{"id":"palomar/PALOMAR-2026-09-10-000004","url":"https://github.com/shaikidris/prime-cover-power-band-spectra-palomar/tree/72288e1f32ba491a873e63647023b70d195f7139","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-10-000004-v1.json","repo":"https://github.com/shaikidris/prime-cover-power-band-spectra-palomar","tags":["math.CO","math.NT","msc:05C50","msc:11N05","spectral-graph-theory","prime-numbers","eigenvalues","analytic-number-theory"],"score":0,"title":"Density-one prescribed-rank spectra of prime-cover graphs","trust":"high","domain":"math.CO","source":"palomar","authors":["Idris Ali Shaik"],"reviews":0,"subject":"math.CO","summary":"For finite prime-cover graphs, the ordered adjacency eigenvalues at prescribed arithmetic ranks obey density-one squared-error estimates in every fixed power range 0 < theta < 1/2. Relative to the prime-counting main term and explicit first-exit correction, the error is bounded by a constant times a/log(X) + log(X)^D sqrt(X/(a log(X))), with quantitative terminal exception counts and zero limiting failure density. This gives a positive power improvement over the sqrt(X)/log(X) block bound. Separate sharp every-terminal-centre statements on 21/61 < theta < 1/2 at error O(a/log(X)) remain open…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-10","reactions":{},"formalized":true,"formalizes":[{"title":"Prescribed Eigenvalues in Power Bands of Finite Prime-Cover Graphs","identifier":"https://papers.ssrn.com/abstract=7441718","relationship":"formalizes"},{"title":"Ordered comparison and density-one power-band estimates","identifier":"","relationship":"formalizes"}]}