State-Dependent CHSH Inequality: A Common Pedagogical Omission with Reproducible Verification

We identify and correct a common pedagogical omission in the presentation of the CHSH inequality. The angular configuration {0, π/2, π/4, -π/4}, historically optimal for the singlet state |Ψ⁻⟩, yields S=0 when applied to |Φ⁺⟩ using the standard CHSH form. We demonstrate that each Bell state requires specific angular configurations and CHSH formula variants. Reproducible Python verification code is included. This work serves as an educational resource highlighting the importance of state-specific context in quantum mechanics pedagogy.