HEXAGONbeta

Arithmetic coding against unknown small-prime signs

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorymath.IT — Information Theory

Contributed by Brice Pouly ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

We encode binary messages in completely multiplicative sign functions and read them along the three sequences n2+1n^2+1, n2+2n^2+2 and n2−2n^2-2. Two observations with the same exact cofactor remove the unknown smaller-prime signs when multiplied. Allocating shared large primes before pairing embeds ordinary binary error-correcting codes in each trace. Under an explicit large-factor mass hypothesis, a computable schedule gives one infinite encoding with positive prefix rate and a linear errors-and-erasures allowance. A first-occurrence filter makes its raw readings depend causally on a linear message prefix, while encoding and decoding remain polynomial in the numerical observation length. Its supporting primes have arbitrarily small harmonic cost. A finite construction, independent of the analytic hypothesis, stores 273273 bits in 87368736 readings of each trace and corrects every pattern of 499499 reading errors. Complementary constructions give persistent sign prescriptions, uniform laws for the active coordinates of words, and compact multiplicative subgroups whose nontrivial quotients have positive quadratic entropy. The word-law and entropy conclusions are not asserted for the polynomially decoded encoding. A separate matrix construction gives another causal positive-rate map. Fixed-scale balancing provides an unconditional quarter-sign bound against every small-prime assignment. All coding statements concern deliberately constructed families, not recovery of an arbitrary multiplicative source. The sparse constructions at superlinear cutoffs keep their large-factor mass input explicit.

Provenance statement

The following declaration describes the research and the published source edition: GPT-6 Astra assisted the research notes and initial consolidation; GPT-6 Astra Pro assisted the subsequent proof exposition, reviews, code and finite checks, bibliography and manuscript revisions. The author is responsible for the article. No complete independent human verification, Lean formalisation or independent external replication is claimed. The implementation appendix identifies the reproducible finite checks and their limits; they do not establish the stated large-factor mass hypothesis. For this Hexagon deposit, OpenAI Codex assisted source packaging, compilation checks and the preparation of submission metadata. This preparation does not constitute independent mathematical peer review. The published edition used for this deposit, including its supplementary source material, is available at https://doi.org/10.5281/zenodo.23319574. At the time of preparation for Hexagon, the submitter declares the text primarily AI-generated and declares that no part of the contribution has been checked and understood by a human.

References

  1. J ó zsef Beck and Tibor Fiala. ``Integer-making'' theorems . Discrete Applied Mathematics , 3(1):1--8, 1981. doi:10.1016/0166-218X(81)90022-6 (https://doi.org/10.1016/0166-218X(81)90022-6). The classical floating-variable proof is also given in Nikolov's Lecture 3.DOI
  2. Vitaly Bergelson, Joanna Kułaga-Przymus, Mariusz Lema ń czyk, and Florian K. Richter. Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics. Ergodic Theory and Dynamical Systems , 39(9):2332--2383, 2019. doi:10.1017/etds.2017.130 (https://doi.org/10.1017/etds.2017.130). Author version: arXiv:1611.08392v2 (https://arxiv.org/abs/1611.08392v2), Theorem 1.7.arXivDOI
  3. G é za Bohus. On the discrepancy of 3 permutations. Random Structures & Algorithms , 1(2):215--220, 1990. doi:10.1002/rsa.3240010208 (https://doi.org/10.1002/rsa.3240010208).DOI
  4. R é gis de la Bret è che and Sary Drappeau. Niveau de r é partition des polyn ô mes quadratiques et crible majorant pour les entiers friables. Journal of the European Mathematical Society , 22(5):1577--1624, 2020. doi:10.4171/JEMS/951 (https://doi.org/10.4171/JEMS/951). Locators: Theorem 1.2 and Section 8.4 of arXiv:1703.03197v3 (https://arxiv.org/abs/1703.03197v3), 7 May 2019.arXivDOI
  5. Jean-Marc Deshouillers and Henryk Iwaniec. On the greatest prime factor of n2+1n^2+1 . Annales de l'Institut Fourier , 32(4):1--11, 1982. Section 8. https://www.numdam.org/item/AIF_1982__32_4_1_0/.link
  6. Graham Everest and Glyn Harman. On primitive divisors of n2+bn^2+b . In James McKee and Chris Smyth, editors, Number Theory and Polynomials , volume 352 of London Mathematical Society Lecture Note Series , pages 142--154. Cambridge University Press, 2008. doi:10.1017/CBO9780511721274.011 (https://doi.org/10.1017/CBO9780511721274.011). Proposition 1.3; author preprint arXiv:math/0701234 (https://arxiv.org/abs/math/0701234) (2007).arXivDOI
  7. G. David Forney, Jr. Generalized minimum distance decoding. IEEE Transactions on Information Theory , 12(2):125--131, 1966. doi:10.1109/TIT.1966.1053873 (https://doi.org/10.1109/TIT.1966.1053873).DOI
  8. Nikos Frantzikinakis, Mariusz Lema ń czyk, and Thierry de la Rue. Furstenberg systems of pretentious and MRT multiplicative functions. Ergodic Theory and Dynamical Systems , 45(9):2765--2844, 2025. doi:10.1017/etds.2024.140 (https://doi.org/10.1017/etds.2024.140). Broader context; not a proof input here.DOI
  9. Stefano Galatolo, Mathieu Hoyrup, and Crist ó bal Rojas. A constructive Borel--Cantelli lemma. Constructing orbits with required statistical properties. Theoretical Computer Science , 410(21--23):2207--2222, 2009. doi:10.1016/j.tcs.2009.02.010 (https://doi.org/10.1016/j.tcs.2009.02.010). Author version: arXiv:0711.1478v2 (https://arxiv.org/abs/0711.1478v2). Context for effective statistical selection, not a black-box input.arXivDOI
  10. Lasse Grimmelt and Jori Merikoski. On the greatest prime factor and uniform equidistribution of quadratic polynomials. arXiv:2505.00493v2 (https://arxiv.org/abs/2505.00493v2), 30 May 2025, 2025. Theorem 1.1. Preprint cited for context, not as an input to the coding theorem.arXiv
Show all 25 referencesShow fewer references
  1. Venkatesan Guruswami. Introduction to coding theory. Notes 6 : Reed--Solomon , BCH , Reed--Muller , and concatenated codes. Carnegie Mellon University, February 2010, 2010. Scribes: Eric Blais and Venkat Guruswami. Sections 1 and 4. https://www.cs.cmu.edu/~venkatg/teaching/codingtheory/notes/notes6.pdf.link
  2. Venkatesan Guruswami. Introduction to coding theory. Notes 7 : Justesen codes, Reed--Solomon and concatenated codes decoding. Carnegie Mellon University, March 2010, 2010. Scribes: Venkat Guruswami and Balakrishnan Narayanaswamy. Sections 3.3 and 5. https://www.cs.cmu.edu/~venkatg/teaching/codingtheory/notes/notes7.pdf.link
  3. Oleksiy Klurman. Correlations of multiplicative functions and applications. Compositio Mathematica , 153(8):1622--1657, 2017. doi:10.1112/S0010437X17007163 (https://doi.org/10.1112/S0010437X17007163). Locator for the large-prime argument: arXiv:1603.08453v1 (https://arxiv.org/abs/1603.08453v1), Section 2, proofs of Lemma 2.1 and Proposition 1.1.arXivDOI
  4. Runbo Li. On the largest prime factor of quadratic polynomials. arXiv:2406.07575v2 (https://arxiv.org/abs/2406.07575v2), 31 May 2025, 2025. Theorem 1.1. This title and statement are those of version 2, not the different version 1 of 2024; context only.arXiv
  5. Colin McDiarmid. On the method of bounded differences. In Surveys in Combinatorics, 1989 , pages 148--188. Cambridge University Press, 1989. doi:10.1017/ CBO9781107359949.008 (https://doi.org/10.1017/CBO9781107359949.008).DOI
  6. National Institute of Standards and Technology . Digital Library of Mathematical Functions , section 27.11: Asymptotic formulas: Partial sums. https://dlmf.nist.gov/27.11. Equation (27.11.11) and the fixed-progression prime number theorem. Accessed 27 September 2026.link
  7. Alantha Newman and Aleksandar Nikolov. A counterexample to Beck's conjecture on the discrepancy of three permutations. arXiv:1104.2922v1 (https://arxiv.org/abs/1104.2922v1), 14 April 2011, 2011.arXiv
  8. Aleksandar Nikolov. Prague Summer School: Discrepancy Theory. Lecture 3 , 2018. Theorem 2, pages 1--2. https://www.cs.toronto.edu/~anikolov/Prague18/Lectures/lect3.pdf.link
  9. Alexandru Pascadi. Large sieve inequalities for exceptional Maass forms and the greatest prime factor of n2+1n^2+1 . Forum of Mathematics, Pi , 14:e8, 2026. doi:10.1017/fmp.2026.10025 (https://doi.org/10.1017/fmp.2026.10025). Locators refer to arXiv:2404.04239v3 (https://arxiv.org/abs/2404.04239v3), 15 January 2026, Sections 6.1 and 6.3, equation (6.20).arXivDOI
  10. Brice Pouly. Arithmetic reconstruction and error recovery from three quadratic traces. Zenodo, 2026. doi:10.5281/zenodo.23317860 (https://doi.org/10.5281/zenodo.23317860). Theorems 1.1--1.2.DOI
  11. Brice Pouly. Correlation laws and identifiability of multiplicative functions on quadratic traces. Zenodo, 2026. doi:10.5281/zenodo.23318918 (https://doi.org/10.5281/zenodo.23318918). Section 7.DOI
  12. Brice Pouly. Selective normality and algebraic rigidity of multiplicative traces. Zenodo, 2026. doi:10.5281/zenodo.23318326 (https://doi.org/10.5281/zenodo.23318326). Section 5, especially Proposition 5.1.DOI
  13. Maxim Raginsky. Concentration inequalities. Lecture notes, 29 August 2013, 2013. Theorem 2, page 6. https://maxim.ece.illinois.edu/teaching/fall13/notes/concentration.pdf.link
  14. Irving S. Reed and Gustave Solomon. Polynomial codes over certain finite fields. Journal of the Society for Industrial and Applied Mathematics , 8(2):300--304, 1960. doi:10.1137/0108018 (https://doi.org/10.1137/0108018).DOI
  15. Ravi Teja Sukhavasi and Babak Hassibi. Linear error correcting codes with anytime reliability. In 2011 IEEE International Symposium on Information Theory Proceedings , pages 1748--1752, 2011. doi:10.1109/ISIT.2011.6033847 (https://doi.org/10.1109/ISIT.2011.6033847). Author record: https://authors.library.caltech.edu/records/4jyv2-64h44.DOI

Version history

  1. v1Submitted by Brice PoulyInitial depositCurrentOct 18, 2026