Derive HPC from AD^+ with no mouse pairs having long extenders

Establish that the HOD Pair Capturing (HPC) principle—equivalently, that strategies of mouse pairs are Wadge-cofinal in Suslin co-Suslin sets of reals—follows from AD^+ together with the assumption that there are no mouse pairs with long extenders. This would remove the need for stronger anti-large-cardinal hypotheses (such as the non-existence of a measurable limit of Woodin cardinals) currently used to obtain HPC and enable HOD analysis in broader determinacy contexts.

Background

The paper relies on the HPC principle to carry out HOD analysis, a central technical tool for constructing and identifying determinacy models from below. While HPC is established in the paper under the stronger anti-large-cardinal hypothesis that there is no mouse pair with a measurable limit of Woodin cardinals (NMLW), the authors note a widely held conjecture that HPC should already follow from AD^+ assuming only that there are no mouse pairs with long extenders.

Proving this conjecture would strengthen the foundation for HOD analysis by reducing the large cardinal anti-assumptions needed and could extend the reach of the construction to broader classes of determinacy models. The conjecture is cited as Conjecture 1.7.6 in the referenced literature and remains a key open direction in the area.