  Loop amplitudes are conveniently expressed in terms of master integrals whose
coefficients carry the process dependent information. Similarly before
integration, the loop integrands may be expressed as a linear combination of
propagator products with universal numerator-tensors. Such a decomposition is
an important input for the numerical unitarity approach, which constructs
integrand coefficients from on-shell tree amplitudes. We present a new method
to organise multi-loop integrands into a direct sum of terms that integrate to
zero (surface terms) and remaining master integrands. This decomposition
facilitates a general, numerical unitarity approach for multi-loop amplitudes
circumventing analytic integral reduction. Vanishing integrals are well known
as integration-by-parts identities. Our construction can be viewed as an
explicit construction of a complete set of integration-by-parts identities
excluding doubled propagators. Interestingly, a class of 'horizontal'
identities is singled out which hold as well for altered propagator powers.
