Formal interface between discrete quantum networks and low-energy effective field theory

This paper constructs a formal interface between microscopic quantum networks and low-energy effective field theory (EFT). Motivated by developments in network-based quantum geometry and causal set theory, we introduce a phenomenological scale ΛN below which an approximately continuous description of spacetime may cease to be adequate. We define a mutual-information-based proxy for locality and emphasize that standard EFT assumptions—such as analyticity, locality, and approximate isotropy—can be fragile in discrete settings and therefore should be stated as explicit conditions rather than assumed implicitly. The objective of this study is to provide a conservative foundation for a phenomenological framework in which an approximately continuous spacetime description arises as a low-energy effective account of quantum dynamics on an underlying discrete network. We stress that ΛN is an organizing parameter whose empirical value must be constrained (or refuted) in concrete implementations. We formulate minimal interface hypotheses: (i) correlation patterns can induce a coarse-grained notion of locality, and (ii) standard lattice gauge-theory ingredients provide a robust representation of gauge redundancy on a discrete substrate. As numerical diagnostics (not proofs), we report second-order consistency tests of centered discretizations (confirming O(a2) scaling for the lattice Laplacian in 1D and 3D, and for the lattice momentum p^2), and a finite-size 3D Poisson test with monopole-inspired boundary conditions showing small interior residuals consistent with double-precision solver tolerances and far-field scaling consistent with 1/r and 1/r2, up to finite-volume and binning effects.