Future-Sufficient Control Quotients Robust Closure, Operational Reduction, and Physical Realization

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A task-relative theory of what may be forgotten, what may be ignored in practice, and what need not be physically maintained Reduced representations can be sufficient in at least three inequivalent senses: they may close under a declared family of future continuations, support near-optimal decisions for a specified objective, or admit economical physical realization. We formalize these as structural, operational, and physical future sufficiency. For finite-horizon controlled Markov systems, exact quotient conditions yield Bellman factorization. In finite-dimensional linear systems, backward propagation of task observables gives the minimal robust future-relevance family, with a continuous-time moving-null criterion characterizing exact closure. For finite-horizon linear-quadratic regulation, we define a strong closure residual, construct a nearby exactly closed surrogate problem, and obtain a local quadratic policy-regret certificate. A controlled perturbation experiment reproduces the predicted first-order gain deviation and second-order regret scaling. Exact counterexamples and a diffusion-control comparison show that strong closure is nevertheless not necessary for near-optimal control: performance-oriented reductions can tolerate substantial structural nonclosure and attain much smaller operational order. We then distinguish reduced control dimension from physical realization burden. Effective-support and mobility bounds show that low operational rank need not imply localized or inexpensive implementation, while a Clifford-circuit construction exhibits rank-one observable relevance with extensive Pauli support. The resulting framework treats closure as a robustness guarantee rather than a universal compression optimum and identifies the additional assumptions required to convert informational reduction into physical resource advantage. Keywords: future sufficiency; controlled quotients; model reduction; optimal control; LQR; state aggregation; bisimulation; physical realization; quantum control; coheroputation Scope and claim discipline This paper does not claim that task-oriented model reduction, state aggregation, bisimulation, balanced truncation, reduced Riccati control, observable backpropagation, or a posteriori reduced-control certification are new. Those are mature areas with substantial prior art [3–13]. The contribution is narrower: the paper places robust structural closure, task-performance sufficiency, and hardware-relative physical sufficiency in one explicit hierarchy; develops a particular strong-closure residual and exactly closed surrogate for finite-horizon LQR; and proves no-go separations showing why reduced informational dimension alone cannot be promoted to a physical resource claim.