On $p$-Adic Valuation Trees Arising from Laplace Transform of Fibonacci Polynomials

We study the $p$-adic valuation trees associated with the sequence \[ e_n=\int_0^\infty e^{-t}F_n(t)\,dt, \] obtained by evaluating the Laplace transforms of the Fibonacci polynomials at $1$. We answer an open question of Givens and Moll in the negative by constructing an explicit counterexample showing that even internal branches need not be simple. We then reformulate the Givens--Moll lifting criteria for the even and odd subsequences in terms of the roots and derivatives of two associated polynomials. In particular, simple roots modulo $p$ give rise to simple internal branches. Finally, we develop a recursive lifting procedure that determines the unique surviving residue class at each level of a branch associated with a simple root and computes the $p$-adic valuations of the corresponding terms.