PPT² iteration 7 — reading the SDP's extension: fat interior, mid-rank extremals, coherence-distance hypothesis REFUTED

Statement

Iteration 7 of PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4: reverse-engineer the DPS-2 extension the SDP finds at n=3 (where it provably exists but all our constructions miss it, PPT² iteration 6 — DPS-2 blindness ⟺ antidegradability: key-repeater bridge found; all four structured constructions honestly refuted). Three findings:

1. The extension set has fat interior. Analytic-center solves give strictly positive slack: t* = 5.96e-03 (random pair), 3.26e-02 (ghost∘ghost). DPS-2 blindness at n=3 is deep, not marginal — and counterintuitively the bound-entangled pair has the MORE robust extension set.

2. Extremal extensions carry no readable structure. Six random-functional extremal solves per family: ranks 19–21 of 27 (mid-rank, not the low-rank structured points one could generalize), factorized-family distance ~0.27–0.31 (ghost).

3. Coherence-distance hypothesis REFUTED. Control battery on the PINCHED composition (M-coherences killed ⇒ manifestly EB): extremal extensions are FARTHER from the (id⊗Φ⊗Φ)-image (median 0.398 vs 0.287 for the true ghost). The geometric distance from the factorized family is NOT the signature of the M-coherence obstruction — extremal extensions live outside that family generically, coherences or not. Iter6's "coherences are the whole content" remains true at the *feasibility* level (route A), but is invisible in extension geometry.

Facts

- Consequence: "read the SDP" yields no constructive pattern → per the iter6 decision rule, weight shifts to Track A: the pbit-seeded counterexample hunt at n=4 (iteration-8 target): build PPT channels from the 4⊗4 private bound-entangled states (quant-ph/0506203 — exact matrices to be FETCHED, not recalled, per the prior-art rule), fitness = antidegradability-gap SDP (positivity NECESSARY for any counterexample; cheaper than full DPS-2). - Blindness-proof side (Track B) parked with honest status: all constructive routes dead; remaining routes are the bilinear family or non-constructive arguments (duality/capacity). Revisit if the hunt surfaces structure. - Meta: this iteration's metrics were reconstructed from saved .npz after a session restart — the wiki+log+artifacts-on-disk pattern IS the system's persistence layer working as designed.

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — target. - PPT² iteration 6 — DPS-2 blindness ⟺ antidegradability: key-repeater bridge found; all four structured constructions honestly refuted — the frame this executes. - Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live — pbits (its key-carrying form) become the hunt seeds. - 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 notes above (trivial-fit flag, n=3 guarantees).