1. Purpose and perspective

The final test function in the paper is compact enough that its origin can be easy to miss. In its finished form it is

where \(L=\operatorname{span}\{p_1,\ldots,p_m\}\), \(s_L(y)=\abs{\pi_{L^\perp}y}\). Throughout, we call \(p_1,\ldots,p_m\) the prescribed directions; they are precisely the zeros of the test function. The term pole is reserved for the multipole construction, its factors, and its numerical configurations. The cutoff \(\phi\) is linear near zero and becomes a \(2/m\)-power away from zero. Read backwards, every part of (1.1) has a reason. The background \(s_L^2\), the product structure, the exponent \(2/m\), the linear core, the order in which \(T\) and \(k\) are selected, and the normal-rank restriction were each forced by a different calculation.

The purpose of this note is not only to explain the mathematical origin of this formula, but also to describe how intelligent assistance contributed to the research process. Our principal AI-assisted environments during the original search were a GitHub Copilot subscription and Cursor’s Composer 2. Starting from hand calculations, we used these tools to generate and revise numerical tests quickly, compare alternative implementations of long formulas, scan large parameter families, and concentrate computation in the singular regimes where a proposed test function was most likely to fail.

This assistance was particularly important for numerical validation. It made the cycle from a geometric idea to an executable test sufficiently short that we could examine many more candidates than would otherwise have been practical. A negative numerical value often exposed the exact scale or region in which an ansatz broke down; a positive scan identified a candidate worthy of further analysis. Neither outcome was confused with a proof. The choice of meaningful asymptotic regimes, the interpretation of numerical failures, and the conversion of observed patterns into rigorous estimates remained the mathematical work of the researchers.

This is also a historical account of a rapidly changing technology. In the few months since the original experiments, the intelligence and reasoning ability of leading AI models have improved qualitatively. A comparable search undertaken now may be completed more quickly and with more capable assistance in symbolic reasoning, numerical design, adversarial testing, and proof organization. The workflow described here should therefore not be read as a benchmark for current AI systems. We nevertheless describe it in detail because it gives a concrete example of how AI assistance contributed to this research and of how such tools can help push mathematical understanding further.