7. Summary and outlook

The construction developed through four successive ideas: the capillary prototype suggested a coercive background plus a small detector; numerical screening isolated the balanced exponent \(2/m\); the one-pole calculation forced a linear model at each zero; and the logarithmic ODE supplied the transition between these two regimes. The computations guided the search, while the final conclusions still required rigorous mathematical proof.

AI-assisted symbolic and numerical tools greatly accelerated the testing, comparison, and correction of candidate formulas. We expect increasingly capable AI systems to contribute even more to mathematical research—not by replacing mathematical understanding, but by helping us explore ideas more quickly and deepen that understanding.