AI 中文总结
研究双倾斜CHSH泛函在临界线处NPA层次精确性的相变，通过量化对称切片上的机制，在超临界侧证明层次精确性，亚临界侧验证前四个层次，确定极限障碍为Motzkin多项式，借助定理给出相位边界几何解读并记录勘误。
AI 中文摘要
吉根纳等人证明了双倾斜CHSH泛函\(B_{\alpha\beta}=\alpha\langle A_0\rangle+\beta\langle B_0\rangle+\mathrm{CHSH}\)的精确量子最大值，并观察到达到该值所需的NPA层次朝着临界线\(\alpha+\beta=2\)无明显界限地增长。本文在对称切片\(s=2-\alpha-\beta\)上量化了该机制：量子值以三次方形式离开局部界限；每个NPA层次以二次方形式超调；所需精确层次的发散等同于单序列\((a_k)\)的正性。在超临界侧，通过三个明确的有理证书实现仿射恒等式，证明了层次的精确性。在亚临界侧，通过精确算术验证了前四个层次。确定了精确机制：缩放到临界角时，极限障碍是Motzkin多项式。通过Nie的有限收敛定理和Marshall的边界Hessian条件，相位边界有精确的几何解读。还记录了吉根纳等人发表的多项式系统中的三个已验证勘误。
英文摘要
Gigena et al. [npj Quantum Inf. 11, 82 (2025)] proved the exact quantum maximum of the doubly-tilted CHSH functional $B_{αβ}=α\langle A_0\rangle+β\langle B_0\rangle+\mathrm{CHSH}$ and observed that the NPA level required to reach it grows without evident bound toward the critical line $α+β=2$. We quantify the mechanism on the symmetric slice $s=2-α-β$: (i) the quantum value leaves the local bound cubically, $c_Q=4-s+s^3/6-s^4/36+O(s^5)$; (ii) each NPA level overshoots quadratically, $c_k(s)=4-s+a_k s^2+O(s^3)$, with the almost-quantum coefficient computed exactly, $a_{1+AB}=3/64$; (iii) the divergence of the required exact level is equivalent to positivity of the single sequence $(a_k)$ - proven for every $k$ in the companion paper. We prove the supercritical side completely: for all $α,β\ge 1$ and every level the hierarchy is exact, via three explicit rational certificates realizing an affine identity. The hierarchy's exactness thus undergoes a phase transition at the critical line. On the subcritical side we certify the first four levels in exact arithmetic (rational pseudo-moments beating $c_Q$, confirmed by Sturm's theorem). We identify the exact mechanism: rescaled to the critical corner, the limiting obstruction is the Motzkin polynomial, the classical nonnegative-but-not-sum-of-squares form, so the finite-level failure sits in the restricted-certificate regime. The phase boundary has a precise geometric reading via Nie's finite-convergence theorem and Marshall's boundary Hessian condition: a self-tested optimum is finitely NPA-certifiable whenever its boundary Hessian is nondegenerate (contact order two), which holds for the single tilt and fails exactly at the doubly-tilted cubic touch. Three verified errata in the published polynomial system of Gigena et al. are documented.
Comments11 pages, Companion to "No finite level of the NPA hierarchy is exact for the doubly-tilted CHSH functional near the critical tilt", submitted simultaneously. Certificates and verification code at https://github.com/tohafrit/npa-nonexactness
