ProofAtlas

Analytic number theory · lattice-point discrepancy · exponential sums

Gauss Circle Problem

Collaboration beta

How closely does the number of integer lattice points in a growing disk track the disk's area at the conjectured square-root boundary scale?

P(X)≪ϵX1/4+ϵ
Known results and sources
A dark lattice field meets one luminous circular boundary, with filled points inside, hollow points outside, and an unresolved amber boundary band.
The Gauss circle problem compares the discrete count of lattice points in a disk with its continuous area; the finest possible boundary-scale discrepancy remains open.

Research problem

Exact mathematical statement

For X≥1X\ge 1, let

P(X)=#{(a,b)∈ℤ2:a2+b2≤X}-πX.P(X)=\#\{(a,b)\in\mathbb Z^2:a^2+b^2\le X\}-\pi X.

The Gauss circle conjecture asks for the bound

P(X)≪ϵX1/4+ϵP(X)\ll_\varepsilon X^{1/4+\varepsilon}

for every ϵ>0\varepsilon>0. Equivalently, in the geometric radius variable RR, the disk discrepancy should satisfy E(R)≪ϵR1/2+ϵE(R)\ll_\varepsilon R^{1/2+\varepsilon}. The retained source develops a route toward this endpoint but keeps it open.

Problem infographic

Problem at a glance

A wide problem-first view compares integer lattice points inside growing circular disks with their smooth area, emphasizing changing boundary layers and leaving the finest discrepancy scale visibly unresolved.
As a disk grows, its area changes smoothly while lattice points enter discretely near the boundary. The conjecture asks for quarter-exponent control in squared radius, and the retained research route does not yet reach it.

Current mathematical picture

Where work on Gauss Circle Problem stands

Open problem

The cumulative v6 source reports an exact mixed determinant packet, canonical midpoint lift, packet mass, collision and low-lcm control, a parity-correct tree normal form, quotient valuations, and signed midpoint incidence. The Gauss circle conjecture remains open: centered exact-defect control, generated local operator cancellation, 2-primary/composite gluing, the global norm budget, and all-fixed-order growth remain unresolved.

Leading routeExact mixed midpoint-tree operator route

Primary source-reported route through the exact mixed packet, midpoint tree, local completion, alignment control, and global Hilbert-norm gluing.

Route status · Active route
Main reductionFixed-moment endpoint hierarchy

The source reports that all fixed H_r imply the conjectural endpoint and that H_3 gives only exponent 4/13, so a cubic theorem is a milestone rather than the endpoint.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeClose or rigorously avoid DET-H0-CENTER.Task status · Ready to work on
Latest mathematical updatev6 narrows the exact-defect and midpoint-tree frontier

The cumulative v6 source reports exact determinant, midpoint, collision, atom, tree, quotient, and incidence advances while retaining centered, operator, gluing, all-order, and endpoint gaps.

Retained source record

Work mapped so far

Gauss Circle Problem in numbers

1.6kretained lines of mathematical investigation1,246 in the current working snapshot
Argument development
1,417 · 86%
Explored or eliminated routes
51 · 3%
Computational analysis
9 · 1%
Open obligations
97 · 6%
Definitions and setup
75 · 5%
8inventoried working statements4routes investigated4open 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.

Status checkedAug 26, 2026
Source recordedAug 27, 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

16 selected steps

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

16 selected steps

Scroll horizontally to explore the route

Working route overview for Gauss Circle ProblemA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Gauss circle quarter-exponent target remains open — Depends on missing premiseGauss circlequarter-exponent targetremains…Fixed-moment endpoint hierarchy — Depends on missing premiseFixed-moment endpointhierarchyRadial-annular normal form and rectangular control — Depends on missing premiseRadial-annular normal formand rectangular controlv6 annular midpoint-tree reduction — Depends on missing premisev6 annular midpoint-treereductionCentered exact-defect boundary — Depends on missing premiseCentered exact-defectboundaryExact mixed determinant and midpoint packet — Depends on missing premiseExact mixed determinant andmidpoint packetExact-defect-corrected midpoint-tree cubic target — Depends on missing premiseExact-defect-correctedmidpoint-tree cubic targetRetained base moments and fixed-order transfer — Depends on missing premiseRetained base moments andfixed-order transferExact mixed midpoint-tree operator route — activeExact mixed midpoint-treeoperator routeSpectral, reciprocal-dual, squarefree, and tail route watch — activeSpectral, reciprocal-dual,squarefree, and tail routewatchModel edge or Hilbert-Schmidt bounds as the cubic theorem — stoppedModel edge orHilbert-Schmidt bounds asthe…Independent quotient, odd-prime, or centered-energy shortcuts — stoppedIndependent quotient,odd-prime, orcentered-energy…Close or rigorously avoid DET-H0-CENTER. — OpenClose or rigorously avoidDET-H0-CENTER.Prove TREE-COMPLETE and TREE-OP for the exact generated local star form. — OpenProve TREE-COMPLETE andTREE-OP for the exactgenerated…Prove TREE-2ADIC and TREE-GLUE while preserving quotient and lcm identities. — OpenProve TREE-2ADIC andTREE-GLUE while preservingquotient…Extend the midpoint-tree mechanism to arbitrary fixed order or find a route-changing tail theorem. — OpenExtend the midpoint-treemechanism to arbitrary fixedorder…
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.

Active routeExact mixed midpoint-tree operator route

Primary source-reported route through the exact mixed packet, midpoint tree, local completion, alignment control, and global Hilbert-norm gluing.

Route status · Active route
Active routeSpectral, reciprocal-dual, squarefree, and tail route watch

Parallel route-change options remain open, but the source does not report that any currently removes the centered, gluing, all-order, and endpoint barriers.

Route status · Active route

Explored alternatives

Other routes

2 recorded
Useful but insufficientUnsigned balanced near-energy counting

The source states that unsigned near-energy counting is polynomially too large in the balanced range and that the signs must be preserved. Exploit primitive character weights, bandpass centering, and signed oscillation before absolute values, either directly through TT* or after an exact product-centered packet formula.

Route status · Useful but insufficient
Useful but insufficientPremature determinant or Kloosterman import

Historical v1 route: DET-PACKET had not yet been derived. The selected v6 source now derives the exact mixed determinant packet; the remaining live limitation is that no imported Kloosterman or operator theorem closes the centered, tree, 2-adic, gluing, and outer-sum interfaces.

Route status · Useful but insufficient

Route statements and reductions

Statements the next route can inspect and build on

Route statementGauss circle quarter-exponent target remains open

For X ≥ 1 and every ε > 0, the target for the disk lattice-point discrepancy in the squared-radius variable X is P(X) ≪_ε X^(1/4+ε); the v6 source does not supply a proof.

Source-reported route statement · dependencies incomplete
Route statementv6 annular midpoint-tree reduction

The source reduces the first new fixed moment to an exact mixed determinant packet and a midpoint-tree operator problem, while retaining centered exact-defect, local cancellation, 2-primary/composite gluing, and outer norm summation as open interfaces.

Source-reported route statement · dependencies incomplete
Route statementExact-defect-corrected midpoint-tree cubic target

The current cubic target requires closing or rigorously avoiding DET-H0-CENTER and proving TREE-COMPLETE, TREE-OP, TREE-2ADIC, and TREE-GLUE for the exact generated packet; this remains open.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Close or rigorously avoid DET-H0-CENTER.Suggested move: Audit the exact-defect decomposition and either prove the centered estimate or retain only a checked ordering that never invokes it.
Ready to work on
02
Prove TREE-COMPLETE and TREE-OP for the exact generated local star form.Suggested move: Freeze one cubic star and derive the actual completed local exponential sums without dropping the common defect, quotient phase, or packet symbols.
Ready to work on
03
Extend the midpoint-tree mechanism to arbitrary fixed order or find a route-changing tail theorem.Suggested move: Formulate the arbitrary-tree analogue before treating the cubic theorem as an endpoint route.
Ready to work on
04
Prove TREE-2ADIC and TREE-GLUE while preserving quotient and lcm identities.Suggested move: Separate odd-prime, 2-primary, quotient, and composite gluing while preserving the exact packet Hilbert norms.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 26, 2026
Current statusOpen problem

The quarter-exponent Gauss circle conjecture remains open in this bounded primary-source review. Li and Yang's preprint improves the squared-radius upper exponent to about 0.314483 while explicitly retaining 1/4 as the conjecture, and Lester–Wigman study boundary-point statistics under the corresponding still-conjectural radius exponent.

[2][3]
External progress

What the literature has established

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

  1. Peer reviewedLester and Wigman prove autocorrelation and independence results for suitably separated lattice points near circles, supporting study of the boundary statistics without proving the discrepancy endpoint.[3]
  2. PreprintLi and Yang report the squared-radius upper-bound exponent 0.3144831759741..., improving Huxley's 131/416 while explicitly retaining one quarter as the conjectural endpoint.[2]
  3. Peer reviewedHuxley proved a discrepancy exponent 131/208 in the geometric radius variable, equivalently 131/416 in the squared-radius variable used by the packet.[1]
  4. Peer reviewedSoundararajan obtained strengthened omega results for the circle and divisor error terms, supplying lower-bound context rather than an endpoint upper bound.[4]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusGauss circle problem
Equivalent formulationHardy's radius-variable circle discrepancy conjecture

The squared-radius quarter-exponent formulation and Hardy's geometric-radius one-half-exponent formulation describe the same disk discrepancy after substituting squared radius for radius.

[2][3]
Related problemDirichlet divisor problem

The Dirichlet divisor problem shares the quarter-exponent benchmark and several exponential-sum methods, but it remains a distinct arithmetic error term.

[1][2]
Related problemdistribution of lattice points near circles

Statistics and correlations of lattice points close to circular boundaries probe the heuristic structure behind the conjecture but do not imply its pointwise discrepancy bound.

[3]

Formalization opportunities

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

  • Formalization targetA statement-aligned formalization would need the exact integer-lattice disk count, the continuous area term, both radius conventions, and the quantified epsilon-dependent asymptotic bound.
  • Formalization targetA formal treatment of the current best upper bounds would require substantial analytic-number-theory infrastructure for exponential sums, smoothing, and lattice-point discrepancy estimates.
  • Formalization targetThe current work's radial–annular, exact-resonance, and balanced near-energy objects remain source-reported research interfaces rather than externally checked formal resources.

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.

v6 narrows the exact-defect and midpoint-tree frontierThe cumulative v6 source reports exact determinant, midpoint, collision, atom, tree, quotient, and incidence advances while retaining centered, operator, gluing, all-order, and endpoint gaps.

Changed the research frontierLater mathematical revision

Cumulative v6 source ingested; not a claim of mathematical occurrence time

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 inventory covers all currently cataloged mathematical statements in the research notes.

6 standing statements2 proposed statements4 open questions
Statements by mathematical role8 mapped statements
  • theorem candidate1 of 81
  • reduction2 of 82
  • lemma3 of 83
  • equivalence1 of 81
  • negative result1 of 81
Complete mathematical inventory1 mathematical clusters
v6 exact-defect and midpoint-tree frontierThe source-reported v6 frontier connects the retained endpoint and annular reductions to an exact determinant packet and a narrower open midpoint-tree operator/gluing program; the conjecture remains open.14 displayed rows · 2 routes included
  • retained route statementGauss circle quarter-exponent target remains open
  • retained route statementv6 annular midpoint-tree reductionintermediate
  • retained route statementExact-defect-corrected midpoint-tree cubic targetintermediate
  • retained route statementFixed-moment endpoint hierarchyintermediate
  • retained route statementRetained base moments and fixed-order transferintermediate
  • retained route statementRadial-annular normal form and rectangular controlintermediate
  • retained route statementExact mixed determinant and midpoint packetintermediate
  • retained route statementCentered exact-defect boundaryintermediate
  • Research targetClose or rigorously avoid DET-H0-CENTER.open
  • Research targetProve TREE-COMPLETE and TREE-OP for the exact generated local star form.open
  • Research targetProve TREE-2ADIC and TREE-GLUE while preserving quotient and lcm identities.open
  • Research targetExtend the midpoint-tree mechanism to arbitrary fixed order or find a route-changing tail theorem.open
  • Active routeExact mixed midpoint-tree operator routePrimary source-reported route through the exact mixed packet, midpoint tree, local completion, alignment control, and global Hilbert-norm gluing.
  • Active routeSpectral, reciprocal-dual, squarefree, and tail route watchParallel route-change options remain open, but the source does not report that any currently removes the centered, gluing, all-order, and endpoint barriers.
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 bridgeClose or rigorously avoid DET-H0-CENTER.

The current research map records this as an open mathematical step.

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

Contribute

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.

Read-only beta · actions unavailable
Prepared starting pointClose or rigorously avoid DET-H0-CENTER.

Gauss Circle Problem · 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

How closely does the number of integer lattice points in a growing disk track the disk's area at the conjectured square-root boundary scale?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references4 cited works · next context review by Nov 26, 2026

The mathematical context was checked on Aug 26, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Exponential Sums and Lattice Points IIIpeer reviewed result · M. N. Huxley · Proceedings of the London Mathematical Society · 2003-10-23 · DOI 10.1112/S0024611503014485 · accessed Aug 26, 2026
  2. 2
    An improvement on Gauss's Circle Problem and Dirichlet's Divisor Problempreprint · Xiaochun Li, Xuerui Yang · arXiv · 2023-09-14 · ARXIV 2308.14859 · DOI 10.48550/arXiv.2308.14859 · accessed Aug 26, 2026
  3. 3
    Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circlespeer reviewed result · Stephen Lester, Igor Wigman · Journal of the London Mathematical Society · 2024 · ARXIV 2305.03549 · DOI 10.1112/jlms.12977 · accessed Aug 26, 2026
  4. 4
    Omega results for the divisor and circle problemspeer reviewed result · K. Soundararajan · International Mathematics Research Notices · 2003-01-01 · DOI 10.1155/S1073792803130309 · accessed Aug 26, 2026

Important qualifications

  • Open status is a conservative inference from current primary papers that still state the quarter exponent as a conjecture and report weaker bounds; a bounded search cannot establish the absence of every later claim.
  • Li and Yang's 2023 improvement remains identified as a preprint by the arXiv record and the author's current publication list; it is not promoted here to peer-reviewed evidence.
  • The radius-variable and squared-radius exponents are kept distinct: 131/208 in radius is 131/416 in squared radius, while the conjectural exponents are 1/2 and 1/4 respectively.
  • No packet attachment, submitted URL, source-reported audit, or source-reported computation was used as independent external status authority.
  • No statement-aligned formalization, checked certificate, or independently reproduced computation was established by the scoped search; empty resource lists do not assert nonexistence.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces

Expanded visual

Open original image