MATH.23:1 - Problem frame

Use this pattern when a mathematical construction works in some cases, fails after a change, or reveals a relation that could support a new theorem. You need to formulate the next claim and a way to investigate it.

A conjecture is a mathematical assertion proposed for investigation. Developing it includes choosing its objects, assumptions, quantifiers and conclusion. The result can become a theorem, a refuted claim, or a more useful question. Its value lies in what the answer would let someone construct, infer, explain or change.

First useful move: change one mathematically consequential part of the construction and compare what happens. Replace a parameter by a variable, combine two outputs, remove a used assumption, or ask the same operation to preserve another property. Try the changed construction and identify the equation, condition or obstruction that controls its result.

The reader needs the mathematical operations used by the starting construction and elementary proof reasoning. The examples explain their additional operations; the last example develops a limit argument. The method can be performed by a person, an AI agent or cooperating contributors with the needed preparation.

Use an established result when it already answers the question. MATH.19 helps prove a sufficiently clear claim. B.5.QD develops questions more generally. Here the work develops a mathematical assertion through construction, variation, proof analysis and counterexamples.