diff --git a/fano_cartier_sections_v2.txt b/fano_cartier_sections_v2.txt
index 7065887..7a44b29 100644
--- a/fano_cartier_sections_v2.txt
+++ b/fano_cartier_sections_v2.txt
@@ -131,7 +131,7 @@ denote by sigma the dual cone C-dual in Hom(M, R), and put K = Frac(R)^T. The co
 
     Delta_P = {y in Hom(M, R) : P(f) + <y, m> >= 0 for every f z^m in R}.
 
-Finitely many homogeneous algebra generators suffice to define Delta_P, and Delta_P is nonempty; adding a large positive grading handles every inequality. For rational y in Delta_P the Gauss rule
+Finitely many homogeneous algebra generators suffice to define Delta_P, and Delta_P is nonempty; adding a large positive grading handles every ineuality. For rational y in Delta_P the Gauss rule
 
     (P, y)(sum_m f_m z^m) = min over f_m nonzero of (P(f_m) + <y, m>)
 
@@ -171,7 +171,7 @@ Consequently the neutral functions of cost b in nonnegative integers are precise
 
     {g in K : g h^b in R_double} = S(b D).
 
-The proof restricts to the torus quotient and observes that the character of g xi^q h^{q m} is trivial by construction; the two inequalities are then matched by the maximum-pole description of semiample divisors.
+The proof restricts to the torus quotient and observes that the character of g xi^q h^{q m} is trivial by construction; the two ineualities are then matched by the maximum-pole description of semiample divisors.
 
 Corollary (characters for a tilted height). Suppose there are boundedly many invariant rational eigenfunctions Y_j^+ and Y_j^- of weights k_j and -k_j respectively, with
 
@@ -222,7 +222,7 @@ The compactness proof applies the retraction-error argument at psi = H. Every ze
 
     A(v) = sum_i a_i A(E_i) = 1
 
-cuts out a compact simplex, because A(E_i) > 0. The additional inequalities are linear on an adapted resolution, and finitely many cones occur. Function evaluations are monomial minima and agree on faces by localization. Uniform centering follows from the strong separation between the set {A >= 1} and the closed subset of valuations with v(m) = 0.
+cuts out a compact simplex, because A(E_i) > 0. The additional ineualities are linear on an adapted resolution, and finitely many cones occur. Function evaluations are monomial minima and agree on faces by localization. Uniform centering follows from the strong separation between the set {A >= 1} and the closed subset of valuations with v(m) = 0.
 
 Lemma (volume interpolation and a strict gap). Let w be centered, v over x, and suppose u >= t w + (1 - t) v on R for 0 < t < 1. For f nonzero with w(f) > 0,
 
@@ -235,13 +235,13 @@ If v is also centered, with t = 1/2 and v(f) >= (1 + gamma) w(f) > 0 for 0 < gam
 where c(n, gamma) > 0 depends only on n and gamma. Consequently two valuations whose volumes are within relative o(1) of the minimum on the same A = 1 common slice are multiplicatively 1 + o(1) equal on all regular functions, and a sufficiently small fixed relative error gives any prescribed fixed multiplicative closeness to one.
 
 ./pdf/li-minimizing-normalized-volumes-2018.pdf
-./pdf/li-xu-normalized-volume-inequalities-2018.pdf
+./pdf/li-xu-normalized-volume-ineualities-2018.pdf
 
 The gap proof adapts a basis to the w-flag and the v-flag simultaneously. Lifts of a homogeneous complement of a_{gamma N/4}(w), multiplied by f raised to the power floor(N / w(f)), have w-value less than (1 + gamma/4) N and v-value at least (1 + 2 gamma/3) N. These contribute at least
 
     (e^{-(1 + gamma/4)/2} - e^{-(1 + 2 gamma/3)/2})^2
 
-to a sum of squares bounding twice the u-Laplace sum, whose normalized limit is vol(u). Dividing by N^n, letting N tend to infinity, then T tend to infinity, gives the claimed c(n, gamma); the interpolation inequality for the volume functional, and the strict gap along a segment of valuations, are due to Li, and the normalized volume inequalities applied here are those of Li and Xu.
+to a sum of squares bounding twice the u-Laplace sum, whose normalized limit is vol(u). Dividing by N^n, letting N tend to infinity, then T tend to infinity, gives the claimed c(n, gamma); the interpolation ineuality for the volume functional, and the strict gap along a segment of valuations, are due to Li, and the normalized volume ineualities applied here are those of Li and Xu.
 
 Lemma (boundary encoding). Fix the common slice Q. There is a rational klt pair (S, Delta) on the germ with A_{S, Delta}(v) = 1 for every v in Q, and every v in Q remains log canonical against (S, Delta). Consequently normalized-volume minimization against A_{S, Delta} coincides with normalized-volume minimization over the common slice Q.
 
@@ -339,7 +339,7 @@ Proposition (high-scale pin). Consider a high stage with pre-pinning slice, work
 
 (i) The resulting slice contains a centred valuation. The added tests are exact on that slice; all old tests and proportionalities remain in force; and the accumulated positive symbols remain algebraically independent at the exact minimiser and at the working valuation of (iii).
 (ii) The selected values either vanish or lie on a single rational line. In the line case their proportions obey the bounded-cost bound with L_- = L. An old anchor remains the anchor; if a new anchor is chosen, its scale is L_1 = L.
-(iii) The baseline height is retained unless an anchor is first chosen, in which case it may change to a rational baseline with the bounded-baseline inequality on the new slice. At its exact minimiser v' and at an arbitrarily close rank refinement w', the anchor, when present, satisfies
+(iii) The baseline height is retained unless an anchor is first chosen, in which case it may change to a rational baseline with the bounded-baseline ineuality on the new slice. At its exact minimiser v' and at an arbitrarily close rank refinement w', the anchor, when present, satisfies
 
     |v'(Y_0^+)| >= c L_1 and |w'(Y_0^+)| >= c L_1.
 
@@ -376,7 +376,7 @@ Third clause. The saturation hypothesis demands T_i(s) >= m for every regular se
 Conclusion of the obstruction. No sequence can exist, and the pinning obstruction is proved.
 
 ./pdf/xu-zhuang-stable-degenerations-2025.pdf
-./pdf/li-xu-normalized-volume-inequalities-2018.pdf
+./pdf/li-xu-normalized-volume-ineualities-2018.pdf
 
 Completion of the birational theorem. Combine the reduction to the local model (first paragraph of the proof) with the pinning obstruction. The reduction produces, for any failing sequence of birational klt complement problems, a sequence satisfying all hypotheses of the pinning obstruction. The pinning obstruction excludes all such sequences. Hence no failing sequence exists, and there is a uniform n = n(d, epsilon) for every d and epsilon.