Response Structure versus Information Structure in Bounded Control: Finite-Resolution Branching, Realisability, and Requisite Variety

Abstract What changes when a controller must organise its responses according to distinctions it can actually make? We study that question in a deterministic finite-horizon control model at a declared response resolution. For each valid policy, the realised response set has a minimum metric-covering number. If at most $M$ branch labels may be assigned directly to the hidden cases, the optimal worst-branch cover is elementary: a minimum global response cover can be divided as evenly as possible, giving the exact fixed-policy baseline $Q_M^{\mathrm{abs}}(\pi)=\left\lceil\frac{n_\pi}{M}\right\rceil,\qquad n_\pi:=N_Y(\Psi_\pi(\Omega_0),\varepsilon).$ A real controller cannot in general label cases by hidden identity. At a declared branch-commitment time, branch labels must factor through information available to the controller. The corresponding information fibres are indivisible routing units. We define the resulting decision-realisable minimax covering problem $Q_M^{\mathrm{dr}}(\pi)$ and show that the information constraint can create strict and arbitrarily large logarithmic excess over the ideal baseline. Refining decision information weakly decreases the fixed-policy residual, and exact endpoint formulas expose the roles of branch budget and fibre complexity. Combinatorially, the fixed-policy problem assigns each member of a family of response subsets whole to one of at most $M$ branches, minimising the largest $\varepsilon$-covering number of the union assigned to any branch. The zero-excess problem has a sharp budget structure. Let $n_\pi$ be the size of a minimum global response cover. Information is free at **every** branch budget if and only if it is free at the single critical budget $M=n_\pi$, equivalently if the information-fibre response images are jointly compatible with one minimum global $\varepsilon$-cover. In contrast, simple fibre-wise coverability characterises the sufficiently-large-budget regime. A five-point line-metric witness shows that local fibre coverability need not imply all-budget freeness and that the excess relative to the ideal baseline need not be monotone in $M$. The computational distinction has two different forms. At a specified branch budget $M$, zero-excess realisability is strongly NP-complete in the separated-discrete finite subclass, with hardness already for pairwise-disjoint fibre images. In that disjoint restriction, branch load is ordinary cardinality and the strong 3-PARTITION reduction is the identical-machine makespan mechanism; with overlapping fibre images, the load is a union/coverage load instead. At the critical budget, the corresponding minimum-cover-compatibility problem has a different optimum-membership mechanism: under an explicit finite distance-matrix encoding, CRITICAL-COVER-COMPATIBILITY is $\Theta_2^p$-complete, with hardness already for $\varepsilon=1$, a $\{0,1,2\}$-valued metric, SOFT, and exactly one non-singleton fibre image of cardinality two. Thus the specified-budget routing hardness and the critical-budget optimum-cover hardness are distinct phenomena. Upstream, uniformly forced response packings provide a finite-resolution requisite-variety obstruction: the best valid policy's global response cover cannot be smaller than the largest response packing that every valid policy must separate. Downstream, joint optimisation over valid policies exposes a trade-off between response-cover economy and alignment with the information structure. A complementary strategy-cell specialisation retains graph and hypergraph views of forced fusion, pairwise co-servability, global spanning burden, and the Fusion Gap. These are used as structural projections and calibrations rather than as independent novelty claims.