Information hiding and quantum recovery with nearly minimal locally stable sets

We construct distributed encodings with matching upper and lower bounds on hidden message length, quantum communication, and initial quantum memory. Among n laboratories of fixed local dimension, orthogonal product states remain locally stable against every coalition of fixed maximum size. Above their exact minimum cardinality, a budget of R additional states permits at most O(log(R + 2)) hidden bits at vanishing leakage. Our family attains q = Θ(log R) against arbitrary-round block LOCC for R → ∞ and R = o(√n). Its message occupies one of G = Θ(√n/R) disjoint candidate subsystems. Recovery costs Θ(q) communicated qubits after the location is revealed, or Θ(Gq) initially stored qubits when resources are placed beforehand and subsequent quantum communication is unavailable. The storage bound allows arbitrary multipartite entanglement. Both lower bounds already apply to fixed-advantage prediction of any balanced message bit. Complete 98-page manuscript dated 26 October 2026, including the main text, Appendices A–M, and Supplementary Proofs, Parts I–VII. Author contributions (as stated in the manuscript): Yiming Liu is the sole author. AI tools assisted mathematical reasoning, bibliographic research, code and figure generation, drafting, and language editing.