The current work retains a finite cyclic counterexample and marks the generic theorem refuted. Exact lattice shifts and model-specific matched transport identities remain viable local tools.
Route status · Narrowed routeProbability · interacting particle systems · random growth · KPZ fixed point
Strong KPZ Universality
Collaboration betaMany one-dimensional random growth models are expected to share one profile-valued scaling limit, the KPZ fixed point. TASEP provides an established fixed-point theorem, while the published Quastel–Sarkar claim for the KPZ equation and finite-range asymmetric exclusion is withdrawn on arXiv v7 with an author-reported proof gap; the packet does not establish the requested broad nonintegrable universality theorem.
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Known results and sources 
Research problem
Exact mathematical statement
Strong KPZ universality predicts that a broad natural class of genuinely nonintegrable one-dimensional random growth models converges, after model-specific centering and 1:2:3 scaling, to the same profile-valued Markov evolution: the KPZ fixed point.
A complete theorem must specify checkable microscopic assumptions, derive its constants, prove process tightness and finite-dimensional convergence, identify the limiting evolution, cover broad initial data, and exclude extra relevant modes.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Strong KPZ Universality stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
Show that third-order current remainders are negligible at KPZ scale, then identify the surviving quadratic conservative core with the KPZ fixed point and prove profile/path tightness. In the developed negative-binomial lane, the immediate source-reported target is a long-time smoothing estimate for at most three conservative discrepancies.
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 We removed a duplicate or outdated task or route step.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
Strong KPZ Universality in numbers
1.6kretained lines of mathematical investigation1,557 in the current working snapshot
- Argument development
- 1,304 · 84%
- Explored or eliminated routes
- 9 · 1%
- Computational analysis
- 1 · 0%
- Open obligations
- 89 · 6%
- Definitions and setup
- 154 · 10%
8selected 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
Prove long-time finite-discrepancy transport smoothing.
Suggested move: Establish exact L4 transport bounds for one, two, and three Pólya discrepancies, with time decay, block dependence, dyadic summability, and the forward-density L2 factor.
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
The current work retains a finite cyclic counterexample and marks the generic theorem refuted. Exact lattice shifts and model-specific matched transport identities remain viable local tools.
Route status · Narrowed routeThe source explicitly states that compact containment and temporal modulus estimates remain necessary. A corrected-observable or semigroup modulus estimate may prove temporal tightness.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Prove long-time finite-discrepancy transport smoothing.Suggested move: Establish exact L4 transport bounds for one, two, and three Pólya discrepancies, with time decay, block dependence, dyadic summability, and the forward-density L2 factor.
Ready to work on
Identify the universal quadratic conservative core.Suggested move: Formulate and prove a non-circular uniqueness or comparison theorem that identifies the profile-valued quadratic-core evolution with the KPZ fixed point.
Ready to work on
Upgrade finite-time distributions to a broad process theorem.Suggested move: Prove temporal tightness and compact containment, extend initial data, and verify checkable assumptions across a genuinely nonintegrable microscopic class.
Ready to work on
Sourced mathematical context
The known mathematical landscape
Current statusOpen problem
Strong process-level KPZ universality for a broad natural nonintegrable microscopic class remains open. TASEP fixed-point convergence is established. The Quastel–Sarkar KPZ-equation and finite-range-exclusion convergence claim has JAMS publication history, but official arXiv v7 is withdrawn with an author-reported proof gap and is not treated here as an established theorem.
[2][3][4]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
Authoritative summaryOfficial arXiv v7 is withdrawn by Jeremy Quastel. Its comments report that a missing shift invalidates the following averaging argument and leaves the main results with a present proof gap, even though the…[4] Peer reviewedThe Quastel–Sarkar convergence claim appeared in Journal of the American Mathematical Society 36(1), 251–289. This records peer-reviewed publication history; it does not override the authors' later withdrawal…[3] Peer reviewedDauvergne, Ortmann, and Virag constructed the directed landscape as the scaling limit of Brownian last-passage percolation, a central integrable random geometry in the KPZ class.[5] Peer reviewedMatetski, Quastel, and Remenik constructed the KPZ fixed point as the TASEP 1:2:3 scaling limit with general initial conditions and formulated strong universality and uniqueness questions.[2]
Mathematical neighborhood
Related results and reusable starting points
Current focusStrong KPZ Universality
TASEP convergence constructs the fixed point but does not prove convergence for every natural nonintegrable model.
[2]The claim has JAMS publication history, but official arXiv v7 is withdrawn and identifies a proof gap affecting the main results. It is not counted as an established solved special case.
[3][4]The directed landscape is the universal metric-composition geometry constructed for Brownian last-passage percolation and related integrable limits; identifying it for broad microscopic classes remains separate work.
[5]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetA formal statement must define a non-circular microscopic model class, profile topology, centering and scaling constants, admissible initial data, and process-level convergence to a pinned definition of the KPZ fixed point.
- Formalization targetExisting specific-model theorems need exact hypothesis and state-space alignment before they can support a broader formal theorem.
- Formalization targetThe Quastel–Sarkar KPZ-equation and finite-range-exclusion claim cannot be used as established theorem evidence while official arXiv v7 remains withdrawn with the stated proof gap.
- Formalization targetNo checked proof-assistant formalization of the broad strong universality target was located in the scoped review; this is not evidence of nonexistence.
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
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.
3 standing statements5 proposed statements3 open questions2 narrowed routes
- theorem candidate
1 of 8 1 - reduction
2 of 8 2 - lemma
4 of 8 4 - negative result
1 of 8 1
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.24 displayed rows · 2 routes included
- retained route statementDo broad nonintegrable growth models share the KPZ fixed point?
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementExact broad universality targetintermediate
- retained route statementExact lattice smoothing repairintermediate
- retained route statementOne-time to finite-dimensional bootstrapintermediate
- retained route statementCubic Hahn transport reductionintermediate
- retained route statementGeneric diagonal route refutedintermediate
- 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
- 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 failureGeneric diagonal mixed-norm recenteringreported failure
- Useful failureFinite-dimensional convergence as path convergencereported failure
- Research targetProve long-time finite-discrepancy transport smoothing.open
- Research targetIdentify the universal quadratic conservative core.open
- Research targetUpgrade finite-time distributions to a broad process theorem.open
- Research targetLong-time smoothingsuperseded
- ComputationThe current work reports exact symbolic rechecks of its cubic Meixner/Hahn identities and numerical spot checks, without external independent review.The source reports that the cubic residual reduces to one-, two-, and three-particle transport terms with explicit block prefactors; it downgrades the higher-tail lemma and long-time closure where full proofs are absent. · reported unreproduced
- Narrowed routeGeneric diagonal mixed-norm recenteringThe current work retains a finite cyclic counterexample and marks the generic theorem refuted. Exact lattice shifts and model-specific matched transport identities remain viable local tools.
- Narrowed routeFinite-dimensional convergence as path convergenceThe source explicitly states that compact containment and temporal modulus estimates remain necessary. A corrected-observable or semigroup modulus estimate may prove temporal tightness.
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 pointProve long-time finite-discrepancy transport smoothing.
Strong KPZ Universality · ready to start
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Research contextPrepared context for any AI agent
Many one-dimensional random growth models are expected to share one profile-valued scaling limit, the KPZ fixed point. TASEP provides an established fixed-point theorem, while the published Quastel–Sarkar claim for the KPZ equation and finite-range asymmetric exclusion is withdrawn on arXiv v7 with an author-reported proof gap; the current work does not establish the requested broad nonintegrable universality theorem.
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Sources and references6 cited works · next context review by Nov 14, 2026
The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.
- 1Dynamic Scaling of Growing Interfacesoriginal source · Mehran Kardar, Giorgio Parisi, Yi-Cheng Zhang · Physical Review Letters 56(9), 889–892 · 1986 · DOI 10.1103/PhysRevLett.56.889 · accessed Aug 14, 2026
- 2The KPZ fixed pointpeer reviewed result · Konstantin Matetski, Jeremy Quastel, Daniel Remenik · Acta Mathematica 227, 115–203 · 2021 · ARXIV 1701.00018 · DOI 10.4310/ACTA.2021.v227.n1.a3 · accessed Aug 14, 2026
- 3Convergence of exclusion processes and KPZ equation to the KPZ fixed pointpeer reviewed result · Jeremy Quastel, Sourav Sarkar · Journal of the American Mathematical Society 36(1), 251–289 · 2022-03-08; JAMS 36(1), 2023 · ARXIV 2008.06584 · DOI 10.1090/jams/999 · accessed Aug 14, 2026
- 4Convergence of exclusion processes and KPZ equation to the KPZ fixed pointpreprint · Jeremy Quastel, Sourav Sarkar · arXiv v7 (withdrawn; author-reported proof gap) · 2020; v7 withdrawn 2025-05-07 · ARXIV 2008.06584v7 · DOI 10.48550/arXiv.2008.06584 · accessed Aug 14, 2026
- 5The directed landscapepeer reviewed result · Duncan Dauvergne, Jánosch Ortmann, Bálint Virág · arXiv / Acta Mathematica · 2018 preprint; published 2022 · ARXIV 1812.00309 · accessed Aug 14, 2026
- 6Integrable fluctuations in the KPZ universality classsurvey or monograph · Daniel Remenik · Proceedings of the International Congress of Mathematicians · 2022 · ARXIV 2205.01433 · accessed Aug 14, 2026
Important qualifications
- This metadata is external administrative context and grants no proof, review, credit, publication, or deployment authority.
- The broad model class is intentionally not treated as a settled formal definition; specific theorems retain their exact dynamics, scaling, topology, and initial-data hypotheses.
- The TASEP fixed-point theorem is kept separate from the JAMS-published Quastel–Sarkar claim, whose official arXiv v7 is withdrawn and reports a proof gap affecting the main results.
- Publication history is not treated as overriding a later author-reported gap, and the gapped finite-range-exclusion/KPZ-equation claim is not presented as a proved special case.
- Scoped searches found no checked proof-assistant formalization of the broad claim; negative search results do not establish nonexistence.
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