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

Abstract

The chronological shared realization complexity (CRC) is the smallest normalized stochastic state dimension $C_{\rm seq}$ needed to reproduce a collection of future-response menus using one shared family of controlled transition dynamics. We focus on the rank-tight regime in which both the local realization dimension and the independent-query static carrier width equal $K$, thereby isolating the additional dimensional and computational constraints imposed by chronological consistency. This framework also serves as the classical baseline for state-dimension bounds in sequential quantum processes, where stochastic dynamics generalize to completely positive maps. We establish three results in this regime. First, an explicit payload--delay family exhibits an unbounded multiplicative state blow-up: $C_{\rm loc}=C_{\rm stat}=k$ while $C_{\rm seq}=k(L+1)$, isolating the intrinsic state cost of shared temporal pullbacks. Second, for explicitly listed rational finite menus over a fixed five-letter alphabet with a single Boolean terminal effect, exact shared realizability is $\exists\mathbb{R}$-complete, and the zero-versus-inverse-polynomial defect promise problem is $\mathsf{PromiseNP}$-complete, with local and static optima fixed at $K$. Third, a total five-letter chronology compiler translates bounded-rational Intermediate Simplex instances into a polynomially specified regular geometric family over the same fixed alphabet, preserving exact local and static width $K$. The compiled family yields strong $\mathsf{PromiseNP}$-hardness for shared realization on strongly bounded-rational inputs, together with an $\exists\mathbb{R}$ upper bound certified by an exact polynomial-size finite core. Together, these results show that exact local and static optimality need not compose under chronological sharing, even in the rank-tight regime.