Effective Scalar--Tensor Gravity with Compact Field Sector and Horizon Information Origin

We construct an effective scalar--tensor theory of gravity derived from the conformal decomposition of the metric in Euclidean quantum gravity. The conformal degree of freedom emerges as an effective scalar field after gauge fixing and quantum measure corrections. We propose that the scalar sector is constrained by the finite information content of de Sitter causal horizons, naturally leading to a compact field space and a periodic effective potential. In a semiclassical approximation, this compact structure can be represented by an instanton sum over topological sectors, yielding a cosine potential. The resulting theory reproduces General Relativity in the infrared while introducing small stochastic corrections to cosmology and gravitational wave propagation.