Columns Through Fourteen and Eventual Fixed-Column: q-Fibonomial Unimodality

This paper proves two advances concerning the conjectured unimodality of q-Fibonomial coefficients. These polynomials are q-analogues of Fibonomial coefficients formed by replacing the ordinary integers in factorial quotients with q-integers indexed by Fibonacci numbers. First, the paper proves that every q-Fibonomial coefficient is symmetric and unimodal whenever the smaller of its two parameters is at most 14. This extends the previously established complete range from columns at most 3 through columns at most 14. For columns 5 through 14, the coefficients increase strictly through the rising half, apart from the equality of the two central coefficients forced by symmetry when the degree is odd. Each fixed column is analyzed through an exact restricted-partition kernel. A root-of-unity decomposition separates the dominant polynomial component from lower-degree oscillatory packets and bounded periodic remainders. The numerator of the q-Fibonomial quotient becomes a finite Fibonacci shift stencil. Exact Bernstein-basis derivative certificates, endpoint estimates, parity certificates, and fully active factorizations then establish positivity of all coefficient differences on the rising half. The new fourteenth-column argument separates a degree-12 principal polynomial, a cubic alternating packet, a quadratic order-3 packet, linear packets of orders 4, 5, 8, and 13, and a bounded simple-pole remainder. Its simple-pole components have a joint period of 7,368,166,325,520. The proof avoids enumerating this enormous period by decomposing it into coprime components and inspecting only 17,890 component entries. The column-14 proof uses a 320-shift numerator stencil, exact Bernstein-basis derivative certificates for 316 ordinary activation regions, 319 endpoint and parity error certificates, a separate concavity certificate for one exceptional region, and a final fully active factorization. The accompanying exact computation also checks more than 18.8 million rising-half coefficient differences in column 14. Second, the paper proves eventual strict unimodality in every fixed column of index at least 4. For each fixed column, there is an effectively computable threshold such that all sufficiently large values of the other parameter produce strict increase through the rising half and strict decrease through the falling half, apart from the symmetry-forced two-term central plateau. The asymptotic proof identifies the principal restricted-partition contribution with the derivative of a convolution of uniform densities. This derivative is uniformly positive before the midpoint and has a simple zero at the midpoint. Every nontrivial root-of-unity contribution is smaller by one full power of the relevant Fibonacci number. Consequently, the full two-parameter conjecture is reduced, separately in every fixed column, to checking finitely many remaining parameter values. The results also extend the associated q-FiboCatalan coefficient-positivity consequence through column 14 whenever the required integrality condition holds. All computational certificates use exact integer and rational arithmetic. The general theorems do not depend on finite numerical experimentation.