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

Source abstract

We study packings of annuli ("rings") of a common width into a disk, where a ring may nest inside the hole of a strictly larger one, a selection-oriented relative of the Recursive Circle Packing Problem. The two natural objectives, cardinality and contact area, genuinely diverge. For superincreasing radii (each exceeding the sum of all smaller ones) we prove that the descending greedy maximizes every positive, increasing, superadditive objective. Our main structural theorem shows more: the placement rule is irrelevant - any choice among feasible containers yields the lexicographically maximal feasible set, for containers of arbitrary shape and in every dimension. Both hypotheses are sharp: placement irrelevance holds for at most three rings and fails at four, and twin instances rule out every rule that is a function of the observable state. Write $ρ=\max_i(\sum_{j>i}r_j)/r_i$ for the violation of superincreasingness. The additive relaxation has universal threshold exactly $ρ=1$. In the geometric model we prove, with no tangency idealization, that the rigid four-ring family has infimum exactly the Tribonacci constant $T\approx1.83929$. Yet $T$ is not the global threshold: an explicit golden family breaks placement obliviousness at $ρ=\varphi+3\varepsilon$ for every small $\varepsilon>0$, proving $τ\le\varphi<T$ for the geometric threshold $τ$ and refuting the natural Tribonacci-threshold conjecture. The matching bound $τ\ge\varphi$ remains conjectural; we prove it for pair profiles and outside an explicit heavy region. We also give a phase diagram for this divergence and split hardness into a geometric layer and a combinatorial (subset-sum) layer, of which superincreasingness eliminates exactly the latter. The main theorems carry complete written proofs; every computer-assisted closure carries an epistemic label and a script in the verification map.