ProofAtlas

Linear algebra · finite fields · additive combinatorics

Alon–Jaeger–Tarsi Conjecture

Collaboration beta

Given an invertible matrix over a field with at least four elements, can one choose a vector whose coordinates and transformed coordinates are all nonzero?

∀F(|F|≥4)∀M∈GLn(F),∃x∈(F*)n:Mx∈(F*)n
Known results and sources
An abstract invertible matrix carries a constellation of nonzero input coordinates to nonzero output coordinates across a dark finite-field lattice, presented as an open question rather than a proof.
The cover isolates the conjecture’s central search: avoid every zero coordinate before and after an invertible linear transformation.

Research problem

Exact mathematical statement

∀F(|F|≥4)∀n≥1∀M∈GLn(F),∃x∈(F*)nsuch thatMx∈(F*)n.\forall \mathbb F (|\mathbb F|\ge 4)\; \forall n\ge1\; \forall M\in GL_n(\mathbb F),\quad \exists x\in(\mathbb F^*)^n\text{ such that }Mx\in(\mathbb F^*)^n.

The selected public snapshot develops a prime-field group-ring route and repeatedly warns that its intermediate quotient products do not by themselves transport the primitive class needed to close AJT.

Problem infographic

Problem at a glance

Scientific explainer for the open Alon–Jaeger–Tarsi Conjecture. A highlighted point x avoids the input coordinate-zero hyperplanes, an invertible matrix M maps it to Mx, and the output point also avoids every coordinate-zero hyperplane. The plate shows the exact hypotheses M ∈ GLₙ(𝔽), |𝔽| ≥ 4 and asks whether such an x always exists.
The Alon–Jaeger–Tarsi Conjecture asks whether every invertible matrix over a field with at least four elements admits a vector whose coordinates, and whose transformed coordinates, are all nonzero. The coordinate hyperplanes show what must be avoided. This figure presents the general statement as an open conjecture and does not claim a proof or counterexample.

Current mathematical picture

Where work on Alon–Jaeger–Tarsi Conjecture stands

Partially resolved

The public-selected Revision 15 snapshot retains the AJT reduction architecture and its earlier open selectors while adding exact Three-Basis-Frame support and weight bookkeeping, a source-reported independent cutoff audit, and a sharper first unconstructed mapping-cone frontier. No complete proof or counterexample is claimed.

Strongest supported footholdCoordinate density yields a large admissible circuit

Revision 11 strengthens flat primitivity into two-sided support expansion and a contraction with at least p-1 admissible deletions.

Evidence posture · Reported result
Leading routeLarge-circuit PCDS selector

Use coordinate density to choose a contraction with at least p-1 admissible opposite labels, then combine common residues and the multi-deletion simplex to force a zero residue or an ancestor-closing terminal.

Route status · Active route
Useful failureDense boundary as rank-stuck

Revision 11 eliminates the interpretation that the dense boundary merely lacks another product-level quotient; the live obstruction is primitive and defect control.

Route status · Eliminated route
Main reductionUniversal line descent corrects the dense boundary

At a Three-Basis-Frame node with no proper support-spanned subflat, all pairwise transitions have full support; exact scalar-weight splitting and the matching projector then preserve the relevant Euler-product support, while the first dense-deletion face has the required shifted weight.

Evidence posture · Reported reduction
Completed special caseCompressed packet in every odd characteristic

For every odd prime there is an explicit quadratic-residue rank-two modular-zero packet with neither p parallel labels nor a complete projective line.

Evidence posture · Source-reported route statement
Priority open bridgeConstruct the shifted weight-zero TBF mapping cone

Build the exact projective-unit-gauged shifted weight-zero Z/p² mapping cone that combines periodic corrections with the dense deletion simplex.

Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Alon–Jaeger–Tarsi Conjecture in numbers

9.1kretained lines of mathematical investigation9,125 in the current working snapshot
Argument development
7,733 · 85%
Explored or eliminated routes
395 · 4%
Computational analysis
139 · 2%
Open obligations
366 · 4%
Definitions and setup
492 · 5%
44selected mapped statements13routes investigated10reported milestones8open questions8contribution-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 byNoga Alon, François Jaeger, Michael TarsiProposed1981Status checkedAug 2, 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

28 selected steps

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

28 selected steps

Scroll horizontally to explore the route

Working route overview for Alon–Jaeger–Tarsi ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Alon–Jaeger–Tarsi conjecture — Depends on missing premiseAlon–Jaeger–Tarsi conjectureDense one-basis primitive/defect selector — Depends on missing premiseDense one-basisprimitive/defect selectorPaired Contraction–Deletion Selector — Depends on missing premisePaired Contraction–DeletionSelectorPrimitive periodic norm lift-back — Depends on missing premisePrimitive periodic normlift-backAccepted sufficient architecture — Depends on missing premiseAccepted sufficientarchitectureAncestral flat restart — ActiveAncestral flat restartConditional second-rung exposure — ProposedConditional second-rungexposureDimension-free basis-block descent — ActiveDimension-free basis-blockdescentExact pair-residue bridge — ActiveExact pair-residue bridgeLow-defect exact-cluster lift-back — ActiveLow-defect exact-clusterlift-backOne-basis line descent — ActiveOne-basis line descentOne-basis purification — Depends on missing premiseOne-basis purificationLarge-circuit PCDS selector — activeLarge-circuit PCDS selectorAxis-sparse dense primitive/defect selector — activeAxis-sparse denseprimitive/defect selectorPrimitive periodic norm lift-back — activePrimitive periodic normlift-backResidual second-rung catalecticant seams — activeResidual second-rungcatalecticant seamsTreat every p-divisible quotient product as transporting the original primitive Bockstein value 1. — stoppedTreat every p-divisiblequotient product astransporting…Iterate line stripping and declare the resulting rank-one packet harmless. — stoppedIterate line stripping anddeclare the resultingrank-one…Classify every rank-two zero packet as parallel or a complete pencil. — stoppedClassify every rank-two zeropacket as parallel or acomplete…Use proper-flat primitivity and rank descent without tracking primitive or nullity data. — stoppedUse proper-flat primitivityand rank descent withouttracking…Prove the large-circuit PCDS selector — OpenProve the large-circuit PCDSselectorProve the axis-sparse dense selector — OpenProve the axis-sparse denseselectorProve primitive periodic norm lift-back — OpenProve primitive periodicnorm lift-backExtend lift-back to overfull clusters — OpenExtend lift-back to overfullclustersClose the residual small-prime catalecticant seams — OpenClose the residualsmall-prime catalecticantseamsAudit load-bearing exact and high-risk inputs — OpenAudit load-bearing exact andhigh-risk inputsResolve the Three-Basis Frame terminal — OpenResolve the Three-BasisFrame terminalConstruct the shifted weight-zero TBF mapping cone — OpenConstruct the shiftedweight-zero TBF mapping cone
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 routeLarge-circuit PCDS selector

Use coordinate density to choose a contraction with at least p-1 admissible opposite labels, then combine common residues and the multi-deletion simplex to force a zero residue or an ancestor-closing terminal.

Route status · Active route
Active routeAxis-sparse dense primitive/defect selector

The dense boundary is no longer rank-stuck: the active task is to order line descents without losing defect or primitive control, or force packet, flat, compressed-plane, or paired-residue structure.

Route status · Active route
Active routePrimitive periodic norm lift-back

Build a filtered mapping cone for the periodic corrections so the original unit class cannot terminate in the rank-one all-factor unit line without a prior closing event.

Route status · Active route
Active routeResidual second-rung catalecticant seams

The single-catalecticant computation localizes rather than closes the lower-prime problem; work now targets only the recorded residual nullity intervals with stronger flattenings or geometry.

Route status · Active route

Explored alternatives

Other routes

9 recorded
Eliminated routeDense boundary as rank-stuck

Revision 11 eliminates the interpretation that the dense boundary merely lacks another product-level quotient; the live obstruction is primitive and defect control.

Route status · Eliminated route
Narrowed routeDimension-free one-basis corridor

Basis-block descent and purification give exact rank reduction through defect p-2, then meet a packet, complete pencil, or the corrected dense primitive/defect boundary.

Route status · Narrowed route
Narrowed routeLow-defect exact-cluster lift-back

Exact p-clusters in the first post-PQIT rung are closed for p≥11; the surviving packet analysis begins with overfull or sufficiently compressed clusters.

Route status · Narrowed route
Browse 6 more explored routes
Route held in reserveLarge-prime second-rung exposure

TAC47 and DFE2-47 provide a candidate exposure route from p=47 onward, but the exact arithmetic and downstream iteration remain nonpublication-load-bearing until independently audited.

Route status · Route held in reserve
Refuted routeRank-one stripping alone

Line stripping reaches a genuine primitive p-packet and therefore does not by itself prove AJT or close the ancestor.

Route status · Refuted route
Refuted routeParallel-or-pencil-only rank-two classification

Explicit compressed packets refute this route above the low-predecessor threshold; only the quantified h≤(p-1)/2 subrange remains valid.

Route status · Refuted route
Not yet justifiedAutomatic primitive functoriality through quotients

Product-level p-divisibility survives the exact transfers, but the primitive Bockstein can change through norm corrections.

Route status · Not yet justified
Not yet justifiedQuotient zero without ancestor closure

A quotient packet or pencil does not automatically kill the original integral product; exact lift-back, ancestral restart, or a direct original-product terminal is mandatory.

Route status · Not yet justified
Useful but insufficientSingle catalecticant as universal closure

The exact inequality exhausts a prime-specific range but cannot classify the residual seams or certify existence at the first unexcluded nullity.

Route status · Useful but insufficient

Route statements and reductions

Statements the next route can inspect and build on

Route statementPaired Contraction–Deletion Selector

For a minimum two-basis failure, select a nonparallel contraction c and an admissible deletion d with vanishing paired residue, or reach a genuine ancestor-closing terminal, so that the nullity-lowering route can continue.

Source-reported route statement · dependencies incomplete
Route statementLarge admissible fundamental circuit

Some c in E has at least p-1 nonzero coefficients when expanded in the opposite basis A, yielding at least p-1 admissible second deletions after contraction.

Source-reported route statement
Route statementRank-one packet remains all-factor primitive

A p-parallel packet is nonunit in its colored Tor module, but its all-factor module (z)/(z^p) contains a class with Bockstein value 1.

Source-reported route statement
Route statementPrimitive periodic norm lift-back

The original primitive two-basis class cannot reach the canonical rank-one unit line through periodic norm corrections without an earlier zero residue, ancestor-closing packet, proper support-spanned restart, or nullity-lowering paired deletion.

Source-reported route statement · dependencies incomplete
Route statementDense one-basis primitive/defect selector

At defect p-1, find a return to defect at most p-2, a p-packet or complete pencil, or a basis-line descent that preserves enough primitive data for a paired deletion; the hard no-packet branch has at most one opposite-color label on every basis line.

Source-reported route statement · dependencies incomplete
Route statementOne-basis line descent

For each basis direction b, either its saturated line transfers and changes the second-color defect by e'=e+1-a_b, or that line already contains a true p-packet.

Source-reported route statement
Route statementPrime-specific catalecticant cutoff table

The exact basis-plus-one inequality excludes the full second-rung range from p=47 onward and leaves only nullity 88 at p=43, 83–84 at p=41, and 74–76 at p=37 among the top near-threshold examples listed.

Source-reported route statement

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

8 featured tasks
01
Prove primitive periodic norm lift-back

Construct the filtered mapping cone from the original unit class through the first and paired periodic corrections and force a zero-residue boundary before the canonical rank-one all-factor unit line.

Suggested move: Incorporate the large admissible circuit and low-defect exact-cluster closure, leaving only overfull, sufficiently compressed, or full-preimage/non-support-spanned packet stages.
Ready to work on
02
Prove the large-circuit PCDS selector

Use LFC-1 to choose c with at least p-1 admissible opposite labels, then force a vanishing paired residue or classify the all-nonzero family with a genuine ancestor-closing terminal.

Suggested move: Write every admissible residue in both Delta_(c,d) and eta_(c,d) form and combine the common quotient polynomial, multi-deletion simplex, and support expansion.
Ready to work on
03
Prove the axis-sparse dense selector

At the defect-(p-1) one-basis boundary, either return to controlled defect, find a packet or pencil, or preserve enough primitive data through line descent to obtain a paired deletion.

Suggested move: Separate basis axes with one opposite label from empty axes and the p-1 non-axis excess labels, then choose a descent order whose recurrence never exceeds p-1 or classify every escape.
Ready to work on
04
Close the residual small-prime catalecticant seams

Treat only the nullities not excluded by the exact B1CUT table, beginning with 88 at p=43, 83–84 at p=41, and 74–76 at p=37.

Suggested move: Use simultaneous flattenings, two-color factorization, or closure-collision geometry instead of rerunning the exhausted single-catalecticant range.
Ready to work on
05
Construct the shifted weight-zero TBF mapping cone

Build the exact projective-unit-gauged shifted weight-zero Z/p² mapping cone that combines periodic corrections with the dense deletion simplex.

Suggested move: Choose projective orientations, record the Euler-factor units, assemble the cubical periodic tensor complex, and prove equivariance after the required character shifts.
Ready to work on
06
Resolve the Three-Basis Frame terminal

Exclude the bounded Three-Basis Frame terminal or extract from it a deletion, pencil, proper subflat, nonunit Bockstein, or nullity-lowering paired deletion that advances the ancestor.

Suggested move: Audit the three-basis frame algebra at rank p-1, with the characteristic-five punctured-pencil exception retained explicitly.
Ready to work on
07
Audit load-bearing exact and high-risk inputs

Independently audit transfer, flat primitivity, basis and line descent, rank-two moments and lift-back, periodic complexes, the cutoff reproducer, and the high-risk large-prime circuit theorem before publication use.

Suggested move: Follow the Revision 11 audit order, starting with FTD/EPT and FPR/CFD, and retain exact scripts and digest-bound artifacts.
Ready to work on
08
Extend lift-back to overfull clusters

Classify or transfer a saturated quotient direction containing more than p labels, which lies outside LPB-1's exact p-cluster hypothesis.

Suggested move: Extend the rank-two moment analysis while retaining predecessor multiplicity, color, and integral ancestor data.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026
Current statusPartially resolved

The conjecture is proved for every non-prime finite field of size at least four and, since 2025, for every prime field of order p>61 except p=79. The remaining small prime fields and p=79 are not settled by these results.

[3][5][2]
External progress

What the literature has established

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

  1. PreprintA new group-ring-identity conjecture was formulated whose truth would imply the remaining AJT cases; this is a reduction, not a full solution.[2]
  2. Peer reviewedNagy and Pach proved a stronger forbidden-coordinate version for every prime p>61 except p=79, removing the dimension restriction in that range.[5]
  3. PreprintYu proved the prime-field conjecture for n×n matrices when n<2^(p−2).[1]
  4. Peer reviewedAlon and Tarsi proved the conjecture whenever the field size is a proper prime power.[3]
5 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusAlon–Jaeger–Tarsi conjecture
Logical consequenceKahn permanent-rank conjecture

The permanent-rank conjecture for the block matrix built from M implies AJT through the Alon–Tarsi polynomial argument.

[1]
Stronger or generalized formDeVos matrix choosability conjecture

Matrix choosability asks for coordinate-wise choices from larger allowed sets and implies AJT in the relevant specialization.

[1]
Dependency or reductioninteger/mod-p group-ring identity implication

For every prime p>3, the proposed group-ring identity implication would yield AJT; it is the active route for the remaining primes.

[2]

Later mathematical changes

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

Gauge-invariant residue and bounded dense terminal addedThe new snapshot makes the Tate-corrected carrier residue canonical and replaces an overbroad dense-boundary conclusion with an explicit Three-Basis Frame terminal and selector obligation.

Changed the research frontierLater mathematical revision

Research stage 12

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

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. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Cited passages corrected
Research-record correctionWe corrected the cited passages. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Cited passages corrected
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
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
Research-record correctionWe corrected the cited passages. We clarified what the cited material supports. 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.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

11 mapped milestonesretained argument map

Browse all 11 mapped stages

  1. stage 1Normalized group-ring obstruction and sufficient route
  2. stage 2Rank-one exposure and exact colon algebra
  3. stage 3Exact rank-two p-cluster moments
  4. stage 4Flat primitivity and ancestral restart
  5. stage 5Compressed packets refute the unrestricted two-way classification
  6. stage 6Periodic Bocksteins correct the rank-one terminal
  7. stage 7Catalecticant second-rung range widened
  8. stage 8Dimension-free basis-block corridor
  9. stage 9Coordinate density yields a large admissible circuit
  10. stage 10Universal line descent reinterprets the dense boundary
  11. stage 11First-rung exact clusters close and catalecticant seams localize
Normalized group-ring obstruction and sufficient routeThe current work fixes a primitive two-basis obstruction and the conditional PCDS→PTB→RST⇔NP2→AJT route.

Mapped research milestoneInitial research sequence

Research stage 1
Rank-one exposure and exact colon algebraRank-one stripping and an exact colon formula exposed a p-packet terminal, initially accompanied by an overclaim that closure-collision was complete.

Mapped research milestoneInitial research sequence

Research stage 2
Exact rank-two p-cluster momentsMoment equations and slope-support rigidity gave a quantified classification of exact rank-two p-clusters.

Mapped research milestoneInitial research sequence

Research stage 3
Flat primitivity and ancestral restartProper support-spanned flats in a minimum failure became exact-transferable and bypassable directly from the ancestor.

Mapped research milestoneInitial research sequence

Research stage 4
Compressed packets refute the unrestricted two-way classificationAn explicit quadratic-residue family produced rank-two zeros that are neither p-parallel nor complete pencils.

Mapped research milestoneInitial research sequence

Research stage 5
Periodic Bocksteins correct the rank-one terminalThe norm corrections became explicit periodic Bocksteins and the rank-one packet was corrected to colored-nonunit but all-factor primitive.

Mapped research milestoneInitial research sequence

Research stage 6
Catalecticant second-rung range widenedStronger basis-plus-one derivative bounds widened candidate two-arboricity exclusions and yielded a high-risk full second rung for p≥47.

Mapped research milestoneInitial research sequence

Research stage 7
Dimension-free basis-block corridorBasis-block descent and one-basis purification produced an exact rank corridor through defect p-2 and isolated the dense boundary.

Mapped research milestoneInitial research sequence

Research stage 8
Coordinate density yields a large admissible circuitTwo-sided support expansion in a minimum failure supplies a PCDS contraction with at least p-1 admissible labels.

Mapped research milestoneInitial research sequence

Research stage 9
Universal line descent reinterprets the dense boundaryProduct-level line descent is always available, so the dense boundary is corrected from a rank-exposure problem to a primitive/defect selector.

Mapped research milestoneInitial research sequence

Research stage 10
First-rung exact clusters close and catalecticant seams localizeLow-defect lift-back removes exact first-rung p-clusters for p≥11, while the reproduced cutoff table isolates the remaining lower-prime seams.

Mapped research milestoneInitial research sequence

Research stage 11

Detailed research inventory

Claims, milestones, and routes in the current map

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

39 standing statements5 proposed statements10 mathematical milestones8 open questions2 narrowed routes18 conditional results1 completed special cases
Statements by mathematical role44 selected mapped statements
  • theorem candidate4 of 444
  • reduction11 of 4411
  • definition1 of 441
  • lemma24 of 4424
  • negative result1 of 441
  • counterexample1 of 441
  • equivalence1 of 441
  • computational claim1 of 441
Selected mathematical clusters8 mathematical clusters
Conjecture and sufficient architectureThe unresolved AJT statement, normalized obstruction, accepted conditional chain, and first selector theorem.6 displayed rows · 1 route included
  • retained route statementAlon–Jaeger–Tarsi conjecture
  • retained route statementAccepted sufficient architectureconditional
  • retained route statementPaired Contraction–Deletion Selectorconditional
  • retained route statementNormalized two-basis group-ring failureintermediate
  • Research targetProve the large-circuit PCDS selectoropen
  • Active routeLarge-circuit PCDS selectorUse coordinate density to choose a contraction with at least p-1 admissible opposite labels, then combine common residues and the multi-deletion simplex to force a zero residue or an ancestor-closing terminal.
Exact transfer, flat primitivity, and support densityThe exact and split transfer interfaces, proper-flat restart, generalized weights, coordinate expansion, and large fundamental circuit.9 displayed rows · 1 route included
  • retained route statementSplit-product thresholdintermediate
  • retained route statementSplit-initial flat transferintermediate
  • retained route statementExact-product transferintermediate
  • retained route statementFlat primitivityconditional
  • retained route statementAncestral flat restartconditional
  • retained route statementGeneralized-weight lower boundintermediate
  • retained route statementCoordinate-flat densityconditional
  • retained route statementLarge admissible fundamental circuitconditional
  • Active routeLarge-circuit PCDS selectorUse coordinate density to choose a contraction with at least p-1 admissible opposite labels, then combine common residues and the multi-deletion simplex to force a zero residue or an ancestor-closing terminal.
Rank-one and rank-two packet algebraExact line colon algebra, the corrected primitive packet terminal, moment classification, compressed counterexamples, and low-defect lift-back.16 displayed rows · 3 routes included
  • retained route statementExact rank-one colon formulaintermediate
  • retained route statementFull closure-collision completion by rank-one exposureconditional
  • retained route statementCorrected closure-collision reductionconditional
  • retained route statementRank-one packet remains all-factor primitiveintermediate
  • retained route statementRank-two p-cluster moment criterionintermediate
  • retained route statementNaive parallel-or-pencil classificationintermediate
  • retained route statementParallel, complete, or compressed slope trichotomyintermediate
  • retained route statementCompressed packet in every odd characteristicspecial case
  • retained route statementRank-two scalar shorteningconditional
  • retained route statementLow-defect exact-cluster lift-backconditional
  • ChallengeRank-one stripping reaches a genuine p-packet whose all-factor module has a unit Bockstein; it does not prove primitive lift-back or the full closure-collision terminal theorem.overclaimed scope · reported resolved
  • ChallengeThe explicit quadratic-residue family is a rank-two modular-zero packet in every odd characteristic with neither p parallel labels nor a complete projective line.counterexample · reported resolved
  • Research targetExtend lift-back to overfull clustersopen
  • Refuted routeRank-one stripping aloneLine stripping reaches a genuine primitive p-packet and therefore does not by itself prove AJT or close the ancestor.
  • Refuted routeParallel-or-pencil-only rank-two classificationExplicit compressed packets refute this route above the low-predecessor threshold; only the quantified h≤(p-1)/2 subrange remains valid.
  • Narrowed routeLow-defect exact-cluster lift-backExact p-clusters in the first post-PQIT rung are closed for p≥11; the surviving packet analysis begins with overfull or sufficiently compressed clusters.
Periodic norm corrections and primitive lift-backThe two explicit periodic Bocksteins, exact paired-residue bridge, and unresolved mapping-cone control of the original unit class.8 displayed rows · 2 routes included
  • retained route statementFirst norm correction is a periodic Bocksteinintermediate
  • retained route statementPair-deletion correction is a periodic Bocksteinintermediate
  • retained route statementExact pair-residue bridgeintermediate
  • retained route statementPrimitive periodic norm lift-backconditional
  • Useful failureTreat every p-divisible quotient product as transporting the original primitive Bockstein value 1.reported failure
  • Research targetProve primitive periodic norm lift-backopen
  • Active routePrimitive periodic norm lift-backBuild a filtered mapping cone for the periodic corrections so the original unit class cannot terminate in the rank-one all-factor unit line without a prior closing event.
  • Not yet justifiedAutomatic primitive functoriality through quotientsProduct-level p-divisibility survives the exact transfers, but the primitive Bockstein can change through norm corrections.
Dimension-free rank descent and dense frontierBasis blocks, one-basis purification, universal line descent, the superseded rank-stuck interpretation, and the active primitive/defect selector.11 displayed rows · 2 routes included
  • retained route statementDimension-free basis-block descentintermediate
  • retained route statementOne-basis purificationconditional
  • retained route statementDense boundary as a rank-exposure problemconditional
  • retained route statementDense one-basis primitive/defect selectorconditional
  • retained route statementTwo-partitionable line transferconditional
  • retained route statementOne-basis line descentconditional
  • retained route statementProduct-level descent to a p-packetconditional
  • ChallengeOBL-1 supplies further product-level rank descent unless a p-packet appears, so the dense boundary is about primitive transport and defect control rather than rank exposure.overclaimed scope · reported resolved
  • Research targetProve the axis-sparse dense selectoropen
  • Narrowed routeDimension-free one-basis corridorBasis-block descent and purification give exact rank reduction through defect p-2, then meet a packet, complete pencil, or the corrected dense primitive/defect boundary.
  • Active routeAxis-sparse dense primitive/defect selectorThe dense boundary is no longer rank-stuck: the active task is to order line descents without losing defect or primitive control, or force packet, flat, compressed-plane, or paired-residue structure.
Catalecticant bounds and residual seamsThe strengthened basis-plus-one bound, superseded early range, broader challenged claims, exact cutoff computation, and seam-specific work.11 displayed rows · 3 routes included
  • retained route statementStrengthened basis-plus-one catalecticant boundintermediate
  • retained route statementEarly two-arboricity exclusion through p+4conditional
  • retained route statementTwo-arboricity exclusion through 2p-7conditional
  • retained route statementFull second-rung circuit exclusion for p≥47conditional
  • retained route statementConditional second-rung exposureconditional
  • retained route statementPrime-specific catalecticant cutoff tablecomputational
  • ComputationSource-reported exact arbitrary-precision evaluation of the basis-plus-one lower and upper formulas for the 17 listed primes. The retained source includes an embedded reproducer, but this projection has no independently bound source report artifact.The source material reports complete second-rung coverage from p=47 onward and residual seams including 88 at p=43, 83–84 at p=41, and 74–76 at p=37. ProofAtlas has not established an independent digest-bound rerun of those rows. · reported unreproduced
  • Research targetClose the residual small-prime catalecticant seamsopen
  • Active routeResidual second-rung catalecticant seamsThe single-catalecticant computation localizes rather than closes the lower-prime problem; work now targets only the recorded residual nullity intervals with stronger flattenings or geometry.
  • Route held in reserveLarge-prime second-rung exposureTAC47 and DFE2-47 provide a candidate exposure route from p=47 onward, but the exact arithmetic and downstream iteration remain nonpublication-load-bearing until independently audited.
  • Useful but insufficientSingle catalecticant as universal closureThe exact inequality exhausts a prime-specific range but cannot classify the residual seams or certify existence at the first unexcluded nullity.
Corrected, refuted, and insufficient routesThe retained witnesses preventing repeated use of automatic primitive transport, rank-one or block descent alone, naive rank-two classification, quotient-zero lift-back, and one-test catalecticant closure.13 displayed rows · 5 routes included
  • Useful failureTreat every p-divisible quotient product as transporting the original primitive Bockstein value 1.reported failure
  • Useful failureIterate line stripping and declare the resulting rank-one packet harmless.reported failure
  • Useful failureClassify every rank-two zero packet as parallel or a complete pencil.reported failure
  • Useful failureUse proper-flat primitivity and rank descent without tracking primitive or nullity data.reported failure
  • Useful failureIterate coordinate basis blocks and treat dimension-free rank descent as a complete proof.reported failure
  • Useful failureTreat a zero quotient descendant as automatically killing the original ancestor product.reported failure
  • Useful failureTreat the dense defect-(p-1) boundary as merely lacking another rank quotient.reported failure
  • Useful failureUse the one basis-plus-one catalecticant inequality as a universal second-rung classification.reported failure
  • Refuted routeRank-one stripping aloneLine stripping reaches a genuine primitive p-packet and therefore does not by itself prove AJT or close the ancestor.
  • Refuted routeParallel-or-pencil-only rank-two classificationExplicit compressed packets refute this route above the low-predecessor threshold; only the quantified h≤(p-1)/2 subrange remains valid.
  • Not yet justifiedAutomatic primitive functoriality through quotientsProduct-level p-divisibility survives the exact transfers, but the primitive Bockstein can change through norm corrections.
  • Not yet justifiedQuotient zero without ancestor closureA quotient packet or pencil does not automatically kill the original integral product; exact lift-back, ancestral restart, or a direct original-product terminal is mandatory.
  • Useful but insufficientSingle catalecticant as universal closureThe exact inequality exhausts a prime-specific range but cannot classify the residual seams or certify existence at the first unexcluded nullity.
independent checking boundaryThe exact chat-derived packet remains provisional; publication use requires the listed algebraic, symbolic, and computational audits.2 displayed rows · 1 route included
  • Research targetAudit load-bearing exact and high-risk inputsopen
  • Route held in reserveLarge-prime second-rung exposureTAC47 and DFE2-47 provide a candidate exposure route from p=47 onward, but the exact arithmetic and downstream iteration remain nonpublication-load-bearing until independently audited.
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 bridgeBuild the exact projective-unit-gauged shifted weight-zero Z/p² mapping cone that combines periodic corrections with the dense deletion simplex.

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.

  • Construct integral chain maps with all projective-unit gauges and cubical signs explicit.
  • Prove the total cone is Gamma-equivariant and identify the primitive generator in total weight zero.
  • Show any surviving same-weight extension routes to an existing ancestor-closing outcome.

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Prepared starting pointConstruct the shifted weight-zero TBF mapping cone

Alon–Jaeger–Tarsi Conjecture · ready to start

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Research contextPrepared context for any AI agent

Given an invertible matrix over a field with at least four elements, can one choose a vector whose coordinates and transformed coordinates are all nonzero?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references5 cited works · next context review by Nov 2, 2026

The mathematical context was checked on Aug 2, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
  2. 2
  3. 3
    A nowhere-zero point in linear mappingsoriginal source · accessed Aug 2, 2026
  4. 4
  5. 5
    The Alon–Jaeger–Tarsi conjecture via group ring identitiespeer reviewed result · accessed Aug 2, 2026

Important qualifications

  • Do not describe AJT as wholly unresolved: the only field sizes left after combining the 1989 and 2025 results are prime sizes at most 61 and 79, subject to any separately proved small-prime cases.
  • The scoped search did not verify a public proof-assistant formalization of the exact conjecture.
  • Empty formalization or computation lists mean that none was verified in this scoped search, not that none exists.

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