(Part VI-part 2) Multiscale Numerical Invariants and Fractal Properties of the Genetic Code: Internal Constraints and Multiscale Packet Distributions Revealing a Universal Grammar

Note: The present work has been deposited on Zenodo, HAL (technical record for timestamp), and Archive.org. Copyright protection has been registered in several countries. These deposits establish proof of authorship and priority. Please cite accordingly. + peer review in progress. Data are available on this link: https://zenodo.org/records/17306204 Abstract : Following the first study entitled “Multiscale Numerical Invariants and Fractal Properties of the Genetic Code: A Combinatorial and Atomic Analysis” (under editorial and peer review process), this second part deepens the investigation by exploring the hypothesis of a quaternary code. The numerical invariants previously identified reappear as expected, but now interlock directly within the codon matrix, revealing a structural continuity marked by the recurrent emergence of specific constants [1, ≈96-97, 128]. The analysis of packet groupings highlights two complementary signatures: a truncated triangular progression [1, 3, 6, 9], related to Pascal’s triangle and the sequence of triangular numbers, and a descending stepwise law [5, 4, 3, 2, 1] observed in the quaternary code and further extended to duplets and bases. These regularities express a broken fractal symmetry, in which a universal combinatorial law is modulated by an internal constraint perceptible from subatomic particles to the quaternary code itself. This organization recalls the electronic quantization in atoms, where electrons distribute into successive shells [s, p, d, f] according to precise multiplicities [1, 3, 6, 10…], reflecting the same logic of stepwise growth and intrinsic limitation. Finally, this recurrence opens a new perspective in exobiology: it suggests that the identified numerical grammar could constitute a universal signature of life, extending beyond Earth, and may serve as a criterion to anticipate alternative genetic codes in other planetary environments. Above all, these findings reinforce the notion that life is not a product of chance, but a recurrent mathematical organization of matter — a universal transition of information, as suggested by many authors (2, 3, 4, 5, 6, 7). These observations align with the works of Wheeler (1990), Prigogine & Stengers (1984), and Chaitin (2012), who considered life not as a random occurrence but as a recurring mathematical organization of matter governed by the circulation and transformation of information. They also resonate with the hypotheses of Kauffman (1995) and Davies (2019), according to which life necessarily emerges from a universal principle of self-organization embedded in the very dynamics of the cosmos. Complete Primary Data, Computational Materials and Bibliography Kayser-Cuny, V. (2025). Meta-Genesis. Towards a Biology without Matter, based on Pure Logic. Multi-Scale Numerical Invariants and Fractal Properties of the Genetic Code (Abstract and compilation). Zenodo. https://zenodo.org/records/17525084 Kayser-Cuny, V. (2025). (Part 1) Multi-Scale Numerical Invariants and Fractal Properties of the Genetic Code: A Combinatorial and Atomic Analysis. Zenodo. https://zenodo.org/records/17068843 Kayser-Cuny, V. (2025). (Part 2) Multiscale Numerical Invariants and Fractal Properties of the Genetic Code: Internal Constraints and Multiscale Packet Distributions Revealing a Universal Grammar. Zenodo. https://doi.org/10.5281/zenodo.17272500 Kayser-Cuny, V. (2025). (Part IV-part 3) Multi-Scale Numerical Invariants and Fractal Properties of the Genetic Code: A Unified Theory of Biological Information, from Stars to Codons. Zenodo. https://doi.org/10.5281/zenodo.17370443 Kayser-Cuny, V. (2025). (Part VI-part 3) Multi-Scale Numerical Invariants and Fractal Properties of the Genetic Code: Data Availability [Data set]. Zenodo. https://doi.org/10.5281/zenodo.17306204 Kayser-Cuny, V. (2025). Data Availability Part 2 [Data set]. Zenodo. https://doi.org/10.5281/zenodo.17368936 Kayser-Cuny, V. (2025). (Part III) The Mirror-Twin Paradox: A New Approach to DNA Understanding the Implications of an Inverted Genome and Its Applications in Molecular Genetics, Neuroscience, and Medicine. Zenodo. https://doi.org/10.5281/zenodo.15390489 Kayser-Cuny, V. (2025). Meta-Genesis. Towards a Biology Without Matter. From Boolean Algebra to the Expansion of Life: Binary Arithmetic and Multi-Dimensional Projections of the Genetic Code. Zenodo. https://doi.org/10.5281/zenodo.17494922 The author 2023: Elected Fellow of the Linnean Society of London (Biology);2023: Elected Fellow of the Royal Anthropological Institute of Great Britain and Ireland;2024: Elected Full Member of the Genetics Society;2025: Affiliate Member of the Royal Society of Chemistry. Molecular geneticist by training, with specialization in particle physics, chemistry of Life, paleogenetics/evolutionary genetics, and astro/exobiology. Accepted for 2026 in a post-graduate program in molecular cytogenetics at a medical school. Academic and research trajectory spanning Canada, the United States, France, and Switzerland (Educational Outreach Internship at CERN, 2018).

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