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Bending one way spreads electrons out too much; bending the other way mistreats stretched bonds.","posed_since":"","precise":"Construct an exchange-correlation approximation, at cost no higher than a global hybrid, whose energy $E(N, S_z)$ is piecewise linear in fractional electron number N (no delocalization error) and constant in fractional spin at fixed N (no static correlation error), as tested on H2+ and $\\mathrm{H_2}$ dissociation, fractional-charge atoms and the stretched-bond benchmark sets, while keeping main-group thermochemistry errors below 3 kcal/mol. An answer is such a functional, or a proof that no functional of a given class can satisfy both conditions.","problem_ref":"a754ae190f8564c40dd02646a107d8f5d95bab993397c1b6076b67fb059fbeb2","references":"","settled_by":"A functional passing fractional-charge, fractional-spin and standard thermochemistry benchmarks simultaneously, or a no-go theorem for a defined functional class.","status_note":"In 2021 the DM21 learned functional, trained on fractional-charge and fractional-spin data, reduced both errors for small systems; no functional satisfies both conditions in general.","title":"A practical functional that obeys the flat-plane 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