Canonical Formulation and Conservation Laws of an AbelianInformation–Gauge EFTCoupled to a Scalar Potential

We formulate a conservative effective field theory (EFT) in which an Abelian information–gaugesector U(1)Λ, carried by a connection Λµ, couples to a coarse-grained local entropy/nonequilibriumpotential σ(x) and to a conserved gauge-source current Jµinfo. Here σ(x) is a platform-defined coarsegrained state variable; mappings like σ = ln(T/T0) or ln(n/n0) are optional operational proxies forspecific protocols only. Nonlocal entanglement entropy is used only as semiclassical motivation;locality, power counting, and symmetry control are implemented through gauge-invariant operatorsbuilt from the curvature Fµν = ∇µΛν −∇νΛµ and the covariant gradients ∇µσ.From a covariant action principle we obtain the IR/EFT metric field equation Gµν + Λccgµν =8πGc4T(m)µν +T(Λ)µν +T(σ)µν , where T(Λ)µν is the manifestly gauge-invariant Maxwell stress tensor andthe interaction gΛJµinfoΛµ is kept in the matter/entropy sector. Variation with respect to Λµyields the Maxwell-like equation ∇µFµν = gΛJνinfo, and its divergence implies the Noether identity∇µJµinfo = 0 in the minimal anomaly-free EFT. A controlled nonrelativistic reduction reproducesSchrödinger dynamics in which Λµ acts as a phase connection; accordingly, Wilson phases and thecurvature Fµν are the primary gauge-invariant observables.Near dynamical criticality we parameterize state-/environment-dependent scaling by γth(ξ) =(ξ/ξ0)∆η, interpreted as an in-medium scaling factor within a specified near-critical/open-systemregime rather than a universal modification of vacuum Lorentz invariance. In the nonrelativisticsector, minimal coupling to the U(1)Λ connection induces a magnetic-type symplectic structure,equivalently controlling the noncommutativity of kinetic momenta through [ˆ Πi, ˆ Πj] = iℏgΛFij.