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Bayesian Weak Commutativity Does Not Guarantee Attainability of the Nagaoka–Hayashi Bound

Primarily AI-generated textHuman understanding: some partsmath.MP — Mathematical Physics

Contributed by Qihang Wang ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

Bayesian weak commutativity has been suggested as the most likely condition for reaching the Nagaoka–Hayashi (NH) bound in multiparameter estimation. We show that it is not sufficient. For a full-rank four-level ensemble obeying it exactly, the effects required for NH equality cannot sum to the identity. We find the optimal risk exactly: it lies strictly above the bound and needs seven outcomes, while projective readout on the four levels adds over seven times the NH gap. A robust normalization witness bounds the gap from below across a family including truncated oscillator ladders.

Provenance statement

The manuscript text was primarily AI-generated. OpenAI GPT-5.6 Sol, Anthropic Claude Fable 5, Grok 4.6, OpenAI GPT-6.1 Sol, OpenAI GPT-6 Astra, and Anthropic Claude Opus 5.5 were used during the development and preparation of the work. Their combined contributions included model and example construction, analytical derivations, proof and certificate development, exact-arithmetic checking, literature analysis, physical interpretation, figure preparation, LaTeX drafting, and revision. The verification programs and algebraic certificates are available at https://doi.org/10.5281/zenodo.23489570.

Tools used

OpenAI
Codex
Anthropic
Claude Fable
Anthropic
Claude Opus

References

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Version history

  1. v1Submitted by Qihang WangInitial depositCurrentOct 24, 2026