A scalar Herglotz criterion for the even-simplicity hypothesis in the localized Weil quadratic form

The strategy of Connes-Consani-Moscovici (arXiv:2511.22755) and Connes-van Suijlekom (arXiv:2511.23257) toward the Riemann Hypothesis rests on a structural hypothesis on the truncated/localized Weil quadratic form Q_W: that its smallest eigenvalue be simple with even eigenvector (even-simplicity, CCM Def. 5.3). In those works even-simplicity is assumed and verified only numerically; Suzuki (arXiv:2606.09096) proves it for the continuum operator only in the small-interval limit. We give an exact reduction of this hypothesis to a single scalar inequality between two Herglotz functions. Writing the pole term of the explicit formula as the rank-two operator W_{0,2} = 2|C><C| - 2|S><S| (C(x)=cosh(x/2) even, S(x)=sinh(x/2) odd), the parity decomposition turns each sector into a rank-one perturbation of the pole-free operator: A_even = B_e + 2|C><C| on the even sector and A_odd = B_o - 2|S><S| on the odd sector, where B_e has a single negative eigenvalue (the Perron ground of the pole-free part) and B_o > 0. The two sector ground states are the smallest roots of the rank-one secular equations m_e(lambda) = <C,(B_e-lambda)^{-1}C> = -1/2 and m_o(lambda) = <S,(B_o-lambda)^{-1}S> = +1/2. Our main result is that even-simplicity is equivalent to the pole-localization condition lambda_0^even < lambda_0(B_o) together with m_o(lambda_0^even) < 1/2, i.e. <S,(B_o-lambda_0^even)^{-1}S> < 1/2, a single comparison of two resolvent forms at one energy. We verify the full chain (the rank-one sign structure, both secular equations, the pole-localization condition, and the criterion) to machine precision for c = lambda^2 in {5,13,53}. This isolates the CCM hypothesis as one explicit scalar inequality and identifies it as the precise residual; we make no claim of proving even-simplicity itself, and no claim about the Riemann Hypothesis.