  This is the third paper in a series of four in which a renormalisation flow
is introduced which acts directly on the Osterwalder-Schrader data (OS data)
without recourse to a path integral. Here the OS data consist of a Hilbert
space, a cyclic vacuum vector therein and a Hamiltonian annihilating the vacuum
which can be obtained from an OS measure, that is a measure respecting (a
subset of) the OS axioms.
  In the previous paper we successfully tested our proposal for the
two-dimensional massive Klein-Gordon model, that is, we could confirm that our
framework finds the correct fixed point starting from a natural initial naive
discretisation of the finite resolution Hamiltonians, in particular the
underlying Laplacian on a lattice, and a natural coarse graining map that
drives the renormalisation flow. However, several questions remained
unanswered. How generic can the initial discretisation and the coarse graining
map be in order that the fixed point is not changed or is at least not lost, in
other words, how universal is the fixed point structure? Is the fix point in
fact stable, that is, is the fixed point actually a limit of the
renormalisation sequence? We will address these questions in the present paper.
