Proof. Choose a klt log Fano boundary \(\Delta\) over \(Z\) and a positive integer \(r\) with \(-rK_X\) Cartier. The divisor \(D=-rK_X\) is relatively nef, and for a positive integer \(a\), \[aD-(K_X+\Delta)=-arK_X-(K_X+\Delta)\] is relatively ample. The relative klt base-point-free theorem (Fujino 2009b, Theorem 1.3) makes \(-K_X\) semiample over \(Z\). Let \(g\colon X\to X^{\mathrm a}\) be its relative ample model. Since \(X\to Z\) is birational, so is \(g\). Choose canonical divisors from the same rational top form. For a Cartier divisor \(H\) on \(X^{\mathrm a}\) ample over \(Z\) and a sufficiently divisible \(m\), write \(-mK_X=g^*H+\mathop{\mathrm{Div}}_X(\phi)\). Birationality identifies the function fields, so pushing forward gives \(-mK_{X^{\mathrm a}}=H+\mathop{\mathrm{Div}}_{X^{\mathrm a}}(\phi)\). Pulling back this identity gives \[K_X=g^*K_{X^{\mathrm a}}.\] The morphism is crepant; the new variety is \(\epsilon\)-lc and has ample anti-canonical divisor over \(Z\). In particular it is of Fano type there. Every divisorial valuation has the same discrepancy on these two models, and its value on the ideal from \(Z\) is the same. The valuative formula for the threshold proves the assertion. ◻

Proof of Corollary 2. We check item (2) of (Chen 2025, Theorem 1.6) in every birational dimension. For its boundary-zero, relatively nef data, use the crepant ample model \(g\colon X\to Y\) of Lemma 6. Apply Theorem 1 to \(Y\to Z\), decreasing \(\epsilon\) to a fixed positive rational number when needed. After shrinking about \(z\), its complement boundary \(B_Y\) is \(\mathbb Q\)-Cartier, since both \(K_Y\) and \(K_Y+B_Y\) are \(\mathbb Q\)-Cartier there. On this neighbourhood, \(B_X=g^*B_Y\) is effective and \[K_X+B_X=g^*(K_Y+B_Y).\] The pair upstairs is klt, and the same complement index makes its adjoint divisor Cartier and trivial. Take the effective closure of \(B_X\) outside this neighbourhood if necessary. In particular the multiple used to construct the ample model does not change the complement index. This proves Chen’s item (2) for every base-dimension bound. The implication (2)\(\Rightarrow\)(1) of his theorem gives the assertion, with the stated dependence on \(I\). ◻

Proof. Choose functions with rationally independent values spanning the value group over \(\mathbb Q\), and functions of value zero whose residues form a transcendence basis of the residue field. Resolve the divisors of the first functions and the maps to \(\mathbb P^1\) defined by the second, together with the prescribed data. The latter functions are regular and invertible at the centre. Consequently the dimension of the centre is at least the residue transcendence degree. At least as many transverse boundary components as the rational rank pass through it: the independent values of the first functions are integral combinations of their orders. Abhyankar equality forces equality in both bounds. The centre is therefore the generic point of their transverse intersection, and the normal parameters have rationally independent values.

In the completed local ring at that generic point, each nonzero function has a unique least-weight normal monomial with nonzero residue coefficient. The monomial ideal of strictly higher weight is finitely generated. The congruence to that monomial times a unit contracts to the local ring by faithful flatness. This proves the monomial formula.

For a restriction, add the Abhyankar inequalities for the restricted valuation and the valued field extension. Equality for the full field forces equality for the restriction. A nonzero subgroup of the value group \(\mathbb Z\) of a prime divisor is \(j\mathbb Z\), \(j>0\). The restricted Abhyankar valuation of rational rank one is therefore \(j\) times a prime-divisor valuation. ◻

Proof. Relative ampleness gives finite generation of a sufficiently divisible Veronese and finite generation of each shifted-degree module over it. Thus it gives finite generation in all degrees. The ring is the intersection of \(k(X)[t]\) with the valuation rings imposing \(\mathop{\mathrm{ord}}_P(g)-mP(K_X)\geq0\) at strict prime divisors \(P\) of \(X\). Here and below, a strict prime of a specified model means a prime divisor on that model itself; a prime over the model may lie on any higher birational model. Thus a divisor extracted on a new model is strict on that model. The displayed intersection is therefore normal. Over the birational generic point of \(Z\) its field is \(k(X)(t)\).

Consider \[\widehat W=\mathop{\mathrm{Spec}}_X\bigoplus_{m\geq0}\mathcal O_X(-mK_X)t^m .\] For a sufficiently divisible \(r\), the analogous relative spectrum in degrees divisible by \(r\) is the total space of a line bundle. It is proper over the Veronese cone: evaluation by generating sections embeds it, relative to \(X\), into a product with affine space over \(Z\), and \(X\to Z\) is proper. The original relative spectrum is finite over this line-bundle model. Since \(W\) is separated over its Veronese cone, the resulting map \(\widehat W\to W\) is proper. It is birational by the fraction-field calculation. The same prime inequalities give normality of \(\widehat W\).

Over the smooth locus of \(X\), if \(s\) is a local rational function with \(\mathop{\mathrm{Div}}(s)=K_X\), the regular fibre coordinate is \(st\), and the displayed form is a nowhere-vanishing top form. The complement of this locus in \(\widehat W\) has codimension at least two, as is also seen by the finite map to the line-bundle model.

For completeness, the local singularities of \(\widehat W\) are klt. Trivialize \(rK_X\) locally and take the normalized cyclic cover adjoining \(s\) with \(s^r\) equal to its rational trivializing function. Because the coefficients of the Weil divisor \(K_X\) are integral, this cover is quasi-etale. Finite crepant pullback preserves klt singularities. On the index cover, form the line-bundle algebra (the normalized reflexive pullback) with coordinate \(y=st\). The deck group acts on \(y\) with the same character as on \(s\), keeping \(t\) invariant; the form \(\Omega=(\omega/s)\wedge dy\) is invariant. Its cyclic invariants recover the original algebra, by the codimension-one regularity inequalities. The cover of total spaces is again quasi-etale. The finite-cover discrepancy formula consequently proves that \(\widehat W\) is klt. We have \(\mathop{\mathrm{Div}}_{\widehat W}(\Omega)=0\), and pushing forward gives \(\mathop{\mathrm{Div}}_W(\Omega)=0\). Thus the proper birational map is crepant, and \(W\) is klt as well. ◻

Proof. The value formula follows from the assigned weight of \(t\). The divisor \(\mathop{\mathrm{Div}}_X(g)-mK_X\) is effective and rational Cartier, so the values are nonnegative. A Gauss extension is Abhyankar; wedging with \(d\log t\) preserves form discrepancy, while multiplication by \(t\) adds \(q-P(K_X)\). Since \(A_\omega(P)=A_X(P)+P(K_X)\), this proves the discrepancy formula. The rational-weight assertion follows from Lemma 7.

An invariant valuation evaluates a sum of different torus characters by their minimum: finite torus translates separate the character components. It is therefore Gaussian. Properness gives a centre on \(X\) for its restriction, and Lemma 7 applies. The same formulas hold for this restriction and for the trivial one. The corresponding \(q\) is nonnegative: a sufficiently divisible relatively free anticanonical section can be chosen with zero divisor order at the centre, and its value on \(W\) is its degree times \(q\). The lc assertion reduces to \[\frac1n P\bigl(|-nK_X|\bigr)\leq A_X(P),\] which is our choice of \(n\). Invariant divisors suffice: take an equivariant log resolution of the invariant ideal; the connected torus fixes its finitely many prime components.

We shall also use general members of these systems. On a log resolution, the fixed part is the ideal’s order and the residual system is free. A general member with coefficient \(1/n<1\) is transverse to the fixed support and preserves the lc inequalities. The same choice in a finite-dimensional generating subspace can therefore be made lc on both \(X\) and \(W\). ◻

Proof. Take a plt blowup extracting only \(E\) over \(x\), as in (Chen 2025, Lemma 2.27). For its strict boundary \(G_Y\), \(-E\) and \(-(K_Y+G_Y+E)\) are relatively ample rational Cartier divisors. Decreasing the coefficient of \(E\) slightly gives a klt relative log Fano pair, so \(Y\) is of Fano type over \(U\). The coefficients of \(G_Y\) belong to the finite \(1/N\)-grid below one. The relative monotone lc complement theorem (Birkar 2019, Theorem 1.8) gives a bounded-index lc complement of \(G_Y+E\) near \(x\). Its coefficient at \(E\) remains one.

Push the complemented boundary down. The bounded-index principal identity pushes down as well and makes the pullback crepant, so the result is an lc boundary \(G^+\geq G\) with \(E\) an lc place. For a bounded common multiple \(L\) of its complement index and \(N\), the divisor \(H_x=L(G^+-G)\) is effective Cartier. It passes through \(x\): its order at \(E\) is \(L A_{(U,G)}(E)>0\). Taking \(\tau=1/L\), and then a common smaller positive choice over the finitely many bounded indices, proves the claim. ◻

Proof. Choose \(N\) divisible by \(n\), tending to infinity slowly enough that the bad divisors in the contradiction sequence satisfy \[\frac{P(\mathfrak m_z)}{A_X(P)}>\frac{N}{\tau(p,N)}.\] This is a diagonal choice: each fixed \(N\) is allowed before passing sufficiently far along the bad sequence. For each individual algebra, choose \(M>N\) sufficiently divisible that the degree-\(M\) elements cut out the irrelevant locus set-theoretically. Set \[I=(\mathcal A_N,\mathfrak m_z,\mathcal A_M).\] Its radical is \(\mathfrak m_o=\mathfrak m_z\mathcal A+\mathcal A_{>0}\). Since \(I_n^{N/n}\subset I\), Lemma 9 implies \((W,I^{1/N})\) lc.

Suppose it were klt near \(o\). A general element \(g\) of a finite generating space of \(I\) would give \(G=N^{-1}\mathop{\mathrm{Div}}(g)\) klt there. Indeed the moving part has coefficient \(1/N<1\) on a log resolution. The canonical generator of \(W\) shows that \(N(K_W+G)\) is Cartier. Lemma 10 supplies an effective Cartier hypersurface \(H_o\) through \(o\) for which \((W,G+\tau(p,N)H_o)\) is lc.

Choose rational \(q\) with \[\frac{A_X(P)}{\tau(p,N)}<q\leq\frac{P(\mathfrak m_z)}N .\] The Gauss extension \(\widetilde P\) has positive value on every positive-degree element and on \(\mathfrak m_z\), hence is centred at \(o\). It has value at least \(Nq\) on \(I\), whereas its bare discrepancy is \(q+A_X(P)\). Consequently \[A_{(W,G)}(\widetilde P)\leq A_X(P).\] Moreover \(\widetilde P(\mathfrak m_o)\geq q\): each positive-degree homogeneous term has value at least its degree times \(q\), and each degree-zero term vanishing at \(z\) has value at least \(Nq\). A local equation of \(H_o\) therefore has value at least \(q\). This contradicts lc, since \[A_{(W,G)}(\widetilde P)\leq A_X(P) <\tau(p,N)q\leq\tau(p,N)\widetilde P(H_o).\]

Thus \(I^{1/N}\) is lc but not klt at \(o\). An lc divisor on a log resolution is centred there, because \(I\) is supported at \(o\) and \(W\) is klt elsewhere. For this divisor, \[A_W(u^0)=\frac{u^0(I)}N \leq\frac{u^0(I_n)}n\leq A_W(u^0).\] Both inequalities are equalities. Since \(\mathfrak m_z\subset I\), the second inequality in (5) follows too. ◻

Proof. The relative expected-vanishing criterion (Xu and Zhuang 2021, Lemma 3.3, Definition 3.4 and Theorem 3.10) gives \(A(u)\geq S(b;u)\). On \(Q_m=\mathcal O_{U,x}/\mathfrak a_m(b)\), let \(W_m(b;u)\) be the sum of the jumping weights of the induced \(u\)-filtration, with integer rounding if desired. Then \[\frac{S(b;u)}{A(b)} =\lim_{m\to\infty}\frac{W_m(b;u)}{W_m(b;b)}.\] The induced filtration is the image filtration: its dimension at level \(q\) is \[\dim_k\frac{\mathfrak a_q(u)} {\mathfrak a_q(u)\cap\mathfrak a_m(b)}.\] Filtering this image by the \(b\)-grading gives the same dimension, so no compatibility of bases is being assumed.

Suppose \(b(F)>0\), and put \(k_m=\lfloor m/(2b(F))\rfloor\). Multiplication by \(F^{k_m}\) embeds \(\mathcal O_{U,x}/\mathfrak a_{m-k_m b(F)}(b)\) into \(Q_m\). Every vector in this image has induced \(u\)-level at least \(k_m u(F)\). The image dimension is asymptotic to \(2^{-p}\dim_k Q_m\), whereas \(W_m(b;b)\leq m\dim_k Q_m\). Thus \[\frac{S(b;u)}{A(b)} \geq 2^{-p-1}\frac{u(F)}{b(F)}.\] The integer-rounding errors vanish in this limit. This proves (6), for example with \(C_p=2^{p+1}\). A unit has value zero at both valuations. ◻

Proof. From ideals to ordinary klt minimizers. Choose a common log resolution of \(U,I_0,J,\mathfrak m_x\). Replace the order of each ideal by the average orders of sufficiently many general elements in a prescribed finite-dimensional generating space. Write the resulting functions as \[C=A_U-B_0,\qquad a_D=b_0A_U-B_J,\] where \(B_0\) and \(B_J\) are rational nonnegative combinations of the divisors of the chosen regular equations, including the coefficients \(c\) and \(c'\). The resolving support and these general moving hypersurfaces form a simple normal crossing divisor \(\Phi\): choose successively transverse members on its finitely many strata. Dividing each coefficient among enough members makes every new moving coefficient sufficiently small.

The old component orders are unchanged. The simple normal crossing discrepancy tests therefore give \[C\geq0,\qquad a_D\geq A_U/2,\qquad a_D\leq b_0A_U.\] They also give \(C=0\) exactly on the original ideal-lc places and \(a_D=a\) there. Indeed the discrepancy expression for \(C\) has nonnegative residual term and component terms, strictly positive on every added moving component, including those used for \(J\). An ideal-lc place has \(C=0\) by comparison with the fixed ideal orders; conversely, \(C=0\) forces all these moving orders to vanish and recovers the original ideal equality. Every chosen equation then has exactly its ideal’s order.

For positive rational \(t'\), sufficiently small, the function \[\frac{C+t'a_D}{1+t'b_0}\] is the discrepancy of the effective rational klt pair with boundary \[\Delta_{t'}=\frac{B_0+t'B_J}{1+t'b_0}.\] Let \(u=u_{t'}\) minimize its normalized volume, with \(u(\mathfrak m_x)=1\). Comparison with one fixed centred lc place and \(a_D\geq A_U/2\) bounds \(A_U(u)^p\mathop{\mathrm{vol}}(u)\) independently of \(t'\). Properness bounds \(A_U(u)\); log Izumi consequently bounds \(\mathop{\mathrm{vol}}(u)\) away from zero. The same comparison now gives \(C(u)=O(t')\).

Retraction to the lc strata. Let \(\Sigma\) consist of the coefficient-one components for \(C\) on the fixed resolution. Use the monomial retraction of (Jonsson and Mustaţă 2012, Lemma 4.7 and Corollary 4.8): retract \(u\) to the generic stratum of \(\Sigma\) through its centre, retaining the same positive normal weights, and call the retraction \(w\). The component formulas give \[\begin{aligned} d_{\log}:=A_{(Y,\Sigma)}(u)&\leq C_1C(u),& C(w)&=0,\\ a_D(w)&\leq a_D(u),& w(\mathfrak m_x)&\geq1-C_1C(u). \end{aligned}\] To see the first bound, express \(C\) as \(A_{(Y,\Phi)}\) plus the component values multiplied by their discrepancies. These coefficients are zero exactly on \(\Sigma\), and positive on the omitted components. For \(d_{\log}\), the omitted coefficients are instead one. The formula for \(a_D\) has a nonnegative residual term and positive component coefficients. The maximal ideal is a fixed component sum. These observations give all four comparisons. In particular, for small \(t'\), \(w\) is nontrivial and centred at \(x\).

Retraction has removed the lc defect and has not increased \(a_D\). To compare the two objectives, we must also control its possible increase in volume. We prove the following quantitative claim for the minimizers \(u=u_{t'}\) and their retractions \(w\) constructed above: \[ u(F)\leq(1+C_2C(u))w(F) \qquad(0\ne F\in\mathcal O_{U,x}). \tag{7}\] The constant \(C_2\) is independent of \(F\) and of sufficiently small \(t'\) on this fixed resolved germ. We first isolate a polynomial’s least normal terms by a differential operator of controlled order, then bound the value lost under differentiation. A local truncation will give the claim for every regular \(F\).

Polynomial isolation. Cover the relevant strata by finitely many etale-coordinate charts \((x_1,\ldots,x_s,y_1,\ldots,y_{p-s})\), with \(x_i=0\) the components of \(\Sigma\) through the centre and \(w_i=u(x_i)>0\). For a polynomial \(G\) of degree at most \(M\) in fixed affine generators of \(k[U]\) vanishing at \(x\), we claim \[ u(G)-w(G)\leq C_3(1+M)d_{\log}. \tag{8}\] We will obtain a differential expression with \(u\)-value exactly \(w(G)\), using \(O(1+M)\) tangential and logarithmic normal derivatives. The order bound must remain independent of the smallest positive normal weight; otherwise it would not control the minimizers as their weights approach a face.

Expand \(G\) normally in the \(x_i\) at the generic stratum. Etaleness identifies the coefficient field as the finite separable residue-field lift over \(k(y)\). Its normal Taylor coefficients are restrictions of regular lifted \(x\)-derivatives divided by factorials; they are regular on the stratum. Lifted \(y\)-derivatives differentiate these coefficients without changing normal exponents. At a closed point this agrees with the expansion in \(x\) and translated tangential coordinates.

Let \(H\) be the sum of the least-weight normal terms of \(G\). We first bound every active exponent by \(O(1+M)\), independently of the smallest positive weight. The extension \(k(U)/k(x_1,\ldots,x_s,y_1,\ldots,y_{p-s})\) is finite. Fixed multiplication matrices for the affine generators give a relation \[\sum_j a_j(x,y)G^j=0\] of fixed degree in \(G\), with polynomial coefficient degrees \(O(1+M)\). Indeed entries in a degree-\(M\) polynomial of these matrices have denominators bounded by powers of fixed denominators; the characteristic polynomial and denominator clearing preserve a linear degree bound.

For a positive normal weight vector \(\lambda\), write \(f(\lambda)=\min_{\alpha\in\operatorname{Supp}_x(G)} \langle\alpha,\lambda\rangle\). At least two different powers \(j,j'\) tie at the least value in the relation. Choose normal exponents \(\beta,\beta'\) in the finite supports of \(a_j,a_{j'}\) that attain their respective minimum weights. Here the supports retain only coefficients nonzero in the stratum field. The tie gives \[f(\lambda)= \frac{\langle\beta-\beta',\lambda\rangle}{j'-j}.\] Thus \(f\) takes its values among finitely many linear forms whose coefficients are \(O(1+M)\): the coefficient degrees bound \(\beta,\beta'\), and \(|j'-j|\geq1\). The Taylor minimum is locally finite on the positive orthant, since the coordinates of \(\lambda\) stay bounded below in a small neighbourhood and all normal exponents are nonnegative. Let \(\mathcal A(\lambda)\) be the finite set of active Taylor exponents. For every coordinate \(i\) and all sufficiently small \(h>0\), local finiteness gives \[\frac{f(\lambda)-f(\lambda-he_i)}{h} =\max_{\alpha\in\mathcal A(\lambda)}\alpha_i.\] On this segment, \(f\) agrees with the restriction of one of the finitely many linear forms above, so the quotient is \(O(1+M)\). It bounds every active coordinate, including tied exponents. The permissible \(h\) may shrink as the weights approach a face, but the bound does not: no division by a normal weight has occurred.

Take the least parts of the relation at the given weights and divide out the lowest power of \(H\). The resulting polynomial relation in \(H\) has a nonzero constant term, an \(x\)-polynomial with coefficients polynomial in \(y\) of degree \(O(1+M)\). At a general closed point of the centre, not every coefficient of \(H\) can vanish to order exceeding that bound. Otherwise the relation would force every coefficient of its nonzero constant term to vanish to that order, impossible for a nonzero polynomial of the stated degree in etale tangential coordinates. All other coefficients in the relation are regular there, so their possible vanishing cannot invalidate this implication.

A tangential derivative of order \(O(1+M)\) therefore makes one active coefficient a unit. Let its normal exponent be \(\alpha\). For each normal coordinate apply the product of \(x_i\partial_{x_i}-j\) over the integers from zero to the active coordinate bound other than \(\alpha_i\). This isolates the chosen active monomial, with total differential order \(O(1+M)\). Both sorts of derivatives preserve normal exponents; lower-weight coefficients remain zero. To interpret the resulting congruence in the local ring, work at the chosen closed point of the centre and put \[I_{>}=\bigl(x^\beta:\textstyle\sum_i\beta_iw_i>w(G)\bigr).\] This monomial ideal is finitely generated for the individual positive weights. In the completion, the output is congruent to \(x^\alpha\) times a coefficient whose residue on the stratum is a unit. Lift that residue coefficient from \(\mathcal O_Y/(x_1,\ldots,x_s)\) to an ordinary local unit \(h\). Since \(x^\alpha(x_1,\ldots,x_s)\subset I_{>}\), the output minus \(x^\alpha h\) lies in the completed ideal \(I_{>}\), hence in \(I_{>}\) by faithful flatness. Localize this congruence at the generic centre of \(u\). Every ideal generator has \(u\)-value strictly larger than \(w(G)\). The isolated output consequently has exactly \(u\)-value \(w(G)\). The positive gap here may depend on the valuation; its size is never used in an estimate.

The differential loss. Let \(\lambda_0=\bigwedge_i d\log x_i\wedge\bigwedge_jdy_j\). For a prime divisor \(E\) centred in the chart, \(\mathop{\mathrm{ord}}_E(\lambda_0)=A_{(Y,\Sigma)}(E)-1\). Replacing a factor by \(dG_*/G_*\) multiplies the wedge, up to sign, by \(x_i\partial_{x_i}G_*/G_*\) or \(\partial_{y_j}G_*/G_*\). The new wedge has order at least \(-1\): at a uniformizer of \(E\), each logarithmic differential has only a possible simple logarithmic pole, and a nonzero wedge can contain that normal logarithmic factor at most once. Pullbacks of regular tangential differentials have no poles. Therefore each indicated derivation lowers value by at most \(A_{(Y,\Sigma)}(E)\), including over a proper subcentre of the stratum. Approximate real monomial weights on a higher adapted chart by rational divisorial weights. Continuity of the fixed function values and discrepancy gives the same inequality for \(u\). The operator \(x_i\partial_{x_i}-j\) obeys it too, because constants have value zero and \(d_{\log}\geq0\). Composing the bounded number of factors loses at most their total order times \(d_{\log}\). The exact value of the isolated output proves (8).

From polynomials to local functions. For a nonunit \(F\in\mathcal O_{U,x}\), take \(M=\lceil3w(F)\rceil+1\). Modulo \(\mathfrak m_x^M\), the local ring is spanned by polynomials of degree less than \(M\) in the fixed affine generators. In particular a unit denominator is inverted by its finite geometric series in this quotient. Choose such a polynomial \(G\) representing \(F\). For small \(t'\), \(w(\mathfrak m_x)\geq1/2\), so the remainder has \(u\)-value at least \(M\) and \(w\)-value at least \(M/2>w(F)\). Thus \(w(G)=w(F)\). Also \(w(F)\geq1/2\), so \(1+M\leq C'w(F)\). Equation (8) and \(d_{\log}\leq C_1C(u)\) now show that \[u(G)\leq(1+C_2C(u))w(F)<M\] for all sufficiently small \(t'\). Hence \(u(F)=u(G)\), proving (7). Units have value zero at both valuations.

Exactness of the minimum. We now compare the objectives at \(u\) and \(w\). Put \(K=1+C_2C(u)\). Equation (7) gives \(\mathfrak a_m(u)\subseteq\mathfrak a_{m/K}(w)\), and therefore \(\mathop{\mathrm{vol}}(u)\geq K^{-p}\mathop{\mathrm{vol}}(w)\). On the fixed germ, properness has bounded \(a_D(w)\leq b_0A_U(u)\leq M_0\). Choose \(t'\) so small that \(C_2t'M_0<1\). If \(C(u)>0\), then \[(C+t'a_D)(u)-K\,t'a_D(w) \geq C(u)\bigl(1-C_2t'a_D(w)\bigr)>0.\] Together with the volume inequality this contradicts minimality of \(u\), because \(C(w)=0\). Thus \(C(u)=0\) exactly.

On the constraint, \(a_D=a\), so every sufficiently small parameter gives a constrained minimum. Conversely any other constrained minimum would also minimize that ordinary klt objective, whose ray is unique. The ray is consequently independent of the small parameter. Stable degeneration gives finite generation. A torus preserving the ideals and point preserves the constrained objective and its unique ray; an invariant normalization, for instance by \(u(\mathfrak m_x)\), makes the valuation invariant. This proves the proposition. ◻

Proof. Use the rationally independent normal monomial chart, including the divisors of the finitely many probes in the monomialization. A basis of logarithmic normal differentials together with logarithmic differentials of a separating unit tangential basis has wedge of form discrepancy zero. Each probe is a unit times a Laurent normal monomial. In this basis its logarithmic differential has coefficients of nonnegative value. Their degree-zero parts are exactly the coefficients of its initial logarithmic differential in the initial variables: a logarithmic normal derivative of a unit reduces to zero, and a tangential derivative differentiates its restriction. Lifted etale coordinate derivations give these assertions in the local ring.

The determinant relating the probe wedge to this fixed wedge has nonzero reduction, since the initial probes are algebraically independent and the field has characteristic zero. Its value is therefore zero. This proves the discrepancy formula. Comparing two sets of probes by the same determinant computation proves the independence of the leading form. The construction commutes with field automorphisms preserving the valuation. Applying it to an integral tensor power and then dividing discrepancies by that power gives the fractional-form assertion. ◻

Proof. Add lifts of finitely many graded generators to any affine generating set, subtracting constants from the latter. All generator weights are positive. The minimum of a polynomial relation vanishes on their initials, giving one inclusion. Conversely, a homogeneous relation between these initials lifts to a function of strictly higher value. Subtract a polynomial matching that new initial, and continue with strictly higher-weight polynomials. Only finitely many monomial weights in these generators occur below any fixed bound, so the residual value can be made arbitrarily large. Log Izumi then puts the residual in an arbitrarily high power of the maximal ideal of the local ring. The same holds in the affine ring: the power of the closed-point maximal ideal is primary. That ideal is generated by the \(x_i\), so a polynomial expression in its sufficiently high power has all monomials of strictly higher weight than the starting relation. This ends subduction with an actual polynomial relation of the desired initial. The identical argument, starting from a function instead of a relation, gives adapted representatives. Local unit denominators have constant initial, so the affine and local graded rings agree; taking fractions gives the final assertion. ◻

Proof. Choose finite relations whose minima generate the initial ideal in Lemma 15. Add adapted representatives of the finite tests, including full probes and the coefficient \(g\) in (9). Their ties and strict weight comparisons form a finite system of rational linear equalities and strict inequalities. Approximate the generator weights, up to common scaling, by positive integers \(m_i\) preserving that system. Consider the domain \[\mathcal R=k[s,s^{-m_i}x_i] \subset k[U][s,s^{-1}].\] It is flat over \(k[s]\), since it is torsion-free. Its fibre at zero has the minimum-weight initial presentation for these integer weights. Indeed substitution and homogenization give one inclusion; conversely, separating the powers of \(s\) in a homogeneous expression lying in \((s)\) expresses its upstairs value by terms of strictly higher weight.

The chosen initial relations show that this fibre’s ideal contains the prime presenting \(R_b\). The zero-coordinate section makes the fibre nonempty. Every minimal prime of the principal ideal \((s)\) has height one in the finite-type domain \(\mathcal R\), so the fibre has dimension \(\dim U\). The old initial prime has that same dimension, since its fraction field is \(K_b\). A strict enlargement would have smaller dimension. Thus the initial ideals coincide and the special fibre is exactly \(\mathop{\mathrm{Spec}}R_b\), with the required test initials.

All fibres are normal: the nonzero ones are copies of \(U\), and the zero fibre is normal by hypothesis. Flatness over the smooth curve therefore gives normality of the total space. The form \(\Omega_U\wedge ds\) has zero divisor away from the special fibre. That fibre is irreducible, reduced and Cartier, so multiplication by a power of \(s\) gives a rational form with zero divisor on the total space, hence a canonical generator on its smooth locus.

We identify its relative restriction explicitly. At the generic point of the special fibre, write the tested probes as \(s_i=s^{e_i}S_i\) and \(g=s^{e_g}G\), with residues \(\bar s_i\) and \(\bar g\). Because a factor involving \(ds\) disappears after wedging with \(ds\), \[\Omega_U\wedge ds =s^{e_g}G\bigwedge_i d\log S_i\wedge ds.\] The initial logarithmic probe wedge is nonzero. The normalizing power is consequently exactly \(-e_g\), and the relative restricted form is \(\bar g\bigwedge_i d\log\bar s_i=(\Omega_U)_b\), up to sign. At every codimension-one point of the normal special fibre that fibre is smooth; flatness and geometric smoothness there make the family smooth over the base. Its relative generator restricts to a generator there. Thus \((\Omega_U)_b\) has zero divisor on \(\mathop{\mathrm{Spec}}R_b\), as claimed. This argument uses only codimension-one restriction, and makes no canonical-sheaf base-change assumption at singular points. ◻

Proof. For a principal ideal, valuation multiplicativity makes its associated graded ideal principal, generated by the equation’s initial. We verify that component ideals can be preserved at the same time. For each of the finitely many component ideals, use its polynomial preimage in the adapted presentation. Its minimum initial ideal maps onto the associated graded ideal: the adapted representatives in Lemma 15 prove this also for each element of the ideal.

Homogenize all these polynomial ideals with one extra variable of weight zero. Each homogeneous minimum ideal has finitely many generators that are minima of homogeneous elements of the original homogenized ideal. Preserve these generators, and the ring’s initial ideal and the equation tests, in one positive integral weight approximation. Inclusion of the old minima in the new minima is then automatic. There can be no extra elements, since both homogeneous minimum ideals have the original Hilbert function, obtained by filtering each finite-dimensional total-degree piece. Dehomogenization preserves the equality: distinct terms of one homogeneous polynomial cannot coalesce after setting the extra variable to one.

The component closures in the resulting weight family have these specialized ideals, as follows by using the analogous family algebras of the quotient rings. Rescale each boundary equation by its parameter power. Its divisor has no vertical component and restricts to the divisor of the nonzero initial equation. Intersection of codimension-one cycles with the special-fibre parameter is additive. This can also be checked at the regular codimension-one points of the normal special fibre, where the total family is smooth over the base. Decomposing the horizontal equation divisors into their prime components therefore gives precisely the same restricted boundary cycles and multiplicities. ◻

Proof. First use the lexicographically ordered valuation \[V(F)=(b(F),u(\operatorname{in}_b F)).\] On each first-weight slice its second graded calculation is exactly the one for \(u\). Invariance makes the \(u\)-filtration split over first degrees. Hence its associated graded on \(k[U]\) is precisely \(R'\), with the joint grading. Choose lifted joint homogeneous generators, including the additional generators needed for \(R\) and \(k[U]\). Take them homogeneous for the stipulated old grading and vanishing at the point. In the lexicographic subduction procedure there are only finitely many possible values with bounded first coordinate: the first graded pieces are finite-dimensional and only finitely many occur up to that bound. Subduction therefore works as in Lemma 15, using log Izumi for the first coordinate. In particular the lexicographic minimum ideal is the prime presenting \(R'\), and adapted polynomial and fractional representatives exist.

Choose \(\xi>0\) small enough to preserve the minima of finitely many relations generating that initial prime, and avoid all rational ties on the joint character lattice. The flattened initial ideal contains this prime. Suppose it contained an extra initial relation. Approximate the positive flattened generator weights by positive integral weights, preserving that extra minimum together with the previous finite minima. The weight-family argument of Lemma 16 then gives a nonempty special fibre of dimension \(\dim U\), whose ideal strictly contains the prime of \(R'\), also of dimension \(\dim U\). This is impossible. Thus the flattened minimum ideal is exactly the desired prime.

For completeness, the resulting real quotient filtration does define a valuation on the original ring. Its weight on a function is the maximum of the minimum weights of all polynomial representatives. Positivity of the finitely many generator weights makes the weights locally finite. If the minima of representatives could be arbitrarily high, every monomial in those representatives would have arbitrarily large ordinary total degree, putting the function in every maximal-ideal power, hence making it zero. The maximum is thus attained for every nonzero function. At an attained maximum, polynomial minima give the graded quotient by the minimum relation ideal: an improvement of a representative differs from it by a relation with that minimum. The quotient being a domain proves multiplicativity, and the triangle inequality is immediate. Extending by fractions yields \(b_\xi\). Its full graded field has transcendence degree \(\dim U\), so it is Abhyankar. Avoidance of rational ties identifies its value torus with the full joint torus. The homogeneous presentation preserves the old grading.

To preserve any finite test list, choose its lex-adapted polynomial representatives and denominators and retain their leading comparisons. For uniform comparison on all regular functions, take \(\xi\) smaller until every generator’s flattened weight lies between half and twice its first weight. A \(b\)-adapted representative gives \(b_\xi(F)\geq b(F)/2\). Conversely, a \(b_\xi\)-adapted representative gives \(b(F)\geq b_\xi(F)/2\). This uses the attained maximum description separately for the two valuations, and so includes functions whose initially chosen representatives have cancellation.

Finally choose first-stage probes as follows. Use rational torus variables and functions of the invariant field whose restricted \(u\)-initials form a full transcendence basis there. Gaussianity and Abhyankar equality show that these have full independent first and double initials: the second-stage torus initials adjoin independent variables to the full restricted initial field. The invariant probes are ratios of equal-first-weight homogeneous functions, so they lift to \(k(U)\), as do the torus variables. Include these probes and the coefficient of \(\Omega_U\) in their logarithmic basis among the finite tests. Applying (9) first to \(b\), then to \(u\), and then to the flattening gives coefficient values \(1\), \(1\), and \(1+\xi\), respectively. The same formula identifies the flattened leading form with the iterated leading form. This proves the last two assertions. ◻

Proof. Impose nonnegativity on finitely many eigenfunction generators of \(R\). For an extension of \(P\) these are linear inequalities in the rational torus-variable weights, with the one weight on \(z^\chi\) fixed to \(P(L)\). The criterion for a solution is that every nonnegative combination eliminating the other weights has nonnegative right-hand side. By rational polyhedrality it suffices to test rational combinations. Clear denominators and multiply the corresponding eigenfunctions. Their total weight is a multiple of \(\chi\); it is a nonnegative multiple because the occurring cone is pointed. The required inequality is exactly (12) pulled back to \(P\). The rational linear system therefore has a rational solution. Its Gaussian valuation has rational rank one and satisfies Abhyankar equality, so it is divisorial up to scaling. Formula (9) gives its discrepancy \[A_B(P)=a_{\omega_S}(P)+P(L)>0,\] where positivity follows from klt of \(R\). Thus \((S,B)\) is sub-klt; we next verify effectivity and the coefficients.

For a strict prime \(P\) on \(S\), first lift \(P\) to a homogeneous prime divisor \(P'\) of the affine ray cone. Use the punctured cone for a degree-one generated Veronese, and then the finite map from the full ray cone. The affine subtorus quotient \(\mathop{\mathrm{Spec}}R\to\mathop{\mathrm{Spec}}R_\chi\) is surjective: projection to invariant summands splits the inclusion as an \(R_\chi\)-module. Localize at the generic point of \(P'\) and pull back its uniformizer. A height-one minimal prime of this nonzero parameter ideal gives a prime divisor \(E\) on \(\mathop{\mathrm{Spec}}R\) dominating \(P'\). Its contraction is the maximal ideal of that discrete valuation ring. A connected torus preserves each of these finitely many components, so \(E\) is invariant. Its restriction is \(jP\) for an integer \(j>0\), by Lemma 7.

Choose a section in a sufficiently divisible degree whose corresponding section of \(L\) is nonvanishing at the generic point of \(P\). It has order zero at the ray-cone divisor and at its dominating lift, which gives \(E(z^\chi)=jP(L)\). As \(E\) is a strict prime of the variety \(\mathop{\mathrm{Spec}}R\) with zero canonical divisor, \(A_R(E)=1\). The leading-form formula now yields \[1=jA_B(P).\] The coefficient of \(B\) at \(P\) is therefore \(1-1/j\), proving effectivity and the asserted standard coefficients. Together with all prime tests already proved, this makes \((S,B)\) klt. Finally \(-K_S-B=L\) is ample, so it is log Fano. ◻

Proof. Otherwise pass to a subsequence carrying lc primes \(P\) and effective divisors \(H'\sim_{\mathbb Q}H\) with \(P(H')>(1+c_*)P(H)\), for a fixed \(c_*>0\). Here \(P(H)=A_B(P)>0\). The lift from Lemma 19 has bare discrepancy \(P(H)\) and discrepancy zero for \(\mathop{\mathrm{Div}}(\bar h)/n\). Normalize it to bare discrepancy one. By our general choice of \(h\), it is also an lc place of \(I_R^{1/n}\). Average this normalized lift and the grading valuation \(\tilde v\) on eigenfunctions, extending by the Gaussian minimum rule. The resulting valuation \(u_*\) restricts to \(P/(2P(H))\) on the invariant field. It is quasi-monomial, has bare discrepancy one by (9), and is an lc place of \(I_R^{1/n}\). For the last assertion, the core weight is pure: every core function has value at least \(n\) at the lift and exactly \(n\) at the grading, with equality for \(\bar h\). Also \(u_*\geq\tilde v/2\) on regular functions, so it is centred at the vertex and \[\mathop{\mathrm{vol}}(u_*)\leq2^p\mathop{\mathrm{vol}}(\tilde v).\]

Write \(H'=L+\frac1m\mathop{\mathrm{Div}}_S f'\) with \(m>0\) integral, clearing rational denominators. By (12), \(g=f'z^{m\chi}\in R_{m\chi}\). Its value divided by \(m\) is \(P(H')/P(H)>1+c_*\) at the normalized lift, and one at the grading. Hence \[u_*(g)/m>1+c_*/2.\]

Choose rational \(1<c_0<1+c_*/2\) and a fixed rational \(\lambda>0\) large enough that \[1+\lambda(c_0-1)>2C,\] where \(C\) is the old uniform constant in (11). On \(\mathop{\mathrm{Spec}}R\), minimize subject to the same lc constraint, now with residual discrepancy \[a(u)=(1+\lambda c_0)A_R(u) -\lambda\min\{u(g)/m,c_0u(I_R)/n\}.\] This has the ideal form of Proposition 13. Explicitly take an integer \(q_0\) for which \(q_0/m\) and \(q_0c_0/n\) are integral and put \(J=(g^{q_0/m})+I_R^{q_0c_0/n}\); the subtracted term is \((\lambda/q_0)u(J)\). Since the constraint ideal is lc, \(u(I_R)/n\leq A_R(u)\), and consequently \[A_R(u)\leq a(u)\leq(1+\lambda c_0)A_R(u).\] The ideals are \(T\)-invariant, so the exact minimizing valuation is \(T\)-invariant. Normalize it by \(A_R(u)=1\).

This \(u\) cannot be a cocharacter valuation in \(T\). For any centred cocharacter competitor with bare discrepancy one, pure weights give \(u(g)=m\) and \(u(I_R)=n\), hence \(a=1+\lambda(c_0-1)\). Such competitors are lc places of the system; (11) gives their volume at least \(C^{-p}\mathop{\mathrm{vol}}(\tilde v)\). Their objective thus strictly exceeds \(2^p\mathop{\mathrm{vol}}(\tilde v)\), while \(a(u_*)=1\) and the displayed volume bound makes \(u_*\)’s objective at most that number. This contradicts minimality if \(u\) were a cocharacter.

We have obtained a minimizer that is not a cocharacter of the old torus. To contradict maximality, we must realize its joint graded ring on the original \(W\) and preserve the system value \(n\), concentration (10), and the uniform comparison (11).

Let \(R'=\mathop{\mathrm{gr}}_uR\). Stable degeneration for each sufficiently small auxiliary klt parameter gives finite generation, normality and, after removing boundaries, klt of \(R'\). Use the joint effective torus of \(T\) and the new value torus. By Lemma 16, its canonical generator is \((\Omega_{\tilde v})_u\); call its character \(\chi'\). It restricts to \(\chi\) on \(T\), and \(u(\chi')=1\). Choose the old core equations in \(R_{n\chi}\) for the exact-minimum construction. Their next initials have old weight \(n\chi\) and \(u\)-value \(n\), hence joint weight \(n\chi'\). The old and new weights determine characters of the joint torus, so this is an equality of characters, not only of their evaluations.

Let \(I_{R'}=(R'_{n\chi'})\). The core and normalized minimum remain fixed as the auxiliary parameter tends to zero. Boundary specialization and this pure weight therefore imply that \(I_{R'}^{1/n}\) is lc, exactly as in the initial construction. The same auxiliary klt pairs give a new uniform version of (11), initially relative to grading \(u\). Indeed at a centred lc place \(q\) of the new system, the core term is at most zero; all perturbing boundary subtractions are nonnegative, so the auxiliary discrepancy before common scaling is at most \[t'(1+\lambda c_0)A_{R'}(q).\] At the minimizing grading \(u\) it equals \(t'a(u)\), which is at least \(t'\). Formula (6) cancels the parameter and yields a sequence-uniform comparison with \(u\), since \(\lambda,c_0,p\) are fixed.

On the old ring, (11) applies to the normalized lc place \(u\) and gives \(u(f)\leq C\tilde v(f)\). On each new joint piece this says that its new \(u\)-weight is at most \(C\) times its old \(\tilde v\)-weight: use a homogeneous lift of that piece to \(R\). We now flatten the two successive valuations on the actual original \(W\), by Lemma 18, and divide by \(1+\xi\). This preserves \(D\), the iterated canonical generator, and discrepancy one. The parameter \(\xi\) may be chosen individually for each member; all bounds below use fixed factors.

Include a finite degree-\(n\) generating list for the actual ideal \(I_n\) in the preserved tests. If an old value is greater than \(n\), its normalized flattened value is still greater for small enough \(\xi\). If the old value is \(n\), its first initial lies in \(R_{n\chi}\), so the next value is at least \(n\). Include also a degree-\(n\) lift attaining equality for this system minimum; such a lift exists because \(R_{n\chi}\) is precisely that leading space. The normalized flattened valuation therefore still has \(\tilde v_{\mathrm{new}}(I_n)=n\).

The all-functions comparison in Lemma 18, with \(0<\xi<1\), bounds the new actual valuation below by \(\tilde v/4\). Thus (10) persists. On each new joint weight, its normalized value is \[\frac{\text{old weight}+\xi\,\text{new }u\text{-weight}}{1+\xi} \geq\tfrac12\,\text{old weight} \geq\tfrac1{2C}\,\text{new }u\text{-weight}.\] Both grading valuations use the minimum rule, so the same comparison holds on all elements of \(R'\). The new comparison relative to \(u\) therefore gives (11) relative to the normalized flattened grading, with a uniform enlarged constant. All required sequence properties survive.

The joint torus has strictly larger rank. Every old occurring character still occurs after taking the invariant associated graded, so \(T\) continues to act faithfully. If the \(u\)-grading belonged to this old torus, each nonzero element in an old weight space would have its \(u\)-value determined by that character and one fixed linear functional. Invariance and the minimum rule would then make \(u\) itself a cocharacter valuation on \(R\), which was excluded above. Thus the rank increases. Passing to a constant-rank subsequence gives an admissible sequence of larger rank, contrary to its maximal choice. This proves (15). ◻

Proof. On a projective descent write \(M\sim_\mathbb QD_0+E_0\), with \(D_0\) ample and \(E_0\ge0\). For small rational \(s>0\), use \(sE_0\) as fixed part and the ample class \((1-s)M+sD_0\) as moving part. A still smaller pullback ample reserve can be subtracted from this moving part. Resolve fixed supports and take a sufficiently divisible general free representative of the moving part, divided by its multiplicity. Fixed coefficients change arbitrarily little and new coefficients are arbitrarily small. The snc discrepancy formula therefore retains klt, or \(\delta/2\)-lc, on every higher valuation. Subtracting the effective Cartier b-divisor representing the nef part only lowers the generalized discrepancies. For open bases use a projective compactification and common higher projective descents, testing these inequalities over the required open. The ample reserve proves the Fano-type assertion. ◻

Proof. On a log resolution displaying the collection, retain their crepant coefficients, which lie in \([0,1)\). Raise every other exceptional coefficient to a strictly larger nonnegative rational number below one. The resulting effective pair is klt, its boundary is big over the birational base, and its adjoint is relatively the effective exceptional divisor \(R\) supported exactly on the raised components. Normality gives \(f_*\mathcal O(mR)=\mathcal O_Y\) for divisible \(m\). The good log terminal model preserves these sections by negativity and makes the transform of \(R\) relatively semiample. A free multiple has only the relative generator \(1\), so its effective transform vanishes. The pullback difference is zero at every requested divisor; strict adjoint negativity for contracted divisors therefore prevents their contraction. The MMP extracts no divisor. The original pair’s crepant trace on this extraction is effective and has nef big anti-adjoint, so a small perturbation gives Fano type. ◻

Proof. Fix a prime \(P\) and a lift \(T\) computing the mixture. We will construct a model on which \(T\) is the only strict prime above \(\eta_P\), and on which both individually shifted boundaries are effective. On that model, generalized threshold ACC will force the shifted \(A\)-datum to be lc.

Shifting the two data at the computing prime. Work over an open neighborhood \(U\) of \(\eta_P\) on a smooth base model and put \[T|_{\mathbb C(V)}=m_TP,\qquad \alpha_A=A(T)/m_T,\qquad \alpha_C=C(T)/m_T.\] Subtract \(\alpha_A\pi^*P,\alpha_C\pi^*P\) from the respective discrepancy functions. Their mixture is sub-generalized lc over \(U\) after shrinking: its threshold at \(P\) is zero, and horizontal sub-lc is already known. Both shifted totals are pulled back from \(U\).

Add \(\kappa\pi^*P\) to the shifted mixture discrepancy for a small positive rational \(\kappa\). It becomes sub-generalized klt there, with coefficient \(1-\kappa m_T\ge0\) at \(T\). On a high smooth model displaying \(T\), resolve the boundary, exceptional, fiber and nef data. Raise the coefficients of every other component over \(\eta_P\), and every horizontally exceptional divisor over the original generic fiber, to strictly larger nonnegative rationals below one. Remove unwanted vertical supports away from \(\eta_P\) by shrinking. Horizontal strict primes over the original generic fiber already have effective trace. Thus we obtain effective generalized klt data whose adjoint is relatively an effective divisor \(R_{\mathrm{ex}}\), supported exactly on these raises and omitting \(T\).

A model whose only strict prime above \(\eta_P\) is \(T\). By Lemma 24, replace the big nef part by ordinary effective data while keeping klt and big boundary. Take its good \(\mathbb Q\)-factorial log terminal model over \(U\). The reason every raised divisor disappears is that \(R_{\mathrm{ex}}\) has no relative sections except those from \(\mathcal O_U\). We verify this before using semiampleness on the good model.

For every divisible \(m\), a relative section of \(\mathcal O(mR_{\mathrm{ex}})\) has no nonexceptional horizontal poles on the original normal proper generic fiber. It is therefore a constant on that fiber, hence an element of \(\mathbb C(V)\). A base pole at \(P\) is impossible because its pullback has a pole along \(T\), absent from \(R_{\mathrm{ex}}\). At every other retained base prime, a component missing from \(R_{\mathrm{ex}}\) likewise prevents a pole. Such a component exists above each strict prime of a normal contraction base, by pulling back a local equation near its generic point; other vertical supports were removed by the shrink. Consequently these relative section sheaves are precisely \(\mathcal O_U\).

The ordinary approximation may change the adjoint by a rational principal term and a base pullback. Clear denominators and identify the corresponding section spaces by multiplication by that principal function and local trivialization of the base term. Under this identification the algebra just computed is unchanged. Negativity preserves it on the good model. The effective transform of \(R_{\mathrm{ex}}\) is relatively semiample; a divisible free multiple is generated by the section \(1\), whose zero divisor is that multiple of \(R_{\mathrm{ex}}\). Hence its support is empty. The output must still have a divisor over \(\eta_P\), and the MMP extracts none. Thus \(T\) survives and is the unique such divisor.

The pre-raise, \(\kappa\)-adjusted mixture is crepant on this output, since its adjoint is pulled back from \(U\), and has effective trace. Its big b-nef part gives relative Fano type by Lemma 24. In particular the bare output is klt. The surviving horizontal primes come from the original generic fiber; both individual shifted boundaries are therefore effective after shrinking. Their coefficient at \(T\) is one.

The bounded-coefficient threshold test. We now return to the shifted mixture before the \(\kappa\)-adjustment; it is lc on this same model. Remove \(t\) times the shifted \(C\)-boundary and its corresponding nef datum. Effectivity and negativity show that discrepancies only increase. Thus the generalized threshold on the bare variety of the shifted \(A\)-boundary \(B'_A\) together with \(\mathbf M_A\) lies in \([1-t,1]\); the upper bound comes from \(T\). The horizontal coefficients of \(B'_A\) lie in the finite \(1/r_{\mathrm{ind}}\)-grid in \([0,1]\), because \(r_{\mathrm{ind}}A(F)\) is integral for Calabi–Yau data with \(r_{\mathrm{ind}}\mathbf M_A\) Cartier. The only remaining nonzero vertical coefficient is one. Use \((1/r_{\mathrm{ind}})(r_{\mathrm{ind}}\mathbf M_A)\) as a single nef Cartier summand. These are exactly the effective test and nef Cartier coefficient hypotheses of generalized threshold ACC (Birkar and Zhang 2016, Theorem 1.5). The test scales the boundary and the nef datum together; at parameter one it is precisely the shifted \(A\)-presentation. More explicitly, the initial boundary and nef datum in the threshold theorem are both zero; the effective test boundary is \(B'_A\), and the nef test datum is \((1/r_{\mathrm{ind}})(r_{\mathrm{ind}}\mathbf M_A)\). The latter is required to be nef, with the stated Cartier coefficient; its trace need not be effective. A Cartier divisor descending on the low model does not affect the nef-data discrepancy \(\mathbf M_A-\overline{M_{A,Y}}\), so any auxiliary effective representative obtained by twisting on the affine base does not alter the test.

If thresholds below one occurred for arbitrarily small \(t\), a subsequence would increase strictly to one, contradicting ACC. Thus the threshold is one for all sufficiently small \(t\), with a bound depending only on dimension and \(r_{\mathrm{ind}}\). Closedness supplies the inequalities at the endpoint. The shifted \(A\)-data are therefore lc over \(\eta_P\), proving that \(T\) computes \(\pi_\#A(P)\). ◻

Proof. Use the model constructed in the proof of Lemma 26: its unique strict prime above \(\eta_P\) is \(T\), and the shifted \(A\)-boundary is effective and lc. The shifted \(A\)-data now have ordinary discrepancies, and \[K+B'_A=\alpha_A\pi^*P-M_{A,Y}.\] The pair is lc, its anti-adjoint is nef over \(U\), its coefficient set is finite rational (hence contained in a fixed hyperstandard set), and its underlying variety is of Fano type over \(U\). Apply the bounded relative monotone lc complement theorem near a closed point of \(P\cap U\) (Birkar 2019, Theorem 1.8). The effective complement addition restricts to a rationally trivial effective divisor on the proper generic fiber, so has no horizontal component. It cannot add to \(T\), whose coefficient is already one. Thus the boundary is unchanged near \(\eta_P\). Higher crepant traces are determined by the same adjoint and introduce no new independent addition.

For clarity, if the local complement identity is \(n'(K+B'_A)=\mathop{\mathrm{Div}}(h)\) and \(r_{\mathrm{ind}}M_{A,Y}=\mathop{\mathrm{Div}}(g)\), take \(N=n'r_{\mathrm{ind}}\) and \(\phi=h^{r_{\mathrm{ind}}}g^{n'}\). Thus a bounded integer \(N\) gives \[N\alpha_A\pi^*P=\mathop{\mathrm{Div}}(\phi)\] over a neighborhood still containing \(\eta_P\). The principal adjustment from \(M_A\) is absorbed in \(\phi\). This function has no horizontal poles. Generic-fiber properness and the contraction property imply \(\phi\in\mathbb C(V)\), even though its initial expression was obtained on an open neighborhood. Evaluating at \(T\) gives \[N\alpha_A m_T=m_T\mathop{\mathrm{ord}}_P(\phi), \qquad N\alpha_A=\mathop{\mathrm{ord}}_P(\phi)\in\mathbb Z.\] The fiber multiplicity cancels; neither a norm nor an extension-degree factor is introduced. ◻

Proof. We first assume b-semiampleness. Let \(k=\mathbb C\), and enlarge constants to the algebraic closure \(k'\) of a purely transcendental extension of finite transcendence degree. Use enough independent coefficients to choose a member of a free Cartier system \(|m\mathbf M|\), \(m>1\), generic over \(k\), on a high descent. Divide it by \(m\) and denote its Cartier closure by \(\Delta\). The data \[a^{k'}-\Delta =a_{K,Y}^{k'}+\mathbf M^{k'}-\Delta\] are ordinary pair discrepancies, because the remaining moduli term is rationally principal. Their ordinary boundary adds the trace of \(\Delta\) to the generalized boundary. Bertini on a log resolution, with moving coefficient \(1/m<1\), makes them sub-klt generically. Their horizontal trace on the original low model remains effective.

The rank-one condition in the ordinary canonical bundle formula holds as follows. On a resolution over the generic point, the rounded discrepancy sheaf contains constants by sub-klt singularities. Any additional poles allowed by it are exceptional over the original normal generic fiber, because the low horizontal boundary is effective. Such functions are regular on that proper low fiber and therefore constant. The contraction condition gives rank one. The rational principal adjustment makes the total adjoint pulled back from zero. Ambro’s ordinary klt-trivial-fibration theorem therefore gives a b-nef b-\(\mathbb Q\)-Cartier divisor \[\mathbf N^\Delta =(\pi^{k'})_\#(a^{k'}-\Delta)-a_{K,V}^{k'}.\] Here we use (Ambro 2004, Theorem 0.2); see also (Ambro 2005). Vertical negative boundary components are allowed in this theorem. Alternatively, compensate them with a pulled-back base divisor, which translates thresholds and total divisor compatibly.

For each prime \(P\) over \(k\), a resolution computing its threshold is defined over \(k\). Extension does not change its finite snc calculation, and the generic free divisor meets that calculation transversely, without changing the threshold. Thus, writing \(\mathbf N=\pi_\#a-a_{K,V}\) initially as a coefficient function, \[ \mathbf N^\Delta(P_{k'})=\mathbf N(P). \tag{18}\] This holds for every old prime simultaneously in the requisite sense: each transversality condition is a nonempty open condition defined over \(k\), and the chosen coefficient tuple lies over the generic point of its parameter space. No countable intersection of open subsets of \(k\)-points is being asserted.

Descent of the Cartier presentation. The restriction to old prime extensions of any rational b-Cartier divisor over \(k'\) is rational b-Cartier over \(k\). On a projective descent, taking differences and multiples reduces this to a globally generated Cartier divisor \(J\). For finitely many rational sections \(f_j\), \[Q(J)=-\min_j Q(f_j).\] Put these sections over a common denominator in \(k(V)\otimes_k k'\), and express every numerator and the denominator as a finite sum of coefficients in \(k(V)\) times constants linearly independent over \(k\). At \(P_{k'}\), the value of such a sum is exactly the least coefficient value. Indeed those constants remain independent over \(k(P)\) in \(\mathop{\mathrm{Frac}}(k(P)\otimes_k k')\); geometric integrality over algebraically closed \(k\) makes this tensor product a domain. Hence the restricted function is a difference of minima over finite lists in \(k(V)\). Resolving the corresponding fractional ideals makes it Cartier. Apply this to \(\mathbf N^\Delta\) and (18) to obtain a b-\(\mathbb Q\)-Cartier presentation of \(\mathbf N\) over \(k\).

Testing new primes and nefness. The identity \(\pi_\#a=a_{K,V}+\mathbf N\) remains valid after extension at every prime over \(k'\). To prove this, fix one such prime on a smooth model dominating a descent of \(\mathbf N\). Choose a smooth model upstairs, with descended \(\mathbf M\), on which its threshold is computed by snc data including the pullback of that prime. All models, morphisms, prime components, rational forms and Cartier data in this single test are defined over a finitely generated extension of \(k\). Spread them over an integral parameter variety over \(k\). After shrinking, the models remain projective birational resolutions fiberwise; the required components remain geometrically integral; smoothness and relative snc hold; and the same components dominate the tested prime. Properness and constructibility permit these requirements. The finitely many divisor identities for forms, functions and pullbacks also persist with unchanged multiplicities: keep their supports on the smooth models and remove the parameter locus where a stated divisor identity or dominance fails.

Specialize to a sufficiently general closed \(k\)-point. Both the proposed b-Cartier coefficient and the threshold are computed by the same respective finite numerical lists as before specialization. They agree there by the already proved identity over \(k\). They therefore agree for the chosen prime over \(k'\). This argument uses one finite spread per test prime, not a common resolution of all primes.

Since \(\Delta\ge0\), threshold monotonicity gives the b-effective difference \[\mathbf N^{k'}-\mathbf N^\Delta\ge0.\] Its value at every \(P_{k'}\) is zero by (18). Let \(C\) be a curve on a smooth projective descent of \(\mathbf N\) over \(k\), and take a common higher descent over \(k'\) where \(\mathbf N^\Delta\) is nef. Choose an old prime valuation centered at the generic point of \(C\). Its extended center \(Z\) on the common model dominates \(C_{k'}\). After clearing denominators, if \(Z\) lay in the effective Cartier difference, a local equation would have strictly positive valuation, contrary to its zero value. Thus \(Z\) is not contained in that support. General complete intersections in \(Z\) have a component dominating \(C_{k'}\) and not contained in the support: their restriction to the generic fiber of \(Z\to C_{k'}\) has positive top ample intersection, and a general choice avoids containment in a fixed proper closed subset. On this curve the effective difference has nonnegative degree. The nef term and projection formula show \(\mathbf N\cdot C\ge0\). If the base itself is a curve, use the whole curve on its common smooth model; the avoidance issue is automatic. A zero-dimensional base is immediate. This proves b-nefness in the semiample case.

The nef and big endpoint. On a projective descent write \(M\sim_\mathbb QD_0+E_0\), with \(D_0\) ample and \(E_0\ge0\). For sufficiently small rational \(s>0\), \[a_s=a-s\overline E_0\] retains horizontal klt and effectivity, and its moduli class \(M-sE_0\sim_\mathbb Q(1-s)M+sD_0\) is ample, hence semiample. Fix one such \(s_0>0\). Apply Lemma 26 to \(a,a_{s_0}\), choosing a positive integer \(r_{\mathrm{ind}}\) for which \(r_{\mathrm{ind}}\mathbf M\) is integral b-Cartier. This index may depend on the individual presentation; the interval below is uniform over its test primes. Since \[a_s=(1-s/s_0)a+(s/s_0)a_{s_0},\] the lemma gives one interval \(0<s<s_1\), independent of the prime \(P\), on which every computing lift for \(\pi_\#a_s(P)\) also computes \(\pi_\#a(P)\).

For a fixed \(P\), a computing lift at any one positive parameter exists by Lemma 23. Comparing it with every other primary minimizer shows that it maximizes \(E_0(F)/m\) among them. This maximum is attained and is independent of \(s\). Thus the threshold is exactly \[(\pi_\#a_s)(P) =(\pi_\#a)(P)-s e_P\] on this interval, including its stated value at \(s=0\). The identity holds for every prime. If \(0<s'<s''<s_1\) are rational, the endpoint moduli function is therefore \[\mathbf N_0= \frac{s''\mathbf N_{s'}-s'\mathbf N_{s''}}{s''-s'}.\] The two terms on the right descend on a common model; hence so does \(\mathbf N_0\). All other small positive-parameter terms descend there by the same affine formula and are nef there, since nefness on a higher descent descends by lifting curves and projection formula. Taking the limit on this fixed model proves that \(\mathbf N_0\) is nef. This proves the endpoint assertion. Scaling and translation now follow from Lemma 23. ◻

Proof. The bounded b-index of \(\mathbf M_0\) gives a uniform positive gap for nonzero values of \(A^+\). Choose \(e\) below this gap multiplied by \(1-\gamma\). For a bad prime we therefore have \(A^+(P)=0\), and \[aH(P)<e,\qquad \lambda Q(P)<\frac{es}{b}\] when \(Q\) is present. Since \(H(P)+\lambda Q(P)\ge s\), choosing \(e<b/2\) gives the asserted lower bound; initially it follows directly from \(H(P)\ge s\). At the last level, if the unmodified scale is at least \(s_0>0\), these same formulas give a uniform positive lower bound for \(A'\) on lc primes, in terms of \(a,b,s_0\). Decrease \(e\) further.

For completeness, the finiteness assertion uses more than the finiteness of components on one resolution. On a smooth b-Cartier descent of any generalized klt datum, let \(\Sigma\) be the reduced snc support of its boundary. Its discrepancy at \(P\) is \[A_{(W,\Sigma)}(P)+\sum_D a_D\mathop{\mathrm{ord}}_P(D),\qquad a_D>0.\] The first term is a nonnegative integer. If the total is less than \(e<1\), this term is zero. We recall why this forces \(P\) to be monomial at a stratum of \(\Sigma\). At the generic point of its centre choose regular parameters \(x_1,\ldots,x_\ell\), including the boundary normals, and units \(y_j\) lifting a separating transcendence basis of the centre. The reduced-pair discrepancy is the discrepancy of \[d\log x_1\wedge\cdots\wedge d\log x_\ell \wedge\bigwedge_jdy_j\] plus the orders of those \(x_i\) which are not boundary normals. On a smooth model carrying \(P\), write \(x_i=t^{w_i}u_i\), with \(t\) a uniformizer and \(u_i\) units. The displayed form has at most a simple pole. Equality zero thus forces every \(x_i\) to be a boundary normal and the logarithmic wedge to have nonzero residue. The centre is a stratum, and the wedge of the initials \(\overline u_i\overline t^{w_i}\) and \(\overline y_j\) is nonzero. In characteristic zero these initials are algebraically independent in the required normal directions, also over the centre’s function field, which is algebraic over \(\mathbb C(\overline y_j)\). Expand an element of the local ring to arbitrarily high finite order in the \(x_i\), with coefficients lifted from its residue field. Independence prevents cancellation in its leading polynomial, while the remainder has arbitrarily large value. Its value is therefore the minimum of its monomial values. The stratum and the positive integral weights \(w_i\) determine \(P\). There are finitely many strata, and \(\sum a_Dw_D<1\) bounds all these weights. This proves finiteness.

In the collapsing case choose \(G_*\) attaining the scale, and write \(H_*=H(G_*)\). Then \[H_*\le s,\qquad A'(G_*)\le b+as.\] To extract the indicated finite set, replace \(\gamma\) temporarily by a much smaller positive rational \(\widetilde\gamma<a\lambda\), leaving it zero at the first level. The resulting effective generalized klt datum has ample anti-adjoint. Its discrepancy at every requested prime is less than \(1\): these primes are lc for \(A^+\), their \(H\)-values have the bounds just proved, and \(\widetilde\gamma\) can be chosen for this finite set in the given member. Add a sufficiently small positive multiple of \(H\) to both moduli and total adjoint. This leaves discrepancies unchanged, makes the moduli big, and preserves ample anti-adjoint. By Lemma 24, replace the moduli by ordinary effective klt data with no larger discrepancies. The resulting ordinary pair is log Fano. Lemma 25 extracts precisely the requested exceptional primes on a \(\mathbb Q\)-factorial Fano type model. For the original \(A'\), their discrepancies are either less than \(e\) or at most \(b+as<1\), so its trace remains effective. At these primes \(B_0(P)=1\) and \(H(P)\le\max(e/a,s)\le1\) after the indicated choices. Consequently \(B_0\ge H\) also survives the extraction. ◻

Proof. There is nothing to prove in dimension zero. First extract just the bad primes by Lemma 29, without an additional pivot. Every positive strict coefficient of \(H\) is now at least a fixed positive multiple of \(s\). Moreover the \(A'\)-boundary coefficient at each such prime has a uniform positive lower bound. At a new prime it is greater than \(1-e\). At an old non-lc prime use \(B_0(P)\ge H(P)\ge c_0\) and the effectiveness of the other boundary; at an old lc prime use \(B_0(P)=1\) and \(H(P)\le1\).

Let \(G\) be the sum of the bad primes, which are now strict. With a sufficiently large fixed \(C_2\) and \(k_0=\lceil C_2/s\rceil\), the integral Weil divisor \[N=\lfloor k_0H\rfloor-G\] satisfies \(N\ge k_0H/2\). Indeed the rounding error plus the coefficient of \(G\) is at most \(2\) at each prime of the support, whereas each positive coefficient of \(k_0H\) is uniformly large. Choose a fixed sufficiently small rational \(\sigma>0\). Subtract \(\sigma(\{k_0H\}+G)\) from the \(A'\)-boundary and add \((\sigma k_0-d')\overline H\) to its moduli; choose \(C_2\) large after \(\sigma\) so this last coefficient is positive. The total adjoint is \(\sigma N\), the boundary is effective, and its discrepancy is \[A'+\sigma\overline{(\{k_0H\}+G)}.\] It is uniformly generalized positive-lc: nonbad valuations already have discrepancy at least \(e\), the added divisor is effective Cartier up to a rational multiple on this \(\mathbb Q\)-factorial model, and every bad prime has received an increment at least \(\sigma\).

Run its adjoint MMP on this Fano type model. The Mori dream space property and divisor MMP give a nef semiample transform of the big \(N\); see (Birkar et al. 2010, Corollary 1.3.1) and (Hu and Keel 2000, Proposition 1.11). Denote that transform by \(N'\) on \(Z\). It is an effective nef big integral Weil divisor. Negativity preserves the uniform generalized discrepancy bound; forgetting the effective boundary and b-nef part gives a uniform ordinary lc bound on \(Z\). The model remains of Fano type, so \(-K_Z\) is big. In particular \(N'-K_Z\) is big. Theorem 1.1 of (Birkar 2023), which permits an integral Weil polarization, now makes \(|qN'|\) birational for a bounded positive integer \(q\). Negativity on a common resolution transfers every such section to \(qN\) on the original model: the pullback of the source divisor dominates that of its transform. This applies to the rational Cartier divisors represented by Weil sections as well. Finally \(N\le k_0H\), so \(|qk_0H|\) is birational and \(qk_0\le C_1/s\).

One can also obtain a bounded positive integer \(u\) with \(K_Z+uN'\) big. This additional observation is useful when using the version of the polarization result with both signs of \(K_Z\). Let \(u_0\ge0\) be the pseudo-effective threshold of \(K_Z\) with respect to \(N'\). It is rational by the Mori dream space property. If \(u_0=0\), bigness follows for every positive \(u\). Otherwise take the adjoint MMP at \(u_0\), regarding \(u_0\overline{N'}\) as b-nef big data. Negativity retains a uniform generalized lc bound, and the nef semiample threshold adjoint is nonbig, hence defines a contraction of positive relative dimension. On its geometric generic fibre, ordinary approximation with a small ample reserve gives an effective uniformly positive-lc log Fano pair. Theorem 1.1 of (Birkar 2021) bounds the fibre. For a bounded very ample polarization, the degree of \(-K\) is bounded above by adjunction to a general section curve. The transform of \(N'\) has positive integral degree there: it is effective and integral, and its bigness, preserved by sections, excludes purely vertical support. Relative adjoint triviality gives \(u_0\deg N'=\deg(-K)\), so \(u_0\) is bounded. Taking an integer \(u>u_0\) proves the assertion. ◻

Proof. The threshold and its lower bound. On the chosen extraction set \[x=\min\{r:rH-J\text{ is pseudo-effective on }S'\}.\] Subtracting \(J\) leaves the \(A'\)-boundary effective: its coefficient at the pivot is at least \(1-b-as\), and at the other indicated primes it is greater than \(1-e\). The discrepancy becomes \(A'+\overline J\). Every bad prime has received a fixed positive increment, so this datum is uniformly generalized \(\delta'\)-lc for a fixed \(\delta'>0\).

The threshold \(x\) is positive and finite, because \(J\) is nonzero effective and \(H\) is big. It is rational, and \(xH-J\) is rationally linearly equivalent to an effective divisor, by the Mori dream space property. Adding \(J\) to such a representative and dividing by \(x\) gives an effective representative of the pullback of \(H\) on \(S'\). It descends as \(H+\mathop{\mathrm{Div}}(f)/q\) to an effective representative on \(S\), by pushing its coefficients down. Consequently weak essentiality at the pivot gives \[ \frac{b_*}{x}\le H_*+\eta s\le(1+\eta)s. \tag{24}\] It follows that \(x\gtrsim s^{-1}\) and, after choosing \(b\) small, \(x-d'\gtrsim x\). Thus \(\mathbf M_C\) is b-nef big. Adding \((x-d')\overline H\) to the moduli changes the total adjoint to \(xH-J\) without changing \(A'+\overline J\).

Run the adjoint MMP to a nef semiample transform on \(Y\). Its total adjoint is the trace \(xH''-J''\). This divisor is not big, since \(x\) is the threshold. Its semiample contraction \(\pi:Y\to V\) therefore has positive relative dimension. The transformed boundary is effective, and negativity preserves the uniform generalized \(\delta'\)-lc bound. All the birational models used here remain \(\mathbb Q\)-factorial and of Fano type.

Bounded fibres and the upper bound for the threshold. We next bound the degree of \(xH''\) on a geometric generic fibre. A proper normal generic fibre of a contraction in characteristic zero is geometrically integral and normal. Indeed its global functions are \(k(V)\), so \(k(V)\) is algebraically closed in \(k(Y)\); separability gives the geometric assertions. Canonical forms on the fibre are obtained by dividing out a base volume form. A resolution with snc data shows that generic restriction and extension to an algebraic closure preserve the required discrepancy bounds. Apply Lemma 24 to the b-nef big part, reserving a tiny ample class. On the geometric generic fibre this gives an ordinary effective \(\delta'/2\)-lc log Fano pair. Theorem 1.1 of (Birkar 2021) therefore bounds the underlying fibre. The theorem applies over this algebraically closed characteristic-zero field; one may also transport the data along an isomorphism of the algebraic closure of \(\mathbb C(V)\) with \(\mathbb C\).

Fix a bounded very ample Cartier polarization on these fibres, and write \(\deg\) for intersection with its appropriate power. If \(L\) is this polarization in dimension \(r\ge1\), a general complete-intersection curve avoids the singular locus and gives \[\deg(-K)\le (r-1)L^r+2.\] Thus this degree is bounded above. The trace of any b-nef part has nonnegative degree: upstairs add an arbitrarily small ample rational class, take an effective representative, push its trace down, and then let that class tend to zero. On \(Y\) the CY trace of \(A^+\) still has \(B_0''\ge H''\) and is effective. Every component of \(J''\) is lc for \(A^+\), so its \(B_0''\)-coefficient is \(1\) and \(J''\le B_0''\). The CY equality gives \[ x\deg H''=\deg J''\le\deg B_0''\le\deg(-K)\le C_3. \tag{25}\] Here traces are rational Cartier, since \(Y\) is \(\mathbb Q\)-factorial.

The birational functions of Lemma 30 remain sections satisfying \(\mathop{\mathrm{Div}}(f)+kH''\ge0\) after the non-extracting MMP. If \(sx\) were sufficiently large, (25) and \(k\le C_1/s\) would bound the degree of every possible polar divisor on a geometric generic fibre by a number strictly less than \(1\). An integral nonzero effective divisor has positive integral degree, so these polar divisors would vanish. Properness would then put all the functions in \(k(V)\), contradicting their generation of \(k(Y)\) and the positive relative dimension. Thus \(x\lesssim s^{-1}\). Equation (24) now also gives \(H_*\gtrsim s\).

The pivot is the only positive horizontal component. Consider the coefficients of \(H''-J''/x\). At every positive coefficient other than the pivot, this divisor has coefficient at least \(H''(P)/2\gtrsim s\). Outside \(J''\) this follows from the inductive coefficient gaps and the extraction bounds. At a bad prime use \(H(P)\ge s/2\), \(1/x\le(1+\eta)s/b_*\), and choose \(\nu\) sufficiently small compared with \(b_*\). At the pivot its coefficient is at least \(-\eta s\), by (24). If horizontal, that prime has bounded degree, since its coefficient in \(B_0''\) is \(1\). The total degree of \(H''-J''/x\) on a generic fibre is zero. As \(\eta\to0\), a horizontal nonpivot component would contribute at least a fixed multiple of \(s\), and the pivot could contribute only \(-O(\eta s)\). This is impossible. There must nevertheless be a horizontal component in the positive support of \(H''\), since otherwise the birational functions would again be fibrewise regular. Hence the pivot survives and is the sole such horizontal component. The degree identity forces its coefficient in \(H''-J''/x\) to be zero, proving \(x=b_*/H_*\).

The exact divisor on the base. It follows that \(D_Y=xH''-J''\) is effective and vertical. Semiampleness gives \(D_Y\sim_{\mathbb Q}\pi^*L\) for an ample rational Cartier divisor \(L\) on \(V\). The rational function implementing this equivalence has no horizontal poles, and hence lies in \(k(V)\) by normality and properness of the generic fibre. Absorb its principal divisor into \(L\) and divide by \(x\) to obtain an exact equality \(D_Y=x\pi^*H_V\). Every strict base prime has a strict lift, so \(D_Y\ge0\) implies \(H_V\ge0\). Its rational linear class is ample. Finally, on a common resolution, negativity gives the effective rational Cartier b-divisor \[ \mathbf E_{\mathrm{MMP}} :=x\overline H-\overline J-x\overline{\pi^*H_V}\ge0. \tag{26}\] Since \(J\ge0\), this proves (23). The threshold datum after the MMP has discrepancy \[(A'+\overline J)+\mathbf E_{\mathrm{MMP}} =A'+x\overline H-x\overline{\pi^*H_V}.\] Thus its uniform positive lower bound holds on every prime over \(Y\), as required for the next step. ◻

Proof. Discrepancy presentations and their lower bounds. By (26), the threshold datum on \(Y\) has discrepancy exactly \(C-x\pi^*H_V\). Therefore \(C\) has effective horizontal trace and is uniformly generalized klt over the generic point of \(V\). Its moduli are b-nef big, so Proposition 28, followed by translation by the base divisor, gives a presentation for \(Q_{\rm next}\) with rational b-Cartier b-nef part and total adjoint \(xH_V\).

The CY datum \(A^+\) has effective horizontal trace on \(Y\). Apply Lemma 26 to \(A^+,C\). For all sufficiently small rational \(t>0\), uniformly in the member and the base prime, a lift computing the push of \((1-t)A^++tC\) also computes the push of \(A^+\). On primary minimizers its value is \[(1-t)A^+_{\rm next}(P)+t\frac{C(T)}{m_T}.\] A minimizer at any one such \(t\) minimizes the second summand among all primary minimizers. Thus the push is exactly affine on a common interval including \(t=0\). At positive \(t\) the moduli are b-nef big upstairs and the horizontal trace is klt, so Proposition 28 supplies b-nef rational b-Cartier CY moduli downstairs. Two distinct positive rational parameters give a common descent for their affine endpoint. All smaller positive parameters are nef on that descent; passing to the limit proves b-nefness of the endpoint. This constructs the required CY presentation for \(A^+_{\rm next}\).

Nonnegativity of \(A^+_{\rm next}\) follows from (17). For effectivity, take a strict lift \(F\) of a strict prime \(P\) of \(V\), with restriction \(mP\). The effective trace upstairs bounds its normalized discrepancy by \(1/m\le1\), proving effectivity of the pushed boundary. For \(A^+\) the stronger inequality \(B_0''\ge H''\ge\pi^*H_V\) gives \[A^+_{\rm next}(P)\le\frac{A^+(F)}m \le\frac1m-H_V(P)\le1-H_V(P).\] Its boundary therefore dominates \(H_V\).

To obtain a bound for \(Q_{\rm next}\) on all valuations, apply Lemma 24 to the effective uniformly generalized \(\delta'\)-lc threshold datum on the whole of \(Y\). Use an ordinary effective \(\delta'/2\)-lc representative with the same relative rational log CY class and no larger discrepancies. The Fano type property implies that \(-K_Y\) is big over \(V\). The discriminant bound of (Birkar 2025, Theorem 1.1) therefore gives a uniform positive bound for the valuative push of this ordinary pair, on every model of \(V\). Equation (17) then gives the same bound for \(Q_{\rm next}\). Notice that this application uses the uniform lc bound on the entire total pair, not just its generic fibre.

A model of the initial field and the uniform index. To bound the new index, we retain a model of the initial field throughout. For each exceptional prime of \(S'\to S\), which is lc for \(A^+\), attainment and composition in Lemma 23 give a prime over \(\mathcal Y\) that is lc for \(A_1^+\) and restricts to a positive multiple of the prescribed prime. Extract these primes, if they are not already strict, without any others. To justify this extraction, mix the effective ordinary lc CY boundary of \(A_1^+\) on \(\mathcal Y\) with an effective klt log Fano boundary, giving the latter a sufficiently small positive weight. The mixture is log Fano and klt, and the requested discrepancies lie strictly between \(0\) and \(1\). Lemma 25 applies, producing a \(\mathbb Q\)-factorial Fano type model \(\mathcal Y'\). The trace of \(A_1^+\) remains effective, since each new prime has \(A_1^+\)-discrepancy zero. Every strict prime of \(S'\) now has a strict lift on \(\mathcal Y'\); for the nonexceptional ones use the existing morphism to \(S\). The same is true for every strict prime of \(Y\), because the MMP extracts none.

There need not yet be a morphism \(\mathcal Y'\to Y\), so we construct one without losing these properties. Choose an ample Cartier divisor \(D_Y\) on \(Y\), pull it back on a resolution of the rational map, and let \(D^*\) be its trace on \(\mathcal Y'\). It is rational Cartier. The rational-function sections satisfy, in every nonnegative degree, \[ \{f\in K_1:\mathop{\mathrm{Div}}(f)+mD^*\ge0\} =\{f\in k(Y):\mathop{\mathrm{Div}}(f)+mD_Y\ge0\}. \tag{28}\] Indeed a function on the left has no poles on the normal proper generic fibre of \(\mathcal Y'\to S\): the divisor \(D^*\) has coefficient zero at every prime dominating \(S\), since it comes from the field \(k(S)\). It is therefore a member of \(k(S)=k(Y)\). The strict lifts just constructed test its order at every strict prime of \(Y\), giving one inclusion. Conversely an effective Cartier divisor represented by a section on \(Y\) has effective pullback and trace on \(\mathcal Y'\). This also proves the assertion for the possibly nondivisible degrees, by viewing the conditions as inequalities of rational divisors.

Run a divisor MMP for \(D^*\) on the Fano type variety \(\mathcal Y'\). Its effective rational class is pseudo-effective, and its transform is nef semiample. Negativity preserves the divisible section systems in (28). Their Proj is \(Y\), so the resulting model has a morphism to \(Y\) inducing the given field inclusion. This morphism is a contraction: \(k(Y)=k(S)\) is relatively algebraically closed in \(K_1\), and normality identifies its Stein factor with \(Y\). The model remains \(\mathbb Q\)-factorial of Fano type, and the trace of \(A_1^+\) remains effective under non-extraction. Composing with \(Y\to V\) supplies the next auxiliary model. Valuative composition shows that its push of \(A_1^+\) is exactly \(A^+_{\rm next}\).

A klt log Fano boundary on this auxiliary model supplies an effective generalized klt CY datum with b-nef big part: take minus its own adjoint as the moduli divisor. Apply Corollary 27 to this contraction and the initial principal-moduli datum \(A_1^+\). Its dimension and denominator \(n\) are bounded independently of the level and member. Thus a bounded integer clears every discrepancy denominator of \(A^+_{\rm next}\). On a smooth descent of its already constructed CY moduli, the form discrepancies are integral, so the same integer clears the coefficients of the moduli trace. An integral divisor on that smooth model is Cartier. This proves the required bounded b-Cartier multiple, without introducing any fibre-multiplicity factor.

Weak essentiality on the base. If \(P\) is lc for \(A^+_{\rm next}\), choose a lift \(T\), with restriction \(m_TP\), computing the push of \((1-t)A^++tC\) for a fixed sufficiently small rational \(t>0\). Lemma 26 gives \(A^+(T)=0\). The formulas for \(C\), together with \(x\asymp s^{-1}\), give discrepancy inequalities \[A^+\lesssim C,\qquad x\Lambda\lesssim C.\] For example, \(C=(1-\gamma)A^++\gamma Q+(a+x)H\), all summands are nonnegative, \(1-\gamma\) is bounded below, and \(x\lambda/\gamma=xs/b\) is bounded when \(Q\) is present. Compare the mixture at \(T\) with its value at a lift minimizing \(C/m\). Since \(A^+(T)=0\), this proves \[ \frac{\Lambda(T)}{m_T} \le C_4\Lambda_{\rm next}(P). \tag{29}\] The factor \(t^{-1}\) is harmless because \(t\) was fixed uniformly.

If \(H_V'\ge0\) is rationally linearly equivalent to \(H_V\), apply its rational principal change to \(H\) through the field inclusion. Inequality (23) shows on a common resolution that the new divisor is effective: it is the sum of \(H-\pi^*H_V\) and \(\pi^*H_V'\). Pushing down gives an effective representative \(H'\sim_{\mathbb Q}H\) on \(S\). Its change at \(T\) is exactly \(m_T(P(H_V')-P(H_V))\). Apply (20) at \(T\) and then (29). We obtain \[P(H_V')\le P(H_V)+C_4\eta\Lambda_{\rm next}(P).\] Replacing \(\eta\) by \(C_4\eta\) preserves its convergence to zero.

Invariant-prime inequalities and strict coefficient gaps. For (19), the first two inequalities follow immediately from valuative minimization and (23). If an invariant prime has restriction \(jP\) at the old level and remains nontrivial at the next, its full-multiplicity next \(\Lambda\)-value is at most \[x^{-1}j\bigl((1-\gamma)A^+(P)+\gamma Q(P)+(a+x)H(P)\bigr).\] When \(Q\) is present, (19) gives \[j\gamma Q(P)\le\gamma+\frac bsC_0h_E.\] The constant \(\gamma\) cancels the \(-\gamma\) in the bound \((1-\gamma)jA^+(P)\le(1-\gamma)(1-h_E)\). Using \(jH(P)\le h_E\) and \(x\asymp s^{-1}\), we obtain a bound \(x^{-1}+C'h_E\), exactly the next third inequality. Initially the same calculation simply omits the \(Q\)-term.

It remains to check the strict coefficient gaps and witnesses; they cannot be inferred solely from the preceding inequalities. For a strict prime \(P_V\) take a strict lift \(F\) on \(Y\) with restriction \(mP_V\). The effective threshold boundary and the lower bound for \(Q_{\rm next}\) imply \[0<\delta_{\rm next}\le Q_{\rm next}(P_V) \le\frac{(C-x\pi^*H_V)(F)}m\le\frac1m.\] Hence \(m\) is uniformly bounded. If \(H_V(P_V)>0\) and \(A^+_{\rm next}(P_V)>0\), the lift is non-lc for \(A^+\). It is therefore an old strict prime of \(S\), lies outside \(J\), and satisfies \(mH_V(P_V)=H''(F)\ge c_0\). This proves the absolute positive gap. If instead \(P_V\) is lc and \(H_V(P_V)>0\), the nonpivot coefficient bound for \(H''-J''/x\) gives \(H_V(P_V)\gtrsim s\), since the pivot is horizontal and \(m\) is bounded. Before any terminal replacement of the scale, effectivity of the \(Q_{\rm next}\)-boundary gives \[s_{\rm next}\le H_V(P_V)+\lambda_{\rm next}Q_{\rm next}(P_V) \le H_V(P_V)+\lambda_{\rm next}.\] Since \(\lambda_{\rm next}\asymp s\), this proves the required lower bound by a fixed multiple of \(s_{\rm next}\).

Finally suppose \(H_V(P_V)=0\). The same nonpivot coefficient bound forces \(H''(F)=0\). All extracted primes have positive \(H\)-value, so \(F\) is an old strict prime. Its old witness has \(h_E=0\) and bounded restriction multiplicity. Restricting that witness to \(k(V)\) only multiplies this multiplicity by the bounded \(m\), giving the next witness. The lower bound for \(Q_{\rm next}\) also yields \(s_{\rm next}\gtrsim\lambda_{\rm next}\asymp s\) from the definition of the scale. All invariants have now been proved. ◻

Proof. Use \(s=1\) and the actual lower bound for the unmodified scale to choose \(e\) as in Lemma 29. There are no bad primes, so \(A'\), and hence \(C\ge A'\), has a uniform positive discrepancy bound. Take \(b\ll a\) and then a fixed small positive \(x\). Now \(x-d'=x+a-b>0\) (initially it is \(x+a\)), giving a fixed positive \(H\)-reserve. The boundary of \(C\) is \[(1-\gamma)B_0+\gamma B_Q-(a+x)H \ge(1-\gamma-a-x)H,\] with the usual initial convention. Make the constants small enough that the right side is a fixed positive multiple of \(H\).

For the displayed estimate, the same constant-term cancellation as in the inheritance proof gives \[\begin{split} cC(P)&\le (1-\gamma)(1-h_E) +\gamma+bC_0h_E+(a+x)h_E\\ &=1-(1-\gamma-bC_0-a-x)h_E. \end{split}\] Choose \(\gamma+bC_0+a+x<1/2\); this is uniform since \(\gamma=b\lambda\) and \(\lambda\to0\), or \(\gamma=0\) initially. This proves (30). Ordinary approximation with a tiny ample reserve gives an effective ordinary uniformly positive-lc log Fano pair on \(Y\). Theorem 1.1 of (Birkar 2021) bounds the underlying variety. All assertions in dimension zero are interpreted vacuously. ◻

Proof. Put \(p=p_i>0\). The bare geometric fiber is klt, so the cone theorem and its length bound (Fujino 2011, Theorem 1.1(5)) give \(K+2pD\) nef. Thus \(A=(2p+1)D\) satisfies \(A-\Delta\sim_\mathbb QA+K\) ample. Its degree is bounded, and the polarized threshold theorem (Birkar 2021, Theorem 1.8) gives a uniform positive lower threshold for the \(\mathbb R\)-linear system of \(A\) against the ordinary positive-lc pair. The same cone inequality bounds \(\deg(-K)\).

Every geometric prime of degree \(e\) lies in a Cartier section of \(|qD|\) for some \(1\le q\le e\). Indeed a general basepoint-free projection to \(\mathbb P^p\) is finite, since its pulled-back hyperplane is ample. The image of the prime is a hypersurface of degree at most \(e\), and its pullback contains the prime. Consequently a nonzero \(L'\) is dominated geometrically by a rational weighted sum of such sections, of rational class \(q'D\) with \(q'\le\deg L'\). The difference is effective rationally Cartier, even when the individual geometric primes are not rationally Cartier. Scale this sum to the system of \(A\), apply the threshold bound, and compare the ordinary discrepancy with \(C_i\). Extension of the arithmetic test gives (32).

Negativity makes \(\overline{U_{\rm tr}}-\mathbf U\) b-effective. On a high projective descent add a positive rational ample amount tending to zero. The error of the ample addition is again nonnegative, so this can only increase the error being bounded. The perturbed nef class is ample and has, after a rational principal adjustment, an effective representative upstairs. That adjustment changes neither error. Its nonnegative orders bound the new error by the pullback order of its effective trace. Apply (32) and let the added amount tend to zero. The traces in question are arithmetically rationally Cartier and geometrically pseudo-effective; their degrees are nonnegative. This proves (33). ◻

Proof. Horizontally on \(Y_1\), \(\deg D_{1,\rm tr}\lesssim s_1\). For a nonlast level, \(x_1H_1''=J''\le B_{0,1}''\); at the last level use \(H_1''\le B_{0,1}''\). The Calabi–Yau trace identity and nonnegative moduli degree bound \(\deg B_{0,1}''\) by \(\deg(-K)\). The b-nef divisor \(D_1\) has orders at most those of its effective trace closure. Hence (32) gives \(D_1(P)\le C_0a_1(P)\) on horizontal tests. Subtract \(\tau_1D_1\) from \(a_1\) for a small uniform positive rational \(\tau_1\). The moduli stay nef and big by the \(\mathbf H_1\)-reserve, and the horizontal data stay generalized klt and effective.

Inductively define \[D_{i+1}=\pi_\#a_i-\pi_\#(a_i-\tau_iD_i).\] The minimum formula makes it nonnegative everywhere. Proposition 28 writes it as \(\mathbf U-\mathbf U'\) with both terms rational b-Cartier and b-nef. Their canonical coefficients agree, and \[\mathbf U=s_i\mathbf M_{Q_{i+1}}, \qquad \pi_\#C_i=Q_{i+1}+x_iH_{i+1}.\] The next level has a positive reserve of this \(\mathbf U\). Its effective scaled horizontal Calabi–Yau identity therefore bounds \(\deg U_{\rm tr}\le C_0s_{i+1}\). Nonnegativity of trace degrees and of the trace of \(D_{i+1}\) gives the same bound for \(U'_{\rm tr}\) and \(D_{i+1,\rm tr}\). Apply (32) to the latter trace and (33) to the two trace errors. Thus \[0\le D_{i+1}(P)\le C_0a_{i+1}(P)\] horizontally. For a uniform sufficiently small \(\tau_{i+1}>0\), subtracting \(\tau_{i+1}D_{i+1}\) preserves nef big moduli: it subtracts \(\tau_{i+1}\mathbf U\) from its reserve and adds \(\tau_{i+1}\mathbf U'\). It also preserves horizontal klt and trace effectivity. A level without horizontal tests needs no degree estimate.

For a test continuing nontrivially to \(P_{i+1}\), \[(a_i-\tau_iD_i)(P_i) \ge \pi_\#(a_i-\tau_iD_i)(P_{i+1}) \ge -D_{i+1}(P_{i+1}),\] because \(a_i\ge0\). Rearranging and using the horizontal estimate at the last nontrivial restriction proves (34). For a test trivial on \(K_f\), only \(i<f\) contribute. The tower gives \[A_{i+1}^+(P_{i+1})\le A_i^+(P_i),\qquad \Lambda_{i+1}(P_{i+1}) \le\lambda_{i+1}C_i(P_i)\lesssim a_i(P_i).\] Recursively use \(a_i\asymp s_iA_i^++\Lambda_i\) and \(\Lambda_1=H_1\); the tower length is bounded. This gives (35). ◻

Proof. The bounded family need only contain the varieties under consideration; its other members need not satisfy positivity. After finite stratification and finite base extensions, take simultaneous projective smooth resolutions \(b_t:W_t\to X_t\) with labeled exceptional prime divisors. To construct them, resolve the geometric generic fiber (Włodarczyk 2005, Theorems 1.0.1–1.0.3), spread after a finite extension, shrink, and apply noetherian induction to the complement. Keep an isomorphism on dense opens and spread the components of the complements. Stabilizing their fiber and image dimensions identifies all exceptional prime families and excludes additional exceptional divisors. Smoothness of \(W_t\), normality and geometric integrality of the fibers in use, and geometric integrality of the prime families hold after the corresponding open restrictions. Fixed labeled prime or Cartier data can be included in this process.

On the selected fibers, Kawamata–Viehweg vanishing for a klt log Fano boundary (Fujino 2011, Theorem 3.2) and rationality of klt singularities (Fujino 2009a, Theorem 0.1) give \(H^1(\mathcal O)=H^2(\mathcal O)=0\), both on \(X_t\) and on \(W_t\). The exponential sequence and GAGA (Serre 1956, sec. 12, Theorems 1–3; Section 20, Proposition 18) therefore identify \(\mathop{\mathrm{Pic}}\) with integral \(H^2\) on both. Strict transform and pushforward give the diagram \[H^2(X_t,\mathbb Z)\ \xrightarrow{\,b_t^*\,}\ H^2(W_t,\mathbb Z) \ \longrightarrow\ H^2(W_t,\mathbb Z)/\langle[E_1],\ldots,[E_q]\rangle=\mathop{\mathrm{Cl}}(X_t).\] The image of the first group in the quotient is \(\mathop{\mathrm{Pic}}(X_t)\).

These are locally constant integral group data, including torsion, after stratification: proper topological constructibility gives the integral cohomology local systems and functorial pullback, while relative line bundles give the labeled Chern classes. Here is the finite stratification argument, including compatibility with the markings. For either projective family \(\mathcal V\to T\), let \(Z_1,\ldots,Z_s\subset\mathcal V\) be its finitely many closed marked subsets. Apply (Cataldo and Migliorini 2005, Theorem 3.2.3) to the finite map \[\mathcal V\sqcup\bigsqcup_{i=1}^s Z_i\longrightarrow\mathcal V\] given by the identity and the inclusions. Split the source strata by the open-and-closed summands and their connected components. For each connected target stratum \(S\), the subset \(Z_i\cap S\) is closed. It is also open, since it is a union of images of submersive source strata. Hence it is either empty or all of \(S\), so the finite target Whitney stratification is compatible with every marking.

Work over each irreducible base piece and restrict to a smooth dense open. Remove the closures of the images of all nondominating strata and of the critical loci of the dominating strata. These closures are proper by generic smoothness. The remaining strata map submersively to the base, and the whole family remains proper. Embed the family relatively in a smooth projective ambient space. Thom’s first isotopy lemma (Mather 1970, Proposition 11.1) gives local topological triviality. Intersect the opens for the finitely many families in use. Noetherian induction treats the remaining base pieces by reapplying this construction to their reduced base-changed families and closed markings; no Whitney assertion for arbitrary closed restrictions is needed.

In trivializations over a small contractible base open, the fiber maps vary continuously and hence induce constant cohomology maps. No simultaneous trivialization of the resolution morphism, or trivial monodromy, is required. Monodromy preserves the isomorphism type of this group diagram.

There are consequently finitely many group-diagram types. Their cokernels bound the torsion of \(\mathop{\mathrm{Cl}}\); when \(\mathop{\mathrm{Cl}}/\mathop{\mathrm{Pic}}\) is finite, its exponent is bounded as well. The latter finiteness is exactly rational factoriality here, so it gives the asserted Cartier multiple.

For the norm assertion in dimension \(p\ge2\), fix a relative very ample \(D_W\) on each resolved stratum. After further finite extension and shrinking, choose relative line bundles \(T_1,\ldots,T_r\) spanning \(H^2(W_t,\mathbb Q)\), with a common integer \(l>0\) such that \(lD_W\pm T_j\) are ample. We justify this choice, which contains more information than the abstract local systems. Shrink where the coherent vanishings hold and the Betti ranks are constant. Descend the finite family data to a countable algebraically closed subfield of \(\mathbb C\), and realize its geometric generic fiber as a complex fiber generic over that smaller field. The relevant algebraic closures and \(\mathbb C\) are isomorphic over the embedded defining parameter field, since their remaining transcendence degrees have the same infinite cardinality. On that fiber \(\mathop{\mathrm{Pic}}=H^2(\mathbb Z)\). Hard Lefschetz and the Hodge–Riemann bilinear relations (Cataldo and Migliorini 2005, Theorem 3.1.2(a)–(c)) make the pairing \(TT'D_W^{p-2}\) nondegenerate on \(H^2(\mathbb Q)\). Indeed \(H^2=P^2\oplus D_WH^0\) is an orthogonal decomposition: the primitive summand is nondegenerate by polarization, and the ample summand has nonzero self-pairing \(D_W^p\). Choose spanning line bundles there, spread them after a finite extension, and retain their nonsingular intersection matrix and ample comparisons on an open. Betti-rank constancy proves they continue to span. Noetherian induction treats the other strata.

Use the coordinates \[\bigl(G_W\cdot T_j\cdot D_W^{p-2}\bigr)_j.\] For an effective divisor their absolute values are at most \(l\deg_{D_W}(G_W)\), by the ample comparisons. The integral free lattice embeds in \(\mathbb Z^r\) and contains the fixed full-rank subgroup generated by the \(T_j\)-coordinates. Thus only finitely many intermediate lattices occur. The exceptional classes have fixed coordinates too. Orthogonal projection modulo their real span gives lattices of finitely many types for the free class groups downstairs.

If \(G_W\) is the strict transform of an effective prime \(G\), the containing hypersurface used in the proof of (32) pulls back to an effective Cartier divisor dominating \(G_W\). The fixed family intersection degrees give \(\deg_{D_W}(G_W)\le C\deg(G)\). Linearity and the triangle inequality prove (36) for effective rational Weil divisors. In dimension one use ordinary degree and the Picard description above; dimension zero is vacuous.

A numerically trivial rational Cartier divisor pulls back to a numerically trivial class upstairs. The same nondegenerate pairing and \(\mathop{\mathrm{Pic}}(W_t)=H^2(W_t,\mathbb Z)\) make its rational class zero. Pushforward proves rational linear triviality downstairs. Finally a pseudo-effective rational Cartier class becomes big after adding any small positive rational ample amount, and a big rational Cartier class is rational-linearly effective. Apply the effective bound and pass to the limit. ◻

Proof. A witnessing function is defined over a finite normal extension \(L/K\). For \(\sigma\in\operatorname{Gal}(L/K)\), its conjugate quotient has zero divisor, hence is an invertible global regular function on the proper normal fiber. Geometric integrality makes it a constant in \(L^*\). The resulting scalar cocycle is trivial by Hilbert 90. Rescale the function by the corresponding scalar; its divisor is unchanged and the function is Galois invariant. ◻

Proof. We may test the assertion on a contrary sequence, allowing \(\beta\) and \(L\) to vary with its member, and take subsequences throughout. The first clearing factor \(d\) below may be arbitrary; only the eventual factor \(d_0\) is required to be uniformly bounded. At each individual member, finite generation of the weight monoid gives the assertion with member-dependent constants: rational cone points enter the monoid after a common multiple, by decomposition into simplicial generator cones. For \(m_0\in n\mathbb Z\) a little larger than \(L\), both \(m_0\chi\pm\beta\) lie in the cone by its normalized dual inequalities. Thus \(m_0\lesssim L\), and actual opposite sections exist after a possibly arbitrary common clearing factor \(d>0\): \[\bar h^{dm_0/n}z^{\pm d\beta}r_\pm, \qquad r_\pm\in K_1^* .\] We can take late members of a contrary sequence and round \(L\) up to an integer. Start with \[u_\pm=-\mathop{\mathrm{Div}}(r_\pm)/d,\qquad \ell_E=E(z^\beta),\qquad \mu=0.\] The arbitrary \(d\) occurs in \(u_\pm\); the initial \(\ell_E\) are integral.

After subsequences, stop at the first stage with \(Ls_i\gtrsim1\); at all preceding stages \(Ls_i\to0\). A stop exists because \(s_f=1\). At entry to each reached stage we maintain \(\mu\ge0\), \(\mu=o(1)\), \(\mu\lesssim L\lambda_i\), and \[ \begin{aligned} u_\pm&=\mu a_{K_i}+\mathbf U_\pm, &\mathbf U_\pm&\text{ rational b-Cartier and b-nef},\\ u_++u_-&\le C_0L\Lambda_i,\\ j_iu_\pm(P_i)&\le\pm\ell_E+C_0Lh_E+\mu, &j_iP_i&=E|_{K_i}. \end{aligned} \tag{38}\] The first inequality is on every prime and is homogeneous. The last line has left side zero at a trivial restriction. We allow \(\ell_E\) to change by orders of fractional rational functions from tower fields, with bounded clearing denominators. Such fractional functions are notation for rational principal b-divisors, not field extensions. Initially the product of the two sections and (12) give \[\mathop{\mathrm{Div}}(r_+r_-)+2dm_0H_1\ge0,\] which is the sum bound; their regularity gives the last line.

Removing the class obstruction before the stopping scale. Consider a stage before the stop, so \(Ls_i\to0\). On its arithmetic fiber every strict prime other than the pivot \(G_*\) comes from a strict prime of \(S_i\) with \(H_i(P)=0\): the pivot alone among the positive support survives horizontally, and every extracted prime had positive \(H_i\). Choose a witness \(j_iP=E|_{K_i}\) from (19) for each such prime, with bounded positive \(j_i\) and \(h_E=0\), and set \[c_P=\ell_E/j_i,\qquad \Delta_{\rm col}=\sum_{P\ne G_*}c_PP.\] This arithmetic divisor has finite support and bounded denominator. Indeed \(\ell_E\) is initially an integral character order, and subsequently only the orders of previously bounded fractional gauges are added. There are dimension-many stages and bounded restriction integers. The original arbitrary \(d\) never enters \(\Delta_{\rm col}\). A rational function has nonzero orders at only finitely many strict primes of \(\mathop{\mathrm{Spec}}R\), proving the support assertion.

Write \(B_0''=B_{0,i}''\), \(\mathbf M_0=\mathbf M_{0,i}\), and \(c_\pm^*=u_\pm(G_*)\). At nonpivot strict primes, \[(\mathbf U_\pm-\mu\mathbf M_0)(P) =u_\pm(P)-\mu A_i^+(P) \le\pm c_P+\mu B_0''(P).\] At the pivot \(A_i^+(G_*)=0\), while \(c_+^*+c_-^*\le C_0Ls_i\). Hence the divisors \[W_\pm=\pm\Delta_{\rm col}+c_\pm^*G_*+\mu B_0'' -(U_\pm-\mu M_0)_{\rm tr}\] are effective on the arithmetic fiber. Their degrees and those of the pseudo-effective traces \(U_{\pm,\rm tr}\) are \(O(Ls_i)\): add the two identities to obtain \[W_++W_-+(U_++U_-)_{\rm tr} =(c_+^*+c_-^*)G_*+2\mu(B_0''+M_{0,\rm tr}).\] All terms on the left have nonnegative degree. The degrees of \(B_0''\), \(M_{0,\rm tr}\) and \(G_*\) are bounded by Calabi–Yau effectivity; the pivot has coefficient one in \(B_0''\). Also \(\mu\lesssim Ls_i\).

On the geometric fiber, Lemma 36 applied to these identities gives \[\|\pm[\Delta_{\rm col}]+c_\pm^*[G_*]\|_{\rm cl} \le C_1Ls_i .\] The nonzero integral vector \([G_*]\) has bounded length; it is nonzero already by its positive degree. Bounded-denominator lattice vectors projected modulo its line are uniformly discrete, because there are finitely many lattice types and finitely many possible bounded pivot vectors. As \(Ls_i\to0\), we obtain the exact equality \[[\Delta_{\rm col}]=q[G_*],\qquad |c_\pm^*\pm q|\lesssim Ls_i .\] The denominator of \(q\) is bounded. In fact if \([G_*]=h g\) with \(g\) primitive, bounded length bounds \(h\); if \(Q[\Delta_{\rm col}]\) is integral, then \(Qqh\in\mathbb Z\). After also killing the bounded torsion, a bounded positive multiple of \(-\Delta_{\rm col}+qG_*\) is principal geometrically. Lemma 37 descends the witness without an extension-degree multiplier. Let \(\phi\) be a fractional function in \(K_i\), with bounded clearing denominator, having this divisor on the fiber.

Replace \(u_\pm\) by \(u_\pm\pm\mathop{\mathrm{Div}}(\phi)\), and \(\ell_E\) by \(\ell_E+E(\phi)\). These principal changes preserve (38). They give \(u_\pm(P)\le\mu\) on strict nonpivot primes and \(u_\pm(G_*)\le C_1Ls_i\). The new signed trace of \(\mathbf U_\pm-\mu\mathbf M_0\) is \[(c_\pm^*\pm q)G_*+\mu B_0''-W_\pm.\] Its positive and negative parts have degrees \(O(Ls_i)\), and the trace degrees of \(\mathbf U_\pm\) are unchanged. All these arithmetic traces and parts are rationally Cartier. Apply (32) to the signed trace and (33) to the errors of \(\mathbf U_\pm\) and \(\mathbf M_0\), in the identity \[u_\pm=\mu A_i^++\mathbf U_\pm-\mu\mathbf M_0.\] Since \(A_i^+\ge0\), this proves, on every horizontal test, \[u_\pm\ge-C'La_i.\]

Pushing the corrected inequalities. The horizontal lower bound just proved will give klt discrepancies after adding a large multiple of \(La_i\). The strict-prime upper bounds give effective horizontal trace. An additional multiple of \(Ls_iA_i^+\) supplies the needed allowance at the pivot. For sufficiently large fixed positive rational \(K',T'\), push \[u_\pm+K'La_i+T'Ls_iA_i^+\] to the next field and use these pushes as the new \(u_\pm\). Their canonical coefficient is \(\mu_{\rm next}=\mu+(K'+T')Ls_i>0\). They have big nef moduli and horizontal generalized klt discrepancies by the preceding lower bound. They are effective in horizontal trace as well: on strict nonpivot primes use \(u_\pm\le\mu\) and \(C_i,A_i^+\le1\); at the pivot use \(u_\pm\le C_1Ls_i\), \(A_i^+=0\), and the extra slack \(T'Ls_i\). Proposition 28 therefore applies. We have \(\mu_{\rm next}=o(1)\) and \(\mu_{\rm next}\lesssim L\lambda_{i+1}\).

For the sum bound, evaluate both signs on one common lift minimizing \(C_i\). Use \(\Lambda_i+s_iA_i^+\lesssim a_i\) and \(s_i\pi_\#C_i\lesssim\Lambda_{i+1}\). For the witness bound, if the projection remains nontrivial, evaluate on the old \(j_iP_i\). By (31) and (19), the newly added constants are exactly \((K'+T')Ls_i\), while the other terms are \(O(Lh_E)\). They are therefore precisely absorbed in \(\mu_{\rm next}\). If the projection becomes trivial, the expression just pushed has positive value at the old horizontal test, so the same upper bound holds with left side zero. An already trivial projection continues to satisfy the bound. This proves the induction.

Completing the canonical coefficient and obtaining regular sections. At the stopping stage \(Ls_i\gtrsim1\), let \(d_0\) be a bounded positive integer clearing all accumulated gauges. There are only boundedly many stages, and eventually \(d_0\mu<1\). The positive coefficient \(1-d_0\mu\) will complete the canonical coefficient to one; the same equality will give the one-unit allowance in the final integral regularity test. For each sign we claim that some \(f_\pm\in K_i^*\) satisfies, at every prime, \[ P(f_\pm)+d_0u_\pm(P)+(1-d_0\mu)A_i^+(P)+C_2LH_i(P)>0 . \tag{39}\] If \(S_i\) is a point, take \(f_\pm=1\). Otherwise the middle terms have the presentation \[a_{K_i}+\mathbf N,\qquad \mathbf N=d_0\mathbf U_\pm+(1-d_0\mu)\mathbf M_{0,i},\] with b-nef \(\mathbf N\).

On a high smooth descent, (22) provides a big free integral Cartier divisor \(D'\le kH_i\) in actual orders, with \(k\lesssim1/s_i\lesssim L\). Indeed choose a list \(1,f_1,\ldots,f_t\) with poles bounded by \(kH_i\) giving the birational system, resolve its projective map, and pull back its first coordinate hyperplane. Freeness makes the order of this divisor \(-\min(0,P(f_1),\ldots,P(f_t))\), which is at most \(kH_i(P)\). Resolve the fractional support of \(N=\mathbf N_{\rm tr}\) too, and use the canonical divisor \(K\) of the same chosen form. For \(j=1,\ldots,\dim S_i+1\), Kawamata–Viehweg vanishing (Fujino 2011, Theorem 3.1) applies to \[K+\lceil N\rceil+jD',\] with snc klt boundary \(\lceil N\rceil-N\) and nef big adjoint difference \(N+jD'\). Riemann–Roch makes its Euler characteristic a polynomial in \(j\) of degree \(\dim S_i\) with positive leading coefficient. It cannot vanish at all these integers; vanishing identifies its values with section dimensions. Choose a nonzero section at one of them. The associated function \(f_\pm\) makes the left side of (39), with \(jD'\) in place of \(C_2LH_i\), the sum of the positive discrepancy of the snc pair and the order of the section’s effective divisor. This proves the claim.

For a nontrivial \(j_iP_i=E|_{K_i}\), multiply (39) by \(j_i\) and substitute (38), \(j_iA_i^+(P_i)\le1-h_E\), and \(j_iH_i(P_i)\le h_E\). The constant allowance is exactly \[d_0\mu+(1-d_0\mu)=1.\] Consequently, for \(m\in n\mathbb Z_{\ge0}\) sufficiently large with \(m\lesssim L\), both signs satisfy \[E(f_\pm)\pm d_0\ell_E+mh_E>-1.\] At trivial restrictions this follows directly from (38) and \(d_0\mu<1\). Multiply \(f_\pm\) by \(z^{\pm d_0\beta}\), by the corresponding cleared gauge functions, and by \(\bar h^{m/n}\). The displayed orders are integers, hence nonnegative. These are nonzero eigenfunctions. Any polar divisor of an eigenfunction is torus invariant, so the invariant-prime tests and normality make them regular. Their weights are \(m\chi\pm d_0\beta\), proving (37) along every alleged contrary subsequence and therefore uniformly. ◻

Proof. Write \(R_\mu=z^\mu U\) for a finite-dimensional coefficient space \(U\subset K_1\); positive finite generation ensures finiteness. Choose rational \(m'>\|\beta\|\) with \(m'\lesssim B'\), and a nonzero opposite section of weight \(d'(m'\chi-\beta)\), where \(d'>0\) is an arbitrary clearing factor as at the start of the twist proof. Increase \(d'\) so that \(M=d'(m+m')\in n\mathbb Z_{>0}\). If \(g'\) is its coefficient, every nonzero \(g\in U\) gives \(z^{M\chi}g^{d'}g'\in R\). By (12), \[D_1=H_1+\frac1M\mathop{\mathrm{Div}}_{S_1} \bigl(g^{d'}g'/f_h^{M/n}\bigr)\ge0.\]

Assume \(\dim S_1>0\). Construct compatible smooth projective high models with morphisms down the tower, resolving rational maps successively from the bottom, and include descents for all displayed data. Choose compatible general closed points where the morphisms are smooth. Lift relative parameters to nested Taylor systems with \(p_i\) parameters at level \(i\). Avoid the supports of the fixed forms, boundary, moduli and \(H_1\). Make \(g',f_h\) units and a fixed basis of \(U\) regular there. Give the parameters at level \(i\) rational weights \(1/s_i\). This monomial test \(P_1\) is divisorial up to scale: it is discrete up to scaling, and ratios of equal-weight monomials give the required residue transcendence degree. Its restriction \(P_i\) uses exactly the parameters at levels \(\ell\ge i\). At the chosen points the smooth monomial formula gives \[a_i(P_i)=s_i\sum_{\ell\ge i}p_\ell/s_\ell \lesssim1,\] using the bounded tower length and \(s_{\ell}\gtrsim s_i\). Proposition 35 therefore implies \[P_1(g)\le CM/d'\le C'B'\] for every nonzero \(g\in U\). Taylor truncation at this weighted bound is injective on \(U\); the number of allowed monomials is at most \(C\prod_i(1+B's_i)^{p_i}\). If \(\dim S_1=0\), use \(K_1=k\). This proves (41).

At fixed physical degree, (40) bounds each integer coordinate by \(|k_j|\le CS'/L_j\). Counting these weights and using (41) proves (42). Weight zero contributes only constants. The angular translation following (15) identifies this count with the associated graded truncation of \(V_m\). There are only finitely many costs below a fixed cutoff, so it is also the dimension modulo higher costs. ◻

Proof. The birational system in (22), containing \(1\), gives \(p_i\) functions algebraically independent over \(K_{i+1}\) whose poles are bounded by \(kH_i\), \(k\lesssim1/s_i\). For any one of them, (23) and pullback orders give \[\mathop{\mathrm{Div}}_{S_1}(g)+kH_1\ge0.\] A trivial restriction evaluates both the lower function and lower divisor at zero. Increase \(k\) to \(m\in n\mathbb Z_{\ge0}\) with \(m\lesssim1/s_i\). Then (12) gives \(g\bar h^{m/n}\in R_{m\chi}\). Its angular lift lies in \(V_m\) with initial \(g\) and cost \(m\).

For a torus axis use (37) with \(L=C\max(1,L_j)\), where the fixed uniform constant \(C\ge1\) is chosen using (40) so that \(\|\beta_j\|\le L\). The resulting opposite sections lift to actual angular functions after dividing their initials by \(\bar h^{m_j/n}\). Their costs satisfy the asserted bound because \(\widetilde v\in Q_T\).

Tower independence first makes the residual initials algebraically independent. The positive torus initials then have independent Laurent charges over \(K_1\), so adjoining them preserves algebraic independence. For \(S'\ge1\), take monomial exponents through \[\left\lfloor \frac{cS'}{\max(1,W_j)} \right\rfloor\] at each axis, for sufficiently small fixed \(c>0\). Their degrees sum to at most \(S'\); nest the products into a common allowed degree, rounding up to a multiple of \(n\). Their costs are \(O(S')\), and their independent initials ensure the same upper cost bound for nonzero combinations. This proves the probe and box assertions. ◻

Proof. The effective Calabi–Yau datum \(C_f\) has a uniform positive discrepancy bound, a boundary dominating a fixed positive multiple of \(H_f\), and a big nef b-part with a fixed positive \(\mathbf H_f\)-reserve. Choose a fixed rational \(C_0\) such that \[\mathbf T_j=\operatorname{div}(g_j)+C_0\mathbf H_f\] is the b-Cartier closure of an effective divisor on \(F\). For sufficiently small fixed rational \(\varepsilon>0\), define guard data by \[C^{\mathrm g}=C_f-\varepsilon\sum_{j\in J}\mathbf T_j.\] Subtract the same terms from the moduli presentation. Equation (32) bounds \(\mathbf T_j(P)\) by a uniform multiple of \(C_f(P)\), so this retains a uniform positive discrepancy bound. Principal terms do not affect nefness, and the reserve absorbs \(\varepsilon C_0|J|\mathbf H_f\). Thus the new moduli remain big and nef, with a positive \(\mathbf H_f\)-reserve, and the trace boundary remains effective.

There is a fixed \(e_0>0\) such that all the prime valuations specified in the statement satisfy \(C^{\mathrm g}(P)\leq1-e_0\). For a projection with \(h_E>0\), this follows from (44), since \(h_E\geq1/n\) and \(c\geq1\). If \(h_E=0\), Equation (19) gives \(P(H_f)=0\). A centre contained in \(\Sigma\) therefore lies generically outside \(\mathop{\mathrm{Supp}}H_f\) and inside a zero divisor of some \(g_j\); its integral order \(P(g_j)\) is at least one. Since \(C_f(P)\leq1\), subtracting \(\varepsilon\mathbf T_j\) gives the required gap. A strict prime of \(F\) in \(\mathop{\mathrm{Supp}}H_f\) has a boundary coefficient bounded below by domination and the strict coefficient gap. A remaining strict prime of \(\Sigma\) has the same positive zero order just used. This also proves the assertion for all components of \(\Sigma\).

Apply Lemmas 24 and 25. The monomial calculation in the proof of Lemma 29 gives only finitely many exceptional prime valuations with \(C^{\mathrm g}(P)\leq1-e_0<1\), including equality at the upper endpoint: their positive integral normal weights satisfy a weighted-sum bound below one, and each coefficient in that sum is positive. Ordinary approximation with a tiny ample reserve does not increase their discrepancies. The extraction construction gives a projective \(\mathbb Q\)-factorial model \(F'\to F\) extracting exactly these primes. The guard trace is effective on \(F'\), retains its uniform positive log discrepancy bound, and has big nef moduli. Ordinary approximation therefore gives an effective big boundary with rationally trivial adjoint and a uniform positive discrepancy bound. By (Birkar 2021, Corollary 1.4), the varieties \(F'\) are bounded; they are also of Fano type by the ample-reserve version of the construction.

Mark every strict prime of \(F'\) whose guard discrepancy is at most \(1-e_0\). Each has guard boundary coefficient at least \(e_0\). The trace degree of the nef b-part is nonnegative by the perturbation argument in (33). Thus the degree of the effective guard boundary is at most the degree of \(-K_{F'}\), which is uniformly bounded for the bounded very ample polarizations by the cone-theorem calculation preceding (32). This bounds both the number and the degrees of the marks. The surviving \(\mathbf H_f\)-reserve gives the asserted bound on the degree of the pullback of \(H_f\) in the same way. ◻

Proof. Use the bounded \(k\) at the final level in (22) and set \(P'=\lfloor kH_f\rfloor\) on \(F'\), using total pullback before rounding. This is effective integral of bounded degree. Its support lies in the marks, since the strict support of \(H_f\) and all exceptional primes of \(F'\to F\) are marked. The functions in (22) still belong to its complete system: their integral orders and positivity of pullback allow the indicated rounding. Thus this system remains birational, and \(P'\) is big.

Choose a small fixed rational \(e'>0\) so that \[\Delta_0=e'\sum_{P\ \mathrm{marked}}P+e'P'\] is at most the guard boundary. Such a uniform choice is possible because the mark coefficients have a fixed lower bound and the coefficients of \(P'\) are bounded by its degree. By ordinary approximation of the big nef b-part, obtain an effective uniformly positive-lc boundary \(\Theta\) with \[K_{F'}+\Theta\sim_{\mathbb Q}0,\qquad \Theta\geq\Delta_0, \qquad \Theta-\Delta_0\ \text{big}.\] Here the added effective representative of the big b-part has big trace: sections on a descent remain sections after pushforward. It supplies the last bigness assertion.

Set \(D'=-(K_{F'}+\Delta_0)\), which is big. Since \(F'\) is a \(\mathbb Q\)-factorial variety of Fano type, it is a Mori dream space by (Birkar et al. 2010, Corollary 1.3.1). Run a \(D'\)-MMP to a \(\mathbb Q\)-factorial model \(Y\) with nef and big transform \(D_Y\). The program extracts no divisors. The rationally trivial log adjoint \(K+\Theta\) is crepant through the program, so \((Y,\Theta_Y)\), and hence \((Y,(\Delta_0)_Y)\), retain the uniform positive discrepancy bound. Moreover \(D_Y=-(K_Y+(\Delta_0)_Y)\) is nef and big. Thus \(Y\) is bounded by (Birkar 2021, Theorem 1.1). It is of Fano type: writing \(D_Y\sim_{\mathbb Q}A+E\) with \(A\) ample and \(E\geq0\), a sufficiently small positive multiple of \(E\) can be added to \((\Delta_0)_Y\) while keeping klt, and the resulting negative log adjoint is ample.

The coefficients of \((\Delta_0)_Y\) have a fixed common denominator. Lemma 36, applied to the bounded \(\mathbb Q\)-factorial varieties \(Y\), gives a bounded integer \(q>0\) with \(qD_Y\) Cartier. Apply (Fujino 2009b, Theorem 1.1 and Remark 1.2) to the klt pair \((Y,(\Delta_0)_Y)\), with \(L=qD_Y\) and \(a=1\). The divisor \[L-(K_Y+(\Delta_0)_Y)=(q+1)D_Y\] is nef and big; for a klt pair there are no additional lc-centre conditions in log bigness. The theorem gives a bounded integer \(N>1\) with \(ND_Y\) Cartier and free. Increase the free multiple if necessary. For a general member \(G_Y=\operatorname{div}_Y\psi+ND_Y\), Bertini on a log resolution shows that \((Y,(\Delta_0)_Y+G_Y/N)\) is klt.

Use the same rational canonical form on all models and define \[\Theta^+=-K_{F'}+\frac1N\operatorname{div}_{F'}\psi.\] On a common resolution with maps \(p\) to \(F'\) and \(q_Y\) to \(Y\), the MMP negativity comparison reads \(p^*D'=q_Y^*D_Y+E'\), with \(E'\geq0\) exceptional over \(Y\). Consequently \[p^*(\Theta^+-\Delta_0)=q_Y^*(G_Y/N)+E'\geq0.\] Thus \(\Theta^+\geq\Delta_0\). Its log adjoint is crepant to that of the klt pair just constructed on \(Y\), so it too is klt. Moreover \(N(K_{F'}+\Theta^+)=\operatorname{div}_{F'}\psi\). Its coefficients belong to \(\frac1N\mathbb Z\cap[0,1)\), and its degree is the bounded degree of \(-K_{F'}\). In bounded projective embeddings, the supports are parametrized by finitely many fixed-dimension, fixed-degree Chow spaces (Kollár 2017, Definition 3.21). Their universal cycles may be taken over normal strata. The finite coefficient set bounds the possible multiplicities, and the degree bound bounds the number of prime components. Taking finite products and labelling their components therefore bounds these data together with the marks and \(P'\). ◻

Proof. We first transport the integral divisor-class relations to the generic fibre. We then transport the crepant snc calculation for \(\Theta^+\) and the section map of a Cartier multiple of \(P'\); these establish klt singularities and bigness, respectively.

Spread a projective smooth resolution of the geometric generic fibre, after a finite extension, and restrict the base so that it resolves every fibre under consideration. Track all exceptional prime divisors and all strict transforms of the finitely many marked primes. This uses the resolution and constructibility argument in Lemma 36: keep a birational isomorphism on dense opens, spread the complementary components, and fix their fibre and image dimensions. Shrink also so that the downstairs fibres are geometrically normal and integral and the prime marking families have integral geometric fibres. Write this simultaneous resolution as \(b_t:W_t\to X^0_t\).

On every selected fibre, vanishing for a log Fano boundary (Fujino 2011, Theorem 3.2) and rationality of klt singularities (Fujino 2009a, Theorem 0.1) give \(H^1(\mathcal O)=H^2(\mathcal O)=0\) both downstairs and upstairs. Semicontinuity and density of the selected points give these vanishings throughout an open base. On its complex fibres, the exponential sequence and GAGA (Serre 1956, sec. 12, Theorems 1–3; Section 20, Proposition 18) identify \(\mathop{\mathrm{Pic}}\) with integral \(H^2\). On a connected dense stratum, topological constructibility gives locally constant integral group data \[H^2(X^0_t,\mathbb Z)\ \xrightarrow{\ b_t^*\ }\ H^2(W_t,\mathbb Z) \ \longrightarrow\ H^2(W_t,\mathbb Z)/ \langle[\text{exceptional primes}]\rangle.\] The last group is \(\mathop{\mathrm{Cl}}(X^0_t)\), and the image of the first group is \(\mathop{\mathrm{Pic}}(X^0_t)\). The exceptional and marked upstairs divisor classes, and the class of the relative canonical bundle of the smooth resolution, are locally constant as well. These are integral diagrams, so their cokernels, including torsion, are controlled; trivial monodromy is not required.

On the selected fibres, the image of \(\mathop{\mathrm{Pic}}\) has finite index in \(\mathop{\mathrm{Cl}}\), and \(N[K+\Theta^+]=0\) in \(\mathop{\mathrm{Cl}}\). The same assertions therefore hold on this entire complex stratum. Here the canonical class downstairs is the pushforward of the canonical class upstairs. They hold also on the geometric generic fibre. For completeness, descend the finite family, markings, resolution and open conditions to a countable algebraically closed subfield of \(\mathbb C\), and choose a complex point generic over that subfield. The geometric generic model is identified, up to an isomorphism of algebraically closed fields preserving these finite data, with this complex fibre: the two algebraically closed extension fields have the same infinite transcendence degree over the embedded defining function field. Thus the integral class conclusions apply there. Finite index of \(\mathop{\mathrm{Pic}}\) in \(\mathop{\mathrm{Cl}}\) gives \(\mathbb Q\)-factoriality, and the second conclusion gives the asserted principal multiple of the log adjoint.

It remains to verify singularities and bigness; neither follows from the integral class diagram alone. Take a log resolution of the geometric generic marked pair, and a relative rational volume form and a rational function witnessing \(N(K+\Theta^+)\sim0\). Spread all these data after a further finite extension. On the smooth resolving family, retain the divisor identities for the form and the witnessing function, the strict transforms and exceptional components, and the snc pullback computation. The coefficients and orders in these finite divisor formulas remain unchanged on every fibre after shrinking. This can be arranged by spreading each prime component of the formulas and retaining its geometric integrality and its image dimension; restricting the formulas on the smooth models then preserves the divisor orders. Pushing down preserves the marked principal relation. The snc crepant boundary coefficients are now constant, and one selected fibre is klt. They are therefore all less than one on the geometric generic resolution, proving that \(\Theta^+_{\bar\eta}\) is klt.

Choose a positive Cartier multiple \(tP'_{\bar\eta}\), and spread its line bundle with a rational section whose divisor is exactly this marked divisor. Retain the divisor identity by the same resolution argument. After further shrinking, formation of its space of sections commutes with base change. The dimension of the image of the associated rational map is generically constant, by constructibility. On each selected fibre, the system \(|tP'_t|\) contains products from the birational system \(|P'_t|\), so its image has dimension \(\dim F'\). The generic system has that same dimension. Hence \(P'_{\bar\eta}\) is big.

Finally \(\Theta^+_{\bar\eta}\geq e'P'_{\bar\eta}\) is big. To see Fano type explicitly, write \(\Theta^+_{\bar\eta}\sim_{\mathbb Q}A+E\) with \(A\) ample and \(E\geq0\). For sufficiently small rational \(t>0\), the effective boundary \((1-t)\Theta^+_{\bar\eta}+tE\) is klt, and its negative log adjoint is rationally equivalent to \(tA\). This is a log Fano boundary. ◻

Proof. By Lemma 45 and (Birkar et al. 2010, Corollary 1.3.1), \(X^0_{\bar\eta}\) is a Mori dream space. Its pseudo-effective cone in numerical divisor space is rational polyhedral. We first show that each facet is tested by a family of curves sweeping a dense open of \(X^0_{\bar\eta}\).

Choose a rational divisor class \(D\) in the relative interior of that facet. The class is pseudo-effective. Its divisor MMP ends in a minimal model \(Y_D\) on which the transform \(D_Y\) is nef and semiample. As \(D\) lies on the boundary of the pseudo-effective cone, \(D_Y\) is not big. A sufficiently divisible multiple defines a contraction \(\varphi:Y_D\to Z_D\) with positive-dimensional general fibres. The birational program has no extraction. There is consequently an open \(U\subset Y_D\), with complement of codimension at least two, isomorphic to an open of \(X^0_{\bar\eta}\). On a common resolution the comparison has the form \[p^*D=q^*D_Y+E_D,\qquad E_D\geq0,\] where the correction is exceptional over \(Y_D\). Enlarge the excluded codimension-two subset to contain its image and all inverse-map indeterminacy. Curves in \(U\) then lift to the original variety and avoid this correction, including the correction from any contracted fixed divisorial part of \(D\).

A general fibre of \(\varphi\) is geometrically integral. Each component of \(Y_D\setminus U\) either does not dominate \(Z_D\), and hence misses a general fibre, or has codimension at least two in that fibre. If the relative dimension is one, the latter case is impossible: a bad component has dimension at most \(\dim Y_D-2<\dim Z_D\). Thus the entire general curve fibre lies in \(U\). In larger relative dimension \(t\), intersect a general fibre with \(t-1\) sufficiently general very ample hyperplanes. Bertini gives an integral curve, and the codimension estimate makes it miss the bad locus entirely.

These curves vary in a projective flat family over an integral parameter space after restriction to a dense open of the fibre-and-hyperplane parameter space. Its evaluation map dominates \(Y_D\). Indeed the incidence obtained by choosing a point and hyperplanes through it is irreducible and dominates \(Y_D\); restriction to the dense open of flat integral curves avoiding the bad locus preserves dominance. The lifted curves therefore sweep a dense open of \(X^0_{\bar\eta}\). Their intersections with divisor classes are constant in this family. The corresponding linear functional is nonnegative on every effective class, because a member can be chosen not contained in the divisor’s support; it is positive on an ample class. Its value on \(D\) is zero by the pullback comparison and contraction. A nonzero supporting functional vanishing at a relative interior point of a facet defines exactly that facet inequality. This constructs the required test for each facet.

There are finitely many facets. Spread the finitely many curve families, their parameter spaces and evaluation maps after a finite extension of the base function field. Spread also fixed Cartier multiples of all marked divisors. After shrinking, the curves remain projective and flat with integral geometric fibres, their intersection vectors with these markings remain fixed, and their evaluation maps dominate dense opens of every selected fibre. To justify the last assertion, the generic evaluation image contains a dense open. Its image is constructible, so after spreading and shrinking one retains an open in that image whose intersection with each fibre is dense; fibre-dimension constructibility excludes the exceptional parameter values. Equivalently one can track the evaluation incidence and its generic dominance before the shrink.

Let \(\sum r_PP\) be the class furnished by an actual candidate on any retained fibre. It has an effective rational Cartier representative. For each of our curve families, density provides a member not contained in that representative’s support. Its intersection with the effective representative is nonnegative, and is the fixed linear combination of the marked intersection vector with coefficients \(r_P\). Thus every generic facet inequality holds. This argument makes no boundedness assumption on the coefficient vector of the candidate.

For the converse, fix one rational solution of the generic test. On the generic Mori dream space, a rational class in the effective cone has an effective rational representative. Numerical and rational linear classes agree for rational Cartier divisors here by Lemma 36. Hence for some positive integer \(q\) there is a rational function \(f\) with \[\operatorname{div}(f)+q\sum r_PP_{\bar\eta}\geq0.\] Replace \((q,f)\) by \((kq,f^k)\) for a sufficiently divisible positive integer \(k\), and retain the notation \((q,f)\). This preserves the effective divisor inequality while making the marked divisor Cartier and all degree and charge exponents integral, with the degree divisible by \(n\). Spread the line bundle \(\mathcal O(q\sum r_PP_{\bar\eta})\), its rational trivialization, and the nonzero regular section represented by \(f\), after a further finite extension. The divisor identity can also be retained on the simultaneous resolution. On a dense open of the parameter, the section is regular and not identically zero on any fibre. Thus its rational-function representatives have exactly the permitted pole bound on each selected fibre; possible additional zeros are harmless. Regularity as a section of this specified line bundle excludes unmarked poles.

The minimal-closure retention property ensures that these open restrictions keep infinitely many selected indices, and eventually all retained ones. Transfer the witness through the marked identifications with the corresponding \(F'\). The converse argument following (46), including its automatic unmarked tests, gives the asserted regular eigenfunctions. Only this one rational solution, its clearing multiple and its effective section were spread. No simultaneous family of the algebras \(R\) is needed, and no equality between the effective cones of different fibres is asserted. ◻

Proof. Suppose first that \(\dim F>0\). Use the stabilized rows of (46) and the generic cone test of Lemma 46. They define a rational polyhedral cone in the variables \((m',a,(r_P))\). Its projection \(\mathcal P\) to \((m',a)\) is again a rational polyhedral cone and is therefore closed. For example, express the original cone by finitely many rational generators and project those generators. This proof of closedness imposes no boundedness condition on the auxiliary coefficients \(r_P\).

Actual candidates lie in \(\mathcal P\). Divide an alleged obstruction sequence by \(\sum|a_j|\). After a subsequence, their charge coordinates converge to \(a_\infty\) of \(\ell^1\)-norm one, whereas their degree coordinates tend to zero. Closedness gives \((0,a_\infty)\in\mathcal P\). The rational face \(\mathcal P\cap\{m'=0\}\) therefore contains a rational point with nonzero charge: one can use its rational generators, or density of rational points in its relative interior. A nonempty fibre of a rational linear inequality system over a rational point has a rational point, so choose rational auxiliary coordinates giving a fixed rational solution of (46) with \(m'=0\) and \(a\ne0\).

By Lemma 46, one fixed clearing multiple and one spread effective witness produce actual nonzero regular eigenfunctions on all sufficiently late retained rings \(R\), of degree zero and fixed nonzero charge coordinates. If \(\dim F=0\), the same conclusion follows directly: pure descent makes \(g\) constant, all restrictions to \(K_f\) are trivial, and the stabilized trivial-projection inequalities in (46) themselves give the rational polyhedral cone. There are no markings or effectivity variables in this case.

The resulting eigenfunctions have positive \(\tilde v\)-cost, since a nonzero charge is nonconstant and the grading is positive. These costs are uniformly bounded: only cheap axes occur, their widths are bounded above, the charge coordinates and the \(d_j\) are fixed or bounded, and \(\tilde v\in Q_T\). Each eigenfunction lifts, in physical degree zero, to a function of \(\mathcal A_0=k[Z]\) with the same positive value. The lift therefore belongs to \(\mathfrak m_z\). Its bounded value contradicts (10), which asserts \(\tilde v(\mathfrak m_z)\to\infty\) uniformly over all nonzero functions in this ideal. This rules out the obstruction sequence. ◻

Proof of Proposition 41. The residual widths \(1/s_i\) are already at least one. If some torus width \(L_j\) tended to zero on a subsequence, choose integers \(k\to\infty\) sufficiently slowly that \(kL_j\) remains bounded and \(ks_{\exp}\to0\). Apply (37) to \(k\beta_j\), using a fixed scalar norm bound at least one. It gives actual opposite weights of bounded physical degree and charges \(\pm d_0k\beta_j\), with \(d_0\) positive and bounded. These weights are supported on a cheap axis, satisfy the slow condition, and have unbounded charge-to-degree ratio. Raising to the earlier bounded common clearing power does not change that conclusion. This contradicts Proposition 47. A further-subsequence argument therefore gives a uniform positive lower bound for every \(L_j\).

Choose \(S_0\to\infty\) so slowly that \[S_0s_{\exp}\longrightarrow0,\qquad S_0/W_j\longrightarrow0\quad\text{for each width tending to infinity}.\] For a weight \(m\chi+\sum\alpha_j\beta_j\) of positive cost at most \(S_0\), the weight comparisons preceding (41) give \[m\lesssim S_0,\qquad \sum_j|\alpha_j|L_j\lesssim S_0.\] An expensive torus coordinate must vanish, since it is integral and its width eventually exceeds every fixed multiple of \(S_0\). The lower width bound then gives \(m+\sum|\alpha_j|\lesssim S_0\), so these weights satisfy the slow condition. Proposition 47 now implies a uniform \(\sum|\alpha_j|\lesssim m\) for all such weights: failure would give an obstruction sequence on late members. The bounded clearing factors and the bounded positive \(d_j\) preserve this equivalence.

Since the remaining widths are bounded above, \[b=m+\tilde v\Bigl(\sum_{j\in J}\alpha_j\beta_j\Bigr) \leq m+\Bigl\|\sum_{j\in J}\alpha_j\beta_j\Bigr\| \lesssim m.\] Thus every initial at fixed degree \(m\) through \(S_0\) already occurs through \(\min\{S_0,C_4m\}\). Apply (42) with this cutoff. The factors for expensive widths are uniformly bounded, while each cheap factor is at most a fixed multiple of \(1+m\). There are exactly \(|J|+p_f=r\) cheap axes. This proves both estimates in (43), including the constant initial at degree zero.

Finally the probe construction after (42) had box denominators \(\max\{1,W_j\}\). The positive lower width bound makes these uniformly comparable to \(W_j\). Hence those lower bounds now match all widths. The residual probes have independent initials along the field tower, and the positive torus probes add independent Laurent charges over \(K_1\). Their combined initials are algebraically independent. The stated \(r\) probes at the cheap axes have bounded degree and cost. ◻

Proof. Take the degree-\(n\) Veronese \(\mathcal A'\) of the actual section algebra \(\mathcal A\). Write \(o'\) for the image of \(o\) in \(\mathop{\mathrm{Spec}}\mathcal A'\). The initial algebra of \(\mathcal A'\) at \(\widetilde v\) is the corresponding Veronese of \(R\): \(\widetilde v\) is degree-torus invariant, and linear combinations of torus translates separate the physical degree characters. This algebra is finitely generated, with value-zero part consisting only of constants. Choose lifts \(s_i=h^{m_i/n}F_i\) of homogeneous initial generators and include ordinary homogeneous algebra generators of \(\mathcal A'\). In degree zero subtract their value at the point \(o'\), so all these generators vanish there and have positive cost.

Every homogeneous \(s=h^{m/n}F\) vanishing at \(o'\) has an exact polynomial representative in these generators, all of whose terms have physical degree \(m\) and cost at least \(b=\widetilde v(s)\). To prove exactness, subduct by the initial generators, retaining physical degree, to arbitrarily high cost. The residual value strictly increases; only finitely many costs occur below a fixed bound, by positivity and finite graded generation. The fixed-germ log Izumi estimate at the bare klt cone point \(o\) then puts a sufficiently high residual in any prescribed power of its maximal ideal. The powers are primary also on the affine ring, so this assertion is global. It may equally be checked after extension to \(\mathbb C\) and contracted.

The same conclusion holds in \(\mathcal A'\). Indeed the radical of its point ideal extended to \(\mathcal A\) is the point ideal of \(o\), as seen by taking degree powers; moreover the Veronese inclusion splits as \(\mathcal A'\)-modules. Thus a sufficiently high residual belongs to any prescribed power of the point ideal in \(\mathcal A'\). Such a power has representatives by sufficiently long monomials in the ordinary generators, and hence by terms of high cost. Project to physical degree \(m\). This finishes subduction by an exact finite representative.

Divide by \(h^{m/n}\). If the finite ratios are bounded below by a positive number \(q\), every term satisfies \(w(\text{term})+B_*m\ge q\,(\text{its cost})\ge qb(F)\). Taking the minimum gives the same bound for \(F\). Zero-cost units cause no additional condition. This proves the asserted finite test. ◻

Proof. Choose functions with rationally independent values spanning the rational value group, and value-zero functions with residues forming a complementary transcendence basis. Resolve the divisors of the first functions and the maps to \(\mathbb P^1\) of the second functions, together with the prescribed finite data. At the centre, the number of crossing components is at least the rational rank: the chosen functions are units times Laurent monomials in their local equations. The closure of the centre has dimension at least the residue transcendence degree, because the chosen residue functions are regular and invertible there. Equality of their sum with the field dimension forces the centre to be generic in a transverse stratum and the normal values to be rationally independent.

Complete the regular local ring at that generic point and expand over a coefficient field. The support of a nonzero series has finitely many componentwise-minimal exponents. Since all normal weights are positive, its least weight is attained among them; rational independence makes the least exponent unique. This also works in the ordered group. The ideal of monomials of strictly higher weight is finitely generated. Faithful flatness contracts the congruence modulo this ideal to the local ring. Hence the function equals its least monomial times a residue-coefficient lift, modulo strictly higher weight. This proves the value formula and the initial-field description. Independence of the normal values also identifies the degree-zero field with the stratum field. ◻

Proof. Include their divisors in the monomialization. Use logarithmic differentials of the normal parameters and of unit lifts of a separating basis on the stratum. Their wedge has form discrepancy zero. Each \(a_i\) is a unit times a normal Laurent monomial. The logarithmic differential matrix has entries of nonnegative value, and its reduction is the corresponding matrix of initial logarithmic differentials. Logarithmic normal derivatives of a unit reduce to zero; tangential derivatives differentiate the residue coefficient. The determinant therefore has value zero exactly when the initial differentials are independent, equivalently when the initials are algebraically independent in characteristic zero. ◻

Proof. The restriction \(P\) is Abhyankar by the relative inequality. The Gauss extension is Abhyankar because its variable initials are independent over the full field \(E_P\). Choose functions \(x\) in \(E\) with full independent initials at \(P\). The coefficient of \(\theta_w^n\) in the \(n\)-th power of \(d\log x\wedge d\log\mathbf z\) is homogeneous, so its value is the same for \(Q\) and \(Q^*\). The logarithmic differential contribution is zero at \(Q\), and is nonnegative at \(Q^*\) by Lemma 50. Equality requires independent initials of \(x,\mathbf z\). As \(E_P\) is algebraic over the field generated by the \(x\)-initials, the \(\mathbf z\)-initials are then independent even over \(E_P\). This gives the termwise Gauss rule at \(Q^*\), proving equality of valuations.

For the last assertion, group a polynomial in the probes by its torus characters. The two valuations agree on each homogeneous part, and \(Q\) takes their minimum. Gaussianity at \(Q^*\) therefore gives the same polynomial minimum at \(Q\). ◻

Proof. Give \(F\) the lexicographically ordered value \((w(F),Q(\overline F))\). Grouping tied first values proves the valuation inequality. Its initials identify with double initials: on each first homogeneous slice the second comparison is exactly the definition, while distinct first degrees remain separated by their \(\mathbf z\)-characters, with coefficients in \(E_P\). Multiplication is preserved. Lifts to \(K\) of the variables \(\mathbf z\) and of the functions \(x\) used above have full independent lex initials. Such lifts exist for all homogeneous elements, including degree-zero residue elements. The lex valuation therefore satisfies Abhyankar equality.

Monomialize it by Lemma 49. Replace each normal weight by its first coordinate plus \(\lambda\) times its second coordinate. These weights are positive for small \(\lambda>0\). For any fixed numerator and denominator, the finitely many componentwise-minimal support exponents show that its former least exponent remains least eventually. The resulting initial with its stratum coefficient and normal variables is unchanged. The normal initial variables remain independent even if the new real values have ties, since a monomial valuation retains all least-weight terms. This proves the value assertion and preserves independence of the lifted full probes. Double initials likewise preserve independence of \(y\) under the two stated Gaussian conditions.

Express \(\theta^n\) in the logarithmic basis of those full probes. Its coefficient computes the discrepancy at \(w\) by (9), and its initial is the coefficient of \(\theta_w^n\). The logarithmic basis contributes zero at \(w_\lambda\) and at \(Q\). Apply the value identity also to this one fixed coefficient and divide by \(n\); the second identity follows. ◻

Proof. At a centre, keep the weights on the crossing normal equations and move to the generic stratum; if there are no such equations, the retraction is zero. Retraction can only decrease the value of a regular function. To see this with the required control at faces, let \(x\) be the normal parameters. A monomial ideal of positive threshold contracts from the generic stratum to the local ring. In its completed quotient there is a finite free module over the formal series in the other parameters. A function outside the stratum prime acts injectively: its reduction in the other parameters is nonzero, and the remaining normal part acts nilpotently. The monomial ideal is therefore primary to the stratum prime, and faithful flatness gives contraction. Membership in this ideal characterizes the threshold at the generic stratum and bounds the value at the original centre.

On an open cone, normal expansion computes values by the minima over finitely many componentwise-minimal support exponents of regular numerators and denominators. At a face apply the same primary-ideal argument to just the remaining positive-weight parameters. It proves that these formulas extend to the face; at the origin both values are zero. This gives continuity, finite piecewise linearity and a Lipschitz bound. Divisors included in the resolution have linear values on each cone. So does \(A_\theta\), using the local logarithmic top generator.

The retraction preserves all resolved divisor values. For a polynomial in \(y\), clear poles by a normal monomial, since each \(y_j\) is a unit times a normal Laurent monomial. Its retracted value is at most its old value and at least its termwise minimum. If \(w\) is Gaussian these agree, proving preservation. Within one open cone the initial formulas of \(y\) in stratum coefficients and independent normal variables do not depend on the positive weights. Gaussianity thus selects whole open cones. It persists on their faces by continuity of all polynomial values.

For the discrepancy comparison, choose supplementary regular functions \(z'\) with étale differentials at the centre, including a separating residue basis, and put \(\lambda_0=d\log x\wedge dz'\). The coefficient of \(\theta^n\) relative to \(\lambda_0^n\) is a unit-monomial. The difference from the retracted discrepancy is \(A_{\lambda_0}(w)\). Switching the regular supplementary factors to logarithmic ones and applying Lemma 50 makes this nonnegative. If the centre is proper in its stratum, a supplementary parameter vanishing there has positive value, making the difference strict.

If the centre is generic in the stratum, use unit lifts of a separating stratum basis. Equality in the determinant test makes those initials and the normal initials independent. The normal initials are independent even over the embedded stratum field, which is algebraic over that basis field. Least normal terms therefore cannot cancel; contracting their congruence modulo strictly higher monomial weight proves that \(w\) itself is the monomial retraction. This proves strictness outside the skeleton. The proof is unchanged over other algebraically closed characteristic-zero constant fields. ◻

Proof. By Lemmas 48 and 53 we may optimize on a finite union of closed polyhedral pieces, with affine projection to \(t\) and piecewise affine objective. Strict retraction excludes every optimizer outside them.

Suppose a sublevel, allowing \(t\) to vary, were unbounded. Normalize an escaping sequence in one closed cone by the norms of its weights and take a limit of norm one, defining a nonzero monomial valuation \(w'\). The barrier and continuity give \(w'(F)\ge0\) for every \(0\ne F\in V_m\), including degree-zero units. The bounded cap in (47) gives \(A_\theta(w')\le0\). Nonnegativity on \(k[Z]\), and properness over the affine base, give a centre on \(X\). Equation (16), valid at such centres by freeness of sufficiently divisible multiples over the affine base, implies \(A_X(w')\le0\). This is impossible: on a monomializing resolution a nonzero quasi-monomial valuation centred on a klt variety has strictly positive discrepancy. The sublevels are compact, proving attainment.

For continuity, let \(w\) be an optimizer with strict barrier slack. Give \(k(\overline y)\) arbitrary rational-direction Gauss weights. Extend this valuation to \(K_w\) by completing to a transcendence basis with freely weighted variables and then extending over the finite algebraic extension. The relative Abhyankar inequality gives equality, and the extended value group is contained in the rational span of the starting real weights, so the extension remains real. Invariantize by Lemma 51 and flatten by Lemma 52. The finite test makes a small segment in the prescribed direction feasible, with cost tending to the optimum. Retraction can only improve it.

A compact skeleton sublevel cut strictly above that optimum projects to a finite union of closed polytopes. The tangent cones at the parameter contain all rational directions and hence, by closedness and density, all directions. Near the point only the active inequalities of the incident polytopes matter. Its projection therefore contains a neighbourhood. Applying this for sublevels arbitrarily close above the optimum proves local finiteness and upper continuity. Compactness proves lower continuity.

Finally subdivide a compact sublevel into finitely many polyhedra with affine cost and fixed active mask and stratum on each relative interior. Projection to parameter and cost gives the piecewise-affine minimum. Avoid its walls and all lower-dimensional parameter projections of domain faces. The affine hull of the relative interior containing any optimizer then projects with full rank. Choose a local affine right inverse through that optimizer, staying in the same relative interior. Its cost minus the optimal value is affine, nonnegative nearby and zero centrally, hence identically zero. This is the required continuation. In the applications below, strict objective bounds maintain strict barrier slack; compactness and continuity then also control limits along nongeneric segments. ◻

Proof. Restrict \(Q^\dagger\) to a valuation \(Q^*\) of \(K_w\). The extension \(L/K_w\) is algebraic, and its enlarged constants contribute \(|g|\) to residue transcendence degree over \(k\). The relative inequality thus shows that \(Q^*\) has Abhyankar equality. The \(g_j\) have value zero and independent initials. Complete them, using functions of \(K_w\), to a list with full independent initials. Their supplementary initials remain independent over \(k'\) at \(Q^\dagger\): the graded-field embedding is injective, and the constant extension is algebraic over \(k(g)\). Expressing the form in \(dg\) wedged with logarithmic supplementary differentials and using (9) gives \[A_{\theta_{w,\mathrm{rel}}}(Q^\dagger) =A_{\theta_w}(Q^*).\] The coefficient is unchanged on dividing by the base wedge; changing \(dg_j\) to \(d\log g_j\) contributes only value-zero factors.

Invariantize \(Q^*\) to \(Q\). It agrees on the homogeneous active entries and preserves zero-value Gaussianity on the old probes \(\overline y\). Equation (49) gives a path \(w_\lambda\) preserving the old Gaussian parameters. No preservation of additional constraints is being required. Let \(\mu\) be the active minimum in (50), including the zero entry if the cap is active. For small \(\lambda>0\), the finitely many test ratios and the cap have minimum \(q(w)+\lambda\mu\). Since \(q(w)>q_{\rm bar}\), this stays above the barrier; Lemma 48 identifies it with \(q(w_\lambda)\) and makes the path feasible for the original optimization. Original objective optimality and (48) prove the inequality.

Equality forces equality in invariantization, hence \(Q^*=Q\), and an exact path of original optimizers \(w_\lambda\). Suppose every angular initial had nonnegative \(Q^\dagger\)-value. Then \(\mu\ge0\), including the possible cap entry. Since the path consists of original optimizers and \(q(w)=q_{\max}\), maximality of \(q\) forces \(\mu=0\). Thus this equality path lies in \(\mathcal M_{\max}\). Some angular value is positive: otherwise all homogeneous initials, obtained as ratios of initials from \(V_0\), have value zero, and the invariant Gauss rule makes \(Q\) trivial. Algebraicity would then make \(Q^\dagger\) trivial as well.

This positive value contradicts the tie rule by differentiation of \(\Psi\). The interchange of sum and derivative is justified as follows. Strict retraction puts every small \(w_\lambda\) on the fixed test skeleton. Monomialize the lex valuation on a model dominating that skeleton. The flattened positive normal weights have fixed centre for small positive \(\lambda\), so their images use one skeleton cone. Values of its finitely many normal equations are affine in \(\lambda\) on a common interval by (49). Their face limit is \(w\), by the pointwise identities and closed-cone continuity. Thus the coordinate distance is \(O(\lambda)\). The summable weighted Lipschitz constants dominate all difference quotients; termwise differentiation follows even though the eventual interval in (49) depends on \(F\). Every derivative is nonnegative and at least one is positive, contradicting maximality of \(\Psi\) on \(\mathcal M_{\max}\). ◻

Proof. Resolve the finite form and active-divisor data on a proper relative model. By strict retraction, the nonzero equality directions in (50) lie on its normal crossing skeleton: retraction preserves the minimum term and strictly lowers discrepancy off the skeleton. A nonzero valuation retracted to zero has strictly positive discrepancy gap and is not an equality direction. Normalize the sum of the nonnegative normal weights to one. The equality locus is closed in the finite compact union of these simplices.

Proposition 55 supplies an open cover of this locus by conditions \(P(\overline U)<0\). Choose a finite subcover. Add field generators: initials from \(V_0\) generate \(K_w\), since \(K=\mathop{\mathrm{Frac}}(V_0)\). Adding the indicated two-element minimum systems with any positive small coefficients makes the inequality strict everywhere, first with coefficients \(\kappa/b_i\).

When a local affine continuation is specified, fix it before choosing these coefficients. It is a continuation by optimizers of the original problem through \(w\); the tie rule is imposed only at \(w\). For each added or active actual function \(F\), fix its assigned degree. On a sufficiently small compact parameter neighbourhood, its absolute values and Lipschitz constant are bounded. Let \(N(F)\) bound the sum of these quantities and its assigned degree. The added terms meet any prescribed positive tolerance \(\eta\) by choosing the \(e_j\) so small that \(\sum_j e_jN(U_j)<\eta\). With no continuation specified, retain just the degree and value bounds needed.

Resolve also the added functions. The strict gap has positive minimum on the normalized enlarged skeleton. Each of the finitely many active evaluations is bounded there, and \[|\min_i a_i-\min_i b_i|\le\max_i|a_i-b_i|.\] Consequently a single sufficiently close rational perturbation of the \(\kappa/b_i\) preserves strictness, including where the minimizing index switches. The cap remains a fixed zero entry. Retraction deals with every remaining direction, including zero retraction. Having chosen the \(e_j\) first, make the rational perturbation still closer so that \(\sum_i|\alpha_i-\kappa/b_i|N(F_i)<\eta\) and the resulting strict margin is retained. Finitely many prescribed tolerances can be met simultaneously by taking \(\eta\) sufficiently small.

After clearing denominators, realize each term \(e_j\min\{0,P(\overline U_j)\}\) by product powers of the two-element system \(\{1,\overline U_j\}\). The ratio \(\overline U_j\), rather than only one of its powers, remains among ratios of the combined system: keep all other factors fixed and replace one \(1\) by \(\overline U_j\). This also supplies its asserted birationality. ◻

Proof. The \(P_0\)-basis expansion has the unique form \[Gf_x^{(P_0-1)/n} =\sum_{0\le\alpha_j<P_0}c_\alpha^{P_0}x^\alpha, \qquad \mathcal C^l_\theta(G)=c_0.\] There is no cancellation between the least-valued terms on the right: the monomials \(\bar x^\alpha\) are linearly independent over \(K_w^{P_0}\). Consequently \[P_0w(c_0)\ge w(G)+\frac{P_0-1}{n}w(f_x).\] The logarithmic form identity (9) says \(A_\theta(w)=-w(f_x)/n\), proving (54). Taking least terms in the same expansion computes \(\bar G\bar f_x^{(P_0-1)/n}\) in the initial \(P_0\)-basis. Its zero-index coefficient is nonzero exactly when the inequality is an equality, and then it is the initial of \(c_0\). The leading-form identity in (9) identifies the fractional form in this calculation with \(\theta_w\). This proves the equality criterion as well as the asserted initial identity. The argument only uses the order, addition of values, and independence over powers, so it also applies to ordered value groups. For nearby positive weights in the same chart the coefficient argument again gives the inequality, with the corresponding numerical discrepancy. ◻

Proof. Choose \(m'\in n\mathbb Z_{\ge0}\) with \(P_0m'+P_0-1\ge M\) and \(m'\le M/P_0+n\). Recall the cone notation \[\Omega=t\,\omega\wedge d\log t,\qquad h=t^nf_h^W,\qquad \theta=\omega/(f_h^W)^{1/n}.\] Since \(\mathfrak p\equiv1\pmod n\), the exponent in \[b=G h^{(P_0m'+P_0-1)/n}\] is integral; \(b\) is regular on \(W\) by \(G\in V_M\) and the displayed inequality. Apply \(b\mapsto\mathcal C^l(b\Omega)/\Omega\). The log-\(t\) formula and Frobenius-linearity give \[\frac{\mathcal C^l(b\Omega)}{\Omega} =h^{m'/n}\mathcal C^l_\theta(G).\] Indeed the \(t\) exponent in \(b\Omega\) is \(P_0(m'+1)\), which trace divides by \(P_0\), and the remaining coefficient is precisely the defining coefficient of \(\mathcal C^l_\theta\). The output is regular in codimension one, because \(\Omega\) is a regular canonical generator there and Cartier trace sends regular top forms to regular top forms. It is therefore regular everywhere by normality. Its physical degree is \(m'\), so division by \(h^{m'/n}\) places the output in \(V_{m'}\). The normal graded algebra of \(W\), its field and its codimension-one canonical generator are retained in the spread. This argument requires no new discrepancy assertion in positive characteristic. ◻

Proof. Write \(\bar u_j'\) for the normal variables of the Taylor initial field; this bar refers to Taylor initial, rather than the original \(w\)-initial. In the logarithmic basis for \(u',s\), write the denominator of \(\theta_w\) as \(f_{u',s}\). Its Taylor initial is \[\operatorname{in}_{\mathrm T}(f_{u',s}) =\prod_j(\bar u_j')^{-n}f_{\mathrm{rel}},\] where \(f_{\mathrm{rel}}\) is the denominator of \(\theta_{w,\mathrm{rel}}\) in the logarithmic basis of the residue coordinates \(s\). This is precisely the effect of first dividing by \(\bigwedge_jdg_j=\bigwedge_jdu_j'\) and only then changing from ordinary to logarithmic normal differentials. The unit coefficient identities are among the retained data at the fiber.

The factors of \(H\) are units at \(\eta\) and restrict to the factors of the nonzero relative trace. If the scalar leading coefficient of \(B\) is \(c\ne0\), the Taylor initial of the coefficient to which Cartier trace is applied is consequently \[c(\bar u')^{P_0a} H|_\eta\, f_{\mathrm{rel}}^{(P_0-1)/n}.\] The normal factor \(\prod_j(\bar u_j')^{-(P_0-1)}\) from the denominator has been canceled exactly by the flags in (55). Extraction in the initial \(P_0\)-basis therefore gives \[c^{1/P_0}(\bar u')^a \mathcal C^l_{\theta_{w,\mathrm{rel}}}(H|_\eta),\] which is nonzero. The ordered-weight version of Lemma 58 identifies it with the Taylor initial of the full trace, proving the asserted order and nonvanishing. ◻

Proof. Models, coordinates, and forms. Descend the finite algebraic data to a finitely generated integral base over \(\mathbb Z\). When geometric generic constants \(k'\) are used, first extend the actual and full initial models by these constants as well, and include the finitely many constants occurring in the constructions in the spread. Spread generators, relations, maps, their inverse identities on birational opens, and rational forms. Shrinking the base retains flatness, geometric integrality, and normality where used. It also retains the smooth charts, chosen transverse integral strata, normal parameters and étale differential or supplementary coordinates. On \(W\) retain the graded algebra, its birational field description, and the fact that \(\Omega\) generates in codimension one: its generating smooth open has complement of codimension at least two, and the same remains true after shrinking. Thus the \(\mathfrak p\)-bases and their initial-field identifications used above coexist with the angular algebra. They may also be imposed together with the finite spreading conditions in the deferred proof of Lemma 57, by a common shrinking of the spread.

Finite initial expressions. Consider a regular numerator or denominator at one of the stratum points. For sufficiently large ordinary normal degree \(N\), successive jets express it modulo the \(N\)-th power of the normal ideal as a finite sum of distinct normal monomials with lifted unit coefficients. Retain their nonzero residue coefficients and the congruence, whose remainder is a combination of degree-\(N\) monomials with regular coefficients. Near fixed positive normal weights, all terms in that remainder have weight greater than the weights under consideration. Consequently the finite expansion determines the value and the initial expression there. Spread this finite expansion and its unit and stratum data. Its minimum and homogeneous initial expression then agree in characteristic zero and in the identified reduced stratum/normal-variable field. Applying this to numerators and denominators handles the specified rational functions. For a finite list it also retains the piecewise leading rules for locally varying positive weights.

Finite initial linear independences are retained by a nonzero evaluation determinant, after adjoining the finitely many points or constants needed to witness it. Finite algebraic independences in characteristic zero are retained by nonzero differential wedges. Include the full initial-field model itself through its stratum/normal-variable identification. This ensures that preservation of the initials is a statement in the full initial field, rather than only a list of numerical equalities.

The geometric generic component and Taylor data. Spread the smooth component chart at \(g=\xi\) as a subvariety of the total initial-field model over the enlarged constants. Retain geometric integrality of the selected component, the field identifications, the local normal equations \(u'=g-\xi\), the differential conditions, and the restrictions of the specified functions and forms to that component. For a specified Taylor leading exponent, the monomial ideal of exponents strictly above it is finitely generated. The required leading expansion is a finite congruence modulo this ideal. A congruence first obtained in the completion contracts to the local ring, by faithful flatness of completion. Thus it too can be spread, together with any required nonzero scalar leading coefficient. Apply this to a numerator and a denominator if necessary. Later shift products only multiply these predetermined leading expressions. Their possibly \(P_0\)-dependent multiplicities cause no new condition on the spread. This retains the relative form, the Taylor flag formula, and the actual-to-initial identification simultaneously.

Entire spaces at finite cutoffs. The angular space \(V_m\) is a finite module over \(k[Z]\). At each of the finitely many tested degrees, choose module generators \(e_{m,t}\) and choose affine algebra generators \(a_1,\ldots,a_s\) of \(k[Z]\) vanishing at \(z\). Both \(v\) and \(w\) are positive on these \(a_j\), by the given barriers and the initial construction. Retain the module generation and these values after constant extension and reduction. There is a common \(\delta>0\) that is a lower bound for all their \(v\)- and \(w\)-values; for \(w\) it remains a lower bound, after a harmless decrease, on a small neighborhood of the chosen positive weights. If \(\mathfrak a=(a_1,\ldots,a_s)\), every element of \(\mathfrak a^NV_m\) has both values at least \[N\delta+\min_t\{v(e_{m,t}),w(e_{m,t})\}.\] Choose \(N\) so this exceeds every relevant upper cutoff. The same estimate holds after reduction, and locally for \(w\). Modulo this tail, the finite span of \(a^\alpha e_{m,t}\) with \(|\alpha|<N\) represents all of \(V_m\).

Choose bases of that finite span adapted separately to the two valuation filtrations. Retain the relations, change-of-basis matrices, values, and linear independence of initials in each same-value slice. The last condition prevents cancellation of a least-valued slice for any nonzero coefficient vector over the extended constants. Therefore the retained finite spans have exactly the same filtration dimensions, gaps, and specified cutoff inclusions. Combined with the tail estimate, these statements hold for every element of \(V_m\) through the specified cutoffs. For the high-\(v\)-cost subspace use its retained finite basis. A lower \(w\)-bound with strict margin persists under a small change of positive weights by continuity of each basis value; the tail remains above the same bound. This proves also the local assertion. Only these finite comparisons are transported; no global optimization or infinite ratio test is asserted in the reduction. ◻

Proof. Choose \(k_I\) so that \(k_I\alpha_i\) and \(k_Ie_j\) are positive integers. On the relative field take the system consisting of \[\bar F_i^{k_I\alpha_i}\prod_j\bar U_j^{l_j}, \qquad 0\le l_j\le k_Ie_j,\] for each active index \(i\); if the cap is active include also the same products with initial active factor \(1\). The minimum of a valuation on this system, divided by \(k_I\), is exactly \[\min_{i\ \mathrm{active}}\alpha_iP(\bar F_i) +\sum_je_j\min\{0,P(\bar U_j)\},\] with zero included in the minimum for the active cap. Its ratios include each \(\bar U_j\), by comparing the exponents \(0\) and \(1\) while holding the other choices fixed. The field-generation provision in (51) therefore makes the system birational. Equation (51) is exactly the hypothesis (52) for this fixed system.

Apply Lemma 57 and choose a surviving product of \(N_I=(P_0-1)/k_I\) generators. If \(N_i\) occurrences use the active index \(i\), put \(\tau_i=N_i/N_I\). Then its \(\bar F_i\) exponent is \(N_ik_I\alpha_i=(P_0-1)\tau_i\alpha_i\), proving (56). A cap occurrence accounts for the possible deficit in \(\sum_i\tau_i\). The bound for the \(\bar U_j\) exponents follows from \(N_Ik_Ie_j=(P_0-1)e_j\).

Lift this product using the same full initial factors and multiply by the Taylor flags of (55), and by any shift prescribed in the application. Each flag expands over the extended constants as a linear combination of \(g_j^a\) with \(0\le a\le P_0-1\). More generally, expand every prescribed shift in its fixed list of homogeneous factors, with the Taylor leading term required by Lemma 60. Lemma 60 supplies full initial-field survival. For every homogeneous summand whose initial trace survives, Lemma 58 produces the actual trace and its initial with equality in (54). Multiplying actual lifts puts the input in \(V_M\), where \(M\) is the sum of their physical degrees, and Lemma 59 gives \(m'\le M/P_0+n\). These surviving initials span the full initial-field output, as explained after Lemma 60.

If, in a later application, every system generator is multiplied by \(\prod_a\bar B_a^{k_I\beta_a}\), with nonnegative rational \(\beta_a\) and integral \(k_I\beta_a\), then every selected product contains \(\prod_a\bar B_a^{(P_0-1)\beta_a}\). These factors also remain in each term after expanding flags and shifts. Selection of a surviving term therefore retains all mandatory factors, including their degree and value contributions.

Finally fix the finite lists of factors and their actual lifts before reduction. Their physical degrees, values at the tested valuations and, when needed, Lipschitz bounds on the given continuation are now known. For any additional shifts, include their separately verified bounds on normalized factor counts among these data. Each multiplicity is then bounded by a fixed constant times \(P_0\). Summing and dividing by \(P_0\), the degree estimate and equality at \(w\) give predetermined output degree and \(w\)-value bounds; the rounding term \(n\) is also fixed. The required \(v\)-filtration cutoffs are chosen before reduction as well. The same accounting is applied separately to every term in the expansions of flags and shifts. Any output spaces, their \(v\)-filtration cutoffs, and high-\(v\)-to-high-\(w\) inclusions used later are retained in their entirety by Lemma 61, before \(\mathfrak p\) and \(P_0\) are chosen. Thus they cover every possible later output coefficient vector, without having to lift that particular vector to characteristic zero.

The normalized contributions of the tiny factors are bounded by the fixed finite data times \(e_j\). Those from replacing \(\kappa/b_i\) by \(\alpha_i\) are bounded by the same finite data times the approximation errors, since the \(\tau_i\) lie in the simplex of (56). As in the construction of (51), first choose the positive \(e_j\) sufficiently small and then approximate sufficiently closely to respect any prescribed finite tolerances. This controls physical degrees, values and Lipschitz changes on the already specified continuation. The strict budget is kept during these choices. Neither uniform comparison for varying pairs nor a global characteristic-\(\mathfrak p\) optimizer is used. ◻

Proof of Lemma 57. The cone and its sub-klt boundary. Introduce an indeterminate \(u\), put \(S=k[Iu]\), and let \(R_c\) be the normalization of \(S\) in its fraction field. The ratios of \(I\) generate \(L\), so \(\mathop{\mathrm{Frac}}(S)=L(u)\). Finite normalization gives a nonnegatively graded normal affine domain \(R_c\) with \((R_c)_0=k\). The conductor is a nonzero homogeneous ideal; multiplying a nonzero homogeneous conductor element by a positive-degree element if necessary, choose \(d_c\) of positive degree with \[0\ne d_c\in R_c,\qquad d_cR_c\subset S.\] Set \[\Theta=\nu\wedge d\log u,\qquad K^\Theta=\mathop{\mathrm{Div}}_{R_c}(\Theta),\qquad \Delta_c=-K^\Theta+\frac1n\mathop{\mathrm{Div}}(f)-\frac1{k_I}\mathop{\mathrm{Div}}(u).\] In particular \(K^\Theta+\Delta_c\) is rational-principal.

We verify that \((\mathop{\mathrm{Spec}}R_c,\Delta_c)\) is sub-klt. An invariant prime valuation \(D\) centered on \(\mathop{\mathrm{Spec}}R_c\) is Gaussian in \(u\) over \(L\). Its restriction to \(L\) is either trivial or a positive multiple of a prime divisorial valuation \(P\). For the latter assertion, the restriction has rational rank one, and the residue-transcendence-degree inequality for \(L(u)/L\), together with Abhyankar equality for \(D\), forces Abhyankar equality for the restriction. We allow the positive multiple in the notation \(P\); (52) is homogeneous and therefore applies to it. The logarithmic differential formula (9) gives \[A_{R_c,\Delta_c}(D)=A_\vartheta(P)+\frac{D(u)}{k_I}.\] Since \(D(Fu)\ge0\) for every \(F\in I\), we have \(D(u)\ge-\min P(I)\), and the displayed discrepancy is positive by (52). If the restriction is trivial, set \(A_\vartheta(0)=0\). Then \(D(u)\ge0\), and equality would make the Gaussian valuation \(D\) trivial; hence again the discrepancy is positive. These tests suffice: an equivariant log resolution of the homogeneous divisor data has invariant divisor components, because a connected torus cannot permute a finite set nontrivially. All the divisors on this resolution consequently have positive log discrepancy, which is the sub-klt criterion.

An effective boundary and multiplier-ideal membership. Choose a homogeneous \(0\ne h_c\in R_c\) such that \[\Delta'=\Delta_c+\mathop{\mathrm{Div}}(h_c)\ge0.\] We record a construction that also makes every coefficient of \(\Delta'\) at most one. Each coefficient of \(\Delta_c\) on \(\mathop{\mathrm{Spec}}R_c\) is strictly less than one. If they are all nonnegative, take \(h_c=1\). In particular, when \(L=k\), the ring is \(k[u]\) and the sole boundary coefficient is \(1-1/k_I\). Otherwise \(\dim R_c\ge2\). At the finitely many negative prime components \(Q_i\) write \(a_i=\lceil-\operatorname{coeff}_{Q_i}\Delta_c\rceil\), and use the homogeneous ideal \[\mathfrak a=\bigcap_i Q_i^{(a_i)}.\] Its minimum order is \(a_i\) at \(Q_i\) and zero at every other height-one prime, as follows by localization. Choose homogeneous generators of \(\mathfrak a\). Multiply each by all products of the degree-one generators \(Iu\) that bring it to one sufficiently large common degree. The resulting finite system retains the same minimum order at every height-one prime. Indeed no such prime contains all of \(Iu\): by the finite extension from \(S\), the inverse image of the vertex has codimension \(\dim R_c\ge2\). A general combination attains the prescribed minimum orders at the finite support of \(\Delta_c\). Away from that support the system has no fixed prime divisor, and Bertini on the smooth basepoint-free open makes any new divisorial zeros reduced. This gives the asserted \(h_c\) and boundary.

Write \(\mathcal J\) for the multiplier ideal, allowing a fractional ideal for a non-effective sub-pair. Sub-klt gives \[R_c\subseteq\mathcal J(R_c,\Delta_c).\] For example, this containment follows on a log resolution by rounding up the relative discrepancy divisor, whose coefficients are greater than \(-1\). Cartier translation therefore yields \[\mathcal J(R_c,\Delta') =h_c\mathcal J(R_c,\Delta_c)\supseteq h_cR_c.\] Only the membership of \(h_c\) is needed. The fractional multiplier ideal of a non-effective sub-klt boundary need not equal the structure sheaf.

Comparison and the full family of Cartier maps. Apply the fixed-pair multiplier/test-ideal comparison to the effective pair \((\mathop{\mathrm{Spec}}R_c,\Delta')\). Its hypotheses hold because \[K^\Theta+\Delta' =\frac1n\mathop{\mathrm{Div}}(f)-\frac1{k_I}\mathop{\mathrm{Div}}(u)+\mathop{\mathrm{Div}}(h_c)\] is rational-principal, with index dividing \(nk_I\); neither klt of \(\Delta'\) nor an independent \(\mathbb Q\)-Gorenstein hypothesis on \(R_c\) is required. The comparison is (Takagi 2004, Theorem 3.2); the dense-open form with the big test ideal is recalled in (Fernex et al. 2015, Theorem 11), applied with \(Z=0\). Thus, on a sufficiently general reduction to a finite residue field of characteristic \(\mathfrak p\equiv1\pmod{nk_I}\), the image of \(h_c\) belongs to the big test ideal \(\tau_b(R_c,\Delta')\).

We use the following characterization: on an \(F\)-finite normal domain the big test ideal of an effective rational divisor pair is the smallest nonzero ideal preserved by all restricted maps \[\mathop{\mathrm{Hom}}_{R_c}\!\left( F_*^eR_c\bigl(\lceil(\mathfrak p^e-1)\Delta'\rceil\bigr),R_c\right), \qquad e\ge1.\] This is (Schwede 2011, Definitions 3.15–3.16 and Theorem 3.18), with the divisor-pair identification in its Remarks 3.10 and 3.17. The log-\(\mathbb Q\)-Cartier comparison with the usual test ideal is also recorded in (Takagi 2004, Theorem 2.8(2)) and (Schwede 2011, Remark 6.1).

In the reduction define \[T^e(b)=\frac{\mathcal C^e(b\Theta)}{\Theta},\qquad L_e=R_c\bigl((1-\mathfrak p^e)(K^\Theta+\Delta')\bigr).\] We use the convention \(R_c(D)=\{b\in\mathop{\mathrm{Frac}}(R_c):\mathop{\mathrm{Div}}(b)+D\ge0\}\), so explicitly \[L_e=f^{(\mathfrak p^e-1)/n}u^{-(\mathfrak p^e-1)/k_I}h_c^{\mathfrak p^e-1}R_c.\] All the allowed maps of degree \(e\) are exactly \(T^e(r\,\cdot)\), \(r\in L_e\). Here every \((\mathfrak p^e-1)\Delta'\) is integral, because \(\mathfrak p-1\) is divisible by \(nk_I\); thus this statement describes the full allowed family in every Frobenius degree, with no rounding or omitted degrees. This is the divisor–Frobenius-map correspondence (Schwede 2009, Corollary 3.10 and Theorems 3.11, 3.13). We check it in the chosen trivialization to fix the normalization. On the fraction field a nonzero Frobenius-linear functional generates all such functionals by input multiplication. At a smooth codimension-one point, Cartier trace for a regular canonical generator in ordinary coordinates extracts the monomial with all exponents \(\mathfrak p^e-1\). Multiplication and trace give the perfect pairing on the Frobenius basis, also after an étale change of coordinates. Changing the canonical generator to \(\Theta\) twists the multiplier by the \((1-\mathfrak p^e)\)-th power of its coefficient. Allowing the poles prescribed by \((\mathfrak p^e-1)\Delta'\) gives precisely \(L_e\) at that height-one point. Normality makes these codimension-one conditions sufficient globally.

A compatible ideal and homogeneous extraction. Set \(T^0=\mathop{\mathrm{id}}\), \(L_0=R_c\), and \[J=\sum_{l\ge0}T^l(d_ch_cL_l)\subset R_c.\] This is an ideal: ordinary multiplication can be moved inside \(T^l\) by \(aT^l(b)=T^l(a^{\mathfrak p^l}b)\). It is nonzero, since its degree-zero Frobenius summand is \(d_ch_cR_c\). It is preserved by every allowed map. Indeed, if \(a_s\in L_s\) and \(a_l\in L_l\), composition gives \[T^s\bigl(a_sT^l(d_ch_ca_l)\bigr) =T^{s+l}\bigl(d_ch_ca_s^{\mathfrak p^l}a_l\bigr),\qquad L_s^{\mathfrak p^l}L_l\subseteq L_{s+l}.\] In particular the conductor factor stays present. The minimality characterization therefore gives \(\tau_b(R_c,\Delta')\subseteq J\). A comparison localized at the homogeneous vertex would already suffice: compatibility localizes, because Hom localizes and denominators move inside trace after taking Frobenius powers.

The ideal \(J\) is homogeneous. Indeed log-\(u\) trace kills a homogeneous input whose degree is not divisible by \(\mathfrak p^l\), and otherwise divides its degree by \(\mathfrak p^l\). Thus local membership of the homogeneous element \(h_c\) also gives global membership: multiply by a vertex unit, and project the result to degree \(\deg h_c\); the unit’s nonzero constant part is the only contribution in that degree. We conclude \(h_c\in J\).

For \(P_0=\mathfrak p^l\) the input factor \(h_cL_l\) contains \(h_c^{P_0}\). Dividing the finite trace identity for \(h_c\in J\) by \(h_c\), using Frobenius-linearity in every summand, gives \[1\in\sum_{l\ge0}T^l\left( f^{(\mathfrak p^l-1)/n}u^{-(\mathfrak p^l-1)/k_I}d_cR_c\right).\] Project to homogeneous degree zero. The term with \(l=0\) contributes nothing, since \(d_c\) has positive degree and \(R_c\) is nonnegatively graded. At least one term with \(l\ge1\) has nonzero degree-zero output. It arises from a homogeneous element \[H u^{(P_0-1)/k_I}\in d_cR_c\subset k[Iu].\] Consequently \(H\in\operatorname{span}(I^{(P_0-1)/k_I})\). The log-\(u\) coefficient formula identifies its nonzero output with (53). Additivity and perfectness of the constants let us choose a single product in that span with nonzero output.

The reduction choices. Spread the finite normalization algebra, its generators and inclusions, its birational field identification, the conductor, the forms, and all the boundary data together. Shrinking the base preserves geometric normality, grading, codimension-one orders, effectivity of \(\Delta'\), and the displayed rational-principal log canonical relation. For the form and divisor identities, spread their rational sections on smooth opens, or on the chosen resolving model, and retain codimension at least two for the complements used in codimension-one tests downstairs. The fixed-pair comparison above then holds on a sufficiently general open of this spread. These requirements are compatible with any finite additional open conditions. A nonempty open of a finite-type integral base of characteristic zero has nonempty fibers at all sufficiently large primes; choose arbitrarily large primes \(\mathfrak p\equiv1\pmod{nk_I}\) and closed points in those fibers. Their residue fields are finite and hence \(F\)-finite. After the trace identity has been obtained, extension to algebraically closed perfect constants preserves it, and preserves its particular nonzero output, by the coefficient formula and faithful scalar extension. This proves the lemma. ◻

Proof. If \(s\) is outside the rational span of the \(t_j\), all \(NL^r\) displayed values are distinct. This contradicts the bound once \(N>A\). Otherwise write \(s=\gamma\cdot t\), uniquely by independence of \(t\). The distinct values are precisely the distinct vectors \(a\gamma+\alpha\). If \(D\) is the order of \(\gamma\) modulo \(\mathbb Z^r\), these vectors occupy \(\min(N,D)\) disjoint lattice cosets, each containing a box of \(L^r\) points. Consequently \(\min(N,D)L^r\le A(NT)^r\le AL^r\); taking \(N>A\) bounds \(D\) by a constant \(D_0\).

Use only \(a=0,D,2D,\ldots,qD\), where \(q=\lfloor(N-1)/D\rfloor\). For a coordinate \(j\), the projections of the corresponding boxes are integer intervals of length \(L\), whose successive displacement is \(D\gamma_j\in\mathbb Z\). Their union has \[L+q\min(L,D|\gamma_j|)\] points. Over each such point, one of the boxes supplies \(L^{r-1}\) points in the remaining coordinates. Hence \(q\min(L,D|\gamma_j|)\le(A-1)L\). Choose \(N_0\) so large, in terms of \(A,D_0\), that \(q>A+1\) and \(qD\ge N/2\). The minimum cannot equal \(L\), and therefore \[|\gamma_j|\le\frac{(A-1)L}{qD}\le C_0T.\] Increasing \(C_0\) gives the asserted norm bound. If \(A<1\), the hypothesis is empty, so this case causes no exception. ◻

Proof. If \(r=0\), take \(w_c=v\) and \(B_0\ge1\); the assertion is immediate. Assume henceforth that \(r>0\). Use (47) with cap \(q_*=1\), barrier \(q_{\rm bar}=1/8\), and budget scalar \(B_*=B_0\). Write \(L(t)=\min\mathcal L\) under the Gaussian constraint \(w(y)=t\). At \(t=v(y)\) the competitor \(v\) has \(A_\theta(v)=0\) and \(q(v)=1\) when \(B_0\ge1\). Since \(A_\theta\ge0\) and \(q\le1\), this gives \(L(v(y))=-\kappa\). We shall move \(t\) along the segment from \(v(y)\) to zero. Work in a fixed bounded open parameter region containing all these segments, and consider the part where \(L(t)<-\kappa/2\). Every optimizer there has \(q>1/2\). Thus it lies strictly above the barrier, and the openness, continuity, and local polyhedral properties of the constrained optimization apply.

Take a full-dimensional cell, a point \(t\) with rationally independent coordinates, and the tiebroken optimizer \(w\) there. Fix its local affine continuation by optimizers with the same active mask. Apply Corollary 56 and Proposition 62 on the geometric generic fibre over the \(w\)-initials \(g=\bar y\), without further Taylor shifts. Select a homogeneous summand with surviving full-initial-field trace and lift its factors to the actual field in a reduction. The continuation is fixed before choosing the tiny exponents and rational approximations to \(\kappa/b_i\). Choose them so that, after division by \(P_0\), their total errors in degree, value, and Lipschitz variation on that continuation are bounded by a fixed small constant. For the resulting nonzero traced output \(F'\in V_{m'}\), put \[\tau=\sum_i\tau_i,\qquad \rho=\sum_i\tau_i\frac{m_i}{b_i}\le C.\] Here the \(\tau_i\) are the active multiplicities in (56); \(\tau\le1\), and \(\tau<1\) is possible only when the cap is active. The degree bound and the equality case of (54) give \[ \begin{split} m'&\le C+\kappa\rho,\\ w(F')&=(1-P_0^{-1}) \bigl(\kappa\tau q-\kappa B_0\rho-A_\theta(w)\bigr)+O(1). \end{split} \tag{57}\] Indeed, every active input satisfies \(w(F_i)=qb_i-B_0m_i\). The Taylor flag factors from (55) have exponents between zero and \(P_0-1\) in the expanded summands. Their normalized degrees and values, and their slopes because \(w_{\widetilde t}(y)=\widetilde t\), are bounded. All formulas used here are among the finite chart and filtration data retained before reduction. The constants in the following counting argument are independent of the large fixed \(B_0,\kappa\).

Set \(T_0=C+1+\kappa\rho\), increasing the constant in (57) if necessary. For a sufficiently large fixed integer \(N\), consider \[(F')^a y^\alpha,\qquad 0\le a<N,\quad 0\le\alpha_j<NT_0,\quad\alpha_j\in\mathbb Z.\] These products lie in a common nested degree \(O(NT_0)\) and have values at most \(C'(\kappa+T_0)N\). The same upper bound, after a fixed enlargement, holds near \(t\) on the chosen continuation. Their number of distinct values is at most \(C''(NT_0)^r\). To justify this last assertion uniformly, before selecting the output transfer the filtration comparisons for every degree up to \(C'''N(1+\kappa)\). At each such degree \(m\), the original high-cost part \(b>S_0\) has \[w>S_0/2-B_0m,\] whereas (43) bounds the dimension of its quotient by \(C_4(1+m)^r\). For fixed \(N,B_0,\kappa\), the right side of the value inequality exceeds all the required upper cutoffs on late members. Retain this implication with slack by the finite-filtration spreading argument, including the zero vector in each high-cost subspace. Its strict margin persists locally on the continuation. The quotient by values above the tested cutoff therefore has the same dimension bound in the reduction. Distinct values below the cutoff give independent classes in this quotient, proving the count.

Lemma 63 now yields \[w(F')=\gamma\cdot t,\qquad \gamma\in\mathbb Q^r,\qquad \operatorname{ord}_{\mathbb Q^r/\mathbb Z^r}(\gamma)\le D_0, \qquad |\gamma|\le C'(1+\kappa\rho).\] The same linear expression holds on a neighborhood of \(t\) on the affine continuation. Indeed, after fixing \(F'\), the same count applies at nearby rationally independent parameters, with the same bounds. There are only finitely many rational vectors of bounded norm and denominator. At \(t\) their evaluations are distinct. Continuity of monomial evaluation therefore excludes all but the central vector on a sufficiently small neighborhood; density of rationally independent parameters extends its formula to that whole neighborhood.

Because \(t\) is bounded, this gives \(w(F')\ge-C'(1+\kappa\rho)\). In (57) use \(A_\theta(w)\ge0\) and \(q,\tau\le1\). Since \(1-P_0^{-1}\ge1/2\), we obtain \[\kappa(B_0-C')\rho\le C''(1+\kappa).\] Thus \(\rho\le C'''/B_0\) once \(B_0\) and then \(\kappa\) are sufficiently large.

We next differentiate the trace inequality, using equality at the central point rather than assuming equality throughout the continuation. Denote its lifted input by \(G\) and its valuations by \(w_{\widetilde t}\). The function \[D(\widetilde t)=w_{\widetilde t}(F') -P_0^{-1}w_{\widetilde t}(G) +(1-P_0^{-1})A_\theta(w_{\widetilde t})\] is nonnegative near \(t\), by (54), and \(D(t)=0\) by the surviving initial trace. Active entries retain \(w_{\widetilde t}(F_i)=b_iq(w_{\widetilde t})-B_0m_i\). Consequently the variation of their normalized contribution is \((1-P_0^{-1})\kappa\,dq\) up to the chosen small Lipschitz error. If \(\tau<1\), the cap is active and \(dq=0\), so the same statement holds. The flag contributions and the remaining errors have a uniformly bounded Lipschitz constant.

The optimizer value \(L\) is affine on this cell and \(w_{\widetilde t}(F')=\gamma\cdot\widetilde t\). Subtracting \(D(t)=0\) from the inequality at \(t+h\) therefore gives \[\gamma\cdot h+(1-P_0^{-1})\nabla L\cdot h\ge-C|h|.\] Using \(-h\) gives the reverse bound. Together with the bounds on \(\gamma\) and \(\rho\), these inequalities imply \[ |\nabla L|\le C_5(1+\kappa/B_0). \tag{58}\] All degree and cutoff bounds used in this deduction are uniform. The individual reductions, Frobenius powers, and approximations may depend on the optimizer; no uniform choice of these is required.

Let \(D_y\) bound the lengths of the segments from \(v(y)\) to zero. Choose first \(B_0\) so that \(D_yC_5/B_0<1/4\), and then \(\kappa\) so that \(D_yC_5<\kappa/4\), with strict room in both inequalities. Integrating (58) keeps \(L\) strictly below \(-\kappa/2\) throughout any portion of the segment reached so far. The bound on generic full-dimensional cells also applies along nongeneric segments by continuity and local polyhedrality. At a first limiting endpoint, compactness of the relevant sublevels supplies a feasible optimizer with the same strict objective bound. Openness then extends the continuation, so the whole segment is reached. At the endpoint \(q\ge1/2\) and \(A_\theta(w_c)=L(0)+\kappa q\le\kappa\), which proves the proposition. ◻

Proof. Suppose that the asserted vanishing fails. Restrict to a subsequence on which \(w_i(F_i)\ne0\), and choose a nonprincipal ultrafilter \(\mathcal U\) on it. Let \(k_{\mathcal U}=\prod_{\mathcal U}k_i\). The nested union \[A=\bigcup_m\prod_{\mathcal U}V_{m,i}\] is a \(k_{\mathcal U}\)-algebra, because the spaces are nested and multiply with addition of degrees. Membership in this union means that the degree is bounded independently of \(i\); it does not permit an unbounded sequence of degrees. Define \[I=\{a=(a_i)\in A: v_i(a_i)\longrightarrow_{\mathcal U}+\infty\}.\] In a fixed degree \(m\), nonnegative cost gives \(v_i(a_i)\ge-m\). This uniform lower bound, the triangle inequality, and additivity on products show that \(I\) is an ideal. It is proper because it does not contain \(1\). It is prime: if neither \(a\) nor \(b\) belongs to \(I\), each has a finite upper valuation bound on a \(\mathcal U\)-large set; their product has such a bound on the intersection of those sets. Thus \(A/I\) is a domain.

Every \(a\in I\) also satisfies \(w_i(a_i)\to_{\mathcal U}+\infty\). This implication uses the gap, not just slow growth of the cutoffs. Indeed, represent \(a\) in one fixed degree \(m\). Divergence gives \(b_i(a_i)>C_4m\) on a \(\mathcal U\)-large set. The gap excludes all costs between this bound and \(S_i\), hence \(b_i(a_i)>S_i\) there. The last hypothesis then gives \(w_i(a_i)>R_i\to\infty\).

The image in \(A/I\) of degree \(m\) has dimension at most \(C_4(1+m)^r\) over \(k_{\mathcal U}\). To see this, take more than this many elements and representatives in \(V_{m,i}\). The finite quotient-dimension bound gives, at each sufficiently large \(i\), a nontrivial linear combination in the high-cost subspace. There are finitely many coefficients; on a \(\mathcal U\)-large set one fixed coefficient can be normalized to \(1\). The resulting coefficient sequences define constants in \(k_{\mathcal U}\), and the combination has \(v_i\to_{\mathcal U}\infty\). It therefore gives a nontrivial dependence modulo \(I\).

Let \(\widehat y_j\) denote the classes of the probes in \(A/I\). They are algebraically independent over \(k_{\mathcal U}\). For a nonzero polynomial with coefficients in \(k_{\mathcal U}\), choose representatives of its finitely many nonzero coefficients. On a \(\mathcal U\)-large set they remain nonzero. Gaussianity at \(v_i\) evaluates that polynomial by the minimum of finitely many fixed monomial weights in the \(v_i(y_{j,i})\). Those weights are uniformly bounded, so the polynomial does not belong to \(I\).

It follows that \(\mathop{\mathrm{Frac}}(A/I)\) is algebraic over \(k_{\mathcal U}(\widehat y_1,\ldots,\widehat y_r)\). Otherwise some element of \(A/I\) would be transcendental over that field. This element and the \(r\) probes lie in some fixed nested degree. Their monomials with every exponent at most \(N\) would be linearly independent, giving \((N+1)^{r+1}\) independent elements in a degree \(O(N)\), contrary to the established \(O(N^r)\) bound. This also proves the assertion when \(r=0\).

The bounded degree and bounded \(v_i\)-value of \(F_i\) show that its class \(\widehat F\) in \(A/I\) is nonzero. Its minimal polynomial over the cheap rational field therefore has nonzero leading and constant coefficients. Clearing denominators yields \[\sum_{a=0}^{d_0}h_a(\widehat y)\widehat F^{\,a}=0, \qquad h_a\in k_{\mathcal U}[Y_1,\ldots,Y_r], \qquad h_0h_{d_0}\ne0.\] Choose representatives of all its finitely many scalar coefficients and lift this relation to \(A\). Its residual belongs to \(I\), so its \(w_i\)-value tends to \(+\infty\) along \(\mathcal U\). For every nonzero \(h_a\), Gaussianity on \(y_i\) over the whole constant field \(k_i\), together with \(w_i(y_i)=0\), gives \(w_i(h_{a,i}(y_i))=0\) on a \(\mathcal U\)-large set. Intersect these finitely many sets. If \(w_i(F_i)>0\), the constant term is the unique least term of the lifted relation, of value zero. If \(w_i(F_i)<0\), its highest-degree term is uniquely least, of negative value. In either case the residual has value at most zero, a contradiction. No positive lower bound on \(|w_i(F_i)|\) is used.

The relation was obtained after forming the ultraproduct, but this does not reverse the order of the choices. Its supports, its coefficients after denominator clearing, and all its products have some fixed finite nested degree \(m_0\). The hypotheses were retained for every \(m\le M_i\), with \(M_i\to\infty\), so they already apply to \(m_0\) on a tail. The proof does not require spreading this newly chosen relation. The contradiction on any contrary subsequence proves ordinary eventual vanishing, as asserted. ◻

Proof. The lower bound follows from (59) and \(m\le Cb(F)\). Degree-zero units have zero value at \(w\) by centering, so they also satisfy the assertion. To prove the upper bound uniformly, consider an arbitrary sequence of tests \(w,F\) with \(b=b(F)>0\). We may pass to subsequences throughout. A contradiction for every sequence violating one uniform upper bound gives the stated uniform conclusion.

Return to the leading construction at \(v\), and let \(\bar F\) be the \(v\)-initial of \(F\), of charge \(\beta\in M_0\). The occurring weight comparisons give \(\|\beta\|\le Cb\), and positive costs are bounded away from zero. By (37), choose an actual angular \(G\) whose initial has charge \(-d'\beta\), for a positive integer \(d'\), and whose degree \(m_G\) and cost are at most \(Cd'b\). Thus \[u=\bar F^{\,d'}\bar G\in K_1,\qquad \operatorname{div}_{S_1}(u)+MH_1\ge0,\qquad M=d'm+m_G\le Cd'b,\] by purity and (12). Here \(M>0\): if \(M=0\), both degrees vanish and the opposite proportional charges have opposite proportional costs, one strictly positive, contradicting nonnegative degree-zero costs.

Work on a geometric generic fibre field of \(K_v\) over the cheap initial probes \(\bar y\). For a prime-divisor valuation \(Q\) there, over the geometric generic constants, restrict first to \(K_v\) and then put \(P=Q|_{K_1}\). The algebraic-extension and relative Abhyankar argument of (50) shows that \(P\) is trivial or prime-divisorial up to scaling. In particular, restriction preserves Abhyankar equality, and a nontrivial subgroup of the discrete value group remains discrete. Moreover, \(P\) is trivial on \(K_f\): the cheap residual probes contain a transcendence basis of \(K_f\), and the remaining extension is algebraic. We have \[ \frac{P(u)}{d'} \le Cb\bigl(H_1(P)+s_{\exp}A_1^+(P)\bigr), \qquad A_{\theta_{v,\mathrm{rel}}}(Q)\ge A_1^+(P). \tag{61}\] The first inequality is (35) applied to the effective pure divisor just constructed. For the second, first apply the relative-form comparison in (50), then invariantize the restriction to \(K_v=K_1({\bf z}_0)\) by (48). The form is \[\theta_v=\pm\frac{\omega_S\wedge d\log{\bf z}_0} {f_h^{1/n}},\] so the Gauss extension of \(P\) has discrepancy exactly \(A_1^+(P)\). Equality in the second inequality of (61) therefore requires the restriction to \(K_v\) itself to be Gaussian over \(P\) in these torus variables.

Fix a large \(\ell\) with \(1/\ell\le\delta\), before taking the sequence limit. We now construct a strict trace budget in which the factors \(\bar F,\bar G\) are mandatory. Give them respective rational exponents \[u_0\in[\ell/b,2\ell/b],\qquad u_0/d'.\] Choose a sufficiently divisible \(D\in n\mathbb Z_{>0}\) for which \(DH_1\) is free Cartier. A basis of this complete pure angular system has minimum order \(-DH_1(P)\) at every \(P\), by freeness and (12). Lift it by charge-zero angular functions and give its minimum a rational exponent in \([K_0\ell/D,2K_0\ell/D]\), where \(K_0\) is a sufficiently large fixed constant. The mandatory contribution equals \(u_0P(u)/d'\) and is at most \(C\ell(H_1(P)+s_{\exp}A_1^+(P))\). The pure system subtracts at least \(K_0\ell H_1(P)\).

For \(P\ne0\), choose \(K_0>C\) and then take a sufficiently late member so that \(C\ell s_{\exp}<1/2\). Since \(H_1(P),A_1^+(P)\) are nonnegative and their sum is positive, these contributions have sum strictly less than \(A_1^+(P)\), and hence less than \(A_{\theta_{v,\mathrm{rel}}}(Q)\). For \(P=0\), both contributions vanish. Failure of strictness would force equality in (61), so the restriction to \(K_v\) would be a Gauss valuation over the trivial valuation of \(K_1\). That restriction is nontrivial, by the algebraic-extension argument in (50); hence some torus basis weight is nonzero. Add finite minimum systems consisting of \(1\) and angular initials of both signs of the basis charges, allowing multiples as supplied by (37). One of their minima is negative in each such direction. Give each an arbitrarily tiny positive rational exponent. Finally add systems \((1,\bar U)\) for angular initials generating the relative field, again with arbitrarily tiny positive rational exponents, as in (51). These extra minima are nonpositive everywhere, make the budget strict also when \(P=0\), and retain birationality through their ratios.

Clear all rational denominators. The resulting finite system \(I\) and integer \(k_I\) satisfy (52). Every member of \(I\) contains the mandatory factors, while products of the finite minimum systems realize exactly the specified remaining minimum orders. Because all terms have the same mandatory factor, retaining the field-generating ratios is unaffected by that factor. Choose the tiny exponents to make their total normalized degree and absolute initial-value contributions bounded by a fixed constant. These choices may depend on the individual test.

Before invoking trace existence, also choose the following finite data in its reductions. Along the sequence take slow cutoffs \[M_i\longrightarrow\infty,\qquad M_i\ll S_i'\le S_{0,i},\qquad S_i'\longrightarrow\infty, \qquad R_i'\longrightarrow\infty.\] For every degree \(m\le M_i\), retain nonnegative costs, the exclusion of all costs in \(C_4m<b\le S_i'\), and the quotient-dimension bound \(C_4(1+m)^r\) from (43). Retain also \[b>S_i'\quad\Longrightarrow\quad w>R_i'.\] This is possible by (59): its lower bound is \(c\delta S_i'-\delta B_0M_i\) on these high parts, and we choose \(R_i'\) below this bound with slack. Here \(\delta\) is fixed and \(S_i'\) dominates \(M_i\). The finite-filtration comparisons from the preceding section retain these statements for whole spaces over the enlarged reduced constant field, using adapted bases and high module tails. There are only finitely many degrees and cutoffs at each index. Slowing the diagonal choices as necessary allows all of them to be retained before choosing \(P_0\) or a traced output.

We also retain Gaussianity on \(y\) at both valuations over the entire new constant field, with bounded \(v(y)\) and \(w(y)=0\). This does not follow merely from preserving finitely many polynomial tests. On each of the two monomial charts, write the probe initials as rational functions of the stratum coordinates and independent normal variables. In characteristic zero these initials are algebraically independent. Retain their finite expressions and a nonzero differential minor witnessing this independence, together with smooth chart data and separating residue coordinates. A sufficiently general large-characteristic reduction preserves that minor and those separating conditions. Over the perfect extended constants the reduced initials are therefore still algebraically independent. This proves the Gaussian rule for every polynomial over those constants, including coefficients chosen later and varying with \(i\). The actual charts at \(v,w\), the form identities, the prescribed input functions, and their needed values are spread simultaneously with these data.

Now apply (53) and (55) at \(v\). A surviving full-initial-field trace, followed by homogeneous expansion and selection of a surviving term, lifts to an actual nonzero traced output \(F'\) in the reduction. All its input degrees and absolute \(v\)-values, after division by \(P_0\), are at most \(C\ell\). Indeed, \(m,|v(F)|\le Cb\) and \(m_G,|v(G)|\le Cd'b\); multiplication by \(u_0\) and \(u_0/d'\) respectively cancels these costs. The pure system has degree \(D\) and initial value zero, while its exponent times \(D\) is at most \(2K_0\ell\). The tiny systems were chosen with bounded normalized contributions, and the Taylor flags contribute bounded amounts. The degree bound for actual trace and the equality case of (54), using \(A_\theta(v)=0\), therefore give \[m'\le C\ell+n, \qquad |v(F')|\le C\ell.\] These are bounds before knowing which surviving term is selected; no bound on the Frobenius iterate is needed.

Apply Lemma 65 to these reduced systems, with \(S_i=S_i'\) and \(R_i=R_i'\). Its hypotheses were retained before trace existence, and \(\ell\) is fixed, so the output degree and initial value are uniformly bounded. It follows that \[w(F')=0\] eventually. In particular, this is an exact statement even if a hypothetical sequence of nonzero output values were tending to zero. The diagonal degree tests and full Gaussianity are precisely what allow the lemma to handle the polynomial relation chosen later.

Finally apply only the lower inequality of (54) at \(w\) to the product traced to this \(F'\). The mandatory input power of \(F\) is exactly \(F^{(P_0-1)u_0}\), by denominator clearing and the chosen characteristic congruence. Every remaining factor has value at least \(-\delta B_0\) times its degree by (59) and the retained comparisons; constants have value zero. Their total degree divided by \(P_0\) is at most \(C\ell\). The chart calculation also retains the value of \(F\) and \(A_\theta(w)\le1\). Hence \[0=w(F')\ge(1-P_0^{-1})\bigl(u_0w(F)-1\bigr) -C\delta B_0\ell.\] Since \(1-P_0^{-1}\ge1/2\), \(u_0\ge\ell/b\), and \(1/\ell\le\delta\), this yields \[w(F)\le b\bigl(\ell^{-1}+2C\delta B_0\bigr) \le C_6\delta b.\] The constants are independent of small fixed \(\delta\) and of the chosen sequence of tests. This proves the upper bound and completes (60). ◻