diff --git a/symmetric_mahler_equality_v2.txt b/symmetric_mahler_equality_v2.txt
index e182d61..fe95d55 100644
--- a/symmetric_mahler_equality_v2.txt
+++ b/symmetric_mahler_equality_v2.txt
@@ -12,7 +12,7 @@ CONTENTS
     3. The holomorphic mass lemma
     4. Feasible bases: a probability that is always at least 1
     5. Extra boundary hits are null
-    6. The real simplex identity (the main inequality)
+    6. The real simplex identity (the main ineuality)
 
 
 ================================================================
@@ -51,7 +51,7 @@ p_A(x) + p_{A'}(x') <= 1 is
     (x, x') = p_A(x)·(x/p_A(x), 0) + p_{A'}(x')·(0, x'/p_{A'}(x')),
 where we suppress a zero coordinate (set that term to 0 if p = 0). This
 is a convex combination of a point of A×{0} and a point of {0}×A',
-of total weight p_A(x) + p_{A'}(x') <= 1. The reverse inequality follows
+of total weight p_A(x) + p_{A'}(x') <= 1. The reverse ineuality follows
 similarly. QED.
 
 LEMMA 1.1 (Polars and volumes of the two sums).
@@ -102,11 +102,11 @@ The polar formulas (1.4), (1.5) exchange ⊕_1 and ⊕_∞ and preserve the
 and preserve P. QED.
 
 Dimension one. Every origin-symmetric convex body is a centered
-interval. The sharp inequality and the equality classification in
+interval. The sharp ineuality and the equality classification in
 dimension one are the base case above.
 
 What is left to do. The rest of Parts 2 through 10 proves the
-inequality in every dimension d, together with the classification of
+ineuality in every dimension d, together with the classification of
 equality. The cubes and cross-polytopes we saw are the simplest
 Hanner bodies; the argument has to handle arbitrary symmetric bodies
 A of arbitrary dimension.
@@ -439,7 +439,7 @@ Define
 
 The "active" coordinates i in I are automatic (they solve to the
 boundary by construction). All P_I are probabilities. The next lemma is
-the analytic core of the main inequality.
+the analytic core of the main ineuality.
 
 LEMMA 4.1 (Boundary feasibility). Under the setup above (any F, any d,
 any spanning nonzero rows),
@@ -591,7 +591,7 @@ to prove.
 ================================================================
 
 Now we translate the feasibility probabilities into real volumes, and
-extract the main inequality.
+extract the main ineuality.
 
 Keep the nonzero, spanning, pairwise nonproportional rows b_1, ..., b_m
 of (4.1). For X in int A, define
@@ -621,7 +621,7 @@ tight at the same point Y_{I, epsilon}(X). The simplex T_{I, epsilon}
 records "which d signed rows were active at this vertex". Summing over
 feasible pairs gives a kind of footprint of L_X on the polar body.
 The footprint is always at least 4^d/d! in integrated volume (up to a
-constant), which is precisely the volume-product inequality.
+constant), which is precisely the volume-product ineuality.
 
 PROPOSITION 6.1 (Feasible-simplex identity).
 Let b_1, ..., b_m be nonzero, pairwise nonproportional real rows
@@ -737,7 +737,7 @@ Proposition 6.1 gives |A||A°| >= 4^d/d! for polyhedral strip bodies.
 We now approximate arbitrary origin-symmetric convex bodies by
 polytopes and pass to the limit.
 
-COROLLARY 7.1 (Symmetric volume-product inequality).
+COROLLARY 7.1 (Symmetric volume-product ineuality).
 For every origin-symmetric convex body K in R^d, d >= 1,
     |K| · |K°|  >=  4^d / d!.                                (7.1)
 
@@ -876,12 +876,12 @@ the mass placed at the simplex vertex 0. Both cases are allowed.)
 By feasibility of (I_k, epsilon_k), there is a witness Y_k in R^d with
     c_{ik} · Y_k  =  lambda(c_{ik} · X_k)        (i = 1, ..., d),     (8.7)
     |v · Y_k|  <=  lambda(v · X_k)              (v a vertex of C_{j_k}).
-The second inequality extends to all h in C_{j_k}: write h = Σ s_l v_l
+The second ineuality extends to all h in C_{j_k}: write h = Σ s_l v_l
 as a convex combination of vertices v_l, so s_l >= 0, Σ s_l = 1, then
     |h · Y_k|  <=  Σ s_l |v_l · Y_k|
               <=  Σ s_l lambda(v_l · X_k)
               <=  lambda( (Σ s_l v_l) · X_k )  =  lambda(h · X_k),
-where the last inequality uses concavity of lambda (Lemma 2.1) and the
+where the last ineuality uses concavity of lambda (Lemma 2.1) and the
 fact that both h · X_k and v_l · X_k lie in [-1, 1]. (That inclusion
 holds because h in C and X_k in B implies |h · X_k| <= 1 by polarity.)
 
@@ -892,14 +892,14 @@ of k. We can pass to a subsequence.
 Convergence to a representation. Compactness: Y_k in a bounded set,
 c_{ik} ∈ Q × {±1} (compact set of vertices), theta_{ik} ∈ [0,1] (compact).
 Pass to a subsequence so that c_{ik} -> c_i, theta_{ik} -> theta_i, and
-Y_k -> Y. Then (taking limits in (8.6), (8.7), and the derived inequality)
+Y_k -> Y. Then (taking limits in (8.6), (8.7), and the derived ineuality)
     c_i ∈ Q × {±1},     theta_i >= 0,       Σ theta_i <= 1,
     e  =  Σ theta_i c_i,
     c_i · Y  =  lambda(c_i · X)    (i = 1, ..., d),
     |h · Y|  <=  lambda(h · X)     (all h in C).                       (8.8)
 
-(For the last "all h in C" inequality, use h_k = t_{j_k} h ∈ C_{j_k} in
-the derived inequality and let k -> infinity.)
+(For the last "all h in C" ineuality, use h_k = t_{j_k} h ∈ C_{j_k} in
+the derived ineuality and let k -> infinity.)
 
 Total mass = 1. Since e ∈ ∂C, p_C(e) = 1. Then
     1  =  p_C(e)  =  p_C(Σ theta_i c_i)
@@ -990,7 +990,7 @@ If two of {0, x, y} agree, that common point is trivially a median.
 Assume 0, x, y are distinct. Set
     p_1 = ||x||,        p_2 = ||y||,        p = ||x - y||,
     s = p_1 + p_2,       a = p_1 - p_2.
-The triangle and reverse-triangle inequalities give s >= p >= |a| > 0.
+The triangle and reverse-triangle ineualities give s >= p >= |a| > 0.
 
 Define
     A_+  =  (p + a) / 2,     A_-  =  (p - a) / 2,     c  =  (s - p) / 2.
@@ -1015,7 +1015,7 @@ x + y = (r_+ - r_-) + (s - p) u for some u in Q. Set
 y + r_- iff x - y = r_+ + r_-.) Then
     2 m  =  x + y - (r_+ - r_-)  =  (s - p) u,
 so ||m|| <= (s - p)/2 = c. Also ||x - m|| = ||r_+|| <= A_+, and
-||y - m|| = ||r_-|| <= A_-. Now the triangle inequality forces every
+||y - m|| = ||r_-|| <= A_-. Now the triangle ineuality forces every
 one of these to be an equality:
     p_1  <=  ||m|| + ||x - m||  <=  c + A_+  =  p_1,
     p_2  <=  ||m|| + ||y - m||  <=  c + A_-  =  p_2.
@@ -1109,7 +1109,7 @@ Extend to all u ∈ K by continuity of all the maps. For nonzero
 v ∈ R^n, put u = v / h_Q(v) ∈ ∂K. By homogeneity of support functions,
 (9.10) at this u gives
     <x + y, v>  <=  h_{D_0}(v)  +  (s - p) h_Q(v).
-At v = 0 the same inequality holds trivially. So
+At v = 0 the same ineuality holds trivially. So
     <x + y, v>  <=  h_{D_0 + (s-p)Q}(v)          for all v in R^n.
 Convex separation (or the separating-hyperplane theorem for compact
 convex sets) then gives
@@ -1185,14 +1185,14 @@ Hanner body. This is the Main Theorem stated in Part 0. QED.
 11. TWO COROLLARIES: FUNCTIONAL AND ENTROPY
 ================================================================
 
-The Main Theorem implies two sharp functional inequalities. Both are
+The Main Theorem implies two sharp functional ineualities. Both are
 already in the literature; here they appear as corollaries of the new
 geometric input. Both proofs reduce to pre-existing theorems that hold
-CONDITIONALLY on the all-dimensional symmetric Mahler inequality, which
+CONDITIONALLY on the all-dimensional symmetric Mahler ineuality, which
 we now have unconditionally.
 
 ----------------------------------------------------------------
-11a. Even functional Mahler inequality.
+11a. Even functional Mahler ineuality.
 ----------------------------------------------------------------
 
 For a proper lower-semicontinuous convex function phi : R^n -> R ∪ {+inf}
@@ -1205,7 +1205,7 @@ COROLLARY 11.1 (Even functional Mahler).
 For every n >= 1 and every even proper lower-semicontinuous convex
 phi : R^n -> R ∪ {+inf} with 0 < ∫ e^{-phi} dx < inf,
     ( ∫ e^{-phi(x)} dx ) · ( ∫ e^{-phi*(y)} dy )  >=  4^n.          (11.1)
-The second integral is allowed to be +inf, in which case the inequality
+The second integral is allowed to be +inf, in which case the ineuality
 is immediate.
 
 PROOF. Set f = e^{-phi}: nonzero integrable log-concave function (a
@@ -1215,14 +1215,14 @@ the polar function
 and the translation-minimized functional volume product
     P(f)   =  ( ∫ f(x) dx ) · inf_{z in R^n} ∫ f^z(y) dy.        (11.3)
 Their Proposition 1(a) states: if the symmetric geometric Mahler
-inequality P(K) >= 4^n/n! holds in EVERY dimension n, then P(f) >= 4^n for
+ineuality P(K) >= 4^n/n! holds in EVERY dimension n, then P(f) >= 4^n for
 every even log-concave f : R^n -> R with 0 < ∫ f < inf. (Reference:
-Fradelizi-Meyer, "Some functional inverse Santalo inequalities",
+Fradelizi-Meyer, "Some functional inverse Santalo ineualities",
 Adv. Math. 218 (2008), 1430-1452.)
 
 Theorem of Part 10 supplies this all-dimensional hypothesis. Then
     ( ∫ e^{-phi} dx ) · ( ∫ e^{-phi*} dy )   =   ( ∫ f dx ) · ( ∫ f^0 dy )   >=   P(f)   >=   4^n.
-The middle inequality uses z = 0 in the infimum defining P(f), and
+The middle ineuality uses z = 0 in the infimum defining P(f), and
     f^0(y)  =  inf_{x : f(x) > 0} e^{-x·y}/f(x)
            =  inf_x exp(-x·y + phi(x))
            =  exp( - sup_x (x·y - phi(x)) )
@@ -1242,7 +1242,7 @@ geometric equality case. For potentials WITHOUT evenness the correct
 sharp constant is e^n (companion paper on general convex bodies).
 
 ----------------------------------------------------------------
-11b. Symmetric entropy-transport inequality.
+11b. Symmetric entropy-transport ineuality.
 ----------------------------------------------------------------
 
 For a log-concave probability measure eta with a density, define the
@@ -1257,7 +1257,7 @@ where psi ranges over proper lower-semicontinuous convex functions
 R^n -> R ∪ {+inf} and psi* is the Fenchel-Legendre transform. The cost
 may be +inf.
 
-COROLLARY 11.2 (Symmetric entropy-transport inequality).
+COROLLARY 11.2 (Symmetric entropy-transport ineuality).
 For every n >= 1, let eta_i(dx) = e^{-V_i(x)} dx, i = 1, 2, be full-
 dimensional log-concave probability measures on R^n, both symmetric
 about the origin, with proper lower-semicontinuous convex potentials
@@ -1281,7 +1281,7 @@ entropy H(eta) = -∫ V d eta is finite.
 
 Gozlan's equivalence (as recorded in Fradelizi-Gozlan-Zugmeyer,
 arXiv:2109.00871, Theorem 1.3) connects the symmetric functional inverse
-Santaló inequality to the symmetric entropy-transport inequality. Apply
+Santaló ineuality to the symmetric entropy-transport ineuality. Apply
 that equivalence to Corollary 11.1 with the sharp constant c = 4. We
 obtain (11.4). QED.
 
@@ -1303,12 +1303,12 @@ CLASSICAL LITERATURE (cited in the proof).
     operators." Ann. Inst. Fourier 28 (1978), 35-65. Contains the norm-
     additive decomposition (median vs. 3.2 IP) criterion.
   - Fradelizi, M.; Meyer, M. "Some functional inverse Santalo
-    inequalities." Adv. Math. 218 (2008), 1430-1452. Proposition 1(a)
+    ineualities." Adv. Math. 218 (2008), 1430-1452. Proposition 1(a)
     is used in Cor. 11.1.
   - Cordero-Erausquin, D.; Klartag, B. "Moment measures." J. Funct.
     Anal. 268 (2015), 3834-3866. Lemma 4 and Proposition 7 in Cor. 11.2.
   - Fradelizi, M.; Gozlan, N.; Zugmeyer, S. "Transport proofs of some
-    functional inverse Santalo inequalities." arXiv:2109.00871v3.
+    functional inverse Santalo ineualities." arXiv:2109.00871v3.
     Theorem 1.3 in Cor. 11.2.
 
 LENS AND MASS LEMMA.
@@ -1340,7 +1340,7 @@ HISTORICAL CONTEXT (background, not prerequisites).
     nonsymmetric 3-D case.
   - Saint-Raymond, J. "Sur le volume des corps convexes symetriques."
     Seminaire d'initiation a l'analyse (1980-1981), exp. 11. Sharp
-    inequality for unconditional bodies.
+    ineuality for unconditional bodies.
   - Meyer, M. "Une caracterisation volumique de certains espaces normes
     de dimension finie." Israel J. Math. 55 (1986), 317-326. Hanner
     equality cases in the unconditional class.
@@ -1356,7 +1356,7 @@ HISTORICAL CONTEXT (background, not prerequisites).
   - Kim, J. "Minimal volume product near Hanner polytopes." J. Funct.
     Anal. 266 (2014), 2360-2402.
   - Kim, J.; Zvavitch, A. "Stability of the reverse Blaschke-Santalo
-    inequality for unconditional convex bodies." Proc. Amer. Math.
+    ineuality for unconditional convex bodies." Proc. Amer. Math.
     Soc. 143 (2015), 1705-1717.
   - Bourgain, J.; Milman, V. D. "New volume ratio properties for convex
     symmetric bodies in R^n." Invent. Math. 88 (1987), 319-340.