  In this paper, we establish a general framework for calculating pionless
matrix elements between $A=3$ bound-states up to next-to-leading-order. This
framework is useful for pionless calculations of electroweak observables, such
as $^3$H,$^3$He magnetic moments and $^3$H $\beta$ decay. Starting from a
Bethe-Salpeter equation, we prove that for a bound-state, the three-nucleon
wave-function normalization can be expressed diagrammatically in a way that is
equivalent to the unit operator between two identical three-nucleon
bound-states. This diagrammatic form of the identity matrix element is the
foundation for constructing an $A=3$ matrix element of a general operator. We
show that this approach can be used to calculate the energy difference between
$^3$H and $^3$He due to the Coulomb interaction, and to calculate the NLO
corrections to the $^3$H and $^3$He scattering amplitudes due to effective
range corrections.
