Private information of the qubit depolarizing channel beyond the single-use threshold: explicit finite-n witnesses up to p = 0.28467, and what one-way key distillation already implies

Short note with supporting code and data on Open Problem Zoo record op_d2813fe3fcdf09ad (private capacity P of the qubit depolarizing channel Lambda_p(rho) = (1-p) rho + p I/2). The record certifies P > 0 only for p ≤ 0.259824 (via the quantum capacity) and lists positivity on (0.259824, 1/3) as open. Two observations. (1) Published one-way key-distillation results already imply more: the one-way six-state rate with noisy preprocessing (Renes-Smith 2007, Kraus-Gisin-Renner 2005) is a single-use private information of this channel and is positive for p < 0.28223 (QBER = p/2), and the rates of Kern and Renes (2008) with repetition codes of length 250 are positive up to QBER 14.5741%, i.e. p = 0.291482; a short in-band lemma (send the publicly announced syndromes through extra channel uses with a classical code; I(S;B) = H(S) ≥ I(S;E) up to Fano terms) turns any such forward-announcement rate into a Devetak private information over finitely many uses, so P(Lambda_p) > 0 for all p ≤ 0.291482 with no public channel. (2) Explicit finite-n witnesses: ensembles of two classical mixed states on n ≤ 8 qubits whose private information I(X;B) - I(X;E) is positive at p = 0.28336, 0.28372, 0.28405, 0.28437, 0.28466 for n = 4, 5, 6, 7, 8 (thresholds 0.283375 ... 0.284671), above the single-use threshold 0.28223. They were found by an autonomous AI research agent (the depositor's EVO harness); the members are distributions over bit strings whose weights depend only on the Hamming weight, with the second member the bit complement of the first. Verification uses an exact closed-form block decomposition of the environment state's spectrum (no 4^n-dimensional objects); every value is recomputed in 30-digit arithmetic and agrees with double precision to about twelve digits. Negative findings of the same searches: the family is locally optimal within classical ensembles, gains ~0.0003 per added qubit, and general (quantum) members at n ≤ 3 give no gain; the finite-n frontier will not reach the key-distillation frontier at n ≤ 10. Claim boundary: lower bounds on the positivity threshold and on finite-n private informations; no capacity value, optimality or priority claim. Files: the note (PDF + LaTeX), the ensembles as JSON, the exact evaluator and verifier, the high-precision certifier, the agent's notes and search code, SHA-256 sums.