Microscopic Motivation of Postulate P4 in QFT-VER: The Vacuum Origin of the Test-Particle Invariant µ = mc

Abstract
We provide a microscopic motivation for Postulate P4 of Vacuum–Energonic Relativity (VER),
according to which a test body carries the invariant
µ ≡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 parameter
Ψ=√φeiϑ,
—the natural invariant of a localized gapped excitation is neither the inertial mass m nor the local
rest energy mc2, but the geometric gap scale
µ =¯hκ,
where κ is the inverse correlation-length / rest-momentum scale of the excitation. In local physical
coordinates this yields the dispersion relation
ω2 =c2(x)(k2 +κ2),
so that
E0(x) = µc(x),
E2 =c2(x)p2 +µ2c2(x),
m(x) = µ
c(x) .
Thus the vacuum preserves the geometric gap scale of the excitation, while the local causal factor
c(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 worldline
action
∫
S(0)
wl =−µ
d˜s,
1
so that P4 is naturally interpreted as the leading reparametrization-invariant statement of the test
body EFT. By contrast, imposing m = const or mc2 = const requires explicit φ-dependence of
the microscopic gap at fixed physical-frame EFT, i.e. a non-minimal deformation of the universal
coupling branch. In this sense, P4 is not an additional ad hoc axiom, but the leading low-energy
worldline consequence of a localized gapped excitation in minimal QFT-VER.