An Explicit Evaluation of a Fibonacci Character Sum for Primes with Full Rank of Apparition

This preprint gives an explicit evaluation of the quadratic character sum S(p) = ∑ₙ₌₁ᵖ⁻¹ χ_p(F_n), where F_n denotes the nth Fibonacci number and χ_p is the quadraticcharacter modulo an odd prime p. Under the full-rank hypothesisα(p)=p-1, where α(p) is the rank of apparition of p in theFibonacci sequence, the paper proves that p ≡ 11 or 19 (mod 20) andthat S(p)=+1 if p≡11{20},(p)=-1 if p≡19{20}. The proof combines the rank--order identity for Fibonacci roots, aprimitive-root parametrization in the full-rank regime, a structuralreduction of the sum to a quadratic character of a root difference, anda classical finite-field character-sum identity. Computational verification scripts and supporting reproducibilitymaterials are archived separately athttps://doi.org/10.5281/zenodo.21431565.