diff --git a/fano_cartier_sections_v2.txt b/fano_cartier_sections_v2.txt
index 69796c4..7065887 100644
--- a/fano_cartier_sections_v2.txt
+++ b/fano_cartier_sections_v2.txt
@@ -22,9 +22,9 @@ Shokurov put the conjecture this way at Oberwolfach in 2004:
 
 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
-./chen-boundedness-klt-complements-2403.01154v4.pdf
+./pdf/shokurov-bounded-complements-oberwolfach-2004.pdf
+./pdf/birkar-boundedness-fano-annals-2021.pdf
+./pdf/chen-boundedness-klt-complements-2403.01154v4.pdf
 
 The remainder is the proof of the birational klt complement theorem. It proceeds by contradiction. Suppose no n(d, epsilon) exists. Then there is a sequence of birational Fano type contractions X_i -> Z_i of dimensions d_i bounded by d, with closed points z_i of Z_i, such that the least positive integer n_i for a klt complement over a neighbourhood of z_i tends to infinity along the sequence.
 
@@ -32,7 +32,7 @@ Birkar's bounded log canonical complements theorem reads as follows.
 
 Theorem (bounded lc complements). For each integer d >= 1, each set R of rational numbers in [0, 1] satisfying the descending chain condition, and each real epsilon > 0, there is a positive integer J = J(d, R, epsilon) such that the following holds. Let (X, B) be a projective log pair of dimension d, with coefficients of B in R, klt and epsilon-lc and of Fano type. Then there is an effective rational divisor Lambda on X with Lambda >= B, for which (X, Lambda) is log canonical and J(K_X + Lambda) is linearly trivial.
 
-./birkar-boundedness-fano-annals-2021.pdf
+./pdf/birkar-boundedness-fano-annals-2021.pdf
 
 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.
 
@@ -60,15 +60,15 @@ Proposition (comparison along long parameter intervals). Let X_i -> S_i be a seq
 
 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
-./ambro-kawamata-viehweg-vanishing-2025.pdf
+./pdf/filipazzi-generalized-canonical-bundle-2020.pdf
+./pdf/birkar-cascini-hacon-mckernan-2010.pdf
+./pdf/ambro-kawamata-viehweg-vanishing-2025.pdf
 
 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 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
+./pdf/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.
 
@@ -84,7 +84,7 @@ where R(H) is the ramification divisor defined by
 
 The divisor D is relatively ample; when H is nef, D is nef; when H is nef and big, D + p^* H is nef and big. Only finitely many terms of R(H) are nonzero.
 
-./kollar-seifert-bundles-2004.pdf
+./pdf/kollar-seifert-bundles-2004.pdf
 
 To construct the bundle, use the two affine charts defined by the algebras of rational monomials g z^j with div(g) + j H >= 0, for j >= 0 and j <= 0 respectively. If l H = div(b) locally, the charts are finite cyclic quotients of ordinary line charts with coordinates b^{1/l} and z/b^{1/l}. The construction gives the two sections, their Q-Cartier property, and the first identity. Restriction of D to itself is numerically H, restriction to E_0 is trivial, and relative ampleness gives the positivity statement. The relative canonical divisor computed via d log z gives the second identity with coefficients exactly 1 - 1/r_P.
 
@@ -95,7 +95,7 @@ Discrepancy on the Seifert bundle. Let B be an effective rational boundary on V
 
 where def(v) is a nonnegative integer vanishing exactly at the Gauss or fibre-torus-invariant valuations. If w = 0 the primitive valuations are the ordinary places of the generic fibre P^1, with discrepancy 1. Consequently klt base data give klt data on Y after dropping the two sections, and adding both sections gives lc data whose non-klt locus is contained in the two sections.
 
-./kollar-2013-singularities-minimal-model-program.pdf
+./pdf/kollar-2013-singularities-minimal-model-program.pdf
 
 The discrepancy identity is proved as follows. If v restricts trivially, the trivial-restriction case gives A(v) = 1 + v(t) where t is the fibre coordinate, so A(v) - 1 equals v(E_0) - v(D) via the divisor of z. If v restricts nontrivially to w, take a sufficiently high model of Y and a sufficiently high model of V with a birational morphism between them. The relative canonical comparison, the transformation of div(z) and the definition of R(H) combine to give the identity, and the congruence follows by evaluating the invariant function z r_P at v, r_P the appropriate ramification index.
 
@@ -109,7 +109,7 @@ Lemma (phase nonvanishing). Let X -> S be a sequence of birational Fano type con
 
 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
+./pdf/kawakita-inversion-adjunction-2007.pdf
 
 Uniform cutoff models. Begin from the data b = a + D and extract via a construction that produces a projective Q-factorial Fano type model U -> V containing every primitive prime with a = 0 and D <= 1, and extracting no other prime, with discrepancy funded by a + D/2. On U the floor primes are exactly the support of D_U, with coefficients in (0, 1]. For rational beta in (0, 1] put
 
@@ -159,7 +159,7 @@ All divisorial lc tests are Gauss. Every nontrivial divisorial valuation nonnega
 
 The proof uses the tie in the discrepancy sum. In a chart where P is realised by a coefficient vector, the discrepancy is a sum of monomial values weighted by the character. Off-floor entries are strict, but floor ties require a subsystem of the initial form with more vanishing than the ambient dimension, contradicting integrality of a. If the restriction to K is trivial, use a separating transcendence basis of K in the same wedge computation: if all angular weights vanished, the unit residues would have transcendence degree at most r_T - 1 over K, and the leading wedge would vanish, so zero log discrepancy forces a nonzero angular weight and algebraic independence of the residues. The tie argument then proves the pure valuation is Gauss. Nonnegativity on R places the weight in sigma.
 
-./xu-zhuang-stable-degenerations-2025.pdf
+./pdf/xu-zhuang-stable-degenerations-2025.pdf
 
 Section symbols. For a rational b-divisor A on a normal projective model, write S(A) for the union of zero and those g in k(V)^* with div(g) + A >= 0, test on which is test on nonzero elements. If A >= 0 the space contains 1, so the generated field is unambiguous. For a rational b-divisor A that descends to a model V write S(A) = {0} union {g in k(V)^* : div(g) + A >= 0}; test on the determining model is equivalent to testing all divisorial orders. This notation extends to signed divisors.
 
@@ -198,7 +198,7 @@ The limit exists by Cutkosky's theorem for graded families of m-primary ideals;
 
 Its nth root is A_S(v) vol(v)^{1/n} and has the same minimizers.
 
-./cutkosky-graded-families-of-ideals-2014.pdf
+./pdf/cutkosky-graded-families-of-ideals-2014.pdf
 
 The algebraic equation. For every rational function g on the germ and fixed embedding into a product of P^1s, there is a fixed-q polynomial equation
 
@@ -234,8 +234,8 @@ 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.
 
-./li-minimizing-normalized-volumes-2018.pdf
-./li-xu-normalized-volume-inequalities-2018.pdf
+./pdf/li-minimizing-normalized-volumes-2018.pdf
+./pdf/li-xu-normalized-volume-inequalities-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
 
@@ -251,8 +251,8 @@ Theorem (exact minimizer via a single rational height). Fix a rational height fu
 
 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
+./pdf/blum-existence-normalized-volume-minimizers-2018.pdf
+./pdf/xu-zhuang-uniqueness-minimizer-normalized-volume-2001.05924v1.pdf
 
 Corollary (domination by the minimizer). For the minimizing ray w and every v in Q,
 
@@ -286,7 +286,7 @@ For every sufficiently nearby linear functional L: Lambda_w tensor R -> R positi
 
 The proof uses subduction: subtract a polynomial in the X_i with the prescribed initial symbol, then repeat at a strictly larger value. Only finitely many values lie in a bounded interval: each is a nonnegative integral combination of the finitely many positive alpha_i. Izumi's upper bound for the fixed valuation w controls remainders, and Nakayama's lemma shows the x_i generate m_x via m_x in (x_1, ..., x_N) + m_x^2. Monomial threshold ideals are finitely generated, and faithful flatness of completion descends the conclusion.
 
-./jonsson-mustata-valuations-quasimonomial-2012.pdf
+./pdf/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; 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.
 
@@ -310,7 +310,7 @@ Compression of two gradings. Suppose w is h-lc and satisfies w(f_j) = 0 for ever
 
 The proof comes from a two-step initial reduction on the weighted presentation. Choosing finitely many generators of the prime ideal in_beta in_alpha I that are iterated initials of elements of I, positivity and finiteness of the relevant leading faces make each iterated initial equal to its (alpha + delta beta)-initial for all sufficiently small positive delta. Thus in_beta in_alpha I is contained in in_{alpha + delta beta} I, and equality of dimensions gives equality of ideals. The weighted-presentation lemma supplies an actual multiplicative valuation w_delta with the required quasi-monomiality, and avoiding countably many delta makes (a, b) -> a + delta b injective on the joint lattice. Canonical forms are then matched across the joint compression: applying the canonical-form-degeneration proposition first to v on S^0 and then to w_delta on S makes the two computations of <w_delta, wt(h)> agree, since each uses no prior lc assumption. Uniform comparison in the definition of w_delta gives the multiplicative limit.
 
-./xu-zhuang-stable-degenerations-2025.pdf
+./pdf/xu-zhuang-stable-degenerations-2025.pdf
 
 Yield and pinning. An invariant rational function Y has cost at most m if h^m Y is regular, with m a nonnegative integer. Costs are subadditive under multiplication, on the germ and on a graded cone with h replaced by its initial symbol. Every homogeneous ratio on a cone of cost m lifts to an actual invariant ratio of the same cost: lift its regular numerator and project to the character chi^m by linear reductivity of the finite group. Test functions and accumulated variables are kept on the original germ while the working valuation and its cone are replaced, so pinning weights to a line does not reduce the transcendence degree of the accumulated symbols; suitable ratios of monomials then carry zero weight and retain the angular information.
 
@@ -361,7 +361,7 @@ Pin consequences. The working valuation w' supplied by the high-scale pin satisf
 
 Finite termination of pinning. Each stage either terminates in a contradiction (a quotient prime with tiny negative discrepancy, excluded by the epsilon-lc hypothesis) or produces a new algebraically independent symbol. Dimension bounds the number of new symbols, so finitely many rounds suffice in each fixed dimension. Because each round reduces the total remaining freedom by at least one dimension, after boundedly many rounds the pinning obstruction's hypothesis becomes false, yielding the contradiction.
 
-./xu-zhuang-uniqueness-minimizer-normalized-volume-2001.05924v1.pdf
+./pdf/xu-zhuang-uniqueness-minimizer-normalized-volume-2001.05924v1.pdf
 
 Proof of the pinning obstruction. Suppose a sequence as in the statement of the obstruction exists. Start with the initial common slice Q_0 for the pair (S_i, div h_i) of a sufficiently large index, with the standard baseline height B_0 and the working valuation w^{(0)} given by the normalized-volume minimizer of B_0 on Q_0. Take the initial symbols Y_1^{(0)}, ..., Y_{s_0}^{(0)} from the eigen-canonical generator omega_i and the canonically associated eigenfunctions of character chi_i^m for bounded m.
 
@@ -375,26 +375,26 @@ 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.
 
-./xu-zhuang-stable-degenerations-2025.pdf
-./li-xu-normalized-volume-inequalities-2018.pdf
+./pdf/xu-zhuang-stable-degenerations-2025.pdf
+./pdf/li-xu-normalized-volume-inequalities-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.
 
 Uniform Cartier sections. Apply Chen's equivalence to the birational klt complement theorem just proved. The equivalence identifies the birational statement at dimension bound d with the existence, through any prescribed base point of a Fano type contraction, of a Cartier section whose pullback with a fixed small coefficient tau is lc. The constant tau = tau(d, epsilon) is Chen's, uniform in the base point, in the Fano type contraction, in the boundary, and in the nef over Z. That proves the uniform Cartier sections theorem.
 
-./chen-boundedness-klt-complements-2403.01154v4.pdf
+./pdf/chen-boundedness-klt-complements-2403.01154v4.pdf
 
 Real boundaries. Over the complex numbers the same argument runs with effective real B and K_X + B real Cartier. Real klt log Fano witnesses on a rationally Gorenstein total space can be chosen rational by density of rationals in the coefficient space, and a real sqrt-Cartier integral Weil divisor is already rationally Cartier on a sufficiently high model. Chen's real-pair form then asserts the same transfer for real B without perturbing the boundary. The constant tau retains its uniform dependence on d and epsilon.
 
-./shokurov-bounded-complements-oberwolfach-2004.pdf
-./birkar-boundedness-fano-annals-2021.pdf
-./birkar-singularity-fano-fibration-2305.18770v1.pdf
-./filipazzi-generalized-canonical-bundle-2020.pdf
-./kollar-2013-singularities-minimal-model-program.pdf
+./pdf/shokurov-bounded-complements-oberwolfach-2004.pdf
+./pdf/birkar-boundedness-fano-annals-2021.pdf
+./pdf/birkar-singularity-fano-fibration-2305.18770v1.pdf
+./pdf/filipazzi-generalized-canonical-bundle-2020.pdf
+./pdf/kollar-2013-singularities-minimal-model-program.pdf
 
 Assembly. Birkar's bounded lc complements produce a uniform index J(d). The signed anticanonical torsor, or the canonical cover in the isomorphism case, reduces each failing example to a normal complex klt germ with cyclic quasi-etale action, eigen-canonical generator of character chi, and eigenfunction h of the same character, admitting a centred quasi-monomial valuation T with A_S(T) = T(h) = 1. The saturation condition transfers to that local model. Pinning obstruction excludes the local model sequence, so the birational klt complement theorem holds with a uniform n(d, epsilon). Chen's transfer upgrades the birational statement to uniform Cartier sections, with the constant tau(d, epsilon). Real boundaries pass through the same transfer.
 
 The chain of reasoning closes the argument. No hypothesis of the theorem is violated and no uniformity is lost along the sequence. The constants depend only on the total dimension d and the discrepancy gap epsilon.
 
-./birkar-boundedness-fano-annals-2021.pdf
-./chen-boundedness-klt-complements-2403.01154v4.pdf
+./pdf/birkar-boundedness-fano-annals-2021.pdf
+./pdf/chen-boundedness-klt-complements-2403.01154v4.pdf