  This paper is the fifth in a series devoted to the development of a rigorous
renormalisation group method applicable to lattice field theories containing
boson and/or fermion fields, and comprises the core of the method. In the
renormalisation group method, increasingly large scales are studied in a
progressive manner, with an interaction parametrised by a field polynomial
which evolves with the scale under the renormalisation group map. In our
context, the progressive analysis is performed via a finite-range covariance
decomposition. Perturbative calculations are used to track the flow of the
coupling constants of the evolving polynomial, but on their own perturbative
calculations are insufficient to control error terms and to obtain
mathematically rigorous results. In this paper, we define an additional
non-perturbative coordinate, which together with the flow of coupling constants
defines the complete evolution of the renormalisation group map. We specify
conditions under which the non-perturbative coordinate is contractive under a
single renormalisation group step. Our framework is essentially combinatorial,
but its implementation relies on analytic results developed earlier in the
series of papers. The results of this paper are applied elsewhere to analyse
the critical behaviour of the 4-dimensional continuous-time weakly
self-avoiding walk and of the 4-dimensional $n$-component $|\varphi|^4$ model.
In particular, the existence of a logarithmic correction to mean-field scaling
for the susceptibility can be proved for both models, together with other facts
about critical exponents and critical behaviour.
