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Pebody's even cyclic four-deck reconstruction conjecture

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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 1, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-assisted. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Pebody's even cyclic four-deck reconstruction conjecture · Pebody's four-deck conjecture

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

GPT-6 Pro
model · ai model contributor · OpenAI

Astra (via ChatGPT)
model · ai model contributor · OpenAI

Oleksiy Babanskyy
human · human collaborator

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VibeMathed record: Pebody's even cyclic four-deck reconstruction conjecture

specialist registry · passed

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.

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Pebody's even cyclic four-deck reconstruction conjecture

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Original conjecture: Pebody (2007), Conjecture 8.1, p. 514

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