Does the perfect matching polytope have a polynomial-size semidefinite lift?

Theorem 1.1: there is $c>0$ such that for every fixed $0<\rho<1$ the odd-cut matrix with entries $|M\cap\delta(U)|-1+\rho$ has real PSD rank at least $2^{cn}$ for large even $n$ , with no symmetry or precision restriction on the factors. Corollary 1.2: every exact semidefinite lift of the perfect matching polytope has size $2^{\Omega(n)}$ . The method follows Lee-Raghavendra-Steurer with a Grigoriev-type parity functional and new averaging lemmas. The endpoint $\rho=1$ has a polynomial factorization, and approximate lifts are not excluded (Kaniewski-Lee-de Wolf give subexponential approximations).

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Oct 5, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Does the perfect matching polytope have a polynomial-size semidefinite lift? · SDP lifts of the matching polytope

Confidence: Not scored

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VibeMathed record: Does the perfect matching polytope have a polynomial-size semidefinite lift?

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 were read against the question. The paper claims $\mathrm{rank}_{psd}A_n(\rho)\ge2^{cn}$ for every fixed $0<\rho<1$ and hence that every exact semidefinite lift of the perfect matching polytope has size $2^{\Omega(n)}$ . Proof not refereed. The challenges ComparatorChallenges/MatchingAffineLift.lean and MatchingPSD.lean are not in lean/formalization.yaml; they are found through lean/docs/126.md and their solution modules (OAI.Combinatorics.MatchingPSD.AffineLift and .Main) exist at the pinned commit. Statements read here: for every $C>0$ and large even $n$ , every affine section of a PSD cone projecting onto the matching polytope has size $>n^C$ , and the PSD rank of the unshifted slack matrix exceeds $n^C$ . This states the headline (no polynomial-size lift); the exponential bound and the shifted matrices are not formalized. Not rebuilt here.