Density-one prescribed-rank spectra of prime-cover graphs
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…
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