{"id":"openai-math/Power-law-violations-of-Yaus-nodal-upper-bound","pdf":"https://github.com/openai/math/blob/main/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026","tags":["spectral-geometry","eigenfunctions","riemannian-geometry"],"score":0,"title":"Power-law violations of Yau's nodal upper bound","domain":"Differential geometry","family":"350","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DG","summary":"We construct a smooth Riemannian metric on $`S^4\\times S^1`$ and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than $`\\lambda^{1/2+\\epsilon_0}`$ for one fixed $`\\epsilon_0\\gt 0`$. This disproves the smooth upper-bound assertion in Yau's nodal-set conjecture and the proposed bound with an arbitrarily small positive power loss.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-23","reactions":{},"formalized":true,"family_title":"Yau’s nodal bounds: surfaces and higher dimensions"}