MATH.23:11 - SoTA-Echoing

The working question is how to develop a mathematical conjecture from constructions and their failures, with a useful route into further work.

Adapt proof analysis. Lakatos’s Proofs and Refutations, Appendix 1 connects counterexamples to the hidden lemmas and concepts in an attempted proof. The adopted move changes :4.4-.5: inspect the failed inference and consider a revised condition or construction. It improves on simply excluding the offending example when that exclusion abandons the intended class. The method can require more work than proving a fixed, adequate claim; use the latter route when its premises and conclusion already serve the inquiry. Lakatos supplies a method for developing the conjecture; its use can still leave the conjecture unresolved.

Adapt construction search and generalization. Georgiev, Gómez-Serrano, Tao and Wagner’s Mathematical exploration and discovery at scale, §§1.3-1.6 and 4 distinguishes search for constructions from finding a program or formula that generalizes, and describes transitions to proof. It also reports evaluator defects and the limits of the tested search methods. This informs :4.1/:4.3-.4: inspect the generator and evaluator, recover a general construction, then establish the claim its use needs. Compared with direct object search, program search can retain reusable structure but adds generation and evaluation cost. A cheap direct search or existing theorem can be preferable. The paper supports these bounded choices, not a universal preference for AI search.

Retain methods as a usable result. The Math and AI declaration of 11 September 2026 argues that counting solved problems can obscure understanding and transmission of methods. Adopt that question in :4.6 and the relation to C.36.RP: preserve what another prepared contributor needs to change and use the construction. The declaration is a position on the purpose of mathematical work, not evidence that transmission must remain exclusively human. The construction-to-proof account above supplies a concrete alternative involving AI contributors.

Reopen the route when a new counterexample changes the admitted class, another representation makes the conjecture simpler, or the receiver needs a result the present construction cannot provide. Improved search tools change the means and cost of investigation; they leave the proposed mathematical assertion and its possible use to be stated.