The constant 1/5 for PPT composites through a qutrit: barriers, and a smaller constant for PPT channels on M₃

The PPT2 conjecture asserts that composites of PPT (completely positive and completely copositive) maps are entanglement breaking. For PPT maps Φ: Md1 → M3 and Ψ: M3 → Md2 let W be the Choi matrix of Ψ ∘ Φ. With the theorem that composites of PPT maps through a qubit are entanglement breaking (the qubit theorem), it is known that W + (1/5) Φ*(I)⊤ ⊗ Ψ(I) is separable, while PPT2 through a qutrit is the same statement with 1/5 replaced by 0. We ask how far 1/5 can be lowered. Certificates built from the qubit theorem independently of the pair cannot lower it, and for d1, d2 ≥ 3 with max(d1, d2) ≥ 4 neither can averages of certificates that apply maps separately to the input, middle and output spaces, even with PPT2 on M3 added. The least λ for which J(Φ) + λ Φ*(I)⊤ ⊗ I is a sum of PPT operators supported on ℂd1 ⊗ V for planes V is the one-sided constant λQD(Φ) ≤ 1/5; by the qubit theorem it is valid for every partner. It is below 1/5 when one compression of the middle qutrit to a qubit is strictly PPT of full rank, and its largest value Λ(d) over d1 = d is below 1/5 unless some extreme PPT channel is rigid, that is, has λQD = 1/5. We prove, without the qubit theorem, that no PPT channel M3 → M3 is rigid, so that some c < 1/5 makes ρ + c I ⊗ I separable for the Choi matrix ρ of every PPT channel M3 → M3. With the qubit theorem, W + c Φ*(I)⊤ ⊗ Ψ(I) is then separable for all PPT maps Φ: M3 → M3 and Ψ: M3 → Md2, and exact rational certificates beat 1/5 at explicit factors with d1 = 4. The exact computations used in the proofs are also checked in Lean 4. This paper develops Section 10, Appendix A and supplement S12-S13 of version 3 of "Diagonal-unitary twirls of composites of PPT maps through a qutrit" (doi:10.5281/zenodo.23675449) with a new introduction and preliminaries; its supplement S1-S2 is S12-S13 there. The statements and proofs are those of that version, renumbered, with references to the companion paper, version 4 of that record (doi:10.5281/zenodo.23680469). Programs, certificates and Lean checks are in the deposit, version 4 (doi:10.5281/zenodo.23680464). Dagger k of this paper is dagger k+24 in version 3 and in the deposit's folders, map files, replay record and Lean targets; the deposit's README uses this paper's numbering. Mathematics Subject Classification (2020): 81P40; 15A69, 47L07.