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Voiculescu's question: is free entropy dimension an invariant of the generated von Neumann algebra?

Voiculescu's free entropy dimension δ(X1,…,Xn)\delta(X_1,\ldots,X_n) of a self-adjoint tuple in a tracial von Neumann algebra, together with its variants δ0\delta_0, δ∗\delta^* and δ⋆\delta^\star, equals nn for a free semicircular nn-tuple generating L(Fn)L(\mathbb F_n). If δ\delta depended only on the von Neumann algebra generated, it would distinguish the free group factors. Does the free entropy dimension of a finite self-adjoint generating tuple depend only on the tracial von Neumann algebra it generates?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Operator algebras: free probability, free entropy
Posed by
Dan Voiculescu (Free entropy, Bull. London Math. Soc. 2002, Section 2.6)
Year posed
2002
Years open
24y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
38 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims a negative answer for δ\delta, δ0\delta_0, δ∗\delta^* and δ⋆\delta^\star: each takes every integer value n≥2n\ge2 on finite self-adjoint generating tuples of the single factor L(F2)L(\mathbb F_2), obtained by transporting semicircular and group-algebra tuples through the new isomorphisms. It does not compute free entropy dimension for other algebras or address the microstates-free equality questions.

What the AI did

The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author. This entry is a corollary in the free group factor manuscript.

Verification

No independent mathematician has checked this yet. Checked here: Corollary (dependence on von Neumann generators) of the TeX source read against Voiculescu's question; for every n≥2n\ge2 it gives generating tuples of L(F2)L(\mathbb F_2) with δ=δ0=n\delta=\delta_0=n and others with δ∗=δ⋆=n\delta^*=\delta^\star=n, a negative answer for all four dimensions. It is a short transport argument that depends entirely on the unrefereed isomorphism theorem. The Lean comparator for the family does not state this corollary.

Sources

Changelog1 change

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