CMP.8:4.3 - Carry error through the operations actually used

Relate the computed intermediate object to the original target. MATH.20 supplies comparison arguments; MATH.21 supplies approximation and convergence. The present task must make their sufficient stage obtainable.

For example, if an intermediate approximation has distance at most e from its target and the next operation G obeys d(G(u),G(v))≤L*d(u,v) on the relevant inputs, this contribution to final error is at most L*e. Add another error term only when it measures a comparable discrepancy and an argument, such as a triangle inequality, permits that addition. A probability of failure and a numerical error are different quantities.

For an optimization construction, compare the original objective of the recovered candidate with the original optimum. Follow every inequality in its proper direction. Rounding values can bound objective loss while leaving constraints unchanged; rounding constraint coefficients needs a separate admissibility argument.

When several approximations are composed, carry their bounds through the actual downstream operations. Allocate finer resolution where it changes the final bound most usefully. An unbounded sensitivity or a lost discontinuous distinction can require another formulation rather than further use of the same family.