More minimal non-$$\sigma $$-scattered linear orders

In a recent paper, Cummings, Eisworth and Moore gave a novel construction of minimal non- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>σ</mml:mi></mml:math> -scattered linear orders of arbitrarily large successor size. It remained open whether it is possible to construct these orders at other cardinals. Here, it is proved that in Gödel's constructible universe, these orders exist at any regular uncountable cardinal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi></mml:math> that is not weakly compact. In fact, for any cardinal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>κ</mml:mi></mml:math> as above we obtain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>2</mml:mn> <mml:mi>κ</mml:mi></mml:msup> </mml:math> many such orders which are pairwise non-embeddable. At the level of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℵ</mml:mi> <mml:mn>1</mml:mn></mml:msub> </mml:math> , their work answered an old question of Baumgartner by constructing from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>♢</mml:mo></mml:math> a minimal Aronszajn line that is not Souslin. Our uniform construction is based on the Brodsky-Rinot proxy principle which at the level of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℵ</mml:mi> <mml:mn>1</mml:mn></mml:msub> </mml:math> is strictly weaker than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>♢</mml:mo></mml:math> .