CMP.Preface:3.2 - Control error and computational cost

CMP.8 turns a permitted approximation into an effective computation with an error and stopping account. Mathematical convergence supplies part of the reasoning; the algorithm still needs usable operations and a finite return condition. CMP.9 constructs sampling and estimation, retaining the target law, dependence and stopping conditions. A random output, a sample distribution and an estimate have different uses.

CMP.10 derives a data representation from the required access and update operations. It exposes costs moved into conversion, maintenance or output. CMP.11 proves a lower bound by finding inputs that remain indistinguishable under the allowed observations but require different answers. It constrains all procedures within that model; a changed access operation or tolerated error can reopen the conclusion.

These methods can change the construction in Part A. A prohibitive shared table can motivate scaling, another representation or a weaker answer. A lower bound can redirect the question rather than motivate another attempt at the same impossible guarantee. A randomized construction remains subject to the output conditions the receiving work needs.