QFT-VER: An Effective Unified Theory of the Vacuum with the Complex Order Parameter Ψ=√iϑ

AbstractWe propose a QFT-motivated effective framework for the vacuum sector of Vacuum–Energonic Relativity(VER), in which the causal limit c(x) and the axion-like phase of the vacuum are unified as the modulusand phase of a single complex order parameter,Ψ=√φeiϑ.In VER, the physical frame of matter is defined by the unique operational metric ˜gµν, while the energonicityfield φ > 0 determines the local causal limit c(x) = φ(x) and enters the conformal relation ˜gµν = φ−1gµν(realization I), where gµν is the variational metric of the vacuum–gravitational sector. Universality of matteris implemented through minimal coupling Sm[ψm,˜g] and the absence of direct couplings to {φ,ϑ} at fixed˜g, which removes the standard units-based criticism of VSL approaches and suppresses any direct fifth-forcechannel from ϑ.Weformulate a covariant vacuum EFT action, derive the universal source of φ from matter through the trace˜T, and fix a minimal working model (Model-0), in which the canonical kinetic term of Ψ rigidly determinesZ(φ) = 1/(2φ) and K(φ) = 2φ, while regime 3 couples ϑ to φ through the potentialV(φ,ϑ) = U(φ)+Λ4(φ) 1−cosϑ .Wethenshowhowthefunction Λ(φ) naturally arises in QFT from hidden confinement and RG transmutationthrough a φ-dependent gauge kinetic function, thereby turning VER–I–ϑ into a compact unified theory of thevacuum in the EFT sense. Finally, we analyze the radiative structure of the model, the weak-field Yukawachannel, dimensionless anti-units invariants, and formulate a map of viability gates without developing fullcosmology.