  We find exact analytic expressions for the energies and wavefunctions of the
charged and neutral excitations above the exact ground states (at rational
filling per unit cell) of projected Coulomb Hamiltonians in twisted bilayer
graphene. Our exact expressions are valid for any form of the Coulomb
interaction and any form of $AA$ and $AB/BA$ tunneling. The single charge
excitation energy is a convolution of the Coulomb potential with a quantum
geometric tensor of the TBG bands. The neutral excitations are (high-symmetry
group) magnons, and their dispersion is analytically calculated in terms of the
form factors of the active bands in TBG. The two-charge excitation energy and
wavefunctions are also obtained, and a sufficient condition on the graphene
eigenstates for obtaining a Cooper-pair from Coulomb interactions is obtained.
For the actual TBG bands at the first magic angle, we can analytically show
that the Cooper pair binding energy is zero in all such projected Coulomb
models, implying that either phonons and/or non-zero kinetic energy are needed
for superconductivity. Since the [Phys. Rev. Lett. 122, 246401] showed that the
kinetic energy bounds on the superexchange energy are less $10^{-3}$ in Coulomb
units, the phonon mechanism becomes then very likely. If nonetheless the
superconductivity is due to kinetic terms which render the bands non-flat, one
prediction of our theory is that the highest $T_c$ would not occur at the
highest DOS.
