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Model theory · groups of finite Morley rank · algebraic groups · incidence geometry

Cherlin–Zilber Algebraicity Conjecture

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Must every infinite simple group of finite Morley rank be an algebraic group over an algebraically closed field?

Ginfinite simple of finite Morley rank⇒∃K,H(Kalgebraically closed,Halgebraic overK,G≅H(K))
Known results and sources
A dark forest-green cover contrasts an abstract orbit network for a simple group with a geometric algebraic-group lattice, separated by an unresolved gap, while profile curves and a small finite loop suggest the source's narrow rank-4 geometry lane.
The Cherlin–Zilber conjecture asks whether every infinite simple group of finite Morley rank is algebraic; the highlighted profile geometry represents one conditional unresolved lane, not a proof.

Research problem

Exact mathematical statement

Let GG be an infinite simple group of finite Morley rank. The Cherlin–Zilber Algebraicity Conjecture asserts that there exist an algebraically closed field KK and an algebraic group H\mathbf H over KK such that

G≅H(K).G \cong \mathbf H(K).

The full universal statement remains open. The retained v2 source studies only one conditional special full-Frobenius rank-4 boundary configuration; success in that lane would not classify all simple groups of finite Morley rank.

Problem infographic

Problem at a glance

Problem-first explainer showing an infinite simple group of finite Morley rank, the model algebraic group H(K), a broken bridge asking whether every such G is isomorphic to some H(K), and four parallel 2-type classification lanes; odd and degenerate type remain open frontiers, while a small subordinate source-context card identifies one conditional unrefereed rank-4 configuration.
The Cherlin–Zilber conjecture asks whether every infinite simple group of finite Morley rank is algebraic over an algebraically closed field. The full conjecture remains open; the current source addresses only one conditional, unrefereed rank-4 configuration.

Current mathematical picture

Where work on Cherlin–Zilber Algebraicity Conjecture stands

Partially resolved

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

Useful failureFalse inversion identity, affine-reflection formulas, and completed rank-4 exclusion

The corrected source identity is X_t t = X_{t^{-1}}. Fixed-coordinate edge maps form only a partial action on composable domains; actual Borel-level loops compress trivially, while a finite quotient-holonomy obstruction can remain. Exact fixed-coordinate path compression and finite relative-position holonomy remain usable only with their partial-domain and quotient boundaries preserved.

Route status · Narrowed route
Main reductionFinite profile separation

The source reports that a generic Borel is algebraic over its profile codes from at most three independent centers when profile dimension is one and at most two when it is two, modulo named standard inputs.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeCertify every external input used by finite profile separation and the rank-4 special-Frobenius setup.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

Cherlin–Zilber Algebraicity Conjecture in numbers

1.1kretained lines of mathematical investigation1,106 in the current working snapshot
Argument development
955 · 86%
Explored or eliminated routes
15 · 1%
Computational analysis
5 · 0%
Open obligations
39 · 4%
Definitions and setup
92 · 8%
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.

Proposed byGregory Cherlin, Boris ZilberProposed1977Status checkedAug 13, 2026
Source recordedAug 13, 2026

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.

14 selected steps

Scroll horizontally to explore the route

Working route overview for Cherlin–Zilber Algebraicity ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Every infinite simple group of finite Morley rank should be algebraic over an algebraically closed field. — Depends on missing premiseEvery infinite simple groupof finite Morley rank shouldbe…Current reduction — Depends on missing premiseCurrent reductionFinite profile separation — Depends on missing premiseFinite profile separationRank-(2,2,3) profile pseudoplane — Depends on missing premiseRank-(2,2,3) profilepseudoplaneClosing target — Depends on missing premiseClosing targetFinite quotient holonomy remains — Depends on missing premiseFinite quotient holonomyremainsFixed-coordinate path compression — Depends on missing premiseFixed-coordinate pathcompressionFalse inversion identity, affine-reflection formulas, and completed rank-4 exclusion — stoppedFalse inversion identity,affine-reflection formulas,and…Unrestricted-trichotomy and naive secant-to-line shortcuts — stoppedUnrestricted-trichotomy andnaive secant-to-lineshortcutsCertify every external input used by finite profile separation and the rank-4 special-Frobenius setup. — OpenCertify every external inputused by finite profileseparation…Control finite relative-position holonomy before passing the edge law to profile composition. — OpenControl finiterelative-position holonomybefore…Prove the open Holonomy–Closure dichotomy and verify the endpoint theorem chain. — OpenProve the openHolonomy–Closure dichotomyand…Universal algebraicity question remains open — OpenUniversal algebraicityquestion remains openHolonomy–Closure Lemma — OpenHolonomy–Closure Lemma
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 routeFalse inversion identity, affine-reflection formulas, and completed rank-4 exclusion

The corrected source identity is X_t t = X_{t^{-1}}. Fixed-coordinate edge maps form only a partial action on composable domains; actual Borel-level loops compress trivially, while a finite quotient-holonomy obstruction can remain. Exact fixed-coordinate path compression and finite relative-position holonomy remain usable only with their partial-domain and quotient boundaries preserved.

Route status · Narrowed route
Narrowed routeUnrestricted-trichotomy and naive secant-to-line shortcuts

Neither shortcut is available without an exact theorem whose hypotheses match the profile structure; they cannot close the field-configuration endpoint. Use stationary profile germs and the narrower Holonomy–Closure dichotomy, then verify the exact field-configuration, internality, and algebraicization inputs separately.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Certify every external input used by finite profile separation and the rank-4 special-Frobenius setup.Suggested move: Locate exact published formulations for the full-Frobenius and Frécon inputs, definability of Morley rank and degree, Morley degree one of G/B, canonical coding in imaginaries, and the invariant-equivalence/intermediate-subgroup correspondence; record hypotheses beside each use.
Ready to work on
02
Control finite relative-position holonomy before passing the edge law to profile composition.Suggested move: Determine whether generic Hol_{s,q} is trivial or uniformly removable by a finite cover, prove the cocycle is parameter-controlled, and restate the fixed-coordinate edge maps as stationary finite-correspondence germs with both projection degrees recorded.
Ready to work on
03
Universal algebraicity question remains open

The source explicitly states that there is no proof of the full Cherlin–Zilber conjecture and that its rank-4 program is only one degenerate-type lane.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Holonomy–Closure Lemma

The decisive open dichotomy is to obtain either a code-rank-two field-configuration family from profile composition or a nontrivial invariant block from profile equality.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Prove the open Holonomy–Closure dichotomy and verify the endpoint theorem chain.Suggested move: For k=2, bound the code rank of a generic profile-composition component by two; for k=1, analyze the hidden-rank variable to obtain either an almost faithful curve family or a generic edge congruence, then check the exact field-configuration, internality, algebraicization, and maximal-torus-normalizer inputs before claiming rank-4 exclusion.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 13, 2026
Current statusPartially resolved

The Cherlin–Zilber Algebraicity Conjecture remains open in full. Major classification lanes are established, including even type under the literature's precise hypotheses, while substantial odd-type and degenerate-type configurations remain unresolved. The current research source studies only one conditional special full-Frobenius rank-4 boundary lane and does not claim a proof of the universal conjecture.

[6][5][7]
External progress

What the literature has established

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

  1. PreprintTent's current survey preprint reviewed the still-open conjecture and explained why sharply 2-transitive groups and the Burnside problem remain relevant to possible counterexamples.[6]
  2. Peer reviewedCherlin's peer-reviewed survey described the remaining algebraicity problem in odd type and identified unfinished classification work rather than a completed universal proof.[5]
  3. Peer reviewedThe classification of simple groups of finite Morley rank of even type established algebraicity in that major lane, leaving the universal conjecture open.[4]
  4. Historical sourceZilber and Cherlin independently formulated the algebraicity program in the late 1970s; the finite-rank form asks whether every infinite simple group of finite Morley rank is algebraic over an algebraically…[1][2]
9 cited sources4 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusCherlin–Zilber Algebraicity Conjecture
Solved special caseEven-type classification

The mixed/even-type classification supplies a major algebraic lane under the literature's precise K-star and type hypotheses; it does not settle odd or degenerate type.

[4][7]
Weaker or relaxed formOdd-type algebraicity program

Odd-type work reduces broad cases to algebraic groups, low Prüfer 2-rank, or minimal-simple configurations under stated hypotheses. These are classification reductions, not the full conjecture.

[5][7]
Related problemSharply 2-transitive groups and the Burnside problem

Sharply 2-transitive groups of finite Morley rank are studied as a possible source of nonalgebraic behavior, connecting the frontier to Burnside-type questions.

[6]
Related problemDegenerate type and bad-group configurations

Degenerate-type and bad-group configurations remain a separate obstruction lane. Progress in one special Frobenius boundary case cannot be identified with a classification of all such groups.

[7][6]

Formalization opportunities

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

  • Formalization targetA statement-aligned formal definition of groups and simple groups of finite Morley rank in a machine-checked model-theory foundation.
  • Formalization targetFormalized structure theory for algebraic groups over algebraically closed fields and an exact bridge from the finite-Morley-rank hypotheses to algebraic-group identification.
  • Formalization targetMachine-checked versions of the classification-by-2-type reductions, including every K-star, connectedness, definability, and rank hypothesis used in the established lanes.
  • Formalization targetExact published formulations and hypothesis checks are still required for the special full-Frobenius, Frécon, Morley-degree, canonical-base, field-configuration, and algebraicization inputs used by the conditional rank-4 work.

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.

5 standing statements2 proposed statements5 open questions2 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction3 of 73
  • lemma2 of 72
  • negative result1 of 71
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 statementEvery infinite simple group of finite Morley rank should be algebraic over an algebraically closed field.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementRank-(2,2,3) profile pseudoplaneintermediate
  • retained route statementFinite profile separationintermediate
  • retained route statementFixed-coordinate path compressionintermediate
  • retained route statementFinite quotient holonomy remainsintermediate
  • 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 failureFalse inversion identity, affine-reflection formulas, and completed rank-4 exclusionreported failure
  • Useful failureUnrestricted-trichotomy and naive secant-to-line shortcutsreported failure
  • Research targetCertify every external input used by finite profile separation and the rank-4 special-Frobenius setup.open
  • Research targetControl finite relative-position holonomy before passing the edge law to profile composition.open
  • Research targetProve the open Holonomy–Closure dichotomy and verify the endpoint theorem chain.open
  • Research targetUniversal algebraicity question remains openopen
  • Research targetHolonomy–Closure Lemmaopen
  • Narrowed routeFalse inversion identity, affine-reflection formulas, and completed rank-4 exclusionThe corrected source identity is X_t t = X_{t^{-1}}. Fixed-coordinate edge maps form only a partial action on composable domains; actual Borel-level loops compress trivially, while a finite quotient-holonomy obstruction can remain. Exact fixed-coordinate path compression and finite relative-position holonomy remain usable only with their partial-domain and quotient boundaries preserved.
  • Narrowed routeUnrestricted-trichotomy and naive secant-to-line shortcutsNeither shortcut is available without an exact theorem whose hypotheses match the profile structure; they cannot close the field-configuration endpoint. Use stationary profile germs and the narrower Holonomy–Closure dichotomy, then verify the exact field-configuration, internality, and algebraicization inputs separately.
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 bridgeCertify every external input used by finite profile separation and the rank-4 special-Frobenius setup.

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.

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Prepared starting pointCertify every external input used by finite profile separation and the rank-4 special-Frobenius setup.

Cherlin–Zilber Algebraicity Conjecture · ready to start

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Research contextPrepared context for any AI agent

Must every infinite simple group of finite Morley rank be an algebraic group over an algebraically closed field?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references9 cited works · next context review by Nov 13, 2026

The mathematical context was checked on Aug 13, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Groups and rings whose theory is categoricaloriginal source · Boris I. Zilber · Fundamenta Mathematicae · 1977 · DOI 10.4064/fm-95-3-173-188 · accessed Aug 13, 2026
  2. 2
    Groups of small Morley rankoriginal source · Gregory Cherlin · Annals of Mathematical Logic · 1979 · DOI 10.1016/0003-4843(79)90019-6 · accessed Aug 13, 2026
  3. 3
    Simple groups of finite Morley ranksurvey or monograph · Gregory Cherlin · Logic Colloquium 2005 · 2005 · accessed Aug 13, 2026
  4. 4
    Simple Groups of Finite Morley Ranksurvey or monograph · Tuna Altınel, Alexandre V. Borovik, Gregory Cherlin · American Mathematical Society · 2008 · accessed Aug 13, 2026
  5. 5
    Around the algebraicity problem in odd typepeer reviewed result · Gregory Cherlin · Model Theory · 2024 · DOI 10.2140/mt.2024.3.505 · accessed Aug 13, 2026
  6. 6
    From the Cherlin-Zilber Conjecture via sharply 2-transitive groups to the Burnside problempreprint · Katrin Tent · arXiv · 2026-06-16 · ARXIV 2606.18207 · accessed Aug 13, 2026
  7. 7
    Simple Groups of Finite Morley Rankauthoritative webpage · Tuna Altınel, Alexandre V. Borovik, Gregory Cherlin · Rutgers University · accessed Aug 13, 2026
  8. 8
    Formal Conjectures repositoryformalization · Google DeepMind · GitHub · accessed Aug 13, 2026
  9. 9
    Mathlib documentation indexformalization · Mathlib contributors · Lean community · accessed Aug 13, 2026

Important qualifications

  • This record concerns the finite-Morley-rank algebraicity conjecture, not Cherlin's broader conjecture for arbitrary simple omega-stable groups and not unrelated conjectures carrying Cherlin's name.
  • The current-status summary uses a 2026 survey preprint together with peer-reviewed and author-maintained sources. The preprint remains labeled as such and is not treated as peer review.
  • The conditional special full-Frobenius rank-4 argument is unrefereed first-party research, not an established result in the external literature.
  • The mixed/even, odd, and degenerate classification summaries are deliberately coarse. They do not inventory every hypothesis, K-star qualification, or later refinement in the literature.
  • The scoped search found no statement-aligned formalization in current mathlib documentation or the Formal Conjectures repository. This does not establish nonexistence in every formal library.
  • No computation, finite dataset, or certificate can by itself settle the universal classification statement, so none is promoted as readiness evidence.

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