B.5:4.3 - Make the result understandable for its use

Explain what was obtained, why the decisive step works, and where the result can be used. The needed depth depends on whether the receiver will apply, criticize, extend or teach the argument.

Recover the argument needed for that use. Begin with the conclusion or proposed change the receiver needs to understand. B.5.RA develops both the main reason for the result and the local transitions needed to use it.

1. Read the claim with its domain and conditions. For a mathematical statement, recover the meanings of its objects and quantifiers.
2. Work backward from that conclusion through the intermediate claims or constructions it uses. At each needed transition, identify the premises, the operation or inference, and what it establishes. Follow a shared premise wherever the conclusion depends on it; keep jointly needed premises together.
3. Reconstruct a transition the receiver cannot follow from the source’s definitions, rules or worked cases. Obtain the missing explanation or specialist contribution when those do not suffice. A conditional argument can be useful while one premise remains to be established; state that premise and the consequence its failure would have for this use.
4. If a premise or requested result changes, use B.5.RR to follow the affected reasoning and derive what still follows. Preserve joint prerequisites and sufficient alternatives. Stop at a sufficient argument for the receiving use or a named unsupported transition. An unchanged, adequately supported part can be reused; a full reproof is needed only when the question calls for it.

For instance, a team expects to recover a drawing because a backup exists. Recovering that conclusion requires the needed data in a readable format and an available way to decode it. If the backup is encrypted and its key is unavailable, the next question concerns access to that key or another copy of the drawing. Repeating the fact that a backup exists leaves that prerequisite unresolved.

When beginning with an unfamiliar theory, reconstruct its concept of use for one working question. State what the practitioner wants to explain, predict, construct or decide; which objects and relations the theory lets them describe; what information and operations the application needs; and how its result answers that question. Use a source application when it answers the question. Otherwise propose a small application from the theory’s stated objects and operations, work it through and test the correspondence. Keep that constructed trial distinguishable from an application already supported by the source. The first result is a usable explanation of that application, or the particular missing premise or operation that prevents it. Use the intended application to choose what to learn next. Study the construction deeply enough to perform or change the contribution the work requires. B.5.TU develops this application-construction method, including recovery of the needed operations and use of a correct result that answers only part of the receiving question.

To compare theories or ways of using them, ask the same question of each account. Preserve each source’s meanings while comparing what is given, what operations are allowed and what answer follows. If the results differ, identify whether the difference comes from the question, assumptions, mathematical structure, inference or proposed physical mechanism. If one account answers a different question, state that complementary use. Choose or change an account for the distinction the receiving work needs. Use C.29 for a proposed mathematical-lens transfer or local candidate choice; domain prediction and intervention claims still need their applicable Methods and evidence. B.5.TC develops the comparison: reconstruct both applications, work a common case, distinguish theoretical differences from approximation or execution, and return the use or inquiry that follows.

When the correspondence between a formal account and its intended use is the difficulty, reconstruct one small instance. Name the variables and their domains, the given inputs, the assumptions and permitted operations, and the statement the calculation or proof establishes. Relate the decisive quantities and conditions to the intended use. A source’s worked case can supply this reconstruction; recover the omitted step if the receiver cannot yet carry it out.

Challenge the correspondence with a case that could change the answer. Look for a formally admissible answer that the intended use excludes, or two situations identified by the representation that require different answers. If this exposes a mismatch, repair the domain, constraint, quantity or correspondence, or use the existing result for a weaker question that it does answer. A relaxation can remain useful as a bound even when its optimizer cannot be enacted. The worked case below makes both uses explicit.

For a physical application, connect the mathematical objects and quantities to the phenomenon, measurement and operating conditions. Ask which approximation, omitted interaction, scale or uncertainty could change the conclusion. A rigorous derivation from a model and evidence that the model applies answer different questions.

Use the explanation to test the formulation: does the result answer the intended question, or only the conveniently formalized one? If a distinction was lost, determine whether the use can tolerate that loss, whether a bound suffices, or whether another representation is needed.