Exact energies of limited-reach low-autocorrelation binary sequences for all lengths

For each fixed reach r=3,...,26, this preprint gives an exact eventually periodic-affine formula for the minimum total energy E(n,r) of an open binary sequence of length n, with the full finite residue data and exact integer checks. The minimum mean energy mu_r is determined exactly. Cycle-and-potential certificates are included for r=3,...,24; normalized-vector recurrence certificates cover r=3,...,26 and are extended to r=27,28. The same finite-range window Hamiltonian was introduced and studied by Liers et al. (2010), who computed finite ground states for R=3,...,6 and discussed transfer-matrix and iterative methods. We compare our values in their normalization, including the reported R=5 energy-gap scale. We make no claim to have introduced the model or finite-range dynamic programming. Among the computed reaches, mu_r equals the least one-window energy e*_r exactly for r=3,4,6.This version adds the sliding-window merit factor of periodic sequences, F*_r=r^2/(2 mu_r), and exact values for r=24,...,28; all five exceed 3.342065. It derives a lower bound of 3.342065... from the Legendre-sequence asymptotics of Jedwab, Katz, and Schmidt. A heuristic search at reaches r>=30 found a best candidate score about 2.90 at r=30, while exact values at r=27 and r=28 are about 3.68 and 3.57. Since these are different reaches, the comparison does not prove that the r=30 candidate is suboptimal at r=30; it does show that the heuristic results cannot inform the limiting value. No finite upper bound or convergence theorem is claimed.