The Geometric Origin of CHSH Inequality Violations: A Unified Derivation from the Qubit Manifold to the Tsirelson Bound

Why does the quantum violation of the CHSH inequality reach exactly 2√2 (i.e., about 0.207 in the CH inequality)? Standard quantum mechanics computes this value from Hilbert-space algebra, but never exhibits its geometric origin.

This paper postulates a spherical virtual space (radius R=2, curvature K=1/4) underlying the qubit state space. Taking the singlet correlation law as input from standard quantum mechanics, the Tsirelson bound 2√2 is obtained by pure spherical-geometric optimization: it is the necessary consequence of the optimal four-vector configuration (coplanar, equal angles π/4) on the sphere. An appendix proves that, given spherical geometry, symmetry, and no-signaling, the Born rule and the cosine correlation law are equivalent — closing the logical chain from geometry to 2√2.

The framework is mathematically equivalent to standard quantum mechanics (no new predictions); its contribution is a geometric reinterpretation: the Tsirelson bound as a geometric necessity rather than an algebraic output. This work is a specific application of Information Dynamics to the foundations of quantum mechanics.