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A Strict Upper Bound for Three-Point Weighted Selection in Planar Vector-Valued Regression

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 $15/8$.

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2026
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arxiv.org/abs/2611.12550
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Abstract

We prove F_{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_{weighted}(2,2,3)<11/6. No numerical value of \varepsilon_0 is claimed.