The Infraparticle Edge

Abstract

I derive the charged-particle spectral edge from the quantum instrument of soft QED. I use two projections of that instrument. Tracing over unresolved photons gives the reduced hard-sector channel. Pushing the outcomes to total energy gives the inclusive energy distribution. Its Laplace exponent is fixed by the diagonal soft intensity. For ${\rm d} N_h(\omega)=\eta_h{\rm d}\omega/\omega+{\rm d} N_{h,\mathrm{reg}}(\omega)$, I obtain $\rho_{\mathrm{inc}}(s)\sim C\theta(s-m^2)(s-m^2)^{-1+\eta_h}$. I retain the coherence kernel and derive hard-sector dephasing and the spectral edge from two contractions of one soft environment. The diagonal coefficient $\kappa_{aa}$ fixes the endpoint exponent, while $\frac12(\kappa_{aa}+\kappa_{bb}-2\operatorname{Re}\kappa_{ba})$ fixes the dephasing exponent between hard alternatives. I then classify infrared energy marginals, derive the finite-resolution residue $Z(\mu)=(\mu/\Lambda)^{\eta_h}$, prove stability under infrared-integrable perturbations, and separate the bath exponent from a hard threshold exponent. For the one-electron spectral measure, the hard threshold factor is regular. The resulting edge has the local power law of a gapped unparticle spectrum, while its exponent remains a response coefficient of the unresolved photon sector.