Level-two exactness interval and transition boundary

Prove an interval of exact level-two certification for the Alice-conditioned tilted-CHSH hierarchy and characterize its boundary using the parameter-dependent optimal-face system, including determining whether the numerically observed unique transition near 1.28427 exists.

Background

The paper establishes exact Alice-conditioned level-two certification at three isolated tilts, namely α=1, 5/4, and 41/32, and proves non-closure on an interval beginning at an exact algebraic coverage boundary α_-≈1.293774. A band between the largest certified exact point and this non-closure interval remains unclassified.

Numerical calculations suggest a unique transition near α≈1.28427 between level-two exactness and strict separation, but the paper does not prove that this transition is unique or that the numerically suggested boundary is exact. The proposed parameter-dependent optimal-face system is intended to characterize this boundary analytically.

References

Prove an interval of level-two exactness and characterize its boundary using the parameter-dependent optimal-face system. A unique transition near $1.28427$ remains a numerical conjecture.

— Unbounded degree overhead for Alice-conditioned quantum Bell certificates  (2609.10162 - Wang, 9 Sep 2026) in Section Literature boundary and open problems, paragraph 'Open problems', item (a)

Determine the full level-three exactness region beyond the interval certified in \cref{sec:interval}.

— Unbounded degree overhead for Alice-conditioned quantum Bell certificates  (2609.10162 - Wang, 9 Sep 2026) in Section Literature boundary and open problems, paragraph 'Open problems', item (b)

Determine whether each fixed $0<\alpha<2$ has a finite exact conditioned certificate. Prove or refute an exact $O((2-\alpha){-1/2})$ upper bound matching the exponent of \cref{eq:degree-lower-growth}, and determine the leading constant.

— Unbounded degree overhead for Alice-conditioned quantum Bell certificates  (2609.10162 - Wang, 9 Sep 2026) in Section Literature boundary and open problems, paragraph 'Open problems', item (d)

A parameter-dependent analytic criterion connecting these observations remains open.

— Unbounded degree overhead for Alice-conditioned quantum Bell certificates  (2609.10162 - Wang, 9 Sep 2026) in Section Qualitative limitations, final paragraph