manufacturing process planners

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.

Abstract: The 11/6 worst-case constant of the Wang-Sitters rounding scheme, which a companion note establishes, can naturally be attributed to the freedom in Step 3, where an arbitrary valid slot matching is permitted. We show that eliminating that freedom does not improve the constant. A minimum-makespan completion oracle still has worst-case constant exactly 11/6 against the optimum; both natural 7/4 statements about it are false; and restricting Step 1 to vertices of the relaxation does not help. The loss therefore cannot be attributed solely to the freedom in Step 3. We also record what structure survives: a reduction confining every overload to two shapes, a seven-machine instance defeating the natural two-phase repair, and a strict 7/4 bound on the generalized three-path family.