C.40.CD:4 - Solution

Recover the problem and an available way, construct a consequential change, perform a revealing target use, and let its result determine the next question or use. Keep the original receiving need connected to any replacement or subsidiary question.

C.40.CD:4.1 - Recover what is being sought and what can already be done

Identify the result being sought, the conditions it must satisfy and the material or situation to which it applies. Distinguish a result someone needs from a constructed challenge used to develop a method. Either can justify inquiry; they give different reasons for continuing it.

Recover an available way far enough to find its inputs, operations, output and supporting conditions. A worked case may be enough. B.5.RC recovers the construction; B.5.TU makes an available theory usable. Start at the point affected by the new question rather than reconstructing an unchanged whole.

Keep any protected condition explicit while it matters. A resource limit, physical condition or required output can be changed in an exploratory problem, but the resulting answer then has to be related back to the original requirement.

C.40.CD:4.2 - Construct a question that the available result makes worth asking

Locate a consequential change and carry it through the question:

A premise or condition failed. Ask which result remains obtainable under the supported conditions, or which missing condition would permit the wanted result.
A construction succeeded. Vary an input, condition or required output to find a useful broader operation or its limit.
A new operation became available. Find a problem whose needed intermediate result that operation can supply; work the connection through to a possible use.

Use B.5.QD to form and examine the new question. Change a named condition, demanded result, available means or composition of operations. Explain how that change affects what an answer would need to establish.

If the opening is an incompatibility, B.5.QD.CF examines the premise that makes the demands conflict and constructs a question about another means or a legitimate distinction. Return its candidate to the retained requirements here; changing an assumption for exploration leaves the corresponding realization question open.

Check the formulation before investing in solving it. Its objects and operations must have meanings, its conditions must be interpretable together, and the proposed answer must be assessable. A question asking whether a configuration is possible can legitimately return an impossibility result. When a task generator promises a solvable challenge, support that promise with a feasible case. Otherwise keep feasibility open.

Select a revealing attainable case. Its result may expose a counterexample, a useful intermediate construction or a missing operation. Leave difficulty relative to the available means and method where it has not been established more generally.

C.40.CD:4.3 - Construct the way for the changed problem

Try a sufficient existing operation first. Where the receiving problem breaks the old construction, locate the failed contribution: input meaning, intermediate result, operation, output condition or applicability premise.

Retain the usable contributions. Transform the input, replace a constituent or compose available operations so that each result supplies what the next step needs. C.39.RO develops that reusable operation and its support. A new physical or professional operation still needs the Method that constructs or performs it.

Trace the changed way on the target material. Distinguish a derived conditional result, an observation from a performed trial and a proposed operation whose effect remains to be obtained. The target question determines which of these can support the next use.

C.40.CD:4.4 - Examine what the target use establishes

Judge the result against the target’s conditions. Recover the failed or satisfied condition and the contribution that produced it. An earlier source success is useful support for trying the operation; the target comparison decides this use.

For a mathematical construction, an argument may establish the result over the stated inputs. For a physical claim, use the model and observations required by that claim. A generated executable challenge may need both a solving construction and a check that an empty response cannot satisfy its result condition. Select the examination that can change the next action; a local inquiry need not become a separate assurance programme.

Keep distinguishable the useful result, the part still unresolved and any attempted transfer that failed. When no attempt was performed, retain the proposed way and its missing prerequisite as such.

C.40.CD:4.5 - Use the result to choose the next problem

Use a sufficient result and stop this development. Otherwise follow the part of the outcome that can change receiving work:

A partial result can support a narrower use while another condition remains open.
A failure witness can support a revised problem, a necessary-condition question or a justified change of operation.
A successful new construction can support a further question about changed inputs, composition or use.
A missing contribution can direct a bounded inquiry or specialist request when its attainable answer warrants the effort.

For each continuation, state what it would permit and which earlier requirement it answers, changes or leaves open. A problem constructed for exploration can be retained for a later possible use without claiming present demand. Its interest lies in the further reasoning or action it may enable.

Retain the actual material and explanation needed to continue. Compare the prospective gain with the cost of the attempt, adaptation and retention, including displaced work. C.11.DUA helps when the value of further information work is unsettled. Close an unwarranted branch without discarding an earlier qualified use of its source method.

C.40.CD:4.6 - Coordinate several attempts only when their results warrant it

Several problems and ways may develop together. Try a transfer because a usable constituent or intelligible adaptation could change the target answer. Characterize the difference needed for that choice: for example, which conditions defeat available methods or which new operations a result makes possible.

Use C.18 when archive, front, lineage or possibility-space claims are needed, and C.19 for a resource-bearing continuation policy. A small problem and one useful attempt need neither a population nor an archive.

People, AI agents and teams can contribute different operations to this work. Preserve the receiving participant’s means and preparation when passing on a construction. A copyable program or policy can supply executable material; learning how to adapt a method may require separate acquisition or practice. C.36.RP coordinates that continuing availability when it becomes the problem.