SynchronoGeometry: A Temporal-Geometric Framework for Cosmic Acceleration — Complete Audit and PPN Compatibility

We present SynchronoGeometry (SG), a scalar–tensor framework in which a dynamical temporal scalar field T(x^μ) is non-minimally coupled to the Ricci scalar through F(T) = 1 − ξT² and subject to an exponential potential V(T) = V₀ exp(−λT). The framework is intended to account for late-time cosmic acceleration without introducing a separate dark-energy component in the stress–energy tensor. We report a complete audit of the background branch 0000_branch_1 (ξ = 3.0137×10⁻⁴, λ = 0.4152, Ω_m0 = 0.8403). The audit establishes: (i) consistency of the background equations with the Friedmann constraint to machine precision; (ii) algebraic consistency of the perturbation coefficients G_S and F_S with the standard scalar–tensor formalism; (iii) the structural relation F_S − G_S = (3/2)Ω_m F²/θ², which yields a corrected scalar propagation speed c_corr² = 1 across the full branch; (iv) absence of ghost and gradient instabilities (F > 0, K_E > 0, G_S > 0, F_S > 0); and (v) compatibility with the Cassini bound, with γ_PPN − 1 ≈ −2.41×10⁻⁶. We emphasize that these results are conditional on the branch and on the approximate treatment of the local scalar field; a full numerical solution of the static scalar equation in the Solar-System environment remains as future work.