Exact Local Optimality Does Not Compose: The Complexity of Chronological Realization

Abstract

We study a controlled, normalized multi-menu positive-realization problem. A normalized realization reproduces declared root--word responses using common row-stochastic transition matrices and a single terminal effect. We compare three realization complexities: independent local realizations, a static shared carrier with independent query effects, and chronological shared realizations. We construct response families for which the local and static optimum widths both equal $k$, isolating the additional cost imposed by chronological consistency. An explicit payload--delay family has exact width $k$ locally and statically but requires exact width $k(L+1)$ under shared chronological coupling, yielding an unbounded multiplicative separation. For explicitly listed rational menus, exact shared realizability is $\exists\mathbb{R}$-complete, while the promise problem of distinguishing zero defect from defect at least inverse-polynomial is $\mathsf{PromiseNP}$-complete. These finite-menu hardness results hold with five control letters and a single Boolean terminal effect. For regular response families generated by a geometric compiler, rank-tight realizability over the full infinite language is equivalent to realizability on a polynomial-size finite core. Consequently, exact rank-tight realizability for the compiled instances has an $\exists\mathbb{R}$ upper bound, complementing a strongly bounded-rational $\mathsf{PromiseNP}$-hardness result.