Birational algebraic geometry · Minimal Model Program · flips · Nakayama–Zariski decompositions

Termination of Flips in Arbitrary Dimension

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Can an arbitrary Minimal Model Program perform infinitely many flips, or must every flip sequence eventually stop in every dimension?

X_{0} ⇢ X_{1} ⇢ X_{2} ⇢ \dots cannot be infinite?

Known results and sources

The termination problem asks whether any sequence of Minimal Model Program flips must eventually stop, regardless of dimension and ray ordering.

Research problem

Exact mathematical statement

Prove that every sequence of flips occurring in the Minimal Model Program terminates in arbitrary dimension:

X_0 \dashrightarrow X_1 \dashrightarrow X_2 \dashrightarrow \cdots \quad\text{cannot be infinite.}

The full target includes arbitrary ray orderings, both pseudoeffective and non-pseudoeffective log divisors, ordinary and generalized pairs, absolute and relative settings, and rational, NQC, and eventually real coefficient data.

The full problem remains open. The source's safest current theorem core is the projective ordinary Q-factorial klt setting. In a pseudoeffective rational subcategory it records exact Nakayama negative-part identities, an endpoint-angle dichotomy, fixed-carrier localization, and a canonical spectral formula for an actual MMP with scaling. Those source-reported results do not settle arbitrary termination, do not transfer scaling order to arbitrary sequences, and do not solve the non-pseudoeffective Mori-fiber branch.

Problem infographic

Problem at a glance

Wide birational-geometry explainer showing a sequence of small birational flips between models, the question of whether the chain can continue forever, and three visibly open mathematical regimes: a D-trivial null face, low walls over a fixed carrier, and a separate non-pseudoeffective Mori-fiber threshold branch. — Exact divisor ledgers organize hypothetical infinite flip sequences into geometric obstruction regimes, but null-face adjunction, fixed-carrier detection, and the non-pseudoeffective branch remain open.

Current mathematical picture

Where work on Termination of Flips in Arbitrary Dimension stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureTreating an arbitrary flip sequence as an MMP with scaling

The source explicitly marks this route invalid and directs arbitrary sequences to universal transfer and endpoint-angle theory instead. Adjunction or nef reduction on the null face, complement/index control on the fixed carrier, endpoint Cartier/NQC descent, and a finite-kink theorem for actual scaling remain viable but unproved closure routes.

Route status · Narrowed route

Main reductionCurrent reduction

In the source-reported pseudoeffective rational klt core, any hypothetical infinite arbitrary sequence falls into a null-face geometricization problem or a fixed-carrier uniformly near-crepant extraction problem.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeRule out arbitrarily deep or unbounded-index active valuations over the fixed carrier, or convert that escape into a lower-dimensional infinite sequence.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Termination of Flips in Arbitrary Dimension in numbers

1.5kretained lines of mathematical investigation1,543 in the current working snapshot

Argument development: 1,327 · 86%
Explored or eliminated routes: 19 · 1%
Computational analysis: 29 · 2%
Open obligations: 19 · 1%
Definitions and setup: 149 · 10%

7selected mapped statements2routes investigated5open questions5contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Rule out arbitrarily deep or unbounded-index active valuations over the fixed carrier, or convert that escape into a lower-dimensional infinite sequence.

Suggested move: In the positive-angle regime, prove a fixed-carrier terminal or exceptionally-noncanonical detector that supplies an active valuation with a normalized discrepancy jump bounded below, unless adjunction lowers dimension.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeTreating an arbitrary flip sequence as an MMP with scaling

Route status · Narrowed route

Narrowed routeClaiming that the pseudoeffective Nakayama budget solves the full problem

The source expressly labels full-problem closure from the pseudoeffective budget false and keeps a threshold-extractor target for the non-pseudoeffective branch. Repeated pseudoeffective-threshold resets may still organize low-wall witnesses into a bounded extractor, a fixed Mori-fiber stratum, or a lower-dimensional sequence.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks

Rule out arbitrarily deep or unbounded-index active valuations over the fixed carrier, or convert that escape into a lower-dimensional infinite sequence.Suggested move: In the positive-angle regime, prove a fixed-carrier terminal or exceptionally-noncanonical detector that supplies an active valuation with a normalized discrepancy jump bounded below, unless adjunction lowers dimension.

Ready to work on

Convert the numerical D-trivial limiting curve class into a fixed geometric locus suitable for adjunction or nef reduction.Suggested move: Geometricize the null-face limit as a fixed proper stratum, contraction, or lower-dimensional generalized pair carrying infinitely many induced modifications.

Ready to work on

Full arbitrary-dimensional target

The full target covers arbitrary ray orderings, pseudo- and non-pseudoeffective branches, ordinary and generalized pairs, absolute and relative settings, and rational through real/NQC data; the source marks it open.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Normalized-jump detector

The highest-leverage open target is a uniform lower bound on one active normalized discrepancy jump over a fixed carrier, unless adjunction lowers dimension.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Prove a threshold-extractor or Mori-fiber adjunction theorem for the non-pseudoeffective branch before claiming full termination.Suggested move: For an actual scaling MMP, prove local finiteness of the canonical wall spectrum near zero; do not use this spectral target for arbitrary ray orderings.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen problem

Termination of arbitrary flip sequences in unrestricted dimension remains open. Primary literature proves important special cases, including termination for four-dimensional pseudoeffective NQC log canonical generalized pairs, and gives conditional reductions through lower-dimensional termination, weak or Nakayama–Zariski decompositions, minimal-log-discrepancy conjectures, bounded Cartier index, or scaling hypotheses. None of these sources establishes the full arbitrary-order, all-dimensional, pseudo- and non-pseudoeffective statement.

[2][3][4]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2025
Peer reviewedLazić and Xie showed, conditional on a natural conjecture about Nakayama–Zariski decomposition under MMP operations, that termination of one sequence of flips for a pseudoeffective projective pair implies…[5]
2024
PreprintKim proved termination for a bounded-index scaling regime when the numerical Kodaira dimension is at least dimension minus one, illustrating a further special branch rather than arbitrary termination.[6]
2023
Peer reviewedChen and Tsakanikas proved termination for four-dimensional pseudoeffective NQC log canonical generalized pairs and a lower-dimensional weak-Zariski-decomposition reduction, while noting unresolved…[3]
2022
PreprintHan and Liu reduced termination to terminal-flip termination together with ascending-chain and lower-semicontinuity conjectures for minimal log discrepancies in terminal and exceptionally non-canonical…[4]

Mathematical neighborhood

Related results and reusable starting points

Current focusTermination of flips in arbitrary dimension

Solved special caseFour-dimensional pseudoeffective NQC lc generalized pairs

Termination holds for four-dimensional pseudoeffective NQC log canonical generalized pairs. The theorem does not extend by omission of dimension, pseudoeffectivity, NQC, or singularity hypotheses.

[3]

Dependency or reductionWeak Zariski decomposition termination engine

Weak Zariski decomposition plus exact lower-dimensional termination hypotheses converts portions of the termination problem into induction on dimension.

[2][3]

Dependency or reductionEasy termination and Nakayama–Zariski behavior

A natural conjecture controlling components of the Nakayama negative part under an MMP would upgrade termination of one pseudoeffective sequence to termination of all sequences in the stated projective generalized-pair setting.

[5]

Dependency or reductionMinimal log discrepancies and enc singularities

Termination is reduced to terminal-flip termination and minimal-log-discrepancy ACC/LSC statements for terminal and exceptionally non-canonical singularities, with important low-dimensional components proved.

[4]

Solved special caseHigh numerical Kodaira dimension with bounded index

Certain flips with scaling terminate under bounded Cartier index and high numerical Kodaira dimension. These hypotheses are not available for a general arbitrary-order sequence.

[6]

Related problemExistence of minimal models and finite generation

Finite generation and existence of minimal models for log-general-type varieties supply indispensable infrastructure for MMP with scaling and big adjoint divisors but are not identical to arbitrary termination.

[1]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA proof-assistant development of normal projective varieties, divisors, numerical equivalence, birational maps, contractions, flips, discrepancies, and singularity classes from terminal through log canonical.
Formalization targetFormal Minimal Model Programs with arbitrary ray choices and with scaling, including preservation of Q-factoriality and exact separation of divisorial contractions from flip-only tails.
Formalization targetFormal Weil and Cartier b-divisors, Nakayama asymptotic multiplicities, positive and negative parts, weak and Nakayama–Zariski decompositions, and relative/NQC variants.
Formalization targetFormal cone and contraction theorems, negativity, finite generation, adjunction, canonical bundle formulas, complements, minimal log discrepancies, and the bounded-dimensional termination theorems used as inputs.
Formalization targetA formal statement matrix that prevents arbitrary-order, scaling, pseudoeffective, non-pseudoeffective, ordinary, generalized, rational, real, absolute, and relative variants from being silently conflated.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

1 standing statements6 proposed statements5 open questions2 narrowed routes

Statements by mathematical role7 selected mapped statements

theorem candidate1 of 71
reduction3 of 73
lemma3 of 73

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 2 routes included

retained route statementMust every sequence of flips terminate?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementExact negative-part peelingintermediate
retained route statementEndpoint-angle null-face branchintermediate
retained route statementPositive-angle fixed carrierintermediate
retained route statementCanonical spectrum for scalingintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureTreating an arbitrary flip sequence as an MMP with scalingreported failure
Useful failureClaiming that the pseudoeffective Nakayama budget solves the full problemreported failure
Research targetRule out arbitrarily deep or unbounded-index active valuations over the fixed carrier, or convert that escape into a lower-dimensional infinite sequence.open
Research targetConvert the numerical D-trivial limiting curve class into a fixed geometric locus suitable for adjunction or nef reduction.open
Research targetProve a threshold-extractor or Mori-fiber adjunction theorem for the non-pseudoeffective branch before claiming full termination.open
Research targetFull arbitrary-dimensional targetopen
Research targetNormalized-jump detectoropen
Narrowed routeTreating an arbitrary flip sequence as an MMP with scalingThe source explicitly marks this route invalid and directs arbitrary sequences to universal transfer and endpoint-angle theory instead. Adjunction or nef reduction on the null face, complement/index control on the fixed carrier, endpoint Cartier/NQC descent, and a finite-kink theorem for actual scaling remain viable but unproved closure routes.
Narrowed routeClaiming that the pseudoeffective Nakayama budget solves the full problemThe source expressly labels full-problem closure from the pseudoeffective budget false and keeps a threshold-extractor target for the non-pseudoeffective branch. Repeated pseudoeffective-threshold resets may still organize low-wall witnesses into a bounded extractor, a fixed Mori-fiber stratum, or a lower-dimensional sequence.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeRule out arbitrarily deep or unbounded-index active valuations over the fixed carrier, or convert that escape into a lower-dimensional infinite sequence.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointRule out arbitrarily deep or unbounded-index active valuations over the fixed carrier, or convert that escape into a lower-dimensional infinite sequence.

Termination of Flips in Arbitrary Dimension · ready to start

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Research contextPrepared context for any AI agent

Can an arbitrary Minimal Model Program perform infinitely many flips, or must every flip sequence eventually stop in every dimension?

Exact question and boundaries
Current routes and known obstacles
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Sources and references6 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
Existence of minimal models for varieties of log general typepeer reviewed result · Caucher Birkar, Paolo Cascini, Christopher D. Hacon, James McKernan · Journal of the American Mathematical Society · 2010 · ARXIV math/0610203 · DOI 10.1090/S0894-0347-09-00649-3 · accessed Aug 14, 2026
2
On weak Zariski decompositions and termination of flipspeer reviewed result · Christopher D. Hacon, Joaquín Moraga · Mathematical Research Letters · 2020 · ARXIV 1805.01600 · DOI 10.4310/MRL.2020.v27.n5.a6 · accessed Aug 14, 2026
3
On the termination of flips for log canonical generalized pairspeer reviewed result · Guodu Chen, Nikolaos Tsakanikas · Acta Mathematica Sinica, English Series / Springer · 2023 · ARXIV 2011.02236 · DOI 10.1007/s10114-023-0116-3 · accessed Aug 14, 2026
4
On termination of flips and exceptionally non-canonical singularitiespreprint · Jingjun Han, Jihao Liu · arXiv · 2022 · ARXIV 2209.13122 · accessed Aug 14, 2026
5
Nakayama-Zariski decomposition and the termination of flipspeer reviewed result · Vladimir Lazić, Zhixin Xie · Épijournal de Géométrie Algébrique · 2025 · ARXIV 2305.01752 · accessed Aug 14, 2026
6
On diminished multiplier ideal and the termination of flipspreprint · Donghyeon Kim · arXiv · 2024 · ARXIV 2405.11902 · accessed Aug 14, 2026

Important qualifications

Termination results vary materially with dimension, singularity class, coefficient set, generalized-pair data, pseudoeffectivity, scaling, and absolute versus relative setting; this record does not conflate them.
The review selects representative primary results and conditional reductions rather than attempting an exhaustive history of threefold, fourfold, special-termination, or MMP-with-scaling theorems.
No intake-packet theorem, packet status label, source-reported computation, embedded bibliography, or submitted URL was used as external-status authority.
The Han–Liu, Lazić–Xie, and Kim results retain the stated conjectural or bounded hypotheses; they are not promoted to unconditional arbitrary-dimensional termination.
No exact problem-level formalization was identified in the scoped review, but this does not establish nonexistence in every proof assistant or private project.
This metadata has no proof, novelty, review, acceptance, credit, visibility, publication, or deployment authority.

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