Exact Local Optimality Does Not Compose: Invariant Simplices and the Complexity of Chronological Stochastic Realization

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 R-complete, while the promise problem of distinguishing zero defect from defect at least inverse-polynomial is-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 R upper bound, complementing a strongly bounded-rational-hardness result.