This question proposes a grand challenge for AI, setting an extremely high bar for the field, intended to measure when AIs will be considered on par with, or having surpassed the brightest humans in the world at mathematics. While competetive mathematics, such as the IMO and Putnam, provide concrete benchmarks for mathematical reasoning, mathematicians have long recognized that there is more to mathematics than competetive, narrow problem solving. Famous mathematicians, such as Gauss, Hilbert, and Grothendieck, have fundamentally reshaped humanity's understanding of mathematics by not only solving some narrow problems, but by introducing new fundamental mathematical theories.

Mathematical theories are systematic frameworks composed of definitions, axioms, and propositions that are used to analyze and interpret the properties and relationships of certain mathematical entities or structures. They are fundamental to all areas of mathematics, and many theories are so foundational that they're often just considered part of the discipline rather than an individual theory. Foundational math theories include number theory, graph theory, group theory, topology, chaos theory, category theory, knot theory, algebraic geometry, and set theory.

Will AIs be widely recognized as having developed at least one new, innovative, foundational mathematical theory before 2035?

Here are some important definitions:

1. New Mathematical Theory: A 'new mathematical theory' for the purposes of this question must be a systematic and original framework of abstract concepts, propositions, and methods that significantly extends existing mathematical knowledge or explores a previously uncharted area of mathematics. It should demonstrate logical coherence and internal consistency, and have aided in the solution of previously unsolved problems that have evaded mathematicians for many years.

2. AI Involvement: The new theory must have been primarily developed by AI systems, where 'development' involves the generation of the main theoretical concepts, propositions, and methods, and not just assistance in computations or the execution of tasks designed by human mathematicians.

3. Innovation: The new theory must be innovative, meaning it should not be a straightforward extension or minor modification of existing theories, but rather should introduce new mathematical objects, concepts, or methods, or provide significant new insights into existing ones.

4. Foundational: The new theory should also be widely recognized as foundational, meaning it has had a substantial impact on the field and has opened up new research avenues. Conclusive proof that this criterion is met will be obtained if the theory is used to solve a problem in the Open Problem Garden within one year that was previously mentioned in the literature at least 50 different times but unsolved for a period of at least five years.

5. Recognition by the Mathematical Community: The theory, including its exposition and applications, must have been published in at least three highly respected, peer-reviewed mathematical journals. Furthermore, it should have been cited in subsequent mathematical literature a significant number of times, at least 500 times across all publications, within five years of publication, and before 2035, in non-self-referential works.

6. Independent Verification: The theory's validity and its utility in providing mathematical insight should be confirmed by independent and credible testimonials from more than ten mathematicians represented at American and European universities, including at least two living fields medalists.

If a new, innovative, foundational mathematical theory is developed primarily by AIs meeting these criteria before 2035, as determined by the question author in his best judgement, potentially upon the advice of experts, then this question resolves to YES. Otherwise, it resolves to NO.