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])

Statement

Iteration 11 of PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — the razor's verdict and two structural windfalls.

1. RAZOR VERDICT: FEASIBLE at joint slack exactly 0. DPS-3-with-cuts on pbit∘pbit admits an extension; PPT² survives its sharpest test. The composition is extremal at level 3 too — the level-2 extremality (PPT² iteration 10 — the DSE theorem (first proven partial result); the key∘key hard core is DPS-2-extremal; the DPS-3 razor) is a persistent geometric fact, not an artifact. The extension lives in the 816-dimensional (Pauli-lift × S₃) fixed algebra with marginal-pinned coordinates algebraic on sight (1/64, x²/32).

2. WINDFALL 1: Ψ∘Ψ is a two-qubit PAULI CHANNEL. C_Ψ carries a twisted Pauli symmetry (u, θ(u)) whose involution squares to id ⇒ the composed Choi is invariant under all 16 diagonal Paulis ⇒ exactly Bell-diagonal. Closed-form weights in x = √2−1 (verified 5.3e-16): Bell-projector weights {II: 4x², XI/XX/XZ/YY: x, IZ/ZZ: 2x², IX/IY/ZX/ZY: x², XY/YI/YX/YZ: x³, ZI: 0}, summing to 4 via the exact identity x+3x²+x³ = 1.

3. WINDFALL 2: the pbit absorbs transposition. C_Ψ^{T_B} = C_Ψ and C^{T_B} = C exactly (bitwise 0.0): Γ swaps the p₁X₁ blocks into p₂X₂ blocks identically. C has rank 15 with kernel |B_ZI⟩ — it sits ON the PSD and PPT boundaries simultaneously, which explains the everywhere-extremal behavior in one stroke.

CORRECTION (2026-07-04, iter12) — extremality deflated

The "extremal at every level" reading is WRONG as geometry: w_ZI = 0 means λ_min(C) = 0, so any extension X with Tr_{B′B″}X = C has λ_min(X) = 0 and the joint DPS slack is trivially ≤ 0 at every level — rank-deficiency, not deep structure. The razor's FEASIBLE verdict and all certificates stand; the *interpretation* of zero slack is downgraded. See 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, where the hard core is closed outright (Ψ_θ∘Ψ_θ proven EB for the whole family).

Facts

- The reformulation that changes the attack: "is pbit∘pbit EB?" = "is this specific Bell-diagonal 4⊗4 state (a MAGIC SIMPLEX point with Z[√2] weights) separable?" — a question with a developed literature (Baumgartner–Hiesmayr–Narnhofer line) and possibly an ANALYTIC answer. An explicit separable decomposition would prove PPT² ITSELF on the hard core — strictly stronger than antidegradability. - Iteration-12 targets (two tracks): (a) *X(x) family*: solve the reduced SDP over other pbit unitaries U (varying x), fit the 816 coordinates polynomially in x, verify symbolically → the first exact degree-2 certificate on the key-channel core; (b) *magic-simplex separability* (prior-art gate FIRST: is separability decided in that literature for this simplex region?): attempt an explicit separable decomposition of the Bell-diagonal composition — direct EB proof of the hard core. - Cut-reduction lemma recorded: the 14 PPT cuts of A|B|B′|B″ reduce to {T_A, T_{B″}, T_{B′B″}} via complements + S₃ orbit equivalence. - Semi futuri: whether EVERY pbit-family composition is a Pauli channel (twisted-symmetry algebra suggests yes for the whole HPHH family); the kernel vector |B_ZI⟩ as the algebraic handle on the boundary.

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — target; hard core now reduced to a magic-simplex point. - PPT² iteration 10 — the DSE theorem (first proven partial result); the key∘key hard core is DPS-2-extremal; the DPS-3 razor — DSE covers generic channels; this attacks the rest. - PPT² iteration 8 — pbit-seeded hunt: one composition destroys the key structure; even the necessary condition fails on the best possible seeds — the pbit construction whose algebra pays off here. - Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live / Peres–Horodecki (PPT) criterion — partial transpose positivity is necessary for separability, and sufficient only in 2×2 and 2×3 — the boundary geometry. - Bisociation (Koestler 1964) — creative insight as the collision of two habitually unrelated frames of thought — fourth domain enters: Pauli-channel/magic-simplex theory. - Dual-oracle verification — certify every claimed object by two independent oracles, count-certify exhaustive runs, and scan all prior-art branches before claiming novelty — three-solver agreement + numpy two-sided pin + exact anchors.