The All-11q Fibonacci Case: A Cyclotomic Descent and Elliptic-Chabauty Closure

This paper closes the all-11q case of the odd-exponent first-appearance divisor problem for Fibonacci numbers. It proves that, for every prime q, the Fibonacci number with index eleven times q has a prime divisor other than five whose rank of apparition is exactly eleven times q and whose exponent in that Fibonacci number is odd. Exact local certificates first reduce any hypothetical failure to two residue classes modulo 37,837,800, together with a quadratic-residue condition modulo 89. The remaining square obstruction is then expressed as a norm equation in the maximal real subfield of the eleventh cyclotomic field. Class number one, full unit signatures, and the factorization of 89 reduce all possible solutions to a single explicit square class. Multiplying the conjugate equations produces a genus-one quartic with a rational parameter. Its elliptic model has exact Mordell-Weil rank one and torsion of order four. An explicit non-torsion point is certified to be two-saturated, proving that the supplied Mordell-Weil subgroup has odd index. Elliptic Chabauty and the Mordell-Weil sieve then show that the only rational parameter value is the trivial boundary value three. This contradicts the Lucas-number values arising from every remaining prime q and completes the proof. The computation is reproducible in Magma and uses exact rank bounds, saturation, local Chabauty, and Mordell-Weil-sieve certificates. It does not rely on the Generalized Riemann Hypothesis, analytic-rank assumptions, or a bounded point search. The argument adapts the established real-cyclotomic elliptic-Chabauty strategy to a ramified 89-twist; the specific descent, elliptic quotient, certified Mordell-Weil data, and all-11q application are the new components.