Algebraic geometry · algebraic cycles · motives · arithmetic geometry

Bloch–Beilinson Conjectures

Collaboration beta

Do rational Chow groups carry a canonical finite filtration whose layers reflect topology, motives, and regulators?

{CH}^{p} (X)_{Q} = F^{0} \supseteq F^{1} \supseteq \dots \supseteq F^{p + 1} = 0

Known results and sources

Algebraic cycles are expected to organize into a finite filtration whose successive layers connect geometry with cohomological and motivic information.

Research problem

Exact mathematical statement

For every smooth projective variety $X$ over a characteristic-zero subfield $k\subseteq\mathbf C$ , and every codimension $p$ , construct a descending filtration on the rational Chow group

CH^p(X)_{\mathbf Q}=F^0\supseteq F^1\supseteq\cdots\supseteq F^{p+1}=0.

The expected package places homologically trivial cycles in $F^1$ , makes pullback, pushforward, correspondences, and products compatible with the filtration, and relates the graded pieces to cohomological or motivic Ext data through cycle-class and regulator maps. These expectations comprise a family of connected conjectures; no unconditional construction with the full package is known in general. Chow–Künneth projector formulations additionally require algebraicity and compatible lifting inputs.

Problem infographic

Problem at a glance

Scientific problem explainer for the Bloch–Beilinson conjectures: codimension-p cycles on a smooth projective variety enter the rational Chow group, a finite descending filtration places homologically trivial cycles in its expected first step, graded pieces face motivic Ext and regulator realizations through open relations, and canonical construction, graded-piece control, projector algebraicity, and the full functorial package are marked not established in general. — The conjectural family asks for a canonical finite filtration on rational Chow groups, compatible with natural operations and controlled by cohomological or motivic realizations; the general construction remains open.

Current mathematical picture

Where work on Bloch–Beilinson Conjectures stands

Open problem

The v14 packet keeps the Bloch–Beilinson filtration package open while reporting a substantial conditional reduction. Under explicit Nori, coniveau, continuity, and motivic-comparison inputs, it defines a candidate filtration, identifies its terminal step with the kernel of the Nori cycle map, reduces injectivity to residue-field odd cohomology, and expresses those groups through an exterior defect module generated by a quartic piece. The preferred open route is to establish actual quartic saturation, the equivalent weight-minus-four Ext comparison, or Tate nonresonance over every relevant residue field. Algebraicity of cohomological Künneth projectors remains an independent requirement for the stated Murre consequences. Included exact-arithmetic model computations are source-reported and were not reproduced by ProofAtlas.

Leading routeQuartic saturation

Current priority: compute the actual Nori comparison defect and prove D_F^2=0 through the full linear-plus-quadratic saturation identity, retaining double-weight-two cross terms.

Route status · Active route

Useful failureTwo-jet and first-prolongation screens

Retain weight-two injection and first-prolongation injection as efficient sufficient screens, but not as necessary targets; quartic saturation is more permissive.

Route status · Narrowed route

Main reductionConiveau obstruction tower identified

Injectivity is conditionally reduced to vanishing of explicit fieldwise odd Nori groups, with the local/global distinction and low-dimensional base cases kept explicit.

Evidence posture · Reported reduction

Completed special caseCurves and codimension one close in the formal setup

The current work reports that p=0 is trivial, p=1 has no higher incoming differential, and curves have only p=0,1; hence the candidate filtration is complete for curves within the formal setup.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeCompute the actual quartic defect

Determine V_F, K_2,F, K_3,F, the linear map ell_F, the full cross-term quotient B_2,F, and q_F in finite Nori diagrams, then test quartic saturation as an R_F-module identity.

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

Bloch–Beilinson Conjectures in numbers

1.1kretained lines of mathematical investigation1,096 in the current working snapshot

Argument development: 1,014 · 93%
Explored or eliminated routes: 7 · 1%
Computational analysis: 8 · 1%
Open obligations: 23 · 2%
Definitions and setup: 44 · 4%

13selected mapped statements6routes investigated5reported milestones6open 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 bySpencer Bloch, Alexander BeilinsonProposed1979Status checkedAug 6, 2026

Source recordedAug 6, 2026

Recommended next task

Compute the actual quartic defect

Determine V_F, K_2,F, K_3,F, the linear map ell_F, the full cross-term quotient B_2,F, and q_F in finite Nori diagrams, then test quartic saturation as an R_F-module identity.

Suggested move: Apply rank-one, finite-rank, Koszul, and low-weight Stallings screens before calculating the full R_F-module defect.

What would count as progress

All actual finite-diagram modules and maps are identified.
Cross terms in B_2,F are retained.
Quartic saturation is proved or a surviving R_F-module class is exhibited.

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

26 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.

Active routeQuartic saturation

Current priority: compute the actual Nori comparison defect and prove D_F^2=0 through the full linear-plus-quadratic saturation identity, retaining double-weight-two cross terms.

Route status · Active route

Active routeWeight-minus-four derived comparison

Prove weight-one Ext agreement together with Ext^3 surjectivity and Ext^4 injectivity for every pure weight-minus-four coefficient over every relevant residue field.

Route status · Active route

Active routeTate-visible nonresonance

If the quartic module survives, use the actual reductive representation to prove its exterior descendants contain no relevant Tate summands.

Route status · Active route

Active routeIndependent projector algebraicity

In parallel with the no-phantom problem, establish algebraicity of the cohomological Künneth projectors; only then use nilpotent lifting and separately address self-duality.

Route status · Active route

Explored alternatives

Other routes

2 recorded

Route held in reserveGlobal quartic-symbol reciprocity

Deferred fallback: if a Tate-visible defect survives, construct the filtered coniveau complex and prove its terminal transgression is rationally trivial by a global reciprocity law.

Route status · Route held in reserve

Narrowed routeTwo-jet and first-prolongation screens

Retain weight-two injection and first-prolongation injection as efficient sufficient screens, but not as necessary targets; quartic saturation is more permissive.

Route status · Narrowed route

Route statements and reductions

Statements the next route can inspect and build on

Route statementRelative exterior-defect module

When the graded Lie comparison is an isomorphism in weight one, all relative excess-one defects form a graded module D_f over the exterior algebra on the common weight-one dual, generated in degree zero, with an exact linear-plus-quadratic presentation that retains cross terms.

Source-reported route statement

Route statementQuartic propagation theorem

Exterior multiplication surjects from every degree-q defect to each higher defect, so D_f^2=0 forces D_f^{2r−2}=0 for every r≥2. The condition D_f^2=0 is the explicit quartic saturation identity.

Source-reported route statement

Route statementExact local phantom formula

Under weight-one agreement and the stated continuity, invariant-exactness, image-reduction, and motivic comparison inputs, the current work identifies A_N^{2r−1}(F,Q(r)) with the R_F-invariant Tate part of D_F^{2r−2}.

Source-reported route statement · dependencies incomplete

Route statementWeight-minus-four comparison criterion

After weight-minus-one Ext^1 agreement, quartic saturation is equivalent to Ext^3 surjectivity and Ext^4 injectivity for all pure coefficients of weight −4; this reduces the fieldwise tower to ordinary and first higher Chow-group comparison data.

Source-reported route statement · dependencies incomplete

Route statementMurre consequences after projector algebraicity

Conditional on the separated filtration package and algebraicity of the cohomological Künneth projectors, the current work derives nilpotence, orthogonal Chow-projector lifting, Murre B–D, and equality of the induced projector filtration with F_N. Self-duality requires a further compatible lifting argument.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

6 featured tasks

Compute the actual quartic defect

Determine V_F, K_2,F, K_3,F, the linear map ell_F, the full cross-term quotient B_2,F, and q_F in finite Nori diagrams, then test quartic saturation as an R_F-module identity.

Suggested move: Apply rank-one, finite-rank, Koszul, and low-weight Stallings screens before calculating the full R_F-module defect.

Ready to work on

Test Tate-visible nonresonance

If the quartic defect is nonzero, determine whether any exterior descendant contains the Tate summand Q(−r) and prove uniform nonoccurrence when possible.

Suggested move: Decompose D_F^2 as an R_F-module and use actual weights, Hodge types, polarizations, parity, and monodromy rather than dimension counts.

Ready to work on

Prove the quartic derived comparison

For every pure weight-minus-four coefficient, prove Ext^3 surjectivity and Ext^4 injectivity from the reference category to the Nori category, after the weight-one comparison is established.

Suggested move: Translate the comparison to the relevant Künneth summands of ordinary and first higher Chow groups where possible.

Ready to work on

Solve the independent projector problem

Establish algebraicity of the cohomological Künneth projectors, then apply nilpotent lifting; add a separate transpose-compatible argument if a self-dual decomposition is claimed.

Suggested move: Keep this workstream separate from cycle-map injectivity and state precisely which Murre conclusions its completion unlocks.

Ready to work on

Instantiate the residue-field comparison framework

Specify coherent reference and Nori categories over every coniveau residue field, prove pure equivalence, continuity, exact invariants, weight splittings, field functoriality, and weight-one comparison with correct variance.

Suggested move: Build the coherent comparison system N6a and M1–M5 over one controlled finite diagram and audit every map direction before generalization.

Ready to work on

Build the chain-level global fallback

Only if a Tate-visible local quartic defect survives, construct the filtered coniveau model, derive the corrected higher differential, globalize a defect generator, and prove terminal cycles are rationally trivial through quartic-symbol reciprocity.

Suggested move: Defer this route until an actual Tate-visible quartic survivor is established.

Blocked by the current route

Sourced mathematical context

The known mathematical landscape

Context collected Aug 6, 2026

Current statusOpen problem

Open in general. The conjectures predict a canonical finite, functorial, multiplicative filtration on rational Chow groups whose first step is homological triviality and whose graded pieces are controlled by cohomology or motivic Ext data. Murre's Chow–Künneth-projector formulation and several important classes of varieties furnish substantial partial results, but no unconditional construction with the full expected package is known for all smooth projective varieties.

[6][7][10]

External progress

What the literature has established

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

2025
PreprintPelaez and Reyes described the Bloch–Beilinson filtration as still conjectural and proved that an unconditional orthogonal filtration recovers incidence equivalence in the stated zero-cycle setting, with additional results over finite fields.[10]
2020
Peer reviewedFu and Vial established a section-property framework and proved it for broad specified classes with motives of abelian type, supplying multiplicative Chow–Künneth structures in further special families without resolving the universal filtration conjectures.[9]
2016
Peer reviewedShen and Vial constructed a motivic Fourier decomposition of the Chow ring for the Hilbert square of a K3 surface and for the variety of lines on a very general cubic fourfold, providing a concrete Bloch–Beilinson-type splitting in important hyperkähler examples.[8]
2007
Peer reviewedJannsen proved the full Murre package for further special finite-dimensional motives while emphasizing that the strongest form was known only for restricted classes and not for arbitrary smooth projective varieties.[7]

Mathematical neighborhood

Related results and reusable starting points

Current focusBloch–Beilinson conjectures

Equivalent formulationMurre's conjectures A–D

Murre's conjectures ask for algebraic Chow–Künneth projectors, their prescribed action on Chow groups, independence of the induced filtration, and identification of its first step. For all smooth projective varieties, the complete Murre package provides the projector formulation of the Bloch–Beilinson filtration program.

[3][5]

Dependency or reductionmixed motives and motivic Ext descriptions

The expected mixed-motive category and its Ext groups motivate the graded pieces of the filtration. Constructing a candidate filtration inside one motivic realization does not by itself prove the full expected comparison, independence, or separatedness statements.

[5][6]

Dependency or reductionalgebraicity of Künneth projectors and the standard conjectures

Murre's projector route requires algebraic lifts of cohomological Künneth components. Their existence is part of the Chow–Künneth problem and is entangled with Grothendieck's standard conjectures; it cannot be inferred from filtration separatedness alone.

[3][6]

Logical consequenceBloch's conjecture on zero-cycles of surfaces

The general filtration would imply Bloch's prediction for zero-cycles on complex surfaces of geometric genus zero. Proven instances of that surface conjecture are evidence for only the corresponding special cases, not for the full filtration program.

[7][13]

Solved special casespecial varieties satisfying Murre's conjectures

The strongest Murre package is established for curves, rational surfaces, Brauer–Severi varieties, and certain additional motives covered by Jannsen's hypotheses; weaker subsets are known for more surfaces and threefolds.

[7]

Related problemBeauville splittings and distinguished-cycle decompositions

Beauville-type splittings, Fourier decompositions, and distinguished-cycle section properties construct graded Chow structures for abelian or hyperkähler-type families. They model expected Bloch–Beilinson behavior but are not a universal replacement for it.

[8][9]

Weaker or relaxed formorthogonal filtration on Chow groups

The orthogonal filtration is unconditionally finite and satisfies several analogous properties; its identified second step in the stated settings is a concrete partial structure, not an identification with the complete conjectural filtration in all codimensions and fields.

[10]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

formal library support · partial resource linkedMathlib algebraic-cycle infrastructure
Mathlib defines algebraic cycles on schemes as locally finite coefficient functions on points and provides a basic pushforward API. This is useful groundwork only; it is not a definition of rational Chow groups, a Bloch–Beilinson filtration, or a formal statement or proof of the conjectures.
[11]

Formalization opportunities

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

Formalization targetA statement-aligned formalization needs rational equivalence and Chow groups for smooth projective schemes, intersection products, pullback and proper pushforward, correspondences, and cycle-class maps into an appropriate Weil cohomology.
Formalization targetThe Murre formulation additionally requires Chow motives, algebraic correspondences with composition, cohomological Künneth projectors, orthogonal idempotent Chow lifts, and their actions on Chow groups.
Formalization targetThe Ext-graded formulation requires a precise formal category or realization of mixed motives, Tate twists, weight structures, derived Ext groups, and comparison maps; none may be replaced silently by a finite toy category.
Formalization targetA formal candidate filtration must separately establish functoriality, multiplicativity, finite length or separatedness, independence from choices, and the exact first-step identification. Defining a descending family of subgroups is not statement alignment.
Formalization targetNo problem-level formal statement or proof was identified in the scoped current Mathlib documentation and public Formal Conjectures repository search.

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

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.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

Open Bloch–Beilinson target and quartic comparison frontierThe source reports a conditional Nori-filtration construction and compresses the no-phantom problem to an actual quartic motivic-comparison or Tate-nonresonance theorem over every relevant residue field, while projector algebraicity remains separate.

Mapped research milestoneInitial research sequence

Research stage 1

Detailed research inventory

Claims, milestones, and routes in the current map

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

11 standing statements2 proposed statements5 mathematical milestones6 open questions1 narrowed routes7 conditional results1 completed special cases

Statements by mathematical role13 selected mapped statements

theorem candidate2 of 132
definition1 of 131
lemma6 of 136
reduction3 of 133
equivalence1 of 131

Selected mathematical clusters3 mathematical clusters

Conjectural filtration and scopeThe open filtration target, its conditional Nori construction, finite-length reduction, and low-dimensional base cases.4 displayed rows

retained route statementBloch–Beilinson filtration package
retained route statementCandidate Nori filtrationconditional
retained route statementTerminal filtration step equals the cycle-map kernelconditional
retained route statementCurves and codimension one close in the formal setupspecial case

Coniveau and local defect reductionThe residue-field obstruction, corrected chain fallback, relative defect module, propagation, and exact local phantom formula.11 displayed rows · 4 routes included

retained route statementConiveau reduction to residue-field odd groupsconditional
retained route statementCorrected intrinsic d2 representativeintermediate
retained route statementOne-sided two-jet transferintermediate
retained route statementRelative exterior-defect moduleintermediate
retained route statementQuartic propagation theoremintermediate
retained route statementExact local phantom formulaconditional
ComputationThe current work supplies six exact-rational-arithmetic programs for finite graded Lie-algebra and residue models, plus stored outputs and a runner that reportedly compares outputs byte-for-byte.The source reports that all six audits pass and support the corrected residue, two-jet, prolongation, and quartic-defect model calculations. ProofAtlas did not execute the scripts or independently reproduce the outputs by ProofAtlas. · reported unreproduced
Active routeQuartic saturationCurrent priority: compute the actual Nori comparison defect and prove D_F^2=0 through the full linear-plus-quadratic saturation identity, retaining double-weight-two cross terms.
Active routeWeight-minus-four derived comparisonProve weight-one Ext agreement together with Ext^3 surjectivity and Ext^4 injectivity for every pure weight-minus-four coefficient over every relevant residue field.
Active routeTate-visible nonresonanceIf the quartic module survives, use the actual reductive representation to prove its exterior descendants contain no relevant Tate summands.
Route held in reserveGlobal quartic-symbol reciprocityDeferred fallback: if a Tate-visible defect survives, construct the filtered coniveau complex and prove its terminal transgression is rationally trivial by a global reciprocity law.

Open mathematical frontierThe conditional no-phantom endpoint, independent projector route, six concrete obligations, and narrowed approaches that should not be repeated unchanged.19 displayed rows · 6 routes included

retained route statementWeight-minus-four comparison criterionconditional
retained route statementConditional master no-phantom theoremconditional
retained route statementMurre consequences after projector algebraicityconditional
Research targetInstantiate the residue-field comparison frameworkopen
Research targetCompute the actual quartic defectopen
Research targetProve the quartic derived comparisonopen
Research targetTest Tate-visible nonresonanceopen
Research targetBuild the chain-level global fallbackblocked
Research targetSolve the independent projector problemopen
Useful failureProve all local odd Nori Ext groups vanish using abstract nilpotent Lie theory.reported failure
Useful failureAssume coniveau is represented by a strict one-step Cousin bicomplex.reported failure
Useful failureRequire the full weight-two kernel K2 to vanish.reported failure
Useful failureInfer algebraic Chow–Künneth projectors from separatedness alone.reported failure
Active routeQuartic saturationCurrent priority: compute the actual Nori comparison defect and prove D_F^2=0 through the full linear-plus-quadratic saturation identity, retaining double-weight-two cross terms.
Active routeWeight-minus-four derived comparisonProve weight-one Ext agreement together with Ext^3 surjectivity and Ext^4 injectivity for every pure weight-minus-four coefficient over every relevant residue field.
Active routeTate-visible nonresonanceIf the quartic module survives, use the actual reductive representation to prove its exterior descendants contain no relevant Tate summands.
Route held in reserveGlobal quartic-symbol reciprocityDeferred fallback: if a Tate-visible defect survives, construct the filtered coniveau complex and prove its terminal transgression is rationally trivial by a global reciprocity law.
Active routeIndependent projector algebraicityIn parallel with the no-phantom problem, establish algebraicity of the cohomological Künneth projectors; only then use nilpotent lifting and separately address self-duality.
Narrowed routeTwo-jet and first-prolongation screensRetain weight-two injection and first-prolongation injection as efficient sufficient screens, but not as necessary targets; quartic saturation is more permissive.

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 bridgeDetermine V_F, K_2,F, K_3,F, the linear map ell_F, the full cross-term quotient B_2,F, and q_F in finite Nori diagrams, then test quartic saturation as an R_F-module identity.

1 approach has 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.

All actual finite-diagram modules and maps are identified.
Cross terms in B_2,F are retained.
Quartic saturation is proved or a surviving R_F-module class is exhibited.

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Prepared starting pointCompute the actual quartic defect

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Do rational Chow groups carry a canonical finite filtration whose layers reflect topology, motives, and regulators?

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Sources and references13 cited works · next context review by Nov 6, 2026

The mathematical context was checked on Aug 6, 2026. Status can be refreshed sooner after a material result or claim.

1
Lectures on Algebraic Cyclesoriginal source · Spencer Bloch · Duke University; second edition Cambridge University Press · Duke lectures 1979; original 1980 · DOI 10.1017/CBO9780511760693 · accessed Aug 6, 2026
2
Height pairing between algebraic cyclesoriginal source · Alexander A. Beilinson · Springer · 1987 · DOI 10.1007/BFb0078364 · accessed Aug 6, 2026
3
On a conjectural filtration on the Chow groups of an algebraic variety: Part I. The general conjectures and some examplespeer reviewed result · Jacob P. Murre · Indagationes Mathematicae · 1993-06-21 · DOI 10.1016/0019-3577(93)90038-Z · accessed Aug 6, 2026
4
On a conjectural filtration on the Chow groups of an algebraic variety: Part II. Verification of the conjectures for threefolds which are the product of a surface and a curvepeer reviewed result · Jacob P. Murre · Indagationes Mathematicae · 1993 · DOI 10.1016/0019-3577(93)90039-2 · accessed Aug 6, 2026
5
Motivic sheaves and filtrations on Chow groupspeer reviewed result · Uwe Jannsen · American Mathematical Society · 1994 · accessed Aug 6, 2026
6
Lectures on the Theory of Pure Motivessurvey or monograph · Jacob P. Murre, Jan Nagel, Chris A. M. Peters · American Mathematical Society · 2013 · accessed Aug 6, 2026
7
On Finite-dimensional Motives and Murre's Conjecturepeer reviewed result · Uwe Jannsen · Cambridge University Press · 2007 · DOI 10.1017/CBO9781107325968.008 · accessed Aug 6, 2026
8
The Fourier Transform for Certain HyperKähler Fourfoldspeer reviewed result · Mingmin Shen, Charles Vial · Memoirs of the American Mathematical Society · 2016 · accessed Aug 6, 2026
9
Distinguished cycles on varieties with motive of abelian type and the Section Propertypeer reviewed result · Lie Fu, Charles Vial · Journal of Algebraic Geometry · 2020 · ARXIV 1709.05644 · DOI 10.1090/jag/729 · accessed Aug 6, 2026
10
Incidence equivalence and the Bloch-Beilinson filtrationpreprint · Pablo Pelaez, Araceli Reyes · arXiv · 2025-01-31 · ARXIV 2501.19147 · accessed Aug 6, 2026
11
Mathlib.AlgebraicGeometry.AlgebraicCycle.Basicformalization · Mathlib community · accessed Aug 6, 2026
12
List of conjecturesencyclopedia · Wikimedia Foundation · accessed Aug 6, 2026
13
Chow groupencyclopedia · Wikimedia Foundation · accessed Aug 6, 2026

Important qualifications

The name Bloch–Beilinson covers several closely related formulations. This record focuses on the functorial finite filtration on rational Chow groups and its Murre Chow–Künneth-projector formulation; it does not identify every Beilinson, Beilinson–Bloch, Bloch–Kato, height-pairing, or special-value conjecture with that statement.
The historical year 1979 records Bloch's Duke lectures as reported by the 2013 Murre–Nagel–Peters monograph. Beilinson developed an independent formulation in work published later; the record does not claim a single-day priority chronology.
The general conjectures remain open, while many special varieties and individual parts of Murre's package are known. The milestone list is representative rather than a comprehensive catalogue of such cases.
Pelaez and Reyes (2025) is an arXiv preprint and remains in the current research map at preprint posture. Its unconditional orthogonal filtration has some expected Bloch–Beilinson properties but is not a solution of the general conjectures.
The scoped recognition search found a Wikipedia list/reference but no current Clay, Hilbert, Smale, Erdős, Epoch/FrontierMath, named-prize, or comparable selective-list designation. Absence from this bounded record is not a global nonexistence claim.
The scoped formalization search checked current Mathlib algebraic-cycle documentation and the public Google DeepMind Formal Conjectures repository. It found partial algebraic-cycle infrastructure but no problem-level Bloch–Beilinson statement or proof; this does not establish nonexistence in every proof assistant or private project.
No canonical external computation, dataset, or certificate for the general conjectures was identified. The unreviewed source material's finite Lie-algebra scripts were not executed and are source-asserted packet material, not external administrative evidence.
No submitted mathematical claim, contributor probability estimate, or packet computation was used as external authority. This record has no proof, novelty, review, acceptance, or publication authority.

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