Start with a uniformly shuffled sequence and turn it into a path that moves up and down. A 2021 paper conjectured that the number of edges lying above zero follows a discrete arcsine law, a familiar U-shaped probability pattern that favors very small or very large occupation times. The missing piece was a proof for this permutation-based walk. ^[1]

According to the preprint, GPT-6 Astra proposed representing the shuffled walk as one built from independent, continuous and symmetric random increments. Once that representation is in place, the desired result follows from the classical Sparre–Anderson theorem. ^[1]

The AI did not supply experimental data or a numerical approximation. Its contribution was the unexpected representation and a short proof. The human author selected, presented and checked the argument, including the link to the existing theorem on which it depends. ^[1]

In mathematics, a compact argument can expose the hidden structure of a problem better than a long calculation. Here, the claimed advance is not speed alone: it is the bridge from a walk defined by a random permutation to a setting covered by a well-known theorem. ^[1]

This is a single-author arXiv preprint. It has not been peer reviewed, and this discovery review did not provide an independent line-by-line verification. The model name and reported generation time do not establish correctness; other mathematicians must be able to check every reduction and assumption in the written proof. ^[1]