Unimodality of (q)-Fibonomial Coefficients and A Rank-Two Cyclotomic Reduction for Fibonacci Semiprimes: Resolution of the (q)-Fibonomial conjecture and a semiprime reduction

This paper studies two unrestricted problems involving Fibonacci numbers. The first part proves the Bergeron-Ceballos-Kuestner unimodality conjecture for every q-Fibonomial polynomial. For every pair of positive integer indices, the coefficient sequence is symmetric and unimodal. The proof combines a weighted complete-intersection realization, an exact fixed-shell recurrence, a lower-shell single-crossing theorem, root-of-unity Fourier analysis, and cancellation over complete Fibonacci blocks. Columns through ten are treated by exact finite arguments, columns eleven through thirty-four by uniform terminal absorption, and all columns from thirty-five onward by a quantitative Fourier theorem. The unbounded proof requires only twenty-four exact scalar inequalities; the large coefficient computation through total index thirty-nine is retained solely as an independent audit. The second part concerns odd-multiplicity first-appearance prime divisors of Fibonacci numbers at semiprime indices. For every fixed odd prime multiplier at least eleven, the problem is reduced to finitely many prime partners. For multiplier eleven, the exceptional collision at the prime 89 is resolved, a high-density congruence certificate is established, and the remaining obstruction is converted into an explicit genus-two integral-point problem. A reproducible Magma computation certifies that the associated Jacobian has trivial torsion and Mordell-Weil rank two, identifying rank-equals-genus integral quadratic Chabauty as the remaining step in this paper. The q-Fibonomial unimodality problem is completely resolved here. The separate all-11q Fibonacci problem is reduced but not claimed as closed in this paper.