Equiconsistencies at subcompact cardinals

We present equiconsistency results at the level of subcompact cardinals. Assuming SBH_δ, a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both □(δ) and □_δ fail, then δ is subcompact in a class inner model. If in addition □(δ⁺) fails, we prove that δ is ({Pi_1^2}) subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary we also see that assuming the existence of a Woodin cardinal δ so that SBH_δ holds, the Proper Forcing Axiom implies the existence of a class inner model with a ({Pi_1^2}) subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to δ is δ⁺⁽ⁿ⁾ supercompact for all n < ω. We state some results at this level, and indicate how they are proved.