OpenAI · 126-01 · 2026-10-05 · Theoretical computer science
Exponential PSD rank of positively shifted matching matrices
preprints/Exponential-PSD-rank-of-positively-shifted-matching-matrices-October-5-2026
Family 126. Exponential semidefinite complexity of perfect matching.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves that every exact semidefinite lift of the perfect matching polytope has exponential size, answering Rothvoss's polynomial-size lift question negatively. The bound holds even for the positive semidefinite rank of its odd-cut slack matrix after any fixed shift , allowing arbitrary real positive semidefinite factors.
For every fixed , the matrix indexed by odd vertex sets U and perfect matchings M of Kn, with entries , has real positive semidefinite rank as even n tends to infinity. Here is the edge cut of U. Consequently, every exact semidefinite lift of the perfect matching polytope has exponential size.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 126: Exponential semidefinite complexity of perfect matching. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.