V12 reports top-map injectivity and H̃₅ = 0 for PSp₈(5), closing the top-degree QD route while leaving ordinary homology to other certificates.
Evidence posture · Reported resultFinite group theory · subgroup posets · rational homology
Rational Homological Quillen Conjecture at p = 2
Collaboration betaFor a finite group with no nontrivial normal 2-subgroup, the conjecture predicts nonzero rational reduced homology in the poset of its nontrivial elementary abelian 2-subgroups.
Known results and sources
Research problem
Exact mathematical statement
For a finite group , let be its largest normal 2-subgroup and let be the poset of nontrivial elementary abelian 2-subgroups. The conjecture is
The displayed homology is reduced homology with rational coefficients.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Rational Homological Quillen Conjecture at p = 2 stands
Version 15 remains the governing source and explicitly supersedes the v13 constant-valence and Δ_n arguments. The v13 lower bound 118|𝓔₄ᶠʳ| and the claim that 11,400 is a failed high-rank source-target comparison are historical only. The current field-five formulation is the repaired fixed-P crosscut packet: at n=4 the selected crosscut source kills private and exceptional residues, K° kills the pure-Pauli residues, fixed literal Pauli-plane packets cancel root-gain residues, and commutator-rank isolation prevents a rank-six boundary. Subject to the stated promotion dependencies, this supplies the source-reported degree-four working class. All claims remain provisional or conditional as stated; ProofAtlas did not independently rerun the computations, and the conjecture remains open.
Determine the theorem-numbered component contract, certificate types it accepts, actual split elementary extensions, characteristic-three relevance, and complete residual-family ledger before expanding local exception work.
Route status · Active routeThe proposed degree-five-to-degree-four attachment route is eliminated because commutator rank forbids the required incidence.
Route status · Refuted routeThe formal splice combines a strong marker block with one commuting residual finite-order element whose fixed Euler value is nonzero modulo the marker prime.
Evidence posture · Reported reductionAt n=4 the retained intersection is the explicitly defined selected crosscut source: it kills the exceptional residues and suffices for the reported degree-four class, but it is not asserted to be the complete private-wall kernel.
Evidence posture · Source-reported route statement · dependencies incompleteIndependently reconstruct the projective endpoint classification, selected-crosscut/residue theorem, Latin relative chain model, fixed-P double centralizer, panel counts and signs, orthogonal moment bound, rank-six wall audit, and the distinct top-injectivity package required by v15.
Task status · Prerequisites still openWe 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 unchangedWork mapped so far
Rational Homological Quillen Conjecture at p = 2 in numbers
13kretained lines of mathematical investigation12,958 in the current working snapshot
- Argument development
- 10,833 · 84%
- Explored or eliminated routes
- 541 · 4%
- Computational analysis
- 540 · 4%
- Open obligations
- 452 · 3%
- Definitions and setup
- 592 · 5%
22selected mapped statements10routes investigated7reported milestones9open 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.
Recommended next task
Retain independently reproducible certificate artifacts
Create the deterministic endpoint, wall, Fourier-orbit, contraction, and marker-scope artifacts listed by v13 before any source-reported package is promoted.
Suggested move: Implement the named deterministic tests for endpoint commutator ranks, relative sharedness, n = 6 Fourier counts, contraction maps, and marker-peeling scope, retaining inputs, outputs, and digests.
What would count as progress
- Every promoted computation has exact input, implementation, output, and certificate digests.
- Independent reruns distinguish reproduced computations from packet-reported values.
- Group-theoretic classification dependencies are cited and separately audited.
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
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.
Determine the theorem-numbered component contract, certificate types it accepts, actual split elementary extensions, characteristic-three relevance, and complete residual-family ledger before expanding local exception work.
Route status · Active routePromote the v15 selected-crosscut, Latin-boundary, fixed-P completion-cancellation, and commutator-rank-isolation degree-four packet together with the distinct rank-six top-injectivity certificate; do not reuse the v13 quantitative bound.
Route status · Active routeAudit the family-specific Chevalley and full-automorphism inputs that turn the formal strong-marker identity into arbitrary-layer theorems for the listed field-five families.
Route status · Active routeExplored alternatives
Other routes
Retain the complete v15 local Fourier tables as regression evidence and promote cancellation among distinct frame completions with one fixed literal Pauli plane; no source-only local orbit remains to be found.
Route status · Narrowed routePeel the field-five block, close every residual code orbit detected by one retained cyclic witness, and reserve new bi-filtered or multi-prime machinery for the common vanishing locus.
Route status · Narrowed routeRetain the exact fixed-poset identity and rank-one code case as a fallback and independent topology problem, but do not resume entangled diagonal codes while the strong Sylow-five marker remains viable.
Route status · Route held in reserveBrowse 4 more explored routes
The proposed degree-five-to-degree-four attachment route is eliminated because commutator rank forbids the required incidence.
Route status · Refuted routeV15 forbids the old Δₙ incidence formula and the interpretation of 11,400 as a failed source-target inequality; this route is historical only.
Route status · Eliminated routeA strong marker cannot be combined automatically with an arbitrary no-source class, a distinct marker prime, or an unrelated noncyclic marker system.
Route status · Not yet justifiedKeep the explicit Clifford-equivariant finite map paused unless the exact source audit shows that characteristic three remains in the residual component ledger.
Route status · Route held in reserveRoute statements and reductions
Statements the next route can inspect and build on
The current work's working minimal-counterexample reduction directly consumes the Quillen dimension property only for each exact occurring split elementary outer 2-extension LB ≅ L ⋊ B; strong markers, fixed-Euler certificates, no-source classes, and lower-degree homology require a separate component or layer theorem.
Source-reported route statement · dependencies incompleteSubject to the retained rank-six endpoint and wall-centralizer audits, the top map is injective and H̃₅(𝒜₂(PSp₈(5)); ℚ) = 0, so the simple group fails (QD)₂.
Source-reported route statement · dependencies incompleteNo rank-five frame–Pauli endpoint in PSp₈(5) lies in a rank-six elementary abelian subgroup: restriction from a nondegenerate rank-six alternating form has rank four on every hyperplane, while the frame–Pauli commutator form has rank two.
Source-reported route statement · dependencies incompleteFor a complete isotypic block M = Lʳ with coordinate product Q of a strong odd-prime marker, every intermediate N ≤ H ≤ Aut(N) satisfies 𝒜₂(H)^Q = 𝒜₂(C_H(M)).
Source-reported route statementThe current work reports strong Sylow-five marker packages for arbitrary isotypic layers of PSp₂ₙ(5) and PΩ₂ₙ₊₁(5) for n ≥ 3 and PΩ⁺₂ₙ(5) for n ≥ 5, with the minus-type family still conditional on the exact relative-root and full-automorphism audit.
Source-reported route statement · dependencies incompleteAt n=4 the retained intersection is the explicitly defined selected crosscut source: it kills the exceptional residues and suffices for the reported degree-four class, but it is not asserted to be the complete private-wall kernel.
Source-reported route statement · dependencies incompleteSubject to the v15 promotion dependencies, the repaired crosscut, Latin-boundary, and fixed literal Pauli-plane packet reports H̃ₙ(𝒜₂(PSp₂ₙ(5)); ℚ) ≠ 0 for n≥3 with n≠4; at n=4 it reports H̃₅(𝒜₂(PSp₈(5)); ℚ)=0 and H̃₄(𝒜₂(PSp₈(5)); ℚ)≠0. The n=4 source is selected rather than a claimed complete private-wall kernel. No old Δₙ incidence formula or v13 118|𝓔₄ᶠʳ| bound is used.
Source-reported route statement · dependencies incompleteMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Retain independently reproducible certificate artifacts
Create the deterministic endpoint, wall, Fourier-orbit, contraction, and marker-scope artifacts listed by v13 before any source-reported package is promoted.
Suggested move: Implement the named deterministic tests for endpoint commutator ranks, relative sharedness, n = 6 Fourier counts, contraction maps, and marker-peeling scope, retaining inputs, outputs, and digests.Ready to work on
Promote global fixed-P completion cancellation
Use the complete v15 local Fourier tables as regression evidence and prove the global cancellation among distinct frame completions with one fixed literal Pauli plane.
Suggested move: Construct the global fixed-P completion-cancellation chain and its residue signs; do not search again for a source-only local Fourier orbit.Ready to work on
Resolve the q=9 split outer-lift table
Settle the q=9 diagonal-field split-lift question before assigning that extension a QD status.
Suggested move: Either exclude an involutory split lift in the exact projective group or analyze the exact extension directly, and audit the order-41 class in every actual outer lift.Ready to work on
Audit the unitary type-A obstruction set
Calculate the exact unitary duality, field-graph, diagonal, and central-quotient obstruction labels for PSU_n(q).
Suggested move: Treat unitary groups in a separate theorem rather than importing the PSL_n(q) graph argument.Ready to work on
Promote the repaired PSp₈(5) fixed-P crosscut package
Independently reconstruct the projective endpoint classification, selected-crosscut/residue theorem, Latin relative chain model, fixed-P double centralizer, panel counts and signs, orthogonal moment bound, rank-six wall audit, and the distinct top-injectivity package required by v15.
Suggested move: Promote the nine-part v15 work packet C without using the old Δₙ formula or calling the n=4 selected source the full private-wall kernel.Prerequisites still open
Audit the exact source and actual-extension contract
Identify the exact Piterman–Smith theorem numbers and posets, determine which certificate types the component reduction consumes, and enumerate actual split elementary outer 2-extensions and residual families.
Suggested move: Record exact theorem citations, source posets, admissible local inputs, characteristic-three relevance, and A/B/C/D including D₄ plus exceptional, sporadic, Suzuki/Ree, and low-rank rows.Prerequisites still open
Complete the residual family and low-rank ledger
Close or explicitly route the odd type-A, defining-characteristic, exceptional, sporadic, Suzuki/Ree, low-rank B₂ = C₂, and D₄ triality rows required by the audited component theorem.
Suggested move: After the source audit fixes the required level and actual extensions, turn V13-8 into a theorem-numbered table with no inherited lower-layer current labels.Prerequisites still open
Promote the field-five strong-marker theorem
Reconstruct the Chevalley lower-central-series, simple-quotient root generators, and full-automorphism pointwise centralizers needed by each asserted field-five marker family.
Suggested move: Check diagonal and graph automorphisms, the relative Bₙ₋₁ model for Dₙ⁻, and the explicit C₃, B₃, D₅⁺, and D₅⁻ hostile cases.Prerequisites still open
Close the residual marker/no-source splice
After peeling the PSL₄(5) block, test each retained cyclic fixed-Euler witness on every residual PSL₃(4) graph-field code orbit and isolate only the orbits where every mark vanishes.
Suggested move: Enumerate the residual code orbits, record order-three, order-five, and order-seven fixed Euler values modulo five, and design a new theorem only for the common vanishing locus.Prerequisites still open
Sourced mathematical context
The known mathematical landscape
Current statusOpen conjecture
The rational-homological strong version remains open at p=2. A July 2026 paper proves rational-homology nonvanishing for groups with a component of Lie type in characteristic p under mild inductive assumptions, excluding those components from a minimal counterexample, but explicitly identifies p=2 as still open.
[3]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
PreprintUnder mild inductive assumptions, a component that is a simple group of Lie type in characteristic p forces nonzero rational homology of the Quillen poset and therefore cannot occur in a minimal counterexample.[3] Peer reviewedPiterman and Smith extend the Aschbacher-Smith main theorem from p>5 to p=3 and p=5 and record partial results toward p=2.[7] PreprintPiterman proves the Quillen dimension property at p=2 for 2-extensions of exceptional finite simple groups of Lie type in odd characteristic, with finitely many exceptions, narrowing possible components of a minimal counterexample.[2] Peer reviewedPiterman introduces centralizer attachment methods, proves equivalence of the original conjecture with its integral-acyclic version, and establishes the rationally acyclic strong version for groups of p-rank at most 4.[6]
Mathematical neighborhood
Related results and reusable starting points
Current focusRational-Homological Quillen Conjecture at p=2
Nonzero reduced rational homology rules out contractibility, so the rational-homological formulation is the strong Q-acyclic version studied by Piterman.
[6]The original contractibility conjecture is equivalent to the integral-acyclic formulation.
[6]Nonzero homology in maximal possible degree implies the rational-homological conclusion and hence Quillen's conjecture for that group.
[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.
PSp₁₂(5) Fourier supplement is made reconstructibleVersion 15 records the eleven target orbits, all eight fine transition rows over fifteen roots, exact regression totals, and source-reported byte-for-byte rerun status for the PSp₁₂(5) Fourier certificate.
V15 reproducibility repairChanged the research frontierLater mathematical revision
Actual-action and family scopes are correctedVersion 15 adds the missing inner-rigidity premise and narrows the exceptional-rank, type-A, and q=9 claims to their justified scopes.
V15 scope and premise correctionsChanged the research frontierLater mathematical revision
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. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
Research-record correctionWe removed a duplicate or outdated task or route step. The mathematical claims and their status did not change.
Claim connections clarifiedCorrected the research recordCorrection note
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.
9 mapped milestonesretained argument map
Browse all 9 mapped stages
- stage 1Exact actual-extension working contract retained
- stage 2Field-five arbitrary-layer marker packages reported
- stage 3PSp₈(5) top degree closed in v12
- stage 4Four-gain QD cases reported for PSp₆(5) and PSp₁₀(5)
- stage 5Marker peeling narrowed to one residual Euler witness
- stage 6PSp₈(5) degree-four packet corrected to surviving homology
- stage 7Four-gain construction and endpoint exhaustion scopes separated
- stage 8Order-313 diagonal-code seam retained only as fallback
- stage 9High-rank symplectic frontier refined to the n = 6 sector map
Exact actual-extension working contract retainedThe current work organizes the open conjecture around the exact QD status of each actual occurring split elementary outer 2-extension.
Research stage 1Mapped research milestoneInitial research sequence
Field-five arbitrary-layer marker packages reportedThe retained v12 layer reports strong Sylow-five marker exits for broad field-five classical isotypic layers while preserving separate local-QD and audit requirements.
Research stage 2Mapped research milestoneInitial research sequence
PSp₈(5) top degree closed in v12The retained v12 layer reports H̃₅ = 0 and failure of local QD for the simple group, while at that stage treating the positive degree-four packet as only a candidate.
Research stage 3Mapped research milestoneInitial research sequence
Four-gain QD cases reported for PSp₆(5) and PSp₁₀(5)The current work reports positive four-gain top classes for n = 3 and n = 5, conditional on the shared endpoint and residue audits.
Research stage 4Mapped research milestoneInitial research sequence
Marker peeling narrowed to one residual Euler witnessThe current work reports that one strong marker block can be combined with one commuting residual finite-order element whose fixed Euler value is nonzero modulo the marker prime.
Research stage 5Mapped research milestoneInitial research sequence
PSp₈(5) degree-four packet corrected to surviving homologyV13 reports that commutator-rank isolation makes the rank-five frame–Pauli family inclusion-maximal, so the positive restricted kernel survives and H̃₄ ≠ 0.
Research stage 6Mapped research milestoneInitial research sequence
Four-gain construction and endpoint exhaustion scopes separatedV13 retains the frame–Pauli construction for all n ≥ 3 but restricts maximum-endpoint exhaustion to n ≠ 4 and uses relative sharedness at n = 4.
Research stage 7Mapped research milestoneInitial research sequence
Order-313 diagonal-code seam retained only as fallbackV13 restores the exact fixed-poset identity and rank-one code case but keeps entangled diagonal codes off the main line while the strong marker remains viable.
Research stage 8Mapped research milestoneInitial research sequence
High-rank symplectic frontier refined to the n = 6 sector mapV13 replaces another aggregate dimension attempt with exact character-orbit, coefficient-system, and contracted-wall calculations for PSp₁₂(5).
Research stage 9Mapped research milestoneInitial research sequence
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 statements2 proposed statements7 mathematical milestones9 open questions2 narrowed routes9 conditional results7 completed special cases
- theorem candidate
1 of 22 1 - reduction
4 of 22 4 - negative result
4 of 22 4 - lemma
11 of 22 11 - computational claim
2 of 22 2
Selected mathematical clusters6 mathematical clusters
Conjecture boundary and component contractThe open conjecture, exact actual-extension working reduction, source-contract challenge, and theorem-numbered residual-family obligations.9 displayed rows · 2 routes included
- retained route statementRational homological Quillen conjecture at p = 2
- retained route statementExact actual-extension component contractconditional
- retained route statementPSp₈(5) actual-extension status splitspecial case
- Recorded relationshipThe current work organizes a minimal-counterexample proof by determining which exact local extension theorem the component reduction consumes.reduces to · reported by source
- ChallengeThe current work explicitly leaves the exact theorem numbers, posets, actual split outer-extension tables, and admissible certificate types as publication dependencies, so the working global contract has not been independently sourced here.unsupported step · open
- Research targetAudit the exact source and actual-extension contractopen
- Research targetComplete the residual family and low-rank ledgeropen
- Active routeExact source and actual-extension auditDetermine the theorem-numbered component contract, certificate types it accepts, actual split elementary extensions, characteristic-three relevance, and complete residual-family ledger before expanding local exception work.
- Route held in reservePSp₈(9):2 conditional finite mapKeep the explicit Clifford-equivariant finite map paused unless the exact source audit shows that characteristic three remains in the residual component ledger.
PSp₈(5) corrected degree pictureThe v12 candidate and v13 quantitative route are historical; v15 retains the conditional field-five theorem only through its repaired fixed-P crosscut packet, with the distinct reported degree-five vanishing at n=4.32 displayed rows · 2 routes included
- supersededHistorical v12 PSp₈(5) degree-four candidatespecial case
- supersededSuperseded v13 PSp₈(5) quantitative degree-four claimspecial case
- retained route statementReported PSp₈(5) top-degree vanishingspecial case
- retained route statementPSp₈(5) endpoint commutator typesspecial case
- retained route statementCommutator-rank isolationintermediate
- supersededSuperseded v13 four-gain kernel boundcomputational
- retained route statementInclusion-maximal full-chain injection
- retained route statementPSp₈(5) actual-extension status splitspecial case
- Recorded relationshipV15 invalidates the v13 Δ-based premise or routing interpretation used by this historical edge. It is not a current support edge; use the repaired fixed-P crosscut formulation.refutes · contested
- Recorded relationshipThe incompatible commutator ranks of the two endpoint geometries are the classification input for inclusion-maximality.supports · reported by source
- Recorded relationshipV15 invalidates the v13 Δ-based premise or routing interpretation used by this historical edge. It is not a current support edge; use the repaired fixed-P crosscut formulation.supports · contested
- Recorded relationshipV15 invalidates the v13 Δ-based premise or routing interpretation used by this historical edge. It is not a current support edge; use the repaired fixed-P crosscut formulation.supports · contested
- Recorded relationshipV15 invalidates the v13 Δ-based premise or routing interpretation used by this historical edge. It is not a current support edge; use the repaired fixed-P crosscut formulation.supports · contested
- Recorded relationshipV15 invalidates the v13 Δ-based premise or routing interpretation used by this historical edge. It is not a current support edge; use the repaired fixed-P crosscut formulation.supports · contested
- Recorded relationshipTop-degree vanishing is the reason the trivial outer extension fails the local QD contract.supports · reported by source
- Recorded relationshipV15 invalidates the v13 Δ-based premise or routing interpretation used by this historical edge. It is not a current support edge; use the repaired fixed-P crosscut formulation.supports · contested
- DerivationScalar commutator forms restrict under subgroup inclusion. A rank-five hyperplane in a nondegenerate six-dimensional alternating space has commutator rank four, whereas a frame–Pauli endpoint has rank two, so such an endpoint cannot lie in a rank-six endpoint.active reported
- DerivationHistorical v13 derivation only. Its 118|𝓔₄ᶠʳ| kernel premise comes from the superseded Δ₄ calculation, so it is not a current derivation of degree-four homology. V15 replaces it with the fixed-P selected-crosscut route.invalidated
- DerivationThe simple group has ordinary degree-four homology but zero top degree, so it fails QD; the separately retained diagonal tensor package assigns QD to PGSp₈(5), making the actual induced outer subgroup decisive.active reported
- ChallengeV13 reports that commutator rank forbids any rank-five frame–Pauli endpoint from lying in a rank-six Pauli endpoint, so the attachment mechanism invoked by v12 does not exist.counterexample · reported resolved
- ChallengeThe challenged v13 derivation has been invalidated because its Δ₄ kernel premise is superseded. The separate v15 fixed-P crosscut claim remains provisional and is governed by the repaired promotion obligation.unsupported step · withdrawn
- Useful failureProve the local QD input for the trivial outer extension PSp₈(5) in maximum degree.reported failure
- Useful failureCompute an attachment map from rank-six Pauli endpoints into the rank-five frame–Pauli four-gain packet.reported failure
- Research targetPromote the repaired PSp₈(5) fixed-P crosscut packageopen
- ComputationHistorical v13 constant-valence four-gain calculation, retained only as an invalidated computation.V15 states that the Δ₄=118 calculation and 118|𝓔₄ᶠʳ| lower bound are not exact group-incidence results and must not be used. The current degree-four formulation is the repaired fixed-P selected-crosscut packet. · reported unreproduced
- ComputationReported complete top-map injectivity calculation for the rank-six triple-Pauli endpoint family of PSp₈(5).The current work reports that every rank-five wall has a unique rank-six completion and concludes H̃₅ = 0. The underlying endpoint/wall classification and finite sector checks were not independently regenerated here. · reported unreproduced
- retained route statementRepaired fixed-P field-five symplectic working theoremspecial case
- Recorded relationshipThe selected crosscut and fixed-P cancellation kill the relevant residues, while commutator-rank isolation excludes a rank-six boundary.supports · reported by source
- Recorded relationshipThe repaired source-reported degree-four class supplies the conjectured ordinary rational homology conclusion for this one simple group, conditional on the v15 promotion dependencies.supports · reported by source
- Recorded relationshipV15 retains the degree-four conclusion only through the repaired fixed-P crosscut theorem and expressly supersedes the v13 quantitative bound.refutes · reported by source
- Active routeRepaired PSp₈(5) fixed-P crosscut promotionPromote the v15 selected-crosscut, Latin-boundary, fixed-P completion-cancellation, and commutator-rank-isolation degree-four packet together with the distinct rank-six top-injectivity certificate; do not reuse the v13 quantitative bound.
- Refuted routeRank-six attachment to frame–Pauli endpointsThe proposed degree-five-to-degree-four attachment route is eliminated because commutator rank forbids the required incidence.
Fixed-P crosscut family route and historical four-gain comparisonsV15 supersedes every Δ-based v13 homology derivation. The current all-rank field-five working theorem uses the repaired crosscut and fixed-P packet, with global completion cancellation replacing the source-only Fourier search.20 displayed rows · 2 routes included
- retained route statementFrame–Pauli construction used by the repaired fixed-P packetconditional
- retained route statementMaximum-endpoint exhaustion excludes n = 4conditional
- supersededSuperseded v13 small-rank Δ-based derivationspecial case
- supersededSuperseded v13 high-rank comparisonconditional
- retained route statementFourier sector dimensions and n = 6 regressioncomputational
- Recorded relationshipThis historical support edge uses the superseded Δ-based derivation. The retained group-level conclusion is supported only by the repaired v15 fixed-P crosscut theorem.supports · contested
- Recorded relationshipThis historical support edge uses the superseded Δ-based derivation. The retained group-level conclusion is supported only by the repaired v15 fixed-P crosscut theorem.supports · contested
- Recorded relationshipV15 invalidates the v13 Δ-based premise or routing interpretation used by this historical edge. It is not a current support edge; use the repaired fixed-P crosscut formulation.reduces to · contested
- DerivationHistorical v13 derivation only. Its positive Δ₃ and Δ₅ premises use the formula that v15 says is not an exact group-incidence formula. The group-level conclusions survive only through the repaired fixed-P crosscut theorem.invalidated
- ChallengeV15 supplies the complete target and transition tables and reports a byte-for-byte rerun, resolving the source-level reproducibility omission. ProofAtlas still has not independently rerun the computation, and no global theorem follows from the local tables alone.unsupported step · reported resolved
- Useful failureTreat the frame–Pauli construction as the complete maximum-endpoint classification in every rank n ≥ 3.reported failure
- Useful failureUse the old Δₙ constant-valence source-target comparison for PSp₂ₙ(5).reported failure
- Research targetPromote global fixed-P completion cancellationopen
- ComputationHistorical n=6 local Fourier counts recorded as regression data and superseded by the complete v15 target-orbit supplement.The counts 93, 930, and 1953 remain regression data. V15 reports that every source orbit contracts nontrivially and forbids another source-only-orbit search; the live mechanism is global fixed-P completion cancellation. · reported unreproduced
- retained route statementRepaired fixed-P field-five symplectic working theoremspecial case
- Recorded relationshipV15 forbids the old Δ_n comparison and reinterprets 11,400 as the exact combined pure-residue rank; global fixed-P completion cancellation is the live route.refutes · reported by source
- ComputationV15 records eleven Fourier target orbits, all eight fine transition rows, and exact regression totals including 11,400 as the combined pure-residue rank rather than a failed source-target deficit.The source reports a byte-for-byte rerun and that every local source orbit contracts nontrivially. ProofAtlas did not independently rerun it; the remaining mechanism is global fixed-P completion cancellation. · reported unreproduced
- Recorded relationshipV15 retains the small-rank group-level conclusions through the repaired fixed-P theorem while forbidding their v13 Δ-based derivation.refutes · reported by source
- Narrowed routeGlobal fixed-P completion cancellation at n = 6Retain the complete v15 local Fourier tables as regression evidence and promote cancellation among distinct frame completions with one fixed literal Pauli plane; no source-only local orbit remains to be found.
- Eliminated routeSuperseded Δₙ high-rank comparisonV15 forbids the old Δₙ incidence formula and the interpretation of 11,400 as a failed source-target inequality; this route is historical only.
Strong markers and heterogeneous splicingThe formal relative fixed-poset identity, reported field-five applications, common-prime product marker, exact one-witness peeling theorem, and the narrowed mixed residual seam.17 displayed rows · 3 routes included
- retained route statementRelative strong-marker fixed-poset identity
- retained route statementReported field-five arbitrary-layer strong markersconditional
- retained route statementCommon-prime heterogeneous marker theoremconditional
- retained route statementMarker peeling with one residual Euler witness
- retained route statementResidual mixed marker/no-source seamconditional
- Recorded relationshipThe relative fixed-poset identity is the formal mechanism by which a promoted strong marker controls an arbitrary isotypic layer.supports · reported by source
- Recorded relationshipThe reported field-five packages supply a common prime across the listed component blocks, which is exactly the hypothesis of the product-marker theorem.supports · reported by source
- Recorded relationshipMarker peeling closes exactly the residual code orbits detected by one cyclic Euler witness and isolates the genuinely new seam.reduces to · reported by source
- DerivationOnce the family-specific deep-unipotent subgroup satisfies the strong-marker conditions in the full automorphism group, the relative fixed-poset identity gives the reported arbitrary-layer Euler congruence. Those family-specific checks remain unpromoted inputs.active reported
- DerivationA product of strong markers at the same prime has empty fixed Quillen poset across all complete blocks, so the Lefschetz congruence is nonzero modulo that prime.active reported
- ChallengeThe general marker identity is distinct from the family-specific assertion that the chosen deep subgroup has the required normalizer and odd pointwise automorphism centralizer; those Chevalley and full-automorphism checks remain unpromoted.unsupported step · open
- Useful failureCombine a strong marker automatically with any no-source class or with an unrelated noncyclic marker system.reported failure
- Research targetPromote the field-five strong-marker theoremopen
- Research targetClose the residual marker/no-source spliceopen
- Active routeField-five strong-marker promotionAudit the family-specific Chevalley and full-automorphism inputs that turn the formal strong-marker identity into arbitrary-layer theorems for the listed field-five families.
- Narrowed routeResidual marker/no-source splicePeel the field-five block, close every residual code orbit detected by one retained cyclic witness, and reserve new bi-filtered or multi-prime machinery for the common vanishing locus.
- Not yet justifiedUnrestricted marker/no-source combinationA strong marker cannot be combined automatically with an arbitrary no-source class, a distinct marker prime, or an unrelated noncyclic marker system.
Order-313 fallback and diagonal codesThe exact reported fixed-poset reduction, one-dimensional repetition-code result, unreproduced torus-centralizer input, and paused entangled-code branch.7 displayed rows · 1 route included
- retained route statementOrder-313 diagonal-code fixed-poset identityconditional
- retained route statementOrder-313 rank-one diagonal code casespecial case
- Recorded relationshipThe diagonal-code reduction and the distinct simple/diagonal attachment behavior feed the retained tensor argument for the repetition-code case.supports · reported by source
- DerivationThe fixed-poset identity isolates a binary diagonal code. In dimension one, the retained tensor law compares the simple and diagonal local attachment maps and yields a nonzero rational-homology repetition-code extension.active reported
- Useful failureUse the order-313 marker alone for every full-automorphism repeated PSp₈(5) layer.reported failure
- ComputationReported negative-cycle torus centralizer calculation for a regular C₃₁₃ subgroup of PSp₈(5).The current work reports C_Aut(PSp₈(5))(P) ≅ C₆₂₆, with two-part generated by one diagonal outer involution, leading to the binary diagonal-code fixed-poset identity. · reported unreproduced
- Route held in reserveOrder-313 diagonal-code fallbackRetain the exact fixed-poset identity and rank-one code case as a fallback and independent topology problem, but do not resume entangled diagonal codes while the strong Sylow-five marker remains viable.
Evidence posture and reproducibilityThe invalidated v13 calculation remains historical, while the repaired fixed-P crosscut claim remains narrative-only and promotion-dependent; v15 source-reported reruns are not represented as independent ProofAtlas reproduction.16 displayed rows · 3 routes included
- ChallengeThe current work explicitly leaves the exact theorem numbers, posets, actual split outer-extension tables, and admissible certificate types as publication dependencies, so the working global contract has not been independently sourced here.unsupported step · open
- ChallengeThe challenged v13 derivation has been invalidated because its Δ₄ kernel premise is superseded. The separate v15 fixed-P crosscut claim remains provisional and is governed by the repaired promotion obligation.unsupported step · withdrawn
- ChallengeThe general marker identity is distinct from the family-specific assertion that the chosen deep subgroup has the required normalizer and odd pointwise automorphism centralizer; those Chevalley and full-automorphism checks remain unpromoted.unsupported step · open
- ChallengeV15 supplies the complete target and transition tables and reports a byte-for-byte rerun, resolving the source-level reproducibility omission. ProofAtlas still has not independently rerun the computation, and no global theorem follows from the local tables alone.unsupported step · reported resolved
- Research targetPromote the repaired PSp₈(5) fixed-P crosscut packageopen
- Research targetPromote the field-five strong-marker theoremopen
- Research targetPromote global fixed-P completion cancellationopen
- Research targetRetain independently reproducible certificate artifactsopen
- ComputationHistorical v13 constant-valence four-gain calculation, retained only as an invalidated computation.V15 states that the Δ₄=118 calculation and 118|𝓔₄ᶠʳ| lower bound are not exact group-incidence results and must not be used. The current degree-four formulation is the repaired fixed-P selected-crosscut packet. · reported unreproduced
- ComputationReported complete top-map injectivity calculation for the rank-six triple-Pauli endpoint family of PSp₈(5).The current work reports that every rank-five wall has a unique rank-six completion and concludes H̃₅ = 0. The underlying endpoint/wall classification and finite sector checks were not independently regenerated here. · reported unreproduced
- ComputationHistorical n=6 local Fourier counts recorded as regression data and superseded by the complete v15 target-orbit supplement.The counts 93, 930, and 1953 remain regression data. V15 reports that every source orbit contracts nontrivially and forbids another source-only-orbit search; the live mechanism is global fixed-P completion cancellation. · reported unreproduced
- ComputationReported negative-cycle torus centralizer calculation for a regular C₃₁₃ subgroup of PSp₈(5).The current work reports C_Aut(PSp₈(5))(P) ≅ C₆₂₆, with two-part generated by one diagonal outer involution, leading to the binary diagonal-code fixed-poset identity. · reported unreproduced
- retained route statementRepaired fixed-P field-five symplectic working theoremspecial case
- Active routeRepaired PSp₈(5) fixed-P crosscut promotionPromote the v15 selected-crosscut, Latin-boundary, fixed-P completion-cancellation, and commutator-rank-isolation degree-four packet together with the distinct rank-six top-injectivity certificate; do not reuse the v13 quantitative bound.
- Active routeField-five strong-marker promotionAudit the family-specific Chevalley and full-automorphism inputs that turn the formal strong-marker identity into arbitrary-layer theorems for the listed field-five families.
- Narrowed routeGlobal fixed-P completion cancellation at n = 6Retain the complete v15 local Fourier tables as regression evidence and promote cancellation among distinct frame completions with one fixed literal Pauli plane; no source-only local orbit remains to be found.
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
2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.
A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.
- The projective endpoint classification and selected/full shared-normal scope are independently checked.
- The crosscut–residue theorem, Latin relative chain identification, and fixed-P panel counts and signs are reconstructed.
- The orthogonal moment theorem and exact PSp₈(5) rank-six wall audit are checked.
- The full-automorphism rigidity and complete-block application are stated with exact dependencies.
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Rational Homological Quillen Conjecture at p = 2 · ready to start
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Research contextPrepared context for any AI agent
For a finite group with no nontrivial normal 2-subgroup, the conjecture predicts nonzero rational reduced homology in the poset of its nontrivial elementary abelian 2-subgroups.
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Sources and references8 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.
- 1Acyclic 2-dimensional complexes and Quillen's conjecturepreprint · accessed Aug 2, 2026
- 2Maximal subgroups of exceptional groups and Quillen's dimensionpreprint · accessed Aug 2, 2026
- 3Components in characteristic p and Quillen's conjecturepreprint · accessed Aug 2, 2026
- 4Quillen's Conjecture | QUILCON | Project | Fact Sheetauthoritative webpage · accessed Aug 2, 2026
- 5Homotopy properties of the poset of nontrivial p-subgroups of a grouporiginal source · accessed Aug 2, 2026
- 6An approach to Quillen's conjecture via centralisers of simple groupspeer reviewed result · accessed Aug 2, 2026
- 7Some results on Quillen's Conjecture via equivalent-poset techniquesauthoritative webpage · accessed Aug 2, 2026
- 8Homotopy properties of the poset of nontrivial p-subgroups of a grouporiginal source · accessed Aug 2, 2026
Important qualifications
- This is the subgroup-poset conjecture, not the unrelated Quillen-Lichtenbaum or Bass-Quillen conjectures.
- Empty formalization or computation lists mean that none was verified in this scoped search, not that none exists.
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