On the Logarithmic Barrier in the Beurling-Nyman-Báez-Duarte Criterion and its Acceleration via Sum-Zero Constraints

Abstract:
This paper presents strong numerical and analytical evidence that the logarithmic decay barrier in the Beurling-Nyman-Báez-Duarte (BNBD) criterion can be broken via sum-zero constraints. We prove analytically that the diagonal elements of the Gram matrix decay as Gn,n∼C/nG_{n,n} \sim C/nGn,n∼C/n, where the constant has the exact closed-form expression C=ln⁡(2π)−γ≈1.26066C = \ln(2\pi) - \gamma \approx 1.26066C=ln(2π)−γ≈1.26066. This causes the trace to grow logarithmically as Tr(GN)∼Clog⁡N\text{Tr}(G_N) \sim C \log NTr(GN)∼ClogN.By imposing ∑cn=0\sum c_n = 0∑cn=0, we transform the decay from dN2∼O(1/log⁡N)d_N^2 \sim O(1/\log N)dN2∼O(1/logN) to a polynomial law dN,const2∼A⋅N−α(log⁡N)−βd_{N,const}^2 \sim A \cdot N^{-\alpha} (\log N)^{-\beta}dN,const2∼A⋅N−α(logN)−β. With β=1.5\beta = 1.5β=1.5 fixed theoretically, we obtain a highly stable fit with α=0.577±0.017\alpha = 0.577 \pm 0.017α=0.577±0.017. The spectral correction δ=α−1/2≈0.077\delta = \alpha - 1/2 \approx 0.077δ=α−1/2≈0.077 is consistent with δ≈1/(4π)\delta \approx 1/(4\pi)δ≈1/(4π), suggesting a second-order spectral correction arising from a rank-one perturbation.Key Contributions:Analytical proof of the Gram matrix diagonal decay constant.Derivation of the exact finite-size correction matching numerical data.Breaking the logarithmic barrier via a constrained optimization subspace.Documentation of 15 methodological pitfalls to guide future research.