Phyllotactic Reversal Fractions in Fibonacci p-Number Sequences: Farey Chains, Exterior-Square Dynamics, and the Exceptional p = 4 Pathway

Rational divergence angles and changes of parastichy handedness are classical features ofmathematical phyllotaxis. We study rational reversal pathways generated by Fibonacci p-numberrecurrences and ask why second-order pathways form Farey-neighbor chains whereas higher-orderpathways can jump or repeat reduced rational nodes. For a divergence fraction α and nearbyreduced rational a/b, the signed residual 2π(bα − a) records the local winding sense of the b-stepfamily.For the order-m = p + 1 recurrence, the determinant governing adjacent scalar fractionsis a Pl¨ucker coordinate of Xn ∧ Xn+1 and therefore evolves under the exterior square of thecompanion matrix. We derive its exact spectral expansion, prove nonvanishing of every paircoefficient for the canonical positive sequence, and show that Farey adjacency after reductionoccurs precisely when gcd factors cancel the evolving minor. The limiting reversal fraction isαp = qp+1p = 1 − qp.The case p = 4 is exceptional. Although the cyclotomic factor of x5 − x4 − 1 is classical,we prove that the canonical reversal signature is exactly + + + − −− from n = 6 onward. Ageneral trinomial root-location theorem further implies that m = 5 is the unique order withm ≡ 5 (mod 6) for which the sixth-root pair dominates every other non-Perron root. Historicalorder-five branching models provide biological context, while the period-six signature remains amathematical prediction requiring independently indexed biological data.