AI 中文总结
研究普罗米斯洛群P及斐波那契群H_4 = F(3,4)中非唯一积集。通过精确整数模型和约束求解器，研究非UP集最小规模、双边最小值等，给出相关计算结果并证明有限直径原理，还猜想了D(n)的形式，贡献了P内部数据和结构。
AI 中文摘要
设P为普罗米斯洛群，即三维可定向汉茨谢 - 温特 Bieberbach群，它是普罗米斯洛经典非唯一积集以及加尔丹对单位猜想的反证的基础。有限集A若满足A·A不包含唯一表示元素，则称其为非UP集；此类集是卡普兰斯基零因子和单位问题中的组合障碍。我们对P内的非UP集进行了全面验证的计算和结构研究。在精确整数模型中，我们（i）展示了一个明确的由14个元素组成的非UP集，其最小字半径为3及其完整的重合模式；（ii）通过精确约束求解器运行到不可行证明，证明对于3≤r≤6，半径为r的球内非UP子集的最小规模恰好为14；（iii）通过排序论证单独隔离这些球受限边界不能推广到整个P的结构原因。利用P嵌入到D_infinity^3中，我们证明了一个有效的有限直径原理：如果存在非UP的n集，则在明确半径D(n)≤4^n poly(n)的球内存在一个，因此P的最小值是有效可判定的。我们猜想D(n)=O(n^{1/3})，在此猜想下我们半径为6的计算将已经证明14是P中的最小非UP基数；14是否为此最小值仍未解决。我们还计算了双边最小值，即A·B为非UP时最小的|A| + |B|：在半径为3内它等于24，所以它在[16,24]内，下限是尼尔森 - 索尔伯格定理。作为伴随情况我们处理了斐波那契群H_4 = F(3,4)：它对称地不满足唯一积性质，在半径为4的球内最小对称规模恰好为16，而其在半径为3的球内的双边最小值为22。约束求解器方法并非新方法；我们的贡献是P内部的数据和结构。
英文摘要
Let P be the Promislow group, the orientable Hantzsche-Wendt group of dimension 3, which underlies Promislow's non-unique-product set and Gardam's counterexample to the unit conjecture. A finite subset A of a group is non-UP if every element of A.A has at least two representations ab with a, b in A. Working in an exact integer model of P, we determine the least size of a non-UP set inside word-balls of the standard generators: it is 14 for every radius from 3 to 6, so a smaller non-UP set of P, if one exists, is not contained in the radius-6 ball. The non-existence half of this statement is certified by machine-checked DRAT and VeriPB proofs. Inside the radius-3 ball there are exactly 16 minimal witnesses, all of point-group distribution (2,6,0,6) up to the swap symmetry. The symmetric non-UP property is not translation invariant, so ball searches cannot be recentred; instead we prove an effective finite-diameter principle: if P contains a non-UP n-set, it contains one inside the ball of explicit radius D(n) = 24(n+1)3^n + 10, so the minimum non-UP cardinality of P is computable in principle. Writing rho(n) for the least word-radius of a non-UP n-set, re-realization experiments on witnesses lead us to conjecture rho(n) = O(n^{1/3}); the bound rho(n) <= 6 (whenever finite) for 8 <= n <= 13 would already show, by the radius-6 computation, that Promislow's 14 is that minimum. We also compute, over balls, the two-sided minimum min(|A|+|B|), the profile beta(m) and the unique-product staircase u(n), and compare with the Fibonacci group H_4 = F(3,4): its least symmetric witness over the radius-4 ball has exactly 16 elements, and its two-sided minimum over the radius-3 ball is 22. Over the stated balls the two groups are ordered oppositely by the symmetric and two-sided invariants (14 < 16 but 24 > 22).
Comments18 pages. v2: journal version: complete proof of the finite-diameter theorem with explicit bound; H_4 result placed in the context of Strojnowski's theorem; ball-isometry group corrected. Ancillary: code, logs, verifier
