diff --git a/four_dim_kakeya_hausdorff_v2.txt b/four_dim_kakeya_hausdorff_v2.txt
index 1156fcf..bedb078 100644
--- a/four_dim_kakeya_hausdorff_v2.txt
+++ b/four_dim_kakeya_hausdorff_v2.txt
@@ -312,7 +312,7 @@ balls that almost-covers K.
 
 Third, we count. The density of selected points at each resolution
 scale gives us a "density exponent" f(r). The geometry of K
-imposes inequalities between these exponents. The result is a
+imposes ineualities between these exponents. The result is a
 class of probabilistic problems -- the "cap class C(d)" for some
 d < 1 -- with positive "horizon".
 
@@ -390,24 +390,24 @@ N_0 large enough, it is also bounded by N_0^{alpha + eps}. We
 are sloppy about them everywhere. When two quantities are equal
 up to subpower factors, we write A ~ B.
 
-The "cap class C(d; S, eta)" is defined by two inequalities. Fix
+The "cap class C(d; S, eta)" is defined by two ineualities. Fix
 d in (0, 1) and S, eta > 0. The problem is in the class if, for
 all 0 <= r <= L,
 
     f(L) - f(r) <= d (L - r),
 
 and the joint density of (p_L, [V]_{a_u}), at angular depth u in
-(0, eta L], is at most n(L) N_0^{-Su+o(1)}. The first inequality
+(0, eta L], is at most n(L) N_0^{-Su+o(1)}. The first ineuality
 is the one that matters. It says: refining from depth r up to
 terminal depth L loses at most d per unit of depth. The factor
 d < 1 is the "compression budget" the problem allows. The second
-inequality is an angular cap: the velocity V of a terminal
+ineuality is an angular cap: the velocity V of a terminal
 observation is confined to balls of angular width N_0^{-u}, with
 density at most N_0^{-Su+o(1)} times the terminal density. This
 restrains how much incidence mass is concentrated near any one
 direction, at depth u.
 
-What are these inequalities for, intuitively? The cap inequality
+What are these ineualities for, intuitively? The cap ineuality
 f(L) - f(r) <= d (L - r) says the density profiles are almost
 linear. As we refine the hidden observation, we lose at most d
 per unit of refinement. Since d < 1, the loss is strictly smaller
@@ -678,26 +678,26 @@ to subpower. Nothing is lost yet in a serious way.
 
 Let f(r, u) be the joint density exponent of (observation at
 depth r, velocity at depth u), and put g = r - f. Then three
-inequalities hold:
+ineualities hold:
 
     g(1, 0) >= sigma,
     g(0, 0) <= 0,
     g(r, u) <= r + 2 - 2u.
 
-First inequality. An enlarged ball uses O(1) hidden w-bins over
+First ineuality. An enlarged ball uses O(1) hidden w-bins over
 base (t, y)-volume O(delta^3). The total state volume is at most
 C delta^{-(4 - sigma)} * delta^3 = C delta^{-(1 - sigma)}. Since
 mass is inverse-subpower, n(1, 0) is at least delta^{1 - sigma}
 up to subpower, so f(1, 0) <= -(1 - sigma) and g(1, 0) >= sigma.
 
-Second inequality. The projected y-lines satisfy a full-time
+Second ineuality. The projected y-lines satisfy a full-time
 plank cap at every dyadic mesh: any spatial plank of half-widths
 a rho, b rho contains projected y-traces of total prior mass at
 most K_0 a b rho^2 (this is the "planar" fact observed in the
 original). If n(0, 0) grew by a fixed positive power, the weighted
 plank estimate (Lemma 2.7) would break this cap, so g(0, 0) <= 0.
 
-Third inequality. The velocity marginal has bounded range and
+Third ineuality. The velocity marginal has bounded range and
 prior density at most K_0. At fixed t, the conditional y-density
 is at most C delta^{-2}, because the smoothed segments are of
 width delta. So the joint density is at most K_0 N_0^{2 - 2u};
@@ -710,7 +710,7 @@ maximize g(r, u) - Rr + Su on [0, 1] x [0, 4]. The maximizer
 (r_0, u_0) satisfies r_0 > 0, u_0 < 4, g(r_0, u_0) >= sigma/2.
 
 Restrict the problem to the maximizer's scales. The maximization
-inequality acts like this: the local problem lies in C(1 - R) at
+ineuality acts like this: the local problem lies in C(1 - R) at
 angular parameters S, eta = (4 - u_0) / (2 r_0). Refining from
 depth r up to r_0 loses at most (1 - R)(r_0 - r), and the angular
 cap holds. Length becomes r_0.
@@ -788,7 +788,7 @@ when we differentiate, iterate, or compose the fits.
 
 Everything is done in the complex plane, because that is where
 polynomial roots behave nicely. The main lemma is a "Bernstein-
-Remez"-type inequality for algebraic functions: it bounds a
+Remez"-type ineuality for algebraic functions: it bounds a
 sheet's norm on a comparison ball by its norm on a smaller piece
 of it. That is exactly the rigidity we need: agreement on a
 large set forces agreement everywhere.
@@ -859,7 +859,7 @@ The first two parts say: a normalized sheet has bounded
 derivatives (of any fixed order) and cannot be locally
 flat-to-all-orders -- some derivative of order at most D is
 nonzero at each point. The final estimate is the Remez
-inequality: a sheet's norm on a smaller ball is at least a
+ineuality: a sheet's norm on a smaller ball is at least a
 polynomial fraction of its norm on a bigger ball, provided the
 smaller ball has positive measure.
 
@@ -984,7 +984,7 @@ Proof idea. The proof uses an "entropy" argument.
 
   (c) That map's image has small measure but at least a
       Jacobian-determined amount of "state mass." Comparing the
-      state count with the image size gives a strict inequality
+      state count with the image size gives a strict ineuality
       on state counts. Summing over scales forces d >= 1.
 
 The details involve information-theoretic entropy chains and
@@ -1668,7 +1668,7 @@ Idea of proof. Two stages.
        controlled by comparing values at two shared times; that
        gives A(u) via a limit. Then check that J_1 (the residual
        after subtracting A y) has small entropy using the
-       triangle inequality for entropy.
+       triangle ineuality for entropy.
 
 The role of the "prepared labels" is technical: before fitting
 each component, we restrict to times where the current labels
@@ -1901,7 +1901,7 @@ possibilities" property means the polynomial fit at a coarse cell
 predicts the polynomial fit at fine sub-cells, and this propagates
 all the way to the whole Q-block.
 
-This is one of the places where the "<=" inequalities have to be
+This is one of the places where the "<=" ineualities have to be
 done carefully. The choices of fiber polynomials, the coarse-to-
 fine propagation, and the Remez step all lose a constant factor
 subpower in N_0. Those losses add up. The final choice of eps_T
@@ -2918,7 +2918,7 @@ dimension-4 lower bound in R^4. Combining:
     every Nikodym set Omega subset of M has dim_H Omega = 4.
 
 This resolves Conjecture 1.4 of Gao-Liu-Xi in dimension four.
-The reverse inequality dim_H Omega <= 4 holds trivially for any
+The reverse ineuality dim_H Omega <= 4 holds trivially for any
 subset of a 4-manifold.
 
 11.3 Fixed-chart curved Kakeya sets