B.5.TC:4 - Solution

Fix the working question → construct each account’s application → align the case and the meaning of its answer → locate the consequential difference → use what the comparison establishes, or pursue the difference that matters.

The sequence is revisable. Reuse an established application or correspondence. Return to the question when the accounts reveal that it combines different needs. Stop when the available result suffices for the intended use.

B.5.TC:4.1 - State the answer the work needs

Say what you want to explain, predict, construct or decide, and what would count as an answer. Specify the conditions that can alter that answer: for example, the initial state and time of a prediction, the allowed routes and quantity to minimize, or which intervention is contemplated.

Keep the account and its proposed use separately visible. Comparing a theory’s possible reach with a rival’s already implemented calculation is a comparison of unequal contributions. Either construct the relevant application of both or state which contribution is still unavailable. An unavailable application can be a reason to use another account now while retaining the first for development.

If the accounts answer different questions, identify their shared question, if any. For the remaining questions, compare their usefulness as complementary contributions. A newly expressible question can itself be a valuable result: explain what inquiry it opens and why that inquiry matters. B.5:4.4 connects this result to problem development.

B.5.TC:4.2 - Recover an application of each account

For each account, recover the objects and relations used, what is supplied or assumed, the operation that obtains the answer, and how that answer is interpreted. Follow the reasoning far enough to reproduce the step on which the comparison turns. B.5.TU, B.5.RC and B.5.RA supply application, construction and argument recovery when needed.

Preserve the source’s distinctions during this work. If two accounts use state differently, explain which information each state retains and what continuation it determines. If a symbol has no counterpart, say what the account does with it before proposing a correspondence. Translate the required objects and operations; a word-to-word substitution alone cannot settle their relation.

An account may need information the present work lacks. Separate a supplied premise, a proposed premise and a derived result in the explanation. Continue conditionally when that is useful: “Under this independence assumption the minimum is 3; if continuations depend on the incoming route, we must retain that dependency.”

A collaborator or tool can supply a calculation or proof. Recover the interpretation and conditions needed to compare its contribution. Learn or request the missing operation when you cannot yet follow the decisive step; a broad study of both theories is unnecessary when a smaller contribution settles the use.

B.5.TC:4.3 - Work the same case and align the answers

Choose a small case that exercises the disputed operation or distinction. Give each account the corresponding inputs, derive its answer, and express the answers in the terms of the working question. Explain the correspondence when the inputs or outputs have different forms.

For a mathematical application, compare the retained distinctions and operations. C.29.1 supplies the preservation argument when a result is to be transferred. For a physical application, keep the phenomenon, operating conditions and meaning of the predicted quantity aligned. For a computation, separate the mathematical formulation from approximation, numerical tolerance and execution.

Begin with an understood case so a transcription or implementation error can be found. Then use a changed case that exercises the claimed difference: a shared resource, a constraint, an intervention or another relevant variation. Derive a general relation when the intended conclusion ranges beyond the cases. A finite example establishes its own result; a counterexample can refute a general assertion whose premises it satisfies.

Agreement of reported numbers is informative only after their meanings and conditions are aligned. Conversely, different intermediate objects can lead to the same answer through an explained correspondence. Record only the derivation or explanation the receiving use needs; the comparison does not require a standard table or separate report.

B.5.TC:4.4 - Locate what causes the difference

Follow each answer back to the first step where the applications cease to agree. Use the following distinctions where they resolve that case.

Several differences can interact. Aligning one premise can remove an apparent conflict; it can also expose a remaining one. Keep the remaining question named. If the answer depends on a missing correspondence or inaccessible operation, return that missing contribution rather than ranking the theories from their summaries.

Additional evidence is useful when it can change the needed conclusion. First use an available derivation, counterexample, observation or established result. If the difference remains consequential, compare the possible gain from a discriminating inquiry with its cost and feasibility under C.11.DUA. A difference that cannot change the present use can remain unresolved. When it does change the use and cannot be settled, retain the conditional answers or obtain a sufficient weaker result.

B.5.TC:4.5 - Choose the use and preserve the reason

Return the comparison in the form the work can use. A short explanation can state the question, the decisive construction, what agrees or differs and the resulting continuation.

Common outcomes are:

• One sufficient account. Use the account that answers the question under acceptable conditions and effort. State the limitation that would make another account necessary.
• Agreement for a stated range. Reuse either account for that result, choosing by effort, available tools or the further construction it makes easier. Preserve the relation that supports the agreement.
• Complementary uses. Give each account its question and contribution. When combining their results, establish the compatibility needed at the connection.
• A consequential unresolved difference. Keep the conditional answers and the premise or observation that would discriminate them. Continue with an unaffected result, a sufficient bound or the worthwhile inquiry selected for that difference.

Accuracy, reach, computational effort, learning effort and the ability to change the construction can matter differently across uses. Compare the values that can change this choice. If their trade-off is unresolved, C.11 supplies the decision method; a single “best theory” score would hide it.

Reopen when the question, premises, interpretation, available operation or relevant evidence changes. B.5.RR helps revise the affected reasoning. Reuse the comparison where its dependencies remain intact.

B.5.TC:4.6 - Separate recognition from assurance

Recognize the need from an unresolved disagreement or a choice between accounts. One worked comparison can already reveal an omitted premise or a useful complementary answer.

Qualify the result for its intended use. A mathematical equivalence needs a derivation over its claimed domain. A physical prediction needs the corresponding model and observation basis. An implemented approximation needs an error account adequate for the question. Use the relevant subject Methods and B.3 for these claims. Broader reliance is a reason to establish the broader claim, while an ordinary local use can stop with its sufficient local result.