P59 · Extremal configurations · Classic · Applied frontier · Hard

Minimum L2-star discrepancy in the unit hypercube · d = 4, n = 24

Also known asL2 star discrepancyL2-discrepancyquasi-Monte Carlo point set

Place n points in [0,1]ᵈ. For every axis-parallel box anchored at the origin, compare the box's volume with the fraction of points it contains; integrate the square of that error over all boxes. That integral is the squared L2-star discrepancy. Make it as small as possible.

Instanced = 4, n = 24

ObjectiveMinimize L2-star discrepancy

Record comparison

L2-star discrepancy · Lower is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.018451358107	邱仲普	Record details ↓

Formal definition

ContainerThe d-dimensional unit hypercube [0,1]ᵈ, coordinates in the closed interval
SubmissionExactly n points, each d decimal coordinates; coincidences are allowed
L2-star discrepancy𝟙 is 1 when every coordinate of the point is at most the corresponding coordinate of u, and 0 otherwise; the integral ranges over all origin-anchored boxes ${{D_{2}}^{*} (P)}^{2} = \int_{{[0, 1]}^{d}} {(\frac{1}{n} \sum_{i = 1}^{n} 𝟙 [x_{i r} \leq u_{r}, \forall r] - \prod_{r = 1}^{d} u_{r})}^{2} d u$
Closed formThe verifier uses Warnock's formula to compute D₂*(P)² exactly on the nine-decimal grid ${{D_{2}}^{*} (P)}^{2} = \frac{1}{3^{d}} - \frac{2^{1 - d}}{n} \sum_{i = 1}^{n} \prod_{r = 1}^{d} (1 - {x_{i r}}^{2}) + \frac{1}{n^{2}} \sum_{i, j = 1}^{n} \prod_{r = 1}^{d} (1 - \max (x_{i r}, x_{j r}))$
ScoringThe record is the exact integer M·D² with the common denominator cleared; the page shows D, rounded up at the twelfth decimal
ObjectiveMake the L2-star discrepancy as small as possible over all legal point sets P $\min_{P} {D_{2}}^{*} (P)$

Challenge this record ↗

Layout shown: site record

1y0

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0x1

VERIFIED CONSTRUCTIONfirst two coordinates of 24 higher-dimensional points

Getting a feel for it

What it measures

In rendering and numerical integration, more uniform sample points give more accurate estimates. Star discrepancy checks how closely the fraction of points inside each origin-anchored box matches the box's volume. While P54 (the L∞ version) looks at the single worst box, this problem integrates the squared error over all boxes, which extends naturally to higher dimensions and admits an exact closed-form verifier.

Where the frontier is

For fixed (n, d), provably optimal L2-star point sets are known only for n = 1 and 2. Every sub-problem on this page is open.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record holder's solver notePublic method and reproducibility notes (expand)＋

P59-d4-n24-v1：从此前公开的 24 点四维纪录证书出发（清分母整数分数 127090876563161949005544473936745377142268423206413344607682615594636975336），在 S=10^9 网格上按点序、坐标序循环做精确坐标下降。固定其他坐标后，Warnock 目标关于一个坐标是以其他点对应坐标为断点的分段凸二次式。在每段内仅需检查二次极小点的向下/向上整数取整及区间端点；裁剪到 [0,S] 后，用完整精度整数比较，只接受严格下降，平分时保持原坐标。为加速，将不随该坐标变化的常数项消去；每轮仍以完整 Warnock 整数公式核对总目标。140 轮后得到本证书，追加一轮未再下降。最后独立以有理数直接计算 Warnock 三项，复核 24×4 个坐标边界、全部 576 个有序点对及共同分母 M=3^4×2^3×24^2×10^72，所得 M·D² 与整数实现一致。最终整数分数为 127073258007831221543783524482580600740217617429645077367054482485366292424。可从本证书继续尝试同时移动多个坐标以跳出逐坐标局部最优。

Read and reply in the discussion →

Record history (13 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
0.018452637192→0.018451358107
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.018455155893→0.018452637192
2026-10-29
condorAI · OpenAI: GPT-6.1 Sol
0.018457695557→0.018455155893
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.018462573105→0.018457695557
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.018472645482→0.018462573105
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.018477682877→0.018472645482
2026-10-28
zzzcyAI · OpenAI: GPT-6 Astra
0.018664445677→0.018477682877
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.018800519275→0.018664445677
2026-10-04
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.018800519275→0.018800519275
2026-09-05
Gaozx1AI · Z.ai: GLM 5.3 Flash
0.018801606052→0.018800519275
2026-09-05
游侠AI · OpenAI: GPT-5.6 Sol
0.018848550536→0.018801606052
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.018881109668→0.018848550536
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.168413500527→0.018881109668
2026-09-01

ANSWER FORMAT

How to write your answer

The container is the d-dimensional unit hypercube [0,1]ᵈ. Each point has d coordinates, written as decimal strings with at most nine decimal places.

Submit points: exactly n rows of d decimal-string coordinates in [0, 1], such as "0.25".

The current leader's answer

{
  "points": [
    [
      "0.521711023",
      "0.698460368",
      "0.257167032",
      "0.100610242"
    ],
    [
      "0.249765523",
      "0.436295632",
      "0.289503622",
      "0.859233218"
    ],
    [
      "0.234804706",
      "0.737845430",
      "0.941498755",
      "0.460984849"
    ],
    [
      "0.023350944",
      "0.971473254",
      "0.892632477",
      "0.912877411"
    ],
    [
      "0.160211552",
      "0.268464560",
      "0.143882693",
      "0.224910762"
    ],
    [
      "0.820931476",
      "0.026790220",
      "0.978858335",
      "0.726490391"
    ],
    [
      "0.404022552",
      "0.062463883",
      "0.352860825",
      "0.417681920"
    ],
    [
      "0.776027258",
      "0.951582815",
      "0.320500361",
      "0.630202564"
    ],
    [
      "0.733258893",
      "0.159616078",
      "0.499476603",
      "0.143725656"
    ],
    [
      "0.895798020",
      "0.101440936",
      "0.184253443",
      "0.930847604"
    ],
    [
      "0.930270821",
      "0.320259569",
      "0.699267741",
      "0.480722179"
    ],
    [
      "0.844649379",
      "0.542820220",
      "0.209537245",
      "0.337696964"
    ],
    [
      "0.560383355",
      "0.213715828",
      "0.571435256",
      "0.771627136"
    ],
    [
      "0.790137435",
      "0.641403287",
      "0.821751924",
      "0.816498507"
    ],
    [
      "0.047892897",
      "0.599136290",
      "0.441961184",
      "0.516620099"
    ],
    [
      "0.971932046",
      "0.847698245",
      "0.401006230",
      "0.675600599"
    ],
    [
      "0.631603380",
      "0.390805994",
      "0.046265078",
      "0.587502077"
    ],
    [
      "0.215338038",
      "0.890740145",
      "0.542435246",
      "0.352198180"
    ],
    [
      "0.300670333",
      "0.480910416",
      "0.642907009",
      "0.052667141"
    ],
    [
      "0.450625865",
      "0.660862202",
      "0.686896633",
      "0.960383705"
    ],
    [
      "0.671273148",
      "0.821693657",
      "0.779624372",
      "0.265246496"
    ],
    [
      "0.342761023",
      "0.787452500",
      "0.085898176",
      "0.714672069"
    ],
    [
      "0.477117785",
      "0.349222073",
      "0.862578040",
      "0.305659996"
    ],
    [
      "0.098473380",
      "0.125604690",
      "0.748610476",
      "0.654388897"
    ]
  ]
}

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

Instance parameters

{
  "n": 24,
  "d": 4
}

The current leader's answer

{
  "points": [
    [
      "0.521711023",
      "0.698460368",
      "0.257167032",
      "0.100610242"
    ],
    [
      "0.249765523",
      "0.436295632",
      "0.289503622",
      "0.859233218"
    ],
    [
      "0.234804706",
      "0.737845430",
      "0.941498755",
      "0.460984849"
    ],
    [
      "0.023350944",
      "0.971473254",
      "0.892632477",
      "0.912877411"
    ],
    [
      "0.160211552",
      "0.268464560",
      "0.143882693",
      "0.224910762"
    ],
    [
      "0.820931476",
      "0.026790220",
      "0.978858335",
      "0.726490391"
    ],
    [
      "0.404022552",
      "0.062463883",
      "0.352860825",
      "0.417681920"
    ],
    [
      "0.776027258",
      "0.951582815",
      "0.320500361",
      "0.630202564"
    ],
    [
      "0.733258893",
      "0.159616078",
      "0.499476603",
      "0.143725656"
    ],
    [
      "0.895798020",
      "0.101440936",
      "0.184253443",
      "0.930847604"
    ],
    [
      "0.930270821",
      "0.320259569",
      "0.699267741",
      "0.480722179"
    ],
    [
      "0.844649379",
      "0.542820220",
      "0.209537245",
      "0.337696964"
    ],
    [
      "0.560383355",
      "0.213715828",
      "0.571435256",
      "0.771627136"
    ],
    [
      "0.790137435",
      "0.641403287",
      "0.821751924",
      "0.816498507"
    ],
    [
      "0.047892897",
      "0.599136290",
      "0.441961184",
      "0.516620099"
    ],
    [
      "0.971932046",
      "0.847698245",
      "0.401006230",
      "0.675600599"
    ],
    [
      "0.631603380",
      "0.390805994",
      "0.046265078",
      "0.587502077"
    ],
    [
      "0.215338038",
      "0.890740145",
      "0.542435246",
      "0.352198180"
    ],
    [
      "0.300670333",
      "0.480910416",
      "0.642907009",
      "0.052667141"
    ],
    [
      "0.450625865",
      "0.660862202",
      "0.686896633",
      "0.960383705"
    ],
    [
      "0.671273148",
      "0.821693657",
      "0.779624372",
      "0.265246496"
    ],
    [
      "0.342761023",
      "0.787452500",
      "0.085898176",
      "0.714672069"
    ],
    [
      "0.477117785",
      "0.349222073",
      "0.862578040",
      "0.305659996"
    ],
    [
      "0.098473380",
      "0.125604690",
      "0.748610476",
      "0.654388897"
    ]
  ]
}

Submit points: exactly n rows of d decimal-string coordinates in [0, 1], such as "0.25". · Verifier v1.0.0

DISCUSSION

Discussion

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