HEXAGONbeta

Arithmetic reconstruction and error recovery from three quadratic traces

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 study completely multiplicative sign functions hh through their values at positive integers of the forms n2+1n^2+1, n2+2n^2+2 and n2−2n^2-2, with observed indices 1≤n≤N1\le n\le N. First occurrences of prime factors give a canonical information set and a complete, prefix-compatible family of parity relations, each involving at most ⌊log⁡2N⌋+3\lfloor\log_2N\rfloor+3 readings. These relations yield exact bounded-distance repair, including corrupted information coordinates, in time NO(1)(4+⌊log⁡2N⌋)tN^{O(1)}(4+\lfloor\log_2N\rfloor)^t for tt errors; erasures enter through an explicit rank parameter. We then construct arithmetic redundancy without estimating a quadratic regulator. If h(p)=−1h(p)=-1 and K=log⁡N/log⁡max⁡(3,p)K=\log N/\log\max(3,p), the three traces contain at least exp⁡(exp⁡(clog⁡K/log⁡log⁡K))\exp(\exp(c\log K/\log\log K)) negative readings for every fixed 0<c<log⁡20<c<\log2 and sufficiently large KK, uniformly in the prime pp and all other prime signs. The same number of each sign occurs on one trace between NαN^\alpha and NN, for each fixed α<1\alpha<1. The proof combines short reconstruction certificates, Lehmer divisibility and balanced divisors. Coprime blocks give compact access to these families without factoring large recurrence terms. Under independent reading noise with fixed erasure probability 0≤a<10\le a<1 and conditional flip probability 0≤b<1/20\le b<1/2, every source prefix P=⌊exp⁡((log⁡N)δ)⌋P=\lfloor\exp((\log N)^\delta)\rfloor, for fixed 0<δ<10<\delta<1, is recoverable from O(P)O(P) readings with polynomially small failure probability in PP and preparation polynomial in the numerical PP and log⁡N\log N. A fixed positive erasure rate nevertheless leaves exponentially many assignments of observed large-prime signs. No positive-density inverse for unrestricted sources is asserted.

Provenance statement

The following declaration describes the research and the published source edition: GPT-6 Astra assisted mathematical exploration, calculations, drafting, bibliographical work, code, rechecking and editorial preparation. The author is responsible for the manuscript. No independent human specialist review, human replication or Lean formalisation of this article is claimed, nor does this declaration assert that a complete unaided human verification has taken place. Detailed provenance and the scope of each computational check are recorded in the archive. 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.23317860. 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.

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Version history

  1. v1Submitted by Brice PoulyInitial depositCurrentOct 18, 2026