Proof of the level-one two-regime formula

Prove the numerically identified two-regime algebraic formula for the Alice-conditioned level-one value of the tilted-CHSH functional over the parameter range described in Section 4.1, including its critical point and claimed series behavior.

Background

The paper reports, based on high-precision KKT analysis and algebraic identification, a candidate two-regime formula for the Alice-conditioned level-one value ω₁^os(F_α), with critical point α_c=1−1/√3. The formula predicts a vanishing right derivative at α=0 and a specific expansion near that endpoint.

These results are explicitly described as numerical guidance rather than a formal theorem. A proof would turn the observed algebraic expression into a rigorous characterization of the level-one relaxation.