Sum Rules and Sign Thresholds for Twisted Sector Sums. An exact sum rule over all 2^d twist sectors, and a proved threshold: the regularized sum turns positive exactly when the twisted directions are at least as many as the untwisted ones

Assign to each direction of a d-dimensional lattice a twist label t_i in 0,1 and consider the regularized alternating lattice sum Z_t(s)=sum'_n in Z^d(-1)^n. t|n|^-s — the family of 2^d sector sums that the framework's sector calculus produces on a d -torus with periodic or antiperiodic gluing in each direction [1]. Individual members of this family are classical and have been computed many times; what appears to be missing are the laws that bind the family together. This article supplies two. (T1, the sum rule) The 2^d sector sums add up to a single rescaled copy of the untwisted one: sum_tZ_t(s)=2^d-sZ_0(s), an exact identity for every admissible s, with a two-line proof from character orthogonality. At the value s=d+1 the right-hand side is exactly one half of the untwisted sum; in d=1 this is the familiar -tfrac112+tfrac124=-tfrac124. (T2, the sign threshold) Writing k=sum_i t_i for the number of twisted directions and E(t)=-Z_t(d+1), we prove that E is strictly increasing in k, that the untwisted sector is negative in every dimension, and that 2k >= d implies E(t)>0; these give a unique threshold k^*(d)