A Cycle Criterion for CAT(0) Signed Permutation Mapping Tori

Let F be freely generated by a,b_1,...,b_m and let phi(a)=a^epsilon and phi(b_i)=a^{p_i} b_{sigma(i)}^{delta_i} a^{q_i}, where epsilon and delta_i are signs, sigma is a permutation and p_i,q_i are integers. We give an explicit necessary and sufficient cycle criterion for F semidirect_phi Z to be CAT(0). A positive cycle has return data (P_C,Q_C,l_C). If epsilon=1, the ratios (P_C-Q_C)/l_C must agree on positive cycles with nonzero P_C+Q_C. If epsilon=-1, each even positive cycle must satisfy P_C^2=Q_C^2. Negative cycles impose no additional condition. In the positive cases a compact two-dimensional locally CAT(0) graph of spaces realizes the original group; failure obstructs every proper semisimple CAT(0) action. Positive cycles become HNN edges of a torus or Klein-bottle group, and negative cycles become square-root attachments. We also prove power-invariance within this family and give a polynomial-time arithmetic test. The unsigned identity-permutation subcase is the theorem of Wu and Ye. The full AIM characterization of CAT(0) free-by-cyclic groups remains open. This is an AI-assisted, self-audited, unrefereed preprint; no absolute priority is asserted.