Every recovery channel's recovered symbol is quasi-free feasible

Status: proved — a complete structured proof.

Area: Optimality of recovery channels (O8, Phases 3 and 3b) · Card type: theorem · Knowledge-base id: all-channels-symbol-feasible

Statement

For every normal unital CP even beta: A(BC) -> A(B), with xi_g = beta(a(g)) Omega, Kadison-Schwarz for a(g) and a*(g) gives Z >= Gram(xi) and 1 - Z >= Gram(S_B xi); projecting with Pi := P_- P_1 onto the one-particle charge -1 sector preserves these (the projections commute with Delta_B, Baumgaertel-Jurke-Lledo Cor. 5.4); D(S_B) on one-particle vectors is exactly the localised ones (BJL Prop. 5.3, closed graph Lemma 4.10). Hence the PROJECTED vector satisfies Pi xi_g = a(X* g) Omega with X* g in h_B (thm:range <1>1-<1>2), and (X, Z) in F with Q^beta = Q_rec(X, Z), Z = Q^beta_DD. The unprojected xi_g need not be a one-particle vector (only Pi xi_g is used), so xi_g = a(X* g) Omega itself is not claimed.

How to verify

rigor/optimality_all_channels.tex Sec. 3 with screenshots rigor/shots/BJL_p*.png; lattice test numerics/optimality_all/symbol_in_F_test.py. (the screenshots are page excerpts of the cited publications and are not redistributed; see THIRD-PARTY.md)

Evidence

• rigor/optimality_all_channels.tex — document, at thm:range
• refs/BJL_twisted_duality_math-ph-0204029.pdf — source, at Prop5.3; held privately, not redistributed

Related claims

• Second-order problem over quasi-free channels is a convex SDP — proved

Review

Review record: passed 2026-09-23 by REF-DEP-CFT-A3.

The referee's report and the card's event log stay in the private knowledge base; only the verdict, the date and the referee are published here.

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