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

Statement

Iteration 13 of PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — from two theorems to a coverage map.

1. The pbit island is FULLY closed. Ψ_θ∘Ψ_θ′ is exactly Bell-diagonal and kernel-LP separable — EB certified — for 87/87 pairs (grid + off-grid + both orders; the family commutes exactly, Choi diff ≤ 5.6e-17, unexplained — derivation seed). Closed forms in (t,t′) with the jewel w_ZI = (t−t′)²/(2(1+t)(1+t′)): the kernel that produced iters-6–11's "extremality" exists ONLY on the self-composition diagonal — mixed points are interior AND separable, so the family's EB-ness is not a boundary artifact.

2. Twisted-square LEMMA (corrected and generalized). For τ a linear involution of Z₂⁴ (symplecticity NOT needed; the pbit involution is the non-symplectic generalized transvection τ_{ZZ,IZ}): Φ is τ-covariant ⟺ C_Φ ∈ span{σ_{τ*(h)}⊗σ_h} (16 real dims, checkable by projection), and any composition of two τ-covariant channels is exactly a Pauli channel. 24 new non-Pauli twisted-covariant PPT channels constructed (extreme points of the 15-dim covariant spectrahedra, 3 classes); 18 have certified bound-entangled Choi — new n=4 BE channels outside the pbit family; all squares and within-class cross pairs certified EB. CZ corollary: cross-class pairs with symplectic product involution become Bell-diagonal after a one-sided CZ — 6/6 certified EB, including a non-DSE∘non-DSE pair.

3. DSE census: the generic sector is DSE. 30/30 random PPT channels and 6/6 named specimens are DSE (Theorem 1 applies); the pbit family is non-DSE at every θ; and BE-Choi ⇏ non-DSE (tw_adj_ext0 is BE yet DSE) — refines iter10. Everything non-DSE found so far is τ-covariant for some τ ≠ id: non-DSE appears to BE the twisted class.

4. The residual set — named, small, structured. Residual = both factors non-DSE AND no common τ (non-symplectic product): concrete members pbitH∘tw_symp_ext0 (both orders), tw_adj_ext5∘tw_symp_ext0. Not Bell-diagonal (off-diag 0.038–0.049) but the Choi lies EXACTLY (≤ 8.7e-18) in the shifted span of the non-symplectic pairing {ū⊗π(u)}. No entanglement certificate fires (consistent with PPT²); no separability certificate exists yet.

Facts

- Coverage after 13 iterations: DSE theorem (generic) ∪ kernel polytope (pbit island, squares+mixed) ∪ twisted classes + CZ corollary ∪ — residual: the named non-symplectic pairs. HONEST note: coverage is specimen-based; the general theorem still needs a proof that the classification is EXHAUSTIVE over all PPT pairs (iter15+ target), not just over our zoo. - Iteration-14 target: the twisted kernel polytope. Build the analogue of the Lagrangian-coset polytope for the twirl {ū⊗π(u)} with NON-symplectic π: identify the EB channels covariant for that pairing, their Chois' polytope, and LP-test the residual specimens. Closing them closes every named point of the zoo. - Semi futuri: symbolic 2-var proof of the (t,t′) closed forms; the commutation of the family (why?); LU-check of tw_symp_ext0 vs the Müller-Hermes cross state (CCNR = 1.5000 exactly — flagged, unchecked); exhaustiveness classification of τ-covariance among non-DSE PPT channels.

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — target; the map toward the full n=4 theorem. - 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 — Theorem 2, extended here to mixed pairs and new classes. - PPT² iteration 10 — the DSE theorem (first proven partial result); the key∘key hard core is DPS-2-extremal; the DPS-3 razor — Theorem 1; census refines its boundary. - PPT² iteration 11 — razor verdict: DPS-3 feasible at slack exactly 0; and the hard core is secretly a PAULI CHANNEL (Bell-diagonal, Z[√2]) — symmetry reading corrected (non-symplectic transvection). - Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live — 18 new BE channels enter the corpus. - 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 claim specimen-certified; honest pendings flagged. - Bisociation (Koestler 1964) — creative insight as the collision of two habitually unrelated frames of thought — symplectic/stabilizer algebra now the campaign's main engine.