PPT² iteration 10 — the DSE theorem (first proven partial result); the key∘key hard core is DPS-2-extremal; the DPS-3 razor

Statement

Iteration 10 of PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — the proof front's first summit.

1. THEOREM (DSE ⇒ composition antidegradability) — proven, new. Call a PPT channel Φ DSE if its Choi admits a BB′-symmetric extension decomposable across the input:(outputs) cut: W = P + Q^{T_M}, P,Q ⪰ 0, Tr_{B′}W = C_Φ. Then for every PPT pair (Ψ,Φ) with at least one DSE factor, Φ∘Ψ is antidegradable — hence zero forward private capacity. Proof: 4 lines of link positivity (each PPT-ness used exactly once; mirror version uses Φ's twice). Covariance-free, SDP-checkable per channel, one certificate covers ALL PPT partners simultaneously. DSE holds for every generic specimen tested (ghost, idlean, randoms, EB anchor — numpy-certified); prior-art scan: appears nowhere.

2. The hard core is exactly key∘key. DSE fails precisely on the pbit channel; kernel-relaxed J1 fails; hybrid H rescues pbit∘rand but NOT pbit∘pbit. The private-capacity obstruction has now appeared four times (Lemma L, Lemma D, DSE, H) — it IS the shape of the difficulty.

3. Extremality discovery. DPS-2-with-cuts extensions of pbit compositions exist (PPT² survives on the hard core) but with joint slack EXACTLY zero (t* ≈ −1e-12 vs +2.8e-02 for generic pairs): pbit compositions are extremal points of the DPS-2-with-cuts body. This explains at once why every interior-type certificate misses them and why iter9's adversarial climb never crossed the boundary — the PPT∘PPT image runs ALONG it.

4. Dead ends honestly closed: no universal degrader exists (optimal face diameter ~2.5–3.0 relative); Petz cannot be corrected (provably optimal within the whole Θ∘P family at rel 6.1e-02 — the family is too small); the gap-SDP dual vanishes at optimum (no certificate there); the spectral route λ_max ≤ 1/n is void (every PPT channel passes it, including non-antidegradable Horodecki channels).

Facts

- Iteration-11 target — THE RAZOR: DPS-3-with-cuts on pbit∘pbit. At zero DPS-2 slack this test cuts cleanly: INFEASIBLE ⇒ the composed Choi is NOT separable ⇒ PPT² is FALSE (counterexample certified by the hierarchy); FEASIBLE at zero slack ⇒ reconstruct the boundary-rigid extension exactly from the pbit ±-block algebra (symmetry transplant) — the missing degree-2 certificate and the template for the general proof. Honesty anchor: same test at n=3 must be feasible (theorem). - Publication shape already in hand: DSE theorem + certified examples + the honest boundary (key∘key core) + 129-composition evidence (iters 6–9). - Semi futuri: degree-2 decomposable certificates (X = Σ link(C⊗C, W_i)); fresh-agent tools to fold in (k-extendibility ladder — one level per PPT factor; explicit Γ-identity J(Φ∘Ψ)^Γ = (id⊗(T∘Φ))(J(Ψ))); bosonic-symmetric reduction for DPS-3 scale.

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — target; first proven structural theorem at unrestricted n≥4. - PPT² iteration 9 — adversarial gap maximization pins at the solver floor: the counterexample front is exhausted; campaign commits to the proof — the boundary-running now explained. - PPT² iteration 8 — pbit-seeded hunt: one composition destroys the key structure; even the necessary condition fails on the best possible seeds — the key channels that define the core. - PPT² iteration 6 — DPS-2 blindness ⟺ antidegradability: key-repeater bridge found; all four structured constructions honestly refuted — the dictionary this builds on. - Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live / Entanglement witness — a Hermitian W with Tr(Wσ)≥0 on all separable states but Tr(Wρ)<0 on some entangled ρ — objects and duals. - Dual-oracle verification — certify every claimed object by two independent oracles, count-certify exhaustive runs, and scan all prior-art branches before claiming novelty — proof re-verified independently; coverage gap #4 caught. - Bisociation (Koestler 1964) — creative insight as the collision of two habitually unrelated frames of thought — the certificate ladder is link-algebra ⨯ convex geometry ⨯ capacity theory.