Maximum periods of orientable sequences: exact values, formulas for orders 4–8 and leading-order asymptotics

We determine maximum periods of orientable sequences at orders four through eight, with the previously known odd-alphabet order-four formula attributed to Mitchell and Wild. Integer dual certificates, odd cuts and explicit pattern constructions establish the formulas, including M_k(8) = (k^8 − 7k^4 + 8k^2 − 2k)/2 for odd k ≥ 3. We collect the binary exact values at orders seventeen through twenty and fifteen closed entries of an earlier comparison table. A palindrome-return bound, mixed edge–vertex counting and a sparse parity-repair construction, together with published constructions of Mitchell and Wild, determine the leading deficit for every fixed order n ≥ 4. The coefficient is (n−1)/2 for odd n, and is n−1 or n/2 for even n according as the alphabet size is odd or even. Complete finite proof data, sequences and exact checkers accompany the manuscript. Independent specialist review is pending.The supplementary ZIP includes the LaTeX source, full sequences, integer certificates, standalone checkers, proof notes and SHA256 manifests. Article and data: CC BY 4.0; original code: MIT.Review scope: the author confirmed independent checking by Claude of the earlier results through E1, E2 and even-alphabet order eight. The new odd-alphabet order-eight section was checked with exact symbolic code, an independent equality-bit implementation and full cyclic-window checks; a second-AI review of that new section is not claimed. See the manuscript for the full AI disclosure.Author contact: [email protected].