Conflicting Conjectures and the Search for Common Ground: A Meta-Theorem Approach

When two famous math ideas seem to contradict each other, maybe there’s a hidden reason they both "almost" work—let's hunt for a unifying principle that explains their overlap and differences.

Research Question: When two mathematical conjectures or theorems appear to conflict, is it possible to synthesize a meta-theorem that explains their domain of agreement and divergence, thereby resolving the apparent contradiction?

Hypothesis: Many unresolved or conflicting conjectures in mathematics arise from overlooked structural assumptions or hidden symmetries; by explicitly modeling these, a more general meta-theorem can be formulated that encompasses both cases as special instances.

Experiment Plan: - Select a pair of conflicting or notorious conjectures (e.g., from Hilbert’s problems or easier variants of hard problems as per Heule, 2021).

• Analyze their assumptions, proof strategies, and failure modes.
• Attempt to construct a general framework (possibly using p-adic methods or other modern techniques) that unifies the two results.
• Formulate and test a meta-theorem explaining the underlying structure leading to apparent contradiction.

References:

• Heule, M. J. H. (2021). Easier variants of notorious math problems.
• Dasgupta, S. (2009). Stanford Department of Mathematics Colloquium: A p-adic approach to Hilbert’s 12th problem.