Greedy Packing of Nested Rings: Placement Rules, a Golden Counterexample, and a Tribonacci Floor

Abstract

We study packings of annuli of a common width, allowing each ring to nest inside the hole of a larger one. The objectives of maximizing contact area and cardinality diverge: area is superadditive in the radius, cardinality is not. Under superincreasing radii, every descending greedy maximizes every positive, strictly increasing, superadditive objective. More strongly, any choice among feasible containers yields the lexicographically maximal feasible set, for containers of arbitrary shape in every dimension. This placement irrelevance holds unconditionally for at most three rings and fails at four in disks and squares; twin instances exclude every universal rule based only on the observable state. Write $ρ=\max_i(\sum_{j>i}r_j)/r_i$. The additive model has threshold exactly $1$. For disks we prove the exact global threshold $τ=φ$, with no failure at $ρ\leφ$, for every finite inventory, even with independent hole radii. The key geometric theorem states that, under golden tail bounds, an entire disk list fits a circular container if and only if its three largest disks fit; this supplies the uniform exchange of parents that the threshold proof needs. The Tribonacci constant $T\approx1.83929$ remains the exact floor of a rigid subfamily. A dimension-reduction lemma transfers spherical sharpness results to all dimensions $d\ge2$, and a separate argument proves the golden threshold for at most five rings in those dimensions. For square pans, a Cartesian confinement criterion gives twins and a family proving $τ_{\square}\le Y\approx1.6845$; its optimality is open. For independent holes, the exact universal area guarantee under $ρ\leκ<1$ is $\min(1,κ^{-2}-1)$, with threshold $1/\sqrt2$. The repository has 122 Lean theorems. Euclidean geometry, forest assembly and continuity remain written proofs; numerical checks do not substitute for them.