AI 中文总结
本文为整群环的扭曲酉元证明紧致性定理，结合SAT认证与镜像对称，将Promislow群B(4)窗口的theta-酉单位消失问题归约为单个下界，并给出数值估计与结构障碍分析。
AI 中文摘要
设G是一个无挠群，其实群代数R[G]无零因子，并设u -> u^{*theta}是R[G]上的一个l2-等距反自同构（即反转对合与环自同构及符号特征的复合）。我们证明了一个紧致性定理：对于每个有限的*theta-闭支撑窗口W，常数mu*(W) = min{ ||w^{*theta}w||_2: ||w||_2 = 1, supp(w) in W }严格为正，并且每个支撑在W中的实theta-酉元（满足u^{*theta}u = 1）都满足||u||_2 <= mu*(W)^{-1/2}。特别地，支撑在W中的整数theta-酉元构成一个有限且可有效枚举的集合：单位搜索中先验无限的“高度”方向坍缩为单个实常数。对于Promislow（Hantzsche-Wendt）群P以及承载Gardam在F_2上单位猜想反例的窗口B(4)，我们将此与精确的SAT认证高度阶梯（每层高度<= 31，更大高度处有DRAT认证的主单元）、无半径深度尾部定理以及+-层之间的镜像对称相结合，将所有非平凡theta-酉单位u = +-1 (mod 2)（supp(u) in B(4)）的消失归结为对mu*(B(4))的单个认证下界。我们报告数值估计mu*(B(4)) ~ 1.4e-3，远高于所需阈值，并证明了关于剩余认证问题的三个结构结果：不存在线性（Cauchy-Schwarz）对偶证书，因为P承载theta-反酉平凡元素；并且，数值上，在直接公式和理想乘子公式中，二级平方和松弛都被边界钉住且具有显式斜率——障碍是一个没有任何测度能实现的虚假伪矩。我们还将该机制与Z[D_infinity]进行对比，其中挠性产生零因子，mu* = 0，以及真正无界的扭曲酉单参数族。
英文摘要
Let G be a torsion-free group whose real group algebra R[G] has no zero divisors, and let u -> u^{*theta} be an l2-isometric anti-involution of R[G] (a composition of the inversion involution with a ring automorphism and a sign character). We prove a compactness theorem: for every finite *theta-closed support window W the constant mu*(W) = min{ ||w^{*theta}w||_2 : ||w||_2 = 1, supp(w) in W } is strictly positive, and every real theta-unitary element (u^{*theta}u = 1) supported in W satisfies ||u||_2 <= mu*(W)^{-1/2}. In particular the integer theta-unitary elements supported in W form a finite, effectively enumerable set: the a priori infinite "height" direction of the unit search collapses to a single real constant. For the Promislow (Hantzsche-Wendt) group P and the window B(4) that hosts Gardam's counterexample to the unit conjecture over F_2, we combine this with exact SAT-certified height ladders (heights <= 31 per stratum, DRAT-certified master cell at larger heights), a radius-free depth-tail theorem, and a mirror symmetry between the +- strata, reducing the vanishing of all nontrivial theta-unitary units u = +-1 (mod 2) with supp(u) in B(4) to a single certified lower bound on mu*(B(4)). We report numerical estimates mu*(B(4)) ~ 1.4e-3, well above the required threshold, and prove three structural results about the remaining certification problem: no linear (Cauchy-Schwarz) dual certificate exists, because P carries theta-anti-unitary trivial elements; and, numerically, the level-2 sum-of-squares relaxation is boundary-pinned with an explicit slope, both in the direct and in the ideal-multiplier formulation -- the obstruction being a spurious pseudo-moment that no measure can realize. We also contrast the mechanism with Z[D_infinity], where torsion produces zero divisors, mu* = 0, and genuinely unbounded unipotent families of twisted unitaries.
Comments9 pages. Ancillary files: verification scripts, DRAT certificates and run summaries (34 files)
