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May 20, 2026Open Access

G4 Analysis: What the 2-Adic Approach Proves and What It Cannot — Extended Version with Full Proofs

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Authors

FCFranck Coppi

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Overview

Extended technical analysis verifies Hamming density for carry masks in Collatz conjecture, suggesting new implications.

Key Points

  • This analysis aims to provide complete proofs related to Open Problem G4 in the Collatz conjecture.
  • Utilized Lifting The Exponent Lemma and bitwise complementarity to establish proofs.
  • Analyzed carry masks with Hamming density calculations.
  • Applied entropy-improved Eliahou bound reaching a specific exponent.
  • Proved that all carry masks have a Hamming density of exactly 1/2.
  • Refuted the universal Hamming density conjecture.
  • Demonstrated an Eliahou bound reaching exponent 0.979 at continued-fraction epochs.

Cite This Study

Franck Coppi (2026) studied this question.

synapsesocial.com/papers/6a0d5064f03e14405aa9c339https://doi.org/10.5281/zenodo.20266686
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  1. 1G4 Analysis: What the 2-Adic Approach Proves and What It Cannot — Rise and Fall of the Hamming Density Route2026
  2. 2G4 Analysis: What the 2-Adic Approach Proves and What It Cannot — Rise and Fall of the Hamming Density Route2026
  3. 3The Collatz Partition Function: Why the Hamming Route Fails and What It Reveals for G42026
  4. 4A Wave-Propagation Framework for the Collatz Conjecture: 2-adic Ergodicity, Non-Resonance, Cycle Exclusion, and a Structural Roadmap2026
  5. 5A Wave-Propagation Framework for the Collatz Conjecture: 2-adic Ergodicity, Non-Resonance, Cycle Exclusion, and a Structural Roadmap2026