A geometric resolution limit from vacuum entanglement: Topological Structure and Particle-Wave Asymmetry

Abstract

We establish that the entanglement entropy of the electromagnetic vacuum, when regulated by weak spacetime curvature, imposes a fundamental lower bound on the radial resolution available to quantum excitations. Starting from the area law of vacuum entanglement, we demonstrate how linearized gravity introduces a natural ultraviolet regulator via the perturbative Green's function. An exact geometric projection from tangential to radial resolution yields a minimal radial scale $Δr_{\min} \propto r_s^2/R$, where $r_s$ is the Schwarzschild radius of the enclosing body and $R$ is the boundary radius. We analyze how particle propagation regimes depend on the relation between the Compton wavelength $λ_C$ and $Δr_{\min}$, showing that consistency $λ_C \gtrsim Δr_{\min}$ defines a global, environment-dependent mass scale $m_{\text{geo}} \equiv \hbar R/(c r_s^2)$. Measured particle masses satisfy $m = αm_{\text{geo}}$, where $α$ is a dimensionless factor encoding the vacuum's informational structure. In this extended version, we show that the vacuum resolution limit affects massive and massless excitations asymmetrically. For massive particles with $λ_C \ll Δr_{\min}$, the vacuum appears smooth and transparent, yielding classical geodesic motion. For photons with $λ\ll Δr_{\min}$, the vacuum cannot sustain phase coherence, leading to decoherence or dispersion. We argue that this asymmetry follows naturally if $λ_C$ is interpreted not as a wave scale but as the core size of a topologically stable excitation (knot, vortex, or soliton) in the quantum vacuum field. This framework reframes mass as a probe of vacuum-imposed resolution limits set by global geometry and entanglement, offering a structural perspective on the hierarchy problem and yielding distinct, falsifiable predictions for high-energy propagation.