HEXAGONbeta

Selective normality and algebraic rigidity of multiplicative traces

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorymath.PR — Probabilitymath.DS — Dynamical Systems

Contributed by Brice Pouly ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

A completely multiplicative sign function may be changed so that two quadratic traces remain identical at every index while a third becomes normal. We classify this freedom for the traces n2+1n^2+1, n2+2n^2+2 and n2−2n^2-2. After two traces and a finite prefix have been protected, a monic irreducible polynomial can be made normal from every prescribed source exactly when the field generated by one of its roots contains neither protected quadratic field. In the complementary case, every value of the polynomial is already fixed. A single decidable set of primes of relative density zero supports a continuous, relatively computable completion treating all admissible polynomials jointly, with uniform bounds for each fixed observation family. The proof combines weighted root counts with a sieve for moments whose order increases with the observation scale, followed by a selection using exact finite normalisers. We also give a sufficient condition on prime availability, explicit sparser reservoirs, and a divergent reservoir on which some sources have no completion with a convergent ordinary mean, despite convergence of all fixed word laws in probability. For larger protected families, the obstruction is governed by odd products of quadratic square classes. An explicit example distinguishes certificates for every numerical value from a rational norm identity in the variable.

Provenance statement

The following declaration describes the research and the published source edition: GPT-6 Astra assisted with the development of the underlying research notes, consolidation of the proofs and exposition, and subsequent revision using mathematical and bibliographic feedback. Assistance also included targeted primary-source checks, finite diagnostic re-execution and preparation of the manuscript and sources. The author retains responsibility for the manuscript. The finite checks concern congruence counts, moment identities, selection costs, algebraic examples and numerical certificates; they are not premises of the asymptotic proofs. No independent specialist review or Lean formalisation of this article is claimed. No diagnostic executes the infinite normalising selector or the infinite obstructive calendar. 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.23318326. 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