Cosmic ray energy spectra exhibit power law distributions over many orders of magnitude that are very well described by the predictions of q-generalized statistical mechanics, based on a q-generalized Hagedorn theory for transverse momentum spectra and hard QCD scattering processes. QCD at largest center of mass energies predicts the entropic index to be q =
11. Here we show that the escort duality of the nonextensive thermodynamic formalism predicts an energy split of effective temperature given byMeV, where T H is the Hagedorn temperature. We carefully analyse the measured data of the AMS-02 collaboration and provide evidence that the predicted temperature split is indeed observed, leading to a different energy dependence of the e + and e − spectral indices. We also observe a distinguished energy scale E * ≈ 50 GeV where the e + and e − spectral indices differ the most. Linear combinations of the escort and non-escort q-generalized canonical distributions yield excellent agreement with the measured AMS-02 data in the entire energy range.Statistical mechanics is a universal formalism based on the maximization of the Boltzmann-Gibbs-Shannon entropy subject to suitable constraints. Despite its universal validity and success for short-range equilibrium systems, the applicability of the Boltzmann-Gibbs formalism has severe restrictions: It is not valid for nonequilibrium systems, it is not valid for systems with long-range interactions (such as gravity), and it is not valid for systems with a very small volume and fluctuating temperature (as probed in scattering processes of cosmic ray particles at very high energies). For these types of complex systems it is useful to generalize the formalism to a more general setting, based on the maximization of more general entropy measures which contain the Shannon entropy as a special case. Probably the most popular one of these generalizations is based on q-entropy (or Tsallis entropy), which leads to power law distributions (the so-called q-exponentials), but other generalized entropic approaches are possible as well [1][2][3][4][5][6][7][8] . In high energy physics, a recent success of the q-generalized approach is that excellent fits of measured transverse momentum spectra in high energy scattering experiments have been obtained 9 , based on an extension of Hagedorn's theory 10-12 to a q-generalized version and a generalized thermodynamic theory [13][14][15][16][17][18][19] . This includes recent experiments in the TeV region for pp and pp collisions 9,20-25 but there is also early work on cosmic ray spectra 26,27 and e + e − annihilation 15,28,29 . In this paper we systematically investigate the relevant degrees of freedom of the q-generalized statistical mechanics formalism at highest center of mass energies and develop an effective theory of energy spectra of cosmic rays, which are produced by scattering processes at extremely high energies (e.g. supernovae explosions). Some remarkable predictions come out of the formalism in its full generality. First of all, the para...