C.29.2:4.5 - Resolve accuracy and computational limits that affect the use

A well-defined mathematical object need not have the finite representation or uniform procedure being assumed. Ask what information the input representation actually provides and which operations are effective on it. Replacing “all real numbers” by finite strings, an evaluation oracle or a family of increasingly accurate approximations changes the computational problem. If a computability or impossibility claim matters, obtain the applicable subject argument rather than inferring it from a failed attempt.

For a finite search domain with an effective test, explicit enumeration can provide a terminating baseline. It may be too expensive, but it separates an obtainable procedure from an open construction. For an unbounded search, failure to find an answer in the allotted time leaves a different result; it does not establish nonexistence.

For numerical work, connect the stopping test to error in the requested output. Locate relevant errors in input representation, algorithmic approximation, arithmetic and output conversion. Allocate tolerance among them only under a justified rule for combining their effects. Physical-model and measurement uncertainty remain separate inputs from the relevant modeling and C.16 methods; numerical convergence does not settle them.

If finite precision can reverse a decisive comparison, increase precision, use a justified enclosure, reformulate the test or return that comparison as unresolved. If an iteration no longer changes its stored state, a limit of the ideal iteration does not show that the implementation will reach the requested tolerance. Return to the representation or stopping rule. Narrow the claim only when the narrower answer remains useful and the change is explicit.