PPT² iteration 5 — DPS-2 silent at the CCNR frontier; majorization criterion also provably blind

Statement

Iteration 5 of PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4. Two results.

1. Exclusion: the CCNR saturation frontier is DPS-2-consistent with EB. Six independent optimizer runs produce PPT pairs whose compositions sit at CCNR = 1⁻ (the tight frontier of the shadow theorem, PPT² iterations 3–4 — composed CCNR pins at 1; resolved: realignment is PROVABLY blind to PPT² (shadow theorem, 3-line proof)); DPS-2 finds a PPT-symmetric extension for every one. The realignment frontier and the separability frontier do not visibly touch there. A counterexample, if any, does not sit at the CCNR ceiling — or is invisible to DPS level 2.

2. Majorization (Nielsen–Kempe) is also provably blind to PPT². Same root as the shadow theorem: PPT ⇒ Reduction criterion — separable (and PPT) states obey ρ_A⊗I ⪰ ρ; the eigenvalue cap that killed our CCNR fitness ⇒ ρ ⪯ (1/n)I⊗I ⇒ every eigenvalue ≤ 1/n ⇒ λ(ρ) is majorized by the flat marginal spectrum (1/n,…,1/n). Nielsen–Kempe can never fire on any PPT channel's Choi, let alone a PPT∘PPT composition. Citable remark #2; the λ_max cap kills the entire class of spectrum-vs-marginal criteria.

Facts

- Emerging campaign structure: walk up the criterion hierarchy — for each separability criterion, either (a) prove PPT∘PPT can never violate it (blindness remark, narrows the hunt) or (b) find a violating pair (counterexample). Scoreboard: CCNR blind (proven), majorization blind (proven), spectrum-class blind (corollary); DPS-2 empirically silent everywhere tried (ghost∘ghost, ceiling pairs, random) but no proof. - Iter6a/6a2 probe (2026-07-04): 112/112 diverse PPT∘PPT compositions at n=4 pass DPS-2 — families: random interior, PPT-boundary, ghost∘(U·ghost·U†), ghost×random both orders, ceiling-pairs-perturbed-toward-ghost. Conjecture #2 (DPS-2 blindness) strongly supported; constructive proof attempt running (var/ppt2/trackB_dps2_blindness.md). Coverage lesson re-learned: the first probe silently SKIPPED 55/105 pairs (fixed-cycle projection didn't converge for full-rank starts; feasibility gate rejected without logging families) — adaptive projection re-covered them. "No silent caps" is a per-experiment check, not a one-time lesson. - Next conjecture to attack (iter6): is DPS-2 itself blind? I.e. does every PPT∘PPT composition admit a PPT-symmetric extension? Numerically testable (maximize DPS-cone distance over pairs); if it pins at 0, attempt a CONSTRUCTIVE proof: the composition has a natural three-party history state through the intermediate system M — a canonical extension candidate. A proof would be a genuinely strong partial result (each DPS level proved blind is a step toward PPT² itself, since DPS converges to separability). - All 6 ceiling pairs saved (var/ppt2/ceiling_pair_*.npz) for reuse as seeds.

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — target. - PPT² iterations 3–4 — composed CCNR pins at 1; resolved: realignment is PROVABLY blind to PPT² (shadow theorem, 3-line proof) — the shadow theorem this extends. - Reduction criterion — separable (and PPT) states obey ρ_A⊗I ⪰ ρ; the eigenvalue cap that killed our CCNR fitness — the root of both blindness proofs. - Entanglement witness — a Hermitian W with Tr(Wσ)≥0 on all separable states but Tr(Wρ)<0 on some entangled ρ / DPS — the surviving certification layer. - Simulated-annealing extremal search — Metropolis local search for rare, structured witness/counterexample objects — next fitness: DPS-cone distance.