bibkey: cao2026sizeindependent authors: Shen Cao; Xingjian Zhang; Fei Shi; Qi Zhao year: 2026 title: “Size-Independent Robustness in Multipartite Bell Self-Testing” doi: 10.48550/arXiv.2608.30851 url: https://arxiv.org/abs/2608.30851v1 claim: “Non-negativity of the explicit five-term lambda expression on [0, kappa]^n for h = 1, 2 and n >= 6.” strata_touched:

• D5/S3/QuantumBounds/MabkSelfTestingPositivity license: citation-only triage: anchor

Size-Independent Robustness in Multipartite Bell Self-Testing

The Supplemental Material, “Efficient Verification of Optimal Lower Bound”, PDF p. 39, states:

For $n\le 5$ this bound is already established analytically [20, 23]; for $n\ge 6$ the two remaining cases ($h=1,2$) are an open conjecture supported by the numerical evidence above.

The TeX source is content_aps/sm_numeric.tex, lines 14–23 (the five-term expression) and line 160 (the quoted sentence, with citation keys Kaniewski_2016,Jha_Singh_Pan_2026). The expression is evaluated for $\kappa =1-1/\sqrt{2}$, with the two complementary index blocks of cardinalities $h$ and $d$. The source assumes $h\le d$ by symmetry; $n\ge 6$ and $h\in \{1,2\}$ imply this condition.

The Lean definition lambdaA preserves all five terms, including the full product multiplied by $((1+\sqrt{2})/\sqrt{2}{)}^n$ and the two square-root cross terms. Indices Fin n correspond to the source’s indices $1,\dots ,n$ by adding one. The block T represents ${\mathcal{T}}_{\vec{a}}$ and its complement represents ${\mathcal{P}}_{\vec{a}}$.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2608.30851
• URL: https://arxiv.org/abs/2608.30851v1
• Source: https://arxiv.org/e-print/2608.30851v1
• PDF: https://arxiv.org/pdf/2608.30851v1, p. 39.