Mathematics > Statistics Theory
Title: A Strict Upper Bound for Three-Point Weighted Selection in Planar Vector-Valued Regression
Abstract: We prove $F_{\mathrm{weighted}}(2,2,3)<11/6$ for minimum-Frobenius-norm least-squares regression with two input and two output coordinates, improving the upper bound $15/8$. More precisely, a universal $\varepsilon_0>0$ satisfies $\mathcal T_3\le(5/6-\varepsilon_0)E$ for every finite residual frame. The proof combines an exact scalar certificate, rigidity of equality in planar selection, and a compactness argument that controls disappearing weights. Together with the known lower bound, this gives $13/8\le F_{\mathrm{weighted}}(2,2,3)<11/6$. No numerical value of $\varepsilon_0$ is claimed.
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