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This analysis examines the open problem G4 and its relationship to the Hamming density conjecture.

May 21, 2026Open Access

G4 Analysis: What the 2-Adic Approach Proves and What It Cannot — Rise and Fall of the Hamming Density Route

Key Points

This analysis examines the open problem G4 and its relationship to the Hamming density conjecture.
Analyzed computational evidence on Hamming density for 2-adic rigidity residues
Proved that all carry masks possess a density of exactly 1/2
Presented an entropy-improved Eliahou bound reaching an exponent of 0.979.
Refuted the universal Hamming density conjecture with the result that all carry masks have density 1/2
Demonstrated the improved Eliahou bound with exponent 0.979 at epochs
Confirmed that the G4 conjecture remains open for further investigation.

Abstract

Concise analysis of Open Problem G4. Computational evidence initially suggested a promising universal positive lower bound on the Hamming density of 2-adic rigidity residues rm(a). We prove that all carry masks have exactly density 1/2, definitively refuting the universal Hamming density conjecture. We also present the entropy-improved Eliahou bound reaching exponent 0.979 at epochs. G4 remains open.

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Franck Coppi (Tue,) studied this question.

synapsesocial.com/papers/6a0ea1c1be05d6e3efb6080e https://doi.org/https://doi.org/10.5281/zenodo.20298125

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