Brauer-Picard Groups and Module Orbits for Twisted Cyclic Categories

For every cyclic group C_n and every associator class omega_t, this preprint explicitly computes the Brauer-Picard group of Vec_(C_n)^(omega_t) and its action on all indecomposable module categories. Module labels are the divisors of gcd(n,t). For p^k exactly dividing n and s=min(v_p(t),k), let U be the units modulo p^k satisfying a^2=1 modulo p^(k-s). The local group is U when s=0, and the inversion semidirect product U with C_2 when s>0. The units fix all module labels; the involution sends the local exponent j to s-j. There are product_p(floor(s/2)+1) orbits, with transitivity exactly when gcd(n,t) is squarefree. The proof computes the center metric group, all Lagrangian subgroups, group generators, orbit invariants and stabilizers, including every 2-primary case.

The calculation uses established categorical theorems and credits earlier cyclic-Lagrangian counts and untwisted orbit formulas. It is a complete theorem for the cyclic family, not a resolution of the broad AIM-OTHER-0057 classification program. The unrestricted proof is supplemented by 383 exact finite isometry cases and 1,556,863 categorical checks in each of normal and optimized Python. This is an AI-assisted, unrefereed preprint; no independent review, formal proof verification or absolute priority is claimed.