Anti-basis theorems for higher Aronszajn lines, part 11

We prove that for every regular kappa above Aleph_1, if square(kappa) holds, then there exists a C-sequence over kappa for which the corresponding T(rho_0) satisfies a strong negative partition relation (see Definition 2.9 of our preprint). To mitigate for the lack of a nonreflecting stationary set, the proof goes through a division into four cases.