diff --git a/planar_xy_spin_field_v2.txt b/planar_xy_spin_field_v2.txt
index 3b0f160..b72591c 100644
--- a/planar_xy_spin_field_v2.txt
+++ b/planar_xy_spin_field_v2.txt
@@ -227,7 +227,7 @@ so the last assertion follows by gauge change. []
 ============================================================
 
 Throughout, graphs are finite with the fixed primary structure.
-The affine inequality of the companion Bessel paper already
+The affine ineuality of the companion Bessel paper already
 includes real observation centers and real numerator tilts; we
 recall its Gaussian-chain proof, then apply it to cuts, relative
 data and released constants.
@@ -259,7 +259,7 @@ a fixed real linear functional ell set
    R(t) = [ sum_{x in z+Lambda} exp( -<x,(T+tQ)x>/2 + ell(x) ) ]
           / [ sum_{x in Lambda} exp( -<x,(T+tQ)x>/2 ) ].
 
-The lattice-Gaussian inequalities of Regev and Stephens-Davidowitz
+The lattice-Gaussian ineualities of Regev and Stephens-Davidowitz
 (Theorem 2.1 and Corollary 3.2) give that the numerator law has
 covariance dominating the centered denominator law (companion
 Bessel paper, Lemma 2.1). With m_{z,ell} and Cov_{z,ell} the
@@ -552,7 +552,7 @@ An operator-norm bound does not control the determinant; a
 summable singular-value bound is needed. For an integer m >= 1,
 restrict f in V_p by requiring its Neumann cosine coefficients on
 every unit cell, for 0 <= k_1, k_2 <= m, to vanish. The
-codimension is at most C N_d m^2, and the cell spectral inequality
+codimension is at most C N_d m^2, and the cell spectral ineuality
 gives |f|_{L^2(Q)} <= C m^{-1} |nabla f|_{L^2(Q)}. Choose nested
 cell-unions N_1 in N_2 in N_3, each containing H, separated from p,
 with fixed-width buffers between successive N_j and between H and
@@ -560,7 +560,7 @@ the relative boundary of N_1 (neighbourhoods may meet free walls).
 Put E_j(f) = sum_{Q in N_j} ( a^{-1} integral_Q |nabla f|^2 + f_Q^2 ).
 Use cutoffs equal to one on N_j, supported in N_{j+1}, with
 uniformly bounded gradients. The cutoff estimate and the spectral
-inequality give, with f the original function in both tests,
+ineuality give, with f the original function in both tests,
 
    E_1(f) <= C m^{-2} E_2(f) <= C m^{-4} E_3(f) <= C m^{-4} |f|_d^2.
 
@@ -1600,7 +1600,7 @@ X is symmetric and bounded, so its Lee-Yang product representation
 
 with B >= 0, sum_j y_j^{-2} < inf, B + sum_j y_j^{-2} = (1/2) E X^2,
 zeros listed as pairs +-i y_j. Since uniform rotation gives
-E X^2 = v/2, the inequality 1+u <= e^u yields
+E X^2 = v/2, the ineuality 1+u <= e^u yields
 
    E e^{tX} <= e^{t^2 v/4},   t in R.          (10.3)
 
@@ -1920,7 +1920,7 @@ squared Sobolev weights is (1+|nu|^2)^{-1} -> 0.
 Given eta > 0 choose R_j < inf so the sum of Markov bounds from
 (10.14) with thresholds R_j^2 is less than eta. With probability at
 least 1 - eta all | chi_j S_n |_{H^{-2}} <= R_j hold. The set of
-distributions satisfying these inequalities is relatively compact in
+distributions satisfying these ineualities is relatively compact in
 H^{-3}_loc(D): compact embedding gives a convergent subsequence for
 each localized field, a diagonal choice converges for all j, and the
 identities chi_j (chi_{j+1} S_n) = chi_j S_n pass to limits since
@@ -1991,7 +1991,7 @@ REFERENCES
      superfluid density of two-dimensional superfluids. Phys. Rev.
      Lett. 39 (1977), 1201-1205.
 
-[10] C. M. Newman. Inequalities for Ising models and field theories
+[10] C. M. Newman. Ineualities for Ising models and field theories
      which obey the Lee-Yang theorem. Comm. Math. Phys. 41 (1975),
      1-9.
 
@@ -2001,7 +2001,7 @@ REFERENCES
 [12] C. M. Newman and W. Wu. Lee-Yang property and Gaussian
      multiplicative chaos. Comm. Math. Phys. 369 (2019), 153-170.
 
-[13] O. Regev and N. Stephens-Davidowitz. An inequality for Gaussians
+[13] O. Regev and N. Stephens-Davidowitz. An ineuality for Gaussians
      on lattices. SIAM J. Discrete Math. 31 (2017), 749-757.
 
 [14] D. van Engelenburg and M. Lis. An elementary proof of phase