Least cardinality of a non-sober meet-continuous complete lattice

Determine whether ZFC proves that the least cardinality of a meet-continuous complete lattice with a non-sober Scott space is [?]aleph_1, equivalently whether there exists such a lattice of cardinality [?]aleph_1.

Background

The paper proves that every countable meet-continuous complete lattice has a sober Scott space, establishing an [?]aleph_1 lower bound for the least cardinality at which non-sobriety can occur. It also proves that the corresponding spectrum is upward closed and that, under the Continuum Hypothesis, the least cardinality is [?]aleph_1.

The unresolved issue is whether this [?]aleph_1 threshold follows in ZFC without assuming the Continuum Hypothesis. An affirmative answer would, by upward propagation, produce non-sober meet-continuous complete lattices at every uncountable cardinality and witnesses to failure of the square Scott-product identity at all such cardinalities.

References

Does ZFC prove that $s=\aleph_1$? Equivalently, must there exist a meet-continuous complete lattice of cardinality $\aleph_1$ with a non-sober Scott space?

— Every countable meet-continuous lattice is Scott sober  (2609.19612 - Xu et al., 17 Sep 2026) in Question 1, Section 7.2, "Questions beyond countability"

Must $s=s$ hold in ZFC, or is it consistent with ZFC that $s<s$?

— Every countable meet-continuous lattice is Scott sober  (2609.19612 - Xu et al., 17 Sep 2026) in Question 2, Section 7.2, "Questions beyond countability"

Is it consistent with ZFC that $s>\aleph_1$?

— Every countable meet-continuous lattice is Scott sober  (2609.19612 - Xu et al., 17 Sep 2026) in Question 3, Section 7.2, "Questions beyond countability"