Probabilistic polynomial time machine interface

A fixed prefix of j bits occurs in $2^{p(n)-j}$ words, hence has mass $2^{-j}$. More generally, prescribing any j distinct bit positions leaves $p(n)-j$ free coordinates and has the same count. Each marginal is one half, and the joint probabilities factor, exactly as in F1. The empty prescription has mass one.

For a fixed supplied word, the simulator and the PTM take the same first table, and agreement through transition j makes them use the same table and configuration at the next transition. Thus they agree up to the halt by induction. A branch halting after j steps corresponds to precisely the words with that consumed prefix, of total mass $2^{-j}$; prefixes for distinct terminal branches are disjoint. Summing their masses proves agreement with fresh fair-bit semantics, even when the stopping time depends on previous bits. Unused suffixes have been counted, not assumed independent of a stopping rule. For p(n)=0 the space is the singleton containing the empty word; in a model with nonhalting start such a bound is impossible on that input, which is consistent with the definition.