  We extend Gor'kov theory to address superconducting pairing at high magnetic
fields and general temperatures with arbitrary attractive interaction strength.
This analysis begins with a new interpretation of the high-field Gor'kov gap
equation which we associate with an instability in a generalized
particle-particle ladder series. Importantly, this interpretation of the
non-linear gap equation enables a treatment of pairing which is distinct from
condensation. We also show how to consolidate two distinct fermionic pairing
schemes in real and momentum space, both corresponding to an Abrikosov lattice.
Numerical results for the fermionic local density of states demonstrate that
gapless structure in a field is robust and presumably relevant to quantum
oscillation experiments. We find that despite their differences, both pairing
schemes contain very similar physics. Our formalism is designed to explore a
variety of magnetic field effects in the so-called pseudogap phase and
throughout the BCS-BEC crossover.
