Negative orientable sequences: exact periods at orders three and four, finite certificates and even-order asymptotics

We study negative orientable sequences over Z_k with modular negation. We give explicit integer dual certificates and matching constructions for the maximum period at order three for every even k>=4, and at order four for every k>=3. A negative-palindrome return bound, combined with the construction of Mitchell and Wild, yields the optimal leading deficit m k^m+O_m(k^(m-1)) at each fixed even order 2m>=4. Two separately implemented integer checkers verify 62 baseline finite cases, closing 27 entries in the cited Mitchell–Wild table, and four additional order-four cases at k=17–20. The package includes sequences, certificates, reproducible checkers and a stepwise adversarial review of the general arguments. Finite mechanical verification is distinguished from analytical proofs; independent specialist review is pending. Higher-order exact families and the odd-order leading deficit remain open.