PPT² iteration 9 — adversarial gap maximization pins at the solver floor: the counterexample front is exhausted; campaign commits to the proof

Statement

Iteration 9 of PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — the counterexample front's last stand. Adversarial optimization of the antidegradability gap over the PPT∩CPTP cone: the gap is pinned at the numerical floor everywhere (best 1.5e-06 vs threshold 1e-4, consistent with solver-noise exploitation, not signal).

Cumulative exhaustion of the counterexample front (all dual-oracle validated): 112 diverse compositions (iter6a/6a2) + 6 CCNR-ceiling pairs (iter5) + 11 pbit-seeded pairs both orders (iter8) + adversarial climb from 5 warm starts (this page) — zero live directions at any level of the criterion hierarchy.

Standing conjecture, now with optimizer status

every composition of two PPT channels is antidegradable ("one PPT hop kills forward key"). Combined with the criterion-hierarchy blindness theorems, the evidence that PPT² is TRUE is now overwhelming at n=4. The campaign commits to the PROOF front.

Facts

- Side observation worth a lemma attempt: the PPT∩CPTP projection maps (1−ε)·id + ε·noise to id-fidelity EXACTLY 1/4 = 1/n for all ε — i.e. the cone's nearest points to the identity have completely-depolarizing overlap. A cheap-to-check spectral statement (likely reduction-criterion-adjacent: λ_max(Choi) ≤ 1/n caps id-fidelity at 1/n... check: F_id = ⟨Ω|ρ_Choi|Ω⟩ ≤ λ_max ≤ 1/n ✓ — actually a ONE-LINE consequence of Reduction criterion — separable (and PPT) states obey ρ_A⊗I ⪰ ρ; the eigenvalue cap that killed our CCNR fitness; record it as such, verified numerically here). - Iteration-10 target (the proof front opens): READ THE DUAL. The antidegradability-gap SDP has value 0 on every PPT∘PPT — so its DUAL optimal solution is a certificate that exists universally. Extract dual certificates at several pairs (the SDP machinery already solves them), look for universal structure (a canonical degrading map D as function of (Ψ,Φ); note iter8's Petz was 1.2e-02 wrong — the true D is near-Petz but corrected). In parallel: adversarial prior-art scan on "PPT channel antidegradable", "sufficiently noisy channels antidegradable", "degradable PPT compositions" — the λ_max ≤ 1/n cap suggests a spectral antidegradability criterion may already exist or be provable. - Proof ladder (unchanged): antidegradability of PPT∘PPT (first summit, genuine partial result) → PPT cuts (full DPS-2 blindness) → higher DPS levels → EB (the conjecture). Untried heavy tools if the dual is opaque: SDP+symmetry transplant (arXiv:2512.06551), second-order free probability, Positivstellensatz (var/openproblems/ppt2-scan.md census).

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — target; counterexample front closed honestly, proof front open. - PPT² iteration 8 — pbit-seeded hunt: one composition destroys the key structure; even the necessary condition fails on the best possible seeds — the seeds this climb started from. - PPT² iteration 6 — DPS-2 blindness ⟺ antidegradability: key-repeater bridge found; all four structured constructions honestly refuted — the frame being proven. - Reduction criterion — separable (and PPT) states obey ρ_A⊗I ⪰ ρ; the eigenvalue cap that killed our CCNR fitness — yields the id-fidelity ≤ 1/n one-liner observed here. - Simulated-annealing extremal search — Metropolis local search for rare, structured witness/counterexample objects — the technique's last act on this front. - Dual-oracle verification — certify every claimed object by two independent oracles, count-certify exhaustive runs, and scan all prior-art branches before claiming novelty — floors quoted; optimizer-noise reading applied.