Proposition 9.5

Proposition 9.5 (Feedback programs compile). Let a persistent program, at each input letter, start from a fixed number $s$ of roots and perform a fixed finite composition of strict purely functional sequence operations satisfying the hypotheses of Theorem 9.4 on opaque pointer elements. In addition it may create a bounded number of fresh fixed-arity records whose layouts are determined by the same old observation and whose fields target old records, resulting sequence roots, or records in that fresh finite batch. At most one designated fresh owner’s pointer is stored as a base-type element of those sequences. Suppose the complete per-letter computation, including those sequence operations, follows old logical fields only within some fixed dereference radius $r$. Then the program is a bounded simultaneous record transducer, and it has an exact letter-synchronous implementation by a finite scaffolding automaton of degree $s+Aq$ and distance at most $r+1$ for suitable finite $A,q$.