Fibonacci Square Sums: Rank Descent and Sparse Roots

This work studies the Diophantine equation Fₙ + Fₘ = z² for indices of opposite parity. Three complementary structures organize the analysis: prime ranks constrain the common index divisor, quadratic norms isolate the index gap, and signed Fibonacci expansions reveal restrictions on the square root. The principal classification concerns roots whose unique far-difference expansion contains at most two alternating runs, with no restriction on the length of either run. Exactly the five known triples occur. Together with uniform square inequalities for separated expansions, this yields an analytical classification of all roots of signed Fibonacci length at most three, without enumerating candidate solutions. A telescoping identity turns each alternating run into two endpoints after doubling, giving effective index bounds in terms of the number of runs. Explicit Lucas-based multipliers extend this compression to repeated arithmetic progressions. An exact congruence also simplifies recovery of Fibonacci solutions from the associated quartic curves. The main paper presents the essential proofs independently of finite-search certificates. A companion develops further arithmetic and geometric results and preserves reproducible computational classifications. The unrestricted classification remains open; the papers identify precise upper bounds on root complexity or quartic height that would suffice for finiteness.