Worst-case analysis, the setting of the $P \stackrel{?}{=} NP$ question, measures the cost of a problem as a function of input size alone. Yet problems that are exponential in general often become polynomial, even linear, once a checkable property of the input is known: Partition, for instance, is decidable in linear time on inputs whose elements are all equal. We develop a framework that makes such knowledge an explicit parameter $K$ of complexity statements. Costs are measured for fixed algorithms; we distinguish the promise cost from the guarded cost that also pays for checking $K$, and introduce the residual uncertainty $Ψ(n,K)$, the number of input cells that, together with $K$, determine the input. An Uncertainty Reduction Theorem decomposes the guarded cost into checking $K$, locating and reading the residual information, and solving an explicit reduced problem. The classes $P_\upsilon[K]$ and $NP_\upsilon[K]$ are restrictions to polynomial-time promises, made informative by the coverage of $K$. We show that small $Ψ$ forces sparsity but not tractability, characterize FPT as uniform membership in graded families $P_\upsilon[K_k]$, prove a free-variable threshold for SAT, and show that under the factoring assumption no checkable property of non-negligible coverage makes factoring easy. We relate the framework to promise problems, kernelization, average- and generic-case complexity and beyond-worst-case analysis, illustrate it on Partition, Graph Isomorphism, SAT, Vertex Cover and TSP, and apply it to guarded algorithm dispatch with latency contracts and optimal guard ordering, to solver routing in agentic AI, and to the role of promises in quantum speedups.

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