The failure of square at all uncountable cardinals is weaker than a Woodin limit of Woodin cardinals

TL;DR

This paper demonstrates that the failure of the square principle at all uncountable cardinals is a weaker assumption than the existence of a Woodin limit of Woodin cardinals, showing its consistency relative to certain inner model hypotheses.

Contribution

It establishes that the failure of square at all uncountable cardinals can be achieved in models with strong inner model properties, specifically under the HOD Hypothesis, without requiring large cardinal assumptions.

Findings

Square_kappa fails at all uncountable cardinals in the model

Every regular cardinal is omega-strongly measurable in HOD

Failure of square is consistent with the HOD Hypothesis

Abstract

We force the Axiom of Choice over the least initial segment of a Nairian model satisfying ZF. In the forcing extension, square_kappa fails at all uncountable cardinals kappa, and every regular cardinal is omega-strongly measurable in HOD, as witnessed by the omega-club filter. Thus the failure of square everywhere is within the current reach of inner model theory, and the HOD Hypothesis is not provable in ZFC.