Using downward self-reducibility to place Shattering in PLS^Σ2^P

Show that the TFΣ3^P problem Shattering (derived from the Sauer–Shelah lemma) is in PLS with access to a Σ2^P oracle by developing a downward self-reduction aligned with the divide-and-conquer structure of the Sauer–Shelah proof.

Background

Shattering is a canonical TFΣ3^P problem whose totality follows from the Sauer–Shelah lemma. Prior work places Shattering in PPP^Σ2^P via a hashing-based approach.

Given the paper’s general framework linking downward self-reducibility to PLS-membership (with appropriate oracles), it is natural to ask whether the divide-and-conquer flavor of the Sauer–Shelah lemma can be turned into a downward self-reduction to place Shattering in PLS^Σ2^P.