The entanglement of formation has a supporting affine functional at every pure state

A. S. Holevo and M. E. Shirokov, Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation, arXiv:2609.12667v3, Introduction:

It is somewhat surprising that the above problem is nontrivial even in the case when the state $\rho$ is pure. We and GPT-5.6 have found neither a general proof nor counterexamples which would confirm or refute the hypothesis of the existence of a supporting affine functional for the EoF at any pure bipartite state.

Section 3.1 restates the finite-dimensional question. The verbatim passages are in the literature note.

Issue #19308 fixes the reading: for all finite a, b and every unit ψ ∈ ℂ^a ⊗ ℂ^b, there is a Hermitian W with Tr(W P_ψ) = E_F(P_ψ) and Tr(Wσ) ≤ E_F(σ) for every density matrix σ. This is proved.

A supporting affine functional at ρ is equivalent to a one-sided Lipschitz lower bound for E_F at ρ. The authors' earlier work (arXiv:2608.27363) shows that such functionals fail at some mixed boundary states, and the source settles pure states with equal Schmidt coefficients.

The literature check before any Lean is recorded in #19308:

• Song–Zong (arXiv:2609.31097) settles two-qubit states only;
• arXiv searches find no other work on the question;
• the repository had no record of it.

not-found-in-searched-scope.

Let R be the marginal of P_ψ, c = Σ_i r_i^{−1} over its positive eigenvalues r_i (the module's choice; the smaller c = 1/r_min also works) and K = 2bc, and put W = −(log₀ R) ⊗ I − K(I − P_ψ).

1. Scalar bound. −t ln t ≥ −t log₀ r − (t − r) − c(t − r)² for each eigenvalue r of R and each t ∈ [0, 1].
2. Matrix bound. Averaging with the doubly stochastic overlaps of the two eigenbases gives S(T) ≥ −Tr(T log₀ R) − c‖T − R‖²_HS for every density T.
3. Pure states. ‖Tr_B X‖²_HS ≤ b‖X‖²_HS, and ‖Q − P‖²_HS = 2(1 − Tr PQ) for pure Q. Together they give S(Tr_B Q) ≥ Tr(WQ), with equality at Q = P.
4. Convex roof. Averaging over pure ensembles gives Tr(Wσ) ≤ E_F(σ).

The theorem would not settle the question if the module's E_F differed from the source's. The module uses the frozen E_F of MoreauYosidaFormationSelectiveLoccRefutation, the infimum over finite pure ensembles of the average marginal entropy. Because E_F takes values in [0, ∞], the inequalities are stated through ENNReal.ofReal. This is equivalent to the real inequality because E_F ≥ 0.

The canonical source is D5/S3/Quantum/PureState/PureStateFormationSupportingFunctional.lean (429 lines).

• Proof. It proves result : claim.
• Axioms. It uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom.
• Escape registration: Reg/D5/S3/Quantum/PureState/PureStateFormationSupportingFunctional.lean (result).

Frozen state: Golden/Frozen/state/D5/S3/Quantum/PureState/PureStateFormationSupportingFunctional.lean.json, statement sha256:8155f1408fe70ea5113d2eae46a554ef73a9c52920d480bdc79485e97feeab70.

Declaration statements:

• result: sha256:f23e4348040e7cfac128c5c1ff0fb51bd369df0e61d9e4106fb8818155b378a9;
• claim: sha256:6034f7041d2ed70eb8a87196c23ca01a47e89da6d6df4e1f364ac0e221dcce55.

Tier 1 open question of a 2026 paper, preregistered in issue #19308 before any Lean. theorem; resolution proved.

The module is admitted under the open-problem-resolution basis of CLAUDE.md §3.2; the per-declaration proof shapes are listed in the delivery PR. utility: none. There is no digestion atom.

What the settlement shows

Proved by result: every pure state of a finite-dimensional bipartite system has a supporting affine functional for the entanglement of formation.

Argued, not formalized.

• Mechanism. The entropy of the marginal is controlled from below by the relative-entropy tangent at R plus a quadratic correction. The correction is quadratic in the Hilbert–Schmidt distance, and for pure states that distance is linear in 1 − Tr(PQ), which is affine in Q. The constant K = 2bc grows as the least positive Schmidt coefficient tends to zero. Dividing E_F(P) − E_F(σ) by ‖P − σ‖₁ is therefore bounded at every pure state, though not uniformly.
• Scope. Mixed boundary states remain excluded, consistent with the earlier counterexamples.

Effect on the paper. The pure-state question of the Introduction and Section 3.1 is answered positively for all finite dimensions.

The bounded literature check does not establish exhaustive worldwide novelty or priority. The statements in "Argued, not formalized" are not formalized.