Proof. Here everything over a generic DVR can be carried out by shrinking about a generic point of a prime in a smooth model over \(k\). Only assertions over the generic fibre and that DVR are needed. A model retaining the selected fibre components. Write \(R\) for the trait ring. Fix a nonempty finite collection of divisors computing \(b_*(P)\) and choose a log resolution \(Z\) over \(\mathop{\mathrm{Spec}}R\) on which they occur. For the adjusted generalized data \(A_*-b_*(P)e\), they have boundary coefficient one. We first construct an ordinary klt pair on \(Z\) whose minimal model contracts every other special-fibre component and every horizontal divisor exceptional over \(F\).

Subtract a sufficiently small positive multiple of the fibre from the adjusted boundary. The resulting generalized subpair is sub-klt, and its coefficients on the selected components remain positive. Mix it with a sufficiently small amount of auxiliary sub-CY data having an ample b-part on \(Z\) and effective nonexceptional horizontal trace. Such data exist because \(F\) is Fano: a small pushed ample contribution can be subtracted from \(-K_F\), and vertical terms can be adjusted freely over the trait. The pushforward is rationally Cartier on the \(\mathbb Q\)-factorial total space \(X_1\). Replace the mixed ample b-part by general relatively free representatives. Make these choices on a common resolution and retain a general big contribution avoiding the components whose coefficients will be changed.

Increase the coefficients of all horizontal divisors exceptional over \(F\) and all special-fibre components outside the selected collection, strictly and to values in \([0,1)\). We obtain an effective ordinary klt boundary \(\Delta\), big over the trait, and \[K_Z+\Delta\sim_{\mathbb Q,R}E,\] where \(E\ge0\) is supported precisely on the divisors just increased. In particular at least one special-fibre component is absent from \(\mathop{\mathrm{Supp}}E\).

Here the relative fixed part can be computed directly. For divisible \(m\), a rational section of \(\mathcal O_Z(mE)\) restricts on the generic fibre to a rational function with poles only along divisors exceptional over the normal projective variety \(F\). It descends to a regular function on \(F\), and is therefore a constant in the trait field. The section condition on a retained special-fibre component, whose coefficient in \(E\) is zero, forces that constant to have nonnegative DVR order. Conversely every element of \(R\) gives such a section. Thus the relative section module is \(R\), and the relative fixed part of \(|mE|\) is exactly \(mE\).

The good minimal model theorem for klt pairs with big boundary now applies, after spreading over an open neighbourhood of the generic point of \(P\). Every curve in a negative extremal ray lies in \(\mathop{\mathrm{Supp}}E\), so no divisor outside this support is contracted. The section ring is preserved by the MMP. On its semiample output the surviving trace of \(E\) must therefore vanish, since otherwise it would remain a nonzero fixed part. Consequently every component of \(E\) is contracted. Call the output \(Y\). Its boundary remains big and klt and its log canonical divisor is rationally trivial over the trait, so \(Y\) is of Fano type there. Its generic fibre agrees with \(F\) in codimension one. The original adjusted generalized data are crepant through these birational operations. Their trace on \(Y\) is effective: the only special-fibre components are the retained ones, with coefficient one, and the surviving horizontal coefficients are the effective nonexceptional coefficients over \(F\).

Connectedness of the minimizing locus. Let \(\pi:\widetilde Y\to Y\) be a log resolution on which the nef part of the original adjusted data descends. Write \(B^{\log}\) for its boundary and \(H=(B^{\log})^{=1}\). The negative boundary is \(\pi\)-exceptional because the trace on \(Y\) is effective. Moreover \(-(K_{\widetilde Y}+B^{\log})\) is nef over \(Y\), and is big over \(Y\) because \(\pi\) is birational. The birational connectedness lemma therefore makes \(H\) connected over each fibre of \(\pi\).

One can see the relevant structure-sheaf assertion directly. Put \(G=\lceil-(B^{\log})^{<1}\rceil\), an effective exceptional divisor. Relative Kawamata–Viehweg vanishing for \(\mathcal O_{\widetilde Y}(G-H)\) gives a surjection \[\mathcal O_Y\longrightarrow\pi_*\mathcal O_H(G).\] The image factors through \(\pi_*\mathcal O_H\), which injects into \(\pi_*\mathcal O_H(G)\). Hence \(\pi_*\mathcal O_H\) is the structure sheaf of the reduced image. This image is the entire reduced special fibre of \(Y\), since all its components have coefficient one. The special fibre is connected by Stein factorization. Thus \(H\) is connected, and the prescribed divisors lie in one connected snc coefficient-one locus. Generic klt ensures that every component of this locus is vertical. Monomial valuations along its intersections join its divisorial orders continuously; the uniformizer has positive value throughout. Evaluations of any additional discrepancy data are linear on these cones after taking a common resolution, so their ratios to \(e\) interpolate continuously. Rational weights recover the required divisorial interpolants.

An ordinary complement preserving the chosen place. For a tailored ordinary \(\Omega\), use the model above (or horizontally, extract the horizontal place using a CY slightly perturbed towards klt as funding). Write \(T_0\) for the resulting prime, which has coefficient one. Run a \(-(K+T_0)\)-MMP; this divisor is pseudo-effective over the trait (respectively generically) by the effective generalized CY containing \(T_0\). On every model the pair with strict \(T_0\) is lc by comparison with the generalized data and b-nef negativity. At the nef model take a bounded ordinary lc complement (coefficients only 0 and 1, Fano type; in the relative case use Birkar’s complement over a closed point of the relevant prime/open, after appropriate shrinkings as above, containing the generic trait/field in the resulting neighbourhood). Indeed over the trait/field the final nef divisor is semiample by Fano type; generation of a multiple spreads out, as does a Fano type witness (relative ampleness and klt) and lc of the indicated pair via resolution, shrinking by projectivity to ensure the conditions for complements over an open. Its crepant pullback boundary on the model with \(T_0\) still dominates \(T_0\): it adds the pull of the effective complement addition and the effective MMP discrepancy of \(-(K+T_0)\). In particular contraction of \(T_0\) in a negative step would even contradict lc. Push generically to \(F\), preserving effective boundary there. All complement indices use only the absolute dimension bound. This proves both assertions. ◻

Proof. Bounded marked fibres and descent of an embedding. The pairs \((F,{\rm Supp}\,\Omega_F)\) are in bounded embedded families over the geometric field. Indeed boundedness gives a very ample class \(\mathcal H\) of bounded degree and embedding dimension on each bounded \(F\). One can use the complete system here (dimension still bounded by degree for the nondegenerate image). To bound the anticanonical degree, let \(f=\dim F\) and take a smooth complete-intersection curve \(C\) of \(f-1\) general members of the very ample system \(\mathcal H\), avoiding the singular locus. Adjunction gives \[-K_F\cdot\mathcal H^{f-1} =(f-1)\mathcal H^f+2-2g(C) \le (f-1)\mathcal H^f+2 .\] For \(f=1\), the same formula is the usual degree formula on the smooth curve \(F\). The nonzero coefficients of \(\Omega_F\) are at least \(1/n\), so \[\deg_{\mathcal H}\mathop{\mathrm{Supp}}\Omega_F \le n\deg_{\mathcal H}\Omega_F =-n K_F\cdot\mathcal H^{f-1}.\] The support therefore has bounded degree. In the DVR case we can secure the embedding over a bounded degree extension of \(k(W)\). The geometric Picard group is free abelian of bounded rank: it injects by normality into the Picard group of a smooth projective resolution, which is rationally connected (rational connectedness for klt Fano) and has bounded Betti numbers for a suitable resolution, by resolving the bounded families after stratification. We use here the usual characteristic zero Picard consequences of smooth rational connectedness (in particular trivial connected Picard and no torsion). The finite image of Galois on the discrete Picard lattice has bounded order. One way to see the uniformity is to reduce a finite subgroup of \(\operatorname{GL}_r(\mathbb Z)\) modulo \(3\): the principal congruence subgroup at \(3\) is torsion free, so this reduction is injective, and \(\lvert\operatorname{GL}_r(\mathbf F_3)\rvert\) bounds its order. After bounded extension the chosen \(\mathcal H\) is invariant; its complete linear system descends projectively as a projective-space form of bounded dimension (the ambiguity on section maps is scalar), which splits after another bounded extension (Severi–Brauer index bound). This suffices. Bounded-degree embedded varieties and their marked reduced divisors as above are parametrized by finitely many Chow families (Kollár 2017, Definition 3.21, (3.21.1)–(3.21.3)). Here we use effective cycles and their reduced supports, with the universal cycles restricted to smooth characteristic-zero strata.

Toroidalization of the fixed families. Apply marked weak toroidalization (Abramovich et al. 2013, Theorem 1.1) to these families. Its source and base modifications can be taken projective and birational; allowing a fixed alteration would give the same construction after a bounded extension. Here are the uniformity and chart details we use. By compactification and noetherian stratification, for a dense open of each parameter stratum we can use a single proper toroidalized family over a modification (an alteration would also suffice) of its compactified base. Over the dense open (which can always be shrunk, treating its complement by noetherian induction) the new fibres give birational models of the original varieties. Here one works stratum by stratum starting with geometrically integral generic fibres, retaining these and birationality in the spread-out open (in case of base alteration it suffices to modify the base-changed family birationally). The hypotheses on the fibres parametrized, such as geometric integrality and marked support of lower dimension, can be imposed by ordinary Hilbert stratification. The compactification of a stratum serves to extend traits whose field point is in that dense open; their closed points need not parametrize good pairs at all. We include horizontal markings as follows: first resolve the generic fibre with its marked support over this stratum, spread out over an open, and mark the strict boundary and the exceptional locus; arrange their preimage in the toroidalization to be in the boundary. Thus on the resulting generic-fibre models relevant to us any positive crepant coefficient of \(\Omega_F\) is supported on the toroidal boundary: outside the old marking one had smooth with crepant subboundary \(\le0\). We arrange the dense open to be interior on the toroidal base of the family. We may use smooth strict toroidal models (also by toroidal resolution); all modifications here are of fixed finite type families. Any base alteration only requires bounded degree field extension to lift the field point on the chosen dense open. Thus after bounded degree extension of \(k(W)\), say to \(K_1\), and extension of the place, we have a lift of the whole new trait to this base by properness, with its generic point interior. Write \(R\) for the new trait ring, \(t_*\) for its uniformizer, \(\rho\) the ramification index over the original DVR (bounded). We always use log pullback over the finite extension. One may view the trait as the localization at a prime divisor in a smooth model over \(k\), working close to the generic point of that divisor (and spreading the map to such a neighbourhood, shrinking to pull the base boundary only to the closed-point boundary there). In particular the divisorial log structure of the trait comes from a log-smooth \(k\)-pair locally.

Full character rank and the log chart. We verify log smoothness at the strata, including the case in which the map on boundary exponent groups has a kernel. In a completed toroidal description use the full character lattices of the toric models. These include invertible torus characters as well as the characters cutting out the boundary. Dominance gives an injective map of full character lattices after tensoring with \(\mathbb Q\). In their logarithmic differential bases its matrix has constant integral entries. It therefore has full target rank at the closed stratum as well as at the generic point. Any lattice torsion is tame in characteristic zero. The unit characters are essential here: two target boundary characters can have the same boundary exponents on the source while their quotient pulls back to a nonconstant unit, whose differential supplies the missing direction.

To detail the algebraic chart check, at closed points in these smooth strict models write the target boundary equations \(y_j,\ 1\le j\le s\), as pulling to \(u_j x^{\alpha_j}\) with \(x_1,\ldots,x_m\) source boundary parameters and \(u_j\) units. The map of log differential fibres has full target rank (as above in the completed toroidal models, where boundary equations match up to units). For \(\alpha:\mathbb Z^s\to\mathbb Z^m\) the exponent map, let \(K=\ker(\alpha)\). The quotient \(\mathbb Z^s/K\) is the free group \(\operatorname{im}(\alpha)\), so \(K\) is saturated and admits an integral projection \(p:\mathbb Z^s\to K\). The unit homomorphism \(u\) from \(\mathbb Z^s\) factors as its kernel-direction contribution via this projection, times a homomorphism on \({\rm im}(\alpha)\). The latter extends to \(\mathbb Z^m\) after adjoining unit roots étale locally, so can be absorbed by rescaling the \(x_\ell\) by units. We now have the chart \(\mathbb N^s\to \mathbb N^m\oplus K\) using \(\alpha\) and the projection, with the free group summand mapping on the source by unit functions \(u|_K\). This is injective on groupifications with tame cokernel torsion. The induced map to the product of the target with the source toric chart scheme over \(\mathbb A^s\) is smooth near the point: besides the rescaled \(x_\ell\), one needs independent ordinary differentials for those kernel units and the pulled target transverse coordinates along the source stratum. Exactly these are independent there by full log differential rank (their boundary-weight entries are zero). This proves the smoothness required in Kato’s chart criterion (Kato 1989, Theorem 3.5 and Remark 3.6); see also the statement and proof in (Kato 1994, Theorem 4.1, Remark 4.2 and Section 6). In particular the required rank at a closed stratum follows from the constant full-character matrix; it does not require specializing an arbitrary ordinary differential minor.

The trait and its integral contact cone. Take the fine saturated log fibre product with the trait equipped with its closed-point divisorial log structure. Log smoothness is preserved by fs base change and composition (Illusie et al. 2013, sec. 1.1). Spread the trait to the log-smooth \(k\)-pair described above and form the same fs fibre product there. Its source is log smooth over \(k\) with the trivial log structure. It therefore admits smooth local charts over affine toric varieties (Kato 1994, Theorem 4.7 and Section 7). In these charts the monomial stratum quotient is regular, and its dimension plus the characteristic-monoid rank is the ambient dimension. Thus the spread-out source is log regular; localization gives this property for the fs fibre product over the trait. Its main component is proper over the trait and gives the desired generic model. Indeed the map from the fs fibre product to the ordinary one is proper here (in charts, integralization and finite toric saturation); the trivial-log open is dense in a log regular model, and maps to the generic point of the trait. At a stratum with boundary generators from the total family and from the trait, the cone of nonnegative orders is given precisely by \(Mx=r e_*,\ x,e_*\ge0\). Here rows come from the toroidal base boundary equations, pulled to total boundary monomials times units with multiplicities \(M\), and to the trait with contacts \(r\); thus \(M\), unlike \(r\), comes from fixed chart types. These are the matching conditions from the pushout chart (the total family and its base before this log base change can have smooth strict toroidal structures, hence boundary-coordinate charts). After spreading near the trait the base has also ordinary residue/transverse coordinates, with the prime divisor as its single boundary near the generic point in use; only its boundary weight thus enters these equations. Units have weight zero; at the point used only boundary generators vanishing at its images in the factors are counted. No further independent zero conditions on weights arise from units of the saturated monoid becoming units at the point: a positive multiple of such an element comes from the original generators and if it is a unit cannot involve any generator vanishing at the point. An integral match defines an integer-valued homomorphism on the groupification of the pushout monoid. Saturation takes place in that same groupification, so this homomorphism remains integer-valued on the saturated monoid. We may work in strict étale local charts at a geometric stratum and then restrict valuations. Thus nonzero rational weights give monomial divisorial evaluations (and the zero vector the trivial evaluation); integral ones restrict to integral multiples of primitive divisorials when nontrivial. Indeed in the split toroidal chart one is taking the dual cone of the characteristic monoid at the stratum. A nonzero rational ray even on a face can be introduced in a toroidal subdivision (over the corresponding generization stratum for a face); the primitive vector for its toric lattice then gives a boundary-divisor order. Matched integral weights as just described are integral in this lattice. For \(e_*>0\) the restriction to \(K_1\) is \(e_*{\rm ord}_{R}\). This description is equivalently obtained by fs pulling back the log-smooth monomial charts and subdividing toroidally. Use the family of monomial valuations for one chosen lift of the stratum in a split local chart (cones/subdivisions there, then restrict to the original function field); integral points give integral orders since they already do so in such a chart. No degree bound on further strict étale local charts is needed here: restriction of an integer-valued valuation still has value group a subgroup of \(\mathbb Z\), hence is an integral multiple of a primitive divisorial order. It is never divided by the chart degree. Only the initial field extension contributes the separately bounded ramification index \(\rho\). The rational functions used in the metric estimate below are taken on the fixed total family itself.

All special components are log boundary on this model. The subboundary of the crepantly pulled ordinary \(\Omega\) is bounded above by the reduced log boundary (coefficient bound by lc, horizontal positive support marked as above). Thus its chosen lc place is a place also of the reduced-log pair, since the difference is effective \(\mathbb Q\)-Cartier. Such lc places are monomial on the log regular model: pass to a toroidal snc subdivision and use the lc place description for an snc pair. We take a cone as above containing the extended chosen place. For a horizontal place as in the first bullet one just takes a bounded stratified log resolution of the generic pairs \((F,{\rm Supp}\,\Omega_F)\); same reasoning yields an ordinary snc monomial cone (over the generic field, or its extension).

Logarithmic coordinates and Gauss restriction. For the later metric estimate, choose rational log coordinates \(z_1,\ldots,z_f\), \(f=\dim F\), on charts of the fixed toroidalized families, such that \(d\log z_j\) form a basis of relative log differentials near the stratum in question. Indeed the latter are locally free by log smoothness, and one can choose among boundary parameters and invertible functions giving generators from the smooth total chart. Finitely many choices of genuine rational functions on opens of smooth strict charts suffice (Cartier local boundary equations and local units generate logarithmic differentials). Their weights are bounded linear forms in the cone coordinates (boundary parameters or units), and the basis property persists under fs base change and toroidal resolution. Moreover the \(z_j\) have bounded pole degrees on the embedded \(F\)’s for these charts: they are chosen on fixed families birational to the embedded ones, so on a sufficiently shrunk dense open of each stratum they give fixed bounded-degree rational maps on the embedded fibres. This shrinking can be built into the stratification (express each function by fractions of bounded-degree embedded coordinates on the generic fibre of the family and spread). Use the analogous absolute log bases in the horizontal generic case.

A rational toroidal valuation here with \(e_*>0\) restricts to a Gauss valuation on \(K_1(z_1,\ldots,z_f)\). To check, rescale to a primitive divisor on a toroidal subdivision. Write \(\nu_j=v(z_j)\in\mathbb Z\), \(e_*=v(t_*)>0\). At this divisor the residues of \(z_j^{e_*}t_*^{-\nu_j}\) have independent log differentials. Indeed relative log cotangent there, modulo the divisor, is identified with residue differentials over the base residue field: \(t_*=\pi^{e_*}\cdot(\text{unit})\), for uniformizer \(\pi\), eliminates \(d\log\pi\), while base unit differentials restrict as usual. This also follows from the exact sequences for log differentials at the generic smooth divisors (and separability). The indicated unit log differentials are \(e_*\) times the chosen basis in relative log cotangent. The residues are thus algebraically independent. In particular a lattice basis of the weight ties (monomials \(z^\gamma\) with \(\nu\cdot\gamma\) a multiple of \(e_*\), adjusted by that power of \(t_*\)) has algebraically independent residues too, so no cancellation occurs at equal coefficient-adjusted weights. Horizontally one keeps \(d\log\pi\), without a base uniformizer; for a nontrivial monomial valuation of integral rank one weights, a basis of the kernel characters of the weight gives independent residues by the same log basis calculation. This again gives Gauss, now for the trivial base valuation. ◻

Proof. Piecewise linearity and faces. On these cones rational functions have continuous homogeneous finite piecewise linear values, including faces. For example subdivide a split chart toroidally to snc (subfaces using the corresponding generization strata in the subdivision) and use the monomial order formula at the generic stratum (after splitting a chart if necessary), by expanding at the snc parameters: minima on power series supports are finite polyhedral by monomial noetherianity. The same minima using weights zero off a subcollection of parameters give the values on its face, i.e. at the more general stratum. Indeed the monomial ideals in the retained parameters test order there; membership is still tested in the power series ring localized along the prime of those parameters (faithful flatness and the monomial formula; a positive-threshold monomial ideal for the retained parameters only is primary to their prime, as seen already in completion). This gives continuity and compatibility across subdivisions. In a local split toroidal chart the subdivisions are toric subdivisions at that stratum; one is using the corresponding monomial order family throughout (weight ideals of boundary monomials), so that subfaces common to two cones use the same such orders.

A degree bound for angular slopes. Suppose a nonzero rational function \(g\) on \(F\) over the extended field has pole degree \(\le m,\ m\ge1\), in the bounded embedding. It satisfies a polynomial equation over the \(z\)’s of bounded degree in \(g\) and degree \(O(m)\) in the \(z\)’s. Use the equation of the image hypersurface of \((z,g)\) in a product of projective lines. Its multidegrees are bounded by intersecting general level divisors of all but one coordinate on \(F\) on the locus of definition: individual divisor degrees are bounded by their pole degrees, so projective Bézout bounds isolated general solutions by a constant times the product bound (characteristic zero). Equivalently intersect the product of those level-divisor supports with diagonal conditions of bounded degree in a product of projective spaces, counting isolated points (the product variety has Segre degree bounded by a fixed multiple of the product of individual degrees). Write this equation as \[\sum_{j=0}^q c_j(z)g^j=0, \qquad q\le C,\qquad \deg_z c_j\le Cm .\] By the Gauss property, \(v(c_j)\) is a minimum of affine functions whose \(x\)-slopes have norm at most \(Cm\). Their constant terms can involve arbitrary orders of coefficients in the trait field, but at fixed \(e_*\) these terms do not vary. At least two nonzero terms of the equation attain its least valuation. Consequently \(v(g)\) is one of the finitely many quantities \[\frac{v(c_i)-v(c_j)}{j-i}\qquad(i\ne j).\] Each is \(Cm\)-Lipschitz after enlarging the fixed constant. Along a segment the continuous, finite piecewise linear function \(v(g)\) selects from this finite list; subdividing at its finitely many switches gives the same Lipschitz bound on the entire segment. Rational weights are dense, so the estimate holds also for real weights. Horizontally the argument applies on the entire cone. If the pole degree itself is \(<1\) the function is constant along the generic fibre. ◻

Proof of Proposition 5. Lemma 6 supplies the chart, the prescribed monomial place and the integral orders. We first extend the piecewise-linearity assertion of Lemma 7 from rational functions to the discrepancy data. Evaluations of \(\mathbb Q\)-Cartier b-divisors are also finite piecewise linear: locally over an affine neighbourhood of the stratum write a determining Cartier multiple on a projective model as a difference of relatively generated divisors, then it uses minima on finite sets of rational functions. Generalized discrepancies differ on this cone from the reduced-log discrepancy (identically zero) by such Cartier b-evaluations. Piecewise-linearity here need not have bounded combinatorial complexity.

Two-point replacement of the nef part. Fix two rational valuations on one cone with the same \(e_*\), and choose a common high model on which their centres and the determination of the nef part are visible. On that model let \(L\) be an ample divisor, relative to the trait, and choose a rational \(\varepsilon>0\). After twisting by a pullback from the base, take general effective representatives \[E_+\sim_{\mathbb Q}\mathbf M+\varepsilon L,\qquad E_-\sim_{\mathbb Q}\varepsilon L\] which contain neither centre. The ample perturbation is selected after the high model: its coefficient can therefore be made so small that its pushed horizontal degree is arbitrarily small. Add \(E_+-E_-\) to the crepant generalized boundary and remove the nef part. Since both representative divisors have order zero at both chosen valuations, the resulting ordinary divisor data have exactly the prescribed discrepancies at these two valuations.

Let \(\Theta\) be its horizontal trace on the bounded fibre \(F\). Write degrees with respect to the bounded very ample class. The original trace \(B_F\) is effective, and the trace of the nef part has nonnegative degree. Relative triviality gives \[\deg B_F+\deg\mathbf M_F=-\deg K_F .\] Negative exceptional crepant coefficients on the high model disappear when pushed to \(F\). Hence, if \(L_F\) denotes the pushed ample contribution, then \[\deg\Theta^+\le -\deg K_F+\varepsilon\deg L_F, \qquad \deg\Theta^-\le \varepsilon\deg L_F .\] Both degrees are uniformly bounded. This estimate uses no bound on the height of the determination or the denominator of \(\mathbf M\). For divisible \(l\), relative rational triviality supplies a rational \(l\)-pluriform \(\sigma\) for these ordinary data. Its divisor on the generic fibre is \(-l\Theta\); the possible pullback term from the trait has constant value on a slice with fixed \(e_*\). Divide by the \(l\)-power of the log form \(\psi\) from wedging our relative \(d\log z_j\)’s with a base log volume near this DVR. At either evaluation the discrepancy is \(v(\sigma/\psi^l)/l\) up to the base term constant at fixed \(e_*\), since \(\psi\) is a log generator at the toroidal divisors. The generic pole degree of this ratio is \(O(l)\): the log volume of the \(z_j\) has controlled poles (on the regular codimension one locus of \(F\)), hence controlled zeros by the canonical degree (nonpositive here by log CY effectivity horizontally), and the horizontal boundary degree was just bounded. Lemma 7 applied to \(\sigma/\psi^l\), followed by division by \(l\), proves the uniform metric assertion. The integer \(l\) may be arbitrarily large. Continuity extends the estimate from the two rational endpoint valuations to any two real valuations on the slice. Thus neither the nef-part index nor the vertical contact enters the constant. Horizontal traces and these canonical-degree computations on the generic fibre can use forms divided by a separating base volume (the absolute log order at a horizontal divisor is the fibre log order, by taking independent base residues). Horizontally use exactly the same replacement and log volume over the generic field. ◻

Proof. For (i), tailor the complement on the Fano type model using the klt perturbation in Lemma 4, and then push the complement and the effective CY data crepantly to the bounded normal small model. Their total divisors are \(\mathbb Q\)-linearly trivial. This supplies the bounded embedded marked family and all the degree estimates in Proposition 5; the anticanonical degree is bounded by the same curve slicing argument.

For (ii), use the two-point construction for each of the two b-nef parts, taking the pushed ample errors to be \(o(\xi)\). As above replace those parts by general effective representatives after the same arbitrarily tiny ample addition on a high determination, avoiding both centres. The residual rational principal term for \(N\) now uses functions with pole degree divided by denominator \(O(\xi)\) on \(F\); apply Lemma 7 (using constancy if pole degree \(<1\)). ◻

Proof. For (i), pass diagonally so that for every integral character \(\gamma\), the numbers \(\langle\gamma,u_i\rangle\) either converge in \(\mathbb R\) or diverge in absolute value. The characters with bounded values form a subgroup \(\Gamma\). Their limits are additive and extend to evaluation on a vector \(u_*\). For a fixed interval \(I=[a,b]\subset(0,\infty)\), the probability distribution of \(s(u_i-u_*)\) modulo the lattice, with \(s\) uniform on \(I\), tends to Haar measure on \(\Gamma^\perp\). Indeed its Fourier coefficient at a character in \(\Gamma\) tends to one; for a character outside \(\Gamma\), integration of the exponential bounds the coefficient by \[\frac{2}{2\pi(b-a) \lvert\langle\gamma,u_i-u_*\rangle\rvert}\longrightarrow0.\] Trigonometric approximation proves weak convergence. Every neighborhood of zero in \(\Gamma^\perp\) has positive Haar measure. Choose intervals tending to zero sufficiently slowly and neighborhoods of radius \(o(s_i)\), while also requiring \(s_i\lvert u_i\rvert\to\infty\). A diagonal choice gives an integral \(n_i\) with \(\lvert n_i-s_i(u_i-u_*)\rvert=o(s_i)\), proving (3). The error is eventually less than one in each coordinate. Since \(s_i u_i\ge0\), an integral coordinate of \(n_i\) cannot be negative. Its norm tends to infinity, so \(n_i\ne0\).

For (ii), pass to a subsequence with \(M_i=M\). Elementary integer linear algebra gives a rational \(x_{0,i}\) with \(Mx_{0,i}=r_i\) and \(l x_{0,i}\) integral for a positive integer \(l\) depending only on \(M\). For example, solve using a fixed maximal nonzero minor and take \(l\) divisible by its determinant. Put \(H_0=\ker M\), \(\Lambda=H_0\cap\mathbb Z^m\), and \(U_i=x_i-x_{0,i}\). On the dual lattice of \(\Lambda\), pass diagonally so that every normalized distance \[\frac{\operatorname{dist} (\langle\gamma,lU_i\rangle,\mathbb Z)}{l\alpha_i}\] converges to a finite value, with a signed residual, or tends to infinity. The characters in the first case form a subgroup \(\Gamma\). Their signed residual limits are additive; choose \(u_*\in H_0\) whose evaluations give these limits.

Sample positive integers \(j\) for which \(s=jl\alpha_i\in[a,b]\), with \([a,b]\subset(0,\infty)\) fixed. The distributions of \(jl(U_i-\alpha_i u_*)\) in \(H_0/\Lambda\) tend to Haar measure on \(\Gamma^\perp\), which may be disconnected. For \(\gamma\in\Gamma\) the frequency, reduced modulo \(\mathbb Z\), is \(o(\alpha_i)\), so its Fourier coefficient tends to one. Otherwise the geometric-series bound is at most \[\frac{C\alpha_i} {\operatorname{dist} (\langle\gamma,l(U_i-\alpha_i u_*)\rangle,\mathbb Z)} \longrightarrow0 .\] Use neighborhoods of zero and a slow diagonal choice as before. There are \(j_i>0\) and \(k_i\in\Lambda\) such that, with \(s_i=j_i l\alpha_i\to0\), \[\lvert k_i-j_i l(U_i-\alpha_i u_*)\rvert=o(s_i).\] Set \(E_i=l j_i\) and \(n_i=E_i x_{0,i}+k_i\). These are integral matches and \(\lvert n_i-E_i x_i\rvert\le C_*s_i\). The same coordinate argument proves \(n_i\ge0\).

For (iii), apply simultaneous Dirichlet approximation to \(lU_i\) on the fixed kernel torus. It supplies a positive \(j_i\le J_0(\gamma)\) with \(\operatorname{dist}(j_i lU_i,\Lambda)\le\gamma\). Use the same definition of \(n_i\) and \(E_i=l j_i\). Matching is exact, positivity follows when \(\gamma<1\), and the bound on \(E_i\) is independent of \(r_i\). ◻

Proof of Lemma 1. We induct on the dimension of the total space; the birational case was settled in Subsection 1.3.

Extraction and effectivity. Extract using \(a_0\) with \(\delta_*\) fixed small and take the Mori step \(X_1\to W\). Effectivity of all parameters on the extraction holds by monotonicity, or in the two-sided case by the ratio hypothesis taking \(\delta_*<\min(\delta,1)/3\), after trimming \(L_i\to\infty\) more slowly. Indeed concavity using the opposite endpoint of a fixed interval controls interior upper ratios too. For example, if \(0\le t\le T\), then concavity at zero gives \[a_t(T')\le (1+t/T)a_0(T')-(t/T)a_{-T}(T').\] The hypothesis on all test subsequences makes the right side \((1+o(1))a_0(T')\), uniformly on the extracted divisors. The other sign is identical. Apply this for each fixed rational \(T\), then choose a slowly increasing interval by diagonalization. At subsequent Mori steps the discrepancy families are discriminants of the original family. They need not have a common global piecewise affine presentation. For each fixed prime \(P\), however, resolve the original family over its generic DVR and include the fibre in the snc support. In the trimmed range, original divisorial valuations trivial on the current function field already have positive discrepancy. The discriminant is therefore the minimum of the finitely many vertical component ratios on that resolution. Minima compose under the nested field restrictions, so the finite rational parameter transitions used below persist at each step. Generic klt over the next base can likewise be tested on a resolution of the original family over its generic field.

Generic klt over the next base. We claim generic klt over \(W\) in the Mori step for all bounded parameters eventually. Otherwise take a sign and a first horizontal lc threshold \(t_*\ne0\) with \(|t_*|\) bounded, and a horizontal minimizing divisorial \(T\). This first threshold is rational by resolution; if tested on the original family, a place giving it restricts nontrivially to the current field (by the previous generic klt), hence gives an lc place there too. Tailor the chart using \(a_{t_*}\). Concavity at this place gives \[a_{\pm L}\le (1-L/|t_*|)a_0\] for arbitrarily large fixed rational \(L\), sign as chosen. Let \(w\) be the chart vector representing \(T\). Put \(u=w/a_0(w)\), and bound \(\lvert t_*\rvert\) by a fixed \(T_0\). If \(\lvert u\rvert\) were bounded along a subsequence, the uniform homogeneous Lipschitz bound would bound \(\lvert a_{\pm L}(u)\rvert\) independently of fixed \(L\), whereas \(a_{\pm L}(u)\le1-L/T_0\). Choosing \(L\) large gives a contradiction. Thus, after a further subsequence, \(\lvert u\rvert\to\infty\). Lemma 9(i) gives \(\lvert n_i-s_i u_i\rvert\le C_*s_i\), \(n_i\ne0\), and hence \[0<a_0(n_i)\le (1+CC_*)s_i,\qquad a_{\pm L}(n_i)\le(1-L/T_0+CC_*)s_i .\] The same \(C\) works for every effective CY discrepancy \(a_{\pm L}\). Choose \(L>T_0(1+CC_*)\) after fixing the subsequence. The restriction of \(n_i\) is \(k_i\mathop{\mathrm{ord}}_{T_i}\), with \(k_i\ge1\). Therefore \(a_0(T_i)\to0\), while \(a_{\pm L}(T_i)<0\), contradicting the appropriate ratio hypothesis. This proves the claim; trim again to keep generic klt throughout and induce effective generalized data on \(W\).

Transfer of the ratio hypothesis. We must pass the corresponding ratio hypothesis to \(b_t(P)\) on \(W\). Suppose there is failure with \(b_0(P)\to0\), respectively with \(b_0(P)<\delta'\) where \(\delta'>0\) is fixed sufficiently small below. Concavity (also using the other sign in the two-sided case) gives a bounded parameter with \(b_t\le0\). Move from zero along a sign to the first \(b_t=b_0/2\); we may take \(b_s\le2b_0\) up to that time: if there is a higher peak use the opposite sign, where the drop is then forced. In the one-sided case this uses monotonicity instead.

There is a rational parameter \(p\) on this segment and a simultaneous minimizer over \(P\) with \[a_0(T)/e(T)\to0\quad (\text{respectively }\ a_0(T)/e(T)\le 10\delta'),\qquad a_p(T)\le \tfrac12 a_0(T).\] Indeed take a cutoff radius \(\sqrt{b_0}\) or \(8\delta'\). At zero all minimizers have ratio \(a_0/e=b_0\). At the half-drop any minimizer below the cutoff works. If minimizers exit the cutoff first, at a transition interpolate where a minimizer below and one at/above tie, to obtain a ratio near the cutoff, where the desired discrepancy drop holds since \(a_p/e=b_p\le 2b_0\). It suffices to track the finite minimizing candidates from log resolution (or the divisorial restrictions of the finite candidates minimizing in the original family at each parameter; restriction of any minimizing original candidate gives a current minimizer by composition); use the LC-place connected interpolation above at the rational transition parameter.

Tailor the vertical chart to this \(T\). In its notation \(\alpha=\rho\,a_0(T)/e(T)\) and concavity gives a decrease as above by arbitrarily large factors at a fixed signed \(L\). If \(\alpha\to0\), kernel approximation and uniform Lipschitz at fixed uniformizer value give again \[a_0(n,E)=O(s),\qquad a_{\pm L}(n,E)<0\] for sufficiently large fixed \(L\), impossible. Otherwise we are in the two-sided case with \(\alpha\le C\delta'\) but bounded below after passage to a subsequence. Use the bounded Dirichlet approximation with \(\gamma\) so small that the Lipschitz error for \(a_0\) is \(<\delta/2\). The constants here (also ramification and \(J\)) are uniform for this Mori step, so choose \(\delta'\) small enough that \(E\alpha<\delta/2\). A sufficiently large signed \(L\) fixed on the subsequence still makes the discrepancy negative by the uniform Lipschitz bound even for \(a_{\pm L}\). This contradicts the strong ratio hypothesis. We have thus transferred that hypothesis with fixed \(\delta'>0\) (and the one-sided hypothesis without a fixed cutoff as required). Iterate using \(\dim W<\dim X\), stopping if birational over \(S\). This proves Lemma 1. ◻

Proof of Lemma 2. Absorb the input multiple into \(H\). Use the same Mori and discriminant steps with fixed klt CY \(a\). For \(b(P)\to0\) at a step choose a minimizer, so in the vertical chart \(\alpha=a(w,e_*)/e_*=\rho b(P)\to0\). Here write \(h=v_P(H_W)\) where \(H_W\) is the base pullback. Use the kernel argument on the augmented torus with sampling vector \(l(U,\rho h)\). Write \((u_*,z_*)\) for an extension of the signed limits divided by \(l\alpha\) on the bounded-frequency subgroup \(\Gamma\). The same sampling equidistributes modulo the augmented lattice after subtracting \(s(u_*,z_*)\).

Every vector in the Lie space of \(\Gamma^\perp\) must have last coordinate zero. Otherwise choose its sign with negative last coordinate and target a small multiple \(\xi v\) with \(\xi\to0^+\) on that torus. By taking intervals \(0<s\ll\xi\to0\) sufficiently slowly and approximating the target to \(o(\xi)\) along a diagonal subsequence, the integral matched valuation has angular error \(O(\xi)\), hence \(a(n,E)=O(\xi)\), but signed phase \(\xi v_{\rm last}+o(\xi)\), contradicting the lower sign bound. The sign bound applies to integral multiples whenever their total discrepancy tends to zero. Indeed, for \(v_i=k_i\mathop{\mathrm{ord}}_{T_i}\) with \(a(v_i)\to0\), the primitive discrepancies also tend to zero. If their phase lifts are \(r_i=O(a(T_i))\), \(r_i\ge-\varepsilon_i a(T_i)\), with \(\varepsilon_i\to0\), then \(k_i r_i\) are lifts with \[k_i r_i=O(a(v_i)),\qquad k_i r_i\ge-\varepsilon_i a(v_i).\] They tend to zero, so no phase wrapping affects the sign. The projection of the compact subgroup \(\Gamma^\perp\) to its last circle now has zero Lie algebra, hence finite image. Some \(q>0\) annihilates this image; by character duality \((0,q)\in\Gamma\). Consequently \({\rm dist}(q l\rho h,\mathbb Z)=O(l\alpha)\). Its normalized residual limit is nonnegative: indeed \(z_*\ge0\) by the same sign test targeting zero on the torus with precision \(o(s)\), angular error \(O(s)\). Pass to a subsequence with \(l,\rho\) fixed.

For every test sequence \(b(P)\to0\) we thus have a subsequence and a fixed multiple there giving both the \(O(b)\) bound and the lower \(-o(b)\). There exists a single fixed multiple giving the upper size bound for all such sequences. Otherwise choose diagonal violations with \[{\rm dist}(k!h,\mathbb Z)>k!\,k\,b(P),\qquad b(P)\longrightarrow0.\] The subsequence property gives a fixed good multiple \(M\) on a further subsequence, with \({\rm dist}(Mh,\mathbb Z)\le Cb(P)\). For large \(k\), \(M\mid k!\), and thus \[{\rm dist}(k!h,\mathbb Z) \le (k!/M){\rm dist}(Mh,\mathbb Z) \le (k!/M)C b(P),\] a contradiction. Fix such a multiple \(M_0\). If its small lift has a negative ratio to \(b(P)\) bounded away from zero along a subsequence, apply the subsequence property again and take a common multiple of \(M_0\) and the resulting good multiple. Both small lifts, multiplied by fixed positive integers, tend to zero and represent the same phase, so eventually they are equal. Their incompatible lower signs give a contradiction. This proves the required lower \(-o(b(P))\) bound for \(M_0\). This gives the required induction and proves Lemma 2. ◻

Proof of Lemma 3. Use the Mori step extracting by \(a+t_0d\). Since \(a\) has bounded denominator and \(d\ge0\), all extracted divisors with this discrepancy tiny have \(a=0,\ d\) tiny. Thus in the extra-data case too, effectivity on the extraction holds uniformly, by the upper bounds on \(a=0\).

Horizontal signs and generic klt. First in the extra-data case, \(h<0\) on a horizontal \(a\)-lc place over \(W\) is impossible eventually. Tailor using \(a\). The homogeneous comparison cells for \(a,h\) come from finitely many rational cones. On a cell containing a witness, the face \(a=0\) is rational and \(h\) is linear. Some extremal ray of that face has \(h<0\). The possible primitive integral ray generators form a bounded finite set, so there is a bounded nonzero integral witness \(a=0,\ h<0\). But \(c\) on it is bounded by Lipschitz (again by fixed parameters), and \(h\) quantized and \(\ge-\eta_i c\), impossible. Next generic klt for \(a+\lambda(h+\theta c)\) holds uniformly for small \(\lambda>0\), \(0<\theta\le\theta_0\): otherwise take \(\lambda_i\to0\) equal to a first positive lc threshold with the corresponding fixed \(\theta_i\) in each example. There is such a first threshold by common resolution, since on horizontal \(a=0\) we have \(h\ge0,\ c>0\). Tailor using this threshold place. In the homogeneous chart distance from its vector \(w\) to \(a=0\) (including the zero vector) is bounded by \(C_1 a(w)\) by the slope/polyhedral bound. On that set \(h+\theta_i c\ge0\), by the divisorial conditions and rational approximation there. Thus \(h+\theta_i c\ge-C_2 a(w)\) at \(w\), inconsistent with the threshold tending to zero (\(a(w)>0\)).

Lexicographic minima and their denominators. For \(P\) over \(W\) denote the claimed lexicographic minima by \(a',d'\) (and \(a',h',c'\)). They exist: on a common resolution over the DVR use the vertical components and the formula for discrepancies by monomial orders plus the nonnegative reduced-log discrepancy. Horizontal \(a\) is nonnegative, and in the extra-data case on horizontal \(a=0\), \(h\) is nonnegative and \(c>0\). Explicitly include the fibre among the divisors on the resolution, so \(e\) is a nonnegative linear combination of vertical orders only. At a generically klt parameter a minimizer has reduced-log discrepancy zero (on the resolution) and all its positive orders in those coordinates lie on minimizing vertical components, by subtracting the minimum times \(e\); thus tracking these finitely many components captures departures from the faces in the argument. For each example and \(P\), sufficiently small strictly positive parameters in lexicographic order thus have only lexicographically minimizing components, and give a simultaneous minimizer for tailoring using a generically klt parameter. On that chart the minimum \(a/e_*\) is \(\rho a'\), and likewise for \(h\) on the minimizing locus if used. By linear optimization at cell vertices on the slice \(e_*=1\), both of these minima have bounded denominator. Thus so do \(a'\) and \(h'\), uniformly in \(P\).

Uniformity of the chambers. If minimizers of \(a+t d\) leave the \(a\)-minimum for some \(t\) tending to zero, take the first (strictly positive rational) transition where minimizers of both types tie. Connected interpolation gives a simultaneous minimizing place with \((a/e)-a'(P)\) arbitrarily small positive. Tailor at that place for the transition discrepancy. On the cell of its normalized weights (value \(e_*=1\)), the affine function \(a-\rho a'(P)\) is nonnegative. Its minimum is attained at a vertex, because the cell is pointed. Both the vertex values of \(a\) and \(\rho a'(P)\) lie in a fixed rational grid. A strictly positive minimum would therefore be bounded below by a fixed positive number. Taking the interpolated excess smaller than this number forces the face \(a=\rho a'(P)\) in that cell to be nonempty. Let \(\varepsilon=a(w)-\rho a'(P)>0\). The distance estimate in Claim 10 gives \(w_0\) on that face with \(\lvert w-w_0\rvert\le C_1\varepsilon\). If \(C_2\) is the Lipschitz constant of \(d\), then \[(a+td)(w_0)-(a+td)(w) \le-\varepsilon+tC_1C_2\varepsilon<0\] for a uniformly small \(t>0\). This contradicts minimality; continuity permits the comparison at real weights and rational approximation recovers divisorial tests. This proves uniform exact minimization on the \(a\)-face for \(a+t d\). Identical reasoning with \(h+\theta c\) yields uniform small \(\lambda\) with all minimizers still on the \(a\)-face for every \(0<\theta\le\theta_0\) in the extra-data case. Fix a uniformly positive small such \(\lambda\). If first positive \(\theta\)-transitions away from the \(h\)-minimum within the \(a\)-minimum approach zero, now interpolate with tiny positive excess of \(h/e\) above \(h'\); all simultaneous minimizers stay on the \(a\)-minimum. Use joint \(a,h\) cells with the additional affine face \(a=\rho a'\); the grid and distance argument within this face applied to \(h\) contradicts minimality for \(h+\theta c\). Within the \(a\)-minimum the minimizer condition no longer depends on the magnitude of \(\lambda>0\), only on \(\theta\). Thus the full formulas hold uniformly on a smaller sector.

Induction on the base. For induction \(a'+t d'\) gives klt data for positive \(t\) by the minimum formula, and data at zero by the endpoint limit already explained. In the extra-data case on \(a'=0\), \(d'\le c'\), \(h'\ge-\eta_i c'\), and testing at a \(d'\)-minimizer at fixed uniform small \(\theta>0\) gives \(h'+\theta c'\le C_3 d'\) by the exact formula. Hence \(c'\le C_4 d'\) and \(h'\le C_3 d'\) there. All hypotheses now hold at the next step. Any negative \(h\) at \(a=0\) vertical in this step marks a negative \(h'\) at \(a'=0\), by restriction. Induct to the base to obtain all assertions. ◻

Proof. The two affine charts are the relative spectra of the algebras of rational monomials \(gz^j\) satisfying \(\mathop{\mathrm{div}}(g)+jH\ge0\), with \(j\ge0\) and \(j\le0\), respectively. If \(\ell H=\mathop{\mathrm{div}}(b)\) locally, these charts are finite cyclic quotients of the ordinary line charts obtained using \(b^{1/\ell}\) and the coordinate \(z/b^{1/\ell}\). They glue to the stated normal projective bundle. This description gives the two sections, their \(\mathbb Q\)-Cartier property, and the first equality in (10). The restriction of \(D\) to itself is numerically \(H\), while its restriction to \(E_0\) is trivial; this and relative ampleness give the positivity assertions.

At a prime \(P\) of \(V\) the reduced vertical divisor has multiplicity \(r_P\). Computing the relative canonical divisor by \(d\log z\) therefore gives the second equality in (10), including exactly the coefficient \(1-1/r_P\).

For the discrepancy formula, write \(w=e\mathop{\mathrm{ord}}_T\) on a smooth base model and choose a uniformizer \(x\) at \(T\). Wedge a base logarithmic volume form with \(d\log z\). The additional logarithmic order is a nonnegative integer; it is zero exactly when the residue of the valuation-zero monomial \(z^e/x^{v(z)}\) is transcendental over \(\kappa(T)\). This is the Gauss condition. Removing the two sections from the boundary adds \(v(E_0)+v(D)=|h(v)|\), since their supports are disjoint. Evaluating the first identity in (10) gives the congruence. The same calculation on a determination of the nef part proves the generalized formula. For trivial restriction the assertion is the usual discrepancy calculation on \(\mathbb P^1_{k(V)}\). ◻

Proof. Take a small \(\mathbb Q\)-factorialization. The class \(-(K_V+R(H))\) is pseudo-effective, because it is the sum of \(B-R(H)\) and the trace of a b-nef divisor. Moreover \((V,R(H))\) is ordinarily klt: its discrepancies dominate the given generalized discrepancies, by \(B\ge R(H)\) and b-nef negativity. Run its anticanonical MMP, using Fano type to obtain a nef model. Ordinary lc comparison holds along this MMP by the negativity argument of Section 1. The coefficients of \(R(H)\) belong to the fixed hyperstandard set \(\{1-1/r:r\in\mathbb N\}\). The bounded ordinary lc complement theorem (Birkar 2019, Theorems 1.7–1.8), followed by pullback through this MMP, gives an ordinary lc boundary \(\Delta\ge R(H)\) with \(n(K_V+\Delta)\sim0\), where \(n\) is bounded in terms of dimension. The pullback dominates \(R(H)\) because the MMP has the anticanonical sign; this is the same complement pullback used in Lemma 4.

If \(r_P>n\), the \(1/n\)-quantization of \(\Delta\) forces \(\operatorname{coeff}_P(\Delta)=1\). Multiply \(H\) by the least common multiple of the integers at most the bound for \(n\). All smaller denominators disappear. Every remaining component of \(R(H)\) is therefore contained in \(\lfloor\Delta\rfloor\). Let \(\Theta_0\) be any klt log Fano boundary on \(V\). For a sufficiently small positive rational \(\rho\), chosen separately for each variety, \[\Theta=(1-\rho)\Delta+\rho\Theta_0\] is klt, dominates the new \(R(H)\), and has ample negative log canonical class. No uniform lower bound for \(\rho\) is needed.

On the bundle use boundary \(p^*(\Theta-R(H))+(1-\rho')(E_0+D)\), with \(\rho'>0\) sufficiently small. Lemma 12 proves klt. If \(A=-(K_V+\Theta)\), its negative log canonical class is \(p^*(A-\rho'H)+2\rho'D\), which is ample for small \(\rho'\) by relative ampleness and base positivity. Thus the bundle is of Fano type. ◻

Proof. The contraction exists by semiampleness of the infinity divisor. One can also construct it by the finite fibre-power map to the ordinary compactified bundle for a very ample Cartier multiple of \(G_i\), contracting its zero section and then Stein factorizing. The bundle identity gives \[K_Y+(1-I_i)E_0+(1+I_i)D\sim_{\mathbb Q}0.\] The pair with subboundary \((1-I_i)E_0\) is sub-klt by (11); the discrepancy of the contracted zero section is \(I_i>0\). This proves the assertions about the cone and its canonical divisor. Near infinity the discrepancy formula reads, for \(u=v(D)>0\), \[ A_C(v)=A_{F,R(G)}(w)+u+\operatorname{def}(v), \qquad u\equiv w(G)\pmod\mathbb Z. \tag{14}\]

If (13) held, apply Lemma 1 over a point to the nonincreasing family with discrepancies \(A_C-tv(D)\), boundary \(tD\), and nef part \((I_i+1-t)D\) on a range tending to infinity. The phase-free ratio hypothesis is exactly (13). The comparison lemma would make this family klt on every fixed bounded parameter range, whereas at \(t=1\) the prime \(D\) has discrepancy zero. This is a contradiction. ◻

Proof. Apply Lemma 13 to obtain, after a further bounded scaling, ordinary klt log Fano data dominating \(R(P_i)\). This does not affect boundedness of the original principalization orders. A small \(\mathbb Q\)-factorialization preserves all the data and these orders. Suppress the index and suppose for contradiction that \(q\to\infty\). Write \[\mathop{\mathrm{div}}(\phi)=qP, \qquad \widetilde F=\operatorname{Norm}_F\bigl(k(F)(r)\bigr), \qquad r^q=\phi.\] The degree is \(q\) by minimality of \(q\) and the Kummer criterion over a field containing all roots of unity. The ramification in codimension one is encoded by \(R(P)\). In particular the log pullbacks of both the ordinary log Fano boundary and the generalized boundary are effective and klt.

Form the diagonal quotient \[ X=(\widetilde F\times\mathbb P^1_y)/\mu_q \longrightarrow\mathbb P^1_{y^q}, \tag{15}\] where \(\mu_q\) acts inversely on \(y\) and \(r\), so that \(yr\) is invariant. The map from the product to \(X\) is quasi-étale: a nonidentity element has a fixed locus of codimension at least one on \(\widetilde F\) and only the two fixed points on \(\mathbb P^1\). The generic fibre of (15) is connected. Log pulling the ordinary log Fano data to the product and descending proves that \(X\) is of Fano type over the curve.

Pull back the generalized data to the product and descend them. They give effective generalized klt Calabi–Yau data over the curve. The invariant nef b-divisor descends on a sufficiently high finite quotient; its \(\mathbb Q\)-Cartier property and relative rational triviality descend by taking norms. For a primitive valuation over zero put \(u=v(y^q)/q>0\), and let \(w\) be its restriction to \(k(F)\). The product discrepancy calculation gives \[ A(v)\ge a(w)+u, \qquad u\equiv-w(P)\pmod\mathbb Z, \tag{16}\] where \(a(w)=0\) if \(w=0\). The congruence follows by evaluating the invariant function \(yr\). These formulas can equally be computed upstairs and divided by the ramification index, using log pullback and homogeneous discrepancies.

If \(A(v_i)\to0\), trivial restriction is impossible because it would make \(u_i\) a positive integer. For nontrivial restriction, (9) and (16) give \(u_i=o(A(v_i))\): all terms tend to zero, so there is no wrap, and the lower bound for the signed phase bounds the positive number \(u_i\) from above.

Apply Lemma 1 over the curve to \(A-t\,v(f^*[0])/q\). These are effective generalized Calabi–Yau data obtained by adding \(tf^*[0]/q\) to the boundary; the nef part is unchanged. The parameter range may tend to infinity. The reduced zero fibre has discrepancy \(1\) and multiplicity \(q\), hence its initial discriminant discrepancy satisfies \(0<b_0(0)\le1/q\). Since the perturbation is a pure base pullback, \[b_t(0)=b_0(0)-t/q, \qquad b_1(0)\le0.\] This contradicts the ratio conclusion of Lemma 1, because \(b_0(0)\to0\). ◻

Proof of Lemma 11. Choosing the phase corrections. We argue by dimension induction. The assertion for a point is immediate. If the assertion fails in positive dimension, pass to a sequence and choose integers \(N_i\to\infty\) such that \[ |\lfloor mH_i\rfloor|=\varnothing \quad\text{for all integers }1\le m\le N_i. \tag{17}\] Choose rational \(k_i\to\infty\) sufficiently slowly relative to \(N_i\) and \(L_i\). Use the phase hypothesis at \(k_i\) and Lemma 13, pass to a subsequence, and scale \(H_i\) by fixed positive integers. Reparametrize the given families by these same factors, and decrease \(N_i\) by bounded factors. We retain (17), obtain the phase hypothesis without a displayed scaling at \(k_i\), and have Fano type Seifert bundles. These bounded changes never replace the given discrepancy families by different ones.

Fix a sufficiently small constant \(\eta>0\). Extract precisely the exceptional primes \(T\) with \(a(k_i,T)<\eta\) whose small signed phase \(f_T\) of \(H(T)\) is positive. Include the original primes with the same property in the list of corrections. This is a finite list: monotonicity gives \(a(0,T)\le a(k_i,T)<\eta\), and the extraction and finiteness argument of Section 1 applies to the klt data \(a(0)\). Write the resulting Fano type model as \(U\to V\). For small fixed \(\eta\) we have, eventually, \[ 0<f_T\le C a(k_i,T)\le C\eta, \qquad C_U=\sum_T f_TT, \qquad P_U=H_U-C_U, \tag{18}\] where \(H_U\) is the pullback of \(H\). We first prove that \(P_U\) is not big, eventually.

The threshold contradiction under bigness.

Suppose instead that \(P_U\) is big. On the Seifert bundle let \(A\) be the discrepancy of the data at \(k_i\) without the sections. Choose a small fixed \(0<\delta<1/4\), sufficiently small relative to \(\eta\) and the phase bound. Extract every exceptional prime with \(A<\delta\), and let \(\mathcal S\) be the reduced sum of all primes with \(A<\delta\) on this extraction, including original primes. Lemma 12 shows that they are Gauss valuations. Their extraction is funded by the effective zero-parameter generalized Calabi–Yau data with both sections, mixed with a sufficiently small ordinary klt log Fano contribution as in Section 1. Thus the extraction is of Fano type, and the trace boundary of \(A\) is effective on it.

Put \(\mathbf H=p^*H\), and denote by \(\mathbf D\) the b-pullback of \(D\). The b-divisor \[\mathbf J=\mathbf D+\mathbf H\] is nef and big and has integral Weil trace on the original bundle: the multiplicities \(r_P\) clear the coefficients of \(H\) at all vertical primes. Let \(\tau\) be the pseudo-effective threshold for subtracting \(\mathcal S\) from its trace on the extraction. We claim that \[ \mathcal S\ne0, \qquad k_i\tau_i\longrightarrow0. \tag{19}\]

To prove the claim, if it fails pass to a subsequence and choose integers \(k_i,\tau_i^{-1}\ll m_i\ll N_i\); omit \(\tau_i^{-1}\) when \(\mathcal S=0\). Round the coefficients on the extracted primes to obtain \[Q=\sum_{T\subset\mathcal S}q_TT, \quad 1\le q_T\le2, \qquad N=m\mathbf J_U-Q\] big and integral. All nonintegral coefficients of \(m\mathbf J_U\) are on the extracted primes, so this rounding suffices. When \(\mathcal S=0\), use \(N=mJ\). Run the \(N\)-MMP to a nef and big model. Here and in what follows a prime denotes the output trace, and a star denotes its Cartier b-pullback to a common resolution. Set \[e_H=(H')^*-\mathbf H,\qquad e_D=(D')^*-\mathbf D,\qquad e_J=e_H+e_D.\] B-nef negativity and MMP negativity give \[ e_H,e_D\ge0, \qquad m e_J\le(Q')^*-Q^*, \qquad (Q')^*\le2(\mathcal S')^*. \tag{20}\] The total logarithmic divisor for \(A\) has trace-pull error \((k_i+1)e_H-2e_D\), by (12). Its effective trace boundary dominates \((1-\delta)\mathcal S'\). Comparing generalized and ordinary discrepancies on the output therefore gives \[\begin{align*} A_{\mathrm{ord}}' &\ge A+(1-\delta)(\mathcal S')^*-(k_i+1)e_H\\ &\ge A+ \left(1-\delta-\frac{2(k_i+1)}m\right)(\mathcal S')^* \ge A+\frac12(\mathcal S')^* . \tag{21}\end{align*}\] The comparison also discards only a nonnegative trace-pull error of the nef part.

This gives a gap at contracted primes as well as surviving ones. Indeed, every primitive \(T\) with \(A(T)<\delta\) was put on the extraction, and hence \(Q^*(T)=q_T\ge1\), even if the subsequent MMP contracts \(T\). Equation (20) implies \[ (Q')^*(T)\ge Q^*(T)\ge1, \qquad (\mathcal S')^*(T)\ge\frac12. \tag{22}\] Such a prime has \(A_{\mathrm{ord}}'(T)\ge1/4\) by (21); all other primitive primes have \(A_{\mathrm{ord}}'(T)\ge A(T)\ge\delta\). Thus the output is ordinarily \(\min\{\delta,1/4\}\)-lc.

Fano type makes \(-K'\) pseudo-effective, so \(N'-K'\) is pseudo-effective. Integral-Weil effective birationality (Birkar 2023a, Theorem 1.1) supplies a nonzero section of a bounded multiple of \(N'\). MMP negativity pulls it back to the same multiple of \(N\), and then to a bounded multiple of \(m\mathbf J\). The fibre torus acts on the latter section space, so it contains a weight vector \(gz^j\). A section of \(\ell J\) satisfies \[ 0\le j\le\ell, \qquad \mathop{\mathrm{div}}(g)+(j+\ell)H\ge0. \tag{23}\] Indeed the coefficients of its divisor along \(E_0,D\) are \(j,\ell-j\), respectively, by (10). Thus it gives a section of \(\lfloor (j+\ell)H\rfloor\) in positive degree at most \(2\ell=O(m_i)\ll N_i\), contrary to (17). This proves (19).

Run the MMP of \(J-\tau\mathcal S\) to its semiample nonbig model and let its Iitaka fibration be \(Y'\to Z_\tau\). The threshold is rational because the extraction is of Fano type. Negativity now gives \[ 0\le e_J\le\tau\bigl((\mathcal S')^*-\mathcal S^*\bigr). \tag{24}\] Choose a small fixed \(c\) with \(0<c<1-\delta\). On this output use the boundary \(B_Y'-c\mathcal S'\) and nef b-divisor \[ \mathbf M^\# =\mathbf M(k_i)+ \left(2+\frac c\tau\right)\mathbf D+ \left(\frac c\tau-k_i-1\right)\mathbf H. \tag{25}\] This boundary is effective. Both displayed coefficients in the nef part are eventually nonnegative by (19); in particular the nefness is absolute, and uses no Cartier-index bound. Its total logarithmic divisor is \(\mathbb Q\)-linearly equivalent to \((c/\tau)(J'-\tau\mathcal S')\). The discrepancies are \[ A^\#=A+c\left((\mathcal S')^*-\frac{e_J}{\tau}\right) \ge A+c\mathcal S^*. \tag{26}\] Every primitive prime with \(A<\delta\) has \(\mathcal S^*(T)=1\), including a prime contracted along the MMP. The remaining primes have \(A\ge\delta\). Thus \(A^\#\) has a uniform generalized gap \(\min\{c,\delta\}\).

On a geometric generic fibre \(F\) of \(Y'\to Z_\tau\) these are effective generalized Calabi–Yau data. Restriction can be computed using horizontal divisors on all higher models over the generic field, and then separable extension of constants; it is crepant and preserves the displayed gap. The fibre is of Fano type and rationally connected. Apply (Birkar 2023b, Theorem 1.7) to \(A^\#\) on \(F\) and take a bounded normal model isomorphic to it in codimension one. Normalization of the bounded models is still bounded after stratification and spreading, and does not add divisors to a codimension-one isomorphism.

The effective boundary \(B_Y'-c\mathcal S'\) has coefficients at least \(1-\delta-c\) on \(\mathcal S'\). Intersecting its Calabi–Yau equality with a fixed bounded polarization bounds its degree: the trace of the nef b-part is pseudo-effective, and the canonical degree is bounded in the bounded family. Consequently \[ \deg\mathcal S'=O(1),\qquad \deg J'=O(\tau),\qquad \deg\mathbf H_F,\ \deg\mathbf D_F=O(\tau). \tag{27}\] Here \(J'\sim_{\mathbb Q}\tau\mathcal S'\) on the fibre, and the last two degrees are nonnegative and sum to that of \(J'\).

We now use the assumed bigness of \(P_U\). It gives an effective rational divisor \(\Sigma\sim_{\mathbb Q}H\) such that for every \(v\) listed in \(\mathcal S\), with restriction \(w\), one has \[ w(\Sigma)>v(D)-v(E_0). \tag{28}\] For completeness, if \(v(D)>0\), write \(w=e\mathop{\mathrm{ord}}_T\). The discrepancy formula gives \(ea(k_i,T)<\delta\) and \(v(D)<\delta\). Since \(e|f_T|\le Cea(k_i,T)\) and \(\delta\) is sufficiently small, the congruence has no wrap and \(v(D)=ef_T\). Thus \(f_T>0\), and this prime was corrected in \(P_U\). An effective representative of the big class \(P_U\) may be chosen to vanish strictly at these primes and at any of the remaining finitely many projected centres. Every listed valuation has nontrivial angular restriction, since trivial restriction would give \(A=1\); its centre on \(U\) is therefore proper. Reserve a tiny ample summand and choose a sufficiently divisible member through all these centres. Adding back \(C_U\) gives (28). Pushing down yields an effective representative of \(H\) on \(V\); the representative on \(U\) is its pullback, since it differs from \(H_U\) by a rational principal divisor.

Every homogeneous normalized threshold section has divisor \[ (1-b)D+bE_0+p^*\Xi, \qquad 0\le b\le1, \qquad \Xi\sim_{\mathbb Q}(1+b)H. \tag{29}\] If \(b<1\), replace \(b\) by \(b+\alpha\) and \(\Xi\) by \(\Xi+\alpha\Sigma\) for a sufficiently small positive rational \(\alpha\). The divisor remains effective, while every \(\tau\mathcal S\) constraint improves strictly by (28). The finite list of strict improvements would increase the pseudo-effective threshold, a contradiction. Hence \(b=1\) for every such section. All these systems are torus invariant, because the listed valuations and the original \(J\) are torus invariant. Equal-degree ratios therefore lie in \(k(V)\), by their weight decompositions and \(b=1\). The valuation \(v_0=\mathop{\mathrm{ord}}_{E_0}\) is trivial on this field, so it is horizontal over the Iitaka base \(Z_\tau\).

We obtain a contradiction on the geometric generic fibre. Tailor a bounded ordinary lc complement preserving \(v_0\) as an lc place, using the effective zero-parameter generalized Calabi–Yau data with the two sections and the Fano type funding of Lemma 4. This may first be done on a small \(\mathbb Q\)-factorialization of the fibre, using the horizontal lift of \(v_0\), and then transferred to the bounded model. Work on its homogeneous horizontal chart. Proposition 5 bounds the Lipschitz constant of \(A^\#\). The difference \(A^\#-A\) is the order of \[\frac c\tau\left(\mathbf J-(J'-\tau\mathcal S')^*\right).\] The parenthesized b-divisor has effective trace \(\tau\mathcal S'\) of degree \(O(\tau)\) on this fibre and is b-nef up to a rational principal divisor. The shrinking-degree estimate of Corollary 8 therefore gives an \(O(1)\) Lipschitz bound for the difference, and hence for \(A\).

The same estimate gives an \(O(\tau)\) Lipschitz bound for \(h_0(v)=v(E_0)\). Indeed \(E_0\) has effective trace and is b-linearly equivalent to \(\mathbf D-\mathbf H\), whose two nef terms have degree \(O(\tau)\) by (27). In applying the estimate one replaces these nef terms by effective representatives with arbitrarily small error, avoiding the two centres in question. The function-degree argument on the bounded chart then applies: a rational function with pole degree strictly below \(1\) is constant. This is why the Lipschitz bound tends to zero, rather than merely remaining bounded.

Let \(w_0\) be a chart vector representing \(v_0\). We have \(h_0(w_0)=A(w_0)>0\). The shrinking Lipschitz bound implies \(|w_0/A(w_0)|\to\infty\). The horizontal approximation in Lemma 9, applied to this ray, gives integral divisorial vectors \(n\) and numbers \(s\to0^+\) such that \[A(n)=O(s)\longrightarrow0, \qquad h_0(n)\ge s/2.\] On the other hand, any such small-discrepancy valuation with \(h_0>0\) has \(v(D)=0\). Formula (11) and the lower signed phase bound at \(a(k_i)\), including (9), imply \(h_0=o(A)\); trivial restriction would instead make \(h_0\) a positive integer and is impossible. This contradicts the two displayed estimates. We have proved that \(P_U\) is not big.

Running the correction MMP.

Diagonalize the preceding conclusion to choose \(\eta_i\to0^+\) sufficiently slowly so that \(P_U\) is still not big. Run its MMP. All parameters used from now on may grow more slowly, remain below \(k_i\), and be \(o(1/\eta_i)\). On the output put \(\mathcal T'\) equal to the reduced sum of surviving correction primes. With \(\mathbf H\) now denoting the original angular b-pullback, MMP negativity and b-nefness give \[ 0\le C_U^*\le E:=\mathbf H-(P')^* \le(C')^*, \qquad C'\le O(\eta_i)\mathcal T'. \tag{30}\] More explicitly, if \(e_H=(H')^*-\mathbf H\ge0\), then \(E=(C')^*-e_H\), and \(P\)-negativity gives \((C')^*-C_U^*\ge e_H\).

Excluding a Mori fibre output. Suppose first that \(P_U\) is not pseudo-effective. On the geometric generic fibre \(F\) of the final anti-\(P'\) Mori contraction, put \(G=-P'\), which is ample. Choose a new slow parameter \(s_i\to\infty\), with \(s_i\eta_i\to0\), and obtain the phase condition at \(a(s_i)\) after a fixed scaling. Scale \(H,P_U,C_U\) together. The correction coefficients remain \(O(\eta_i)\), and \(P_U\) is integral at correction primes: subtracting their phases was precisely what made those coefficients integral. Thus \(R(G)\) is supported away from \(\mathcal T'\), and the pushed boundary satisfies \[B(s_i)'\ge R(G)+(1-o(1))\mathcal T'.\] Comparing with ordinary discrepancies on \(F\), the only adverse total-divisor error is \(s_ie_H\), which is at most \(s_i(C')^*\). B-nef negativity for the moduli part and (30) therefore give \[ A_{F,R(G)} \ge a(s_i)+\tfrac12(\mathcal T')^*. \tag{31}\] In particular \((F,R(G))\) is klt. A uniform ordinary gap is not asserted or needed in this case.

The relative Picard number of the Mori contraction is one. The classes used here, including \(R(G)\), are restrictions from this model. Horizontal prime coefficients restrict unchanged over the generic field and split with unchanged multiplicities after a separable constant extension, so the same class comparisons hold on the geometric generic fibre. There we have \[G+H'=C'\le O(\eta_i)\mathcal T',\] where \(H'\) is pseudo-effective. The zero-parameter generalized Calabi–Yau equality also shows that \(-(K_F+R(G))\) dominates a fixed positive multiple of \(\mathcal T'\) in this one-dimensional space of restricted classes: its boundary contributes \((1-o(1))\mathcal T'\), and the moduli trace is pseudo-effective. Consequently \[ -(K_F+R(G))\sim_{\mathbb Q}I_iG, \qquad I_i\longrightarrow\infty. \tag{32}\] Initially this is a numerical proportionality, with \(I_i\gg1/\eta_i\). It is rational linear proportionality after clearing denominators because \(F\) is rationally connected: pull numerically trivial Cartier multiples to a smooth rationally connected resolution, where they are torsion, and descend the resulting rational equivalence.

Consider the compactified cone for \((F,G)\) and a primitive divisorial valuation with ordinary discrepancy \(A\to0\) and infinity height \(u>0\). Its angular restriction \(w\) has \(A_{F,R(G)}(w)\le A\) and \(u\le A\) by (14). Moreover (30) and (31) imply \[ 0\le E(w)\le O(\eta_i)A_{F,R(G)}(w)\le O(\eta_i)A. \tag{33}\] Divisorial tests on the geometric fibre lift to homogeneous horizontal tests on the original model. Thus the signed phase bound for \(H\) applies, and \[u\equiv w(G)\equiv-w(H)+E(w)\pmod\mathbb Z.\] All three quantities have small signed representatives. There is no wrap, so the lower phase bound and (33) give \(u=o(A)\). If \(w=0\), the congruence makes \(u\) a positive integer, already impossible. Valuations with \(u=0\) satisfy the same conclusion trivially. This contradicts Lemma 14 and (32).

Descent from the semiample output. We are left with a semiample nonbig \(P'\) and its contraction \(g:U'\to W\). For a newly trimmed parameter range tending to infinity, with \(t\eta_i\to0\) uniformly on that range, use boundary and nef part \[ \widetilde B(t)=B(t/2)'-tC', \qquad \widetilde{\mathbf M}(t)=\mathbf M(t/2)+(t/2)\mathbf H. \tag{34}\] Their total logarithmic divisor is \(tP'\) up to rational linear equivalence, and their discrepancies are exactly \[ \widetilde a(t)=a(t/2)+tE. \tag{35}\] To see the equality, first pushing the old total divisor contributes \(-(t/2)e_H\); adding the nef term and subtracting \(tC'\) contributes another \(-(t/2)e_H+t(C')^*\). Their sum is \(tE\).

The new boundary dominates \(R(P')\). At correction primes \(P'\) is integral, the old boundary has coefficient \(1-o(1)\), and the amount subtracted is \(O(t\eta_i)=o(1)\). Away from them the old denominator condition is unchanged. The nef part is b-nef; (35) and \(E\ge0\) show that the discrepancies are increasing and klt, with effective Calabi–Yau data at zero.

This family has the required signed phase condition for \(P'\). Indeed, at a new growing parameter, apply the original condition to \(a(t/2)\) and scale all the relevant divisors by the resulting fixed integer. If \(\widetilde a(t,T)\to0\), then \[a(t/2,T)\le\widetilde a(t,T), \qquad E(T)\le\widetilde a(t,T)/t =o(\widetilde a(t,T)).\] Since \(P'(T)=H(T)-E(T)\), subtracting this negligible nonnegative order preserves the lower \(-o(\widetilde a)\) bound and the upper \(O(\widetilde a)\) bound for the phase.

On the geometric generic fibre of \(g\), the divisor \(P'\) is \(\mathbb Q\)-linearly trivial. Its integral principalization order is bounded by Lemma 15: along any putative unbounded-order subsequence choose one growing parameter with the just-proved phase bound, use the restricted effective generalized klt Calabi–Yau data, and clear the remaining bounded small denominators as in that lemma. These operations use only fixed multipliers. A small \(\mathbb Q\)-factorialization of the geometric fibre preserves all the divisor orders and permits the cyclic-cover construction.

The principalization descends from the geometric generic fibre to its field of definition. Indeed being Cartier descends faithfully flatly, and the geometrically trivial line bundle is trivial by Hilbert 90; equivalently its one-dimensional space of trivializing rational functions descends. Thus for a bounded positive integer \(q\) and some \(\phi\in k(U')^*\), the divisor \(qP'-\mathop{\mathrm{div}}(\phi)\) has no horizontal terms. There is then an exact equality \[ qP'-\mathop{\mathrm{div}}(\phi)=g^*H_W \tag{36}\] for an ample \(\mathbb Q\)-Cartier rational divisor \(H_W\). To justify exactness, semiampleness already gives \(P'\sim_{\mathbb Q}g^*A_W\) with \(A_W\) ample. After clearing denominators the difference between the left side of (36) and \(qg^*A_W\) is principal and vertical. A rational function with vertical divisor restricts to a nowhere vanishing regular function on the projective generic fibre, hence belongs to \(k(W)^*\) because \(g\) is a contraction. Absorb this rational principal divisor into \(qA_W\) to obtain \(H_W\).

If \(W\) is a point, (36) and \(E\ge0\) already give a section of a bounded multiple of \(H\). Otherwise \(W\) has strictly smaller dimension than \(V\). It is of Fano type. Apply the generalized canonical bundle formula to (34), and replace the old parameter \(s\) by \(s=qt\). The induced data on \(W\) are effective generalized data with total \(tH_W\), klt at zero and with nondecreasing discrepancies. Their nef parts are b-nef and \(\mathbb Q\)-Cartier.

We check the denominator condition on \(W\) explicitly. For a prime \(P\) on \(W\), choose a nonexceptional vertical component \(T\) above it with multiplicity \(e\). Let \(r\) be the denominator of \(\operatorname{coeff}_T(P')\) and let \(r'\) be that of \(\operatorname{coeff}_T(qP'-\mathop{\mathrm{div}}(\phi))\). Then \(r'\mid r\). The denominator \(r_P\) of \(H_W\) divides \(er'\), by (36). Since the upstairs boundary dominates \(R(P')\), the discriminant discrepancy at \(P\) satisfies \[b_t(P)\le\frac1{er}\le\frac1{er'}\le\frac1{r_P}.\] Thus the discriminant boundary dominates \(R(H_W)\).

For each growing parameter, the pure pullback phase condition descends by Lemma 2, after its allowed fixed multiple. Every hypothesis of Lemma 11 is now satisfied in smaller dimension. Induction gives bounded-degree sections of \(H_W\). They pull back through (36) to bounded-degree sections of \(P'\), and the b-divisor inequality \(\mathbf H\ge(P')^*\) in (30) carries them to sections of the same bounded multiples of \(H\) on \(V\). This contradicts (17) and completes the induction.

The contradiction excludes every sequence of failures with \(N_i\to\infty\), and hence proves boundedness for the original sequence. Any finite initial segment omitted in an eventual assertion can also be included: each of its big divisors has a section in some positive degree. Taking a common multiple of the bounded degrees gives the common degree in the statement. ◻

Proof. On \(U'\to W_\beta\) consider \[ d=\frac{\mathbf D-\mathbf P}{\beta}+\mathbf P. \tag{42}\] It is nonnegative and strictly positive on \(a=0\). The discrepancies \(a+td\) have effective traces for \(0\leq t\leq1\): at a surviving prime in \(I_\beta\) their trace increment is \(tD(Q)/\beta\leq1\), and at an unremoved floor prime it is \(tD(Q)\leq1\). Realize the family by a nef increment \(t\mathbf D/\beta\) and total log canonical divisor equal, up to a rational principal divisor, to the pullback of \(t(1/\beta-1)D_W\). Lemma 3 gives, on a fixed smaller interval, discriminants \[a'+td'\] with \(a'\) an effective generalized lc Calabi–Yau discrepancy over the point and with uniformly bounded denominator. These are klt for positive \(t\).

We first make explicit a lifting consequence needed here. If \(a'(E_i)=0\), \(d'(E_i)\to0\), and \(k_i>0\) tends to zero, then, after a subsequence, there are primitive lifts \(T_i\) with \[ T_i|_{k(W)}=e_i\mathop{\mathrm{ord}}_{E_i},\qquad a(T_i)=0,\qquad d(T_i)\to0,\qquad e_i k_i\to0. \tag{43}\] At one Mori step, choose a minimizer of \(d\) on the \(a\)-minimum by computing the discriminant at a sufficiently small positive parameter. Its tailored vertical chart from Proposition 5 has normalized values \(0,\rho d'\) on the slice \(e_*=1\). Lemma 9, with a fixed small error \(\gamma\), gives an integral point with \(e_*=lj\) bounded in terms of \(\gamma\), by rounding \(ljU\) in the kernel lattice. The error in \(a\) is less than the gap of its value lattice; nonnegativity and quantization therefore give \(a=0\) exactly. The slope bound in the same chart gives \[d\leq e_*\rho d'+C\gamma.\] First fix \(\gamma\) and go sufficiently far in the sequence, then let \(\gamma\downarrow0\) diagonally slowly enough that also \(e_*k_i\to0\). Primitivizing the resulting order only decreases the orders and the restriction multiplicity. Iterate through the bounded chain of Mori steps, at each step including the already accumulated multiplicity times \(k_i\) among the quantities required to be negligible. Birational steps preserve the discrepancy functions. This proves (43) without asserting a uniform bound for \(e_i\).

There cannot be a sequence of primes \(E\) with, for one fixed sufficiently small \(t>0\), \[ (a'+td')(E)\longrightarrow0, \qquad 0<D_W(E)/\beta\longrightarrow0. \tag{44}\] The bounded denominator gives \(a'(E)=0\) eventually. Apply (43) with \(k_i=D_W(E)/\beta\). It gives \[a(T)=0,\qquad \frac{D(T)}{\beta}\leq d(T)+\frac{eD_W(E)}{\beta}\longrightarrow0.\] Thus \(T\in I_\beta\), where \(\mathbf P(T)=0\), whereas \(\mathbf P(T)=eD_W(E)>0\), a contradiction. Nor can an original prime \(E\) of \(W\) have \(0<D_W(E)/\beta\to0\). A prime \(Q\) above it on \(U'\) satisfies \[D(Q)=\mathbf P(Q)=eD_W(E)>0,\quad a(Q)=0, \quad a'(E)=0,\quad d'(E)\leq D_W(E).\] It would again give (44).

Choose fixed rational \(0<t\leq1/4\) in the chamber. Extract every exceptional prime of \(A=a'+td'\) below a sufficiently small fixed \(\delta_*<1/4\). This finite extraction is funded by mixing \(a'\) slightly with an ordinary klt Calabi–Yau pair, as in Subsection 1.3; the extracted primes have \(a'=0\) and positive boundary coefficient. The preceding exclusions imply a uniform positive lower bound on all nonzero coefficients of \[E_0=(D_W/\beta)|_{\widehat W}.\] Here and below the notation means the rational pullback. The ordinary discrepancies on the extraction dominate those of \(A\), by b-nef negativity and \(\mathbb Q\)-factoriality. Every remaining exceptional primitive prime has \(A\geq\delta_*\), and every extracted prime has ordinary discrepancy one. This proves a uniform ordinary gap on \(\widehat W\).

For completeness, we verify the additional ordinary gap needed to apply polarized birationality to an integral Weil divisor. Let \(B_A\) be the trace boundary of \(A\) on \(\widehat W\). On every component of \(E_0\) its coefficient is at least \(1/2\). For an original component the covering-prime calculation above gives \(a'=0\) and \(d'\leq D_W(\cdot)\leq1\); extracted components have \(A<\delta_*\). Choose a fixed integer \(m>4/\delta\) and an integral divisor \[N=mE_0-Q_0,\] where the coefficients of \(Q_0\) belong to \([1,2]\) on \(\mathop{\mathrm{Supp}}E_0\) and are zero elsewhere. Such rounding is possible coefficient by coefficient. Since \(mE_0-Q_0\) dominates a positive multiple of \(E_0\), \(N\) is big. Run an \(N\)-MMP to a nef and big integral Weil divisor \(N^+\) on a \(\mathbb Q\)-factorial model. With stars denoting pullbacks to a common determination, b-nef negativity and MMP negativity give \[ 0\leq e_E=(E_0^+)^*-E_0^*,\qquad me_E\leq(Q_0^+)^*-Q_0^*. \tag{45}\] The total generalized log canonical divisor of \(A\) is \(t(1-\beta)E_0\) up to a rational principal divisor. Comparing its nef part before and after the MMP, ordinary discrepancies on the output are at least \[ A+(B_A^+)^*-t(1-\beta)e_E \ \geq\ A+\left(\frac14-\frac tm\right)(Q_0^+)^*. \tag{46}\] Indeed \(B_A^+\geq Q_0^+/4\) and \(e_E\leq(Q_0^+)^*/m\) by (45). If a primitive prime with \(A<\delta_*\) is in \(\mathop{\mathrm{Supp}}E_0\), it has \((Q_0^+)^*\geq Q_0^*\geq1\). If it is not in this support, then it is already a prime on \(\widehat W\) and cannot be contracted by the \(N\)-MMP: \(N\) has coefficient zero there, the output pullback is effective, and a contraction would require strictly positive MMP error. Its ordinary discrepancy on the output is one. All other primitive orders have \(A\geq\delta_*\). Formula (46) therefore gives a uniform ordinary gap on the output. Also \(N^+-K^+\) is pseudo-effective, since the output is of Fano type. Birkar’s integral-Weil polarized birationality theorem (Birkar 2023a, Theorem 1.1) now gives birational sections of a bounded multiple of \(N^+\). Negativity identifies these with sections of that multiple of \(N\), hence with functions in \(S(CD_W/\beta)\) for bounded \(C\). This proves the field-generation assertion.

Choose algebraically independent \(\phi_j\) in this space. If \(\omega=\bigwedge_jd\phi_j\), define on the base \[ H(E)=a'(E)-\nu_{\log,E}(\omega),\qquad \nu_{\log,E}(\omega)=1+\mathop{\mathrm{ord}}_E(\omega). \tag{47}\] The Calabi–Yau identity shows that \(H\) is the moduli b-divisor after a rational principal adjustment. It is thus b-nef and \(\mathbb Q\)-Cartier, and its values have bounded denominator because \(a'\) does and the log orders are integral. If \(T|_{k(W)}=e\mathop{\mathrm{ord}}_E\) and \(a(T)=0\), the discriminant minimum gives \(a'(E)=0\). Pole bounds for the \(\phi_j\) on every model give \[\nu_{\log,E}(\omega)\geq-CD_W(E)/\beta.\] If all \(E(\phi_j)\geq0\), the form is regular at that discrete valuation ring and its logarithmic order is at least one. These prove (i). For (ii), if restriction is nontrivial, then \(D_W(E)=0\) by \(\mathbf P\leq\mathbf D\), so the functions are regular and \[H(Q)=e a'(E)-e\nu_{\log,E}(\omega) \leq a(Q)-e\leq0.\] The trivial restriction gives \(H(Q)=0\) directly.

On \(W\) itself we also have \(H_W\leq C D_W/\beta\): use \(a'\leq1\), the lower coefficient bound, and the same log-order estimate. For any arbitrarily small positive rational \(\lambda\), choose on a determining model an effective \[\Sigma\sim_\mathbb QH+\lambda\mathbf D_W/\beta;\] this is possible because the displayed class is nef and big. The rational principal order \[\chi=\mathbf\Sigma-H-\lambda\mathbf D_W/\beta\] has poles at most \(C D_W/\beta\) on \(W\), and therefore at most \(C\mathbf D/\beta\) upstairs. Condition (41), applied to \(D+(\beta/C)\chi\), gives \(\chi(T)\leq C\eta D(T)/\beta\). Since \(\mathbf\Sigma\geq0\), \[H(T)\geq-C\eta D(T)/\beta-\lambda\mathbf P(T)/\beta.\] Let \(\lambda\downarrow0\) to prove (iii). Finally, for \(T\in I_\beta\), \(\mathbf P(T)=0\), so each \(\phi_j\) has nonnegative order there. If the restriction were nontrivial, (i) and (iii), together with \(D(T)\leq\beta\), would give \(-1\geq H(T)\geq-C\eta_i\), which is impossible eventually. ◻

Proof. The hypothesis on nonfloor primes also holds on \(U\), since every additional extracted prime is on the floor. If \(c_g>0\), condition (41) applies to \(G/(c_gL)\). If \(c_g=0\), the effective rationally trivial divisor \(G\) is zero. Consequently, at every floor order to which the smallness hypothesis applies, \[ G(T)\leq c_gL(1+\eta_i)D(T). \tag{50}\]

Suppose the conclusion fails. Factorial diagonalization gives a subsequence and primitive targets \(T\) with \[ a(T)=0,\quad\beta=D(T),\quad L\beta\longrightarrow0,\qquad \frac{\operatorname{dist}(q\ell(T),\mathbb Z)}{L\beta} \longrightarrow\infty \quad\text{for each fixed }q\in\mathbb N_{>0}. \tag{51}\] To justify the simultaneous last assertion, at stage \(i\) choose a failure for the multiplier \(i!\) with ratio tending sufficiently quickly to infinity. For \(q\mid i!\), \(\operatorname{dist}(i!x,\mathbb Z)\leq(i!/q) \operatorname{dist}(qx,\mathbb Z)\); choosing the failure ratio larger than all these factors gives the claimed diagonal sequence. Among such sequences maximize the eventually constant \(\mathop{\mathrm{trdeg}}K_\beta\). Lemma 16 says that \(T\) is horizontal over this field. Let \(\tau\) be the first positive cutoff for which \(K_\tau=K_\beta\). It exists because there are finitely many coefficient cutoffs and the field \(K_{0^+}=k(V)\) cannot have a nonzero horizontal divisorial valuation. In particular, \[ 0<\tau\leq\beta, \qquad L\tau\longrightarrow0. \tag{52}\] After multiplying \(\ell\) and \(G\) by one fixed integer, maximality implies a uniform bound \[ \operatorname{dist}(\ell(Q),\mathbb Z)\leq CLD(Q) \quad(a(Q)=0,\ D(Q)<\tau). \tag{53}\] Indeed, a failure of this assertion for every fixed multiple would, by the same factorial diagonalization, produce a sequence at a cutoff strictly before \(\tau\), hence at strictly larger transcendence degree.

Proof of the field assertions. Because \(J\) is integral and \(L_0=O(L)\), choose a fixed small \(c>0\) so that \[a(T)=0,\quad LD(T)\leq c \quad\Longrightarrow\quad J(T)=0.\] There is, after a subsequence, a single field plateau containing two cutoffs \[ 0<\beta_0\leq\beta_1\leq c/L,\qquad L\beta_0\longrightarrow0,\qquad L\beta_1\geq c'>0,\qquad K_{\beta_0}=K_{\beta_1}=K_*. \tag{74}\] Here is a precise reason for this choice. Pass to limits of the finitely many change locations multiplied by \(L\), adding dummy endpoints if the number of changes varies. The last change tending to zero lies below a cutoff \(\beta_0\) with \(L\beta_0\to0\); choose it decreasing slowly enough to lie beyond that change. The next change either has a positive limiting scaled location or lies beyond \(c/L\). Choose \(\beta_1\) strictly before it, with \(L\beta_1\) fixed positive and at most \(c\). This gives (74). Passing again fixes \(p=\mathop{\mathrm{trdeg}}K_*\).

All \(I_{\beta_1}\) belong to \(\mathcal L\), so Lemma 16 makes them horizontal over \(K_*\). It follows, for every rational \(m>0\), that \[ S(mP_{\beta_0})=S(mP_{\beta_1}). \tag{75}\] Indeed a function in the left side belongs to \(K_*\): enlarge \(m\) to an integer and use the section-field description. Its order at every removed prime in \(I_{\beta_1}\) is zero by horizontality. Removing those pole allowances therefore leaves it in the right side. The reverse inclusion follows from \(P_{\beta_1}\leq P_{\beta_0}\). For fixed \(C_1\), a function in \(S(C_1LD)\) has no pole at \(I_{\beta_0}\) eventually, because its order is integral and \(C_1L\beta_0<1\). Hence \[S(C_1LD)\subset S(C_1LP_{\beta_0}) =S(C_1LP_{\beta_1})\subset K_*.\] This proves (i). The cutoff generation statement at \(\beta_1\), and \(\beta_1\geq c'/L\), place a birational field-generating space for \(K_*\) inside \(S(B_0LD)\) for fixed \(B_0\). Enlarging \(B\) will therefore make \(S(BLD)\) generate \(K_*\); a basis contains \(p\) algebraically independent elements, by greedily choosing a transcendence basis from any field-generating set.

We prove that some fixed \(B\) forces the asserted exact triviality. Otherwise choose \(B_i\to\infty\) slowly and \(T_i\in\mathcal L\) that are nonnegative on \(S(B_iLD)\) but nontrivial on \(K_*\). Since \(K_1=k\), there is a change at \(\beta\geq\beta_1\) where \(T\) is trivial on \(K_\beta\) but nontrivial on the preceding field \(K_{\beta_-}\). Take \(\beta_-\) sufficiently close to \(\beta\) from below that \(\beta_->\beta/2\). Use \(H,\phi_j\) from Lemma 16 at \(\beta_-\). Their pole bounds are \(C\mathbf D/\beta\), and \(1/\beta=O(L)\); thus \(T(\phi_j)\geq0\) eventually. Its nontrivial restriction gives \[ H(T)\leq-1. \tag{76}\]

If \(L_0\beta\to\infty\), then \(L_0\to\infty\) because \(\beta\leq1\). The same bounded-cost field-generating space for \(K_{\beta_-}\) is contained in \(S(L_0D)\) eventually. The hypothesis of triviality on the field generated by the latter space contradicts the choice of \(T\). It remains to treat a subsequence on which \(L_0\beta\) is bounded.

Restrict to the geometric generic fibre of the cutoff contraction \(U^\beta\to W_\beta\). There \(T\) is horizontal and \[D'\leq\beta\sum_{I_\beta}Q'.\] Apply the extra-data part of Lemma 3, over a point, with \[ \widetilde a=a+\kappa J,\qquad d=c_* =\mathbf D/\beta,\qquad h=H, \tag{77}\] where \(\kappa>0\) is fixed rational and sufficiently small; \(c_*\) denotes the additional nonnegative direction of that lemma. We verify all its hypotheses. The orders \(\widetilde a\) have bounded denominator because \(a\) and \(J\) are integral. The order \(H\) has bounded denominator by the cutoff lemma. Both \(J\) and \(\mathbf D/\beta\) are b-nef, and \(H\) is pulled back from a b-nef divisor on \(K_{\beta_-}\); after a birational determination it restricts to a b-nef divisor on this fibre. Their Calabi–Yau realizations hold the total class rationally trivial. At floor primes on this fibre, \[H\leq C D/\beta,\qquad J\leq C D/\beta,\] where the second bound uses boundedness of \(L_0\beta\). At other primes on the model, \(a=1\), \(D=0\), \(J=0\), and \(H\leq0\). These inequalities give effective traces on uniform small positive ranges in (77), including the family with \(\lambda H+sc_*\) for \(0<s\leq\lambda\theta_0\). The family \(\widetilde a+td\) is klt for \(t>0\) because \(D\) is positive on \(a=0\). Finally, \[\widetilde a=0\quad\Longleftrightarrow\quad a=J=0, \qquad -C\eta_i c_*\leq H\leq Cc_* \quad\text{on this zero set}.\] The lower bound is part (iii) of the cutoff lemma, using the assumed uniform (41). Thus the exact sign conclusion of Lemma 3 over a point prohibits \(H<0\) at the lift of \(T\), contradicting (76). This proves (ii).

The proof of (iii), including its common normalization \(v_i\), is given below after establishing the adjoint lattice-point statement. For the extension to rational monomial subcones, take any finite list of tested functions, resolve their divisors, and subdivide the cone so their orders are linear on each piece. Divisorial rational rays are dense in each rational piece. The exact zero assertions extend by continuity; applying this separately to each function gives triviality on the whole field. ◻

Proof. Write \(q:\mathcal X\to X\) for the coarse morphism. For each prime \(B\) of \(X\), let \(P\) be its reduced inverse-image divisor and \(e_B\) its generic stabilizer order. Thus \(q^*B=e_BP\). Rational divisors on \(\mathcal X\) descend uniquely to rational divisors on \(X\); use a subscript \(c\) for the descended divisor. The local normal-direction equation \(t=z^{e_B}\) gives \[K_{\mathcal X}=q^*(K_X+R),\qquad R=\sum_B(1-e_B^{-1})B.\] Choose compatible canonical divisors using one rational top form. Set \(\Delta=R+\Theta_c-\{D_c\}\). If the coefficients of \(D\) and \(\Theta\) at \(P\) are \(d\in\mathbb Z\) and \(\theta\in[0,1)\), write \(d=e_Bn+s\) with \(0\le s<e_B\), also when \(d\) is negative. Then \[\operatorname{coeff}_B\Delta =\frac{e_B-1+\theta-s}{e_B}\in[0,1).\] In particular \(\Delta\) is effective.

The pair \((X,\Delta)\) is klt, including at higher-codimensional quotient singularities. Indeed, etale locally on \(X\) there is a finite quotient presentation \([U/G]\), with \(U\) smooth and \(f:U\to U/G\) finite. Its log pullback boundary is \(\Theta_U-f^*\{D_c\}\). This possibly negative boundary is bounded above by the simple normal crossing boundary \(\Theta_U\). Thus for every divisorial valuation \(F\) over \(U\), \[A(F;U,\Theta_U-f^*\{D_c\}) =A(F;U,\Theta_U)+F(f^*\{D_c\})>0.\] The finite-map discrepancy formula multiplies downstairs log discrepancies by positive ramification indices, so the downstairs pair is klt. This argument requires only local quotient charts.

The coarse pushforward is \[q_*\mathcal O_{\mathcal X}(D)=\mathcal O_X(\lfloor D_c\rfloor).\] To check it, a rational section \(h\) is regular along \(P\) precisely when \(e_B\operatorname{ord}_B(h)+d\ge0\), equivalently \(\operatorname{ord}_B(h)+\lfloor d/e_B\rfloor\ge0\). Both sides extend these codimension-one conditions across codimension at least two: on a smooth finite quotient chart this follows by normal extension and then taking invariants. The stack is tame in characteristic zero, so coarse pushforward is exact and identifies the cohomology groups.

Quotient singularities are locally \(\mathbb Q\)-factorial. Consequently \(M=\lfloor D_c\rfloor\) is an integral \(\mathbb Q\)-Cartier Weil divisor, and \[M-(K_X+\Delta) =D_c-(K_X+R+\Theta_c)\] is nef and big. Kawamata–Viehweg vanishing for a klt pair and an integral \(\mathbb Q\)-Cartier Weil divisor gives the result. We use (Fujino 2024, Theorem 3.2.4, manuscript version of 9 October 2023); the additional log-bigness condition on log canonical centres is vacuous for a klt pair. The characteristic-zero statement also follows from its complex form by descending the finite-type data to a finitely generated field and using invariance of nefness, bigness and cohomology under algebraically closed field extension. ◻

Proof. The shift by \(\mathbf1\) supplies the adjoint correction in this argument. A weighted blowup will convert it into the canonical discrepancy, and the same shift will place the required monomial in a multiplier ideal on the exceptional divisor. Vanishing then lifts that monomial to a global section.

Preparing sections with weighted initials. Resolve \((X,\Delta)\) by \(f:Y\to X\), isomorphically over a neighborhood of the chosen point \(x\). Put \(N_Y=f^*N\) and define the log pullback \(\Delta_Y\) by \[K_Y+\Delta_Y=f^*(K_X+\Delta).\] The exceptional coefficients of \(\Delta_Y\) may be negative. This will cause no difficulty: our vanishing boundary will be a fractional rounding correction, not \(\Delta_Y\). Both \(N_Y\) and \(\Delta_Y\) have zero divisor support near \(x\) after compatible local trivialization. Choose canonical divisors with no support near \(x\) as well.

Let \(C\) be the closed cone generated by the actual pairs \((j,\nu(s))\). It has slice \(K\) at degree one. Choose rational \(0<t_0<t\) with \((t_0,\alpha+\mathbf1)\in\operatorname{int}C\). There is a cone generated by finitely many actual pairs that already contains this point in its interior: approximate the vertices of a small simplex around the point by finite nonnegative combinations of actual pairs. Sufficiently close approximations still surround the point.

Choose positive primitive integral weights \(u_1,\ldots,u_p\) that replace the Taylor order by scalar weight on this finite list, each listed section having its specified initial as a single weight initial. Also require \[\gamma\text{ precedes }\alpha\quad\Longrightarrow\quad u\cdot\gamma<u\cdot\alpha.\] Here is the finite-order justification. Take an integer \(d\) at least the total degree of every listed initial and of \(\alpha\). Weights sufficiently close to \((1,\ldots,1)\) preserve every strict total-degree comparison through degree \(d\), and satisfy \((d+1)\min u_i>d\max u_i\), so all higher-degree monomials are larger than every relevant initial. Successive sufficiently small rational perturbations realize lexicographic comparisons within the finitely many monomials of degree at most \(d\). Clearing denominators and dividing by the common divisor preserves all strict comparisons. This also proves the assertion for formal Taylor expansions.

Put \(p_0=u\cdot\alpha\) and \(U=u\cdot(\alpha+\mathbf1)\). The finite cone meets the plane with degree \(t_0\) and weight \(U\) in a neighborhood of \((t_0,\alpha+\mathbf1)\) relative to that entire plane. Choose finitely many rational points surrounding it there. Write these points as nonnegative rational combinations of the cone generators and clear denominators. Taking products, and then common powers as necessary, gives a sufficiently divisible positive integer \(l\) and sections of the Cartier divisor \(lt_0N_Y\) with single weight initials \(z^{\beta^{(1)}},\ldots,z^{\beta^{(r)}}\) such that \[u\cdot\beta^{(a)}=lU,\qquad \alpha+\mathbf1\in \operatorname{relint}\operatorname{conv} \{\beta^{(1)}/l,\ldots,\beta^{(r)}/l\}.\] The relative interior contains a neighborhood in the whole plane \(u\cdot v=U\). Pulling these sections to \(Y\) is legitimate after clearing denominators: the effective divisor of each product is \(\mathbb Q\)-Cartier and its rational pullback is effective.

The weighted modification and the vanishing divisor. Let \(\pi:\mathcal Z\to Y\) be the smooth stack weighted blowup at \(x\) with weights \(u\), and let \(E\) be its exceptional Cartier divisor. On the coordinate model it is \[\left[(\mathbb A^{p+1}_{x_1,\ldots,x_p,\tau} \setminus\{x_1=\cdots=x_p=0\})/\mathbb G_m\right], \qquad \rho\cdot(x_i,\tau)=(\rho^{u_i}x_i,\rho^{-1}\tau), \qquad z_i=\tau^{u_i}x_i.\] Its exceptional divisor is the weighted projective stack \(\mathbb P(u)\). This description pulls back along an etale parameter map near \(x\) and glues to the identity away from \(x\). Equivalently use the weighted ideal filtration generated by monomials of weight at least \(k\); each ideal is supported on a finite infinitesimal neighborhood of \(x\) and extends as the unit ideal outside \(x\). The finitely generated weighted Rees algebra makes the coarse modification projective. The stack is smooth with finite stabilizers and trivial generic stabilizer. On the displayed atlas the Jacobian gives \[K_{\mathcal Z}=\pi^*K_Y+(\textstyle\sum_i u_i-1)E.\] For \(p=1\) this construction is the identity modification and \(E=x\); the same formulas and restriction sequence apply.

Write \(\Delta_{\mathcal Z}=\pi^*\Delta_Y\), supported away from \(E\), and define \[P_0=\pi^*(tN_Y)-p_0E,\qquad H_0=\pi^*(lt_0N_Y)-lUE.\] The divisor \(P_0\) is rational globally but integral Cartier near \(E\); \(H_0\) is Cartier globally. The selected sections, after their common weighted order is removed, are sections of \(H_0\). Their base ideal \(\mathfrak b\) does not vanish identically on \(E\), and its restriction to \(E\) is generated by their monomial initials. Take a projective log resolution \(\sigma:\mathcal Y\to\mathcal Z\) of these data and all rational divisor supports used below, with strict transform \(\widetilde E\). Arrange that the total support is simple normal crossing, that \(\sigma\) is an isomorphism at the generic point of \(E\), and that \(\tau:\widetilde E\to E\) resolves \(\mathfrak b|_E\). First resolve the additional divisor supports, then principalize the ideal with the resulting simple normal crossing boundary, including the transform of \(E\), on smooth charts (Włodarczyk 2005, Theorems 2.4.1–2.4.3). Smooth functoriality identifies these constructions on chart overlaps, so they descend to the required smooth stack; the sequence of blowups has projective coarse space. Write \[\mathfrak b\mathcal O_{\mathcal Y}=\mathcal O_{\mathcal Y}(-G).\] Then \(G\ge0\) does not contain \(\widetilde E\), and \(\sigma^*H_0-G\) is globally generated.

Set \[Q=P_0-K_{\mathcal Z}-\Delta_{\mathcal Z}-E, \quad R=\sigma^*Q-G/l, \quad A=K_{\mathcal Y}+\lceil R\rceil, \quad\Theta=\lceil R\rceil-R.\] Thus \(A\) is an integral Cartier divisor, and \(\Theta\) is effective with simple normal crossing support and coefficients in \([0,1)\). The weighted canonical identity, \(U-p_0=\sum u_i\), and \(K_Y+\Delta_Y\sim_{\mathbb Q}0\) give \[Q-H_0/l\sim_{\mathbb Q}(t-t_0)\pi^*N_Y.\] Consequently \[A-(K_{\mathcal Y}+\Theta)=R \sim_{\mathbb Q} (t-t_0)\sigma^*\pi^*N_Y+(\sigma^*H_0-G)/l.\] The first summand is nef and big and the second is nef. Lemma 19 therefore gives \(H^1(\mathcal Y,\mathcal O(A))=0\) and the restriction surjection \[H^0(\mathcal Y,A+\widetilde E) \longrightarrow H^0(\widetilde E,(A+\widetilde E)|_{\widetilde E}).\]

The boundary monomial. We now exhibit the required section on the exceptional divisor. Near \(E\) the original boundary vanishes and \(P_0\) is integral. Thus, on a neighborhood of \(\widetilde E\), \[A=K_{\mathcal Y} +\sigma^*(P_0-K_{\mathcal Z}-E)-\lfloor G/l\rfloor.\] Adjunction identifies the restricted bundle with \[\tau^*(P_0|_E)+K_{\widetilde E}-\tau^*K_E -\lfloor(G|_{\widetilde E})/l\rfloor.\] Round-down commutes with this restriction: simple normal crossings prevent two components of \(G\) from restricting to a common prime divisor on \(\widetilde E\), and each disconnected intersection retains the same coefficient. This is the log-resolution expression for \(P_0|_E\otimes\mathcal J_E((\mathfrak b|_E)^{1/l})\).

The weighted-homogeneous monomial \(z^\alpha\) is a global section of \(P_0|_E=\mathcal O_E(p_0)\), after the fixed local frame. On the chart \(x_k\ne0\), use the smooth finite atlas setting \(x_k=1\), with residual group \(\mu_{u_k}\). The ideal becomes the ordinary monomial ideal generated by \(x_{\mathrm{rem}}^{\beta^{(a)}_{\mathrm{rem}}}\) and the proposed section becomes \(x_{\mathrm{rem}}^{\alpha_{\mathrm{rem}}}\). Dropping coordinate \(k\) is an affine isomorphism from the plane \(u\cdot v=U\) to \(\mathbb R^{p-1}\), because \(u_k>0\). Hence \(\alpha_{\mathrm{rem}}+\mathbf1\) lies in the ordinary interior of the convex hull of the projected vectors \(\beta^{(a)}_{\mathrm{rem}}/l\), and therefore in the interior of the scaled Newton polyhedron. Howald’s monomial multiplier-ideal formula (Howald 2000, Main Theorem) gives the required membership on this smooth atlas. It is equivariant for \(\mu_{u_k}\), so descends on every chart. No fractional coordinate exponents are introduced: the root needed to set \(x_k=1\) is already part of the atlas.

Lifting and pushing down. The restriction surjection now lifts this monomial. Near \(E\) its lifting bundle differs from \(\sigma^*P_0\) by \[K_{\mathcal Y}-\sigma^*K_{\mathcal Z} -\sigma^*E+\widetilde E-\lfloor G/l\rfloor.\] The part preceding the last term is exceptional over \(\mathcal Z\); indeed it is effective by the smooth blowup discrepancy calculation for the smooth pair \((\mathcal Z,E)\). At every nonexceptional prime the displayed correction is nonpositive. Therefore the lifted rational section pushes down to a regular section of \(P_0\) near \(E\). Its restriction to \(E\) is \(z^\alpha\), initially generically and then everywhere by regularity. Pushing further to the smooth neighborhood of \(x\) gives a regular section with weight initial exactly \(z^\alpha\).

It remains to check global regularity downstairs, since \(N_Y\) was rational and \(\Delta_Y\) could have negative coefficients. At an original prime \(B\) of \(X\) away from \(x\), write \(n_B=\operatorname{coeff}_B(tN)\in\mathbb Z\), \(\delta_B=\operatorname{coeff}_B\Delta\ge0\), and let \(g_B\ge0\) be the corresponding coefficient of \(G\). Canonical coefficients cancel inside the ceiling, and the allowed divisor coefficient of the pushed lifting bundle is \[n_B+\lceil-\delta_B-g_B/l\rceil\le n_B.\] Thus no forbidden pole appears at any original prime. The point \(x\) was already handled by the local weighted calculation (also when \(p=1\)). Normality now gives a section of \(\mathcal O_X(tN)\). Its weight initial excludes every monomial preceding \(\alpha\) in the original Taylor order, by the choice of \(u\), and the coefficient of \(z^\alpha\) is nonzero. Its Taylor initial is therefore \(\alpha\). ◻

Proof. On a nonempty open set, the map given by \(|N|\) is an immersion and has no base point, because it is birational in characteristic zero. At the chosen general point its values and first derivatives thus span all first jets. Suitable linear combinations give initial exponents \(0,e_1,\ldots,e_p\), proving \(\Sigma\subset K\). The order of vanishing at this point of a section of \(jN\) is at most \(Cj\): intersect its effective divisor with a general complete intersection curve through the point, chosen not to lie in its support. Its degree is \(jN\cdot H^{p-1}\) for a fixed sufficiently ample \(H\), and bounds the local intersection multiplicity. The constant may depend on this individual example, which suffices to make \(K\) bounded.

For a finite-dimensional space of germs, the number of distinct leading exponents is its dimension, by successive elimination of leading coefficients; alternatively use the filtration by the well-ordered monomials, whose nonzero graded pieces are one-dimensional. The exponent set at degree \(n\) is contained in \(nK\cap\mathbb Z^p\). The disjoint half-open unit boxes based at these lattice points lie in \[nK+[0,1]^p\subset nK+p\Sigma\subset(n+p)K.\] Their total volume is their number, proving the upper bound.

For the lower bound let \(y=nx\in nK\). Write \(x_i=a_i+f_i\) with \(a_i\in\mathbb Z_{\ge0}\) and \(0\le f_i<1\). Choose \(n\) integers in coordinate \(i\), namely \(a_i+1\) in exactly \(\lfloor nf_i\rfloor\) of the positions and \(a_i\) in the others. Doing this independently in every coordinate gives \(\alpha^{(1)},\ldots,\alpha^{(n)}\) with \[\sum_{r=1}^n\alpha^{(r)}=\lfloor y\rfloor, \qquad 0<\alpha_i^{(r)}+1-x_i<2.\] When \(f_i=0\) all entries equal \(a_i\), so the strict inequalities still hold. Therefore \(\alpha^{(r)}+\mathbf1-x\in\operatorname{int}((t-1)\Sigma)\): all coordinates are positive and their sum is less than \(2p<t-1\). Since \(\Sigma\subset K\) and \(x\in K\), a neighborhood of \(\alpha^{(r)}+\mathbf1\) lies in \(K+(t-1)K=tK\). Lemma 20 supplies a degree-\(t\) section with each exponent \(\alpha^{(r)}\). Their product has exponent \(\lfloor y\rfloor\).

Consequently all lattice points \(\lfloor y\rfloor\), for \(y\in nK\), occur among product initials. Their unit boxes cover \(nK\), so their number is at least \(\operatorname{vol}(nK)=n^p\operatorname{vol}(K)\). Sections with distinct initials are linearly independent, proving the claimed lower bound. The case \(p=0\) is immediate separately. ◻

Proof of part (iii) of Lemma 18. If \(p=0\), then \(K_*=k\). By part (i), each fixed-cost section space is eventually the one-dimensional constant space. Take \(v_i=1\). Assume \(p>0\) and use the extraction \(\widehat W\) from Lemma 16 at \(\beta_1\) of (74). Let \(D_*\) be the pullback of \(D_W\) to \(\widehat W\). For every rational \(m>0\), the cutoff spaces identify exactly as function spaces: \[ S(mP_{\beta_1})=S(mD_*). \tag{78}\] The negative-MMP error is effective and exceptional, so pushing and pulling the divisor inequality for an individual rational function identifies the space with the one on the semiample output. A function in that space belongs to \(k(W)\); the equality \(P_{\beta_1}'=p^*D_W\) then identifies its inequalities with those downstairs. Pullback to \(\widehat W\) does not change them. This argument works for each rational \(m\), with no requirement that \(mP_{\beta_1}\) be Cartier.

Every positive coefficient of \(LD_*\) is at least \(c'\delta\), because \(L\beta_1\geq c'\) and the coefficients of \(D_*/\beta_1\) are at least \(\delta\). Choose a fixed sufficiently large integer \(M\) and put \[ N=\lfloor MLD_*\rfloor. \tag{79}\] The coefficient lower bound gives \(N\geq LD_*\) once \(M\) is large enough. Increase \(M\) so that \(N\geq C D_*/\beta_1\), with the constant from the field-generation construction. Then \(S(N)\) defines a birational map and \(N\) is big and integral. Run its MMP to a nef and big integral Weil divisor \(N'\) on a \(\mathbb Q\)-factorial projective Fano type variety \(X'\). For every positive integer \(j\), negativity gives \[ S(jN)=S(jN'),\qquad S(jN')\subset S(jMLD). \tag{80}\] Indeed the first equality follows by pushing an effective divisor and, in the reverse direction, adding the effective exceptional MMP error; the second follows from \(N\leq MLD_*\) and (78). For every fixed positive integer \(n\), part (i) and the exact plateau identities give eventually \[ S(nLD)\subset S(nLD_*)\subset S(nN)=S(nN'). \tag{81}\]

Choose an effective ordinary rational klt Calabi–Yau boundary \(\Delta\) on \(X'\). Such a boundary is obtained from a klt Fano type boundary by adding a general sufficiently divisible rational member of its ample anti-log-canonical class. At a general smooth point outside the boundary and divisor supports, let \(K_i\) be the jet body of \(N'\) in Lemma 20. Its volume is positive and finite by Lemma 21. Set \[v_i=\mathop{\mathrm{vol}}(K_i).\] Equations (81) and Lemma 21 imply \[\dim S(nLD)\leq(n+p)^p v_i \leq(p+1)^p v_i n^p.\] Fix once and for all an integer \(t>1+2p\). The same lemma supplies a space of products of \(n\) sections of \(tN'\) of dimension at least \(v_i n^p\). By (80), \(S(tN')\subset S(tMLD)\). Enlarge the constant \(B\) in part (ii) to at least \(tM\). Those products then lie in the product space in (73), proving the lower bound with the same \(v_i\). Both \(t\) and \(M\), and hence \(B\), were fixed before \(n\). This completes the lemma. ◻

Proof. Part (i) of the lemma includes the whole space \(S(C_1L_iD_i)\) at a single eventual index; it does not supply separate indices for its individual elements. Field closure gives \(F_i\subset K_{*,i}\). The cutoff field is relatively algebraically closed, so it contains the stated relative algebraic closure as well. To see the last assertion directly, let \(x\ne0\) be algebraic and choose a polynomial relation for it with nonzero constant term and coefficients in the trivially valued subfield. If the value of \(x\) were positive, the constant term would have uniquely least value; if it were negative, the highest nonzero term would have uniquely least value. Neither is possible in a vanishing sum. Hence its value is zero. This applies to every rational expression and every polynomial degree once the finite generators have been included, rather than by a further limit in that degree. ◻

Proof. Take a uniformizer \(t\) at \(P\) and separating residue coordinates \(s_1,\ldots,s_{p-1}\), where \(p=\mathop{\mathrm{trdeg}}_k\mathcal K\). Up to a unit the base form is \(t^{a(P)}d\log t\wedge ds_1\wedge\cdots\wedge ds_{p-1}\). For a rational Gauss extension, integral powers of the \(z_j\) can be adjusted by powers of \(t\) to obtain angular units with algebraically independent residues. The resulting logarithmic wedge has log order zero. Together with (83), this proves the discrepancy formula. The calculation with trivial restriction to \(\mathcal K\) is the same, using a basis of the kernel of the pure weight. Nonnegativity follows from lc. Integrality is the ordinary integral log order of \(\theta_0\). At a nonzero rational pure weight \(y\), klt of \(X\) gives \(A_X(y)=y(h)=\langle y,\mathbf u\rangle>0\). Positivity on the nonzero rays of \(\sigma\) is equivalent to \(\mathbf u\in\operatorname{int}C\).

Choose a positive integer \(d\) with \(d\mu\in M\) and set \[V_\mu=\mathop{\mathrm{Proj}}\bigoplus_{j\ge0}R_{jd\mu}.\] The ray ring is finitely generated and normal: it is the appropriate diagonalizable invariant subring, followed, if necessary, by a Veronese subring. Its degree-zero part is \(k\), so its Proj is projective. An invariant rational function is a ratio of two homogeneous elements of the same weight. Multiplying their common weight by a suitable occurring weight moves both elements onto a sufficiently divisible multiple of the interior ray. To see the latter assertion directly, express \(\mu\) as a strictly positive rational combination of all the finitely many generating weights and clear denominators; sufficiently high multiples leave room to subtract any prescribed occurring weight. This proves \(k(V_\mu)=\mathcal K\).

Write the algebra generators as \(f_jz^{m_j}\). Linear programming duality gives \[ \min_{y\in\Delta_P}\langle y,\mu\rangle =\max\left\{-\sum_j\lambda_jP(f_j): \lambda_j\ge0,\ \sum_j\lambda_jm_j=\mu\right\}. \tag{90}\] The polytope on the right is compact because the grading is positive. Its rational vertices depend on the weights and \(\mu\), not on \(P\). After clearing their denominators and taking a Veronese generated in degree one, these vertices give actual monomials on the ray. Conversely, a sum of monomials has no worse pole order than its worst summand. Therefore (90) is exactly the maximum allowed pole order in the homogeneous linear system, expressed using \(z^{d\mu}\) as rational frame. On the Proj this system is basepoint-free in sufficiently divisible degree. Its maximum-pole formula on every higher model is the pullback of its ample tautological divisor. This proves (ii), including the b-divisor assertion. Formula (84) identifies \(\mathbf D\) with a rational principal adjustment of this polarization. Since \(h\) is regular upstairs, its invariant powers have nonnegative value at every Gauss extension, so \(\mathbf D\ge0\).

For (iii), choose a homogeneous section \(s\) on the defining ray whose nonvanishing chart contains the generic point of \(P\). The map \(\mathop{\mathrm{Spec}}R_s\to\mathop{\mathrm{Spec}}(R_s)_0\) is the torus quotient, and the latter is the corresponding chart of \(V_\mu\). Localize the base at \(P\). Its uniformizer remains a nonunit upstairs: the invariant ring is a direct summand, and an inverse upstairs would give an inverse downstairs by taking its invariant part. A component of its zero divisor therefore has height one and dominates \(P\), by the principal ideal theorem. Such a component is torus invariant, since a connected torus cannot permute its finitely many irreducible components nontrivially.

The primitive order of a torus-invariant prime is Gauss. Indeed its valuation ideals are torus invariant, hence are direct sums of weight spaces; there is no cancellation between different weights, and on each weight space restriction to \(\mathcal K\) supplies the indicated base order. Write the resulting valuation as \(e(P,y)\). Its restriction to \(\mathcal K\) is an integral multiple \(eP\), and its values on \(z^M\) are integral. Invertibility of \(s\) implies that \(y\) minimizes the ray polarization. A prime on a normal variety has ordinary log discrepancy one. Thus \(1=e\bigl(a(P)+(P,y)(h)\bigr)\), which proves (88). Notice that this argument concerns actual primes on \(V_\mu\); it asserts no such identity at an arbitrary exceptional prime.

On \(V_\mu\) set \(\Omega=-K_{V_\mu}(\theta_0)\). The log discrepancies of \((V_\mu,\Omega)\) are \(a\), and \(K_{V_\mu}(\theta_0)+\Omega=0\). At actual primes (88) gives \(0\le a\le1\); hence \(\Omega\) is effective. Moreover \(a(P)+\mathbf D(P)\le1\) there. At every divisorial \(P\), a rational minimizer for (87) gives a Gauss extension, so klt of \(X\) implies \(a(P)+\mathbf D(P)>0\). Subtracting the trace of \(\mathbf D\) from \(\Omega\) and taking \(\mathbf D\) as nef part gives the claimed generalized klt Calabi–Yau data.

For completeness these data imply Fano type without any uniform index assumption. On a common resolution on which \(\mathbf D\) is semiample, replace it by a general divided member of a sufficiently divisible free system. Added to the generalized boundary this gives an ordinary klt sub-pair; its pushdown has effective boundary and is crepant, with total log canonical divisor rationally trivial. The added divisor is big. Choose an effective ample divisor \(A\) on \(V_\mu\). Bigness permits a second effective representative of \(\mathbf D\) containing a small positive multiple of the pullback of \(A\). Mixing this representative with the general free representative with sufficiently small positive weight preserves klt. The resulting effective boundary downstairs contains \(\varepsilon A\) for some \(\varepsilon>0\). Subtracting that summand leaves an effective klt boundary whose negative log canonical class is \(\mathbb Q\)-linearly equivalent to \(\varepsilon A\), hence ample. Finally, at a non-floor prime the integer \(a(P)>0\) in (88) forces (89). ◻

Proof. Let \(v\) be a primitive divisorial valuation as in the first assertion. By the divisorial restriction statement in Section 1, a nontrivial restriction to \(\mathcal K\) has the form \(eP\), with \(P\) primitive and \(e\) a positive integer. Use the uniformizer \(t\) and separating residue coordinates from the proof of Proposition 23. Every remaining logarithmic differential has nonnegative log order at \(v\). Thus \[ A_{X,H}(v)\ge e a(P). \tag{91}\] More explicitly, pass to the finite-index sublattice of \(M\) on which \(v(z^m)\) is divisible by \(e\), and replace each of its basis monomials by the unit \(z^m t^{-v(z^m)/e}\). If their residues are algebraically dependent over the residue field of \(P\), their leading differential wedge vanishes, so the inequality in (91) is strict. Zero discrepancy therefore forces \(a(P)=0\) and independence of these angular unit residues. For a finite sum of tied terms, divide by a term of least value. All differences of its exponents lie in that finite-index lattice; independence of the adjusted residues prevents cancellation. Thus \(v\) is exactly the Gauss valuation determined by its restrictions to \(\mathcal K\) and \(z^M\).

If the restriction to \(\mathcal K\) is trivial, use a separating transcendence basis of \(\mathcal K\) in the same wedge computation. If all angular weights were zero, the unit residues could have transcendence degree at most \(r_T-1\) over \(\mathcal K\), and the leading top wedge would vanish. Zero log discrepancy therefore forces a nonzero angular weight. It further forces algebraic independence, over \(\mathcal K\), of the residues of a basis of the weight-zero monomials. The same tied-sum argument now proves that \(v\) is a pure Gauss weight. Nonnegativity on \(R\) places that weight in \(\sigma\).

The converse discrepancy assertion was computed in Proposition 23. The center of a Gauss valuation is torus invariant. A positive grading contracts its closure to the vertex, so the vertex lies in that closure; the valuation need not itself be centered at the closed vertex.

For a finite quasi-étale map with log-pulled pairs, the ramification formula makes discrepancies homogeneous under restriction. Hence an upstairs lc valuation restricts to a downstairs lc valuation, and every divisorial extension of a downstairs lc valuation is lc upstairs. Such extensions exist by taking the normalization of a model carrying the downstairs prime. The finite image of an upstairs center closure is the downstairs center closure and contains the vertex. There is only one point upstairs over that vertex: every positive homogeneous semi-invariant generator has an invariant positive power, which vanishes there. Therefore the upstairs center closure contains the upstairs vertex as claimed. ◻

Proof. Regularity implies all Gauss inequalities, hence the implication to the left. Conversely take a sufficiently divisible power so that the semi-invariant becomes invariant, its weight is on the integral ray, and its degree is large enough for the ray ring to agree with the section ring of the tautological polarization in that degree. The divisor inequalities and (90) then put that power in \(R\). The original function belongs to \(\mathop{\mathrm{Frac}}(\breve R)\) and is integral over \(\breve R\), so normality puts it in \(\breve R\). The second formula is the case \(\xi=1\). When \(\mathcal K=k\), the same argument says that an interior weight becomes effective in a divisible degree and then uses normality. ◻

Proof. For \(\mathcal K=k\) the preceding lemma gives the assertion with \(q=1\). Otherwise work first on \(V_\mu\), where \(\mu=k+m\mathbf u\). Use the generalized klt Calabi–Yau discrepancy \(a+\mathbf D\) to extract onto a \(\mathbb Q\)-factorial Fano type model \(U\) all exceptional primes below a fixed small positive discrepancy cutoff, and no other exceptional primes. The extraction statement is supplied by Section 1. All extracted primes are floor primes: \(a\) is integral and nonnegative. On \(U\), the positive coefficients of the trace \(D_U\) are precisely the floor primes, and they are at most one. This holds for surviving primes by (88), and for extracted primes by their small \(a+\mathbf D\) discrepancy.

For each floor prime \(Q\) on \(U\) put \(\rho_Q=\min\{1,L\mathbf D(Q)\}\). Choose a small fixed rational \(c_1>0\), with \(2c_1\le c\), and an effective correction \(R_c\), supported on these primes, such that \[ c_1\rho_Q\le R_c(Q)\le2c_1\rho_Q, \qquad \operatorname{den}\bigl(J_m(Q)-R_c(Q)\bigr)\le C_1/\rho_Q. \tag{96}\] Here \(\operatorname{den}\) is the reduced positive denominator. Such a choice is obtained by taking a rational grid of mesh at most \(c_1\rho_Q\) and denominator at most \(C_1/\rho_Q\) in the interval \([J_m(Q)-2c_1\rho_Q,J_m(Q)-c_1\rho_Q]\). There is no correction outside the floor: no such primes were extracted, and \(J_m\) is integral there.

Minima of sums give the b-divisor inequality \[ \mathbf J_m-\mathbf J_{m_0}\ge(m-m_0)\mathbf D. \tag{97}\] Since \(R_c\le2c_1 L D_U\), the divisor \(P_U=(J_m)_U-R_c\) dominates the trace of \(\mathbf J_{m_0}\) and is big. Run its MMP on \(U\) to a nef big semiample divisor \(P'\) on a \(\mathbb Q\)-factorial Fano type model \(U'\). On a common higher model, denote pullback from a displayed model by a star. Then \[ \mathbf J_m\ge(P')^*\ge\mathbf J_{m_0},\qquad R_c^*\le \mathbf E:=\mathbf J_m-(P')^*\le(R_c')^*. \tag{98}\] Here is justification for both sides, including the comparison with the possibly different ray \(m_0\). The divisor \(J_m\) descends on \(U\), and MMP negativity gives \((P')^*\le P_U^*=\mathbf J_m-R_c^*\). For the middle inequality, every divisible section of \(\mathbf J_{m_0}\) is a section of \(P_U\), hence of \(P'\) by the MMP. The maximum-pole description of semiample divisors, applied on a common determination, gives \((P')^*\ge\mathbf J_{m_0}\). Finally b-nef negativity gives \(\mathbf J_m\le(J_m')^*\); since \(J_m'=P'+R_c'\), this is the last inequality in (98). Also \(\rho_Q\ge D_U(Q)\), so \[\mathbf E\ge R_c^*\ge c_1(D_U)^*\ge c_1\mathbf D.\] The final comparison is again b-nef negativity for \(\mathbf D\).

We apply Lemma 11 to the polarization \(P'\). Choose fixed rational \(\delta_2>0\) small enough for the denominator bound below, and then \(0<\delta_1\le\delta_2c_1\). For \(0\le t\le\delta_2\) use the discrepancy family and nef part \[\begin{align*} a_t&=a+(1-t/\delta_2)\delta_1\mathbf D+t\mathbf E,\\ \mathbf M_t&=(1-t/\delta_2)\delta_1\mathbf D+t\mathbf J_m. \end{align*}\] Their total log canonical divisor is \(\mathbb Q\)-linearly equivalent to \(tP'\). For \(t\ge\delta_2\) keep \(a_t=a+\delta_2\mathbf E\) and add \((t-\delta_2)(P')^*\) to the nef part. This gives families on arbitrarily long parameter intervals. They are nondecreasing, klt, and have nef \(\mathbb Q\)-Cartier b-parts.

We check effectivity and the denominator trace requirement of that lemma. Every prime of \(U'\) survives from \(U\). At a non-floor prime, \(a=1\), \(\mathbf D=\mathbf E=0\), and \(P'\) has integral coefficient. At a surviving floor prime \(Q\), \(\mathbf E(Q)=R_c(Q)\le2c_1\rho_Q\), while \[\frac1{\operatorname{den}(P'(Q))}\ge\frac{\rho_Q}{C_1}.\] Take \(2\delta_2c_1C_1\le1\). Monotonicity gives \(0<a_t(Q)\le\delta_2R_c(Q)\) throughout the family, so \(a_t(Q)\le1/\operatorname{den}(P'(Q))\). Equivalently, its effective boundary dominates the denominator boundary \(R(P')\) used in Section 2.

It remains to check the signed phase hypothesis. At a growing parameter, small discrepancies mean \(a(P)=0\) and \(\mathbf E(P)\to0\). Because \(\mathbf E\ge c_1\mathbf D\), these valuations eventually belong to the fixed-cutoff list already extracted on \(U\). On that list, (96) and (98) give \[\mathbf E(P)\ge c_1\min\{1,L\mathbf D(P)\}.\] Thus \(L\mathbf D(P)\to0\) and \(L\mathbf D(P)=O(\mathbf E(P))\). Combining (95) with (98) yields \[j_P\le(P')^*(P)\le j_P+O(\mathbf E(P)).\] The phase of \((P')^*(P)\) therefore has a nonnegative lift of size \(O(a_t(P))\), which is stronger than the signed phase hypothesis. Lemma 11 supplies a nonzero section in a bounded positive degree of \(P'\). By (98) it lies in \(S(q\mathbf J_m)\), and Lemma 25 finishes the proof. ◻

Proof. Choose an integer \(m_1\asymp L\) with \(m_1>C_0L\), so that both \(\pm k+m_1\mathbf u\) lie in \(\operatorname{int}C\). Bigness of their ray polarizations supplies rational-power functions \(g_+,g_-\) on \(\mathcal K\) such that \[(P,y)(z^{\pm k}h^{m_1})+P(g_\pm)\ge0 \quad\text{for every }P\text{ and }y\in\Delta_P.\] Here a rational-power function means a positive integral root of an element of \(\mathcal K^*\) used only through its \(\mathbb Q\)-principal order; it need not be an element of \(\mathcal K\). Put \[\ell=-\mathop{\mathrm{div}}(g_+),\qquad G=2m_1D+\mathop{\mathrm{div}}(g_+g_-).\] Adding the two Gauss inequalities and taking the height minimum shows that \(G\ge0\) on \(V_{\mathbf u}\), and \(G\sim_{\mathbb Q}2m_1D\). At a non-floor prime \(Q\) of this model, choose the integral minimizing \(y\) in (89). Since its height is zero, the two inequalities give \[0\le\langle y,k\rangle-\ell(Q)\le G(Q), \qquad \langle y,k\rangle\in\mathbb Z.\] Lemma 17 therefore gives a fixed positive integer \(q_0\) and a fixed \(A>0\) such that, on all tiny sequences in the statement, one can choose integers \(j_P\) with \[|q_0\ell(P)-j_P|\le A L\mathbf D(P).\]

Apply Proposition 26 to \(\xi=z^{q_0k}\). For any sufficiently large integer \(m_0=O(L)\) and \(m=2m_0\), the first Gauss inequality gives \[\mathbf J_{m_0}\ge q_0\ell+(m_0-q_0m_1)\mathbf D.\] At a height-minimizing angular vector the second inequality gives \[\mathbf J_m\le q_0\ell+(m-q_0m_1)\mathbf D+q_0\mathbf G \le q_0\ell+(m+q_0m_1)\mathbf D+q_0\mathbf G.\] The order condition (41), applied to \(G/(2m_1)\sim_{\mathbb Q}D\), bounds \(\mathbf G(P)\le2m_1(1+\eta_i)\mathbf D(P)\) on the tiny sequences. Choose \(m_0-q_0m_1\ge AL\). These estimates give exactly (95). A bounded further multiplier \(q_1\) then supplies \(g z^{q_1q_0k}h^{q_1m}\in\breve R\). This is the desired regularization.

Apply the argument to \(-k\) as well, raise the two resulting functions to bounded powers to align their multipliers, and multiply by powers of \(h\) to align their costs. Uniformity in the weight follows by applying the sequence argument to any proposed sequence of exceptions. For an enlarged scale \(L'\ge L\), its tiny sequences satisfy \(L\mathbf D(P)\to0\) as well. All the preceding hypotheses therefore hold with \(L'\) in place of \(L\). A sequence of exceptional pairs \((L',k)\) would contradict the same proof; this proves the stated uniformity, including for growing \(L'/L\). ◻

Proof. If \(W_{t,k}\ne0\), divide any nonzero member by \(h^t\). This is a finite-group invariant rational eigenfunction, so \(k\in M\). Let \(\mathbf J^{\mathrm{test}}\) denote the integral basepoint-free test b-divisor called \(J\) in Lemma 18. That lemma has \(0\le\mathbf J^{\mathrm{test}}\le C L_0\mathbf D\) with \(L_0=O(L)\). At a primitive \(P\) with \(a(P)=0\) and \(L\mathbf D(P)\to0\), its nonnegative integral value is eventually zero. Such \(P\) belong to that lemma’s test set \(\mathcal L\) and hence satisfy (41). We may therefore apply Corollary 27 at the enlarged scale \(nL\).

Choose an opposite invariant eigenfunction \(Y\) of weight \(-qk\), with \(q\) bounded and \(h^bY\) regular for \(b\le C_4 nL\). For \(F\in W_{t,k}\) the neutral function \[\Phi(F)=\frac{F^qY}{h^{qt}}\] satisfies \(h^{qt+b}\Phi(F)=F^q h^bY\in\breve R\). By (94), it belongs to \(S((qt+b)\mathbf D)\subseteq S(C_5nL\mathbf D)\). Although \(\Phi\) need not be linear, it is a homogeneous polynomial map of degree \(q\), nonzero away from the origin, and induces an injective morphism \[\mathbb P(W_{t,k})\longrightarrow \mathbb P\bigl(S(C_5nL\mathbf D)\bigr).\] Indeed proportional images imply that \((F_1/F_2)^q\in k^*\); algebraic closedness of \(k\) then implies \(F_1/F_2\in k^*\). An injective morphism of projective varieties preserves the dimension of its source in its image, so \(\dim W_{t,k}\le\dim S(C_5nL\mathbf D)\). The upper count in Lemma 18, used at an integer at most \((C_5+1)n\), proves (100); \(p\) is bounded by the ambient dimension. The uniform enlarged-scale assertion in Corollary 27 makes \(q,C_4,C_5\) independent of \(n,t,k\). The eventual assertion retains the quantifier that \(n\) is fixed before passing sufficiently far along the sequence. ◻

Proof. Set \(\xi=\prod_j(Y_j^+)^{-r_j}\), which has weight \(-\sum_jr_jk_j\). Regularity and opposite evaluations imply \(|y_j|\le m_jv(h)\) at every test under consideration. Hence, with a fixed constant \(C_7\) controlling the rounding errors, \[ \sum_jr_jy_j \le(M_0\delta+C_7)Lv(h). \tag{103}\] Choose \(M_0\delta\ge2C_7+2\) and an integer \(m_0\asymp M_0\delta L\) strictly larger than \((M_0\delta+C_7)L\), and set \(m=2m_0\). The constants in these choices are independent of \(\delta,M_0\). At pure weights (103) implies strict positivity of the weights of \(\xi h^{m_0}\) and \(\xi h^m\) on \(\sigma\setminus\{0\}\); both rays are therefore interior. At the angular tests it gives \(\mathbf J_{m_0}(P)\ge0\).

At a height-minimizing angular vector, the elementary estimate \[\left|\sum_jr_jy_j\right| \le\sum_j|r_j|m_j\mathbf D(P) \le C_8(M_0+1)L\mathbf D(P)\] also holds. Consequently \(0\le\mathbf J_{m_0}(P)\le\mathbf J_m(P) \le C_9(M_0,\delta)L\mathbf D(P)\) on the tiny sequences. Proposition 26 applies with \(j_P=0\) and produces \(g\xi^qh^{qm}\in\breve R\) for bounded \(q\). Its character, weight, and cost are exactly those in (102). ◻

Proof. Cover the relevant part of \(P\) by finitely many smooth etale coordinate charts. At the centre of \(v\), write the coordinates of the components of \(F\) through that centre as \(z_1,\ldots,z_s\) and the remaining coordinates as \(y_1,\ldots,y_{n-s}\). Thus \(v(z_i)>0\). Choose a closed specialization of the centre in this chart, avoiding the other components of \(F\), and translate the \(y\)-coordinates to vanish there. Both \(v\) and its retraction are nonnegative on this closed-point local ring: the original centre generalizes this point. A unit in that ring consequently has value zero for both valuations. Put \(a_i=v(z_i)\) and expand \(g\) in the completed local ring. The least \(z\)-weight is \[d_0=\min\{\langle a,\alpha\rangle: [z^\alpha y^\beta]g\ne0\text{ for some }\beta\}.\] We first prove a degree bound independent of \(a\).

The function field of the chart is a finite extension of \(\mathbb C(z,y)\). In a fixed basis, multiplication by each of the finitely many affine functions has a matrix with fixed rational-function entries. A polynomial of degree at most \(D\) in these matrices has a common denominator of degree \(O(D+1)\) and numerator entries of the same degree. Its characteristic equation, after clearing denominators, gives \[ \sum_{j=0}^{q}p_j(z,y)g^j=0,\qquad \deg p_j\le C_0(D+1), \tag{107}\] where \(q\) is fixed and the coefficients are not all zero. Translations of coordinates do not change this degree bound. Give all coordinates strictly positive rationally independent weights. Let \(\gamma\) be the leading exponent of \(g\), and let \(\delta_j\) be the leading exponent of a nonzero \(p_j\). At least two distinct powers, say \(j\) and \(k\), have the least weight in (107). Independence of the weights gives \[\delta_j+j\gamma=\delta_k+k\gamma.\] Consequently \(|\gamma|\le C_1(D+1)\), after enlarging the constant. This conclusion holds for every such choice of positive weights.

Now give the \(z\)-coordinates their fixed positive weights and give the \(y\)-coordinates weight zero. There are finitely many \(z\)-exponents of weight \(d_0\), and only finitely many in a sufficiently small fixed neighbourhood of this weight. Every vertex of the convex hull of the tied \(z\)-exponents is exposed by a small generic perturbation of the \(z\)-weights. Give the transverse coordinates sufficiently small positive generic weights afterwards. The leading exponent projects to that vertex: the positive gap to the other \(z\)-weights prevents an exponent outside the least face from entering. The preceding bound therefore bounds every vertex, and convexity bounds every tied \(z\)-exponent by \(C_1(D+1)\). The same perturbation supplies at least one pair \((\alpha,\beta)\) on the least face with \(|\alpha|+|\beta|\le C_1(D+1)\).

For this choice of \(\alpha\), apply the product, over the \(z\)-coordinates, of the operators \[\prod_{\substack{0\le l\le C_1(D+1)\\ l\ne\alpha_i}} (z_i\partial_{z_i}-l).\] It kills every other tied \(z\)-exponent and multiplies the chosen coefficient by a nonzero constant. Apply a transverse derivative \(\partial_y^\beta\) for a bounded exponent occurring in that coefficient. Its new constant term is nonzero. These operators have total order \(O(D+1)\), and do not change any \(z\)-exponent. Their result \(g'\) satisfies \[ g'=u z^\alpha\pmod{I_{>d_0}},\qquad u\text{ a local unit}, \tag{108}\] where \(I_{>d_0}\) is the monomial ideal generated by the \(z\)-monomials of weight greater than \(d_0\). This ideal is finitely generated. The assertion descends from completion by faithful flatness. More explicitly, faithful flatness first gives \(g'=u z^\alpha+q\) with \(u\) regular and \(q\in I_{>d_0}\). In the completion, \(u\) differs from the formal unit coefficient by an element of \((I_{>d_0}:z^\alpha)\). This colon ideal is proper, because \(z^\alpha\) has weight \(d_0\), and hence lies in the local maximal ideal. Thus \(u\) is a unit. Each monomial generator of \(I_{>d_0}\) has value greater than \(d_0\) for both \(v\) and \(r_F(v)\), and its regular coefficients have nonnegative value. Thus \[v(g')=r_F(v)(g')=d_0=r_F(v)(g).\] This also explains why a nonclosed centre causes no problem.

For completeness, a regular logarithmic vector field \(L\) satisfies \[ v(Lf)\ge v(f)-A_{P,F}(v). \tag{109}\] For a prime-order valuation, take a local logarithmic volume form \(\eta\) for \((P,F)\) and use \[(Lf)\eta=f\,d\log(f)\wedge\iota_L\eta.\] At the prime on a higher smooth model, logarithmic coordinate forms and \(d\log(f)\) have nonnegative logarithmic order. Comparing the two sides gives (109). Homogeneity gives it for rational monomial weights, and rational approximation gives it for arbitrary quasi-monomial weights. In this approximation the signs on a fixed chart ring can be retained: the values of its finitely many generators are rational piecewise linear functions on the monomial weight cone, so one approximates in their common sign cells. Discrepancy and the finitely many function values in question are continuous there. The Euler operators and transverse derivatives used above are logarithmic vector fields, and subtracting a scalar does not worsen the estimate. Iterating (109) and using (108) proves the upper bound in (106). The lower bound is the retraction inequality. ◻

Proof. Apply (110) to \(\Psi=H\). Every zero-discrepancy valuation is monomial on the coefficient-one part of the fixed resolution. Conversely those monomial valuations have zero discrepancy for \((S,H)\). Retain the strata whose closed images contain \(x\). This condition is preserved when a stratum is replaced by a larger stratum, so the retained cones include their faces. On a cone with primitive generators \(E_i\) the equation \(A(v)=\sum_i a_iA(E_i)=1\) cuts out a compact simplex, because \(A(E_i)>0\). The additional inequalities are linear on an adapted resolution. There are finitely many cones. Function evaluations are monomial minima on them and agree on faces by localization to the larger stratum, proving compactness and continuity.

We use Li’s Izumi estimate (Li 2018, Theorem 3.1): \[ v(f)\le C_I A(v)\mathop{\mathrm{ord}}_{\mathfrak m}(f) \quad(f\in R\setminus\{0\},\ v\text{ centred}). \tag{112}\] Thus, when \(A(v)=1\), the valuation ideal \(\mathfrak a_t(v)\) is contained in \(\mathfrak m^{\lceil t/C_I\rceil}\). Hilbert–Samuel growth gives the first bound in (111). Choose fixed generators \(z_1,\ldots,z_q\) of \(\mathfrak m\), and let \(z=z_i\) have \(v(z)=\delta=v(\mathfrak m)\). Since \(R\) is a domain, \[(\mathfrak a_t(v):z^j)=\mathfrak a_{t-j\delta}(v).\] Filtering \(R/\mathfrak a_t(v)\) by the images of \(z^jR\) gives \[\operatorname{length}(R/\mathfrak a_t(v)) =\sum_{0\le j<t/\delta} \operatorname{length}\bigl(R/(\mathfrak a_{t-j\delta}(v)+(z))\bigr).\] For \(j\le t/(2\delta)\), Izumi bounds this summand below by \(\operatorname{length}(R/(\mathfrak m^{\lceil t/(2C_I)\rceil}+(z)))\). The latter grows as a positive constant times \(t^{n-1}\); this includes \(n=1\), when it is a positive constant. Since \(z\) belongs to a fixed finite list, the lower constant can be chosen uniformly. Divide by \(t^n/n!\) to obtain the second bound in (111).

To prove continuity, consider valuations converging on one monomial cone to a centred limit. Drop the components whose weights tend to zero, and call the resulting retractions \(r_i\). The retained positive weights converge multiplicatively. On regular functions, the \(r_i\) are therefore multiplicatively \(1+o(1)\) equivalent to the limit. Their maximal-ideal values, and those of the original valuations, are bounded below. The reduced-log discrepancy measuring the dropped weights tends to zero. Lemma 30 upgrades this fact to a multiplicative comparison on all regular functions as follows.

Write \(q=\mathop{\mathrm{ord}}_{\mathfrak m}(f)\ge1\). With \(c_0>0\) a common lower bound for maximal-ideal values and \(C_0q\) the common Izumi upper bound for the values of \(f\), fix \(L\) with \(Lc_0>2C_0\). Represent \(f\) modulo \(\mathfrak m^{N}\), \(N=\lceil Lq\rceil\), by a polynomial \(p\) of degree at most \(N\) in fixed generators of \(\mathfrak m\). Such a representative exists also for localized regular functions by expanding their unit denominators modulo \(\mathfrak m^N\). The error has value greater than both values of \(f\), so \(v_i(p)=v_i(f)\) and \(r_i(p)=r_i(f)\). The jet estimate bounds their difference by \(o(1)q\), uniformly in \(f\). Since \(r_i(f)\ge c_0q\), it follows that \(r_i\le v_i\le(1+o(1))r_i\) on all regular functions. Units have value zero. Inclusion of valuation ideals now proves volume continuity. Every convergent sequence has a subsequence in one of the finitely many cones, which proves the assertion on the whole centred part. ◻

Proof. Use \(N=\lceil L\mathop{\mathrm{ord}}_{\mathfrak m}(f)\rceil\) with the single constant \(L\) chosen in the preceding proof. Equations (106) and (110) give \(v(f)-r(f)\le C e\mathop{\mathrm{ord}}_{\mathfrak m}(f)\le Ce\,r(f)/c_0\). Enlarge \(C\). The ideal inclusions give the volume inequality. ◻

Proof. Fix finitely many functions \(g_i\) whose averaged target values \(s_i=\sum_s p_sv_s(g_i)\) are positive, and choose positive rational \(s'_i\le s_i\). Include \(h\) with the exact target \(1\) and each \(F_j\) with target \(m_j\). Choose \(N\) so that every \(N/s'_i\) is integral, and put \(\mathfrak a=\sum_i(g_i^{N/s'_i})\). We claim that \((S,\mathfrak a^{1/N})\) is not klt at \(x\). The multiplier-ideal summation formula on normal \(\mathbb Q\)-Gorenstein complex varieties gives \[ \mathcal J(\mathfrak a^{1/N}) =\sum_{\substack{t_i\ge0\\\sum_i s'_it_i=1}} \mathcal J\Bigl(\prod_i(g_i)^{t_i}\Bigr). \tag{115}\] This is the mixed-ideal formula of (Takagi 2006, Theorem 3.2) and (Jow and Miller 2008, Corollary 2 and Theorem 3.6); there is no smoothness or Jacobian correction in this particular formula. If the left side were the unit ideal at \(x\), some summand would be the unit ideal in the local ring. The resulting mixed pair would be klt, so for every positive-weight \(v_s\), \(\sum_i t_i v_s(g_i)<A(v_s)=1\). Averaging contradicts \[\sum_i t_i\sum_s p_s v_s(g_i)=\sum_i t_is_i \ge\sum_i t_is'_i=1.\] The failure of klt is witnessed by a prime \(E\) whose centre contains \(x\), and \(A(E)\le\min_i\{\mathop{\mathrm{ord}}_E(g_i)/s'_i\}\). Thus \(u=\mathop{\mathrm{ord}}_E/A(E)\) meets all the lower targets. In particular \(u(h)\ge1\), whereas lc of \((S,H)\) gives \(u(h)\le A(u)=1\). It is already in \(\mathcal Q\).

Here is the geometric mechanism behind (115). Resolve the summand ideals and their sum, with fixed divisors \(D_i,D\), and subdivide \(\{p_i\ge0:\sum p_i=1/N\}\) by the rounding walls of \(\mathcal O_P(\lceil K_{P/S}-\sum_i p_iD_i\rceil)\). The cellular complex indexed by chains of nonempty cells, with the usual alternating face maps and inclusions on removing the first cell, resolves \(\mathcal O_P(\lceil K_{P/S}-D/N\rceil)\). In completed snc coordinates its exactness is coefficientwise: the cells admitting a given Laurent monomial form a convex relatively open subset of the exponent simplex, and their barycentric order complex is contractible when nonempty. This subset is nonempty exactly for monomials in the target. Indeed, locally one summand divided by the invertible sum ideal is a unit, so its fixed divisor there equals \(D\). Local vanishing for each mixed multiplier-ideal term preserves the surjective augmentation under pushforward. This gives the stated sum formula in precisely the singular setting used above.

Let the rational lower targets increase to their averaged values. Compactness from Lemma 31 gives a point meeting the finite list exactly as lower bounds. The closed sets imposing these lower bounds, indexed by all regular functions, have the finite intersection property. Compactness supplies a single \(u\) satisfying (114) for every function. Positivity on generators of \(\mathfrak m\) proves the centring assertion. For unnormalized inputs, put \(a=\sum_s p_sA(v_s)\), apply the normalized assertion with weights \(p_sA(v_s)/a\) to \(v_s/A(v_s)\), and multiply the result by \(a\). The formula for \(b\) and the equality for \(u(h)\) prove its height bound. ◻

Proof. Apply domination to both numerators and use the exact value of \(h\). The two lower inequalities have opposite sides and so must be equalities. ◻

Proof. For a centred filtration, write its Laplace sum as the sum over its jumps \(a\) of \(\dim_\mathbb C(\mathfrak a_a/\mathfrak a_{>a})e^{-a/N}\). Integration of the cumulative dimension function gives \[ \lim_{N\longrightarrow\infty}N^{-n} \sum_a\dim_\mathbb C(\mathfrak a_a/\mathfrak a_{>a})e^{-a/N} =\mathop{\mathrm{vol}}(v). \tag{118}\] There are finitely many jumps below each bound, and polynomial growth controls the tail.

Choose a homogeneous basis of \(\operatorname{gr}_wR/(\operatorname{in}_wf)\), lift it homogeneously, and multiply the lifts by all nonnegative powers of \(\operatorname{in}_wf\). This is a basis of \(\operatorname{gr}_wR\): its independence follows from the domain property, and successive division lowers the degree by \(w(f)>0\), proving spanning. Lift this basis to products \(a_if^j\) in \(R\). They span modulo arbitrary high \(w\)-ideals, hence modulo high \(u\)-ideals since \(u\ge tw\). The \(u\)-value of \(a_if^j\) is at least \(tw(a_i)+j(tw(f)+(1-t)v(f))\). Consequently its Laplace sum is at most the \(w\)-Laplace sum at \(N/t\), multiplied by \[\frac{1-e^{-tw(f)/N}} {1-e^{-(tw(f)+(1-t)v(f))/N}}.\] Use (118) to obtain (116).

For the strict gap, fix \(T>3\) and work in \(V_N=R/\mathfrak m^{\lceil CTN\rceil}\), where \(C\) is large enough for this fixed triple of centred valuations. The indicated power is contained in all their threshold-\(TN\) ideals. Cap both the \(w\) and \(v\) flags at \(TN\) and choose a basis simultaneously adapted to these two flags; denote its capped values by \(W_i,V_i\). Lifts have \(u\)-value at least \((W_i+V_i)/2\). The identity \[e^{-W_i/N}+e^{-V_i/N}-2e^{-(W_i+V_i)/(2N)} =\bigl(e^{-W_i/(2N)}-e^{-V_i/(2N)}\bigr)^2\] therefore bounds twice the capped \(u\)-Laplace sum above by the sum of the other two, less the sum of these squares.

Choose a vector-space complement to \(\mathfrak a_{\gamma N/4}(w)\) in \(R\) and multiply it by \(f^{\lfloor N/w(f)\rfloor}\). Every nonzero resulting element has \(w\)-value less than \((1+\gamma/4)N\) and, for large \(N\), has \(v\)-value at least \((1+2\gamma/3)N\). Multiplication is injective, and this subspace remains injective in \(V_N\) by its upper \(w\)-value bound. Its dimension is asymptotic to \[\frac{\mathop{\mathrm{vol}}(w)}{n!}(\gamma N/4)^n.\] Its projection from the high-\(v\) subspace modulo the high-\(w\) subspace has this rank. Hence at least this many entries of a simultaneously adapted basis satisfy \(W_i<(1+\gamma/4)N\) and \(V_i\ge(1+2\gamma/3)N\). Each contributes at least \[\left(e^{-(1+\gamma/4)/2}-e^{-(1+2\gamma/3)/2}\right)^2\] to the preceding sum of squares. Divide by \(N^n\), first let \(N\to\infty\), and then let \(T\to\infty\). The capped tail errors are bounded by polynomial functions of \(T\) times \(e^{-T}\) and disappear. This proves (117), for example with any smaller positive constant than \[\frac{(\gamma/4)^n}{n!} \left(e^{-(1+\gamma/4)/2}-e^{-(1+2\gamma/3)/2}\right)^2.\]

Finally interpolate two near-minimizers by Lemma 33. Their interpolant has at least the minimum volume. If their values on some regular function differed by a fixed factor greater than one, apply (117), interchanging the valuations if necessary. It gives a fixed relative loss, contrary to near-minimality. This proves the simultaneous all-functions assertion and its fixed-error version. ◻

Proof. Set \(\mathfrak b_j=(h^{m_j},F_j)\). For any valuation, \[\begin{align*} A(v)&-(1-\sum_j\theta_j)v(h) -\sum_j\frac{\theta_j}{m_j}v(\mathfrak b_j)\\ &=A(v)-v(h)+\sum_j\theta_j \left(v(h)-\min\{v(h),v(F_j)/m_j\}\right)\ge0. \end{align*}\] Equality holds exactly on the common lc rays, together with the trivial valuation. Resolve the ideals and choose the \(E_j\) as general members of their pencils. On the resolution the residual free members meet the existing boundary with snc support and contain none of its strata. Their coefficients \(\theta_j/m_j\) are less than one. They therefore introduce no new zero-discrepancy strata, proving the assertion about \(\Psi\). On the coefficient-one monomial skeleton, each free residual member has value zero because it contains no stratum. The value of its pencil member is consequently the fixed-ideal value \(\min\{m_jv(h),v(F_j)\}=m_jv(h)\), proving (120). ◻

Proof. Attainment, uniqueness, and continuity. Compactness gives \(\min_{\mathcal Q}b>0\). A minimizing sequence has bounded volume, so (111) bounds its maximal-ideal values away from zero. A subsequence converges in \(\mathcal Q\) to a centred valuation. Lemma 31 gives volume continuity and hence attainment. If two minimizing rays existed, normalize each by \(b=1\). Their midpoint has a dominating common lc interpolant \(u\) with \(b(u)\le1\). If the minimum value of \(b\mathop{\mathrm{vol}}^{1/n}\) is \(M\), then \(\mathop{\mathrm{vol}}(u)\ge M^n/b(u)^n\ge M^n\). The strict gap (117) contradicts this unless the two height-normalized valuations agree on every regular function. Thus they are the same ray. For heights whose coefficients converge to those of a positive height, the minimum heights stay bounded below and one fixed centred competitor bounds the minimum values above. The same compactness argument shows that every subsequential limit of minimizers is the unique limiting minimizer. This proves continuity. Automorphisms preserving the data preserve the functional, so uniqueness proves equivariance.

Perturbing the boundary. Assume from now on that the height is rational. Choose \(\Psi\) by Lemma 36, and resolve additionally all \(G_\ell\) and \(\mathfrak m\). For small rational \(\lambda>0\), (121) is effective, since the coefficient of \(H\) in (119) is positive. It is klt near \(x\): on each coefficient-one component its new discrepancy is \(\lambda b>0\), while the other finitely many resolved discrepancies remain positive for sufficiently small \(\lambda\). The log-smooth criterion then tests every valuation.

Uniform control of every relevant competitor. Let \(v\) be a centred quasi-monomial valuation with \(A(v)=1\) and \[ A_{S,\Psi_\lambda}(v)\mathop{\mathrm{vol}}(v)^{1/n}\le\lambda M_0, \tag{122}\] where \(M_0\) is the value of \(b\mathop{\mathrm{vol}}^{1/n}\) at one fixed centred point of \(\mathcal Q\). This includes every normalized-volume minimizer of the klt pair. Existence and quasi-monomiality for effective rational klt pairs are supplied by (Blum 2018, sec. 7) and (Xu 2020, Theorem 1.2). The following constants are uniform over all valuations in (122) and all sufficiently small \(\lambda\).

Put \(e=A_{S,\Psi}(v)\ge0\) and retract \(v\) to the coefficient-one part of the fixed resolution, obtaining \(r\). Izumi bounds every fixed function value at \(v\) when \(A(v)=1\); thus \(|b(v)|\le C\). The first bound in (111) and (122) give \[e+\lambda b(v)\le C\lambda,\qquad e\le C\lambda.\] Equation (110) yields \[ A_S(r)=1+O(e),\qquad b(v)=b(r)+O(e). \tag{123}\] For small \(\lambda\), \(r\) is nontrivial and its normalized ray lies in \(\mathcal Q\). Strict positivity of \(b\) on the compact slice therefore gives uniform constants \(\beta,B>0\) with \[\beta\le b(r)\le B,\qquad b(v)\ge\beta/2.\] The sublevel inequality now bounds \(\mathop{\mathrm{vol}}(v)\) above. The second bound in (111) bounds \(v(\mathfrak m)\) below; since \(\mathfrak m\) was resolved, its value at \(r\) differs by \(O(e)\). Thus \(r(\mathfrak m)\) is also bounded below and \(r\) is an admissible centred competitor. Corollary 32 supplies \[ v(f)\le(1+Ce)r(f)\qquad\text{for every }f\in R, \tag{124}\] with a single \(C\) independent of \(f\), \(v\), and small \(\lambda\).

One fixed parameter forces exactness. Use (123) and (124) to obtain \[\begin{align*} &(e+\lambda b(v))\mathop{\mathrm{vol}}(v)^{1/n} -\lambda b(r)\mathop{\mathrm{vol}}(r)^{1/n}\\ &\hspace{1cm}\ge \frac{e\,[1-\lambda C'-\lambda Cb(r)]}{1+Ce} \mathop{\mathrm{vol}}(r)^{1/n}. \tag{125}\end{align*}\] Choose one positive rational \(\lambda\) small enough for all preceding bounds and for \(\lambda(C'+CB)<1\). This choice depends only on the fixed germ, boundary, and height; it is independent of every competitor in (122) and of every regular function. If such a competitor has \(e>0\), (125) is strictly positive, so retraction strictly improves its normalized-volume root functional. An actual klt minimizer consequently has \(e=0\), lies exactly on the common lc slice, and is \(w\) by uniqueness. Every competitor outside (122) already has larger value than the fixed slice competitor. This proves exact realization for this single rational parameter; it is not merely a limiting assertion as \(\lambda\to0\). The argument applies for every smaller positive rational parameter satisfying the same bounds.

Stable degeneration of this actual minimizer. We now apply stable degeneration to the effective rational klt pair \((S,\Psi_\lambda)\) and its actual minimizer \(w\). By (Xu and Zhuang 2025, Theorems 1.1–1.2), its associated graded is finitely generated and defines a normal klt log Fano cone pair \((S^0,\Psi_\lambda^0;w^0)\). The grading minimizes normalized volume on the central pair; ordinary volume and pair discrepancy are preserved, as in (Li and Xu 2018, Lemma 4.10 and Theorem 3.5, arXiv version). The general uniqueness theorem for the klt-pair minimizer is (Xu and Zhuang 2021, Theorem 1.1); the constrained uniqueness needed here was proved before invoking it.

Each component used in (121) is the divisor of a regular function. For a principal ideal \((f)\) and a valuation filtration, multiplicativity gives \(\operatorname{in}_w((f))=(\operatorname{in}_wf)\). Equivalently its Rees homogenization, using its exact order, cuts a Cartier divisor without a fibre component. Hence \(\Psi_\lambda^0\) is an effective rational sum of principal Cartier divisors. Since \(K_{S^0}+\Psi_\lambda^0\) is \(\mathbb Q\)-Cartier, so is \(K_{S^0}\), and removing the effective boundary shows that \(S^0\) is klt. Preservation of pair discrepancy and of all equation orders gives \[A_{S^0}(w^0)=A_S(w)=1,\quad w^0(h^0)=1,\quad w^0(F_j^0)=w(F_j)\ge m_j.\]

The choice of the \(\theta_j\) can be made arbitrarily small. For each such choice the exact minimizer is still the unique \(w\), and thus the graded algebra remains the same. Choose the corresponding \(\lambda\) smaller as necessary. The central effective klt boundaries then contain \((1-o(1))\mathop{\mathrm{div}}(h^0)\). For every prime over \(S^0\), the resulting discrepancy inequalities imply, on taking the coefficient limit, \(A_{S^0}(E)\ge\mathop{\mathrm{ord}}_E(h^0)\). This proves that \((S^0,\mathop{\mathrm{div}}(h^0))\) is lc.

Finally take \(v'\in\mathcal Q^0\). By (120), the initial equation of \(h^{m_j}+a_jF_j\) is either \((h^0)^{m_j}\), or \((h^0)^{m_j}+a_jF_j^0\) when \(w(F_j)=m_j\). In either case its \(v'\)-value is at least \(m_jv'(h^0)\). Therefore \[ 0<A_{S^0,\Psi_\lambda^0}(v')\le\lambda b^0(v'), \tag{126}\] with equality on the right at \(w^0\). The left inequality proves positivity of \(b^0\) on the entire common slice, including its noncentred faces. For centred \(v'\), central normalized-volume minimality and (126) give \[\lambda b^0(v')\mathop{\mathrm{vol}}(v')^{1/n} \ge A_{S^0,\Psi_\lambda^0}(v')\mathop{\mathrm{vol}}(v')^{1/n} \ge\lambda b^0(w^0)\mathop{\mathrm{vol}}(w^0)^{1/n}.\] This is the required constrained minimality on the central slice. ◻

Proof. For the height-normalized pair, the midpoint interpolant has height at most one. Its volume is therefore at least the minimum of volume on the height-one slice. The proof using (117) applies verbatim: any fixed separation on one function would force a fixed relative volume loss. For converging heights \(b_i\), positivity on the compact slice gives \(b_i/b=1+o(1)\) uniformly. Their minimizing rays, normalized by \(b=1\), are thus near-minimizers for \(b\), and the preceding assertion applies. Evaluation on \(h\), which equals \(A\) on the common rays, shows that the factors needed to normalize by \(A=1\) converge as well. ◻

Proof. Normalize both rays by \(b=1\) and interpolate them with weights \(1/2\). The resulting centred \(u\) has \(b(u)\le1\), so minimality gives \(\mathop{\mathrm{vol}}(u)\ge\mathop{\mathrm{vol}}(w)\). For \(w(f)>0\), (116) gives \[1\le2^n\frac{w(f)}{w(f)+v(f)}.\] Thus \(v(f)\le(2^n-1)w(f)\), which implies the stated bound. If \(f\) is a unit both values are zero. The central assertion follows from the same argument and Theorem 37. ◻

Proof. As \(\mu\uparrow\mu_*\), the minimum of \(b_\mu\mathop{\mathrm{vol}}^{1/n}\) tends to zero. To prove this even when an endpoint zero occurs at a noncentred valuation \(v\), interpolate \(v\) with one fixed centred point \(w_1\in\mathcal Q\), giving \(w_1\) weight \(t\). For \(0<\mu_*-\mu\le t\), the interpolant \(u\) has \(b_\mu(u)=O(t)\). Since \(v(h)=w_1(h)=1\), the improved volume estimate (116), applied with \(f=h\), gives \(\mathop{\mathrm{vol}}(u)\le t^{1-n}\mathop{\mathrm{vol}}(w_1)\). Hence \(b_\mu(u)\mathop{\mathrm{vol}}(u)^{1/n}=O(t^{1/n})\). The lower volume bound in (111) then forces the minimizing height to tend to zero. In particular its minimizing valuation eventually has \(g(w)>c\).

If the minimizer at \(\mu=0\) has \(g(w)>0\), take \(\mu=0\). Otherwise continuous dependence in Theorem 37 gives a nonempty open interval of parameters for which \(0<g(w)<c\). Choose a rational parameter in that interval. Then \(b_\mu(w)=1+\mu(c-g(w))\ge1\). Finally \(v(G)\ge0\) gives \(g(v)\ge-1\) on the normalized slice, while \(\mu<\mu_*\le1/c\). Thus \(b_\mu(v)\le1+(c+1)/c=2+1/c\). The same calculation applies centrally because \(G^0\) is regular and \(v'(h^0)=1\) on \(\mathcal Q^0\). ◻

Proof. Subduction subtracts a polynomial in the \(x_i\) with the prescribed initial symbol, then repeats at a strictly larger value. There are only finitely many values in a bounded interval: each is a nonnegative integral combination of the finitely many positive \(\alpha_i\). The Izumi upper bound for the fixed valuation \(w\) implies that remainders whose \(w\)-values tend to infinity tend to zero in the \(\mathfrak m_x\)-adic topology. In particular \(\mathfrak m_x\subset(x_1,\ldots,x_N)+\mathfrak m_x^2\), and Nakayama’s lemma shows that the \(x_i\) generate \(\mathfrak m_x\). Completed subduction proves both assertions in (129) and the assertion about threshold ideals. Monomial threshold ideals are finitely generated. Their completed extensions intersect the local ring in the original ideals, by faithful flatness of completion.

We record why the dimension and primeness arguments used below are valid also for other positive weight vectors. For any such vector \(\gamma\), its monomial filtration is bounded above and below by fixed linear reindexings of the \(\mathfrak m_x\)-adic filtration. Consequently its cumulative quotient lengths grow with exponent \(n\). The associated graded algebra is \(\mathbb C[X]/\operatorname{in}_{\gamma}I\); comparison of its positive weighted filtration with ordinary polynomial degree shows that its growth exponent is its Krull dimension. Thus this dimension is \(n\). If \(\operatorname{in}_{\gamma}I\) is prime, the filtration is multiplicative and separated, and defines a valuation. If its value group has rational rank \(r\), the degree-zero homogeneous fraction field is its residue field and has transcendence degree \(n-r\). This proves Abhyankar equality. Local monomialization of Abhyankar valuations in characteristic zero (Jonsson and Mustaţă 2012, Proposition 3.7) then makes the valuation quasi-monomial.

Take finitely many series in \(I\) whose \(\alpha\)-initials generate \(\operatorname{in}_{\alpha}I\). Their leading faces persist after composition with \(L\): ties are preserved because their fine weights are equal, and positivity leaves only finitely many other terms that could compete in a fixed neighbourhood of \(\alpha\). Therefore \[\operatorname{in}_{\alpha}I\subset \operatorname{in}_{\alpha_L}I.\] The first ideal is prime, and both quotients have dimension \(n\). A nonzero ideal in a finitely generated domain lowers its dimension, so this inclusion is equality.

For the final assertion, use a smooth monomial chart for \(w\) at its stratum and expand the finitely many \(x_i\) over the residue coefficient field. Their leading faces are finite polynomials in the stratum parameters. The weights of these parameters belong to \(\Lambda_w\). Composing them with nearby rational \(L\) preserves the same leading polynomials. The resulting monomial valuation dominates the presentation filtration. The induced map from the presentation graded algebra to the graded ring of this monomial valuation is injective: the polynomial relations among these leading polynomials are exactly the old relations, unchanged by regrading. Hence the two valuations agree on every function. ◻

Proof. First track discrepancy under the rational approximations of Lemma 41. On a fixed monomial chart with stratum parameters \(z_1,\ldots,z_r\) and separating residue coordinates \(y_1,\ldots,y_{n-r}\), write \[\omega=\psi\, \frac{dz_1}{z_1}\wedge\cdots\wedge\frac{dz_r}{z_r} \wedge dy_1\wedge\cdots\wedge dy_{n-r}.\] The quasi-monomial discrepancy formula gives \(A_S(w)=w(\psi)\in\Lambda_w\). Include numerators and denominators of the rational function \(\psi\) among the finite functions whose leading faces are tracked. The same formula on the same chart then gives the exact identity \[ A_S(w_L)=L(A_S(w)). \tag{131}\] In particular this is an identity of linear weights, rather than only a limiting assertion about discrepancies.

Choose rational \(L\) and multiply it by a positive integer so that the presentation weights are positive integers. In this paragraph write \(v\) for this integral scaling of \(w_L\), and \(a_i=v(x_i)\). After shrinking to a \(G\)-stable affine neighbourhood, the \(x_i\) have common zero set \(\{x\}\). Define the extended Rees algebra \[ \mathcal R=\mathcal O(S)[t,x_1t^{-a_1},\ldots,x_Nt^{-a_N}] \subset\mathcal O(S)[t,t^{-1}]. \tag{132}\] The threshold-ideal description proves that its special fibre is the graded algebra in (129); positive-level quotients are supported at \(x\), so the affine and local constructions have the same special fibre. This algebra is of finite type and torsion free over \(\mathbb C[t]\), hence flat. Let \(\mathcal S=\mathop{\mathrm{Spec}}\mathcal R\) and \(E=(t=0)\). The divisor \(E\) is normal, integral, reduced, and Cartier, and \(\mathcal S\setminus E=S\times\mathbb G_m\).

The total space is normal. Away from \(E\) this follows from the normality of \(S\). At points on \(E\), the nonzerodivisor \(t\) and the \(S_2\) property of \(\mathcal R/(t)\) imply \(S_2\) for \(\mathcal R\). At the generic point of \(E\), the quotient by \(t\) is a field, so the total local ring is a regular discrete valuation ring. These observations give \(R_1\) as well.

The order along \(E\) restricts to \(v\) on \(\mathbb C(S)\) and takes value one on \(t\). Indeed, the residues of \(f t^{-v(f)}\) are the nonzero graded symbols of \(f\), and the direct grading prevents cancellation between distinct such leading terms. It is the Gauss extension of \(v\). The monomial log-form computation gives \[ \mathop{\mathrm{div}}_{\mathcal S}\left(\omega\wedge\frac{dt}{t}\right) =(A_S(v)-1)E. \tag{133}\] For example, on a model where the rational valuation \(v\) is \(c\mathop{\mathrm{ord}}_P\), use a uniformizer of \(P\) and separating residue coordinates. The extension has weights \((c,1)\) on that uniformizer and \(t\); the wedge of their logarithmic differentials has log order zero, while the remaining coefficient has value \(cA_S(\mathop{\mathrm{ord}}_P)=A_S(v)\). This proves the coefficient in (133). There are no other divisor coefficients, since off \(E\) the form is the product of the divisor-free \(\omega\) and \(dt/t\).

Since \(K_S\) is Cartier and \(v\) has integral values, \(A_S(v)\) is an integer. Therefore \[\eta=t^{-A_S(v)}\omega\wedge\frac{dt}{t} \qquad\text{satisfies}\qquad \mathop{\mathrm{div}}_{\mathcal S}(\eta)=-E.\] This establishes that the ordinary canonical divisor of the total space is Cartier. At every codimension-one point of \(E\), the fibre is regular and the flat morphism to the line is smooth. The residue of \(\eta\) is consequently a rational top form on \(E\) with neither zeros nor poles at any codimension-one point. It trivializes the reflexive canonical sheaf of \(E\). In particular this establishes ordinary canonical Cartierness, not just Cartierness after adding a boundary.

Give \(t\) weight \(-1\) and \(x_i t^{-a_i}\) weight \(a_i\). The residue has weight \(A_S(v)\) and the same \(G\)-character as \(\omega\). If an old torus is present, choose homogeneous generators; \(t\) is invariant for that torus, so its canonical character is unchanged. In the cone case such homogeneous generators have common zero only at the vertex: their common zero set is invariant and is isolated there, whereas the closure of every positive-grading orbit contains the vertex.

Units of a positively graded affine domain with degree-zero part \(\mathbb C\) are constant. Hence any two divisor-free canonical generators on \(E\) differ by a scalar. The full fine grading torus therefore acts on the generator by a character \(\kappa\), independent of the rational regrading used to construct it. For all nearby rational \(L\), its evaluation is \(L(A_S(w))\) by (131) and the residue computation. Equality on this open set of rational functionals identifies the two lattice weights and proves (130), including its equivariant version.

Finally suppose a nonidentity element of \(G\) acquired a fixed locus of codimension at most one on \(E\). At a general point of that locus the fibre, and hence the morphism \(\mathcal S\to\mathbb A^1\), is smooth. The fixed locus of the cyclic subgroup generated by that element is smooth there. Its tangent space is the invariant tangent subspace, and it maps surjectively to the tangent space of the line: averaging a lift of the base tangent vector gives an invariant lift. Thus this fixed locus is smooth over the line and has the same fibre codimension on nearby nonzero fibres. This contradicts quasi-étaleness of the original effective action. The action on \(S^0\) is therefore quasi-étale. ◻

Proof. Include the finitely many boundary equations in the presentation. Their orders and discrepancy equality persist exactly under rational regrading. In (132), each equation \(f\) has homogenization \(f t^{-v(f)}\), which is a nonzerodivisor and has no component \(E\); it restricts to \(\bar f\) on \(E\). In particular the initial boundary is rationally Cartier. The morphism \(\mathcal S\to S\times\mathbb A^1\) is birational. Comparing the coefficient at its only vertical prime by (133) shows that \[(\mathcal S,E+\Delta_{\mathcal S}) \longrightarrow (S\times\mathbb A^1, \Delta\times\mathbb A^1+S\times\{0\})\] is crepant precisely when \(A_S(v)=v(\Delta)\). Its source is therefore lc, and lc adjunction gives the stated central pair. At codimension-one points of the normal Cartier fibre the morphism is smooth, so the different is exactly the ordinary restriction of the homogenized boundary. Apply this to \(\Delta=H\). Formula (130) and injectivity of evaluation on \(\Lambda_w\) identify the weight of \(\omega^0\) with that of \(\bar h\). Quasi-étaleness and the finite-map discrepancy formula descend klt, lc, and the invariant log form to the quotient. A suitable power of \(\omega^0\) descends to a divisor-free pluricanonical form. ◻

Proof. Use Lemma 36 to encode the common slice. Choose positive rational \(\theta_j\) with \(\sum_j\theta_j<1\) and sufficiently general scalars \(\lambda_j\), and set \[ P_j=h^{m_j}+\lambda_jF_j, \qquad \Psi=(1-\textstyle\sum_j\theta_j)H+ \sum_j\frac{\theta_j}{m_j}\mathop{\mathrm{div}}(P_j). \tag{135}\] The lemma says that \(\Psi\) is effective and lc, with exactly the common rays as its nonzero zero-discrepancy rays. It also gives \(q(P_j)=m_jq(h)\) on every common ray. These two properties let us express a sign inequality as log canonicity of a small boundary perturbation.

Assume the left side of (134). For sufficiently small positive rational \(\delta\) the divisor \[ \Psi_\delta=\Psi+\delta(\mathop{\mathrm{div}}(U)-bH) \tag{136}\] is effective and lc near \(x\). Here is the exact reason for smallness sufficing. The coefficient of \(H\) in (135) has a positive reserve, which preserves effectivity. On a fixed log resolution including \(U\), every coefficient below one stays below one for small \(\delta\); for a coefficient-one prime the discrepancy change is \(-\delta q(g)\ge0\). Checking these finitely many components checks lc on the entire resolution. Restricting the germ if necessary discards centres whose closed image does not contain \(x\). No uniform choice of \(\delta\) in \(U\) is needed.

All equations in (136) have their expected orders at \(w\), and \(A_S(w)=w(\Psi_\delta)\) because \(w(g)=w(f_j)=0\). Corollary 43 gives an lc initial boundary. For every \(q^0\in\mathcal Q^0\), \[q^0(\overline{P_j})\ge m_jq^0(\bar h).\] Consequently \[0\le A_{S^0,\Psi_\delta^0}(q^0) \le-\delta q^0(\bar g),\] which proves the right side of (134).

Conversely assume that right side. Carry out exactly the same boundary construction on \(S^0\), choosing the \(\lambda_j\) general for both the actual and central constructions. The intersection of these finitely many nonempty open conditions is nonempty. The assumed sign and a log resolution of the central equations give a small rational \(\delta>0\) for which \(\Psi_\delta^0\) is effective and lc near \(o\). Choose an integral rational regrading that preserves all the equations involved. Homogenize them in the Rees family. Their rational linear combination \(\Delta_{\mathcal S}\) is an effective rationally Cartier horizontal boundary, disjoint in components from \(E\), and its restriction is \(\Psi_\delta^0\).

The total space is normal, \(E\) is normal and Cartier, and its ordinary canonical divisor is Cartier by Proposition 42. Thus \(K_{\mathcal S}+E+\Delta_{\mathcal S}\) is \(\mathbb Q\)-Cartier. The different is the ordinary restriction, as checked above. Inversion of adjunction for log canonicity (Kawakita 2007, Theorem) now makes \((\mathcal S,E+\Delta_{\mathcal S})\) lc near \(o\). The section with all rescaled coordinates \(x_i t^{-a_i}\) zero specializes to \(o\), so this neighbourhood meets it over a nonzero parameter. Off \(E\) the family and its horizontal boundary are the product with \((S,\Psi_\delta)\). Hence the latter pair is lc near \(x\). For a common actual ray, the initial boundary encoding has zero discrepancy, and the last perturbation subtracts \(\delta q(g)\). It follows that \(q(g)\le0\), proving the converse.

The resolution computations give inequalities on every divisorial ray, and continuity on the monomial cones of the lc skeleton gives them on the full quasi-monomial slices. Additional finite tests are treated in the same boundary and presentation. ◻

Proof. Apply Proposition 44 to \(g=f_j\), and combine its upper bound with the defining lower bound. The equality \(w(f_j)=0\) and injectivity of the fine grading imply that \(\bar f_j\) has weight zero. The invariant log form \(\omega^0/\bar h\) has weight zero, so the pure-weight discrepancy computation of Section 4 gives lc equality for every pure weight. The test ratios then all have value zero. Klt implies that \(\bar h\) is strictly positive on every nonzero such weight, so the stated normalization is possible. ◻

Proof. Choose old-homogeneous elements of \(\breve R\) whose \(v\)-symbols generate \(R^{00}\) and lift them to regular functions \(x_i\) on \(S\). Enlarge the list to generate the first graded algebra and to include \(h,F_j\) and every other specified equation. Use \(G\)-eigenvectors throughout. Put \(\alpha_i=w(x_i)\) and \(\beta_i=v(\bar x_i)>0\). Both successive subductions of Lemma 41 give \[ R^{00}=\mathbb C[X]/\operatorname{in}_\beta (\operatorname{in}_\alpha I). \tag{137}\] Completion of the first ideal does not change its initial ideal for positive \(\beta\).

Choose finitely many generators of the prime ideal on the right that are iterated initials of elements of \(I\). This is possible by first choosing \(\alpha\)-homogeneous elements of \(\operatorname{in}_\alpha I\) with the desired \(\beta\)-initials; such elements lift with their prescribed \(\alpha\)-initial. For each of these finitely many series, positivity and finiteness of the relevant leading faces imply that its iterated initial is its \((\alpha+\delta\beta)\)-initial for all sufficiently small \(\delta>0\). Hence \[\operatorname{in}_\beta\operatorname{in}_\alpha I \subset\operatorname{in}_{\alpha+\delta\beta}I.\] The first ideal is prime and both quotients have dimension \(n\). They are equal. Lemma 41 supplies the actual multiplicative valuation \(w_\delta\) and its quasi-monomiality. Avoiding countably many values of \(\delta\) makes \((a,b)\mapsto a+\delta b\) injective on the joint lattice.

We now prove the lc equality, after constructing the valuation. The first degeneration has canonical character equal to the weight of \(\bar h\). Apply Proposition 42 to \(v\) on that cone, keeping the old torus action. Since \(v\) is \(\bar h\)-lc, the resulting canonical generator has exactly the joint character of \(\bar{\bar h}\). Independently apply the same proposition to the actual valuation \(w_\delta\), whose normal finitely generated graded algebra has just been identified with \(R^{00}\). Its discrepancy is the evaluation of this canonical character under compression. Therefore \[ A_S(w_\delta) =\langle w_\delta,\operatorname{wt}(\bar{\bar h})\rangle =w_\delta(h). \tag{138}\] This uses no prior lc assumption about \(w_\delta\). The included coordinates identify the specified symbols, and the joint degree of each \(\bar{\bar f_j}\) is zero; hence \(w_\delta(f_j)=0\).

For the uniform comparison, if \((1-\varepsilon)\alpha_i\le\alpha_i+\delta\beta_i \le(1+\varepsilon)\alpha_i\) for every \(i\), the same inequalities hold for every monomial weight. The threshold-ideal description, including its intersection with the actual local ring, gives \[(1-\varepsilon)w(f)\le w_\delta(f) \le(1+\varepsilon)w(f) \qquad(0\ne f\in\mathcal O_{S,x}).\] Normalization by \(h\) changes the factors by another quantity tending to one. Finally the joint lattice surjects onto the old lattice. A degree-zero rational function with nonzero \(v\)-value provides a nonzero vector in its kernel. Injective compression therefore strictly increases rational rank. ◻

Proof. The last assertion is the central domination estimate in Corollary 39; equivariance follows from the uniqueness of the slice minimizer. At every stage, Corollary 45 makes all common central test values zero and makes every pure weight common. The \(\bar f_j\) are genuine functions of the angular field \(\mathcal K\). At a height-minimizing Gauss extension of \(T\), regularity of \(\bar F_j\) gives \[ T(\bar f_j)\ge-m_jD(T). \tag{142}\]

Suppose (141) fails along a subsequence. There are a fixed rational \(c>0\), primitive witnesses \(T\), and effective divisors \[\Gamma=D+\frac1m\mathop{\mathrm{div}}(\phi),\qquad m\in\mathbb N_{>0}, \quad\phi\in\mathcal K^*,\] such that \(a(T)=0\), \(T(\bar f_j)\ge0\), and \[ T(\Gamma)-D(T)\ge2cD(T). \tag{143}\] Here \(D(T)>0\) because \(a+\mathbf D\) is klt. After clearing denominators, the ray-ring description in Section 4 makes \(G_0=\bar h^m\phi\) a regular homogeneous function upstairs. Choose a rational Gauss extension minimizing \(\bar h\) at \(T\). It is \(\bar h\)-lc, and its lift to the finite quasi-étale cover cone remains lc. Its centre has the vertex in its closed image: this holds downstairs for a Gauss valuation, and finiteness and the unique point over the vertex give it upstairs. It is common, and (143) reads \[\frac{g(q)}{q(\bar h)}\ge2c, \qquad g(q)=\frac1m q(G_0)-q(\bar h).\]

Apply the one-sided tilt of Lemma 40 to this cone. Its exact minimizer \(v\), normalized by \(v(\bar h)=1\), is centred, \(G\times\mathbb T_w\)-invariant, and has normal finitely generated klt associated graded. Localization at the vertex does not change this graded algebra: a unit has constant initial term. The valuation \(v\) is common, hence exact on the old tests, and the tilt gives \[v(\phi)>0,\qquad b(v)\ge1.\] The tilted height \(b\) is bounded above by a fixed constant on both the old and the new common slice, with constants depending only on \(n_0\) and the fixed \(c\) at this stage. Central domination therefore gives, on the new cone, \[ q^{00}(s)\le C_1v^0(s) \tag{144}\] for common \(q^{00}\) normalized by \(\bar{\bar h}=1\). The joint canonical character equals that of \(\bar{\bar h}\) by Proposition 42, including its old-torus assertion.

Compress these two gradings by Lemma 46. Since \(\phi\) has old weight zero and \(v(\phi)>0\), rational rank strictly increases. The resulting actual valuation is \(h\)-lc, exact on the tests, and can be chosen arbitrarily close to \(w\) uniformly on every regular function. We check balance, which is necessary to repeat the argument. A joint-homogeneous \(s\) has an old-homogeneous lift \(\widetilde s\in\breve R\); the \(v\)-filtration splits into old weight spaces. If its joint degree is \((a,b)\), then \[b=v(\widetilde s)\le Ca\] by (139), since \(v\) is common. In the normalization \(w(h)=v(\bar h)=1\), the compressed degree is \((a+\delta b)/(1+\delta)\). Thus \[v^0(s)=b\le C(1+\delta)w_\delta^0(s).\] Together with (144) this is the next balance estimate, with a fixed constant. A generic compression makes every fine-homogeneous element joint-homogeneous, so this checks precisely all elements required in (139).

If the angular order condition still fails, pass to a subsequence with a fixed positive defect and repeat. Each repetition raises rational rank, which is at most the dimension. There can be at most \(n_0\) repetitions. At a final stage the supremum of the positive normalized defects must tend to zero. This supremum is finite: a height-minimizing Gauss witness and (139), applied to \(G_0\) of grading value \(m\), bound \(T(\Gamma)/D(T)\) by the balance constant. Taking these suprema, or slightly larger positive numbers, gives the single sequence \(\eta_i\) in (141), uniformly for all its displayed tests and divisors. At each of the boundedly many compressions choose the all-functions multiplicative error to tend to zero; their product still tends to one and proves (140). All exact test equalities, equivariance, canonical characters, and quasi-étaleness have been preserved at every step. ◻

Proof. Write \(A_j=\operatorname{in}_wY_j^+\) and \(B_j=\operatorname{in}_wY_j^-\). The pair products \(A_jB_j\) lie in \(K_*\). With an old anchor, the finitely many old relations \[\operatorname{wt}(A_j)=(t_j/d_j)\kappa_0\] give neutral rational functions \(A_j^{d_j}A_0^{-t_j}\). Each is a quotient of two neutral functions of cost \(O(L)\). More explicitly, for \(t_j\ge0\) write it as \[\frac{A_j^{d_j}B_0^{t_j}}{(A_0B_0)^{t_j}};\] for \(t_j<0\) its numerator is simply \(A_j^{d_j}A_0^{|t_j|}\). The required costs are bounded by a fixed multiple of \(d_jL_-+|t_j|L_1=O(L_-)=O(L)\). There are at most \(n\) accumulated variables, and their denominators \(d_j\) are bounded. Thus one fixed constant works for this entire finite collection. The inclusion \(S(CLD)\subset K_*\) puts their numerators and denominators in \(K_*\) at one eventual index. Without an anchor all the old \(A_j\) themselves are neutral and of bounded cost, and belong to \(K_*\) instead.

The new weights are independent modulo the old line, or modulo zero. Consequently a monomial relation of total weight zero has zero net exponent in each new variable. Clearing the bounded old denominators expresses a power of every remaining weight-kernel monomial in the finite neutral generators just described. Every such monomial is therefore algebraic over \(K_*\). This is a simultaneous algebraic statement: once the finite generators have entered \(K_*\), there is no further eventual index depending on the exponents.

Choose a term in the least-\(w\) group, and multiply the whole sum by the opposite factors \(Y_j^-\) corresponding to its \(Y_j^+\) factors. Its \(G_a\) factors already have value zero. The complemented sum has a nonzero \(w\)-initial of weight zero. Each term of that initial is algebraic over \(K_*\) by the preceding kernel argument, so the initial itself is algebraic over \(K_*\). Every common cone test is trivial on this element. Here we use the elementary fact that a valuation trivial on a field is trivial on its algebraic extension inside a valued field: a polynomial equation with nonzero constant term would otherwise have a unique term of smallest value.

The complemented sum is an invariant ratio with a regular numerator after multiplication by some power of \(h\). Its cost is allowed to depend on the chosen polynomial. Proposition 44 has no uniform cost restriction, and yields an actual upper bound zero on the pre-pinning slice. At a point where the original terms tie, opposition and the zero values of \(G_a\) give value zero for every complemented term. The valuation inequality gives the lower bound zero. The complemented sum thus has exact value zero, which proves non-cancellation.

At a later working valuation \(w'\) where the terms tie, this establishes exact value zero before any new bridge is invoked. Its initial symbol there is nonzero. Apply the bridge in the other direction, using only tests exact at \(w'\). At a common test on the new cone where the terms remain tied, the transferred upper bound and the same termwise lower bound again give equality. This proves the onward assertion. ◻

Proof. The rational map defined by \(V\) has a nondegenerate integral projective image of dimension \(a\) in \(\mathbf P^{g-1}\). Its degree-\(N\) homogeneous coordinate space is \(V^{(N)}\). Successively cut by general hyperplanes. In positive dimension the reduced hyperplane section is nondegenerate in its hyperplane. One way to see the required nondegeneracy is to pass to the normalization: a general hyperplane divisor is reduced in codimension one, so another linear form vanishing on its reduced support has a regular ratio with the cutting form, and the ratio is constant. The reduced section is integral until the last curve section by Bertini; the final reduced set of points spans its hyperplane and has Hilbert function at least its dimension in degree one in every higher degree. For the homogeneous coordinate ring, multiplication by a nonzero cutting linear form is injective, and the quotient surjects onto the coordinate ring of the reduced cut. Thus successive differences of Hilbert functions dominate the Hilbert functions of these cuts. Summing these inequalities gives, for \(a>0\), the sufficient bound \[\dim V^{(N)}\ge(g-a)\binom{N+a-1}{a}.\] Since \(g\ge a+1\), one has \(g-a\ge g/(a+1)\), giving the asserted constant. For \(a=0\) the image is a point, \(g=1\), and the assertion is immediate. ◻

Proof. Suppose otherwise. Then, for each fixed bound on normalized coefficient size, every exact integral tie at \(v\) eventually belongs to the old row span. To verify this assertion, a tie with transverse part outside the old span would give the desired row after fixing its bounded integer transverse coefficients. If its transverse part lies in the old span, subtract the corresponding rational combination of the fixed old transverse rows. The remainder is a pure-anchor tie. It is zero because (161) preserves the nonzero anchor value at \(v\). Without an anchor there is no remainder. The claimed eventual assertion for all ties with a given bound follows by choosing a failing tie along a subsequence if necessary.

Let \(\widetilde u_i\) be a unit vector in \(\mathcal R_i^\perp\) tending to \(u\). Choose a large fixed integer \(N\gg qM_0\). In the exponent box \[0\le e_j\le\lfloor NL/L_j\rfloor, \qquad \widehat e=(e_jL_j/L)_j,\] there are at least \(cR_*N^s\) integer points. The range of \(\widetilde u_i\cdot\widehat e\) has length \(O(N)\). Partition this range into intervals of width \(qM_0/4\). One slab contains at least \[ cR_*N^{s-1}qM_0 \tag{166}\] points. For each exponent in this slab, consider on \(\operatorname{gr}_v\) its positive-variable monomial multiplied by \(Z^l\), where \[ 0\le l\le\left\lfloor N/(qM_0\delta)\right\rfloor, \tag{167}\] and by a basis of the products of \(N\) of the \(G_a\) whose \(w\)-initials are independent. Saturation and counting supply at least \(v_iN^p\) such products.

Different powers \(l\) have disjoint grading values in this construction. A collision would give the exact integral tie \[e-e'-q(l-l')r.\] Its normalized size is bounded with \(N,M_0,\delta,q\) fixed. Under our supposition it belongs to the old row span, so its normalized dot product with \(\widetilde u_i\) is zero. The slab bounds the contribution from \(e-e'\) by \(qM_0/4\). But (163) gives \[\widehat r=M_0u_i+O(1), \qquad |\widetilde u_i\cdot\widehat r|\ge M_0/2\] eventually, after increasing the fixed \(M_0\). For \(l\ne l'\) the other contribution has magnitude at least \(qM_0/2\), a contradiction. It is essential here that \(\widetilde u_i\) is orthogonal to the entire old normalized row span.

At any one power \(l\), the slab size times \(v_iN^p\) symbols are linearly independent. Before changing to \(v\), the selected variable weights separate the exponent vectors, and the chosen \(G_a\) products have independent initials at \(w\). Any possible relation after the change can be grouped by its \(v\)-grading, and Lemma 49 rules out cancellation in each group. Multiplication by the nonzero \(Z^l\) preserves this independence. Combining with disjoint grading values for different \(l\) gives at least \[ c\delta^{-1}R_*v_iN^{p+s} \tag{168}\] independent ratios.

Every ratio just counted has cost and absolute grading value at most \(C_0NL\), where \(C_0\) is independent of \(\delta,M_0,q\). The slab monomials cost \(O(NL)\); the \(G_a\) products do too; and (165) multiplied by the exponent bound (167) costs \(O(NL)\). A common \(h\)-denominator gives regular numerators of one finite character and degree \(O(NL)\). By (161), eventually \(v\ge w/2\), so the transferred upper bound (155) has a constant independent of the tilt. It contradicts (168) if \(\delta\) is sufficiently small.

The order of choices is as follows. First fix the current stage constants, the old rows, the saturation constant and the original counting constant. Then fix \(\delta>0\) small enough for this last contradiction. Next fix \(M_0\) sufficiently large compared with \(1/\delta\) and the rounding constants. The character construction then gives a bound \(Q\) for \(q\); pass to a subsequence fixing \(q\), or choose using \(Q\). Only now choose a sufficiently large fixed \(N\), and finally let \(i\) be large enough for all of these fixed-parameter assertions. Thus \(\delta\) never tends to zero with \(i\), and no eventual count is used at a variable product degree. This proves the claim. ◻

Proof. The proposition supplies the new exact tests, rational proportions and, when present, the anchor lower bound. It also gives \(w'\ge c w\) on all regular functions, so the transferred count (155) remains available with a fixed constant.

Use double bars for the initial symbols at \(w'\). Let \(E\) be the field generated by the \(p\) chosen algebraically independent \(\overline{\overline{G_a}}\) and the \(s\) selected variable symbols. These have transcendence degree \(p+s\), by Lemma 49: they were independent together at \(w\), since the selected variable weights were independent and the chosen \(G_a\) were neutral. Write \(E^0\) for its subfield of weight zero. Every common test on the new cone is trivial on \(E^0\). Indeed, the both-sign pinning equations also hold on that cone. If there is an anchor, its nonzero value at \(w'\) ensures that two selected monomials with equal weight at \(w'\) have equal value at every common test; in the all-zero case this is immediate. Lemma 49 then gives the predicted exact value for each nonzero homogeneous polynomial. Ratios of homogeneous polynomials of equal weight generate \(E^0\), proving the assertion.

There is also a space \(V_0\) of invariant ratios on this new cone, of cost and absolute grading value \(O(L)\), with \[ \dim V_0\ge cR_*v_i. \tag{173}\] Take the double-bar symbols of the whole \(G_a\) basis, and, if an old anchor was present, multiply by its powers with exponent from \(0\) to \(\lfloor R_*\rfloor\). Their costs remain \(O(L)\). They are linearly independent at \(w\) by their distinct anchor weights and the basis property, and remain so by Lemma 49.

Choose an enlargement \(V\) of \(V_0\), with the same type of fixed cost and grading bounds, containing \(1\), the specified generators of \(E\), and every neutral function in \(S(LD_{\mathrm{next}})\). We may also include in \(V\) a nonzero homogeneous ratio of weight a bounded positive multiple of any chosen new lattice vector \(k\) of width \(O(L)\). For the latter use Corollary 27 at \(w'\): the refined cone satisfies (41), and tiny \(LD_{\mathrm{next}}\) forces all new tests by the same integrality and cost argument as before. The space \(V\) is spanned by homogeneous ratios. Give all products of \(N\) members of \(V\) one common denominator \(h^t\), where \(t=O(NL)\); their regular numerators then have a single finite character and degree \(O(NL)\). The transferred upper bound (155) applies.

If the field generated by \(V\) had transcendence degree \(a>p+s\), Lemma 50 and (173) would give at least \(cR_*v_iN^a\) independent products, against \(C'R_*v_iN^{p+s}\). Choose one sufficiently large fixed \(N\) and then let \(i\) be sufficiently large. This contradiction shows that the field generated by \(V\) is algebraic over \(E\). Uniformity for the chosen \(k\) follows by applying this argument to any alleged sequence of failing choices.

A neutral element algebraic over \(E\) is algebraic over \(E^0\). In fact, clear denominators in an algebraic equation so that its coefficients are polynomials in the homogeneous generators of \(E\), take a nonzero homogeneous part of the equation, and divide by a nonzero coefficient of that weight. All resulting coefficients have weight zero. Common tests are trivial on \(E^0\), and hence on these algebraic elements. The Gauss description and finite-cover lifting translate this back to every angular test in \(\mathcal L\), proving assertion [it:G-field] for the next scale.

The same algebraic equation argument shows that the weight of an algebraic homogeneous element lies in the rational span of the weights of \(E\). The latter is the anchor line or zero. This proves the first part of assertion [it:G-lattice]. To bound its index in the line case write \(\kappa'_0=\ell k'_0\) with \(k'_0\) primitive. Its width is \(O(L_1)\) by the paired anchor costs. If \(\ell\to\infty\), apply the bounded-multiple construction at the current scale \(L\) to all \(jk'_0\), \(1\le j\le\ell\). Their widths are \(O(L_1)\) and \(L_1\le L\). There are symbols of weights \(q_jjk'_0\), with \(1\le q_j\le Q\) for fixed \(Q\), and cost and absolute grading value \(O(L)\). These bounds are uniform over all these choices: otherwise a sequence of failing \(j_i\) would contradict the arbitrary-sequence version of Corollary 27.

Let \(D_0\) be a fixed common multiple of the bounded denominators of the finitely many pinned variable weights. The subgroup generated by those weights has primitive step \(h_i\ge\ell/D_0\), and \(h_i\) divides \(\ell\). Among the integers \(q_jj\), an integer has at most \(Q\) preimages, and each residue modulo \(h_i\) has at most \(QD_0+1\) representatives in \([1,Q\ell]\). Hence at least \[\frac{\ell}{Q(QD_0+1)}\] distinct residues occur. All weights of the original space \(V_0\) in (173) lie in this subgroup. With an old anchor its basis consists of neutral symbols multiplied by powers of that anchor; when the pin has just chosen a new anchor, \(V_0\) consists only of neutral symbols. Multiplying this original \(V_0\) by symbols representing distinct residues yields a direct sum, since the weight supports are disjoint. Its dimension is an unbounded multiple of \(R_*v_i\), while its costs and degrees are \(O(L)\). The fixed-\(N\) version of (155) forbids this. Thus \(\ell\) is bounded, completing assertion [it:G-lattice]. ◻

Proof of Theorem 48. Assume the excluded sequence exists. The initial data satisfy [it:G-tests]–[it:G-working]. At every stage, either the sole remaining anchor line gives the primitive-discrepancy contradiction above, or a nonempty independent block of new directions can be chosen. A bounded block is pinned directly by (145); a high block is pinned by the finite row procedure, the rational-direction choice, and, if required, the anchor-height adjustment. The next-stage invariants then hold by the field and lattice argument. Each nonterminal stage adds at least one new positive variable, and their initial symbols remain algebraically independent at the next working valuation. A function field of dimension \(n\) cannot contain more than \(n\) algebraically independent elements. Thus the process cannot have more than \(n\) nonterminal stages, whereas all its terminal alternatives have been excluded. This contradiction proves the theorem. ◻

Proof. The coherent sheaf \(f_*\mathcal O_X(-M_iK_X)\) has finitely many generators on the affine base. Resolve their fractional ideal of ratios to a fixed section, together with the relevant divisors. On the resulting smooth model \(p:Y\to X\), their pulled-back divisor system has a common fixed \(\mathbb Q\)-divisor \(F\) and a basepoint-free residual system. This construction is valid even if \(M_iK_X\) is not Cartier: each section divisor is \(\mathbb Q\)-Cartier, their differences are principal, and principalizing the fractional ratio ideal makes the moving differences Cartier. The fixed divisor has order equal to the minimum of the section orders at every prime on the model.

The section \(s_b^{M_i/b}\) belongs to this system. Therefore \(F/M_i\le p^*(\Delta_{s_b}/b)\), and the fixed-order discrepancy \[A_X(E)-\frac{\mathop{\mathrm{ord}}_E(F)}{M_i}\] is nonnegative for every prime \(E\) over \(X\). Take the support on \(Y\) to be simple normal crossings. A general residual member, with coefficient \(1/M_i<1\), is transverse to this support. If every fixed-order discrepancy were strictly positive over a neighbourhood of the specified point, this general member would give a klt \(M_i\)-complement there. The only possible zero discrepancies of the fixed log smooth data come from strata containing a coefficient-one component. There are finitely many such components, and their images in \(Z\) are closed because \(Y\to Z\) is proper. Thus the absence of a klt complement on every neighbourhood implies that one component \(E\) with image containing the specified point has \(\mathop{\mathrm{ord}}_E(F)=M_i A_X(E)\). Set \(w=\mathop{\mathrm{ord}}_E\).

For \(k\mid M_i\) and any section \(s\) in degree \(k\), its power \(s^{M_i/k}\) is in degree \(M_i\). Hence \((M_i/k)w(\Delta_s)\ge\mathop{\mathrm{ord}}_E(F)=M_i A_X(w)\). The same inequality for \(s_b\), together with lc of the chosen complement, gives equality in degree \(b\). These statements also apply to local sections: coefficients in the local base ring have nonnegative \(w\)-order, and shrinking the base only inverts functions of order zero whenever the centre still contains the specified point.

Write \(W\) for the image centre of \(w\) on \(Z\). It is a proper closed subset: \(w(\Delta_{s_b})>0\) and a divisorial centre cannot dominate the birational base. If \(q=\dim W>0\), choose a general linear projection \(Z\to\mathbf A^q\) that is dominant and generically finite on \(W\). Pass to the geometric generic fibre and follow a component of \(E\). The image of that component in the new base is a closed point, and the new dimension is \(\dim X-q\ge1\).

Here are the required preservation details. The prime \(E\) is horizontal over \(\mathbf A^q\). After shrinking its base, the chosen resolution and its finitely many relevant strata are smooth wherever they dominate this base. Dividing a top form by a base volume form therefore restricts compatible canonical divisors to the generic fibre, preserving the log discrepancies and section-divisor orders. Extending the generic constant field separably does not ramify these prime orders: the nonzero discriminant of a finite separable constant extension is a unit in their valuation rings. Geometric integrality can be obtained by choosing the corresponding components after this extension.

Divisorial sheaves restrict to the corresponding divisorial sheaves by localization and separable scalar extension, as is seen on the smooth locus and then in codimension one. Flat base change for the proper morphism shows that the original generators generate the new complete systems; only the finitely many degrees dividing \(M_i\) and the degree \(b\) are needed here. Their inequalities hence hold for every local section at the new image point. Normality, birationality, the discrepancy bound, relative ampleness of \(-K_X\), and a Fano type witness are preserved by this operation. Thus all the conclusions survive the slice. The algebraically closed generic field of the complex base is isomorphic to \(\mathbb C\); alternatively the finite data can again be embedded into \(\mathbb C\). We retain the notation \(X,Z,z,w\) for these new data. ◻

Proof. Let \(r\) be the local Cartier index of \(K_X\) and shrink so that \(rK_X=\mathop{\mathrm{div}}(c)\). Normalize \(X\) in \[t^r=c^{-1},\] with the cyclic group acting on \(t\) by its standard character. The cover is integral of degree \(r\). Indeed a smaller Kummer degree would make \(c^{-1}\) a \(p\)th power for a prime \(p\mid r\), and then \((r/p)K_X\) would be principal, contrary to minimality. At a prime of \(X\), \(K_X\) is Cartier and the exponent of \(c\) is divisible by \(r\). The codimension-one Kummer calculation shows that the cover is quasi-étale. If \(\mathop{\mathrm{div}}(\omega_X)=K_X\), the form \(t\omega_X\) has divisor zero upstairs. Finite crepant pullback proves that \(S_0\) is klt.

A fibre of this finite quotient is a single orbit: invariant functions separate distinct finite orbits. If the fibre over \(z\) had more than one point, regular functions separating its points would have a nontrivial eigencomponent nonzero at one, hence at every, point of that orbit. After shrinking by its invariant norm, it would be an eigenunit. Write it \(s t^j\) with \(0<j<r\). The codimension-one calculation gives \(\mathop{\mathrm{div}}(s)=jK_X\), contradicting the minimal index. Thus the fibre is one fixed point.

For every integer \(k\), the regular functions of character \(k\) are precisely \(s t^k\) with \(s\in\mathcal O_X(-kK_X)\); changing \(k\) by \(r\) is accommodated by \(t^r=c^{-1}\). Their divisors are the pullbacks of \(\Delta_s\). Extend \(w\) to a primitive divisorial valuation \(w_0\) on the cover. Its centre is \(o\). If its restriction is \(e w\), finite crepant pullback gives \(A_{S_0}(w_0)=eA_X(w)\) and \(w_0(s t^k)=e w(\Delta_s)\). Equation (175) proves all the assertions. ◻

Proof. Finite generation and normality. The positive algebra is finitely generated by relative ampleness of \(-K_X\). For the negative algebra, take a small \(\mathbb Q\)-factorialization and a \(K\)-MMP over \(Z\). Fano type gives termination and a semiample output, since every divisor is relatively big for a birational contraction (Birkar et al. 2010). The integral-degree section spaces are preserved in these steps. To check this without assuming a Cartier index in degree one, on a common model write \(p^*K_{X_1}=q^*K_{X_2}+E\) for a \(K\)-MMP step, where \(E\) is effective and \(q\)-exceptional. An effective section divisor \(\mathop{\mathrm{div}}(s)+mK_{X_1}\) pushes to an effective section divisor on \(X_2\). Conversely, the pullback of an effective section divisor on \(X_2\), plus \(mE\), is effective, and pushes to its section divisor on \(X_1\). These are rational pullbacks, valid for every integral \(m>0\) because the divisors involved are \(\mathbb Q\)-Cartier; one can equivalently verify the inequalities after taking a Cartier power. The initial small modification identifies the divisorial sheaves in codimension one. On the semiample output \(Y\), let \(g:Y\to Y^{\mathrm c}\) be the semiample contraction and choose \(rK_Y=g^*L\) with \(L\) relatively ample. For each \(0\le j<r\), the projection formula identifies the residue module with \[\bigoplus_{a\ge0} H^0\bigl(Y^{\mathrm c},g_*\mathcal O_Y(jK_Y)\otimes \mathcal O_{Y^{\mathrm c}}(aL)\bigr).\] The pushforward is coherent by properness, so this is finite over the ample section ring. The Cartier Veronese is finitely generated and the finitely many residue modules are finite over it. The negative algebra is therefore finitely generated. The union of generators of the two half-algebras generates \(R\).

The ring \(R\) is the intersection of \(K[t,t^{-1}]\) with the Gauss valuation rings specified, for every prime \(D\) of \(X\), by \[\mathop{\mathrm{ord}}_D(s)-k\operatorname{coeff}_D(K_X)\ge0 \quad\text{on }s t^k.\] This follows from the codimension-one description of a divisorial sheaf, and proves normality. Its fraction field is \(K(t)\): the fraction field of degree zero is \(K\), and the generic fibre contains every integral degree. In particular \(\dim S_0=\dim X+1\).

The positive chamber and the canonical form. We prove that every prime divisor meets the positive chamber, where the canonical form can be computed on a Laurent bundle. Suppose that a prime divisor \(P\) lies outside it. The connected grading torus preserves every irreducible component of this complement, so \(P\) is homogeneous. It cannot contain all nonzero graded pieces. Indeed the locus where they all vanish is a closed subscheme of \(Z\) missing its generic point, where the signed algebra is \(K[t,t^{-1}]\). That locus has dimension at most \(\dim Z-1=\dim S_0-2\).

Choose a negative homogeneous element \(q\) of degree \(-m\) not in \(P\). Set \[A=R[q^{-1}]=R_{\le0}[q^{-1}],\qquad C=A_0.\] The equality follows by multiplying any positive-degree element by sufficiently many powers of \(q\). The ring \(C[q,q^{-1}]\) is a Laurent polynomial ring: its nonzero powers have different degrees. For a homogeneous \(a\in A_k\) the identity \[ a^m q^k\in C,\qquad a^m=(a^m q^k)q^{-k} \tag{180}\] proves integrality over this Laurent ring. Both formulas are valid for negative \(k\), because \(q\) is a unit on this chart. Finite generation then makes \(A\) finite over \(C[q,q^{-1}]\). The ring \(C\) contains the affine base ring \(R_0\) and lies in \(K\), so \(\mathop{\mathrm{Frac}}C=K\).

The height-one homogeneous prime induced by \(P\) contracts to a height-one homogeneous prime of \(C[q,q^{-1}]\). Since \(q\) is invertible, that contracted prime is generated by its intersection with \(C\) and lies over a prime divisor of \(\mathop{\mathrm{Spec}}C\), a chart of the negative Proj. This also excludes trivial restriction to \(K\): there is no homogeneous prime divisor solely in the invertible Laurent direction. The negative Proj is the relatively ample \(K\)-model obtained by the small modification, the \(K\)-MMP and its semiample contraction. None of these operations extracts divisors. The prime in question therefore has a strict transform \(D\) on \(X\); its restriction has the form \(w'=e\mathop{\mathrm{ord}}_D\), \(e>0\).

There are sections of both sufficiently divisible signs whose \(\Delta\)-orders at \(D\) are zero. For the positive sign this is relative generation of a multiple of \(-K_X\). For the negative sign use a generated multiple of \(K\) on its ample model avoiding the surviving prime. The birational map identifies the two generic DVRs and their compatible canonical coefficients. Its section lifts through the preserved section space, and the effective MMP correction has zero coefficient at this strict divisor. Thus its \(\Delta\)-order on \(X\) is also zero. Relative generation supplies both sections globally over the affine base with the indicated degree signs.

Let \(v=\mathop{\mathrm{ord}}_P\), and put \(\beta=v(t)+w'(K_X)\), where \(w'(K_X)\) means rational pullback order. For a homogeneous element, \[ v(s t^k)=w'(\Delta_s)+k\beta. \tag{181}\] Nonnegativity of \(v\) on the two zero-multiplicity sections forces \(\beta\ge0\) and \(\beta\le0\). Hence \(\beta=0\), and the positive section just chosen has value zero, contrary to \(P\) being outside the positive chamber. This proves the codimension assertion.

On the positive chamber the signed algebra is the usual Laurent divisorial bundle over \(X\). To see this on a Proj chart, localize at a divisible positive section. Poles along its complementary Cartier divisor can be cleared by its powers, identifying every degree piece with the corresponding localized divisorial sheaf on \(X\). A Cartier Veronese is a line bundle with its zero section removed, and the full chart is finite over that bundle by taking powers. In particular the inverse image of a codimension-at-least-two subset of \(X\) has codimension at least two. At a regular codimension-one point write \(K_X=\mathop{\mathrm{div}}(c)\); then \(ct\) is an invertible Laurent coordinate and \[t\omega_X\wedge d\log t =(ct)(\omega_X/c)\wedge d\log(ct).\] This is a nowhere-vanishing top form, and every nonidentity finite scalar acts freely there. The established codimension-two complements extend these conclusions to the assertions about \(\Omega_0\) and quasi-étaleness on \(S_0\).

Discrepancies and the lc boundary. The form also controls valuations whose centres lie outside the positive chamber. We will use its discrepancy estimate both for the boundary defined by \(s_bt^b\) and for the finite quotients in Lemma 57. If a divisorial multiple \(v\) centred on \(S_0\) has nontrivial restriction \(w'\) to \(K\), the relative Abhyankar inequality makes \(w'\) a divisorial multiple: its value group is discrete, and adjoining \(t\) increases residue transcendence degree by at most one, forcing its residue field to have transcendence degree \(\dim X-1\) over \(\mathbb C\). Its centre exists on \(X\) by properness. Relative generation of a positive Cartier multiple gives \(\beta=v(t)+w'(K_X)\ge0\). The top form gives \[ A_{S_0}(v)\ge A_X(w')+\beta, \tag{182}\] with equality for a Gauss extension. One can compute this on a smooth model at the prime defining \(w'\). Choose a uniformizer \(x\) and separating residue coordinates \(y_1,\ldots,y_{\dim X-1}\). The ratio of \(\omega_X\) to \(d\log x\wedge dy_1\wedge\cdots\wedge dy_{\dim X-1}\) has \(w'\)-order \(A_X(w')+w'(K_X)\). After wedging with \(d\log t\), the remaining logarithmic differential has nonnegative log order at \(v\). Indeed, at an upstairs uniformizer \(\rho\), every logarithmic differential has at most a simple \(d\rho/\rho\) pole, at most one such factor survives a wedge, and the differentials of the residue-coordinate units are regular. At a Gauss valuation its residues are independent and this log order is zero. Multiplication by \(t\) yields (182).

If \(w'\) is trivial, work on the smooth generic fibre \(\mathop{\mathrm{Spec}}K[t,t^{-1}]\); a centred valuation there has \(v(t)=0\) and positive discrepancy. Thus \(S_0\) is klt. For a nontrivial restriction, (181), (182), and lc of the actual degree-\(b\) complement give \[A_{S_0}(v)-\frac{v(s_bt^b)}b \ge A_X(w')-\frac{w'(\Delta_{s_b})}b\ge0.\] For trivial restriction, \(v(s_bt^b)=0\). Consequently the displayed boundary pair is lc. It has the lc place constructed next and is not being asserted to be klt.

A centred order with two slopes. It remains to construct the closed point and its saturated valuation. The positive slope will come from a choice of Gauss weight; the negative slope comes from vanishing on the fibre. Every irreducible component of the reduced fibre over \(z\) has positive dimension. An isolated zero-dimensional component would disconnect the fibre unless the whole fibre were finite; a proper birational morphism to a normal base with a finite fibre is an isomorphism after shrinking, the excluded case. A negative-degree section, raised to a Cartier power, restricts on each such projective component to a section of an antiample line bundle. It is therefore zero on every component. Its effective divisor contains the entire reduced fibre, so its order at the specified \(w\) is strictly positive. This argument only concerns reduced supports and also covers nonreduced fibres.

Choose finitely many negative-ring generators \(s_jt^{-m_j}\) over the degree-zero ring and set \[F_w=\min_j\frac{w(\Delta_{s_j})}{m_j}>0.\] Every negative homogeneous section is a homogeneous polynomial in these generators with base coefficients of nonnegative \(w\)-order. It follows that \(w(\Delta_s)\ge |k|F_w\) for \(k<0\). Choose a rational \(0<\alpha<F_w\), and let \(w_0\) be the Gauss extension of \(w\) with \(w_0(t)+w(K_X)=\alpha\). Formula (181) defines it, and gives (179); for the positive sign use effectivity of \(\Delta_s\). All nonzero degrees have strictly positive value, while the degree-zero maximal ideal is \(\mathfrak m_z\). Thus \(w_0\) is centred at the fixed closed point with ideal generated by these elements. A rational Gauss extension of a divisorial valuation is itself divisorial up to scaling. Equality in (182) gives \(A_0=A_X(w)+\alpha\), and (175) now yields (176) in actual positive degrees. ◻

Proof. The product is klt. The action remains quasi-étale: the fixed locus of every nonidentity element has codimension at least two already in \(S_0\). Wedging its canonical generator with \(du_1\wedge\cdots\wedge du_{b-1}\) gives a divisor-free generator of character \(b\). A suitable power descends through the finite quotient, so \(K_Y\) is \(\mathbb Q\)-Cartier and the quotient map is crepant.

Quasi-étaleness alone would not preserve a prescribed discrepancy gap. Let \(E\) be an actual prime divisor over \(Y\), and normalize an extension \(v\) to \(\mathbb C(S)\) so that its restriction to \(\mathbb C(Y)\) is \(\mathop{\mathrm{ord}}_E\). Finite crepant pullback, with homogeneous discrepancies, gives \(A_Y(E)=A_S(v)\). Although \(v\) may take fractional values upstairs, the field \(K=\mathbb C(X)\) is pointwise invariant and is contained in \(\mathbb C(Y)\). If \(w'=v|_K\) is nontrivial, its values are integral. The relative Abhyankar inequality then gives \[w'=e\mathop{\mathrm{ord}}_F\quad\text{for a prime }F\text{ over }X \quad\text{and an integer }e\ge1.\] For the torsor, the log-form calculation above, with the additional coordinate forms, yields \[A_S(v)\ge A_X(w')+\beta+\sum_jv(u_j) \ge e\epsilon\ge\epsilon.\] Here \(\beta\ge0\) by positive generation, each \(u_j\) is regular, and the non-Gauss logarithmic differential defect is nonnegative. For the canonical cover, first restrict to \(\mathbb C(S_0)\) and use the product log-form estimate; finite crepant pullback from \(S_0\) to \(X\) then gives the same lower bound \(A_X(w')\).

If \(w'\) is trivial, the valuation lies over the common generic point of the birational varieties \(X\) and \(Z\). The generic quotient is smooth: the canonical cover there is a cyclic torsor, and the signed construction is a Laurent line with a free scalar action; their products with the coordinate representation have smooth quotients. Generic restriction of the ordinary prime \(E\) is still a primitive prime, since localization at the trivially valued base field does not change its value group or uniformizer. Its discrepancy on this smooth generic quotient is at least one, hence at least \(\epsilon\). This proves the required gap for every actual quotient prime.

Put \(f_b=s_bt^b\) and take a general combination \[ h=a_0 f_b+a_1u_1^b+\cdots+a_{b-1}u_{b-1}^b. \tag{184}\] We check lc using the ideal \(I=(f_b,u_1^b,\ldots,u_{b-1}^b)\). For any divisorial valuation \(v\) with nontrivial restriction \(v_0\) to \(\mathbb C(S_0)\), that restriction is divisorial up to scaling and the product differential calculation gives \[A_S(v)\ge A_{S_0}(v_0)+\sum_jv(u_j) \ge \frac{v(f_b)}b+\sum_jv(u_j) \ge \min\{v(f_b),bv(u_1),\ldots,bv(u_{b-1})\}.\] If the restriction is trivial, \(v(f_b)=0\) and the same ideal inequality holds because \(S\) is klt. Therefore \((S,I)\) is lc. On a resolution of the ideal, a general residual member is smooth and transverse to the fixed log smooth boundary; with coefficient one this proves \((S,\mathop{\mathrm{div}}(h))\) lc. All summands of \(h\) have character \(b\).

Finally extend \(w_0\) by the joint Gauss valuation giving every \(u_j\) weight \(A_0\). It is centred at \((o,0)\) and quasi-monomial, and its product discrepancy is \(A_0+(b-1)A_0=bA_0\). Every summand of \(h\) has that value. Their initials do not cancel for a general choice of the \(a_j\) (the added coordinates are independent), giving (183). When \(b=1\) there are no added coordinates and \(h\) is simply a nonzero scalar multiple of \(f_b\). ◻

Proof. Let \(\chi\) be character \(b\) of the chosen cyclic group; it need not be faithful. Normalize \(T=T_0/(bA_0)\). The preceding lemma gives \[A_S(T)=T(h)=1, \qquad (S,\mathop{\mathrm{div}}(h))\text{ lc}, \qquad S/\mu_r\text{ is }\epsilon\text{-lc},\] and \(h\) and the divisor-free canonical generator both have character \(\chi\). It remains to check, for each fixed positive integer \(m\), that eventually \[ T(a)\ge m \quad\text{for every regular eigenfunction }a\text{ of character }\chi^m. \tag{185}\]

Expand a polynomial eigenfunction in monomials in the added coordinates. A monomial of total degree \(j\) has \(T_0\)-value \(jA_0\). If \(j\ge bm\), this already suffices, since its coefficient is regular and has nonnegative value. Otherwise its coefficient on \(S_0\) has character \[k=bm-j>0.\] In the canonical-cover case, that entire eigenspace is the degree-\(k\) section space described in Lemma 55. Since every fixed \(k\) divides \(M_i\) eventually, its coefficients have value at least \(kA_0\). The monomial therefore has value at least \((j+k)A_0=bmA_0\).

In the signed torsor, coefficients decompose into actual degrees \(k+\ell r\), \(\ell\in\mathbb Z\). For \(\ell=0\) use the same factorial saturation. For \(\ell\ne0\) use both strict slopes in (179). At example \(i\), choose \(r_i>i\) so large that \[ (r_i-i)\min\{\alpha,F_w-\alpha\}\ge iA_0. \tag{186}\] For \(1\le k\le i\), every nonzero residue shift has \(|k+\ell r_i|\ge r_i-i\), so its value is at least \(iA_0\ge kA_0\). This also gives the desired monomial bound. The choice is made after \(\alpha,F_w,A_0\) are known; no uniform lower bound on either slope is needed. For fixed \(m\), eventually \(bm\le i\), and every required unshifted degree divides \(M_i\).

A finite sum cannot have value smaller than the minimum of its terms. The argument therefore proves (185) in the affine coordinate ring. It also proves it in the local ring at the fixed point. Indeed any local function can be written with an invariant unit denominator: multiply an initial denominator by its other group translates and use its invariant norm. For an eigenfunction the resulting numerator has the same character. The invariant denominator has value zero, so the inequality is unchanged.

The dimensions satisfy \[\dim S= \begin{cases} \dim X+b-1,&\text{canonical-cover case},\\ \dim X+b,&\text{signed-torsor case}, \end{cases} \qquad\text{and hence}\qquad \dim S\le d+b.\] They are thus bounded independently of \(i\), and a subsequence has fixed dimension. The valuations are centred and quasi-monomial, the covers are finite cyclic and quasi-étale, and every remaining requirement of Theorem 48 was checked above. ◻

Proof. Shrink \(Z\) to an affine neighbourhood of \(z\), choose generators \(g_1,\ldots,g_N\) of \(\mathfrak m_z\), and resolve both \((X,B)\) and the pulled-back ideal. If \(p:Y\to X\) is the resulting log resolution, write \[K_Y+B_Y=p^*(K_X+B),\qquad \mathfrak m_z\mathcal O_Y=\mathcal O_Y(-F).\] The ideal threshold is the minimum of \(A_{X,B}(E)/\mathop{\mathrm{ord}}_E(F)\) over the finitely many components of \(F\) having positive order. Its interpretation is lc on the inverse image of a neighbourhood of \(z\). Indeed the ideal is the unit ideal away from \(f^{-1}(z)\), and the original pair is already lc there.

For a general linear combination \(g=\sum a_jg_j\), the pullback of \(\mathop{\mathrm{div}}(g)\) is \(F+H\), where \(H\) is a general member of a basepoint-free system. On a further restriction of the base, \(H\) is smooth and transverse to the fixed simple normal crossings support. If \(0<t<1\) is at most the ideal threshold, then \((Y,B_Y+tF+tH)\) is lc, hence so is \((X,B+t f^*\mathop{\mathrm{div}}(g))\). Choose \(g\ne0\). Since \(g\in\mathfrak m_z\), \(D=\mathop{\mathrm{div}}(g)\) is a nonzero effective Cartier divisor containing \(z\). Conversely, a Cartier divisor through \(z\) has a local equation \(g\in\mathfrak m_z\), so that \(v(\mathfrak m_z\mathcal O_X)\le v(f^*g)\) for every valuation. Thus an admissible coefficient for that divisor is also a lower bound for the ideal threshold.

All the varieties, morphisms, rational boundaries, a rational Fano type witness, generators, and the chosen resolutions descend to an algebraically closed subfield \(k_0\) of finite transcendence degree over \(\overline{\mathbb Q}\). Such a field embeds in \(\mathbb C\). Normality, geometric integrality and the identity \(f_*\mathcal O_X=\mathcal O_Z\) are unchanged by algebraically closed extension; the latter also follows from flat base change for the proper morphism. Projectivity and relative ampleness are geometric properties. Relative nefness is geometric as well: any curve on an extension can be spread over a finite-type parameter scheme and specialized to a curve cycle over the smaller algebraically closed field, with the same intersection number. Thus a negative intersection after extension would already yield a negative intersection before extension.

For discrepancies one can use the same log resolution. In the nonvacuous range \(0<\epsilon\le1\), a log smooth sub-boundary with all coefficients at most \(1-\epsilon\) is \(\epsilon\)-lc. At a stratum this follows by comparing with its reduced simple normal crossings support: a primitive monomial order has at least one positive integral weight, and each boundary weight contributes at least \(\epsilon\); a nonmonomial prime has a further nonnegative discrepancy, with an integral contribution at least one when needed away from the strata. This also shows preservation of the discrepancy bound after extension. The coefficients and orders in the displayed resolution formula are unchanged, so the ideal threshold is unchanged. The complex assertion therefore gives the same lower bound over \(k_0\) and over the original field, where the preceding general-member construction finishes the proof. ◻

Proof of Theorem 2. Over \(\mathbb C\), a failure in any fixed dimension and with any fixed \(0<\epsilon\le1\) would, by Proposition 58, give the sequence prohibited by Theorem 48. Therefore a uniform positive integral complement index exists. The range \(\epsilon>1\) in positive dimension is vacuous, since a prime divisor on the smooth locus has log discrepancy one.

The birational conclusion also descends to every algebraically closed field of characteristic zero. Increase the complex uniform index to a multiple greater than one, which preserves the complement assertion. Descend a given example, its Fano type witness, and a resolution to an algebraically closed field embeddable in \(\mathbb C\). For the complex uniform index \(n\), the complete relative system \(f_*\mathcal O_X(-nK_X)\) commutes with these field extensions. Resolve its finitely generated fractional base ideal as in Proposition 54. A klt member over \(\mathbb C\) implies that every fixed-order discrepancy relevant to the fibre is strictly positive. Those discrepancies are unchanged by extension. A general member over the original infinite field consequently gives a klt \(n\)-complement after shrinking, since the general residual coefficient \(1/n\) is less than one. Its section divisor \(\Delta_s\) defines \(\Delta=\Delta_s/n\ge0\), and \(n(K_X+\Delta)=\mathop{\mathrm{div}}(s)\) is Cartier and linearly trivial on the chosen neighbourhood, as required. ◻

Proof of Theorem 1. First work over \(\mathbb C\) and fix the total-space dimension \(d\) and gap \(\epsilon>0\). Theorem 2 in every dimension \(1,\ldots,d\) and for every positive discrepancy parameter, together with its trivial dimension-zero case and the rationalization above, verifies Chen’s assertion (2) for base dimension bound \(\ell=d\). His Theorem 1.6 therefore gives assertion (3), with a positive constant depending only on \(d\) and \(\epsilon\) (Chen 2025). Every base of a contraction from a \(d\)-dimensional variety has dimension at most \(d\), so this applies to the given \(f\). It supplies lc on the full inverse image of a neighbourhood of \(z\). Lemma 59, decreasing the coefficient below one if necessary, transports the uniform constant to an arbitrary algebraically closed characteristic-zero field and produces a nonzero effective Cartier divisor through \(z\) there. This proves the stated neighbourhood assertion. ◻

Proof of Corollary 3. The same application of Chen’s implication (2)\(\Rightarrow\)(3) over \(\mathbb C\) gives his full real-boundary assertion. Only the zero-boundary birational inputs to (2) were rationalized above. The resulting assertion (3) already allows arbitrary effective real \(B\) with \(K_X+B\) real Cartier, and supplies the same dependence on \(d\) and \(\epsilon\). No perturbation of the nef real boundary is involved. ◻