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We show that the Tsirelson bound — the maximal violation of the CHSH inequality compatible with quantum theory — emerges as the unique variational optimum of a persistence functional for agents that rely on nonlocal correlations to solve random-access code (RAC) tasks of unbounded difficulty. Using Information Causality (IC) as a hard viability constraint, we derive a closed-form persistence functional W(ρ) = G(ρ) · S∞(ρ), where G(ρ) is the per-task information benefit and S∞(ρ) is the long-horizon survival probability. The benefit function G is derived entirely from Shannon entropy and the RAC protocol structure, with zero free parameters. Shannon entropy is shown to be the unique measure compatible with the concatenation structure, grounded in the Khinchin–Faddeev theorem and independently confirmed by the Oughton–Timpson result that Rényi-based alternatives yield incorrect bounds. For all task distributions with unbounded support, S∞(ρ) exhibits a sharp phase transition at the Tsirelson point, making it the unique persistence-optimal operating point. The result is universal: it is independent of the specific task distribution, the time horizon, and any relative channel weights. We further establish an exact Benefit–Metric Correspondence linking the curvature of the benefit function to the classical Fisher information of the effective binary symmetric channel induced by the CHSH parameter. This identity provides a geometric reinterpretation of the (2η²)ⁿ scaling identified by Carmi and Moskovich: Information Causality is equivalent to stability of the Fisher–Rao metric pullback under protocol concatenation, with the Tsirelson bound as the critical eigenvalue. The paper includes a global quadratic bound on the IC functional valid for all finite concatenation depths, a Lagrangian equivalence formulation with KKT conditions confirming the active-constraint structure, three independent closure arguments for unbounded task support (algorithmic, maximum-entropy, thermodynamic), integration of communication complexity as a strictly weaker viability channel, and a conditional extension to N-partite Mermin inequalities with explicit acknowledgment of the Gallego–Wolf–Acín impossibility result. All propositions and theorems are verified numerically. Supplementary data, figures, and a reproducibility notebook are included. This paper is part of the Variational Principle of Persistence (VPP) research program at Paradox Systems R&D. The general VPP framework — including foundations, computational validation across biological, physical, and topological systems, and the Systemic Reduction Paradox — is established in the companion paper (doi:10.5281/zenodo.18141677). The present work applies VPP to quantum foundations, deriving the Tsirelson bound as a persistence optimum under Information Causality. Subsequent work addresses sequential measurement admissibility, renormalization group structure, and operational applications. Related preprints will be deposited in Zenodo as the program advances.