5.1 Strong supports
We define a preorder \( \preceq \) on base supports by \( S \preceq T \) if and only if \( \operatorname {\mathsf{im}}S^{\mathcal A}\subseteq \operatorname {\mathsf{im}}T^{\mathcal A}\) and \( \operatorname {\mathsf{im}}S^{\mathcal N}\subseteq \operatorname {\mathsf{im}}T^{\mathcal N}\). For \( \beta \)-supports, we define \( S \preceq T \) if and only if \( S_A \preceq T_A \) for each \( A \).
A \( \beta \)-support \( S \) is strong if:
for every pair of near-litters \( N_1, N_2 \in \operatorname {\mathsf{im}}S_A^{\mathcal N}\), we have \( \operatorname {\mathsf{interf}}(N_1, N_2) \subseteq \operatorname {\mathsf{im}}S_A^{\mathcal A}\); and
for every inflexible path \( I = (\gamma ,\delta ,\varepsilon ,A) \) and \( t : \mathsf{Tang}_\delta \), if there is a near-litter \( N \in \operatorname {\mathsf{im}}S_{{A_\varepsilon }_\bot }^{\mathcal N}\) with \( N^\circ = f_{\delta ,\varepsilon }(t) \), then \( \operatorname {\mathsf{supp}}(t) \preceq S_{A_\delta } \).
If \( S \) is a strong \( \beta \)-support and \( \rho \) is \( \beta \)-allowable, then \( \rho (S) \) is strong.
Interference is stable under application of allowable permutations, and the required supports are also preserved under action of allowable permutations.
For every support \( S \), there is a strong support \( T \succeq S \).
We define a relation \( R \) on pairs \( (A, N) \) where \( A : \beta \rightsquigarrow \bot \) and \( N \) is a near-litter by the following constructor. If \( I = (\gamma ,\delta ,\varepsilon ,A) \) and \( t : \mathsf{Tang}_\delta \), then for any near-litter \( N_1 \) such that \( N_1^\circ = f_{\delta ,\varepsilon }(t) \) and any path \( B : \delta \rightsquigarrow \bot \) and near-litter \( N_2 \in \operatorname {\mathsf{supp}}(t)_B^{\mathcal N}\), we define \( ((A_\delta )_B, N_2) \mathrel {R} ((A_\varepsilon )_\bot , N_1) \). This is well-founded, because if \( (B, N_2) \mathrel {R} (A, N_1) \) then \( \iota (N_2) {\lt} \iota (N_1) \). For any small set \( s \) of such pairs, the transitive closure of \( s \) under \( R \) is small.
Let \( S \) be a support, and let \( s \) be the transitive closure of the set of pairs \( (A, N) \) such that \( N \in \operatorname {\mathsf{im}}S_A^{\mathcal N}\). Generate a support \( T \) from \( S \) and \( s \) using the fact that every small set is the range of some enumeration. This satisfies the second condition of being a strong support.
Now, for any base support \( T \), we define its interference support to be a base support \( U \) consisting of just the atoms in the interference of all near-litters that appear in \( T \). We may extend this definition to structural supports.
Finally, if \( U \) is the interference support of the \( T \) defined above, the support \( T + U \) is strong, and since \( S \succeq T \), we conclude \( S \preceq T + U \).