{"id":"palomar/PALOMAR-2026-10-09-000007","url":"https://github.com/QuanyuTang/polya-neumann-simply-connected-formalization-Palomar-version-v2/tree/f07f1c7051862ba146429db6eda5dc4e8b020f74","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-09-000007-v1.json","repo":"https://github.com/QuanyuTang/polya-neumann-simply-connected-formalization-Palomar-version-v2","tags":["math.SP","math.AP","msc:35P15","msc:35P20","msc:35J05","spectral-theory","polya-inequality","lean-formalization"],"score":0,"title":"Strict Neumann Polya inequality on simply connected planar domains","trust":"high","domain":"math.SP","source":"palomar","authors":["Quanyu Tang","Zuoqin Wang"],"reviews":0,"subject":"math.SP","summary":"A Lean 4 / Mathlib formalization of the strict Neumann Pólya inequality for bounded simply connected Lipschitz domains in the plane. The plane is identified with ℂ; the Neumann eigenvalues μ_0 ≤ μ_1 ≤ ... are defined by the Courant–Fischer min–max formula for the form ∫|∇u|² on H¹(Ω) (weak gradients in L²), indexed from 0 and repeated according to multiplicity, and N_N(E) = #{j ≥ 0 : μ_j |Ω| E / (4π) for every E > 0. The proof development (about 100,000 lines, namespace PolyaNeumann) also identifies the variational eigenvalues with the spectrum of the constructed self-adjoint Neumann…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-09","reactions":{},"formalized":true,"formalizes":[{"title":"The Neumann Pólya inequality on simply connected planar domains","identifier":"https://github.com/QuanyuTang/polya-neumann-simply-connected-formalization-Palomar-version-v2/blob/99037f07a1a3a8eb475133eee1934a2fbb77c35d/paper/N_simply_connected_strict_v3.tex","relationship":"formalizes"}]}