An explicit upper bound for binary orientable sequences via palindrome periods

We give an explicit upper bound M_n ≤ U*_n for the maximum period M_n of a binary orientable sequence of order n. For n = 2m the bound is U*_{2m} = 2^{2m-1} − (m−2)2^{m−1} − m(m−1)/2 − 2, and U*_{2m+1} = 2U*_{2m}. It improves the Dai–Martin–Robshaw–Wild (1993) bound for every n ≥ 11 and has deficit 2^{n−1} − U*_n = Θ(n·2^{n/2}). The even-order proof counts possible return distances between palindromic windows in a cycle decomposition of the de Bruijn graph. A circulation formulation and the binary Lempel map establish exact doubling of the fractional optimum from even to odd order. We also give an explicit dual potential based on even palindromic prefixes and suffixes. Combined with the known construction of Dai et al., this determines the order of the second term: M_n = 2^{n−1} − Θ(n·2^{n/2}), with explicit constants (between 1/4 and 3/4 for even n, between 1/(2√2) and 1/√2 for odd n). Exact-arithmetic checks reproduce the fractional optima for 4 ≤ n ≤ 17; an accompanying finite certificate proves M_17 ≤ 63604, sharper than the general formula at that order. We also exhibit an orientable sequence of order 17 and period 63578, longer than previously tabulated constructions, so that 63578 ≤ M_17 ≤ 63604. The supplement contains solver scripts, a standard-library exact checker, integer certificates for 4 ≤ n ≤ 17, comparison tables (n = 5…40) and checksums. Code: MIT License. Article and data: CC BY 4.0. To the author's knowledge this bound has not appeared before; references to earlier equivalent results are welcome.