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Nusa Dua Studio（努沙杜瓦工作室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过对称层内精确决策与双重验证的认证计算方法，确定了立方体无三点共线问题的若干精确值、下界及平面Guy-Kelly计数的修正常数，并分析了误差与方向谱。
AI 中文摘要
我们报告了关于四个无三点共线问题的认证计算，其中两个在立方体中，两个在平面中，采用同一种方法：在对称层内的精确决策过程，每个见证由第二个不共享代码的程序重新验证，每个数字都追溯到公开期刊。(I) 设a(n)为{0,...,n-1}^3中无三点共线的最大点数（A399138）。我们确定了a(1),...,a(6) = 1, 8, 16, 28, 40, 64，并附有DRAT认证的不可满足性证明，给出了认证下界a(7) >= 73, a(8) >= 94, a(9) >= 116, a(10) >= 138, a(11) >= 164，并证明了对于每个素数p，a(p) >= p^2。最优解共享层结构2n^2 - 2n + 4，该结构在n = 5和n = 7时被证明失效。(II) 对于b(n)，即无四点共面的最大点数（A280537），十九个公开配置给出了9 <= n <= 29的下界，其中四个改进了已知界及其单调闭包（b(12) >= 31, b(21) >= 47, b(22) >= 49, b(27) >= 56）；循环不变子空间在n = 9时最大值为23，在n = 10时为26；四种对称性与该问题不相容；除3n外没有已知上界。(III) 针对平面中的Guy-Kelly一阶矩启发式方法，我们对照精确计数（A000755至n = 20）进行了审计：其修正常数以闭式形式得出，即pi/sqrt(3)，阈值穿越n = 493被重现，并且其误差被证明取决于问题的形状而非仅取决于n，具有无界乘子；残差误差是Theta(n)还是Theta(n ln n)（这决定了常数是否成立）无法通过计数来判断，我们测量了我们离判断有多远。(IV) 测量了2n点解的方向谱，一个无拟合参数的线模型重现了其形状（十五个常数在12%以内，七个预测盲）但未能重现其尺度。被撤回的主张保留在文本中；每个部分都说明了它未确立的内容。
英文摘要
We report certified computations on four no-three-in-line questions, two in the cube and two in the plane, by one method: exact decision procedures inside symmetry strata, every witness re-verified by a second program sharing no code, every number traced to a public journal. (I) Let a(n) be the largest number of points of {0,...,n-1}^3 with no three collinear (A399138). We determine a(1),...,a(6) = 1, 8, 16, 28, 40, 64 with DRAT-certified unsatisfiability proofs, give certified lower bounds a(7) >= 73, a(8) >= 94, a(9) >= 116, a(10) >= 138, a(11) >= 164, and prove a(p) >= p^2 for every prime p. The optima share a layer structure 2n^2 - 2n + 4 that provably fails at n = 5 and n = 7. (II) For b(n), the largest number with no four coplanar (A280537), nineteen public configurations give lower bounds for 9 <= n <= 29, four of which improve the known bounds and their monotone closure (b(12) >= 31, b(21) >= 47, b(22) >= 49, b(27) >= 56); the cyclically invariant subspace has maximum 23 at n = 9 and 26 at n = 10; four kinds of symmetry are incompatible with the problem; no upper bound beyond 3n is known. (III) The Guy-Kelly first-moment heuristic for the plane is audited against exact counts (A000755 to n = 20): its corrected constant comes out in closed form, pi/sqrt(3), the threshold crossing n = 493 is reproduced, and its error is shown to depend on the shape of the question rather than on n alone, with an unbounded multiplier; whether the residual error is Theta(n) or Theta(n ln n), which decides whether the constant survives, cannot be told by counting, and we measure how far from telling we are. (IV) The direction spectrum of the 2n-point solutions is measured, and a line model with no fitted parameter reproduces its shape (fifteen constants within 12%, seven predicted blind) but not its scale. Withdrawn claims are kept in the text; each part states what it does not establish.
Comments27 pages; v1.0 (21 Sep 2026). Consolidates and supersedes three notes declined as submit/8060684, 8062345, 8062503 (arXiv moderation MOD-104121) and one unsubmitted note; earlier versions of the parts at Zenodo (DOIs in Sec. 8); witness archive doi:10.5281/zenodo.22271375; ancillary files: two verifiers, thirteen witness configurations, two data files
