PPT² iteration 8 — pbit-seeded hunt: one composition destroys the key structure; even the necessary condition fails on the best possible seeds

Statement

Iteration 8 of PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — the sharpest counterexample hunt so far, executing iter6's obstruction analysis: seed from the 4⊗4 private bound entangled states (Horodecki–Pankowski–Horodecki–Horodecki pbits), the only known PPT objects engineered to carry key.

Result: every pbit-seeded PPT∘PPT composition is antidegradable to numerical precision (all sdp-gaps ≤ 7.7e-07 against an honest floor of ~1e-07 and a 1e-04 threshold). One composition destroys the key structure: the pbit channel Ψ (Choi = key-carrying bound-entangled state, CCNR 1.086, DPS-2-certified, forward key rate ≥ 0.0213) composes with EVERY partner tried — the n=4 ghost, identity-leaning PPT-frontier channels at four ε values, and itself — into an antidegradable channel, both orders. Composed CCNR collapses to 0.42–0.69.

Even the NECESSARY condition for a PPT²/DPS-2 counterexample fails on the best-engineered seeds. Combined with 118 prior compositions, this is the strongest evidence yet that the conjecture is TRUE — and direct numerical support for the channel-level statement "one PPT hop kills forward key" (PPT² iteration 6 — DPS-2 blindness ⟺ antidegradability: key-repeater bridge found; all four structured constructions honestly refuted).

Facts

- Honest limits: Hadamard-pbit family only; effectively 3 distinct partners (the four idlean_ε projections collapsed to one CCNR=1.000000 frontier point — itself an interesting attractor); gap≈0 = antidegradable, which is consistent with but weaker than EB. - Iteration-9 target (the seed): adversarial gap maximization. Stop testing fixed pairs; OPTIMIZE (Φ,Ψ) directly against the antidegradability gap over the PPT∩CPTP cone at n=4 (perturbation hill-climb, gap SDP as fitness, pbit+ghost warm starts, the iter3/4b projection machinery). Two outcomes: gap escapes 0 → counterexample direction; pins at 0 across the cone → "every PPT∘PPT is antidegradable" earns conjecture-with-optimizer status and the campaign commits to the PROOF front at full strength (the untried angles from var/openproblems/ppt2-scan.md: SDP+symmetry transplant arXiv:2512.06551, second-order free probability, Positivstellensatz — as a multi-agent derivation panel with adversarial verification per lemma). - Semi futuri: paper's §V d>2/non-unitary pbit families as alternative seeds; gap-SDP dual solution as gradient (faster than finite differences); the CCNR=1.000000 attractor point deserves a look (why does PPT-projection of identity-leaning channels land exactly ON the CCNR frontier?).

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — target; evidence for TRUE strengthened again. - PPT² iteration 6 — DPS-2 blindness ⟺ antidegradability: key-repeater bridge found; all four structured constructions honestly refuted — the necessary-condition frame executed here. - PPT² iteration 7 — reading the SDP's extension: fat interior, mid-rank extremals, coherence-distance hypothesis REFUTED — why constructive reading was abandoned. - Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live — pbits: its key-carrying form, now measured to LOSE key in one hop. - Dual-oracle verification — certify every claimed object by two independent oracles, count-certify exhaustive runs, and scan all prior-art branches before claiming novelty — honesty floors quoted throughout.