Unconditional Half-Density at Minimum 2-Class Depth in a Two-Prime Layer III+ Family

Abstract

We prove unconditionally that the minimum-depth event (e=2) has limiting conditional density (1/2) in a two-prime Layer III+ family. The Layer III+ character sum is decomposed into four components. Three ancillary components are realized as tempered quadratic or cyclic-quartic characters, while the remaining mixed obstruction is compressed to the single diagonal Rédei–Artin kernel ([-2A,A,B]). Smith's unweighted bilinear equidistribution theorem gives uniform power savings for all four components. A hyperbolic staircase transfers the rectangular estimates to (AB\le X), and polylogarithmic fixed-slice Chebotarev estimates handle the edge ranges. The result is unconditional but ineffective because of Siegel's theorem. No arbitrary two-sided coefficient estimate or deeper three-way distribution is claimed.