Observational Bounds and Standard Siren Tests of the Unified Scalar-Imprint Theory

The unified scalar-imprint framework proposes that the effective gravitational coupling Geff is a dynamical variable governed by a cosmological scalar field Φ with an exponential coupling function F(Φ) = eβGΦ. The gravitational coupling constant βG and the field’s slow-roll velocity jointly determine the drift parameter λ= βG Φ/H, which is suppressed to observationally consistent values by the sector-saturation constraint N = 17 of the Standard Model. We present the complete quantitative observational tests of this model. Consistency with Lunar Laser Ranging (LLR) requires |βG Φ/H|≲3.2 ×10−5. Big Bang Nucleosynthesis imposes |δYp/Yp|≲2.4 ×10−4, well within current observational uncertainty. The exponential coupling additionally predicts a fixed ratio between the drift of the fine-structure constant α and Newton’s constant G, ˙ α/α = (βα/βG) G/G, providing a second independent and simultaneous test. Most significantly, we derive the gravitational-wave smoking-gun: the ratio of gravitational-wave to electromagnetic luminosity distances follows dGW L /dEML = (1 + z)−λ/2, with λ bounded and fixed by the underlying theory. This prediction is distinct from ΛCDM, Brans-Dicke theory, and f(R) gravity, and is directly testable with LIGO/Virgo standard-siren catalogs and the forthcoming LISA mission.