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The Grothendieck constant represents one of the most profound connections between pure mathematics and quantum physics, relating the optimization power of real Hilbert spaces to that of binary variables. In the context of Bell inequality violations, the CHSH-restricted Grothendieck constant KCHSHG= √2 provides the fundamental ratio between quantum and classical correlation bounds, yet direct experimental verification of this constant on quantum hardware has remained elusive despite its central importance to quantum information theory. Here we report the first systematic, multi-platform measurement of KCHSHG using both trapped-ion (IonQ Forte Enterprise) and superconducting (IBM Torino) quantum processors. Through a comprehensive visibility sweep protocol spanning visibility parameters v ∈ [0.1,1.0], we extract KCHSHG= 1.408 ±0.006 on IonQ and KCHSHG= 1.363 ±0.012 on IBM, achieving deviations of only 0.44% and 3.6% from the theoretical value √2 ≈1.4142, respectively. The IonQ measurement represents, to our knowledge, the most precise systematic measurement of the Grothendieck bound, enabled by the superior coherence properties of trapped-ion systems. Simultaneously, we establish an empirical relationship between the de Finetti error ε and the CHSH parameter S, finding a remarkably precise linear scaling ε = 0.498 ×S −0.468 (for S > Scrit) with R2 = 0.9999, which provides an operational connection between entanglement quantification and Bell inequality violation magnitude that has not been previously demonstrated experimentally. This relationship enables direct estimation of entanglement from Bell measurements without requiring full state tomography. Our cross-platform validation demonstrates that while both architectures achieve similar maximum CHSH values (∼96% of Tsirelson bound), trapped-ion systems exhibit significantly superior precision for fundamental constant measurements, suggesting their preferential use for metrological applications in quantum foundations. These findings have immediate implications for quantum cryptography, device-independent protocols, and the certification of quantum advantage in near-term quantum devices.