Arboreal height of integer-interlaced cubic squares

The arboreal height of a real algebraic integer is the least height of a finite rooted tree having it as an adjacency eigenvalue. Let $\alpha=\lambda^2$ be a totally positive algebraic integer of degree three, and suppose that each open gap between consecutive conjugates of $\alpha$ contains an integer. We prove that $\lambda$ has arboreal height at most three. Together with Schaeffer's height-two criterion, this gives exact height two when the minimal polynomial of $\alpha$ has a root modulo two, and exact height three otherwise. The proof uses interpolation to express additive inverses as finite nonnegative integer combinations of star resolvents. Salez's affine identity supplies inverses at the next branching level, where a cubic polynomial identity completes the construction. All intermediate branches are invertible at the prescribed eigenvalue. A dilation corollary determines the height of every sufficiently large integer multiple of a square root of an arbitrary totally positive cubic algebraic integer.