Three exact reachable transitions impose inconsistent Bellman inequalities on every fixed affine scalar-ladder combination. Labelled M-convex flow, a k=2 primal proof, or resistance-holonomy closure remain viable.
Route status · Narrowed routeOnline algorithms · metric geometry · work functions · competitive analysis
Deterministic k-Server Conjecture
Collaboration betaLocal configurations are sharply controlled, but the proof still lacks a global invariant for changing charts in arbitrary metrics.
Known results and sources
Research problem
Exact mathematical statement
For every metric space and positive integer k, construct a deterministic k-competitive online algorithm for the k-server problem.
This remains open; the source reports partial results and route constraints, not a complete proof.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Deterministic k-Server Conjecture stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
Reduce the conjecture to an endpoint-corrected extended-cost inequality for the work-function algorithm and control the exchange packets created when valid local simplex charts change over time.
Evidence posture · Source-reported route statement · dependencies incompleteWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Reader-facing record corrected; mathematics unchangedWork mapped so far
Deterministic k-Server Conjecture in numbers
1.5kretained lines of mathematical investigation1,451 in the current working snapshot
- Argument development
- 1,212 · 84%
- Explored or eliminated routes
- 39 · 3%
- Computational analysis
- 42 · 3%
- Open obligations
- 58 · 4%
- Definitions and setup
- 100 · 7%
7selected mapped statements1routes 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
Prove or refute the primal two-server Bellman candidate.
Suggested move: Use the exact one-dimensional obstacle update to prove or refute the primal two-server Bellman inequality.
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
Three exact reachable transitions impose inconsistent Bellman inequalities on every fixed affine scalar-ladder combination. Labelled M-convex flow, a k=2 primal proof, or resistance-holonomy closure remain viable.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Prove or refute the primal two-server Bellman candidate.Suggested move: Use the exact one-dimensional obstacle update to prove or refute the primal two-server Bellman inequality.
Ready to work on
Control simplex-chart holonomy through intermediate switches.Suggested move: Model chart changes as time-expanded rank-capacity flow.
Ready to work on
Prove a target-ordering theorem.Suggested move: Derive a uniform target ordering from the local completion requests or find the smallest counterexample.
Ready to work on
Sourced mathematical context
The known mathematical landscape
External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
Preprintk-server-bench: Automating Potential Discovery for the k-Server Conjecture supplies a representative external result or boundary relevant to the problem; it is not treated here as a proof of the full packet target.[1] Peer reviewedDeterministic 3-server on a circle and the limitation of canonical potentials supplies a representative external result or boundary relevant to the problem; it is not treated here as a proof of the full packet target.[2]
Mathematical neighborhood
Related results and reusable starting points
Current focusDeterministic k-Server Conjecture
Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetA formal statement matching the exact public target and all quantifiers.
- Formalization targetFormal libraries for the principal mathematical structures used by the strongest route.
- Formalization targetA checked closing argument for the source-identified open bridge.
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 removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
Research-record correctionWe corrected the cited passages. The mathematical claims and their status did not change.
Cited passages correctedCorrected the research recordCorrection note
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
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.
5 standing statements2 proposed statements3 open questions1 narrowed routes
- theorem candidate
1 of 7 1 - reduction
2 of 7 2 - lemma
3 of 7 3 - negative result
1 of 7 1
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
- retained route statementCan deterministic online service match the factor k?
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementTree and small-metric closureintermediate
- retained route statementCanonical local cellintermediate
- retained route statementSimplex chartintermediate
- retained route statementFirst chart obstructionintermediate
- 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
- 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 failureFixed affine combinations of the scalar ladderreported failure
- Research targetProve or refute the primal two-server Bellman candidate.open
- Research targetControl simplex-chart holonomy through intermediate switches.open
- Research targetProve a target-ordering theorem.open
- Research targetTwo-server Bellman routesuperseded
- Research targetGlobal circulationsuperseded
- Narrowed routeFixed affine combinations of the scalar ladderThree exact reachable transitions impose inconsistent Bellman inequalities on every fixed affine scalar-ladder combination. Labelled M-convex flow, a k=2 primal proof, or resistance-holonomy closure remain viable.
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
The highlighted primal two-server Bellman task is one priority inside a broader open program; even a k=2 result would leave the general-k packet-flow, chart-switch, ordering, and circulation gates.
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.
Continue the mathematics
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Prepared starting pointProve or refute the primal two-server Bellman candidate.
Deterministic k-Server Conjecture · ready to start
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Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.
Research contextPrepared context for any AI agent
Local configurations are sharply controlled, but the proof still lacks a global invariant for changing charts in arbitrary metrics.
- 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 Nov 21, 2026
The mathematical context was checked on Aug 21, 2026. Status can be refreshed sooner after a material result or claim.
- 1k-server-bench: Automating Potential Discovery for the k-Server Conjecturepreprint · Kirill Brilliantov, Etienne Bamas, Emmanuel Abbé · arXiv · 2026 · accessed Aug 21, 2026
- 2Deterministic 3-server on a circle and the limitation of canonical potentialspeer reviewed result · Zhiyi Huang, Hanwen Zhang · Theoretical Computer Science · 2024 · accessed Aug 21, 2026
Important qualifications
- This was a bounded status and identity check, not an exhaustive bibliography, priority review, or legal review.
- Private packet claims were not treated as external authority; submitted links and attachments were not executed or actively rendered.
- No absence claim is inferred from the bounded search, and recent preprints remain subject to ordinary scholarly review.
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