Phenomenological constraints on entropy-flow modifications to expansion and growth from BOSS DR12

We present phenomenological constraints on entropy-flow–motivated modifications to late-time cosmic expansion and structure growth using BOSS DR12 consensus measurements. The model introduces a single effective parameter, α_L2, representing entropy-gradient couplings active at the structure-formation scale, which modifies both background distances and the linear growth rate in an anti-correlated manner. Fitting {H₀, Ω_m, σ₈, α_L2} to the full 9×9 covariance matrix of BAO and RSD observables at z = 0.38, 0.51, and 0.61 yields a best-fit value α_L2 = 0.34⁺⁰·¹⁶₋₀·¹⁶, corresponding to a Δχ² = 3.01 (1.7σ) improvement over ΛCDM. Information criteria (ΔAIC ≈ 1.0, ΔBIC ≈ 0.8) indicate that the additional parameter is not yet required by the data. Separate BAO and RSD fits reveal a sign conflict consistent with the known S₈ tension recast in this parametrization. We emphasize that the modification functions are phenomenological ansätze rather than derivations from a fundamental action. The framework predicts no gravitational slip (η ≡ Φ/Ψ = 1), providing a clear, falsifiable discriminator testable with upcoming Stage IV surveys. These results establish α_L2 as a useful phenomenological parameter for characterizing possible entropy-flow effects, while showing that statistically significant evidence awaits higher-precision multi-probe measurements.