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Math-25 · mathematics · Algebraic geometry

Abundance Conjecture for the minimal model program

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

For a projective klt pair in the source's minimal-model setting, put $D=K_X+\Delta$. A divisor is nef if $D\cdot C\ge0$ for every curve $C$. In the usual rational-divisor formulation it is semiample if for some integer $m>0$, $mD$ is Cartier and its line bundle is generated by global sections. The conjecture is \[ D\text{ nef}\quad\Longrightarrow\quad \exists m>0:\mathcal O_X(mD)\text{ globally generated}. \] The rational-boundary convention is the standard one for this displayed statement; a real-boundary version requires the source's corresponding notion of semiampleness. No bigness hypothesis is added.

Formulation and concept references: [S1].

Short English statement

Must a nef canonical-plus-boundary divisor on a minimal klt variety eventually have enough sections to define a morphism?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Math-25-A2 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-25

Manuscript 1 (family 034, 2026-09-27): The main theorem makes a nef klt adjoint semiample only for compact Kähler fourfolds already assumed to have nonnegative Iitaka dimension; the catalogue omits both restrictions.

Manuscript 2 (family 034, 2026-10-05): The fourfold good-model and nef-endpoint nonvanishing theorems explicitly assume Campana, MMP and effective-abundance inputs and strong Q-factoriality; neither settles the unrestricted unconditional catalogue statement.

Manuscript 3 (family 034, 2026-09-27): The abundance-after-nonvanishing theorem covers projective lc pairs in dimension at most four with nonnegative Iitaka dimension; this problem has neither the dimension bound nor an assumed first section.

Manuscript 4 (family 034, 2026-10-04): The main Kähler theorem explicitly assumes logarithmic Iitaka subadditivity; it gives semiampleness and hence nonvanishing only for nef adjoints under that additional assumption, whereas our catalogue assertions are unconditional.

Manuscript 5 (family 034, 2026-09-24): The log-abundance main theorem proves semiampleness of every nef rational lc adjoint over an algebraically closed characteristic-zero field, strictly extending the catalogue projective klt rational-boundary target.

Manuscript 6 (family 034, 2026-09-27): Zero-Lelong minimal metrics together with the stated fourfold semiampleness interface cover nef complex klt fourfold adjoints after nonvanishing; the theorem is not abundance in every dimension or nonvanishing for all pseudo-effective lc pairs.

Manuscript 7 (family 034, 2026-09-26): The effective Iitaka system concerns dimension-four lc adjoints and describes their section field; even with a nef adjoint this does not cover the catalogue semiampleness assertion in every dimension.

Manuscript 8 (family 036, 2026-09-24): The good-minimal-model corollary treats the numerical-dimension-one smooth canonical case, rather than all projective klt rational adjoints.

Manuscript 9 (family 036, 2026-10-03): The theorem yields a semiample divisor numerically equivalent to the nef adjoint; our formulation requires a generated Cartier multiple of the actual adjoint, not a numerical replacement.

Manuscript 10 (family 036, 2026-10-04): A semiample line with the same real Bott–Chern class is supplied for smooth Kähler rational klt data, but numerical equivalence does not make the actual adjoint line semiample as this problem requires.

Abundance Conjecture for the minimal model program

OpenAI's claim

Manuscript 1 (family 034, 2026-09-27): A nef rational adjoint on a compact Kähler klt fourfold is claimed to be semiample once a nonzero section exists, including triviality of a multiple in Iitaka dimension zero; Manuscript 2 (family 034, 2026-10-05): Under explicit orbifold-subadditivity, fourfold-program, and abundance assumptions, good minimal models are claimed for the specified compact Kähler klt fourfold pairs; Manuscript 3 (family 034, 2026-09-27): An effective adjoint supported on the coefficient-one boundary is claimed to acquire positive Iitaka dimension when its restriction is semiample on the entire reduced support, leading to fourfold abundance after nonvanishing; Manuscript 4 (family 034, 2026-10-04): Assuming the stated logarithmic Iitaka inequality, OpenAI claims semiampleness of nef rational log adjoints on compact Kähler log-canonical spaces in arbitrary dimension. Manuscript 5 (family 034, 2026-09-24): In characteristic zero, nef rational log adjoints of projective log-canonical pairs are claimed to be semiample in every dimension; smooth canonical nonvanishing is another stated consequence; Manuscript 6 (family 034, 2026-09-27): Minimal semipositive metrics of nef klt adjoints are claimed to have no positive Lelong numbers, and zero-Lelong endpoint metrics are claimed to give the stated interior-boundary $H^1$ injection; Manuscript 7 (family 034, 2026-09-26): For log-canonical fourfolds with coefficients from a fixed finite rational set, a uniform rounded adjoint system is claimed to generate the full Iitaka field; a uniform klt Calabi--Yau canonical index is also claimed; Manuscript 8 (family 036, 2026-09-24): A smooth complex projective variety with canonical section-growth numerical dimension one is claimed to admit a terminal factorial minimal model, with good-model consequences using the separate abundance input. Manuscript 9 (family 036, 2026-10-03): For projective characteristic-zero klt pairs, a nef ordinary adjoint plus a nef rational divisor is claimed to be numerically equivalent to a semiample divisor whenever the ordinary adjoint is pseudo-effective; Manuscript 10 (family 036, 2026-10-04): On the stated compact Kähler pairs, a nef adjoint with an additional nef rational line is claimed to have a semiample representative of its Bott--Chern class, allowing a flat change of bundle.

Mathematical statement of the claimed result

Manuscript 1.
Let $X$ be a compact complex space, required to be Kähler, normal and connected, of dimension four and let $\Delta\ge0$ be rational, with $(X,\Delta)$ Kawamata log terminal and actual adjoint $D=K_X+\Delta$ rational Cartier. Assume $D$ is analytically nef: after fixing a Cartier multiple and a Kähler form, its curvature has smooth representatives bounded below by $-\varepsilon$ times that form for every $\varepsilon>0$. Assume also $\kappa(X,D)\ge0$, meaning a nontrivial section exists for at least one positive Cartier multiple. OpenAI claims \[ \exists m>0:\ mD\text{ Cartier},\qquad H^0(X,\mathcal O_X(mD))\otimes\mathcal O_X\twoheadrightarrow\mathcal O_X(mD). \] If $\kappa(X,D)=0$, the multiple may be chosen with $\mathcal O_X(mD)\simeq\mathcal O_X$. This theorem requires nonvanishing as a hypothesis.
Manuscript 2.
This is explicitly conditional on three global premises: $1$ orbifold Iitaka subadditivity in all dimensions for smooth compact manifolds in Fujiki class $\mathcal C$, namely $\kappa(X,K_X+\Delta)\ge\kappa(F,K_F+\Delta|_F)+\kappa(f\mid\Delta)$ for connected-fiber holomorphic fibrations with rational simple-normal-crossing boundary; $\kappa(f\mid\Delta)$ is the Kodaira dimension of the neat orbifold base, using infimum fiber multiplicities; $2$ existence and termination of every negative-extremal-ray minimal model program for pseudo-effective rational klt adjoints on globally strongly rational-factorial compact Kähler fourfolds, with projective divisorial contractions or log-canonical-positive flips and a nef endpoint; $3$ semiampleness of a nef rational klt fourfold adjoint already having nonnegative Iitaka dimension. Given these premises, take a normal connected compact Kähler fourfold $X$ with effective rational $B$, klt pair $(X,B)$, and rational Cartier analytically pseudo-effective $D=K_X+B$. Assume every global rank-one coherent reflexive sheaf has an invertible positive reflexive power. OpenAI claims a bimeromorphic contraction $\phi:X\dashrightarrow Y$ to a fourfold of the same category such that, writing $B_Y=\phi_*B$, \[ D_Y=K_Y+B_Y\text{ is nef and semiample},\qquad a(E;Y,B_Y)\ge a(E;X,B) \] for every prime divisor over the models, strictly for every prime on $X$ contracted by $\phi$. The map extracts no divisor. The three premises are part of the theorem, rather than conclusions proved without assumptions.
Manuscript 3.
Let $(V,C)$ be a normal projective rational-factorial divisorial-log-terminal pair over a characteristic-zero field assumed algebraically closed, with rational effective boundary $C$. Put $A=K_V+C$. Suppose a nonzero effective Cartier divisor $G$ satisfies $G\sim qA$ for a sufficiently divisible integer $q>0$, its support lies in $\operatorname{Supp}\lfloor C\rfloor$, and its line bundle restricted to the whole reduced divisor $G_{\mathrm{red}}$ is semiample. OpenAI claims \[ \kappa(V,A)>0. \] Another theorem asserts that for complex projective pairs $(X,\Delta)$ that are log canonical and have dimension at most four with effective rational boundary and nef rational Cartier $D=K_X+\Delta$, nonvanishing $\kappa(X,D)\ge0$ implies semiampleness, and $\kappa(X,D)=0$ implies $D\sim_{\mathbb Q}0$. The all-dimensional version is conditional: if every smooth connected projective complex $W$ with pseudo-effective $K_W$ has some $H^0(W,mK_W)\ne0$, then every nef rational log canonical adjoint of a projective pair in characteristic zero is semiample.
Manuscript 4.
Assume logarithmic Iitaka subadditivity for every connected-fiber surjective morphism $f:U\to V$ of smooth connected projective complex varieties with reduced simple-normal-crossing divisors $D_U,D_V$ satisfying $\operatorname{Supp}(f^*D_V)\subseteq\operatorname{Supp}D_U$: for a very general smooth fiber $F$ and $D_F=D_U|_F$, \[ \kappa(U,K_U+D_U)\ge\kappa(F,K_F+D_F)+\kappa(V,K_V+D_V). \] Under that premise, OpenAI claims that every normal irreducible compact Kähler complex space $X$ with effective rational $\Delta$, log canonical pair $(X,\Delta)$ and analytically nef rational Cartier $D=K_X+\Delta$ has a positive Cartier multiple generated everywhere by its global sections: \[ H^0(X,\mathcal O_X(mD))\otimes\mathcal O_X\twoheadrightarrow\mathcal O_X(mD). \] The premise covers all dimensions, uses $\kappa=-\infty$ for empty positive systems, and is a hypothesis of this Kähler result.
Manuscript 5.
Let $k$ be algebraically closed of characteristic zero and $(X,B)$ any normal projective pair over $k$, log canonical and with rational effective boundary. For the actual rational Cartier adjoint $D=K_X+B$, OpenAI asserts \[ D\cdot C\ge0\text{ for every curve }C\subset X \quad\Longrightarrow\quad \exists m>0:\ mD\text{ Cartier and }\mathcal O_X(mD)\text{ globally generated}. \] There is no dimension bound and no nonvanishing premise in this main theorem.
Manuscript 6.
For every normal connected projective complex klt pair $(H,\Theta)$ with effective rational $\Theta$ and nef rational Cartier $D_H=K_H+\Theta$, and every projective log resolution $\pi:V\to H$, OpenAI claims that $N=\pi^*D_H$ admits a semipositive singular Hermitian metric of minimal singularities and \[ \nu(h,x)=0\quad\text{for every }x\in V \] for every such metric $h$. Here $\nu$ is the Lelong number; minimal singularities means its weights are no more singular, up to bounded additive error, than the weights of any other semipositive metric on $N$. The components $K_H$ and $\Theta$ need not be separately rational Cartier.
Manuscript 7.
Fix a finite rational set $\Phi\subset[0,1]$. OpenAI claims an integer $m=m(\Phi)>0$ working for every normal integral projective complex fourfold $X$ with effective rational $\Delta$ whose nonzero coefficients lie in $\Phi$, log canonical pair $(X,\Delta)$, and pseudo-effective rational Cartier $D=K_X+\Delta$. Namely, \[ |\lfloor mD\rfloor|\ne\varnothing,\qquad \mathbb C\left(s/t:\ 0\ne s,t\in H^0(X,\mathcal O_X(\lfloor mD\rfloor))\right)=K(D)\subset\mathbb C(X). \] Here $K(D)$ is the field generated by ratios from all positive reflexive systems $H^0(X,\mathcal O_X(\lfloor jD\rfloor))$. Equality of fields asserts the Iitaka fibration itself, rather than only the dimension of its image. The integer is independent of the chosen canonical divisor.
Manuscript 8.
Let $X$ be smooth connected projective over $\mathbb C$, $n=\dim X\ge3$, and suppose $K_X$ is pseudo-effective. Define the section-growth numerical dimension by \[ \kappa_\sigma(X,D)=\max\{k\in\{0,\ldots,n\}:\ \exists\text{ ample Cartier }A,\ \limsup_{m\to\infty}h^0(X,mD+A)/m^k>0\}. \] Assume $\kappa_\sigma(X,K_X)=1$. OpenAI claims finitely many divisorial contractions or flips, each $K$-negative to a normal projective rational-factorial terminal $Y$ with $K_Y$ nef. The inverse map contracts no divisor, and on a common smooth resolution $p:W\to X$, $q:W\to Y$, \[ p^*K_X=q^*K_Y+E,\qquad E\ge0,\quad q_*E=0, \] with the strict transforms of all contracted prime divisors in $\operatorname{Supp}E$. A separate corollary, using the companion log-abundance theorem, claims $K_Y$ semiample. The invariant retains the fixed ample twist $A$; it is not an assumed growth bound for untwisted pluricanonical sections.
Manuscript 9.
For a rational projective pair with klt singularities $(X,B)$ over a characteristic-zero field assumed algebraically closed, assume $K_X+B$ pseudo-effective and $M$ nef rational Cartier. If $D=K_X+B+M$ is nef, OpenAI claims \[ \exists\text{ semiample rational Cartier }L\text{ on }X:\quad D\equiv L. \] Numerical equivalence is equality of degrees on every integral curve. Semiampleness is claimed for a numerically equivalent replacement, not necessarily for the original actual divisor $D$.
Manuscript 10.
Let $X$ be smooth connected compact Kähler, $B\ge0$ rational simple-normal-crossing with coefficients below one, and $M\in\operatorname{Pic}(X)\otimes\mathbb Q$. Assume pseudo-effectivity of $K_X+B$, and nefness of both $M$ and $D=K_X+B+M$, OpenAI claims \[ \exists\text{ semiample rational line bundle }L:\quad c_1(L)=c_1(D)\text{ in }H^{1,1}_{\mathrm{BC}}(X,\mathbb R). \] Nef means the Chern class lies in the closed Kähler cone; pseudo-effective means it contains a positive current. Semiampleness means an actual positive integral multiple is globally generated. Equality is of Bott--Chern classes, not of line bundles.

Outline of the argument

Manuscript 1 (family 034, 2026-09-27): Contractions of threefold null loci and analytic thickenings prepare the boundary components for a fourfold birational reduction. The comparison on floors and lower strata preserves the actual holomorphic line identity, allowing boundary-generated sections to descend to the nef adjoint. Manuscript 2 (family 034, 2026-10-05): Nonvanishing is obtained by fibrations when they exist and by minimal singular metrics in algebraic dimension zero. Spherical transverse structures, reduced-boundary models, gluing of pluricanonical sections, and multiplier-ideal extension address the remaining zero-dimensional-algebraic case before returning to the assumed nef endpoint. Manuscript 3 (family 034, 2026-09-27): Finite-cover invariants lift successive neighborhood layers into divisorial sheaves. Their positive twists provide increasing section counts while the cohomological loss stays bounded, forcing growth without assuming convergence of an infinite formal neighborhood. Manuscript 4 (family 034, 2026-10-04): The induction retains both the nef line and the effective negative part through a restricted birational program. Boundary generation, signed-support root constructions, Hodge-module lifting, and compact cycle deformations produce nonvanishing; affine changes in negative multiplicities make the selected wall crossings finite.

Manuscript 5 (family 034, 2026-09-24): A dimension induction establishes nonvanishing before constructing good models in that same dimension. Cyclic-cover eigenspaces and logarithmic Hodge modules control the boundary lifting obstruction, while the companion subadditivity result supplies the fibration step; descent of the evaluation map recovers the actual line bundle over the original field. Manuscript 6 (family 034, 2026-09-27): An extremal normalized-section argument contradicts a positive Lelong number by improving the minimal weight locally. For injectivity, harmonic forms and local primitives are compared with both endpoint weights; weak convergence in a fixed Hilbert space retains the ordinary coherent-cohomology class. Manuscript 7 (family 034, 2026-09-26): Marked curve chains control degrees of rational quotient leaves with compatibility under projections and finite quotients. Scalar bounds and Frobenius estimates handle the index step; a semiample contraction and the relative big-base theorem then bring uniform floor systems back to the original pair. Manuscript 8 (family 036, 2026-09-24): Positive-boundary programs first reduce the possible exceptional loci. Algebraic codimension-two cycle classes give a finite homological decrease after discrepancy control; a final surface criterion eliminates the remaining obstruction without assuming general minimal-model existence.

Manuscript 9 (family 036, 2026-10-03): Ordinary log abundance and terminating programs supply the birational reductions. The remaining full-nef-dimension Calabi--Yau case is treated with jet and Frobenius estimates, after which the positive part descends as a semiample numerical representative; flat nontorsion bundles explain why numerical equivalence is the conclusion. Manuscript 10 (family 036, 2026-10-04): The induction uses projective positive-part results when algebraic reduction has positive-dimensional image. In algebraic dimension zero, decomposition into torus and generically symplectic pieces, extension estimates, and compact cycle spaces provide the numerical representative while preserving the distinction from an actual line identity.

Formal verification

Family 034 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

Family 036 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

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Math-25-A1 · No claim recorded · LLM-generated · gpt 6 astra pro ·

gpt 6 astra pro attempt on Math-25

Abundance Conjecture for the minimal model program

Definitions and mathematical statement

For a projective klt pair in the source's minimal-model setting, put $D=K_X+\Delta$. A divisor is nef if $D\cdot C\ge0$ for every curve $C$. In the usual rational-divisor formulation it is semiample if for some integer $m>0$, $mD$ is Cartier and its line bundle is generated by global sections. The conjecture is \[ D\text{ nef}\quad\Longrightarrow\quad \exists m>0:\mathcal O_X(mD)\text{ globally generated}. \] The rational-boundary convention is the standard one for this displayed statement; a real-boundary version requires the source's corresponding notion of semiampleness. No bigness hypothesis is added.

Formulation and concept references: [S1].

Short English statement

Must a nef canonical-plus-boundary divisor on a minimal klt variety eventually have enough sections to define a morphism?

Research attempt

Current frontier.
Lazić studies criteria involving metrics with minimal singularities and multiplier ideals, including an equivalence under a nonzero Euler characteristic hypothesis; these additional assumptions are not universal abundance [E1]. Results about anti-canonical nonvanishing in [S4] concern a different divisor. Here the attempt uses the rational-boundary convention of the displayed target and makes no bigness assumption. The aim is to isolate the difference between a nef numerical class, nonzero actual sections, and a section ring whose base ideals ultimately disappear.
Attack route.
A metric approach tries to remove singularities, but even a smooth flat metric can coexist with no sections. A finite-generation approach could turn asymptotic vanishing of actual base orders into semiampleness, provided nonvanishing has first been secured. The latter route is pursued by an explicit base-ideal calculation. The missing geometric steps are then separated from an invalid replacement of actual base orders by numerical orders after an ample perturbation.
Attempt: a base-ideal criterion with all hypotheses exposed.
Let $L$ be a Cartier divisor on a normal projective variety $X$. Assume there is a positive integer $m_0$ such that $H^0(X,m_0L)\ne0$ and the Veronese section ring \[ R^{(m_0)}(L)=\bigoplus_{k\geq0}H^0(X,km_0L) \] is generated by its degree-one piece. Finite generation of the section ring, together with positive-degree nonvanishing, permits such a sufficiently divisible Veronese; one can alternatively retain generation in degree one as the explicit sufficient hypothesis. Let $\mathfrak b$ be the base ideal of $|m_0L|$, defined by the image of its evaluation map after tensoring by $\mathcal O_X(-m_0L)$.

Surjectivity of the multiplication maps from the symmetric powers of $H^0(X,m_0L)$ gives the exact equality of ideal sheaves \begin{equation}\label{wsref-0-65-71-37-38-3a-70-6f-77-65-72-73} \mathfrak b_{|km_0L|}=\mathfrak b^k\qquad(k\geq1). \end{equation} Locally trivialize $m_0L$. Products of $k$ local generators of $\mathfrak b$ are evaluations of products of global sections, giving one inclusion; generation of all degree-$k$ sections by sums of such products gives the other. Thus (eq78:powers) concerns actual ideals, not just set-theoretic base loci.

For a divisorial valuation $v$ of the function field with centre on $X$, write $v(\mathfrak b)$ for the minimum valuation of local generators at its centre. Equation (eq78:powers) gives \begin{equation}\label{wsref-0-65-71-37-38-3a-6f-72-64-65-72} \frac{1}{km_0}v(\mathfrak b_{|km_0L|}) =\frac1{m_0}v(\mathfrak b). \end{equation} If $\mathfrak b\ne\mathcal O_X$, some such $v$ has $v(\mathfrak b)>0$. To see this geometrically, blow up the nonzero ideal $\mathfrak b$ and normalize. Its pullback is an invertible ideal $\mathcal O(-E)$ with $E$ effective. It is not everywhere a unit: over a point of the base locus all local generators lie in the maximal ideal at points above it. Hence $E$ is nonzero. A prime component of $E$ defines the required valuation. If the original ideal is already invertible, its nonzero effective Cartier divisor gives the same conclusion directly.

It follows that the simultaneous hypotheses \begin{equation}\label{wsref-0-65-71-37-38-3a-63-72-69-74-65-72-69-6f-6e} \begin{gathered} H^0(X,m_0L)\ne0,\qquad R^{(m_0)}(L)\text{ generated in degree one},\\ \lim_{k\to\infty}\frac{v(\mathfrak b_{|km_0L|})}{km_0}=0 \quad\text{for every divisorial }v. \end{gathered} \end{equation} force $\mathfrak b=\mathcal O_X$. Thus $m_0L$ is globally generated and $L$ is semiample. The proof uses neither a numerical approximation to $v$ nor an exchange of a limit with a minimum over all valuations: one fixed positive valuation would contradict the last condition.

Attempt: application to the adjoint target.
For the original rational pair choose $q>0$ with $L=q(K_X+\Delta)$ Cartier. If the nonvanishing, generation and actual base-order assertions in (eq78:criterion) can be established for this $L$, the target follows, since a globally generated multiple of $L$ is a globally generated multiple of $K_X+\Delta$. This packages three distinct arithmetic-geometric obligations. It does not assert that nefness alone proves any of them. In particular, a section ring with no positive-degree sections is trivially finitely generated but useless for semiampleness; the first condition excludes this pitfall.

An analytic continuation of the route might compare the actual base ideals with multiplier ideals of minimal metrics, then use vanishing or Euler characteristic information to produce sections. That is exactly where extra hypotheses in the metric literature matter [E1]. No equality between those metric ideals and the actual base ideals is proved here. Without it, substituting metric Lelong numbers for the valuations in (eq78:criterion) is an unproved bridge.

Stress test: a nef flat line bundle with no sections.
Let $C$ be a complex elliptic curve and choose a non-torsion degree-zero line bundle $P\in\operatorname{Pic}^0(C)$. Such bundles exist because the complex torus $\operatorname{Pic}^0(C)$ is uncountable and its torsion subgroup is countable. For every $m>0$, a nonzero section of $P^m$ would have an effective divisor of degree zero, hence no zeros, and would trivialize $P^m$. This contradicts non-torsion. Therefore \[ P\text{ is nef},\qquad H^0(C,P^m)=0\quad(m>0). \] It admits a smooth unitary flat metric; its curvature and Lelong numbers are zero. Also $P+\varepsilon A$ has positive degree for an ample $A$ and $\varepsilon>0$, so numerical vanishing orders after ample perturbation carry no obstruction of this sort. These facts show why numerical positivity and zero metric singularities alone do not create sections.

This is not an abundance counterexample. On an elliptic curve an effective boundary $\Delta$ with $\deg(K_C+\Delta)=0$ must be zero, and $K_C$ is trivial. The non-torsion $P$ cannot be the stipulated adjoint divisor in that degree-zero situation. The missing adjoint input is therefore genuine, not a reason to discard the conjecture.

There is another weak argument to avoid. Along factorial multiples, base loci form a descending family of closed sets whenever the relevant sections are present. Noetherianity can make that family stabilize at a nonempty closed set; it does not make the set disappear. Likewise finding more sections at each step does not imply a section avoiding every remaining base point. The valuation in (eq78:order) detects such a persistent base ideal quantitatively only after the generation hypothesis has been established.

Outcome.
Outcome: Conditional reduction.

A direct base-ideal proof reduces semiampleness to nonvanishing, degree-one generation after a Veronese, and zero actual asymptotic base orders. It isolates an invalid metric-to-section shortcut by an explicit nef flat example outside the adjoint class. The argument supplies no new unconditional abundance case and makes no priority claim for its elementary base-ideal lemma.

The unresolved task is to prove the three properties for arbitrary nef klt adjoint divisors without adding bigness, Euler characteristic, or a lower-dimensional MMP assumption. A real-boundary formulation would also need its own notion of semiampleness; it is not proved by this rational-multiple calculation.

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