Maximum periods of orientable sequences: exact formulas for orders 4 to 9, leading-order asymptotics for all orders, and exact binary values

We determine maximum periods of orientable sequences at orders four through nine, 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, with the binary exceptions stated explicitly. We collect the exact binary values at orders seventeen through twenty, fifteen closed entries of an earlier comparison table, and additional finite exact values and open intervals. Second-order coefficients are determined through order nine. 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 n−1 or n/2 for even n according as the alphabet size is odd or even. Exact palindrome-counting identities, a mixed palindrome/vertex bound, and counterexamples to proposed extrapolations are included. Complete sequences, integer certificates, proof tables and standalone checkers accompany the manuscript. The all-order exact formula and two displayed finite intervals remain open. Independent specialist review is pending. Author: Franciscus G. A. van den Berg, Independent researcher, Bloemendaal, the Netherlands. Contact: [email protected]. ORCID: 0009-0006-4324-0743. Article and data: CC BY 4.0; original code: MIT.