diff --git a/log_sobolev_subgaussian_v2.txt b/log_sobolev_subgaussian_v2.txt
index fd4d35f..07f14fd 100644
--- a/log_sobolev_subgaussian_v2.txt
+++ b/log_sobolev_subgaussian_v2.txt
@@ -46,7 +46,7 @@ For a smooth function f, its Dirichlet energy with respect to mu is
 the average squared steepness of f. It is zero exactly when f is
 constant. It measures how much f wiggles.
 
-1.3 The logarithmic Sobolev inequality (LSI).
+1.3 The logarithmic Sobolev ineuality (LSI).
 
 Saying mu satisfies an LSI with constant R means
 
@@ -59,7 +59,7 @@ information. The smallest such R is written R(mu).
 
 1.4 Why "dimension-free" is the whole point.
 
-Many inequalities in n dimensions carry constants that blow up like n,
+Many ineualities in n dimensions carry constants that blow up like n,
 sqrt(n), or worse as n grows. A constant that is completely independent
 of n is called dimension-free, or absolute if it is a fixed number like
 17. Bounds whose constants grow with n become useless for the
@@ -145,7 +145,7 @@ distance. The relative entropy of alpha with respect to beta is
                   = +infinity            when alpha is not absolutely
                                          continuous with respect to beta.
 
-Corollary 1 (Quadratic transport-entropy inequality). Under the
+Corollary 1 (Quadratic transport-entropy ineuality). Under the
 hypotheses of Theorem 1, every probability nu with finite second moment
 satisfies
 
@@ -153,7 +153,7 @@ satisfies
 
 with the same absolute constant C. This follows from the LSI through the
 Otto-Villani theorem, which converts any LSI into a
-transport-entropy inequality with matching constants; the proof at the
+transport-entropy ineuality with matching constants; the proof at the
 level of this document is in Section 5.7 and only needs an
 approximation argument to remove the smoothness of the density.
 
@@ -166,9 +166,9 @@ linear functions" into concrete concentration statements.
 
 3. HISTORY AND WHY IT MATTERS
 
-3.1 The inequality and the Gaussian.
+3.1 The ineuality and the Gaussian.
 
-The logarithmic Sobolev inequality was introduced by Leonard Gross in
+The logarithmic Sobolev ineuality was introduced by Leonard Gross in
 1975 in his study of hypercontractivity, the property that a semigroup
 pushes functions into higher integrability. For the standard Gaussian
 measure the LSI holds with R = 1, independent of dimension, and Gross
@@ -284,7 +284,7 @@ occurs:
   where phi is an eigenfunction of A_{mu_s} with eigenvalue lambda, and
   eta = mu_s. The same near-eigenfunction relation holds.
 
-In both cases eta satisfies a logarithmic Sobolev inequality with a
+In both cases eta satisfies a logarithmic Sobolev ineuality with a
 constant close to 1, and w is close to a first eigenfunction of its
 diffusion. That is the single object the rest of the proof works with.
 
@@ -469,7 +469,7 @@ squared norm int (f^2 + |Df|^2) dpi. Cutoffs, truncation of values, and
 smoothing on compact sets show this is also the space of functions whose
 weak gradient is square-integrable. A compact-test LSI extends to the
 whole space by lower semicontinuity of the entropy and the Fisher
-information, and by approximation. Every inequality below stated for
+information, and by approximation. Every ineuality below stated for
 smooth compact tests therefore holds on this space; we use this silently.
 
 5.4 The entropy-duality formula.
@@ -478,7 +478,7 @@ For a probability pi, one exact identity underpins most of the analysis:
 
   H(p | pi) = sup over bounded g of { int g dp - log int e^g dpi }.
 
-Jensen's inequality gives the upper bound (log int e^g dpi is an
+Jensen's ineuality gives the upper bound (log int e^g dpi is an
 exponential-moment), and bounded truncations of log(dp/dpi) give
 equality. Three consequences we will use.
 
@@ -544,7 +544,7 @@ at radii -> 0. The affine minorant of U gives a uniform Gaussian-times-
 linear domination, the LSI passes to the limit, and the boundary of a
 full-dimensional convex support has zero Lebesgue measure so boundary
 values do not matter. Inserting f = 1 + eps g and comparing eps^2
-coefficients yields the variance inequality. QED.
+coefficients yields the variance ineuality. QED.
 
 The lemma is the entire content of the curvature method: curvature
 D^2 V >= Id buys an LSI with constant 1. A "minimum curvature kappa"
@@ -655,7 +655,7 @@ p_r = p_0 * gamma_r and q_r = q_0 * gamma_r, then
 
 whenever the left-hand entropy at time a is finite; here gamma_r is the
 centered Gaussian of covariance r Id, and a = 0 is allowed. For the
-standard Gaussian Ornstein-Uhlenbeck semigroup the first inequality is
+standard Gaussian Ornstein-Uhlenbeck semigroup the first ineuality is
 an equality, including at time 0 when the initial entropy is finite.
 All assertions hold after averaging over conditioning labels whose
 noises are independent given those labels.
@@ -759,7 +759,7 @@ is centered and log-concave, smooth.
 
 Next, mu_s is a_s-subgaussian with a_s = a + beta_s and
 beta_s = sqrt(8s/3). For a scalar standard Gaussian Z,
-E exp( z^2 / beta_s^2 ) = 2. The elementary inequality
+E exp( z^2 / beta_s^2 ) = 2. The elementary ineuality
 
   (u+t)^2/(a+beta_s)^2 <= (a/(a+beta_s)) u^2/a^2 + (beta_s/(a+beta_s)) t^2/beta_s^2
 
@@ -780,7 +780,7 @@ s -> 0 at fixed n, with a_s -> a. QED.
 
 The whole proof of the corollary is therefore three moves: regularize
 by Gaussian convolution (which buys smoothness and curvature), invoke
-the main inequality plus Otto-Villani on the regularized law, and let
+the main ineuality plus Otto-Villani on the regularized law, and let
 the regularization vanish. The only inputs are Lemmas 1 and the data
 processing bound.
 
@@ -936,7 +936,7 @@ Gaussian integration by parts followed by Cauchy-Schwarz gives
 Hence the linear-growth entropy correction is bounded by
 delta ||Dv||_2^2 + C_delta ||v||_2^2 for every delta > 0. Gaussian LSI
 and the derivative formula, with another small Cauchy-Schwarz
-parameter, give the defective inequality.
+parameter, give the defective ineuality.
 
 The same moment estimate shows tightness of the L^2(nu) mass of any
 bounded form sequence. On a fixed ball the density is bounded above and
@@ -954,7 +954,7 @@ clipped functions gives
 
 For a nonnegative v with squared norm m <= 1, the negative part of
 v^2 log v^2 has integral at most C sqrt(m) (use -z log z <= C sqrt(z) on
-0 <= z <= 1), and m log m <= 0, so the defective inequality gives
+0 <= z <= 1), and m log m <= 0, so the defective ineuality gives
 
   int v^2 log_+ v^2 dnu <= 2(s+eps)||Dv||_2^2 + C_eps m + C sqrt(m).
 
@@ -991,11 +991,11 @@ first distributionally and then classically.
 
 Proof of existence and the Euler equation. Take nonnegative unit-norm
 maximizing tests f_j. Replacing a test by its absolute value preserves
-entropy and energy, so this loses nothing. The defective inequality of
+entropy and energy, so this loses nothing. The defective ineuality of
 Lemma 8, applied with s + eps < R, bounds the energies ||D f_j||_2.
 
 First exclude energies tending to zero; this is the only place the
-strict inequality R > P is used. Suppose eps_j^2 = ||D f_j||_2^2 -> 0,
+strict ineuality R > P is used. Suppose eps_j^2 = ||D f_j||_2^2 -> 0,
 and write c_j = E_nu f_j, h_j = (f_j - c_j)/eps_j. Poincare gives
 ||h_j||_2^2 <= P, while ||D h_j||_2^2 = 1 and 1 - c_j^2 = eps_j^2
 ||h_j||_2^2, so c_j -> 1. By compactness, after extracting, h_j -> h in
@@ -1039,7 +1039,7 @@ contradicting the maximizing property. So energies do not vanish.
 
 Hence we may extract f_j -> f strongly in L^2 with
 ||D f_j||_2^2 -> E_* > 0. Put E = ||Df||_2^2. The entropy defect and
-the sharp inequality give
+the sharp ineuality give
 
   2R E_* <= Ent(f^2) + 2s(E_* - E) <= 2R E + 2s(E_* - E).
 
@@ -1144,7 +1144,7 @@ distributionally. Its right side is in L^2(eta) because K is bounded and
 w in L^2(eta), so Lemma 3 puts w in the operator domain; no unproved
 higher integrability has been used.
 
-For the inequality, let p = g^2 eta have unit mass with g smooth compact.
+For the ineuality, let p = g^2 eta have unit mass with g smooth compact.
 Direct expansion and integration by parts give
 
   Ical(p|nu) = Ical(p|eta) + int g^2 |w|^2 deta + 2 int D(g^2).w deta
@@ -1156,7 +1156,7 @@ On the other hand H(p|nu) = H(p|eta) + int h dp. Subtracting
 
   Ical(p|nu) - 2 lambda H(p|nu) = Ical(p|eta) - 2 lambda H(p|eta).
 
-Apply the nu-inequality to the compact smooth function g f: its deficit
+Apply the nu-ineuality to the compact smooth function g f: its deficit
 is nonnegative, so the right side is nonnegative. Homogeneity and
 Sobolev approximation give R(eta) <= 1/lambda. Finally, the domain
 identity gives E_eta|Dw|^2 = lambda I - E_eta<w,Kw>, while Poincare for
@@ -1164,7 +1164,7 @@ eta gives the lower bound lambda(I - |E_eta w|^2). Subtract to obtain
 (iii). QED.
 
 The lemma does two things at once. It transfers a logarithmic Sobolev
-inequality from nu to the tilted law eta = f^2 nu (so eta inherits the
+ineuality from nu to the tilted law eta = f^2 nu (so eta inherits the
 good entropy constant), and it writes the score w as an approximate
 first eigenfunction of eta's diffusion. Everything after this point
 happens under eta.
@@ -1293,7 +1293,7 @@ where phi is a centered eigenfunction of A_{mu_s} with eigenvalue
 lambda, and I = lambda E_{mu_s} phi^2.
 
 Proof. Strict branch. Lemma 10 supplies the structural identities and
-the curvature-mean inequality. Apply Lemma 11 to sigma = eta with
+the curvature-mean ineuality. Apply Lemma 11 to sigma = eta with
 F_0 = I and D_0 = I/(2 lambda). The deficit vanishes, so its conclusion
 reads
 
@@ -1303,7 +1303,7 @@ where p = E_eta Y. Because s^{-2} - lambda/s >= 2 on J, and
 |E_eta b_s| <= C_J a sqrt(I) comes from the same proof, the score
 identity E_eta w = (p - E_eta b_s)/s yields |E_eta w|^2 <= C_J sqrt(a) I.
 Now 0 <= K_s <= s^{-1} Id, so K_s^2 <= s^{-1} K_s, and the
-curvature-mean inequality gives
+curvature-mean ineuality gives
 
   E_eta |K_s w|^2 <= s^{-1} E_eta <w, K_s w> <= (lambda/s) |E_eta w|^2.
 
@@ -1320,7 +1320,7 @@ derivative gives
 
 The right side is in L^2(mu_s), so Lemma 3 puts D phi in the
 operator domain. Pairing this display with phi and applying
-Poincare at mu_s reproduces the curvature-mean inequality
+Poincare at mu_s reproduces the curvature-mean ineuality
 E_eta <w, K_s w> <= lambda |E_eta w|^2.
 
 The coordinate function Y also sits in the operator domain, because
@@ -1579,7 +1579,7 @@ coordinate and summing, (7.7) and (7.8) show
 
 By Lemma 5 the left side equals 2 d'(T) for almost every T. The error
 is uniform for T in [0, M], and d(0) = 0. Integrating the resulting
-differential inequality proves (7.3). QED.
+differential ineuality proves (7.3). QED.
 
 
 7.2 Entropy loss under a Gaussian observation.
@@ -1670,7 +1670,7 @@ conditional signal, integrating the Gaussian kernel produces
   || h/phi - 1 ||_{L^2(phi)}^2 = E[exp(delta X_c . X_c') | H_i] - 1,
 
 where X_c' is a conditionally independent copy. The linear term
-vanishes because X_c is centered. The inequality
+vanishes because X_c is centered. The ineuality
 |exp(t) - 1 - t| <= (t^2/2) exp(|t|), together with
 |X_c . X_c'| <= (|X_c|^2 + |X_c'|^2)/2, bounds the last expression by
 C delta^2 times a squared small conditional exponential-square moment.
@@ -1681,7 +1681,7 @@ The preceding bounds therefore give
 This includes the zero-dimensional observed subspace, where the loss is
 zero. Summing the m steps loses O_{N,b,p}(u delta).
 
-Apply the anisotropic curvature inequality of Lemma 1 to the final
+Apply the anisotropic curvature ineuality of Lemma 1 to the final
 posterior pair: on each realized history the covariance of the retained
 observations has curvature at least B. For fixed N and u,
 
@@ -1858,7 +1858,7 @@ moment was proved in Lemma 15. Since H_z <= theta(z) H_0, it gives
 
   int_0^z m(r) dr >= (1 - theta(z))/lambda,   0 < z < 1/4.
 
-The reverse limiting inequality follows from (7.21), 0 <= m <= 1, and
+The reverse limiting ineuality follows from (7.21), 0 <= m <= 1, and
 reverse Fatou. Hence
 
   int_0^z m(r) dr -> 1 - e^{-z}.   (7.22)
@@ -2005,7 +2005,7 @@ and the conditional affine-prediction bound give
 
   I (p_1(r) - m(r)) >= E tr(C_r^2 (Id + C_r)^{-1}).
 
-Proposition 17 makes the left side o(I). The scalar inequality
+Proposition 17 makes the left side o(I). The scalar ineuality
 t^2/(1+t) >= epsilon t/(1+epsilon) again gives (7.27).
 
 Bounding the dimensions of the outlier spaces. Let
@@ -2031,7 +2031,7 @@ Strict branch. Form the joint law by first drawing Y_0 from eta and
 then X from the original posterior kernel. Its entropy relative to the
 original joint law of (X, X + sqrt(s) G) equals H_0 = I/(2 lambda).
 Data processing bounds the entropy of its X-marginal relative to mu by
-H_0. The entropy variational inequality applied to (7.30) gives
+H_0. The entropy variational ineuality applied to (7.30) gives
 
   tr(QD) <= C a^2 (H_0 + l).   (7.31)
 
@@ -2052,7 +2052,7 @@ V_epsilon = {0} suffices. QED.
 8. A SYMMETRIC HIERARCHY OF NEARLY FIRST EIGENFUNCTIONS
 
 The prediction curve measures much more than the size of a conditional
-mean: its Bessel inequality is asymptotically saturated at every fixed
+mean: its Bessel ineuality is asymptotically saturated at every fixed
 derivative order. That saturation yields symmetric tensor fields T_l
 whose derivatives remain nearly first eigenfunctions, and every tensor
 has negligible energy in the curvature directions of mu_s. Throughout,
@@ -2097,7 +2097,7 @@ with rank Pi <= c I,
 The o(.) errors are uniform over fixed l and over the finite lists used
 below.
 
-8.2 Bessel inequality for the prediction.
+8.2 Bessel ineuality for the prediction.
 
 The proof begins with a Gaussian-chaos decomposition of the doubly
 smoothed prediction.
@@ -2121,7 +2121,7 @@ onto that chaos is
 
 where the vector output slot is not contracted. By the Hermite isometry,
 the squared norm of this projection equals r^j |D^j g_r(y)|^2 / j!.
-Bessel's inequality, integrated in y ~ eta, gives (8.8). QED.
+Bessel's ineuality, integrated in y ~ eta, gives (8.8). QED.
 
 
 8.3 Cross-covariance and spectral smoothing.
@@ -2587,7 +2587,7 @@ auxiliary constant used below.
 9.1 The band reduction.
 
 Diagonal choice. For each finite integer K, impose the conclusions of
-Proposition 19 through level 3K, the fixed-projection inequalities for
+Proposition 19 through level 3K, the fixed-projection ineualities for
 rank coefficients at most K, and the Hermite expansion bounds for every
 starting level 0 <= l < 2K and every truncation order 0 <= m <= K, on
 time intervals 0 <= t <= K. Only levels through 3K enter these
@@ -2675,7 +2675,7 @@ so
 
   ||D R||_2^2 >= 1 + o(1).   (9.10)
 
-The upper bound is the elementary inequality
+The upper bound is the elementary ineuality
 ||D R||_2^2 <= ||D Psi||_2^2 = 1 + o(1), following from the chain rule
 D rho = D R · R + R · D R applied to the positive square root and the
 normalized energy of Psi in (9.3). Consequently
@@ -2870,7 +2870,7 @@ Shifting the band by a fixed number of levels changes its endpoints by
 O(1/k), so the first line of (9.45) holds.
 
 Comparing the arrays (H_i R)_i and (R H_i)_i proves the second line of
-(9.45) by the same inequality, using the summed commutator estimate in
+(9.45) by the same ineuality, using the summed commutator estimate in
 (9.21).
 
 Bounded columns. Write H v = (H_i v)_i, S = H^* H, J = (1 + S)^{-1},
@@ -3076,7 +3076,7 @@ scalar Holder gives
     <= prod_{j=1}^m ||X_j||_{p_j, tau},  sum 1/p_j = 1/p.   (9.70)
 
 All bounds use the actual weights beta^2, with no factor depending on a
-block dimension. The finite-matrix Schatten inequality follows by the
+block dimension. The finite-matrix Schatten ineuality follows by the
 same singular-value three-lines argument as in Lemma 25, interpolating
 endpoint trace and operator norms and applying duality.
 
@@ -3249,7 +3249,7 @@ The frozen diffusion law makes the field R(Y_t) close to a matrix
 lognormal of known law. The final contradiction compares two bounds for
 the mutual information that a bounded Gaussian observation of R(Y_t)
 carries about the diffusion path. One bound comes from the logarithmic
-Sobolev inequality of eta. The other comes from the frozen law, via a
+Sobolev ineuality of eta. The other comes from the frozen law, via a
 scalar Gaussian mixture whose scale integral diverges.
 
 10.1 Setup of the observation.