Forcing extensions characterized by finite agreement

Find additional countable models M and forcing notions P in M such that two generic sequences produce the same forcing extension exactly when they eventually agree.

Background

A measurable-cardinal forcing example satisfies the finite-agreement characterization. The problem asks for further models and forcing notions with the same phenomenon.

References

Find more models $M$ and forcing notions $\mathbb{P}\in M$ such that $(*)$ is true.

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 8

Is $\thicksim$ hyperfinite?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 8

Suppose there are a model $M$ and a forcing notion $\mathbb{P}\in M$ such that $(*)$ is true. Can we deduce that $M\models [\exists\ \text{a large cardinal}]$?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 8