KS 型层松弛的认证极限：平方根上限及其失效
The Certification Limits of KS-Type Layer Relaxations: A Square-Root Ceiling and Its Breakdown
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中文总结 AI 辅助
研究 KS 型层松弛的认证极限，证明对于γ > -1/2的情况，认证赤字至多为√N阶，γ = -1/2时有特殊构造，还研究了几何 Lennard - Jones 系统不同成本下的情况，揭示了层松弛认证谱及转变。
中文摘要 AI 辅助
可分离双线性层松弛为每层分配总体成本，为每对层分配非正双线性耦合，当松弛最小值超过参考值时排除轮廓跨度。我们研究库兹涅佐夫 - 萨希尼迪斯型松弛的最佳认证情况。设γ = g(1)为单一层成本。对于γ > -1/2的每个有效松弛、每个有界参考值以及满足两个基本低能量见证条件的每个基础系统，证明认证赤字 N - ρ(N)至多为√N阶，且有仅依赖于参考界和γ的明确界限。在γ = -1/2时构造了具有有界合理参考值和赤字 N - 1的有效松弛。几何 Lennard - Jones 系统表现不同，三原子链在端点恢复平方根上限，足够负的单一层成本允许具有线性赤字的有效松弛。这些情况定义了层松弛的认证谱，揭示了无排除、线性排除和通用平方根上限之间的折返转变。
英文摘要
A separable bilinear layer relaxation assigns a population cost to each layer and a nonpositive bilinear coupling to each pair of layers, then excludes a profile span whenever the relaxed minimum exceeds a reference value. We ask what a Kuznetsov-Sahinidis-type relaxation can certify at best. Let gamma = g(1) be the singleton-layer cost. For every valid relaxation with gamma > -1/2, every reference bounded per particle, and every underlying system satisfying two elementary low-energy witness conditions, we prove that the certified deficit N - rho(N) is at most of order sqrt(N), with an explicit bound depending only on the reference bound and gamma. Thus no relaxation in this class can certify a larger asymptotic deficit, independently of the detailed layer cost, interaction kernel, or reference. The threshold is sharp in the abstract witness class: at gamma = -1/2 we construct a valid relaxation with a bounded sound reference and deficit N - 1. The geometric Lennard-Jones system behaves differently. A three-atom chain restores the square-root ceiling at the endpoint, while sufficiently negative singleton costs allow valid relaxations with linear deficit. For still lower costs, beyond the stability constant, no sound relaxation excludes any span. These regimes define a certification spectrum for layer relaxations and reveal a reentrant transition between no exclusion, linear exclusion, and the universal square-root ceiling.
