Neutrino-Photon Arrival Timing From TXS 0506+056 and Bounds on Energy-Dependent Dispersion

The blazar TXS 0506+056 produced the first compelling association between a high-energy neutrino and a flaring photon source. Because the neutrino, at an estimated 290 TeV, traversed a cosmological baseline from redshift 0.337, its arrival time relative to the gamma-ray flare bounds any energy-dependent dispersion of propagation. This paper computes those bounds with the standard cosmological time-delay integral, whose value for this source is K = 0.359. A conservative coincidence window of ten days yields a lower bound on the linear dispersion scale of 5.519e+16 GeV, which is 0.00452 of the Planck energy; even a Planck-scale linear effect would have delayed the neutrino by only 3.91e+03 seconds, about an hour. A constant, energy-independent refractive offset of 10^-22 would delay it by 1.801e-05 seconds and is wholly unobservable by timing. The honest summary is that a single neutrino with day-scale flare association gives dispersion bounds far weaker than photon-only bounds; its value lies elsewhere. The study is computed from stated assumptions, not measured.