The Schmidt number of PPT states in 3 × 4 dimensions: partial results and the open case

Status. Partial results only. The note studies, without answering it, the question of Yang, Leung and Tang (Linear Algebra Appl. 503 (2016), Question 3.9): is there an indecomposable 2-positive linear map M_3 → M_4? Dually: can a PPT state of a qutrit and a ququart (C^3 ⊗ C^4) have Schmidt number three? Proved. (1) A 2-block-positive operator on C^3 ⊗ C^4 that vanishes on one nonzero product vector is decomposable, so a witness for Schmidt number three among PPT states would have to be strictly positive on all product vectors. (2) For min(M,N) ≥ 2, the range of every PPT state on C^M ⊗ C^N is spanned by its vectors of Schmidt rank at most min(M,N) − 1; hence no range-only test that is valid for all positive operators can show full Schmidt number for a PPT state. (3) With a lemma of Chen, Yang and Tang: every non-extreme PPT state of birank (6,6) on C^3 ⊗ C^4, and every positive operator on C^3 ⊗ C^N of operator Schmidt rank at most seven, has a separable compression to a two-dimensional local subspace, hence Schmidt number at most two. (4) The maps proposed by C. A. Winda (J. Linear Topol. Algebra 14 (2025) 129–136) as indecomposable 2-positive maps M_3 → M_4 are not 2-positive. Open. A birank-(6,6) counterexample would have to be an extreme edge state with product-free kernels and operator Schmidt rank eight or nine. Numerically, every state tested (42 states covering all ten relevant birank classes) has a separable two-dimensional compression; the note describes where the proof attempts for birank (6,6) stopped. Files. ppt-3x4.pdf is the note. ppt-3x4-supplement.zip contains the statements, proofs and blind reviews of the results as written, the exact computation for the Winda maps, the numerical data and scripts (including exact rational birank-(6,6) edge states with separable compressions), the record of failed proof routes, and the unreviewed working files. "python check.py" (NumPy and SymPy) verifies the hashes and repeats the exact and numerical checks (a few seconds on the laptop used to assemble the archive). The written proofs have not yet been verified line by line by a human reviewer. AI tools. The author chose the question and the statements to attack, set the working rules (blind review of every result by the other model family; fixed stopping rules), had the published Winda maps checked and supplied the paper for that, decided when to stop, and is responsible for the content. The research was carried out on 4–5 November 2026 by an AI research system built on Claude Opus 5.5 (Anthropic) and GPT-6 Astra (OpenAI); the text was revised on 9 November 2026. Each proof was written by one of the two models and accepted by a blind review from the other; the Winda computation was done by both independently; Claude Opus 5.5 drafted the manuscript and GPT-6 Astra reviewed it blind. Section IX of the note has the details.