text: RM 
  ω (⃗x) = (RM  ω (⃗xq)) 
  
  where note that each invocation could result in multiple underlying LLM invocations. In the binary router case, however, 
  each invocation results in a single LLM invocation. 
  
  An inference flow policy dictates the control plane designer’s intention regarding use of the underlying models. For 
  example, an application may want to ensure that only a small fraction of queries go to the expensive model Ms. We can 
  define this as a predicate over a sequence of transcripts. In our binary router example, the policy can be more simply 
  defined as a predicate P over (input, model) pairs (⃗x1, i1), . . . , (⃗xq, iq) since this fully defines the sequence of transcripts. 
  For example, a policy might specify that the strong model is used in at most an ϵ fraction of inferences: 
     
  q 
  (cid:88) 
  
  I(ij) 
  q 
  P((⃗x1, i1), . . . , (⃗xq, iq)) =  ≤ ϵ 
     
  j=1 