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

Statement

THEOREM 2 of the campaign (new). For every member Ψ_θ of the HPHH private-bit channel family (quant-ph/0506203; bound-entangled, key-carrying, PPT), the self-composition Ψ_θ∘Ψ_θ is entanglement-breaking — PPT² holds exactly and unconditionally on its hardest known n=4 test family. Certificates are fully explicit and algebraic:

- Hadamard point (x=√2−1): C = Σ_{c=1}^{8} λ_c Σ_{s,t=±1} conj(E_st) ⊗ h_c E_st h_c†, λ ∈ Z[√2], verified symbolically; every identity reduces to x²+2x−1 = 0. - Whole family: exact Bell weights in t = tan(θ/2) (w_XX=w_YY=t, w_XI=w_XZ=(1−t)/(1+t), w_II=(1+t²)²/(1+t)², w_ZI ≡ 0, …) and a single global 14-coset decomposition with λ_c(t) = Λ_c(t)/(1+t)², Λ_c Bernstein-nonnegative — an exact identity in Q[t]: a separable curve in the kernel polytope joining two stabilizer dephasings through the Hadamard point.

Mechanism: each Lagrangian coset of Z₂⁴ gives a dephasing (measure-and-prepare, hence EB) channel whose Choi is an explicitly separable stabilizer state; the composed Choi is a convex combination of 8 (resp. 14) of them.

Honest corrections and guardrails

- Deflation of iters 10–11 "extremality": w_ZI = 0 ⇒ λ_min(C) = 0 ⇒ the joint DPS slack satisfies t* ≤ 0 at EVERY level trivially. The zero-slack "extremality discovery" was rank-deficiency, not deep geometry; the hierarchy was silent because C sits on the PSD boundary itself. (Recorded also as a correction note on 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]).) - The x′-substitution family conjecture is REFUTED (Σ≠4 off Hadamard; θ ↔ π/2−θ is a local-Clifford weight permutation) — replaced by the true t-parametrization above. - The kernel-polytope criterion is not vacuous: it rejects the literature bound-entangled cross state with an explicit Farkas functional.

Facts

- Campaign scoreboard after 12 iterations: Theorem 1 (DSE) — generic PPT channels compose antidegradably; Theorem 2 (this page) — the key-channel family satisfies PPT² exactly. Both new per adversarial prior-art gates. - Iteration-13 targets: (a) mixed compositions Ψ_θ∘Ψ_θ′ (the family's off-diagonal — same kernel-polytope LP, now 2-parameter); (b) classify "which PPT channels have Pauli-diagonal (Bell-diagonal Choi) square" — the twisted-symmetry mechanism as a bridge from the pbit island toward general n=4; (c) compose the two theorems: does DSE ∪ kernel-polytope ∪ (what else?) cover ALL PPT pairs at n=4? Map the residual set. - Fourth domain consolidated (Bisociation (Koestler 1964) — creative insight as the collision of two habitually unrelated frames of thought): stabilizer/symplectic algebra over Z₂⁴ (Lagrangians, cosets, Cliffords) — imported via the magic-simplex bridge, now a workhorse.

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — target; first exact PPT² theorem on a previously-unproven channel class. - 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]) — the Pauli-channel discovery this exploits (and the extremality claim this corrects). - 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, the generic-sector companion. - PPT² iteration 8 — pbit-seeded hunt: one composition destroys the key structure; even the necessary condition fails on the best possible seeds — the family's construction. - Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live — key channels: bound entangled, yet their square is EB. - Dual-oracle verification — certify every claimed object by two independent oracles, count-certify exhaustive runs, and scan all prior-art branches before claiming novelty — symbolic verification + literature cross-control + honest deflation. - Bisociation (Koestler 1964) — creative insight as the collision of two habitually unrelated frames of thought — graph theory → QI → capacity → stabilizer/symplectic algebra.