Sum Rules and Sign Thresholds for Twisted Sector Sums. An exact sum rule over all 2d 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

Abstract

Assign to each direction of a d-dimensional lattice a twist label t_i\{0,1\} and consider the regularized alternating lattice sum Z_t(s)='_{n 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: _{t}Z_t(s)=2^{\,d-s}Z_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 -1{12}+1{24}=-1{24}. (T2, the sign threshold) Writing k=_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) d/2, and and we prove the matching lower bound, so the threshold is exactly d/2: the sum turns positive exactly when the twisted directions are at least as many as the untwisted ones, the threshold being k^= d/2. The proof reduces, via strict monotonicity in k, to the classical theta identity _3_4=_4(2x)^2 together with _4 1. (T3, the mechanism) The large-x expansion of the theta product is 1+2(d-2k)e^{-x}: the first lattice shell carries +1 on untwisted axes and -1 on twisted ones, and the count difference d-2k drives the sign — on the marginal line 2k=d the first shell cancels exactly and the theta product collapses to _4(2x)^d, so positivity there is part of the proved side. All claims are verified deterministically (62/62). The article makes no stabilization claim: it states an exact property of a regularized lattice sum, not a statement about any effective potential (Section 8).