Asymptotic Positivity and Integer Periodicity for the Log-Concavity Operator

This preprint studies asymptotic positivity and periodicity under iteration of the log-concavity operator. It proves that sufficiently large integer powers of a nonnegative polynomial with positive constant, linear, penultimate, and leading coefficients are strictly infinitely log-concave on their original full support. The theorem also allows multiplication by a fixed real polynomial that is positive on the positive real axis and has positive end coefficients, resolving the large-power conjecture of Medina and Straub. Further results classify eventual infinite log-concavity for nonnegative and real polynomial bases, characterize positive-axis positivity through binomial smoothing, and give uniform estimates for entry into a preserved strong log-concavity region. A separate arithmetic theorem proves that every periodic orbit of an integer half-line sequence has period at most two, allowing zeros and negative entries throughout the orbit. The deposit includes the manuscript PDF, LaTeX source, and a computational supplement. Four primary Python computations use only the standard library and reproduce exact certificates for a finite Poisson cone, integer polynomial identities, and fixed polynomial and support examples. An additional independent C++ interval verification is included with compilation instructions. The general analytic and infinite-sequence results are proved in the manuscript; the finite computations certify the specific inputs and identities described there.