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

Hamiltonian Renormalization III. Renormalisation Flow of 1+1 dimensional free scalar fields: Properties