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Binary Reconstruction from the Fourth Deck: Torsion Phases, Local Gluing, and Sign Extinction

Primarily AI-generated textHuman understanding: no partsmath.CO — Combinatorics

Contributed by Oleksiy Babanskyy ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We give a candidate proof of the following statement: for every finite cyclic group CnC_n, the cumulative fourth deck of a binary subset determines it up to translation. Hence rset(Cn)≤4r_{\mathrm{set}}(C_n)\le4 for every nn, and, using the Keleti--Kolountzakis Corollary 1.1 for the lower bound, rset(Cn)=4r_{\mathrm{set}}(C_n)=4 for every even n>10n>10. The proof has four components: finiteness of the binary fibre, cyclotomic torsion of the phases, gluing of primary characters, and extinction of dyadic residues by two odd-denominator projectors. The bridge between ordered analytic correlations and Pebody's cumulative deck is proved explicitly, including stabilizers, zero and repeated offsets, and the empty pattern. The text contains the complete mathematical chain; finite checks are included only as convention and implementation checks, not as premises of the uniform proof.

Provenance statement

The author directed the project and retained responsibility for the mathematical claims, source attributions and publication decisions. The author reports using GPT-6 Pro and Astra via ChatGPT, together with OpenAI Codex. AI assistance included mathematical exploration, source comparison, manuscript drafting and revision, finite sanity checks and repository preparation. These model names are author-reported; individual proof steps are not attributed to particular models. The manuscript presents a candidate proof of the uniform fourth-deck upper bound for binary subsets of finite cyclic groups, up to translation. Together with the lower bound from Keleti--Kolountzakis, this gives the claimed resolution of Pebody's Conjecture 8.1 for even orders greater than 10. The lower bound is credited to its original authors. The companion provides bounded lifting checks and package-integrity checks. These finite checks support conventions and implementation; they are not premises of the all-orders proof. AI-assisted adversarial reviews are not human peer review. For this submission, the author reports that no part of the contribution has been checked and understood by a human. No end-to-end formal verification is supplied. The same manuscript is available as aiXiv 2609.02922 v1: https://aixiv.online/abs/2609.02922v1 Source, companion checks and reproduction instructions: https://github.com/aconsciousfractal/FCIG-Pebody-Even-Cyclic-Four-Deck-Reconstruction/tree/377db7d69564c819628a0049309ded409a241ea7

Tools used

OpenAI
ChatGPTVersion GPT-6 Pro; Astra
Python Software Foundation
Python

References

  1. F. A. Grünbaum and C. C. Moore, The Use of Higher-Order Invariants in the Determination of Generalized Patterson Cyclotomic Sets , Acta Crystallographica Section A 51 (1995), 310--323. DOI: https://doi.org/10.1107/S0108767394009827.DOI
  2. L. Pebody, The reconstructibility of finite abelian groups , Combinatorics, Probability and Computing 13 (2004), 867--892. DOI: https://doi.org/10.1017/S0963548303005807.DOI
  3. L. Pebody, Reconstructing Odd Necklaces , Combinatorics, Probability and Computing 16 (2007), 503--514. DOI: https://doi.org/10.1017/S0963548306007875.DOI
  4. T. Keleti and M. N. Kolountzakis, On the determination of sets by their triple correlation in finite cyclic groups , arXiv:math/0603415 (2006). https://arxiv.org/pdf/math/0603415.arXiv
  5. J. Simon, Reconstructing Colourings of Finite Groups , doctoral thesis, RWTH Aachen University (2017). https://publications.rwth-aachen.de/record/698365/files/698365.pdf.link
  6. J. Mandereau, D. Ghisi, E. Amiot, M. Andreatta and C. Agon, Discrete phase retrieval in musical structures , Journal of Mathematics and Music 5 (2011), 99--116. DOI: https://doi.org/10.1080/17459737.2011.608820.DOI
  7. E. Amiot, On the group of rational spectral units with finite order , arXiv:0907.0857v1 (2009). https://arxiv.org/abs/0907.0857.arXiv
  8. K. Conrad, Cyclotomic Extensions , note, https://kconrad.math.uconn.edu/blurbs/galoistheory/cyclotomic.pdf.link

Version history

  1. v1Submitted by Oleksiy BabanskyyInitial depositCurrentOct 28, 2026