AbstractWe provide a microscopic motivation for Postulate P4 of Vacuum–Energonic Relativity (VER),according to which a test body carries the invariantµ ≡m(x)c(x) = const,Spp = −µ∫d˜s.The central question addressed in this paper is why, within the vacuum-EFT logic of QFT-VER,the natural invariant of the test sector is mc, rather than m or mc2.We argue that in the minimal QFT-VER branch—defined by the physical matter frame ˜gµν,universal matter coupling at fixed ˜g, and the vacuum order parameterΨ=√φeiϑ,—the natural invariant of a localized gapped excitation is neither the inertial mass m nor the localrest energy mc2, but the geometric gap scaleµ =¯hκ,where κ is the inverse correlation-length / rest-momentum scale of the excitation. In local physicalcoordinates this yields the dispersion relationω2 =c2(x)(k2 +κ2),so thatE0(x) = µc(x),E2 =c2(x)p2 +µ2c2(x),m(x) = µc(x) .Thus the vacuum preserves the geometric gap scale of the excitation, while the local causal factorc(x) = √φ(x)converts that fixed scale into inertial and energetic observables.We further show that the WKB reduction of the local matter EFT yields the leading worldlineaction∫S(0)wl =−µd˜s,1so that P4 is naturally interpreted as the leading reparametrization-invariant statement of the testbody EFT. By contrast, imposing m = const or mc2 = const requires explicit φ-dependence ofthe microscopic gap at fixed physical-frame EFT, i.e. a non-minimal deformation of the universalcoupling branch. In this sense, P4 is not an additional ad hoc axiom, but the leading low-energyworldline consequence of a localized gapped excitation in minimal QFT-VER.
Ренат Гафаров (Sun,) studied this question.