Variational Persistence under Information Causality Viability selection for nonlocal correlations in concatenated random-access codes

This version substantially revises and extends the March 2026 preprints on persistence under Information Causality. The manuscript reformulates the analysis around the exact information score of the canonical concatenated binary random-access-code protocol and develops an exact positive-series representation that leads to rigorous finite-depth bounds. These results establish a complete subcritical, critical, and supercritical classification of the concatenated Information-Causality dynamics. Correlations below the Tsirelson point exhibit asymptotic loss of accumulated information; correlations above it eventually violate Information Causality at finite depth; and the Tsirelson point is uniquely characterized by finite, nonzero asymptotic informational persistence. The new version also provides controlled finite-depth corrections at criticality, a rigorous treatment of the critical remainder, a mathematical branching generalization, a local Fisher-information interpretation of the quadratic scaling, and a universality result for smooth symmetric information-divergence measures. Numerical stability near the critical regime is treated explicitly. Compared with the March versions, the present manuscript therefore replaces the primarily asymptotic stability argument with a stronger exact finite-depth formulation and substantially clarifies the mathematical status and scope of the persistence interpretation. The scope remains deliberately restricted to the canonical isotropic concatenated Information-Causality/random-access-code setting. No claim is made that the construction independently characterizes the full quantum correlation set.