diff --git a/fano_cartier_sections_v2.txt b/fano_cartier_sections_v2.txt
index a9bcced..69796c4 100644
--- a/fano_cartier_sections_v2.txt
+++ b/fano_cartier_sections_v2.txt
@@ -16,7 +16,11 @@ Corollary (real boundaries). Over the complex numbers the conclusion of the unif
 
 The conjecture behind these statements asks for bounded complements in the sense of Shokurov's 2004 formulation. The log canonical version was proved by Birkar in 2019: for every dimension bound and every fixed rational coefficient set there is a uniform index J and an effective divisor Lambda at least B with (X, Lambda) log canonical and J(K_X + Lambda) linearly trivial. The klt version is strictly stronger, because a klt complement has strictly positive discrepancies rather than merely nonnegative ones; bigness of an index is compatible with discrepancy having collapsed to zero along some direction.
 
-Bingyi Chen has shown that the klt version of bounded complements is equivalent to uniform Cartier sections for Fano type contractions of bounded base dimension. That equivalence carries the theorem above for pairs into the birational statement, using only base dimension d as the input. No Cartier-index hypothesis on the nef part of any boundary enters the equivalence. A real boundary passes through the same equivalence in Chen's real-pair form; the boundary is not approximated, and real-Cartier integral Weil divisors are already rationally Cartier on the models where the argument is applied.
+Shokurov put the conjecture this way at Oberwolfach in 2004:
+
+    Let d > 0 be a positive integral number and epsilon >= 0 be a real number. Then there exists a finite set N_d(epsilon) of positive integral numbers such that, for any contraction of normal algebraic varieties X -> Z and some complementary index n in N_d(epsilon), if D is a general element of the linear system of -nK, then the log pair (X, D/n) has only epsilon-log canonical singularities in the mobile sense and locally over Z, under the following conditions: (a) dim X <= d; (b) X has only epsilon-log canonical singularities; and (c) -K is nef and big over Z.
+
+Bingyi Chen has shown that the klt version of bounded complements is equivalent to uniform Cartier sections for Fano type contractions of bounded base dimension. That equivalence carries the main theorem for pairs into the birational statement, using only base dimension d as the input. No Cartier-index hypothesis on the nef part of any boundary enters the equivalence. A real boundary passes through the same equivalence in Chen's real-pair form; the boundary is not approximated, and real-Cartier integral Weil divisors are already rationally Cartier on the models where the argument is applied.
 
 ./shokurov-bounded-complements-oberwolfach-2004.pdf
 ./birkar-boundedness-fano-annals-2021.pdf
@@ -30,17 +34,17 @@ Theorem (bounded lc complements). For each integer d >= 1, each set R of rationa
 
 ./birkar-boundedness-fano-annals-2021.pdf
 
-Applied to the failing sequence of the failing sequence just described, this theorem supplies an lc complement Lambda_i on X_i of uniformly bounded index J(d), with coefficients in a fixed finite set, and with a uniform positive discrepancy gap away from the base point z_i. The support of Lambda_i is the correct support, and every coefficient lies in a bounded set. The only defect is that some coefficients may fail the strict positivity required for klt.
+Applied to the failing sequence just described, this theorem supplies an lc complement Lambda_i on X_i of uniformly bounded index J(d), with coefficients in a fixed finite set, and with a uniform positive discrepancy gap away from the base point z_i. The support of Lambda_i is the correct support, and every coefficient lies in a bounded set. The only defect is that some coefficients may fail the strict positivity required for klt.
 
 The birational statement is proved by an induction whose first half reduces each failing example to a local problem and whose second half excludes the local problem outright.
 
-Reduction to the local model. The first normal klt cover of the pair (X_i, Lambda_i), ramified along the support of Lambda_i, has degree bounded by the index J(d) and therefore by a constant depending only on d. Its generic fibre over z_i is a normal complex klt germ x_0 in S_0 of dimension d, and the covering group restricts to a finite group Gamma_0 acting on S_0 by a quasi-etale map with epsilon-lc quotient. A bounded-order further cover replaces Gamma_0 by a strictly cyclic group Gamma and produces a germ S_i of bounded structure. The character of the canonical sheaf is unchanged by the covering. There is a nowhere-vanishing local canonical eigen-generator omega_i of some character chi_i, and a nonzero regular eigenfunction h_i of the same character. The pair (S_i, div(h_i)) is lc, and there is a centred quasi-monomial valuation T_i with A_{S_i}(T_i) = T_i(h_i) = 1. The two cases of birational contractions (nowhere-exceptional and general) both fall under this construction: the first uses the canonical cover, and the second uses the signed anticanonical Z-graded torsor, the total space being the affine cone on the anticanonical section ring with the positive chamber of the torsor realising the germ as a Zariski open subset of codimension at least two.
+Reduction to the local model. The normal klt cover of the pair (X_i, Lambda_i), ramified along the support of Lambda_i, has degree bounded by the index J(d) and therefore by a constant depending only on d. Its generic fibre over z_i is a normal complex klt germ x_0 in S_0 of dimension d, and the covering group restricts to a finite group Gamma_0 acting on S_0 by a quasi-etale map with epsilon-lc quotient. A bounded-order further cover replaces Gamma_0 by a strictly cyclic group Gamma and produces a germ S_i of bounded structure. The character of the canonical sheaf is unchanged by the covering. There is a nowhere-vanishing local canonical eigen-generator omega_i of some character chi_i, and a nonzero regular eigenfunction h_i of the same character. The pair (S_i, div(h_i)) is lc, and there is a centred quasi-monomial valuation T_i with A_{S_i}(T_i) = T_i(h_i) = 1. The two cases of birational contractions (nowhere-exceptional and general) both fall under this construction: the first uses the canonical cover, and the second uses the signed anticanonical Z-graded torsor, the total space being the affine cone on the anticanonical section ring with the positive chamber of the torsor realising the germ as a Zariski open subset of codimension at least two.
 
 Failure of a klt n-complement on Z_i pulls back to a quantitative saturation condition on S_i. For each fixed positive integer m there is an index i(m) such that, for every i >= i(m) and every regular semi-invariant s of character chi_i^m,
 
     T_i(s) >= m.
 
-The valuation is constructed from the two slopes on the signed torsor: the positive slope is a rational Gauss weight of the anti-canonical grading, the negative slope is the order of the diagonal-vanishing negative section, and the canonical discrepancy evaluates to zero on the first and to the character h on the second, so the two slopes match.
+The valuation is constructed from the two slopes on the signed torsor: the positive slope is a rational Gauss weight of the anti-canonical grading, the negative slope is the order of the diagonal-vanishing negative section, and the canonical discrepancy equals one on the first and the h-value equals one on the second, so the two slopes match.
 
 Saturation obstruction (pinning). Fix an integer n > 0 and a real epsilon > 0. There is no sequence, in dimension n, of the following shape. For each index i there is a normal complex klt germ x_i in S_i of dimension n, a finite cyclic group Gamma_i acting on S_i by a quasi-etale map that fixes x_i and has epsilon-lc quotient S_i / Gamma_i, a nowhere-vanishing local canonical eigen-generator omega_i of some character chi_i, a nonzero regular eigenfunction h_i of the same character chi_i, an lc pair (S_i, div h_i), a centred quasi-monomial valuation T_i with A_{S_i}(T_i) = T_i(h_i) = 1, and additionally the saturation condition: for every fixed integer m > 0 there is an index i(m) such that T_i(s) >= m for every regular semi-invariant s of character chi_i^m and every i >= i(m).
 
@@ -50,11 +54,11 @@ Generalized pairs. A generalized pair consists of three pieces X, B, M where X i
 
 A generalized Calabi-Yau pair over a base S is data X, B, M equipped with a contraction X -> S such that K_X + B + M_X is Q-linearly trivial over S. The generalized pair is klt (log canonical, epsilon-lc) when the generalized discrepancies satisfy the corresponding strict/nonnegative/epsilon bounds. A generalized pair is of Fano type over S when there exists an effective rational boundary C on X, klt with C together with B + M_X, and with minus (K_X + B + C + M_X) ample over S; this auxiliary boundary replaces the ordinary Fano condition when the nef part is present.
 
-Discriminant and moduli. Let X -> S be a generalized Calabi-Yau over S, with base S a separated normal variety and X -> S a morphism whose general fibre is klt. The discriminant b-divisor on S has, at a prime P of a sufficiently high model of S, the coefficient equal to the infimum over divisorial valuations T of X restricting to e times ord_P, of the rational number a(T)/e, computed in the discrepancy convention. The infimum is finite and rational. The moduli b-divisor is b-nef and Q-Cartier by the generalized canonical bundle formula, which holds without any Cartier-index hypothesis on the nef part. Vertical trace coefficients exceeding one are handled by composing with a negative effective divisor on the base, which changes the discriminant but leaves the moduli part intact. Composition through birational operations that preserve the Calabi-Yau discrepancy data does not modify the discriminant. Local Kawamata-Viehweg vanishing for the twisted ideal on a model of X -> S is compatible with the same formula, and the twising passes through the moduli part by the projection formula.
+Discriminant and moduli. Let X -> S be a generalized Calabi-Yau over S, with base S a separated normal variety and X -> S a morphism whose general fibre is klt. The discriminant b-divisor on S has, at a prime P of a sufficiently high model of S, the coefficient equal to the infimum over divisorial valuations T of X restricting to e times ord_P, of the rational number a(T)/e, computed in the discrepancy convention. The infimum is finite and rational. The moduli b-divisor is b-nef and Q-Cartier by the generalized canonical bundle formula of Filipazzi, refining the klt-case formula of Ambro and Kawamata, which holds without any Cartier-index hypothesis on the nef part. Vertical trace coefficients exceeding one are handled by composing with a negative effective divisor on the base, which changes the discriminant but leaves the moduli part intact. Composition through birational operations that preserve the Calabi-Yau discrepancy data does not modify the discriminant. Local Kawamata-Viehweg vanishing for the twisted ideal on a model of X -> S is compatible with the same formula, and the twisting passes through the moduli part by the projection formula.
 
 Proposition (comparison along long parameter intervals). Let X_i -> S_i be a sequence of Fano type contractions of bounded dimension. Suppose a_{i, t} are effective generalized Calabi-Yau data over S_i, forming a finite rational piecewise affine family in t on 0 <= t <= L_i, or on -L_i <= t <= L_i with L_i -> infinity. Pieces and endpoints are rational, and a_{i, t}(T) is concave in t at every divisorial valuation T so tested. Assume a_{i, 0} is klt and, when the sequence is modified, one of the following two conditions holds, tested on every subsequence. In the one-sided case: a_{i, t} is nonincreasing in t for every divisorial T, and a_{i, t}(T) / a_{i, 0}(T) -> 1 for every fixed t whenever a_{i, 0}(T) -> 0. In the two-sided case: there is a fixed delta > 0 with a_{i, t}(T) / a_{i, 0}(T) -> 1 for every fixed signed t whenever a_{i, 0}(T) <= delta. Then on every bounded parameter range the data a_{i, t} are eventually klt over the generic point of S_i, and the discriminant discrepancies b_{i, t} on S_i satisfy the corresponding comparison. In the one-sided case, b_{i, t} is nonincreasing and b_{i, t}(P) / b_{i, 0}(P) -> 1 whenever b_{i, 0}(P) -> 0. In the two-sided case, some fixed delta' > 0 gives the same convergence whenever b_{i, 0}(P) <= delta'. The conclusion holds on every subsequence and every birational model of S_i.
 
-The proof extracts all exceptional divisors of a distinguished lc place whose discrepancy against a_{i, 0} drops below a small fixed constant; finiteness of the list uses nonnegativity against the snc structure of the ambient chart and positivity on finitely many strata. The extraction preserves Fano type because it is a birational modification of a Fano type variety over S_i. Concavity at fixed outer parameter together with the one-sided ratio hypothesis forces the extracted divisors to keep positive discrepancy at interior parameters, and the ratio estimate permits no growth of the extracted discrepancy function. Running a K-MMP over S_i on the extracted model then passes to a Mori step X_{i, 1} -> W_{i} over S_i, still Fano type, with base dimension strictly smaller. Discrepancy functions of the discriminant on W_i are discriminants of the Mori step, by the generalized canonical bundle formula. Induction on base dimension supplies the two-sided conclusion. Bounded denominators of the extra data are inspected at the terminal birational case, and no index-boundedness of M is ever required.
+The proof extracts all exceptional divisors of a distinguished lc place whose discrepancy against a_{i, 0} drops below a small fixed constant; finiteness of the list uses nonnegativity against the snc structure of the ambient chart and positivity on finitely many strata. The extraction preserves Fano type because it is a birational modification of a Fano type variety over S_i. Concavity at fixed outer parameter together with the one-sided ratio hypothesis forces the extracted divisors to keep positive discrepancy at interior parameters, and the ratio estimate permits no growth of the extracted discrepancy function. Running a K-MMP over S_i on the extracted model, available by the minimal model theorem of Birkar, Cascini, Hacon and McKernan, then passes to a Mori step X_{i, 1} -> W_{i} over S_i, still Fano type, with base dimension strictly smaller. Discrepancy functions of the discriminant on W_i are discriminants of the Mori step, by the generalized canonical bundle formula. Induction on base dimension supplies the two-sided conclusion. Bounded denominators of the extra data are inspected at the terminal birational case, and no index-boundedness of M is ever required.
 
 ./filipazzi-generalized-canonical-bundle-2020.pdf
 ./birkar-cascini-hacon-mckernan-2010.pdf
@@ -62,13 +66,13 @@ The proof extracts all exceptional divisors of a distinguished lc place whose di
 
 Exhaustion at an lc place. Let a be an effective generalized Calabi-Yau over a DVR S, with klt generic fibre and closed point z_0 of S at the closed fibre. Either (i) a is klt over some open neighbourhood of z_0, and then the closed point itself is not the locus of any sub-klt behaviour; or (ii) all sub-klt behaviour is realised at a single primitive divisorial valuation T with a(T) tiny, and (S, z_0) can be blown up so that the sub-klt locus is a subscheme supported on the closed fibre. The proof extracts every prime of discrepancy smaller than a small constant against the data a, obtains finitely many of them because the extraction is directed by a bounded process, and observes that the extraction is controlled because compressions with respect to the background linear system are. The refinement preserves klt-ness of a over the open locus.
 
-Uniform charts. At the vertical place just constructed, after a bounded-degree extension of the residue field k(W) and a chosen extension of the DVR and of the place, there is a toroidal chart representing the place, with nonnegative integral coordinates x = (x_1, ..., x_m) and a distinguished coordinate e_*, subject to the linear equations M x = r e_*. The matrix M ranges over a finite list of integral matrices of bounded size; the integer r is integral and may be huge; e_* is the value of the new uniformiser, and the original uniformiser has value rho e_* with a bounded positive rho. Nonzero integral solutions on the cone give divisorial valuations that are integral multiples of primitive orders. For horizontal places the chart is an orthant on a bounded generic resolution, without e_*. The charts are constructed by toroidalisation of the morphism family after a bounded base change, and the coefficients of M are bounded because the toroidal structure has bounded complexity, independent of the trait contact r. For any effective generalized Calabi-Yau data A on the total space over S, evaluations on these cones are finite, continuous, homogeneous and piecewise linear, and are Lipschitz in x at fixed e_* with a constant independent of r, of the vertical coefficients, of the nef part index, and of A.
+Uniform charts. At the vertical place just constructed, after a bounded-degree extension of the residue field k(W) and a chosen extension of the DVR and of the place, there is a toroidal chart representing the place, with nonnegative integral coordinates x = (x_1, ..., x_m) and a distinguished coordinate e_*, subject to the linear equations M x = r e_*. The matrix M ranges over a finite list of integral matrices of bounded size; the integer r is integral and may be huge; e_* is the value of the new uniformiser, and the original uniformiser has value rho e_* with a bounded positive rho. Nonzero integral solutions on the cone give divisorial valuations that are integral multiples of primitive orders. For horizontal places the chart is an orthant on a bounded generic resolution, without e_*. The charts are constructed by weak toroidalisation, in the sense of Abramovich, Denef and Karu, of the morphism family after a bounded base change, and the coefficients of M are bounded because the toroidal structure has bounded complexity, independent of the trait contact r. For any effective generalized Calabi-Yau data A on the total space over S, evaluations on these cones are finite, continuous, homogeneous and piecewise linear, and are Lipschitz in x at fixed e_* with a constant independent of r, of the vertical coefficients, of the nef part index, and of A.
 
 ./abramovich-denef-karu-weak-toroidalization-1010.6171v3.pdf
 
 The Lipschitz control is sharpened by a degree argument. A rational function on the chart with pole degree at most m has angular Lipschitz constant at most C m. Clearing the algebraic equation satisfied by such a function over the chart coordinates gives a polynomial equation with coefficients of degree at most C (m + 1). Giving all coordinates strictly positive rationally independent weights and taking the leading exponent gamma of the function, at least two distinct powers attain the least weight, so the phases satisfy an equality among the coefficient degrees, which forces |gamma| <= C (m + 1) uniformly in the choice of positive weights. Differentiating by fixed collections of logarithmic vector fields of total order O(m + 1) then isolates a monomial leading term with a local unit coefficient, descends from completion to the local ring by faithful flatness, and yields a jet of bounded order. This gives the Lipschitz estimate. The horizontal case replaces the vertical contact by a b-order function on a bounded generic resolution and observes the same rule.
 
-Seifert bundle. For a projective normal variety V and a Q-Cartier rational divisor H on V, there is a projective normal compactified Seifert bundle p: Y -> V with disjoint Q-Cartier zero and infinity sections E_0 and D, a rational fibre coordinate z, and compatible canonical divisors, such that
+Seifert bundle. Following Kollar, for a projective normal variety V and a Q-Cartier rational divisor H on V, there is a projective normal compactified Seifert bundle p: Y -> V with disjoint Q-Cartier zero and infinity sections E_0 and D, a rational fibre coordinate z, and compatible canonical divisors, such that
 
     div(z) = p^* H + E_0 - D,
     K_Y + E_0 + D = p^*(K_V + R(H)),
@@ -99,11 +103,11 @@ Cyclic quotient. Let F be a smooth projective variety of Fano type, let G be a Q
 
     K_Y + (1 - I_i) E_0 + (1 + I_i) D ~_Q 0.
 
-The contraction C_i of the zero section in Y over F, when the divisor (1 + I_i) D descends to a model of F, is a projective normal variety of Fano type over a model of F, with discrepancy of the contracted zero section positive and growing with i. The relative contraction exists because the divisor (1 + I_i) D is semiample along the fibres by the bundle identity and the basepoint-free theorem on the model. Uniform products over the z-coordinates, with the operators z_i partial over z_i minus l, bound the poles of invariant rational functions in a bounded slab.
+The contraction C_i of the zero section in Y over F, when the divisor (1 + I_i) D descends to a model of F, is a projective normal variety of Fano type over a model of F, with discrepancy of the contracted zero section positive and growing with i. The relative contraction exists because the divisor (1 + I_i) D is semiample along the fibres by the bundle identity and the basepoint-free theorem on the model. Uniform products over the z-coordinates, with product operators built from the derivations z_i d/dz_i minus integers, bound the poles of invariant rational functions in a bounded slab.
 
 Lemma (phase nonvanishing). Let X -> S be a sequence of birational Fano type contractions between projective normal varieties of bounded total dimension. Let a be an effective generalized lc Calabi-Yau over S with a rational lc effective boundary on X over S; let D be an effective rational nef and big divisor with b = a + D klt, integral on primitive prime orders, of effective trace. Let H be an effective divisor supported in the non-klt locus of b whose trace is very ample on X with generic degree d of order L, with L tending to infinity. Suppose: (i) a is klt for a small fixed positive theta; (ii) the Seifert algebra of L H over d has moduli index bounded by C L/d from above and zero nef discrepancy deficit after a bounded base change; (iii) every floor prime of b with 0 < L D(.) < 1 couples via local vanishing to a prime picked up by the twisted ideal; and (iv) on every sequence of primitive divisorial tests, the signed phase of the Cartier order N = (L/d) H^* satisfies |theta(T)| = O(a(T)) whenever a(T) -> 0. Fix a constant nu. Then after a bounded base change, with a uniform positive lower bound on degrees given by b - (d/L) H^*, the linear H-system contains a section whose twisted form has positive order at every test with a < nu/L.
 
-The proof reduces by induction on the dimension of the total space and of the base. Place a small-discriminant valuation in the uniform chart with the height u = w(H) bounded above by A at fixed angular test. If the restriction to the fibre is trivial, u is a positive integer, which contradicts the unbounded-index obstruction as soon as A is small. So restriction is nontrivial. If u/A is bounded below, the phase is bounded away from zero and the logarithmic Kawamata-Viehweg vanishing on the Seifert bundle already supplies the section from the nef-ness of b. If u/A tends to zero with A, the exceptional divisor correction has matching phase in the next lower Mori step, and induction on dimension of the base applies, giving bounded-degree sections that pull back to the original family. In both cases the required section exists.
+The proof reduces by induction on the dimension of the total space and of the base. Place a small-discriminant valuation in the uniform chart with the height u = w(H) bounded above by A at fixed angular test. If the restriction to the fibre is trivial, u is a positive integer, which contradicts the unbounded-index obstruction as soon as A is small. So restriction is nontrivial. If u/A is bounded below, the phase is bounded away from zero and the logarithmic Kawamata-Viehweg vanishing on the Seifert bundle already supplies the section from the nef-ness of b. If u/A tends to zero with A, the exceptional divisor correction has matching phase in the next lower Mori step, and induction on dimension of the base applies, giving bounded-degree sections that pull back to the original family. In both cases the required section exists; the log canonical adjunction applied on the fibres is Kawakita's inversion of adjunction on log canonicity.
 
 ./kawakita-inversion-adjunction-2007.pdf
 
@@ -237,7 +241,7 @@ The gap proof adapts a basis to the w-flag and the v-flag simultaneously. Lifts
 
     (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).
+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.
 
 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.
 
@@ -245,7 +249,7 @@ The proof chooses an lc boundary supported on the coefficient-one locus of a log
 
 Theorem (exact minimizer via a single rational height). Fix a rational height function b on Q. There are a rational lambda > 0, a klt pair (S, Delta_lambda), and a unique (up to scaling on each ray of Q) minimizing centered valuation w in Q, such that w is the normalized-volume minimizer for (S, Delta_lambda). The minimizer is equivariant for the stabilizer of the data. The same lambda works simultaneously for all competitors: no per-competitor choice of the perturbation parameter is required.
 
-The proof has three parts. Attainment and uniqueness: compactness gives a positive minimum of b on Q; a minimizing sequence has bounded volume, so uniform centering bounds v(m) away from zero, and a subsequence converges in Q to a centered valuation. If two minimizing rays existed, normalize each by b = 1 and interpolate to u with b(u) <= 1; minimality forces vol(u) >= M^n, and the strict gap contradicts unless they agree on every regular function. Equivariance under any automorphism preserving the data follows from uniqueness. Approximation: applying the uniform retraction-and-comparison estimate to a single fixed rational lambda (not just a limiting one) gives a strictly positive excess for any competitor with e > 0, where e = A_{S, Psi}(v) is the extra discrepancy on the common slice. Choosing lambda <= a small positive delta depending on the germ alone, independently of competitors, forces e = 0 for every minimizer, so the actual klt minimizer lies exactly on the common slice and equals w.
+The proof has two parts. Attainment and uniqueness: existence of a minimizer is Blum's theorem, while compactness gives a positive minimum of b on Q; a minimizing sequence has bounded volume, so uniform centering bounds v(m) away from zero, and a subsequence converges in Q to a centered valuation. If two minimizing rays existed, normalize each by b = 1 and interpolate to u with b(u) <= 1; minimality forces vol(u) >= M^n, and the strict gap contradicts unless they agree on every regular function. Equivariance under any automorphism preserving the data follows from uniqueness. Approximation: applying the uniform retraction-and-comparison estimate to a single fixed rational lambda (not just a limiting one) gives a strictly positive excess for any competitor with e > 0, where e = A_{S, Psi}(v) is the extra discrepancy on the common slice. Choosing lambda <= a small positive delta depending on the germ alone, independently of competitors, forces e = 0 for every minimizer, so the actual klt minimizer lies exactly on the common slice and equals w.
 
 ./blum-existence-normalized-volume-minimizers-2018.pdf
 ./xu-zhuang-uniqueness-minimizer-normalized-volume-2001.05924v1.pdf
@@ -284,9 +288,9 @@ The proof uses subduction: subtract a polynomial in the X_i with the prescribed
 
 ./jonsson-mustata-valuations-quasimonomial-2012.pdf
 
-For any positive weight vector gamma the corresponding monomial filtration is bounded above and below by fixed linear reindexings of the m_x-adic filtration, so its growth exponent equals its Krull dimension n. If in_gamma I is prime, the filtration is multiplicative and separated, defining a valuation; if its value group has rational rank r, the degree-zero homogeneous fraction field is the residue field with transcendence degree n - r; this is Abhyankar equality. Local monomialization of Abhyankar valuations in characteristic zero then makes the valuation quasi-monomial.
+For any positive weight vector gamma the corresponding monomial filtration is bounded above and below by fixed linear reindexings of the m_x-adic filtration, so its growth exponent equals its Krull dimension n. If in_gamma I is prime, the filtration is multiplicative and separated, defining a valuation; if its value group has rational rank r, the degree-zero homogeneous fraction field is the residue field with transcendence degree n - r; this is Abhyankar equality. Local monomialization of Abhyankar valuations in characteristic zero then makes the valuation quasi-monomial; the quasi-monomial description, the retraction onto a monomial valuation, and the asymptotic volume of a graded sequence are developed systematically by Jonsson and Mustata.
 
-Canonical forms under degeneration. Under the hypotheses preceding the weighted-presentation lemma, S^0 has a divisor-free reflexive canonical generator omega^0. It has the G-character of omega, is homogeneous for T_w, and its weight kappa satisfies
+Canonical forms under degeneration. Following Xu and Zhuang, under the hypotheses preceding the weighted-presentation lemma, S^0 has a divisor-free reflexive canonical generator omega^0. It has the G-character of omega, is homogeneous for T_w, and its weight kappa satisfies
 
     <w, kappa> = A_S(w).
 
@@ -341,7 +345,7 @@ Proposition (high-scale pin). Consider a high stage with pre-pinning slice, work
 
 Both valuations dominate a fixed positive multiple of w on every regular function; in particular w' >= c w.
 
-Proof of the pin. Use row-reduction of the constrained slice. At stage i, work with the projection x(q) of the slice onto R^s of the selected values. Let R_i be the span of the normalized rows already imposed. Pass to subsequences where R_i and the bounded compact convex sets C_i = [0, 1] K_i converge in Hausdorff distance. If the span of the limiting C_i is strictly smaller than R_inf^perp, then for a rational unit vector u in R_inf^perp perpendicular to that span, u_u_i x(q) = o(1) uniformly on the slice. The thin-case height
+Proof of the pin. Use row-reduction of the constrained slice. At stage i, work with the projection x(q) of the slice onto R^s of the selected values. Let R_i be the span of the normalized rows already imposed. Pass to subsequences where R_i and the bounded compact convex sets C_i = [0, 1] K_i converge in Hausdorff distance. If the span of the limiting C_i is strictly smaller than R_inf^perp, then for a rational unit vector u in R_inf^perp perpendicular to that span, u_i x(q) = o(1) uniformly on the slice. The thin-case height
 
     b(q) = delta B(q) - sum_j u_{ij} y_j / L_j,