Paper5_Quantum Entanglement from the Linearized visible action in the b framework

We investigate whether quantum entanglement can be traced back, in a controlled effectivesense, to the visible action of the α–β–φ framework. Starting from the already establishedvisible action, we linearize the φ and β sectors around a stable visible branch and derive themixed quadratic kernel governed by the visible Hessian μφβ = ∂φ∂βV |0. We then constructthe crossed propagator of the linearized field system and define Gaussian regional subsystemsthrough spatial smearing functions. From the resulting regional covariance matrix, we obtaina nonzero logarithmic negativity whenever the mixed visible Hessian is nonvanishing. We nextextend the analysis beyond the purely Gaussian sector by including the leading quartic mixedinteraction and deriving the first perturbative corrections to both the entanglement entropyand the logarithmic negativity from replica partition functions built directly from the Euclideanaction. The result is established within the linearized, Gaussian and first-order perturbativevisible sector, and does not yet constitute a derivation of fully general quantum entanglementin the complete theory.