Growing evidence suggests that synchrotron radiation plays a significant role in shaping the spectra of most γ-ray bursts. The relativistic jets producing them likely carry a significant fraction of energy in the form of a Poynting flux.The electromagnetic spectra of γ-ray bursts (GRBs) are usually characterized by a mathematical function invoking an exponentially-connected two-power-law function first proposed in a paper by the Burst And Transient Source Experiment (BATSE) team led by David Band (1957-2009) 1 . In the special session in memory of Band at the 2009 Fermi Symposium, Josh Grindlay, his PhD advisor from Harvard University, challenged GRB theorists to "find the physical meaning" of the "Band function" in 10 years.In the GRB field there was never a lack of models to interpret data. The challenge is rather to "identify" than to "find" the physical meaning of the Band function. Indeed, two leading models proposed long ago could both roughly account for the the general shape of the spectra but both models have one critical flaw regarding the low-energy photon index of the Band function, denoted as α. The measured values of α peak around ∼ −1 with a broad distribution 2, 3 . The first model invokes synchrotron radiation 4, 5 of the electrons accelerated in the energy dissipation regions (internal shocks or magnetic reconnection sites) to account for the observed γ-rays. However, for a typical GRB environment the electrons are in the so-called "fast-cooling" regime that demands 6, 7 α = −1.5. This is too "soft" to account for the data. The second model invokes quasi-thermal emission from a relativistic fireball 8, 9 .The model typically predicts 10, 11 α ∼ +1.5. This becomes too "hard".