From Exponent 32 to 8: A Smoothed Refinement of the BIMS Least Quadratic Nonresidue Argument under QRH

This note presents a unified derivation of bounds for the least quadratic nonresidue \(n(p)\), assuming that all nontrivial zeros of Dirichlet \(L\)-functions satisfy \(\Re\rho\le 7/8\). Starting with the BIMS nonresidue detector, it explains the exponent \(32\) displayed in the cited OpenAI preprint, obtains \(16+\varepsilon\) by optimizing the strip parameter, and then introduces a linear smoothing weight to establish \(n(p)\ll(\log p)^8\). The final step follows from a directly derived smoothed explicit-formula estimate and allows zeros on the boundary \(\Re s=7/8\), so no \(\varepsilon\)-loss is required. The contribution is an explicit presentation of these refinements within one detector framework. The smoothing method and resulting conditional exponent are classical; no novelty claim is made for the exponent-\(8\) theorem. The note assumes QRH and does not establish or independently verify it.