Here are my thoughts on the general approach for n>4 dimensions, as it doesn't appear to be explicitly spelled out anywhere.
([j, k]-subpolytopes are defined
here.)
Assuming we've enumerated all CRH polytopes in the previous dimension, these are the possible [-1, n-1]-spyts (i.e. faces, using Wendy's terminology) of any CRH polytope in n dimensions.
Of course this also determines the possible [j, n-1]-spyts for all j (the vertex figures and edge figures etc. of the faces), including the [n-4, n-1]-spyts, which can be considered as spheric polygons.
Then any [n-4, n]-spyt can be considered as a spheric polyhedron, with faces from that known set of polygons. So we try to build polyhedra in 3D spheric space. By Cauchy's rigidity theorem, the polyhedron's dihedral angles are determined by its arrangement of faces. These angles are the [n-2, n]-spyts.
So we know the possible [-1, n-1] and [n-4, n]-spyts. Then we need to fit these subpolytopes together, matching the [n-4, n-1]-spyts between the two types, to build the whole polytope. This task could be broken down further: First enumerate the [n-4, n]-spyts, then enumerate the [n-5, n]-spyts using the known [n-5, n-1] and [n-4, n]-spyts, then enumerate the [n-6, n]-spyts using the known [n-6, n-1] and [n-5, n]-spyts, and so on. In other words, at each step (expanding dimensionally downward), consider faces and verfs to determine the whole subpolytope.
The point is that all the flexibility is confined to 3D. The rest of the problem is discrete/combinatorial.
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