AI 中文总结
研究双倾斜CHSH泛函NPA层次结构，证明对对称临界族，各有限级别在临界点邻域不精确，通过原始构造及计算机辅助证明，给出具体数值界限，表明超调在非量子部分，回答了有限级别是否足够的问题。
AI 中文摘要
吉格纳、潘瓦尔、斯卡拉、阿劳霍、法卡斯和查图尔维迪[《npj量子信息》11, 82 (2025)]确定了双倾斜CHSH泛函的量子最大值，观察到精确性所需的纳瓦斯克斯 - 皮罗尼奥 - 阿辛级别朝着临界倾斜无明显界限地增长，并询问是否有任何有限级别就足够了。我们给出否定答案。对于对称临界族$B_s=(1 - s/2)(\langle A_0\rangle+\langle B_0\rangle)+\mathrm{CHSH}$，我们证明：对于每个$k\geq2$的NPA级别，存在明确的有理数$g_k>0$和$s^*_k>0$，使得在$(0,s^*_k]$上$c_k(s)\geq4 - s+g_k s^2$；由于量子值仅以立方形式离开局部界限，每个有限级别在一个区间上都会超调：在临界点的任何邻域内，没有有限级别是精确的。无条件地，$a_2>1/39$，$a_3>1/188$，$a_4>1/641$。证明是一种原始构造：在每个级别上有一条精确可行的矩曲线，由级别均匀的结构定律和一个与级别无关的有符号见证构建而成——一个封闭形式的类函数$y^*$，在每个级别上都有$N_k^T\Gamma(y^*)N_k=u_k u_k^T$。该机制强制符号：对于$k\geq3$，没有量子态和没有量子模型的光滑曲线能够实现增益方向，因此超调严格存在于NPA切锥的非量子部分。证明在严格意义上是计算机辅助的：具有已证明度数界限的有限精确整数验证是论证的组成部分；该链条已针对独立实现进行了重新验证，包括见证恒等式的符号逐区域证明和仅根据论文文本编写的洁净室实现。唯一的外部输入是吉格纳等人公布的量子值，已交叉检查到十二位数字。
英文摘要
Gigena, Panwar, Scala, Araujo, Farkas and Chaturvedi [npj Quantum Inf. 11, 82 (2025)] determined the quantum maximum of the doubly-tilted CHSH functionals, observed that the Navascues-Pironio-Acin level needed for exactness grows without evident bound toward the critical tilt, and asked whether any finite level suffices. We answer this in the negative. For the symmetric critical family $B_s=(1-s/2)(\langle A_0\rangle+\langle B_0\rangle)+\mathrm{CHSH}$ we prove: for every NPA level $k\ge 2$ there are an explicit rational $g_k>0$ and an $s^*_k>0$ with $c_k(s)\ge 4-s+g_k s^2$ on $(0,s^*_k]$; since the quantum value leaves the local bound only cubically, every finite level strictly overshoots on an interval: no finite level is exact on any neighbourhood of the critical point. Unconditionally $a_2>1/39$, $a_3>1/188$, $a_4>1/641$. The proof is a primal construction: an exactly feasible moment curve at each level, built from level-uniform structural laws and one level-independent signed witness -- a closed-form class function $y^*$ with $N_k^TΓ(y^*)N_k=u_k u_k^T$ at every level. The mechanism forces the sign: for $k\ge 3$ no quantum state and no smooth curve of quantum models can realize the gain direction, so the overshoot lives strictly in the non-quantum part of the NPA tangent cone. The proof is computer-assisted in the strict sense: finite exact-integer verifications with proven degree bounds are constituent parts of the argument; the chain has been re-verified against independent implementations, including a symbolic per-regime proof of the witness identity and a clean-room implementation written from the paper text alone. The one external input is the published quantum value of Gigena et al., cross-checked to twelve digits.
Comments23 pages. Verification code (12 exact-arithmetic verifiers, one-command reproduction) at https://github.com/tohafrit/npa-nonexactness
