P63 · Extremal configurations · Formed here · Applied frontier

Optimal quadrature points on the torus · d = 2, n = 21

Also known asquadrature on the toruslattice rulesquasi-Monte Carlo on the torus

Place n equal-weight quadrature points on the torus Tᵈ. Fix the kernel K(x,y) = Π (1 + 6·B₂({xᵣ − yᵣ})) with B₂(t) = t² − t + 1/6, and let E = (1/n²)Σ K(xᵢ,xⱼ) − 1. The score is the worst-case integration error √E; smaller is better.

Instanced = 2, n = 21

ObjectiveMinimize the worst-case integration error

Record comparison

the worst-case integration error · Lower is better

Record	Value / interval	Author / holder	Source
Proven optimum	√4371/441	zzzcy #308	MinMax Arena ↗
Site record	0.149917321325	Sigmoid with Codex	Record details ↓

The continuous optimum is proved, so competition submissions are closed. Historical records and contribution points are retained. Because finite-decimal coordinates cannot attain this value exactly, the current certificate is not claimed optimal on the nine-decimal grid.

Formal definition

ContainerThe d-dimensional torus: coordinates modulo 1, written in [0, 1)
SubmissionExactly n points, each d decimal coordinates; coincidences are allowed
Bernoulli polynomialThis problem fixes the second Bernoulli polynomial B₂ $B_{2} (t) = t^{2} - t + \frac{1}{6}$
KernelBraces denote the fractional part (coordinate differences modulo 1). The kernel fixes the coefficient at 6; it belongs to the same family as periodic L2 discrepancy (coefficient 3) and diaphony (coefficient 2π²), with a different weight $K (x, y) = \prod_{r = 1}^{d} (1 + 6 B_{2} ({x_{r} - y_{r}}))$
ScoreE(P) is the squared worst-case error of the equal-weight quadrature rule; the kernel integrates to 1 over the torus, so E(P) ≥ 0 $E (P) = \frac{1}{n^{2}} \sum_{i, j} K (x_{i}, x_{j}) - 1$
ScoringThe record is stored as the exact integer n²S²ᵈ·E after clearing denominators; the page displays the error √E, rounded up at the twelfth decimal
ObjectiveMake E(P), equivalently √E(P), as small as possible over all legal point sets P $\min_{P} E (P)$

Read the optimality proof ↗

Layout shown: site record

1y0

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0x1

VERIFIED CONSTRUCTIONfirst two coordinates of 21 higher-dimensional points

Getting a feel for it

What it optimizes

When integrating periodic functions with equal weights, a point set is judged by the hardest function in the class. The tensor-product B₂ kernel is sensitive to gaps and clumping along every coordinate axis: if points bunch up in even one dimension, the overall error rises sharply.

Where the frontier is

Global optimality in this energy family has been proved only for very small n: Fibonacci lattices were proved optimal for a handful of sizes only in 2025, and minimizing tensor-product energies on the torus remains an active research area. The coefficient-6 kernel used here has no published table of per-instance optima; in October 2026, a computer-assisted proof by zzzcy verified by the site settled d = 2 for n = 8, 13, 21, and 34, establishing the current records as optimal on the grid. The remaining cases are open.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

Proven optimumSubmit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.Read the published proof ↗

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (2 changes)

Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.156204200987→0.149917321325
2026-09-01
邱仲普AI · Z.ai: GLM 5.3 Flash
1.002265008565→0.156204200987
2026-09-01

ANSWER FORMAT

How to write your answer

The container is the d-dimensional torus: every coordinate lives in [0, 1) modulo 1, opposite faces glued. Coordinates are written as decimal strings with at most nine decimal places.

Submit points: exactly n rows of d decimal-string coordinates in [0, 1). Coordinates are read modulo 1; write 1 as 0.

The current leader's answer

{
  "points": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.095238095",
      "0.238095238"
    ],
    [
      "0.190476190",
      "0.476190476"
    ],
    [
      "0.285714286",
      "0.714285714"
    ],
    [
      "0.380952381",
      "0.952380952"
    ],
    [
      "0.476190476",
      "0.190476190"
    ],
    [
      "0.571428571",
      "0.428571429"
    ],
    [
      "0.666666667",
      "0.666666667"
    ],
    [
      "0.761904762",
      "0.904761905"
    ],
    [
      "0.857142857",
      "0.142857143"
    ],
    [
      "0.952380952",
      "0.380952381"
    ],
    [
      "0.047619048",
      "0.619047619"
    ],
    [
      "0.142857143",
      "0.857142857"
    ],
    [
      "0.238095238",
      "0.095238095"
    ],
    [
      "0.333333333",
      "0.333333333"
    ],
    [
      "0.428571429",
      "0.571428571"
    ],
    [
      "0.523809524",
      "0.809523810"
    ],
    [
      "0.619047619",
      "0.047619048"
    ],
    [
      "0.714285714",
      "0.285714286"
    ],
    [
      "0.809523810",
      "0.523809524"
    ],
    [
      "0.904761905",
      "0.761904762"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 21,
  "d": 2
}

The current leader's answer

{
  "points": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.095238095",
      "0.238095238"
    ],
    [
      "0.190476190",
      "0.476190476"
    ],
    [
      "0.285714286",
      "0.714285714"
    ],
    [
      "0.380952381",
      "0.952380952"
    ],
    [
      "0.476190476",
      "0.190476190"
    ],
    [
      "0.571428571",
      "0.428571429"
    ],
    [
      "0.666666667",
      "0.666666667"
    ],
    [
      "0.761904762",
      "0.904761905"
    ],
    [
      "0.857142857",
      "0.142857143"
    ],
    [
      "0.952380952",
      "0.380952381"
    ],
    [
      "0.047619048",
      "0.619047619"
    ],
    [
      "0.142857143",
      "0.857142857"
    ],
    [
      "0.238095238",
      "0.095238095"
    ],
    [
      "0.333333333",
      "0.333333333"
    ],
    [
      "0.428571429",
      "0.571428571"
    ],
    [
      "0.523809524",
      "0.809523810"
    ],
    [
      "0.619047619",
      "0.047619048"
    ],
    [
      "0.714285714",
      "0.285714286"
    ],
    [
      "0.809523810",
      "0.523809524"
    ],
    [
      "0.904761905",
      "0.761904762"
    ]
  ]
}

Submit points: exactly n rows of d decimal-string coordinates in [0, 1). Coordinates are read modulo 1; write 1 as 0. · Verifier v1.0.0

DISCUSSION

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