The Cyclotomic Mock-Theta Constant c_0(T_k)=3 as a Mock Gauss Sum

We study the cyclotomic integer T_k arising from Ramanujan's third-order mock theta function at odd prime roots of unity. Writing T_k in the power basis of Z[zeta_k], we prove it is a mock Gauss sum: the Gauss-sum component has vanishing constant coordinate (exact identity via 1+24=25), and conjecturally T_k lifts to a genuine quadratic Gauss sum plus a 3-divisible 0/1 indicator. We prove the character-free cancellation T_k(1) + Tr(T_k) = 3k, the cube-root trace formula, a Galois-invariance obstruction explaining why the coordinate bound is hard, and the Euler-Dirichlet reading c_0 = 1/L(0,chi{-3}) = 3. The central conjecture c_0(T_k)=3 (verified to k