AI 中文总结
本文研究嵌套环打包问题，证明超递增半径下贪婪算法最优且放置规则无关，并发现黄金反例表明几何阈值小于Tribonacci常数，推翻自然猜想。
AI 中文摘要
我们研究将等宽环（“环”）打包进一个圆盘的问题，其中环可以嵌套在严格更大的环的孔内，这是递归圆打包问题的一个面向选择性的变体。两个自然的目标——数量最大化和接触面积最大化——真正地产生分歧。对于超递增半径（每个半径超过所有更小半径之和），我们证明降序贪婪算法最大化任何正的、递增的、超可加的目标函数。我们的主要结构定理表明更多：放置规则是无关的——在任意形状的容器和所有维度中，在可行容器中的任何选择都会产生字典序最大的可行集合。两个假设都是尖锐的：放置无关性最多对三个环成立，在四个环时失败，并且孪生实例排除了任何作为可观察状态的函数的规则。记$\rho=\max_i(\sum_{j>i}r_j)/r_i$为超递增性的违反程度。加性松弛的普遍阈值恰好是$\rho=1$。在几何模型中，我们证明，在没有相切理想化的情况下，刚性四环族的下确界恰好是Tribonacci常数$T\approx1.83929$。然而$T$不是全局阈值：一个显式的黄金族在$\rho=\varphi+3\varepsilon$（对每个小的$\varepsilon>0$）处打破放置无关性，证明几何阈值$\tau$满足$\tau\le\varphi<T$，并反驳了自然的Tribonacci阈值猜想。匹配的界$\tau\ge\varphi$仍然是猜想性的；我们为配对轮廓和显式重区域之外证明了它。我们还给出了这种分歧的相图，并将难度分为几何层和组合（子集和）层，其中超递增性恰好消除了后者。主要定理带有完整的书面证明；每个计算机辅助的闭合都带有认知标签和验证映射中的脚本。
英文摘要
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.
Commentsv2: 73 pages. Proves the global threshold tau = phi for disks, open in one direction in v1, via a criterion reducing an inventory to its three largest disks. Adds dimension transfer, the threshold for five rings in any dimension, square twins with bound Y ~ 1.6845, and independent hole radii with area guarantee min(1, kappa^-2 - 1). 122 Lean theorems. https://github.com/JaviMaligno/calamares
