Minimal Recurrence Orders for Rationally Weighted Binomial Sums

For a positive integer r and a rational function R regular at the origin, this preprint studies a_n(R) = [z^n] R(z)(1-z)^(-rn-1). An elementary specialization of Hermite reduction produces a residual space of dimension s+1, where s is the number of distinct finite poles of R other than 1, irrespective of their multiplicities. The manuscript proves that coefficient extraction is injective on this residual space, so that the first linear dependence among shifted kernels gives the minimal recurrence of the coefficient sequence rather than only a sufficient telescoping relation. A sharpness theorem gives order exactly s+1 under stated hypotheses on the pole images under z(1-z)^r and on the value of R at the critical point. Applications recover the order-one classification for geometric weights and give a uniform third-order recurrence, with a proof of minimality for every positive integer r, for the family sum_k binom((r+1)n-k, n-2k). A Lucas-number identity supplies compact certificates. Explicit applications include OEIS A371742, A371743, A371744 and A385632. The paper situates the construction within established Hermite-reduction and creative-telescoping methods. Companion code, exact rational certificates and a recorded verification log are supplied. AI-assisted research preprint; not independently peer reviewed. The manuscript discloses assistance in mathematical development, literature searching, code construction and English drafting. Computational checks support the displayed identities but do not replace independent review of the analytic proofs or establish first-discovery priority. No claim is made that the work resolves a previously recognized unsolved OEIS conjecture.