text: Inference flow. Given a set of LLMs M, a control plane Rω, and an input x, an LLM inference flow is the sequence of 
  LLM invocations Mij (zj) for 1 ≤ j ≤ m and ij ∈ {w, s} made when executing RM 
  ω (x). Here m is the total number of 
  LLM invocations, and z1, . . . , zm are the queries made to the underlying LLMs. Should R be randomized, the sequence 
  and its length are random variables. An inference flow can be written as a transcript 
  
  T = (i1, z1), (i2, z2), . . . , (im, zm) 
  of pairs of model indexes ij ∈ {w, s} and model inputs zj. Note that for simplicity we ignore the potential for paral- 
  lelization, assuming execution proceeds serially. For binary routers, we have m = 1 and T ∈ {(w, x), (s, x)}. We write 
  submitting a sequence of inferences ⃗x = ⃗x1, . . . , ⃗xq to a control plane as 
  ω (⃗x1), . . . , RM 