Why it is a question
LIT-874 (read in NOTE-679) places a node’s children at hypercube vertices chosen by a Hadamard code, and so embeds any tree in H^r with distortion 1 + ε using O((ℓ/(εr)) log deg_max) bits per coordinate for r up to about log deg_max + 1, and O(ℓ/ε) beyond (Proposition 3.1). The matching lower bound is proved only in the plane, which THEORY-734 now holds with explicit constants. The paper’s remark that no dimension helps beyond log deg_max is a footnote argument. Until 2026-10-19, THEORY-196’s promote_when asked for this extension together with the planar proof. THEORY-196’s title is about the plane, and its body disclaims the extension, so the clause moved here.
What would answer it
A proof that some family of trees embedded in H^r with worst-case distortion 1 + ε forces a node to hyperbolic distance Ω((ℓ/(εr)) log deg_max) from the origin for r ≲ log deg_max, and Ω(ℓ/ε) beyond. The planar argument’s angle-packing step (THEORY-734, step 4) would become a spherical-code bound on S^(r−1). Or a construction that beats Proposition 3.1 in depth or in dimension. Experiments in which deep trees need more bits than bushy ones cannot settle it, because the upper bound already predicts that.