Hall (2010) proved that reproducing all singlet state quantum correlations with a local deterministic model requires giving up at least (√u20602 − 1)/3 = 13.81% of measurement independence, using a variational distance measure on the 2-sphere S² of hidden variable directions. The Displacement Spacetime (DST) framework independently predicts that any measurement performed from inside the displacement condensate is coupled to the condensate with strength gu2080² = (3/8)² = 9/64 = 14.06%, where gu2080 is the same geometric factor that determines the fine structure constant.We show that these two numbers share a common geometric origin on S². Specifically, we derive: (1) the Bell-optimal angle φ = π/4 from the identity f_agree(π/4) = 3/4 = 2gu2080 = N_eff/b_DST, where f_agree is the S² agree fraction; (2) the Tsirelson bound as a function of gu2080, E_max = 4·cos(π(1−2gu2080)) = 2√u20602; (3) Hall's measurement independence minimum as MI_min = (2cos(π(1−2gu2080)) − 1)/3; and (4) the specific measurement-dependent density ρ_XY(λ) from hemispheric projection, lune partition, and singlet weighting — reproducing Hall's optimal density (his Eq. 8) exactly.The DST coupling gu2080² = 14.06% exceeds Hall's minimum by 1.85%. This is not a discrepancy but a derived consequence of the two-observer structure of Bell experiments. Under the structural hypothesis that each observer contributes one insertion of gu2080²/L (where L = ln(m_Pl/m_e) ≈ 51.53) to the effective singlet correlation, the corrected minimum (√u20602(1 + 2gu2080²/L) − 1)/3 equals gu2080² to 0.014% — the same accuracy class as the DST-corrected fine structure constant. Measurement independence becomes the fifth observable closed by the DST self-referential correction, and the first with two insertions. The n-insertion structure is sharply selective: n = 1 and n = 3 both err by ~0.9%, while n = 2 achieves 0.014%. A Bell experiment has exactly two observers.The paper's principal claim is the geometric coincidence, not a specific interpretation of Bell violations. The DST framework is unverified; all results are conditional on its validity. What DST adds is a concrete physical mechanism — the displacement condensate provides a shared vacuum that partially correlates source and detector states — which standard quantum mechanics does not offer.