Gamma-ray emission from interstellar gas can probe Galactic proton and helium spectra down to 400 MeV, while solar modulation restricts the use of direct cosmic-ray measurements to energies above 10 GeV. We use the latest gas emissivity measurements by the Fermi LAT to determine the flux of the local interstellar cosmic-ray spectrum. Particular attention is paid to uncertainties in our knowledge of the hadronic gamma-ray production processes.
A further complication is the flux of bremsstrahlung emitted by primary cosmic-ray electrons and lepton secondaries, which is constrained by synchrotron emission at low energies and direct measurements at high energies. We use the results to verify whether the momentum spectrum is a power-law over the energy range accessible by gamma-ray experiments, and whether there is evidence from the gamma-ray spectrum above 20 GeV for the hardening of the proton and helium spectra above 200 GeV reported by PAMELA. Implications for the physics of Galactic cosmic-ray production and propagation are discussed.