Theorem 11.2

Theorem 11.2 (Opaque-atom staging). Let a strict purely functional sequence operation, parameterized by a base element type $A$, use finite constructor tests and a worst-case bounded number of $\mathsf{car}$, $\mathsf{cons}$, and $\mathsf{cdr}$ operations. It may inspect the representation’s own pairs, buffers, and recursive child structures, but it may not inspect, compare, hash, traverse, or identity-test values of the base type $A$. With new elements replaced by symbolic atoms, its branch and fresh structural allocation graph are determined by old structure alone. The graph is finite, acyclic, and uniformly bounded.

A symbolic element may denote a designated fresh payload record, and a fresh owner may store a fixed tuple of resulting roots. Adding those payload and owner slots to the structural skeleton forms one bounded simultaneous batch. Substitution is sound even when a payload field points back to the owner and creates a cycle.