Mean and variance of finite 2-adic complexity (Question 3.1)
For a uniformly random binary string of length , identified with , let and let , without rounding. Prove and compute . This is Question 3.1 of Lin, Xiao and Chen (2026). The probability space includes all length-N binary strings, including zero.
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-assisted
- Method
- Argument
- Field
- Finite 2-adic complexity, rational approximation and probability
- Posed by
- Hezheng Lin, Lingmei Xiao and Zhixiong Chen, Question 3.1, AIMS Mathematics 11(7) (2026), p.20282, DOI 10.3934/math.2026823.
- Year posed
- 2026
- Years open
- 0y
- Solved
- 2026-10
- Model
- OpenAI GPT models; GPT-6.1 Sol
- Vendor
- OpenAI
- Collaborators
- Oleksiy Babanskyy
- Verification
- Unreviewed
- Publication
- Announced
- Significance
- 4 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
The manuscript claims both requested results: the exact mean upper bound for every N>=1 and an explicit finite arithmetic-logarithmic variance formula obtained from the complete distribution. It also evaluates the limiting variance as 0.892302571204274625294755... . The all-length mean proof combines a written lattice argument with exact finite certificates. The variance and asymptotic arguments use finite counting, marked-lattice equidistribution and uniform integrability. The limiting geometric law is explicitly credited to Boca-Gologan. The finite answer is a sum, not a short elementary expression in N; no convergence rate or raw-height variance theorem is claimed.
What the AI did
OpenAI GPT models assisted mathematical exploration, proof criticism, source comparison, manuscript editing and reproducibility checks under the author's direction. The public manuscript does not identify historical model versions, so none are inferred here. GPT-6.1 Sol was used for the three current author-side reviews of the proof, computations and sources. These reviews shared exposure to the research material. The author retains responsibility for the arguments and attributions.
Verification
Re-run by this site on 4 October 2026: the companion's mean-premise, distribution and constant replays all passed (PASS_FINITE_PREMISES, PASS_FINITE_DISTRIBUTION_AND_MOMENT_PREMISES, PASS_EXACT_RATIONAL_EVALUATION). A separate brute force written here for N = 1 to 14 agrees: the mean stays below N/2 at every N, the gap approaches about 0.414, and the variance rises from 0.869 at N = 8 to 0.8904 at N = 14, consistent with the claimed limit 0.8923. These are finite checks. The all-N mean bound and the limiting variance rest on the written lattice and equidistribution arguments, which were not checked here, and the author-side reviews are by AI models. No independent specialist review.
Sources
Submitted by Oleksiy Babanskyy on