Pebody's even cyclic four-deck reconstruction conjecture
For every even integer , is , where is the least integer such that every subset of the cyclic group is determined, up to translation, by its cumulative -deck?
The cumulative deck records, with multiplicity, the translation classes of all subsets of size at most . Reflections are not identified.
This is Luke Pebody's Conjecture 8.1 in Reconstructing Odd Necklaces (2007), p. 514, with Definitions 1-4 on pp. 503-504. The submitted manuscript writes for Pebody's .
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-assisted
- Method
- Argument
- Field
- Combinatorial reconstruction; Fourier analysis on cyclic groups
- Posed by
- Luke Pebody, Reconstructing Odd Necklaces (2007), Conjecture 8.1, p. 514
- Year posed
- 2007
- Years open
- 19y
- Solved
- 2026-09
- Model
- GPT-6 Pro; Astra (via ChatGPT)
- Vendor
- OpenAI
- Collaborators
- Oleksiy Babanskyy
- Verification
- Unreviewed
- Publication
- Announced
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claimed: r_set(C_n) <= 4 for every n >= 1, that is, the cumulative 4-deck determines every subset of a cyclic group up to translation. With Keleti-Kolountzakis's Corollary 1.1 for the lower bound this gives r_set(C_n) = 4 for every even n > 10, the whole domain of Pebody's Conjecture 8.1. The lower bound is imported, not claimed. The proof works on the ratio of Fourier transforms of two sets with equal decks: finiteness of the phase fibre, cyclotomic torsion of the phases, gluing of primary characters across odd primes, and extinction of the residual dyadic signs by two odd-denominator projectors. Binary subsets and translation-only decks only; no claim for multisets, general integer signals or noncyclic groups. A full-proof claim awaiting authoritative review.
What the AI did
The author reports using GPT-6 Pro and Astra via ChatGPT. OpenAI Codex was also used; its underlying model versions are not specified. AI assistance included mathematical exploration, source comparison, manuscript drafting and revision, finite sanity checks and repository preparation under the author's direction. The author is responsible for the mathematical claims and final decisions. Individual proof steps are not attributed to particular models. This disclosure appears in the manuscript and AI_USE.md.
Verification
Checked here on 22 September 2026. Pebody's Conjecture 8.1 (Combinatorics, Probability and Computing 16 (2007), 503-514, confirmed at Crossref) and the imported lower bound (Keleti-Kolountzakis, arXiv:math/0603415, Corollary 1.1: the 3-deck suffices for Z_n iff n is a power of an odd prime, a product of at most three odd primes, or n in {2,4,6,8,10}) were read. The 11-page manuscript was read: it follows the Grünbaum-Moore route, phases on the Fourier support, Galois covariance making them roots of unity, local characters on generator shells, gluing across odd primes by four-term relations, and removes the remaining dyadic sign by an integrality argument on Fourier projectors, which is exactly the point where Grünbaum-Moore's Theorem 5 needed a nonzero coefficient. The chain is coherent and non-trivial; it was not checked step by step here and no specialist has read it. The site ran an independent brute force: for every n <= 22, any two subsets of C_n with the same cumulative 4-deck are translates, while the 3-deck fails at n = 12, 14, 16, 18, 20, 22 exactly as Keleti-Kolountzakis predict. That is consistent with the theorem and is not evidence for the proof, since Grünbaum-Moore already cover small n.
Sources
Submitted by Oleksiy Babanskyy on