Access thresholds for public quantum records: A finite-depth separation theorem with explicit code instances

We prove a finite-depth access-threshold separation theorem for public quantum records. In a single four-branch many-body environment, one label is encoded in a positive-rate, positive-distance quantum code while a second label is redundantly broadcast in local environmental degrees of freedom. Although unrestricted global decoding selects the encoded record, every public fragment below a distance/light-cone threshold has exactly zero distinguishability for that record, while the broadcast label remains publicly distinguishable with positive density. Conversely, above a complementary high-access threshold, erasure recovery reconstructs the encoded record. The result formalizes a separation between globally decodable environmental information and public objectivity under a declared fragment-access structure.