What was already known
Morris and others progressively reduced prior polynomial-in-d mixing bounds, with an O(d³) upper bound for dyadic decks described in the manuscript. The support argument supplied the linear-order lower scale.
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
Establishes the optimal logarithmic order for full-deck mixing on dyadic deck sizes.
Limitations
The result concerns decks of 2^d cards; it does not determine a sharp leading constant or cutoff, and the cube argument does not address arbitrary even deck sizes.
Technical verification
OpenAI reports a Lean proof matching the selected claim.
Formal checking
The pinned OpenAI Lean scope note identifies this selected claim.
- Checker
- OpenAI-reported Lean 4 / Comparator
- Scope
- For the Thorp shuffle on 2^d cards, the full permutation law after 1600d complete shuffles tends to uniform in total variation as d grows, uniformly over initial decks; a support bound requires at least 2d−O(1) shuffles, giving mixing time Θ(d).
- 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