PPT² iteration 14 — the twisted kernel polytope closes the residual: the ENTIRE named zoo is certified EB; the criterion is COMPLETE on its algebra

Statement

Iteration 14 of PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — the residual set is closed, and the tool that closed it is stronger than its predecessor.

1. All 6 residual pairs are certified EB (both orders of pbitH∘tw_symp, tw_adj∘tw_symp, tw_pbit∘tw_symp): explicit separable decompositions, verified entrywise in numpy at ~1e-16. Column generation (a see-saw separation oracle making the method complete) was implemented and never needed.

2. The twisted kernel polytope is COMPLETE on its algebra. The residual pairings are non-symplectic transvections with fixed algebra A_π ≅ M₂(C)^⊕4 (non-abelian): no local unitary maps it to the Bell span — the "rotate and reuse" escape is provably closed. But A_π is the fixed space of a twirl by 16 LOCAL unitaries, hence SEP∩A_π = conv{twirled products} EXACTLY: membership in the vertex polytope is a decision procedure, not merely a sufficient test. (The Bell kernel polytope never had this.)

3. Literature anchor (honest): tw_symp_ext0's Choi IS the Müller-Hermes cross state up to explicit local unitaries — iter13's count corrects to 17 new BE channels + 1 rediscovery, and the closed residual includes the certified statement "pbit ∘ (unitarily dressed cross-state channel) is EB" about a literature-known object.

4. Sharpest provable statement: every composition of every pair from the named non-DSE inventory {pbit family, tw_pbit_ext3, tw_adj_ext5, tw_symp_ext0 ≅ cross} is certified EB — PPT² holds on the ENTIRE named zoo with no reliance on hierarchy silence. Combined with the DSE theorem (generic sector): what remains for n=4 is populations, not specimens.

Facts

- Bonus numerology flagged UNPROVEN: the pbitH∘tw_symp certificate appears exact in x=√2−1 (m ∈ {1, x/3, x²/3, −√2x/3}; λ ∈ {2x/3, (1−2x²)/6, x²/3, x²/6, 1/6}) — symbolic pass pending. - Iteration-15 target — EXHAUSTIVENESS (the final climb): (a) hunt for a non-DSE PPT channel that is NOT τ-covariant for any linear involution τ (adversarial sampling of extreme PPT channels + both tests); if none exists, conjecture-then-prove "non-DSE ⇒ twisted"; (b) classify involutions/pairings of Z₂⁴ up to local Clifford (finite GF(2) computation) — the twisted polytope construction then covers every class; (c) compose: DSE ∪ (twisted × twisted via complete polytopes) — if (a) holds, this is a DECISION PROCEDURE for all PPT pairs at n=4, and the general theorem reduces to finitely many verifiable class statements. - Also queued: 2-var Bernstein proof of the (t,t′) forms; the family commutation derivation; symbolic Q(√2)/3 pass; write-up consolidation (2 theorems + zoo + completeness property = paper shape).

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — target; the road now runs through exhaustiveness. - PPT² iteration 13 — coverage explodes: 87/87 mixed pbit pairs EB; twisted-square lemma + 24 new channels (18 new BE); the residual set is named — the residual this closes; BE count corrected. - PPT² iteration 12 — THEOREM: the entire HPHH pbit family satisfies PPT² exactly (Ψ_θ∘Ψ_θ is EB, explicit algebraic certificates); the 'extremality' of iters 10–11 honestly deflated — the π=id ancestor of the polytope. - PPT² iteration 10 — the DSE theorem (first proven partial result); the key∘key hard core is DPS-2-extremal; the DPS-3 razor — the generic-sector companion theorem. - Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live — cross-state anchor; 17 new BE channels. - Dual-oracle verification — certify every claimed object by two independent oracles, count-certify exhaustive runs, and scan all prior-art branches before claiming novelty — every certificate numpy-verified; LU claim settled with explicit unitaries. - Bisociation (Koestler 1964) — creative insight as the collision of two habitually unrelated frames of thought — non-abelian fixed algebras: representation theory joins the toolkit.