= C' h, which the Hardy estimate of Step 4 controls exactly
   against ||u||^2 + ||nabla u||^2 on balls.
 
-Hardy, G. H., Littlewood, J. E., and Polya, G., "Inequalities", Cambridge
+Hardy, G. H., Littlewood, J. E., and Polya, G., "Ineualities", Cambridge
 University Press, 1934, Theorem 326. ISBN: 978-0521358804.
 
 Lieb, E. H. and Loss, M., "Analysis", 2nd ed., Graduate Studies in
@@ -1173,7 +1173,7 @@ So the pointwise absolute error is at most C(h^2 + h^4 m(S+1)) plus the
 close-node sum. In particular there is no factor the size of the large
 horizontal area in this bound.
 
-Step 3: the compressed estimate (second inequality). For compression use the
+Step 3: the compressed estimate (second ineuality). For compression use the
 smooth Coulomb fields
 
     K_ij(b) = int |y - b|^{-1} psi_i(y) psi_j(y) dy .                (63)
@@ -1197,7 +1197,7 @@ degree-three cubature (61) to K_ij o G now costs, for each entry,
 because pure fourth derivatives have summable volume bounds as in (62),
 and the other chain-rule terms live only on the exceptional cells. Gram
 correction (via (38)) and operator norms cost at most polynomial factors.
-This proves the second inequality of (60) for the exact nodes.
+This proves the second ineuality of (60) for the exact nodes.
 
 Step 4: the close-node Hardy bound and rounding. The close-node balls have
 bounded overlap also upon any fixed dilation. Around each node one has
@@ -1206,7 +1206,7 @@ bounded overlap also upon any fixed dilation. Around each node one has
       <= C int_{|y - b_alpha| <= 2C'h} ( h^{-1} |u|^2 + h |nabla u|^2 ) .
                                                                      (66)
 
-This follows by a cutoff and the three-dimensional Hardy inequality
+This follows by a cutoff and the three-dimensional Hardy ineuality
 int |u|^2 / |y|^2 <= 4 int |nabla u|^2 (which follows from the divergence
 formula for y/|y|^2 together with Cauchy-Schwarz). Summing (66) over the
 bounded-overlap balls gives the general form bound, since h^2 = O(lambda^{-2/3}).
@@ -1216,7 +1216,7 @@ delta_R changes the inverse distance by at most
 
     delta_R / ( |y - b_alpha| |y - b~_alpha| ) .                     (67)
 
-Cauchy-Schwarz and translated Hardy inequalities bound its form by
+Cauchy-Schwarz and translated Hardy ineualities bound its form by
 4 delta_R ||nabla u||^2. Summing the masses gives the total bound
 
     4 rho |Omega| delta_R ||nabla u||^2 ,     where
@@ -1329,7 +1329,7 @@ forms,
        >= -(beta^2/alpha) ||w||^2
        >= -p(n,D) lambda^{-4/3} ,                                   (76)
 
-using the elementary inequality a s^2 - 2 b s t >= -b^2/a t^2. This proves
+using the elementary ineuality a s^2 - 2 b s t >= -b^2/a t^2. This proves
 the lower bound in (72); the upper bound follows by testing a minimizing
 vector of the compressed operator (which lies in Ran Pi). The lower-bound
 argument applies to every normalized antisymmetric state, in any spin sector,
@@ -1345,7 +1345,7 @@ The pair matrix. For the repulsion R, the pair matrix on the original modes
 (before Gram correction) has spatial entries from products of two transition
 densities psi_i psi_k acting on psi_j psi_l. Keeping only psi_i^2, psi_j^2 as
 the respective densities gives the direct coefficients v_{ij} of (13). Any
-unequal indices within a density give an exponentially small error:
+uneual indices within a density give an exponentially small error:
 
     || psi_i psi_k ||_{L^1} = O(lambda^{-0.8})    for i != k ,       (77)