Complete source-reported rank 3–6 scans have minima 0, 0, 2 and 12; inseparable-pair reduction propagates the lower bounds within the ordered-product class. Sharpness is certified at rank six only. Genuinely different coherent matching laws or parameter changes that alter the minor-sign type are outside this obstruction; the third rank-five transportation type is not excluded by the old two-parent results.
Route status · Narrowed routeReal projective geometry and tope covering
McMullen's projective convex-position problem
Collaboration betaCan every general-position point set below the stated threshold be moved into convex position by a projective transformation? The source studies the equivalent realizable all-rank construction through strict tope codes, with a robust rank-five example and unresolved higher-rank coverage.
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Known results and sources 
Research problem
Exact mathematical statement
Let nu_proj(d) be the largest n such that every general-position n-point set in real affine d-space can be taken by a permissible projective transformation to a set whose points are all vertices of its convex hull. Establish
The source's missing construction is a full-spark real r by 2r matrix for every rank r>=3 whose strict tope code radius-one dominates the entire sign cube. Abstract oriented matroids alone do not suffice. The imported universal lower bound and projective correspondence remain separate premises, and no all-rank family is supplied.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on McMullen's projective convex-position problem stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
The source reduces the missing projective upper construction to a full-spark real r by 2r matrix with strict tope radius-one coverage for every r>=3, together with the separately imported lower bound and projective correspondence. An abstract oriented matroid is insufficient.
Evidence posture · Source-reported route statement · dependencies incompleteWork mapped so far
McMullen's projective convex-position problem in numbers
1.3kretained lines of mathematical investigation1,253 in the current working snapshot
- Argument development
- 912 · 73%
- Explored or eliminated routes
- 83 · 7%
- Computational analysis
- 89 · 7%
- Open obligations
- 66 · 5%
- Definitions and setup
- 103 · 8%
9selected mapped statements2routes investigated3open questions3contribution-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
Produce a real full-spark zero-hole construction with an argument that continues in every rank, not just one finite rank.
Suggested move: Suggested next step: Choose one unrestricted construction interface and write its rank-indexed data and closure conditions, distinguishing an exact next-rank candidate from the separate induction needed for all ranks.
What would count as progress
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
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.
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Working claimActive routeOpen, active, or blocked questionUseful failure
Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.
Explored alternatives
Other routes
Complete source-reported rank 3–6 scans have minima 0, 0, 2 and 12; inseparable-pair reduction propagates the lower bounds within the ordered-product class. Sharpness is certified at rank six only. Genuinely different coherent matching laws or parameter changes that alter the minor-sign type are outside this obstruction; the third rank-five transportation type is not excluded by the old two-parent results.
Route status · Narrowed routeThe source classifies the four real rank-three parent types and reports nine unordered-pair refutations, with the remaining hole-preserving case handled directly. The neighborhood depends on the parents and is not a uniform global rank-six lower bound. Singular-parent limits, larger coupling and arbitrary rank-six matrices remain outside the small-neighborhood exclusion, subject to exact real realization and complete coverage.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Produce a real full-spark zero-hole construction with an argument that continues in every rank, not just one finite rank.Suggested move: Suggested next step: Choose one unrestricted construction interface and write its rank-indexed data and closure conditions, distinguishing an exact next-rank candidate from the separate induction needed for all ranks.
Ready to work on
Make all required repairs coexist in one strict real endpoint rather than in separate witnesses or a more permissive ideal envelope.Suggested move: Suggested next step: Formulate the simultaneous real feasibility and global lost/gained-coverage conditions, retaining every defect whose repair may conflict with another before proposing a height or refinement.
Ready to work on
Establish nonempty successful HEIGHT-GAP or the full centered covering conditions on suitable varying parents in every required rank.Suggested move: Suggested next step: Separate determinant walls from covering inequalities for a proposed parent family and identify the precise nonemptiness statement still missing; allow failed parents where the unrestricted interface requires them.
Ready to work on
Sourced mathematical context
The known mathematical landscape
Current statusOpen problem
The universal projective convex-position threshold is not known in every real dimension. The exact local target ν(d)=2d+1 for d≥2 is the sharp-value conjecture identified with Larman in the cited lecture. It concerns realizable point configurations and permissible projective transformations.
[1][2]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
Peer reviewedGarcía-Colín, Montejano, and Ramírez Alfonsín study guaranteed projective vertex counts and relate them to Radon partitions, tolerance, and hyperplane-arrangement topes.[1] Authoritative summaryThe 2023 lecture attributes ν(2)=5, ν(3)=7, and the general lower bound 2d+1 to Larman's 1972 work.[2]
Mathematical neighborhood
Related results and reusable starting points
Current focusMcMullen's projective convex-position problem
The cited lecture reports thresholds five and seven, respectively; these finite-dimensional cases do not establish all dimensions.
[2]The 2023 paper embeds McMullen's threshold question in a broader study of how many faces can be guaranteed after permissible projective transformations.
[1]The 2023 work develops these connections. An abstract oriented-matroid result requires a realizability bridge before it can answer the real point-set target.
[1]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetA proposed formal statement needs real affine general position, permissible projective maps, convex-hull vertices, and the threshold quantifiers.
- Formalization targetAny sign-vector construction used later must retain real realizability and all-rank coverage; no formal bridge or construction was checked in this collection.
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
7 standing statements2 proposed statements3 open questions2 narrowed routes
- theorem candidate
1 of 9 1 - reduction
2 of 9 2 - lemma
1 of 9 1 - special case
2 of 9 2 - equivalence
2 of 9 2 - negative result
1 of 9 1
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.25 displayed rows · 2 routes included
- retained route statementFind a real, full-spark covering construction in every rank.
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementRealizable full-spark covering interfaceintermediate
- retained route statementMATRIX-15 / COMP-56–57: robust rank fiveintermediate
- retained route statementHEIGHT-GAP: exact scalar success setintermediate
- retained route statementCentered proper-minor normal formintermediate
- retained route statementLIFT-22–24: positive-pair obstructionintermediate
- retained route statementCOMP-49: six-hole envelopes, eight-hole refinementsintermediate
- Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
- Useful failureOrdered-product signed-power charts in higher ranksreported failure
- Useful failureSmall-coupling perturbation of fixed full-spark rank-three block sumsreported failure
- Research targetProduce a real full-spark zero-hole construction with an argument that continues in every rank, not just one finite rank.open
- Research targetMake all required repairs coexist in one strict real endpoint rather than in separate witnesses or a more permissive ideal envelope.open
- Research targetEstablish nonempty successful HEIGHT-GAP or the full centered covering conditions on suitable varying parents in every required rank.open
- ComputationFor the transportation type, the source reports independent reconstruction of 126 capacity assignments, 512 topes and agreement on all 1,024 cube distances, together with exact shifted-minor polynomial checks.A selected library of 29 assignment neighborhoods covers the cube, and 29 is minimal within that fixed library. The associated 428 topes are not claimed to be a minimum-size covering code; the exact coefficient margin supports the stated rank-five perturbation box only. · reported unreproduced
- Narrowed routeOrdered-product signed-power charts in higher ranksComplete source-reported rank 3–6 scans have minima 0, 0, 2 and 12; inseparable-pair reduction propagates the lower bounds within the ordered-product class. Sharpness is certified at rank six only. Genuinely different coherent matching laws or parameter changes that alter the minor-sign type are outside this obstruction; the third rank-five transportation type is not excluded by the old two-parent results.
- Narrowed routeSmall-coupling perturbation of fixed full-spark rank-three block sumsThe source classifies the four real rank-three parent types and reports nine unordered-pair refutations, with the remaining hole-preserving case handled directly. The neighborhood depends on the parents and is not a uniform global rank-six lower bound. Singular-parent limits, larger coupling and arbitrary rank-six matrices remain outside the small-neighborhood exclusion, subject to exact real realization and complete coverage.
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.
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
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Prepared starting pointProduce a real full-spark zero-hole construction with an argument that continues in every rank, not just one finite rank.
McMullen's projective convex-position problem · ready to start
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Research contextPrepared context for any AI agent
Can every general-position point set below the stated threshold be moved into convex position by a projective transformation? The source studies the equivalent realizable all-rank construction through strict tope codes, with a robust rank-five example and unresolved higher-rank coverage.
- Exact question and boundaries
- Current routes and known obstacles
- What a useful result should report
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Sources and references2 cited works · next context review by Dec 7, 2026
The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.
- 1On the number of vertices of projective polytopespeer reviewed result · Natalia García-Colín, Luis Pedro Montejano, Jorge Luis Ramírez Alfonsín · Mathematika 69, 535–561 · 2023-03-23 · DOI 10.1112/mtk.12193 · accessed Sep 7, 2026
- 2Around the vertices of projective polytopesauthoritative webpage · J. L. Ramírez Alfonsín · CMO-BIRS, Oaxaca · 2023-06-05 · accessed Sep 7, 2026
Important qualifications
- This is a bounded external-context collection, not proof checking, an independent mathematical review, legal clearance, or publication authority.
- No source programs, datasets, or certificates were executed. A linked resource does not imply reproduction.
- No exact checked formalization was confirmed in the scoped search; an empty list does not establish nonexistence.
- Unassigned history and recognition fields mean this collection did not establish them. Proposed formal prerequisites are planning tasks, not an exhaustive library inventory.
- The exact local title is descriptive. McMullen's question and Larman's proposed sharp value are distinguished; no priority date is assigned.
- The 1972 paper is cited by the inspected sources but was not directly accessible; originalSourceIds therefore remains empty.
- No claim about an abstract oriented matroid, a single finite rank, or a covering-code cardinality is imported as an all-rank realizable construction.
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