  We apply generalized statistical mechanics developed for complex systems to
theoretically predict energy spectra of particle and anti-particle degrees of
freedom in cosmic ray fluxes, 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 $q=\frac{13}{11}$,
whereas the escort duality of the nonextensive thermodynamic formalism predicts
an energy split of effective temperature given by $\Delta kT =\pm \frac{1}{10}
kT_H \approx \pm 18 $ MeV, where $T_H$ is the Hagedorn temperature. We
carefully analyse the measured primary cosmic ray 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. Moreover, we observe that at larger energies $E$ the
measured $e^+e^-$ flux starts to deviate from our QCD-based statistical
mechanics theory, with a crossover scale of $E^*=(50 \pm 10)$ GeV, which could
be a hint for WIMP decay or other new physics setting in at this mass scale.
Fits using linear combinations of the escort and non-escort $q$-generalized
canonical distributions yield excellent agreement with the measured data in the
entire energy range.
