diff --git a/bi_lipschitz_rcd_coordinates_v2.txt b/bi_lipschitz_rcd_coordinates_v2.txt
index 5e1ffad..38a647c 100644
--- a/bi_lipschitz_rcd_coordinates_v2.txt
+++ b/bi_lipschitz_rcd_coordinates_v2.txt
@@ -36,7 +36,7 @@ PART I. THE QUESTION AND THE MAIN THEOREM
 A metric space is a set of points together with a distance
 function d(x,z) that assigns a nonnegative number to each pair of
 points, is zero exactly when the points are equal, is symmetric,
-and obeys the triangle inequality
+and obeys the triangle ineuality
 
     d(x,y) <= d(x,z) + d(z,y).
 
@@ -104,7 +104,7 @@ Two equivalent routes were developed:
   bound and n the dimension bound.
 
   (Bochner / heat) The same content phrased through the heat
-  flow and a Bochner inequality, by Erbar-Kuwada-Sturm.
+  flow and a Bochner ineuality, by Erbar-Kuwada-Sturm.
 
 The CD condition also allows genuinely non-Riemannian spaces
 (some Finsler-type structures) that behave unlike manifolds. The
@@ -126,7 +126,7 @@ though there is no coordinate chart:
     quadratic Cheeger energy and its generator);
   - a carre du champ Gamma(f,g) = <grad f, grad g>, the "dot
     product of gradients";
-  - a Hessian for nice functions, and a Bochner inequality;
+  - a Hessian for nice functions, and a Bochner ineuality;
   - a heat semigroup and a continuous heat kernel.
 
 These are the only analytic tools the proof uses. None of them
@@ -198,7 +198,7 @@ constant L >= 1 when, for all x,z in U,
 
     L^{-1} d(x,z)  <=  |F(x) - F(z)|  <=  L d(x,z).
 
-The right inequality says F does not stretch distances by more
+The right ineuality says F does not stretch distances by more
 than L; the left says it does not crush them by more than L. A
 bi-Lipschitz map is automatically continuous and injective, and so
 is its inverse on the image, so it is a homeomorphism onto its
@@ -422,7 +422,7 @@ PART II. THE ANALYTIC TOOLBOX
 The proof uses no coordinates on X. It uses five analytic objects
 that exist on every RCD space. Each is covered by a cited theorem,
 used as a black box. This part lists them, defines every symbol,
-and states without proof the inequalities the rest of the tour
+and states without proof the ineualities the rest of the tour
 relies on.
 
 
@@ -453,7 +453,7 @@ solutions are locally Lipschitz.
 Harmonic functions are the natural candidates for coordinates on
 a singular space, exactly as on a smooth manifold where a harmonic
 coordinate chart is a chart whose coordinate functions are
-harmonic. They are chosen because the Bochner inequality gives
+harmonic. They are chosen because the Bochner ineuality gives
 strong control over the squared gradient Gamma(w) of a harmonic
 function -- the "energy" of the coordinate.
 
@@ -487,13 +487,13 @@ degrade as the scale shrinks; R_i(y) need not stay bounded above
 or below.
 
 The tool that measures the size of G is the dimensional Bochner
-inequality, valid on every RCD(K,n) space for test functions. For
+ineuality, valid on every RCD(K,n) space for test functions. For
 a constant-Laplacian solution w with Delta w = a, and with
 kappa = max{-K, 0}, define the matrix measure
 
     M(w) = (1/2) Delta Gamma(w) + kappa Gamma(w) m .
 
-Its content: the Bochner inequality says M(w) is a nonnegative
+Its content: the Bochner ineuality says M(w) is a nonnegative
 measure, locally finite, and dominates the squared Hessian:
 
     M(w) >= |Hess w|^2 m .
@@ -502,7 +502,7 @@ Here Hess w is the Hessian of w, defined through the second-order
 calculus of Gigli; its trace is Delta w, because the space is
 noncollapsed and has dimension exactly n almost everywhere. For a
 constant b (a "shift" toward the metric tensor g) one has the
-shifted inequality
+shifted ineuality
 
     M(w) - (2 b Delta w - n b^2) m  >=  |Hess w - b g|^2 m .
 
@@ -668,8 +668,8 @@ fact used later and justifies them from cited input.
 In plain terms: constant-Laplacian solutions are as smooth as the
 space allows, the Bochner measure M is a genuine nonnegative
 measure even when Hess w is only an L^2 object, the shifted
-inequality always centers correctly, and the Dirichlet boundary
-problem has coercivity (a Sobolev inequality with exponent
+ineuality always centers correctly, and the Dirichlet boundary
+problem has coercivity (a Sobolev ineuality with exponent
 2 d_* > 2). The proof assembles facts from Bacher-Sturm,
 Kell, Jiang, De Philippis-Gigli, Han, Gigli, Connell et al.,
 Mondino-Naber, and Hajlasz-Koskela; the paper makes a point of
@@ -678,7 +678,7 @@ n and 2.
 
 Two subtleties worth noticing:
 
-  - The Hessian inequality for M is stated as an inequality of
+  - The Hessian ineuality for M is stated as an ineuality of
     measure-valued quadratic forms. There is no pointwise Hessian
     to compare against; the "square" |Hess w - b g|^2 lives as a
     density of a nonnegative measure through polarization. Section
@@ -878,7 +878,7 @@ calculus, and it is short enough to keep in view.
     1.3 (normalized Gram -> Id; rescaled curvature -> 0).
   - Positivity of the Bochner measure bounds its mass on the
     inner ball by what the cutoff saw. Polarizing over the fixed
-    finite dimension gives the matrix inequality.
+    finite dimension gives the matrix ineuality.
 
 There is no subtlety about singular parts: the proof only uses M
 on Lipschitz cutoffs, and M is a nonnegative measure there.
@@ -985,11 +985,11 @@ testing center x and every integer k >= 0:
 where Z_i is the unweighted sum of the local D-forms at scale i
 (over the active centers of the patch) and S_i is a weighted sum
 with weights in [0,1] at least one of which equals one and has a
-covering ball. The second inequality is the deep one; without it
+covering ball. The second ineuality is the deep one; without it
 nothing stops H_k from running off to infinity faster than the
 local mass accounts for.
 
-Five smaller lemmas feed into these two inequalities. The order
+Five smaller lemmas feed into these two ineualities. The order
 below is the paper's order.
 
 
@@ -1027,7 +1027,7 @@ strictly positive. H_i is also positive definite.
 The correction term lambda r_i^2 Id is a technical necessity: it
 absorbs the curvature contribution and the cutoff error uniformly,
 without which Q_i need not be positive definite in the strict
-sense that later matrix inequalities need.
+sense that later matrix ineualities need.
 
 
 25. LEMMA 2.1: KERNEL DIFFERENCES AND MEASURE SOURCES
@@ -1065,12 +1065,12 @@ its small-time power is absorbed in the exponential.
 
 Long-time bound. For a in B_r(y), d(a,x) <= (C_0+1) r. Integrate
 the first spatial gradient estimate along a minimizing curve from
-x to a; the inequality d(b,z)^2 >= D^2/2 - d(b,x)^2 absorbs the
+x to a; the ineuality d(b,z)^2 >= D^2/2 - d(b,x)^2 absorbs the
 variation of the Gaussian along the curve. This gives the bound
 pointwise for almost every fixed source pole z. To extend to every
 pole, use symmetry and continuity of the heat kernel: the full-
 measure set of "good" poles is dense, and passing to the limit
-with relaxed Gaussian constants gives the inequality at every z.
+with relaxed Gaussian constants gives the ineuality at every z.
 No gradient representative is ever paired with a singular measure:
 all the integration-by-parts is done against the smooth heat
 kernel at positive time.
@@ -1159,7 +1159,7 @@ possibly be.
 27. LEMMA 2.3: THE DYADIC HEAT COMPARISON
 
 This lemma produces the small series F_i that will appear
-everywhere afterwards and states two inequalities: a decrement
+everywhere afterwards and states two ineualities: a decrement
 estimate for Q_i - Q_{i+1} and a bilinear one relating R_i - Q_i
 to F_i. Both are indispensable for the linear algebra in Section
 2.6.