A compactness theorem for twisted-unitary elements of integral group rings, with a certified route to the theta-unitary Case A at window B(4) of the Promislow group

Abstract

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.