PPT² iterations 3–4 — composed CCNR pins at 1; resolved: realignment is PROVABLY blind to PPT² (shadow theorem, 3-line proof)

Statement

Iteration 3 of PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4: direct optimization of the algebraic objective from PPT² iteration 2 — first n=4 ghost channel; two no-gos (padding, adjoint diagonal); the hunt becomes transfer-matrix alignment (composed CCNR = ||T_Ψ·T_Φ||₁/n) over PAIRS of PPT channels at n=4.

Finding: the optimizer converges to composed CCNR → 1 from BELOW, from every seed, and never crosses. Best feasible value 0.999928 (random seed; ghost seed 0.9892). The maximizers have near-flat transfer spectra σ ≈ 1/n (σ_max = 0.2551 vs flat 0.25). Without the PPT constraint the supremum is n=4 (identity pair, T unitary, all σ = 1); the PPT wall crushes exactly to ≈1.

Shadow THEOREM (resolved 2026-07-04 by the prior-art gate, not by us)

for any two PPT channels on M_n, ||T_Ψ·T_Φ||₁ ≤ n — *the composition of two PPT channels always satisfies the realignment (CCNR) criterion*. Three-line proof from known facts (found adversarially by the scan agent, chain re-verified):

1. PPT ⇒ reduction criterion (Reduction criterion — separable (and PPT) states obey ρ_A⊗I ⪰ ρ; the eigenvalue cap that killed our CCNR fitness, Horodecki–Horodecki 1999, quant-ph/9708015); a channel Choi has A-marginal I/n, so λ_max(ρ) ≤ 1/n, hence ‖ρ‖₂ ≤ 1/√n and ‖T‖₂ = n‖R(ρ)‖₂ ≤ √n (reshuffle preserves the Frobenius norm). 2. Cauchy–Schwarz for Schatten norms: ‖T_Ψ T_Φ‖₁ ≤ ‖T_Ψ‖₂‖T_Φ‖₂ ≤ n. ∎

Corollary: a SINGLE PPT channel has CCNR ≤ √n (tiles ghost 1.115 ≤ √3 ✓).

- Realignment is provably BLIND to PPT²: no counterexample can ever be certified via CCNR, in any dimension — explains the failure of blind numerical searches (Jin 2020, cited in PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4) and kills the CCNR fitness for Track A. The hunt must use finer certificates (DPS symmetric extension, non-decomposable witnesses). - Status per the prior-art scan (var/openproblems/shadow-conjecture-scan.md): expert-obvious once seen, but apparently UNWRITTEN — full-text search of the whole PPT² corpus (1807.01266, 1807.03636, KMP, 1803.00143, 2011.03809, 2110.11825, 2010.15554, Cariello's realignment work…) finds zero mentions of realignment × composition. Honest framing: a citable REMARK ("mild support for PPT²"), not a claimable result. - Our 24/24-seed pinning at 1⁻ is the SATURATION of this inequality — the bound is tight (sup = 1 over PPT pairs, approached from below).

Facts

- Algebra available for a proof attempt (Track B): for C ⪰ 0, R(C) = Σ_k λ_k vec(V_k)vec(V_k*)^T with Σ λ_k V_k†V_k = I (TP); the PPT constraint enters as R(C^Γ) = R(C)·S (S = swap permutation), so C^Γ ⪰ 0 gives a SECOND vec-decomposition of R(C)S. Question: do the two decompositions jointly force ||R(C₂)R(C₁)||₁ ≤ n? - Hardened evidence (iter4b): 24/24 independent random seeds across n=4 AND n=5 pin at 1⁻ (best 0.99998893); the ceiling persists across dimensions and appears TIGHT (sup = 1, approached from below). A projection artifact is now unlikely (same projector reaches 0.9999+, i.e. it does not repel the boundary); remaining caveat: all methods are local ascent. - Methodological lesson recorded: soft penalty blending (iter4a) contracts toward the feasible interior and silently caps the objective (0.547) — a handicapped optimizer 'confirming' a ceiling is void evidence. Always verify the method can reach the known frontier before trusting its failures. - Either outcome is progress: break ⇒ counterexample direction reopens with a concrete mechanism; pin ⇒ the shadow conjecture gains teeth and Track B becomes the priority.

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — the target; this is its CCNR shadow. - PPT² iteration 2 — first n=4 ghost channel; two no-gos (padding, adjoint diagonal); the hunt becomes transfer-matrix alignment — the reformulation this optimizes. - PPT² iteration 1 — WH-mixture exclusion at n=4; first bound-entangled CHANNEL built and composed (n=3 landscape) — seeds. - Peres–Horodecki (PPT) criterion — partial transpose positivity is necessary for separability, and sufficient only in 2×2 and 2×3 — CCNR's sibling criterion. - Simulated-annealing extremal search — Metropolis local search for rare, structured witness/counterexample objects — the search track, now targeted at breaking a specific ceiling. - 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 winner re-certified end-to-end.