What was already known
The precise 2ⁿ growth condition remained open. A later joint paper proves a stronger golden-ratio-growth criterion, which is separate from this output.
AI involvement
The study describes the accepted Aletheia run as autonomous. Domain experts vetted it, Kevin Barreto formalized the original statement in Lean 4, and the human team rewrote and corrected the exposition.
The study explicitly classifies Erdős 1051 as an autonomous Aletheia resolution and distinguishes the accepted second model run from human checking and later collaborative generalization.
Importance
The accepted proof resolves a specific Erdős question, although the study’s experts judged it modest in individual importance.
Limitations
The first model output had a disputed recurrence step. The accepted proof came from a separate run and needed a minor strict-inequality correction; the stronger later theorem is outside this claim.
Technical verification
The paper reports careful human-domain-expert vetting of the accepted corrected output and a Lean 4 formalization by Barreto. The human review and machine formalization concern the original 2^n statement, not the later golden-ratio generalization. The formal run was not reproduced here.
Formal checking
Careful human mathematical vetting after candidate filtering, correction of the strict-inequality slip and comparison with intended problem
- Checker
- Google DeepMind study's internal domain experts
- Scope
- Aletheia's accepted Erdős 1051 irrationality proof under liminf(a_n^(1/2^n)) > 1
- Basis
- Reported by the linked checker
Formalization of the mathematical proof in Lean 4
- Checker
- Kevin Barreto using Lean 4
- Scope
- The accepted Erdős 1051 irrationality solution, distinct from the later generalized threshold
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The human authors rewrote and explained the accepted tail argument, identified the strict-inequality correction, and connect the proof to Mahler's irrationality criterion. This documents understanding of the original mechanism.
Correction and publication history
The accepted proof resolves a specific Erdős question, although the study’s experts judged it modest in individual importance.
Read this published revision