Fibonacci Character Sums at Rank Index Two and Jacobsthal Sums

This preprint studies quadratic character sums over one full rank period of the Fibonacci sequence, $S(p) = sum_{j=1}^{z(p)} (F_j/p)$, where $z(p)$ is the rank of apparition of the prime $p$ and $(./p)$ denotes the Legendre symbol. The paper focuses on the rank-index-two case $i_p = (p - (5/p))/z(p) = 2$. The main results reduce the Fibonacci character sum at rank index two to classical Jacobsthal sums and give explicit split and inert evaluations. In the split case, the sum is expressed in terms of the two-square representation $p = a^2 + b^2$ and a sign depending on $p$ modulo $5$ and $8$. In the inert case, the sum is reduced to one half of a Jacobsthal sum $J(5)$, yielding the alternatives $±b$ for $p$ congruent to $1$ modulo $8$ and $0$ for $p$ congruent to $5$ modulo $8$. The manuscript also records a conditional density corollary for inert index-two primes, derived from the GRH-conditional density theorems of Roskam and Chen. This density result is contextual and is not used in the proofs of the Jacobsthal reductions or the main explicit evaluations. A separate reproducibility archive provides the companion Python verifier, machine-readable numerical outputs, run logs, and checksum evidence. DOI: 10.5281/zenodo.23263042.