What is the situation? We have a functioning field theory, the standard model, in which there are over a dozen numbers that are just input parameters. We have a supposed derivation of a few of those parameters, in terms of quantities associated with the icosahedron. How are we to give these derivations causal significance in a more fundamental theory?
The basic idea is as follows. We have an icosahedron associated with each point in space. (The exact nature of the association doesn't matter at this stage.) Then, we have a field whose expectation value has a specific functional dependence on the properties of the icosahedron. Or rather, we have several such fields, each with its special functional relation. Then, we combine those fields so as to give rise to the standard model parameters. And we're done!
For example, consider the problematic relationship: dihedral angle equals sum of three physical parameters (let's overlook the peculiarity of mr nothing's "GUT angle" for now), plus another quantity. If you were "measuring" an icosahedron through, say, a TQFT, there's simply no reason why the TQFT would directly detect the existence of that decomposition.
So instead, we suppose there is a "dihedral angle field", with a VEV equal to the icosahedron's dihedral angle, and a "Weinberg angle field", which gets its VEV from, say, that DFQ angle (we may suppose that it's a slightly non-Euclidean icosahedron). And then we have a "Cabibbo angle field", whose VEV equals "d.a.f. VEV - W.a.f. VEV - 'GUT angle' - pi/3" - with this relationship being enforced, not by any property of the icosahedron, but simply by algebraic fiat. Voila, we now have the Cabibbo angle as an output from our "icosahedral theory of nothing", ready in turn to be an input to the set of relationships which reproduces the standard model.