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Math-26 · mathematics · Algebraic geometry

Termination of Flips Conjecture in the Minimal Model Program

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

A log pair $(X,\Delta)$ has $X$ normal, $\Delta$ an effective boundary divisor, and $K_X+\Delta$ real or rational Cartier in the chosen MMP category. Log canonicity requires all discrepancies to be at least $-1$; klt requires them to be greater than $-1$. A flip replaces a small negative contraction by the small birational model on which the transformed adjoint divisor is relatively ample. The claim excludes any infinite chain \[ (X_0,\Delta_0)\dashrightarrow(X_1,\Delta_1)\dashrightarrow\cdots \] of the allowed adjoint flips starting from a projective complex log canonical pair. A theorem about one prescribed choice of flips is not automatically termination of every sequence.

Formulation and concept references: [S1].

Short English statement

Must every allowed sequence of flips in the complex minimal model program stop after finitely many steps?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Math-26-A2 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-26

Manuscript 1 (family 056, 2026-10-05): Termination of every maximal ordinary klt rational program on Q-factorial compact Kähler fourfolds; even where arbitrary choices are included, the theorem is restricted to fourfolds and rational boundary, while the catalogue flip-termination target has arbitrary dimension and its broader allowed boundary category.

Manuscript 2 (family 056, 2026-09-24): Termination of every permitted rational lc projective fourfold program in characteristic zero; even where arbitrary choices are included, the theorem is restricted to fourfolds and rational boundary, while the catalogue flip-termination target has arbitrary dimension and its broader allowed boundary category.

Manuscript 3 (family 056, 2026-10-05): Termination of generalized-canonical rational flip sequences on Q-factorial compact Kähler fourfolds; even where arbitrary choices are included, the theorem is restricted to fourfolds and rational boundary, while the catalogue flip-termination target has arbitrary dimension and its broader allowed boundary category.

Manuscript 4 (family 056, 2026-10-05): Termination of every existing generalized lc rational flip sequence on Q-factorial Kähler fourfolds with fixed nef b-data; even where arbitrary choices are included, the theorem is restricted to fourfolds and rational boundary, while the catalogue flip-termination target has arbitrary dimension and its broader allowed boundary category.

Termination of Flips Conjecture in the Minimal Model Program

OpenAI's claim

Manuscript 1 (family 056, 2026-10-05): Every maximal ordinary negative-ray program for the stated rational compact Kähler klt fourfold pairs is claimed to terminate at a nef model or Mori fiber space according to pseudo-effectivity; Manuscript 2 (family 056, 2026-09-24): Every permitted rational-boundary log-canonical fourfold program in characteristic zero is claimed to terminate, including mixed steps and arbitrary ray choices on initially nonfactorial models. Manuscript 3 (family 056, 2026-10-05): Generalized-canonical flip sequences with fixed rational nef data on compact Kähler fourfolds are claimed to terminate even when exceptional discrepancies equal one; Manuscript 4 (family 056, 2026-10-05): Every existing generalized log-canonical flip sequence in the specified globally factorial compact Kähler fourfold category is claimed to terminate, without scaling or pseudo-effectivity assumptions.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $X$ be normal globally Weil rational-factorial compact Kähler of dimension four, and $\Delta\ge0$ rational with $(X,\Delta)$ klt. Global Weil rational-factoriality means every global Weil divisor has a positive Cartier multiple. Fix compatible canonical Weil representatives, or the intrinsic canonical rational line bundle. OpenAI claims every maximal ordinary negative-extremal-ray program beginning exactly on $X$ terminates in finitely many divisorial contractions and flips. Models remain in the same category; contractions are projective with relative line-bundle rank and Bott--Chern dimension one. For $D=K_X+\Delta$, \[ D\text{ pseudo-effective}\ \Longrightarrow\ D_n\text{ nef},\qquad D\text{ not pseudo-effective}\ \Longrightarrow\ X_n\to S\text{ a Mori fiber contraction}. \] The latter has connected fibers, normal compact Kähler base of dimension below four, relative Picard number one and relatively ample $-D_n$. In the first case no discrepancy decreases, while each original contracted prime has a strictly larger discrepancy on the endpoint, and the composite extracts no prime. Global strong rational-factoriality is also preserved when initially imposed.
Mathematical claim 2.
For a projective log canonical fourfold pair $(X,B)$ with rational boundary over a field that has characteristic zero and is algebraically closed, OpenAI claims termination of every permitted negative-extremal-ray program, without rational-factoriality or pseudo-effectivity assumptions. Its permitted birational steps are divisorial contractions, flips, or a divisorial contraction followed by a small positive modification; extra blowups and flops are excluded. It claims all required contractions and positive models exist at each stage, and every maximal program ends with \[ K_Y+B_Y\text{ nef}\quad\text{or}\quad Y\to T\text{ a Mori fiber space}. \] The fiber contraction has connected fibers and normal base of smaller dimension, with its relative Picard number equal to one and its negative adjoint relatively ample. Nef inputs give a zero-step program.
Mathematical claim 3.
Start with a normal irreducible globally Weil rational-factorial compact Kähler fourfold $X_0$, effective rational boundary $B_0$ of coefficients below one, and a fixed rational nef b-divisor $\mathbf M$ represented on a projective birational compact Kähler model. Require generalized log discrepancies positive for every prime and at least one for exceptional primes. For strict transforms $B_i$, traces $M_i$ and rational Cartier $D_i=K_{X_i}+B_i+M_i$, OpenAI claims every sequence of projective small nonisomorphic bimeromorphic diagrams with connected fibers \[ X_i\xrightarrow{f_i}Z_i\xleftarrow{f_i^+}X_{i+1},\qquad -D_i\text{ relatively ample for }f_i,\quad D_{i+1}\text{ relatively ample for }f_i^+, \] is finite, provided every $Z_i$ is normal compact Kähler, all models are normal globally Weil rational-factorial compact Kähler fourfolds, and canonical representatives are compatible. Small means no exceptional prime divisor. This theorem has a generalized-canonical discrepancy hypothesis, not unrestricted generalized log canonicity.
Mathematical claim 4.
Let $X_0$ be normal irreducible globally Weil rational-factorial compact Kähler of dimension four, with effective rational finite-support $B_0$ of coefficients in $[0,1]$ and a fixed analytically nef rational b-divisor represented on a projective bimeromorphic compact Kähler model. Suppose $D_0=K_{X_0}+B_0+M_0$ rational Cartier and all generalized log discrepancies nonnegative. With strict transforms, the same b-divisor and compatible canonical representatives, OpenAI claims finiteness of every sequence \[ X_i\xrightarrow{f_i}Z_i\xleftarrow{f_i^+}X_{i+1},\qquad -D_i\text{ }f_i\text{-ample},\quad D_{i+1}\text{ }f_i^+\text{-ample}, \] with every $Z_i$ normal compact Kähler, whose maps are projective small nonisomorphic bimeromorphic maps with connected fibers and whose models remain normal irreducible globally Weil rational-factorial compact Kähler fourfolds with rational Cartier adjoints. The coefficients and discrepancy bound permit generalized log canonical, not only generalized canonical, pairs.

Outline of the argument

Manuscript 1 (family 056, 2026-10-05): Auxiliary generalized programs supply the comparison with the ordinary sequence. Descent of numerically trivial actual lines preserves Cartier indices, while corrected discrepancy counts and analytic surface homology prove terminal fourfold termination even with real boundary, allowing arbitrary ordinary ray choices. Manuscript 2 (family 056, 2026-09-24): Divisor-cycle ranks discard all steps that lose divisors, and the remaining small tail lifts to crepant terminal junctions. A fixed set of boundary labels is constrained by surface discrepancy ACC; mixed-Hodge support bounds and finite model comparisons make the remaining numerical changes finite.

Manuscript 3 (family 056, 2026-10-05): A corrected finite count of low-discrepancy places first gives ordinary terminality along general points of flipped surfaces. Fixed denominator gaps and analytic surface-cycle comparisons then force a finite decrease, including zero exceptional boundary coefficients and nonnormal boundary components. Manuscript 4 (family 056, 2026-10-05): Relative programs and special termination reduce the sequence to controlled terminal pieces while preserving the nef datum and actual adjoints. Analytic cycle ranks remove divisorial losses, and strict mixed-Hodge weight comparisons bound the residual surface changes, leaving only finitely many possible steps.

Formal verification

Family 056 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-26-A1 · No claim recorded · LLM-generated · gpt 6 astra pro ·

gpt 6 astra pro attempt on Math-26

Termination of Flips Conjecture in the Minimal Model Program

Definitions and mathematical statement

A log pair $(X,\Delta)$ has $X$ normal, $\Delta$ an effective boundary divisor, and $K_X+\Delta$ real or rational Cartier in the chosen MMP category. Log canonicity requires all discrepancies to be at least $-1$; klt requires them to be greater than $-1$. A flip replaces a small negative contraction by the small birational model on which the transformed adjoint divisor is relatively ample. The claim excludes any infinite chain \[ (X_0,\Delta_0)\dashrightarrow(X_1,\Delta_1)\dashrightarrow\cdots \] of the allowed adjoint flips starting from a projective complex log canonical pair. A theorem about one prescribed choice of flips is not automatically termination of every sequence.

Formulation and concept references: [S1].

Short English statement

Must every allowed sequence of flips in the complex minimal model program stop after finitely many steps?

Research attempt

Current frontier.
Lazić--Xie connect arbitrary termination to Nakayama--Zariski decomposition and the existence of terminating sequences under their hypotheses; this is not a theorem that every conceivable sequence already terminates [E1]. Moraga's 2025 theorem concerns pseudo-effective log canonical fourfold flips, not arbitrary dimension [E2]. The distinction between a terminating MMP selected with scaling and termination of every sequence is central. Below a discrete monotone certificate is tested without assuming that any such certificate is available for all pairs.
Attack route.
Canonical rings appear promising because flips are designed by adjoint linear systems, but their sections are largely invariant under small birational changes. Discrepancies can improve instead, suggesting a finite accounting of their total improvement. A third route would control changing centres geometrically and reduce to a finite collection of valuations. The attempt makes that third requirement explicit inside a discrepancy certificate, then checks why boundedness, monotonicity and rational coefficients separately do not suffice.
Attempt: a finite discrepancy ledger.
Work first with a fixed rational-boundary pair and a sequence whose models have a common function field. Let $v_1,\ldots,v_r$ be finitely many normalized divisorial valuations of that field. Write $a_i(j)$ for their log discrepancies at the $j$th model. Suppose that there are an integer $q\geq1$, finite upper bounds $B_i$, and an initial index $j_0$ such that \begin{equation}\label{wsref-0-65-71-37-32-3a-6c-65-64-67-65-72} a_i(j)\in q^{-1}{\mathbb{Z}},\quad a_i(j_0)\leq a_i(j)\leq B_i,\quad a_i(j+1)\geq a_i(j), \end{equation} and every flip after $j_0$ strictly increases at least one $a_i$. Then the number of those flips is at most \begin{equation}\label{wsref-0-65-71-37-32-3a-62-6f-75-6e-64} \left\lfloor q\sum_{i=1}^r(B_i-a_i(j_0))\right\rfloor. \end{equation} Indeed the integer-valued quantity $q\sum_i(a_i(j)-a_i(j_0))$ is nonnegative, is bounded above by the expression before the floor, and increases by at least one at every step. This proves termination under the certificate. It handles arbitrary choices of the subsequent flips, provided the same hypotheses hold for that sequence; it does not merely construct one favourable choice.

The geometric discrepancy monotonicity motivating this setup follows from the negativity comparison for flips, as used in the source [E1]. The statement just proved does not import more from that comparison than monotonicity. In particular, it does not assume that a fixed finite valuation set detects every flip, that the discrepancies of those valuations have upper bounds, or that the indices of all models have a common denominator. Those are precisely the additional conditions that an actual construction of (eq72:ledger) must establish.

One slightly more flexible variant allows the valuations to be introduced in stages, but assigns each newly introduced valuation a nonnegative integer budget and requires a finite total budget for the entire sequence. If every flip spends at least one unit and no unit is reused, the same counting proof terminates the sequence. This restatement does not evade the difficulty: a proof of a finite total budget would be the missing geometric finiteness theorem, not a formal consequence of the existence of only finitely many boundary components on the initial model.

Attempt: why the canonical ring supplies no such clock.
Let $X\dashrightarrow X^+$ be a small birational map of normal projective varieties, and let $D,D^+$ be corresponding rational divisors. At every common sufficiently divisible degree $m$, global sections can be viewed inside the common rational function field as \[ H^0(X,\mathcal O_X(mD))= \{g:\operatorname{div}_X(g)+mD\geq0\}\cup\{0\}. \] Smallness identifies all prime divisors and their valuations, as well as the coefficients of the strict transforms. Thus the effectivity conditions are identical and \begin{equation}\label{wsref-0-65-71-37-32-3a-72-69-6e-67} H^0(X,mD)=H^0(X^+,mD^+) \end{equation} in the common field. For non-Cartier degrees one can use the corresponding reflexive divisorial sheaves; the same codimension-one argument applies. Normality is what allows those divisorial conditions to characterize the sections. Hence the adjoint graded ring is preserved under a flip in this sense, and its Hilbert function does not strictly decrease.

Finite generation of a ring can help organize particular models, but (eq72:ring) shows why the naive proof "each flip simplifies the canonical ring" has no numerical decrease to count. One needs extra control of chambers, centres, or local models. Equality of the rings at every step is fully compatible with a hypothetical infinite chain of different birational presentations; the equality alone says nothing about its length. This is an obstruction to the proposed invariant, not a constructed infinite flip sequence.

Stress test: four distinct missing uniformities.
A strictly increasing bounded rational sequence need not stop: $a(j)=1-2^{-j}$ is a counterexample. Thus the denominator condition in (eq72:ledger) cannot be replaced merely by rationality of every term. Even a single initial Cartier index does not, without proof, bound the indices on every later model. Real-boundary versions need a different well-foundedness mechanism and are not covered by a lattice of rational values.

Conversely, a uniform positive increment and bounded individual coordinates do not suffice with infinitely many coordinates. The vectors with the first $j$ coordinates equal to one and all later coordinates zero increase strictly forever, while each coordinate has range $\{0,1\}$. Each step could be witnessed by a fresh valuation. This disproves a purely formal inference from "every flip improves some discrepancy" to a finite sequence. It is not a valuation construction on an actual variety.

Nor can upper bounds be dropped for a finite ledger: the one-coordinate sequence $a(j)=j$ has all the other properties and never stops. Finally, strict improvement must occur in the chosen ledger at every step; an infinite chain leaving those finitely many coordinates unchanged is not excluded by their boundedness. These examples separate the hypotheses rather than conflating them into a vague appeal to discreteness.

The geometric challenge is therefore to produce a finite detecting family or finite total budget, with uniform value control, after any allowable initial segment. Bounds on one type of singularity, pseudo-effectivity in dimension four, or a prescribed scaling construction are useful partial inputs but do not verify that package in all dimensions and for all lc pairs. No assertion is made here that minimal log discrepancies are known to satisfy all the semicontinuity or chain conditions such a strategy might require. Using an unresolved abundance or singularity conjecture would be an additional conditional input, not a completion.

Outcome.
Outcome: Conditional reduction.

The attempt proves an explicit finite-budget termination criterion and shows that the canonical-section ring is not a decreasing invariant for small flips. Four elementary countermodels identify the distinct ways in which a naive discrepancy argument can fail. They are tests of a proof strategy, not counterexamples to termination.

What remains is to construct the geometric certificate for every allowable sequence in the original scope, including an appropriate replacement for rational discreteness in any real-boundary convention. Neither that construction nor a new unconditional termination case is supplied. The counting lemmas are elementary notebook deductions without a novelty claim.

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