Let σ be a q-rule, where any coalition of size q, from the society of size n, is decisive. Let w(n,q)=2q-n+1 and let W be a smooth ‘policy space’ of dimension w. Let U〖(W)〗^N be the space of all smooth profiles on W, endowed with the Whitney topology. It is shown that there exists an ‘instability dimension’ w*(σ) with 2≦w*(σ)≦w(n,q) such that:
1. (i) if w≧w*(σ), and W has no boundary, then the core of σ is empty for a dense set of profiles in U(W)N (i.e., almost always),
2. (ii) if w≧w*(σ)+1, and W has a boundary, then the core of σ is empty, almost always,
3. (iii) if w≧w*(σ)+1, then the cycle set is dense in W, almost always,
4. (iv) if w≧w*(σ)+2 then the cycle set is also path connected, almost always.
The method of proof is first of all to show that if a point belongs to the core, then certain generalized symmetry conditions in terms of ‘pivotal’ coalitions of size 2q-n must be satisfied. Secondly, it is shown that these symmetry conditions can almost never be satisfied when either W has empty boundary and is of dimension w(n,q) or when W has non-empty boundary and is of dimension w(n,q)+1
