What was already known
The Lang–Plaut problem asked whether every doubling Hilbert subset bi-Lipschitz embeds in some finite-dimensional Euclidean space. Assouad gave a positive result after snowflaking the metric, and earlier obstructions concerned Lp with p > 2, leaving the original Hilbert distance open (manuscript introduction, pp. 1–2).
AI involvement
OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.
OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.
Importance
Answers the Lang–Plaut dimension-reduction question negatively even in Hilbert space and extends the obstruction to all infinite-dimensional real Banach spaces.
Limitations
The explicit constant 76,800 applies to the Hilbert subset. The Lean scope gives a universal but not necessarily identical constant for compact subsets of arbitrary infinite-dimensional Banach spaces. Neither result rules out embeddings after snowflaking or into infinite-dimensional targets.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned Lean note includes the Hilbert counterexample with doubling constant at most 76,800 and a compact counterexample with a universal doubling constant in every infinite-dimensional real Banach space.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- A real Hilbert subset of doubling constant at most 76,800 has no finite-dimensional Euclidean bi-Lipschitz embedding; every infinite-dimensional real Banach space contains a compact doubling subset with a universal doubling bound and no finite-dimensional normed-space bi-Lipschitz embedding.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.
Correction and publication history
Primary manuscript, attribution, prior-work, and scoped checking review for non-notifying local inclusion.
Read this published revision