A Unified Cosmological Analysis of Late-Time Diagnostics and BBN-Safe Early-Time Extensions: Full-Lite Observational Constraints, CMB Acoustic Structure, and the Fundamental rd − rs(z*) Constraint

We present Alirr_eq 2, a stable scalar–tensor effective field theory in which the scalar field is defined as the logarithm of coarse-grained cosmic entanglement entropy, ϕ = ln(SE/S0). The model is constructed within the INAS holographic framework, linking dark energy to horizon thermodynamics through a natural exponential coupling F(ϕ) = eαϕ. A key component of the model is a BBN-safe early-time extension (Scenario B), which introduces a localized Gaussian modification near the drag epoch. This mechanism reduces the sound horizon rd without altering the effective number of relativistic species (ΔNeff = 0), preserving consistency with primordial nucleosynthesis constraints. We present results from two main numerical pipelines. The v8.5p analysis (DESI DR2 BAO + Pantheon+SH0ES, full MCMC sampling) yields rd ≈ 136.55 Mpc and H0 ≈ 73.24 km/s/Mpc, with strong statistical consistency and competitive performance relative to ΛCDM. The extended v9.2 Full-Lite analysis incorporates full Pantheon+ covariance, RSD growth data, and compressed CMB constraints, recovering a corrected growth amplitude σ8 ≈ 0.875–0.889. A central result of this work is the identification of a fundamental acoustic constraint: the same mechanism that reduces rd also suppresses rs(z*) by approximately 8.7%, leading to a mismatch in the acoustic scale ℓA relative to Planck observations. We interpret this as a structural limitation of single-field models rather than a numerical artifact, thereby motivating the transition to a multi-field completion. Alirr_eq 2 therefore serves both as a phenomenologically viable late-time cosmological framework and as a diagnostic bridge toward the two-field Alirr_eq 3.0 model, where this limitation is resolved via a recombination-gated Guardian mechanism.