ProofAtlas

Topological graph theory · graph coverings · projective-planar embeddings

Negami’s Planar Cover Conjecture

Collaboration beta

Negami’s conjecture says a connected graph has a finite planar cover exactly when it can be embedded in the projective plane.

A connected graphGhas a finite planar cover⇔G↪RP2
Known results and sources
Several lifted graph sheets align over a dark projective-plane surface and begin to flatten into an ivory planar disk, introducing finite planar covers without claiming that the conjecture is proved.
A finite family of graph sheets over a projective-plane surface introduces the cover-versus-embedding question.

Research problem

Exact mathematical statement

For every connected graph GG,

Ghas a finite planar cover⇔G↪RP2.G\text{ has a finite planar cover}\quad\Longleftrightarrow\quad G\hookrightarrow\mathbb{RP}^{2}.

The difficult direction is reduced in the retained literature route to proving that K0=K1,2,2,2K_0=K_{1,2,2,2} has no finite planar cover.

Problem infographic

Problem at a glance

Scientific explainer for Negami's open planar cover conjecture. A locally bijective map from a planar covering graph G tilde to a connected base graph G illustrates that every lifted vertex has the same neighbor star as its image. A projective-plane disk model and the exact question finite planar cover if and only if embedding in RP2 appear beside the seven-vertex test graph K_0 = K_1,2,2,2. A hypothetical planar cover of K_0 must have even fold at least 14, while the general converse remains open.
Negami’s conjecture asks whether finite planar covers characterize connected graphs embeddable in the projective plane. Published work concentrates the unresolved direction in the seven-vertex graph K₀ = K₁,₂,₂,₂; any hypothetical planar cover there must have even fold at least 14.

Current mathematical picture

Where work on Negami’s Planar Cover Conjecture stands

Open conjecture

The retained revision-11 packet develops an exact adjusted-source and path-defect program for a hypothetical minimal planar cover of K₁,₂,₂,₂. It derives working-symbolic local and global identities, a six-family low-score path grammar, seven one-junction Y-atoms, a rigid path-free equality core, and a bounded abstract transfer-wheel carrier. It also preserves explicit failures and corrections: graph incidence alone is not a legal replacement disk, wheel-rim contexts need not factor, and weighted defect decrease alone does not certify the intrinsic induction. The present frontier is to establish complete mouths and embedded transfer-star cells, classify path and wheel states, reduce non-pure defect atoms, and prove regional and semantic descent. The full conjecture remains open; no proof or planar counterexample is claimed.

Strongest supported footholdExact adjusted-deficit spectrum

The global equality forces at least six low-adjusted-source tree components and records the exact contribution of every score stratum.

Evidence posture · Reported result
Leading routeExact adjusted-source spine

Audit and use the pointwise zeta bound, exact component and tree source decompositions, adjusted-deficit spectrum, and low-score census.

Route status · Active route
Useful failureIndependent side-stack encoding

Separate left and right nesting stacks are superseded by one circular noncrossing matching carrying LL, RR, LR, and RL pairs.

Route status · Eliminated route
Main reductionBounded abstract transfer carriers

Pure path-free carriers expose a loop, digon, or 8/10/12 graph-theoretic wheel with four possible low-degree centre types.

Evidence posture · Reported reduction
Completed special caseRigid δA = 13 carrier

The path-free equality layer collapses to twelve YQQQ(2) components and two N₆ poles at tree-component level.

Evidence posture · Reported special case
Priority open bridgeClassify the six low-score path families

For every complete realization of P₀, PF, PFF, P₆, PM, and PMF, including arbitrary neutral periods and nested companions, produce a reciprocal, contractible, or compatibility-pure survivor certificate.

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

Negami’s Planar Cover Conjecture in numbers

4.4kretained lines of mathematical investigation4,441 in the current working snapshot
Argument development
3,630 · 82%
Explored or eliminated routes
124 · 3%
Computational analysis
207 · 5%
Open obligations
212 · 5%
Definitions and setup
268 · 6%
21selected mapped statements12routes investigated10reported milestones10open questions4contribution-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 bySeiya NegamiProposed1986Status checkedAug 2, 2026
Source recordedAug 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 Negami’s Planar Cover ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Negami planar cover conjecture — ChallengedNegami planar coverconjectureConditional eT = 6 rows are unbranched — Depends on missing premiseConditional eT = 6 rows areunbranchedConditional square and doubled-radial alternatives — Depends on missing premiseConditional square anddoubled-radial alternativesFour exact wheel-centre types — Depends on missing premiseFour exact wheel-centretypesPath-or-thirteen threshold — Depends on missing premisePath-or-thirteen thresholdReduction to K₁,₂,₂,₂ — Depends on missing premiseReduction to K₁,₂,₂,₂Rigid δA = 13 equality core — Depends on missing premiseRigid δA = 13 equality coreSeven δA = 14 coarse cases — Depends on missing premiseSeven δA = 14 coarse casesSix paths and seven Y-atoms — Depends on missing premiseSix paths and seven Y-atomsUniform 8/10/12 transfer wheel — ChallengedUniform 8/10/12 transferwheelAt least six low-score trees — Depends on missing premiseAt least six low-score treesCircular companion nesting — ChallengedCircular companion nestingExact adjusted-source spine — activeExact adjusted-source spineSix-family complete-state path classification — activeSix-family complete-statepath classificationEmbedded transfer stars and bounded wheels — activeEmbedded transfer stars andbounded wheelsAtomized path-free defect reduction — activeAtomized path-free defectreductionTreat an abstract loop, digon, or transfer wheel as a ready-made replacement cell. — stoppedTreat an abstract loop,digon, or transfer wheel asa…Factor an extracted wheel's exterior context into an independent Cartesian product over rim vertices. — stoppedFactor an extracted wheel'sexterior context into anindependent…Encode companions with independent left-side and right-side stacks. — stoppedEncode companions withindependent left-side andright-side…Require the raw survivor carrier to be connected and cellular before descent. — stoppedRequire the raw survivorcarrier to be connected andcellular…Independently audit exact identities — OpenIndependently audit exactidentitiesProve the complete mouth contract — OpenProve the complete mouthcontractClassify the six low-score path families — OpenClassify the six low-scorepath familiesExtract complete transfer-star cells — OpenExtract completetransfer-star cellsClassify bounded wheels and nonsimple carriers — BlockedClassify bounded wheels andnonsimple carriersReduce every non-pure path-free defect atom — BlockedReduce every non-purepath-free defect atomReconstruct frontier square and radial certificates — BlockedReconstruct frontier squareand radial certificatesProve embedded survivor and regional closure — BlockedProve embedded survivor andregional closure
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 routeExact adjusted-source spine

Audit and use the pointwise zeta bound, exact component and tree source decompositions, adjusted-deficit spectrum, and low-score census.

Route status · Active route
Active routeSix-family complete-state path classification

Expand the six scalar path families into complete states and classify every realization as reciprocal, contractible, or a compatibility-pure survivor.

Route status · Active route
Active routeEmbedded transfer stars and bounded wheels

Convert abstract loop, digon, and 8/10/12 wheel patterns into complete prime-end cells, then classify the four centre families and their bounded joint contexts.

Route status · Active route
Active routeAtomized path-free defect reduction

After the pure wheel stratum, solve the seven δA = 14 coarse cases and then the general lexicographically controlled atomized defect vector.

Route status · Active route
Active routeEmbedded survivor, regional closure, and semantic descent

Assemble actual realized morphisms into an embedded carrier, prove regional zero sums, cellularize, establish orbit purity, and descend to fibrewise cover compatibility.

Route status · Active route
Active routeHistorical finite reconstruction

Reconstruct the missing revision-7/9 local and resource artifacts as conditional frontier support and regression data.

Route status · Active route

Explored alternatives

Other routes

6 recorded
Narrowed routeConditional eT = 6 regression cells

Retain QQ, component–square digons, and the doubled radial template as small tests for the uniform path, wheel, and junction machinery, without treating them as a complete route.

Route status · Narrowed route
Route held in reserveArbitrary atomic-tree fixed-point fallback

The broader rooted-tree relation saturation remains useful only if the prioritized six-path and bounded-wheel lanes leave an irreducible survivor.

Route status · Route held in reserve
Eliminated routeIndependent side-stack encoding

Separate left and right nesting stacks are superseded by one circular noncrossing matching carrying LL, RR, LR, and RL pairs.

Route status · Eliminated route
Browse 3 more explored routes
Not yet justifiedDirect graph-wheel surgery

A transfer loop, digon, or graph-theoretic wheel is only a bounded target until complete prime-end extraction establishes a legal replacement cell.

Route status · Not yet justified
Not yet justifiedIndependent rim-context factorization

Local rim states may be coupled by cross-interval prime-end identities; the default complete object is one bounded joint context.

Route status · Not yet justified
Useful but insufficientScalar-weight-only defect induction

Lowering the atomized weighted sum is too weak unless the replacement also respects the full lexicographic extremal order.

Route status · Useful but insufficient

Route statements and reductions

Statements the next route can inspect and build on

Route statementPointwise zeta nonnegativity

For every apex a, η(a) ≥ t₄(a) + (q(a)−2)₊ + 2((2−q(a))₊−p(a)); equivalently ζ(a) ≥ 0.

Source-reported route statement · dependencies incomplete
Route statementGeneral component source–branch identity

Every component C satisfies 2s*C = rC + 2ℓC + 2ℬC + 3XC + ζC, and summing yields the global source–branch identity.

Source-reported route statement · dependencies incomplete
Route statementExact tree source decomposition

For a tree component C, σ(C) = γC + ℓC + ℬC + 2XC + ζC.

Source-reported route statement · dependencies incomplete
Route statementExact adjusted-deficit spectrum

The low-score spectrum satisfies 4n₀ + 3n₁ + 2n₂ + n₃ = 24 + 6Σ + 𝔫 + 𝔠 + Ξ.

Source-reported route statement · dependencies incomplete
Route statementAt least six low-score trees

Every hypothetical counterexample has at least six tree components with adjusted source score at most three; positive defect terms can only increase the lower bound.

Source-reported route statement · dependencies incomplete
Route statementSix paths and seven Y-atoms

At scalar level, every tree component with αC ≤ 3 is one of six decorated path families or one of seven exact one-junction Y-atoms; complete boundary-state data remain unresolved.

Source-reported route statement · dependencies incomplete
Route statementFace-atomized defect ledger

After eliminating the dependence among Φ, Ω, and 𝔠, the current work rewrites the Path-Defect Identity using individually nonnegative geometrically named atoms.

Source-reported route statement · dependencies incomplete
Route statementSeven δA = 14 coarse cases

A path-free configuration with δA = 14 has Σ = 0 and lies in one of seven coarse triples (N,K,τ𝒴); positive Q₀ is excluded by the atomized cost.

Source-reported route statement · dependencies incomplete
Route statementQuadrilateral transfer embedding

In the equality core and pure stratum, active tree components and quadrilateral corridors form an embedded plane directed multigraph ΓQ with target indegree bounded by the corresponding sector count EC.

Source-reported route statement · dependencies incomplete
Route statementUniform 8/10/12 transfer wheel

Every pure all-quadrilateral path-free transfer graph contains a loop, a parallel-edge digon, or an embedded graph-theoretic wheel with rim length 8, 10, or 12; this is not yet a complete cover cell.

Source-reported route statement · dependencies incomplete
Route statementFour exact wheel-centre types

The pure-stratum low-degree centre census has four realizable types: one 12-wheel path type, one 10-wheel path type, one unbranched 8-wheel path type, and one branched 8-wheel type; the fifth algebraic row violates ζ-nonnegativity.

Source-reported route statement · dependencies incomplete
Route statementZero-labelled cellularization

A finite embedded graph on the sphere whose oriented labels sum to zero over all boundary components of each complementary region can be connected and cellularized by adding only zero-labelled edges.

Source-reported route statement
Route statementRealized-morphism descent

For actual realized A-equivariant edge morphisms on an embedded carrier, regional zero label sums allow a change of torsor origins that makes every original-edge transport zero.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

10 featured tasks
01
Classify the six low-score path families

For every complete realization of P₀, PF, PFF, P₆, PM, and PMF, including arbitrary neutral periods and nested companions, produce a reciprocal, contractible, or compatibility-pure survivor certificate.

Suggested move: Start with QQ and the six ordinary zero-source words while keeping structural zero-source PM and PMF cases separate.
Ready to work on
02
Extract complete transfer-star cells

Lift every selected transfer loop, digon, or wheel star to an actual prime-end disk or controlled annular sector with complete internal and boundary cover-germ accounting.

Suggested move: Construct one actual extracted wheel disk retaining one contiguous exterior interval per rim occurrence and the joint cross-interval prime-end equality partition.
Ready to work on
03
Prove the complete mouth contract

Define complete directed mouths preserving prime ends, cross-boundary equality, both sides, rotations, projected edge types, selected-edge data, arity, transfer sectors, joint rim context, orientation, and synchrony, then prove the one-/two-/junction mouth counts and safe substitution.

Suggested move: Specify the schema and prove geometry-to-word and substitution completeness before any path or wheel enumeration.
Ready to work on
04
Independently audit exact identities

Recheck the local zeta bound, component and tree source identities, adjusted-deficit spectrum, capacity-slack dependence, Path-Defect Identity, and atomized ledger against revision-9 definitions.

Suggested move: Build the exact-integer checker specified in A.11.1 and run every mandatory regression case.
Ready to work on
05
Prove embedded survivor and regional closure

Construct a finite embedded survivor graph representing every cover germ, closure, transfer remainder, region sector, and non-tree component exactly once, with actual realized morphisms and zero total label around every region.

Suggested move: After R/C classification, assemble only actual survivor occurrences and verify regional sums over all boundary components.
Blocked by the current route
06
Classify bounded wheels and nonsimple carriers

After transfer-star extraction, classify the four centre families, all 8/10/12 wheel states, loops, and parallel-edge digons as R/C/S with complete joint exterior contexts.

Suggested move: Finish the complete-mouth and transfer-star prerequisites, then enumerate reduced centre forms and bounded joint rim contexts.
Blocked by the current route
07
Prove compatibility-orbit purity

Show that every realized survivor state lies in an A-torsor whose zero element has the intended projected-rotation and synchrony meanings.

Suggested move: Define the semantic torsor orbits in the complete state schema and reject any relation joining incompatible projected rotations.
Blocked by the current route
08
Descend to actual cover compatibility

Convert zero transport on the realized survivor carrier into one reversal choice per lifted-vertex fibre and one uniform synchronous or antisynchronous sign per base-edge fibre.

Suggested move: After orbit purity and regional closure, prove the Semantic Compatibility Descent Lemma with explicit fibrewise uniformity.
Blocked by the current route
09
Reduce every non-pure path-free defect atom

For each nonzero atom in the exact path-free ledger, produce a bounded marked cell, a valid intrinsic-coordinate descent, or a Gate-2 path outcome, beginning with the seven δA = 14 cases.

Suggested move: Exhaust the seven δA = 14 coarse triples with certificates tied to the exact atom and first differing intrinsic coordinate.
Blocked by the current route
10
Reconstruct frontier square and radial certificates

Rebuild the historical finite inputs and classify QQ, component–square states, and the doubled radial template as regression cases for the uniform machinery.

Suggested move: Reconstruct the missing revision-7/9 generators, verifiers, and certificates before upgrading any conditional frontier conclusion.
Blocked by the current route

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026
Current statusOpen conjecture

The necessity direction remains open: every projective-planar graph has a finite planar cover, but it is not known whether every connected graph with a finite planar cover embeds in the projective plane. The conjecture is equivalent to K_{1,2,2,2} having no finite planar cover.

[5][4]
External progress

What the literature has established

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

  1. PreprintKoizumi, Suzuki, and Tamura gave a new combinatorial proof of the already-known rotation-compatible-cover case; the unrestricted necessity direction remains open.[5]
  2. PreprintAnnor, Nikolayevsky, and Payne proved that K_{1,2,2,2} has no n-fold planar cover for n<14.[8][2]
  3. PreprintAnnor, Nikolayevsky, and Payne showed that any minimal planar cover of K_{1,2,2,2}, if one exists, must be 4-connected and excluded several structural classes.[4]
  4. Peer reviewedHliněný and Thomas consolidated the reduction to K_{1,2,2,2} and showed that, up to specified constructions, there are at most sixteen possible counterexamples.[6]
13 cited sources4 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusNegami’s Planar Cover Conjecture
Equivalent formulationK_{1,2,2,2} has no finite planar cover

The full conjecture is equivalent to nonexistence of a finite planar cover of the single graph K_{1,2,2,2}.

[4]
Solved special caserotation-compatible planar covers

A connected graph with a finite rotation-compatible planar cover embeds in the projective plane.

[11][5]
Related problemFellows’ planar-emulator conjecture

The analogous assertion for planar emulators is false; K_{1,2,2,2} has a finite planar emulator even though existence of a finite planar cover remains open.

[1]
Stronger or generalized formhigh-genus extensions of Negami’s conjecture

Briański, Davies, and Tan formulate surface-cover analogues and prove bounded-genus and decidability results for parts of the extension.

[3]

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

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.

20 standing statements1 proposed statements10 mathematical milestones10 open questions1 narrowed routes9 conditional results2 completed special cases
Statements by mathematical role21 selected mapped statements
  • theorem candidate1 of 211
  • equivalence1 of 211
  • lemma11 of 2111
  • reduction8 of 218
Selected mathematical clusters7 mathematical clusters
Problem boundary and published reductionThe open conjecture, its K₁,₂,₂,₂ reduction, and the explicit no-proof boundary.3 displayed rows
  • retained route statementNegami planar cover conjecture
  • retained route statementReduction to K₁,₂,₂,₂
  • ChallengeThe conjecture remains open: the mouth, extraction, classification, regional closure, and semantic descent gates have not yet been closed.unsupported step · open
Exact source and adjusted-deficit identitiesLocal zeta accounting, component and tree decompositions, the adjusted spectrum, and the resulting abundance of low-score trees.11 displayed rows · 1 route included
  • retained route statementPointwise zeta nonnegativityintermediate
  • retained route statementGeneral component source–branch identityintermediate
  • retained route statementExact tree source decompositionintermediate
  • retained route statementExact adjusted-deficit spectrumintermediate
  • retained route statementAt least six low-score treesintermediate
  • DerivationFor a tree, the incidence identity fixes the terminal count; substituting the summed local residual formula eliminates it and yields the exact source decomposition.active reported
  • DerivationSum the exact tree sources, substitute the global source formula and tree-component count, and collect non-tree and capacity terms into the stated deficit spectrum.active reported
  • DerivationThe exact positive deficit is at least 24, while each α≤3 component contributes at most four, forcing at least six such trees.active reported
  • Research targetIndependently audit exact identitiesopen
  • ComputationRevision-11 exact-integer verifier specified for local source identities, the low-score grammar, path-defect equations, δA = 13 and 14 cases, transfer degrees, and wheel-centre budgets.The current work specifies required checks and filenames but does not retain a runnable implementation or report a reproduced run in this preview. · reported unreproduced
  • Active routeExact adjusted-source spineAudit and use the pointwise zeta bound, exact component and tree source decompositions, adjusted-deficit spectrum, and low-score census.
Low-score paths and path-free defect ledgerThe six path families, seven Y-atoms, path-or-thirteen threshold, rigid equality core, and first δA = 14 perturbation layer.14 displayed rows · 3 routes included
  • retained route statementSix paths and seven Y-atomsintermediate
  • retained route statementExact Path-Defect Identityintermediate
  • retained route statementPath-or-thirteen thresholdconditional
  • retained route statementFace-atomized defect ledgerintermediate
  • retained route statementRigid δA = 13 equality corespecial case
  • retained route statementSeven δA = 14 coarse casesspecial case
  • DerivationSet the small-path weight to zero. Every other term in the exact identity is nonnegative, so the left deficit cannot balance unless δA≥13.active reported
  • DerivationAt equality every nonnegative defect vanishes; the explicit small-Y table forces twelve YQQQ(2) atoms, and the remaining two tree components are N₆ paths.active reported
  • Research targetClassify the six low-score path familiesopen
  • Research targetReduce every non-pure path-free defect atomblocked
  • Useful failureUse decrease of the scalar atomized defect weight alone as the Type-C induction order.reported failure
  • Active routeSix-family complete-state path classificationExpand the six scalar path families into complete states and classify every realization as reciprocal, contractible, or a compatibility-pure survivor.
  • Active routeAtomized path-free defect reductionAfter the pure wheel stratum, solve the seven δA = 14 coarse cases and then the general lexicographically controlled atomized defect vector.
  • Useful but insufficientScalar-weight-only defect inductionLowering the atomized weighted sum is too weak unless the replacement also respects the full lexicographic extremal order.
Quadrilateral-transfer carriersPlane transfer graphs, the bounded wheel theorem, four centre types, and the missing geometric extraction bridge.15 displayed rows · 3 routes included
  • retained route statementQuadrilateral transfer embeddingconditional
  • retained route statementUniform 8/10/12 transfer wheelconditional
  • retained route statementFour exact wheel-centre typesconditional
  • DerivationSaturation of outgoing edges and target-sector indegrees gives the plane degree formulas. A nonsimple graph yields a loop or digon; a simple graph is a triangulation and a low-terminal score-four vertex supplies a bounded wheel.active reported
  • DerivationThe score-four equation gives five algebraic rows, and the pointwise q=3 zeta lower bound excludes the row with d=1, X=1, γ=2, ζ=0.active reported
  • ChallengeThe transfer wheel is an abstract incidence pattern. It does not itself expose all cover germs, complete cuts, rim intervals, prime-end equalities, or a substitution-safe disk.overclaimed scope · open
  • Useful failureTreat an abstract loop, digon, or transfer wheel as a ready-made replacement cell.reported failure
  • Useful failureFactor an extracted wheel's exterior context into an independent Cartesian product over rim vertices.reported failure
  • Useful failureTreat the pure-stratum wheel theorem as a solution of the entire path-free branch.reported failure
  • Research targetExtract complete transfer-star cellsopen
  • Research targetClassify bounded wheels and nonsimple carriersblocked
  • ComputationFinite-state generator and independent verifier specified for loops, digons, four centre types, 8/10/12 wheels, complete rim intervals, and joint exterior contexts.The bounded scalar families are described, but no generator, verifier, certificate archive, or digest is recorded as completed evidence. · reported unreproduced
  • Active routeEmbedded transfer stars and bounded wheelsConvert abstract loop, digon, and 8/10/12 wheel patterns into complete prime-end cells, then classify the four centre families and their bounded joint contexts.
  • Not yet justifiedDirect graph-wheel surgeryA transfer loop, digon, or graph-theoretic wheel is only a bounded target until complete prime-end extraction establishes a legal replacement cell.
  • Not yet justifiedIndependent rim-context factorizationLocal rim states may be coupled by cross-interval prime-end identities; the default complete object is one bounded joint context.
Companion nesting and global semanticsOne circular companion system, regional cellularization, realized affine morphisms, and the remaining orbit-purity and fibrewise semantic obligations.17 displayed rows · 3 routes included
  • retained route statementCircular companion nestingconditional
  • retained route statementZero-labelled cellularizationconditional
  • retained route statementRealized-morphism descentconditional
  • DerivationThis is a proposed program dependency rather than a finished derivation: the circular relation system is intended to represent survivor cells before regional closure and cellularization.proposed
  • DerivationAfter zero-labelled cellularization, the realized labels form a cellular 1-cocycle on S². Vanishing H¹ makes it a coboundary, so changing local origins trivializes every original edge transport.active reported
  • ChallengeThe topological noncrossing theorem is conditional on first proving the complete mouth contract and disjointness at one recursive level; it does not construct those prerequisites.unsupported step · open
  • ChallengeAbstractly trivialized torsor transport is insufficient until compatibility-orbit purity and fibrewise uniform edge synchrony are proved for actual cover semantics.overclaimed scope · open
  • Useful failureEncode companions with independent left-side and right-side stacks.reported failure
  • Useful failureRequire the raw survivor carrier to be connected and cellular before descent.reported failure
  • Useful failureAssign one common A-translation to every element of a coarse Type-S relation.reported failure
  • Research targetProve the complete mouth contractopen
  • Research targetProve embedded survivor and regional closureblocked
  • Research targetProve compatibility-orbit purityblocked
  • Research targetDescend to actual cover compatibilityblocked
  • Active routeEmbedded survivor, regional closure, and semantic descentAssemble actual realized morphisms into an embedded carrier, prove regional zero sums, cellularize, establish orbit purity, and descend to fibrewise cover compatibility.
  • Eliminated routeIndependent side-stack encodingSeparate left and right nesting stacks are superseded by one circular noncrossing matching carrying LL, RR, LR, and RL pairs.
  • Route held in reserveArbitrary atomic-tree fixed-point fallbackThe broader rooted-tree relation saturation remains useful only if the prioritized six-path and bounded-wheel lanes leave an irreducible survivor.
Conditional finite frontierThe unreconstructed eT = 6 finite tables and their bounded square, path, and doubled-radial regression targets.7 displayed rows · 2 routes included
  • retained route statementConditional eT = 6 rows are unbranchedconditional
  • retained route statementConditional square and doubled-radial alternativesconditional
  • DerivationUnder the finite tables, source counting fixes the ordinary and two-absorber assemblies; planar incidence bounds then expose a QQ path, square digon, or doubled-radial template.active reported
  • Research targetReconstruct frontier square and radial certificatesblocked
  • ComputationHistorical revision-7/9 finite reconstruction lane covering the 1,359 augmented local records, eT = 5 and 6 assignments, low-source component rows, and port certificates.The current work reports these finite results conditionally because the original generator, verifier, and certificate archive is missing; this page does not upgrade them. · reported unreproduced
  • Narrowed routeConditional eT = 6 regression cellsRetain QQ, component–square digons, and the doubled radial template as small tests for the uniform path, wheel, and junction machinery, without treating them as a complete route.
  • Active routeHistorical finite reconstructionReconstruct the missing revision-7/9 local and resource artifacts as conditional frontier support and regression data.
Current route dispositionsThe active uniform program, narrowed regression lanes, paused fallback, and explicitly invalid or too-weak shortcuts.19 displayed rows · 12 routes included
  • Useful failureTreat an abstract loop, digon, or transfer wheel as a ready-made replacement cell.reported failure
  • Useful failureFactor an extracted wheel's exterior context into an independent Cartesian product over rim vertices.reported failure
  • Useful failureEncode companions with independent left-side and right-side stacks.reported failure
  • Useful failureRequire the raw survivor carrier to be connected and cellular before descent.reported failure
  • Useful failureAssign one common A-translation to every element of a coarse Type-S relation.reported failure
  • Useful failureUse decrease of the scalar atomized defect weight alone as the Type-C induction order.reported failure
  • Useful failureTreat the pure-stratum wheel theorem as a solution of the entire path-free branch.reported failure
  • Active routeExact adjusted-source spineAudit and use the pointwise zeta bound, exact component and tree source decompositions, adjusted-deficit spectrum, and low-score census.
  • Active routeSix-family complete-state path classificationExpand the six scalar path families into complete states and classify every realization as reciprocal, contractible, or a compatibility-pure survivor.
  • Active routeEmbedded transfer stars and bounded wheelsConvert abstract loop, digon, and 8/10/12 wheel patterns into complete prime-end cells, then classify the four centre families and their bounded joint contexts.
  • Active routeAtomized path-free defect reductionAfter the pure wheel stratum, solve the seven δA = 14 coarse cases and then the general lexicographically controlled atomized defect vector.
  • Active routeEmbedded survivor, regional closure, and semantic descentAssemble actual realized morphisms into an embedded carrier, prove regional zero sums, cellularize, establish orbit purity, and descend to fibrewise cover compatibility.
  • Narrowed routeConditional eT = 6 regression cellsRetain QQ, component–square digons, and the doubled radial template as small tests for the uniform path, wheel, and junction machinery, without treating them as a complete route.
  • Active routeHistorical finite reconstructionReconstruct the missing revision-7/9 local and resource artifacts as conditional frontier support and regression data.
  • Route held in reserveArbitrary atomic-tree fixed-point fallbackThe broader rooted-tree relation saturation remains useful only if the prioritized six-path and bounded-wheel lanes leave an irreducible survivor.
  • Eliminated routeIndependent side-stack encodingSeparate left and right nesting stacks are superseded by one circular noncrossing matching carrying LL, RR, LR, and RL pairs.
  • Not yet justifiedDirect graph-wheel surgeryA transfer loop, digon, or graph-theoretic wheel is only a bounded target until complete prime-end extraction establishes a legal replacement cell.
  • Not yet justifiedIndependent rim-context factorizationLocal rim states may be coupled by cross-interval prime-end identities; the default complete object is one bounded joint context.
  • Useful but insufficientScalar-weight-only defect inductionLowering the atomized weighted sum is too weak unless the replacement also respects the full lexicographic extremal order.
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 bridgeFor every complete realization of P₀, PF, PFF, P₆, PM, and PMF, including arbitrary neutral periods and nested companions, produce a reciprocal, contractible, or compatibility-pure survivor certificate.

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.

  • Every complete path state has an R/C/S certificate.
  • No neutral-length or companion-depth cutoff is used.
  • Type-S occurrences are compatibility-orbit-pure realized affine bijections.

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Prepared starting pointClassify the six low-score path families

Negami’s Planar Cover Conjecture · ready to start

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Research contextPrepared context for any AI agent

Negami’s conjecture says a connected graph has a finite planar cover exactly when it can be embedded in the projective plane.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references13 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
    On high genus extensions of Negami's conjecturepreprint · accessed Aug 2, 2026
  4. 4
  5. 5
  6. 6
    On possible counterexamples to Negami’s Planar Cover Conjecturepeer reviewed result · accessed Aug 2, 2026
  7. 7
    20 Years of Negami’s Planar Cover Conjectureoriginal source · accessed Aug 2, 2026
  8. 8
    K_{1,2,2,2} has no n-fold planar cover for n<14peer reviewed result · accessed Aug 2, 2026
  9. 9
    The spherical genus and virtually planar graphsoriginal source · accessed Aug 2, 2026
  10. 10
    Projective-planar double coverings of graphspeer reviewed result · accessed Aug 2, 2026
  11. 11
  12. 12
    List of unsolved problems in mathematicsencyclopedia · accessed Aug 2, 2026
  13. 13
    Planar coverencyclopedia · accessed Aug 2, 2026

Important qualifications

  • Some sources say ‘proposed in 1986’ while the conjecture's first journal publication is dated 1988; the metadata keeps those as separate proposal and publication dates.
  • Empty formalization or computation lists mean that none was verified in this scoped search, not that none exists.

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