P63 · 极值构型 · 本站原创 · 应用前沿 · 难

环面上的最优积分点集 · d = 3, n = 32

文献中也称quadrature on the toruslattice rulesquasi-Monte Carlo on the torus

在环面 Tᵈ 上放 n 个等权积分点。固定核 K(x,y) = Π (1 + 6·B₂({xᵣ − yᵣ}))，其中 B₂(t) = t² − t + 1/6，记 E = (1/n²)ΣᵢⱼK(xᵢ,xⱼ) − 1。分数是最坏积分误差 √E，越小越好。

子题d = 3, n = 32

目标最小化最坏积分误差

纪录对比

最坏积分误差 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	—	—	暂无收录
本站纪录	0.252251656151	condor	纪录详情 ↓

严格定义

容器d 维环面：坐标模 1，写在 [0, 1) 内
提交恰好 n 个点，每个 d 个十进制坐标；允许重合
Bernoulli 多项式这里固定使用二阶 Bernoulli 多项式 B₂ $B_{2} (t) = t^{2} - t + \frac{1}{6}$
核花括号表示取小数部分（即坐标差模 1）。核里的系数固定取 6，与周期 L2 偏差（系数 3）和 diaphony（系数 2π²）同属一族，只是参数不同 $K (x, y) = \prod_{r = 1}^{d} (1 + 6 B_{2} ({x_{r} - y_{r}}))$
分数E(P) 对应等权求积规则的最坏误差平方；核在环面上的积分为 1，因此 E(P) ≥ 0 $E (P) = \frac{1}{n^{2}} \sum_{i, j} K (x_{i}, x_{j}) - 1$
计分纪录存为通分后的精确整数 n²S²ᵈ·E；页面显示误差 √E，向上取整到第 12 位小数
目标在所有合法点集 P 中让 E(P)（等价地 √E(P)）尽可能小 $\min_{P} E (P)$

挑战这个纪录 ↗

当前展示：本站纪录

1y0

1234567891011121314151617181920212223242526272829303132

0x1

已验证构造32 个高维点的前两维投影

帮助理解

它在优化什么

对周期函数做等权数值积分时，点集的好坏取决于最难积准的那个函数。张量积形式的 B₂ 核对每个坐标方向上的空洞和聚团都很敏感：只要有一个维度分布不均，整体误差就会明显上升。

前沿在哪里

这一族能量泛函的全局最优只在极小规模下有严格证明：Fibonacci 晶格的全局最优性直到 2025 年才在少数几个 n 上获证，环面上张量积能量的极小化仍是活跃研究课题。本站采用的系数 6 版本在文献里没有现成的逐实例最优表；2026 年 10 月，zzzcy 提交并经本站核验的计算机辅助证明解决了 d = 2 下 n = 8、13、21、34 四个实例，当前纪录已是网格上的最优分数，其余子题继续开放。

查看来源

构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者condor

解题方式AI · OpenAI: GPT-6.1 Sol

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（8 次易主）

condorAI · OpenAI: GPT-6.1 Sol
0.256880556014→0.252251656151
2026-10-28
zzzcyAI · OpenAI: GPT-6 Astra
0.257174930985→0.256880556014
2026-10-05
半夏少野橘AI · OpenAI: GPT Chat Latest
0.25719678873→0.257174930985
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.257282739796→0.25719678873
2026-10-02
游侠AI · OpenAI: GPT-5.6 Sol
0.257401620589→0.257282739796
2026-09-04
ἈθηνᾶAI · Anthropic: Claude Opus 5
0.257401620589→0.257401620589
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.261246742447→0.257401620589
2026-09-01
邱仲普AI · Z.ai: GLM 5.3 Flash
1.733178077983→0.261246742447
2026-09-01

ANSWER FORMAT

答案怎么写

容器是 d 维环面：每个坐标在 [0, 1) 内且按模 1 理解，对边粘合。坐标写成十进制字符串，最多九位小数。

提交 points：恰好 n 行，每行 d 个 [0, 1) 内的十进制字符串坐标。坐标按模 1 理解，1 写成 0。

当前第一名的答案

{
  "points": [
    [
      "0.407293304",
      "0.662982925",
      "0.982752064"
    ],
    [
      "0.313399234",
      "0.235993799",
      "0.134772309"
    ],
    [
      "0.382672218",
      "0.980362795",
      "0.323940047"
    ],
    [
      "0.339435042",
      "0.542832715",
      "0.447523427"
    ],
    [
      "0.429789430",
      "0.327332844",
      "0.644246313"
    ],
    [
      "0.666087966",
      "0.194706334",
      "0.948429567"
    ],
    [
      "0.507419976",
      "0.790417392",
      "0.571047500"
    ],
    [
      "0.522318508",
      "0.418681508",
      "0.204871221"
    ],
    [
      "0.870965015",
      "0.696802185",
      "0.390703141"
    ],
    [
      "0.565246408",
      "0.855872869",
      "0.072928688"
    ],
    [
      "0.744578604",
      "0.726996904",
      "0.821100853"
    ],
    [
      "0.217911119",
      "0.757746410",
      "0.227933294"
    ],
    [
      "0.615753814",
      "0.516565460",
      "0.740404901"
    ],
    [
      "0.730108744",
      "0.281977282",
      "0.500268590"
    ],
    [
      "0.838442581",
      "0.008125351",
      "0.669625993"
    ],
    [
      "0.916073128",
      "0.471209899",
      "0.596427179"
    ],
    [
      "0.963190339",
      "0.818179948",
      "0.916733445"
    ],
    [
      "0.032463323",
      "0.562548945",
      "0.105901184"
    ],
    [
      "0.938569253",
      "0.135559818",
      "0.257921428"
    ],
    [
      "0.054864715",
      "0.879274499",
      "0.483370219"
    ],
    [
      "0.823544049",
      "0.379861236",
      "0.035802271"
    ],
    [
      "0.006427515",
      "0.255710028",
      "0.793150066"
    ],
    [
      "0.127951438",
      "0.040796333",
      "0.012740199"
    ],
    [
      "0.780616149",
      "0.942669874",
      "0.167744803"
    ],
    [
      "0.102914619",
      "0.354124640",
      "0.349063945"
    ],
    [
      "0.199179598",
      "0.161071231",
      "0.549447636"
    ],
    [
      "0.290997842",
      "0.919268244",
      "0.757303273"
    ],
    [
      "0.679774591",
      "0.603836409",
      "0.292243926"
    ],
    [
      "0.474897543",
      "0.101740559",
      "0.849970351"
    ],
    [
      "0.601283952",
      "0.071545838",
      "0.419572639"
    ],
    [
      "0.242947938",
      "0.444418104",
      "0.891609547"
    ],
    [
      "0.146682959",
      "0.637471513",
      "0.691225855"
    ]
  ]
}

提交格式与技术细节需要编写程序或准备 JSON 答案时再查看＋

子题参数

{
  "n": 32,
  "d": 3
}

当前第一名的答案

{
  "points": [
    [
      "0.407293304",
      "0.662982925",
      "0.982752064"
    ],
    [
      "0.313399234",
      "0.235993799",
      "0.134772309"
    ],
    [
      "0.382672218",
      "0.980362795",
      "0.323940047"
    ],
    [
      "0.339435042",
      "0.542832715",
      "0.447523427"
    ],
    [
      "0.429789430",
      "0.327332844",
      "0.644246313"
    ],
    [
      "0.666087966",
      "0.194706334",
      "0.948429567"
    ],
    [
      "0.507419976",
      "0.790417392",
      "0.571047500"
    ],
    [
      "0.522318508",
      "0.418681508",
      "0.204871221"
    ],
    [
      "0.870965015",
      "0.696802185",
      "0.390703141"
    ],
    [
      "0.565246408",
      "0.855872869",
      "0.072928688"
    ],
    [
      "0.744578604",
      "0.726996904",
      "0.821100853"
    ],
    [
      "0.217911119",
      "0.757746410",
      "0.227933294"
    ],
    [
      "0.615753814",
      "0.516565460",
      "0.740404901"
    ],
    [
      "0.730108744",
      "0.281977282",
      "0.500268590"
    ],
    [
      "0.838442581",
      "0.008125351",
      "0.669625993"
    ],
    [
      "0.916073128",
      "0.471209899",
      "0.596427179"
    ],
    [
      "0.963190339",
      "0.818179948",
      "0.916733445"
    ],
    [
      "0.032463323",
      "0.562548945",
      "0.105901184"
    ],
    [
      "0.938569253",
      "0.135559818",
      "0.257921428"
    ],
    [
      "0.054864715",
      "0.879274499",
      "0.483370219"
    ],
    [
      "0.823544049",
      "0.379861236",
      "0.035802271"
    ],
    [
      "0.006427515",
      "0.255710028",
      "0.793150066"
    ],
    [
      "0.127951438",
      "0.040796333",
      "0.012740199"
    ],
    [
      "0.780616149",
      "0.942669874",
      "0.167744803"
    ],
    [
      "0.102914619",
      "0.354124640",
      "0.349063945"
    ],
    [
      "0.199179598",
      "0.161071231",
      "0.549447636"
    ],
    [
      "0.290997842",
      "0.919268244",
      "0.757303273"
    ],
    [
      "0.679774591",
      "0.603836409",
      "0.292243926"
    ],
    [
      "0.474897543",
      "0.101740559",
      "0.849970351"
    ],
    [
      "0.601283952",
      "0.071545838",
      "0.419572639"
    ],
    [
      "0.242947938",
      "0.444418104",
      "0.891609547"
    ],
    [
      "0.146682959",
      "0.637471513",
      "0.691225855"
    ]
  ]
}

提交 points：恰好 n 行，每行 d 个 [0, 1) 内的十进制字符串坐标。坐标按模 1 理解，1 写成 0。 · 验证器 v1.0.0

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