Primitive Ternary Relations and Dissociativity in Fibonacci-Type Sequences

Abstract

This preprint studies support-minimal ternary relations among positive integer sequences satisfying the Fibonacci recurrence uₙ₊₂ = uₙ₊₁ + uₙ. It proves that every nonzero support-minimal relation with coefficients in {−1, 0, 1} has, up to an overall change of sign, the unique form uₐ₊₂ᵣ₊₂ = uₐ + uₐ₊₁ + uₐ₊₃ + uₐ₊₅ + ⋯ + uₐ₊₂ᵣ₊₁, for integers a ≥ 1 and r ≥ 0. The classification depends only on the recurrence and not on the initial values. Using this result, the paper gives an exact criterion for a subfamily of {u₁, …, uₙ} to have distinct subset sums and proves that the largest such subfamily has cardinality ⌊n/2⌋ + 1. It also enumerates all dissociated subfamilies by cardinality through a rational generating function. The resulting counting sequence satisfies a third-order linear recurrence and has exponential growth rate 2 cos(π/7). The accompanying verification files independently check finite instances of the primitive-relation classification, the subset-sum criterion, the dimension formula, and the enumerative identities. Computation is not used in the proofs.