Coherence-Based Scalar-Tensor Extension of General Relativity: Variational Formulation, Observational Bounds, and Predictions for High-Eccentricity Orbital Systems

We present a scalar-tensor extension of General Relativity (GR) in which a covariant coherence field Φ is non-minimally coupled to spacetime curvature through a variational action of the form S = R d⁴x √ −g [(1+λΦ)R− ω/2∇μΦ∇μΦ−V (Φ)]/(16πG)+S_m. Variation with respect to the metric yields modified Einstein equations Gμν + Cμν(Φ) = (8πG/c⁴) Tμν, where the coherence tensor C_μν encodes gradients of the scalar field and vanishes identically when Φ → 0, recovering GR exactly. We derive the effective correction to periapsis precession in the weak-field regime and show that it is governed by a single dimensionless combination Ξ = e²(1 − e²)⁻¹ · rg/a, where e is the orbital eccentricity, a the semi-major axis, and r_g = 2GM/c² the gravitational radius. The effective coupling λ_eff is bounded by precision pulsar timing to λ_eff < 1.95, which renders Solar System corrections undetectable at present but predicts corrections of order 10⁻³ for the S2 star orbiting Sagittarius A* — within reach of next-generation interferometric astrometry (GRAVITY+, ELT). The theory constitutes a phenomenological effective framework with a single effective parameter λ_eff , constrained by internal consistency and binary pulsar observations. We outline falsifiable predictions and identify the regimes where screening mechanisms may permit larger deviations, motivating future work on galactic-scale applications.