ProofAtlas

Probability and mathematical statistical mechanics

Three-Dimensional Ising Universality

Collaboration beta

Can critical three-dimensional Ising systems be proved to converge to one universal non-Gaussian continuum field with universal exponents? The exact target remains open.

Φa⇒Φ∈S'(ℝ3),Φnon-Gaussian and universal.
Known results and sources
A critical three-dimensional spin lattice faces a fluctuating continuum field across a dark unresolved scaling gap.
The target is a universal non-Gaussian continuum scaling law; convergence from the critical lattice remains open.

Research problem

Exact mathematical statement

For every admissible ferromagnetic, translation-invariant, finite-range interaction JJ on ℤ3\mathbb Z^3, there should exist universal Δσ,yt,ω\Delta_\sigma,y_t,\omega and a non-Gaussian random tempered distribution Φ\Phi such that

Φa⇒Φ∈S'(ℝ3)(a↓0),\Phi_a\Rightarrow\Phi\in\mathcal S'(\mathbb R^3)\qquad(a\downarrow0),

after model-dependent amplitude and metric normalization, with one law after two-point normalization and the stated universal near-critical family and symmetries.

Problem infographic

Problem at a glance

Two distinct critical cubic spin systems sit inside nested observation scales below one shared candidate continuum random field.
Universality asks whether different admissible microscopic interactions converge, after normalization, to the same non-Gaussian field.

Current mathematical picture

Where work on Three-Dimensional Ising Universality stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureAutomatic inheritance from an abstract fixed point

The current work's false-claims audit separates local chart statements from physical entry and extensive thermodynamic transfer. A chart tied to positive reconstruction and an actual Ising-derived trajectory remains the intended route.

Route status · Narrowed route
Main reductionInfinite-dimensional certificate

The proposed computer-assisted fixed-point theorem must retain an analytically controlled infinite tail.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeConstruct a preserved volume-uniform factor-two/factor-three nonlinear RG chart with positive finite-volume realization.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

Three-Dimensional Ising Universality in numbers

1.5kretained lines of mathematical investigation1,502 in the current working snapshot
Argument development
1,274 · 85%
Explored or eliminated routes
8 · 1%
Computational analysis
48 · 3%
Open obligations
72 · 5%
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.

Proposed byUniversality is a broad statistical-physics program rather than a single recorded proposer claim.Status checkedAug 14, 2026
Source recordedAug 14, 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

11 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

11 selected steps

Scroll horizontally to explore the route

Working route overview for Three-Dimensional Ising UniversalityA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Microscopic three-dimensional Ising systems should share a universal non-Gaussian critical field. — Depends on missing premiseMicroscopicthree-dimensional Isingsystems…Current reduction — Depends on missing premiseCurrent reductionInfinite-dimensional certificate — Depends on missing premiseInfinite-dimensionalcertificateScalar-to-spin transfer — Depends on missing premiseScalar-to-spin transferClosing target — Depends on missing premiseClosing targetContinuum universality target — Depends on missing premiseContinuum universalitytargetExact scalarization — Depends on missing premiseExact scalarizationAutomatic inheritance from an abstract fixed point — stoppedAutomatic inheritance froman abstract fixed pointConstruct a preserved volume-uniform factor-two/factor-three nonlinear RG chart with positive finite-volume realization. — OpenConstruct a preservedvolume-uniformfactor-two/factor-three…Validate a non-Gaussian fixed point and exactly one physical even unstable eigenvalue after gauge removal. — OpenValidate a non-Gaussianfixed point and exactly onephysical…Prove physical source-compatible microscopic entry, convergent gauges, and separate scale-critical tail and pressure transfer. — OpenProve physicalsource-compatiblemicroscopic…
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

1 recorded
Narrowed routeAutomatic inheritance from an abstract fixed point

The current work's false-claims audit separates local chart statements from physical entry and extensive thermodynamic transfer. A chart tied to positive reconstruction and an actual Ising-derived trajectory remains the intended route.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Construct a preserved volume-uniform factor-two/factor-three nonlinear RG chart with positive finite-volume realization.Suggested move: Choose a finite numerical core and analytically controlled infinite tail, then prove uniform differentiability and thermodynamic compatibility.
Ready to work on
02
Validate a non-Gaussian fixed point and exactly one physical even unstable eigenvalue after gauge removal.Suggested move: Produce an approximate core, preconditioner, residual bounds, contraction, Binder gap, and spectral enclosures.
Ready to work on
03
Prove physical source-compatible microscopic entry, convergent gauges, and separate scale-critical tail and pressure transfer.Suggested move: Establish uniform moments, Schwartz-tail control, source cocycles, and pressure-level convergence on a nonzero source annulus.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen problem

High-precision numerical and non-rigorous methods strongly support a three-dimensional Ising universality class, but the exact lattice-to-non-Gaussian-continuum convergence, universality, and exponent package requested here remains open. Rigorous four-dimensional Gaussianity is a neighboring result, not a solution in dimension three.

[3][4][1]
External progress

What the literature has established

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

  1. Authoritative summaryKülske's survey describes three-dimensional exponents as supported by experiments, simulations, and non-rigorous methods, while a fully rigorous foundation remains a mathematical challenge.[4]
  2. Peer reviewedAizenman and Duminil-Copin proved Gaussian scaling limits in four-dimensional Ising-type and φ4 models, clarifying the upper-critical-dimension contrast with the non-Gaussian three-dimensional target.[2]
  3. Peer reviewedThe peer-reviewed JHEP article reports mixed-correlator conformal-bootstrap computations producing high-precision islands for leading three-dimensional Ising CFT data; this assumes the CFT framework and is…[1]
4 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusThree-Dimensional Ising Universality
Related problemthree-dimensional Ising conformal-bootstrap island

Conformal bootstrap yields precise candidate CFT data under structural assumptions but does not prove convergence or universality of the three-dimensional lattice Ising model.

[1]
Solved special casefour-dimensional Ising-type scaling limits

The critical scaling limit is Gaussian in dimension four, not the predicted non-Gaussian three-dimensional universality class.

[2]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

  • computation · not independently reproducedMixed-correlator conformal-bootstrap precision islands

    The published computation gives precise three-dimensional CFT parameter islands. ProofAtlas did not rerun it, and it does not establish the lattice scaling-limit theorem.

    [1]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA formal target needs the infinite-volume Gibbs model, critical temperature, random tempered-distribution convergence, field and metric normalization, non-Gaussianity, symmetries, and universality over an explicit interaction class.
  • Formalization targetNo checked proof-assistant formalization of the exact three-dimensional lattice-to-continuum universality target was located.

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
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.

Corrected the research recordCorrection note

Correction details
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.

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.

5 standing statements2 proposed statements3 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction2 of 72
  • lemma3 of 73
  • equivalence1 of 71
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 statementMicroscopic three-dimensional Ising systems should share a universal non-Gaussian critical field.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementContinuum universality targetintermediate
  • retained route statementExact scalarizationintermediate
  • retained route statementScalar-to-spin transferintermediate
  • retained route statementInfinite-dimensional certificateintermediate
  • 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 failureAutomatic inheritance from an abstract fixed pointreported failure
  • Research targetConstruct a preserved volume-uniform factor-two/factor-three nonlinear RG chart with positive finite-volume realization.open
  • Research targetValidate a non-Gaussian fixed point and exactly one physical even unstable eigenvalue after gauge removal.open
  • Research targetProve physical source-compatible microscopic entry, convergent gauges, and separate scale-critical tail and pressure transfer.open
  • Research targetRG and reconstructionsuperseded
  • Research targetPhysical microscopic entrysuperseded
  • Narrowed routeAutomatic inheritance from an abstract fixed pointThe current work's false-claims audit separates local chart statements from physical entry and extensive thermodynamic transfer. A chart tied to positive reconstruction and an actual Ising-derived trajectory remains the intended route.
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 bridgeConstruct a preserved volume-uniform factor-two/factor-three nonlinear RG chart with positive finite-volume realization.

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.

  • 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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ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

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Prepared starting pointConstruct a preserved volume-uniform factor-two/factor-three nonlinear RG chart with positive finite-volume realization.

Three-Dimensional Ising Universality · ready to start

Mathematical updatesFollow this problem

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

Can critical three-dimensional Ising systems be proved to converge to one universal non-Gaussian continuum field with universal exponents? The exact target remains open.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 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.

  1. 1
    Precision Islands in the Ising and O(N) Modelspeer reviewed result · Filip Kos, David Poland, David Simmons-Duffin, Alessandro Vichi · Journal of High Energy Physics 2016, article 036 · 2016-08-04 · ARXIV 1603.04436 · DOI 10.1007/JHEP08(2016)036 · accessed Aug 14, 2026
  2. 2
    Marginal triviality of the scaling limits of critical 4D Ising and φ4 modelspeer reviewed result · Michael Aizenman, Hugo Duminil-Copin · Annals of Mathematics 194(1), 163–235 · 2021 · DOI 10.4007/annals.2021.194.1.3 · MR 4276286 · accessed Aug 14, 2026
  3. 3
    100 years of the (critical) Ising model on the hypercubic latticesurvey or monograph · Hugo Duminil-Copin · Proceedings of the International Congress of Mathematicians 2022 · 2022 · DOI 10.4171/ICM2022/204 · accessed Aug 14, 2026
  4. 4
    The Ising model: highlights and perspectivessurvey or monograph · Christof Külske · Mathematical Physics, Analysis and Geometry 28, article 20 · 2025-08-02 · DOI 10.1007/s11040-025-09515-1 · accessed Aug 14, 2026

Important qualifications

  • External metadata was collected independently of the unreviewed source material; packet claims are not external evidence.
  • Scoped searches do not establish that no additional result, formalization, or dataset exists elsewhere.
  • No external computation was rerun, and no source-submitted URL or packet attachment was fetched, executed, or rendered.
  • This metadata grants no proof, acceptance, credit, publication, redistribution, deployment, or public-write authority.

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