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Abstract

Let F(z)=ξ((1+iz)/2)/4, G=F'^2-FF'', and let D_0=E_0-Q_0 be the original endpoint correction, where E_0(P)=(4π)^(-1)∫ |(FP)'|^2 and Q_0(P)=(8π)^(-1)∫ GP^2. For every fixed 0<θ<1 and A>0, every sufficiently large degree N, and each polynomial parity, we construct a real subspace with a rational basis, dimension at least c_(A,θ)(log N)^A(log log N)^2, and Q_0≥(2-θ)E_0 on its entire span. The proof combines classical unconditional short-interval zero moments with bounded-population microscopic seed selection, a two-scale root-product budget, seed-averaged logarithmic root energy and higher fixed-order weighted approximation in a derivative and absolute-potential norm. All local complex zeros, multiplicities, exterior roots and mixed polynomial Gram entries are retained. Companion spaces have 0<Q_0/E_0≤θ at the same dimension scale, yielding the corresponding necessary repair-rank bounds for scalar-centered fixed strict contractions. A finite Gaussian zero-ideal approximation theorem and an exact generic current certificate delineate the scope of the method. The constructed correction-negative spaces have positive original Q_0.