Biological Transitions as Multi-Agent Realisations of the Generative Operator Pipeline in TO/TOGT: A Fruit-Fly Connectome Toy Model

Biological Transitions as Multi-Agent Realisations of the Generative Operator Pipeline in TO/TOGT: A Fruit-Fly Connectome Toy Model Version 3, September 2026: what the model does, and corrections Pablo Nogueira Grossi, G6 LLC, Newark, NJ. ORCID: 0009-0000-6496-2186Concept DOI 10.5281/zenodo.19210136; V2 10.5281/zenodo.20128568; V3 (this deposit) 10.5281/zenodo.23029951. Abstract A toy model of a swarm of neural clusters, motivated by the Drosophila connectome. Each cluster is a point in the plane; one step applies G = U∘F∘K∘C (compress, clip, fold with a neighbour, rescale) with coupling α around a ring. Version 3 shows that version 2's contraction claims do not hold, and measures what the model does instead. Corrections G is discontinuous, so it has no Lipschitz constant; version 2's L(α) = 1/2 + |α| was a definition, not derived from G. At α = 0 one agent has four fixed points, so there is no unique fixed point. The plotted bifurcation signature (mean |s|²) is 1 for every α, because U rescales to length 1. Version 2's reference [11] is not registered at doi.org and is withdrawn. What the model does (noise-free, N = 8, 40 starts, 400 steps) Multistable: for α ≤ 0.35 every run settles fast, and 40 starts reach 40 different end states. Cycles: the first appear at α = 0.42; for 0.5–0.65 more runs cycle than settle. Irregular: from α = 0.7 some runs neither settle nor repeat within 400 steps; at 0.95 none do. Lean 4.32.0 / Mathlib v4.32.0, no sorry, axioms propext, Classical.choice, Quot.sound only: C_jumps, U_norm, four_fixed_points. The measurements are numerical and reproducible with python multi_orbit_bioswarm_v3.py --check. Files multi_orbit_bioswarm_v3.pdf, .tex: the revised paper, with figures/ BioSwarmCheck.lean and bioswarmcheck.axioms.txt multi_orbit_bioswarm.py (version 2 operators, unchanged) and multi_orbit_bioswarm_v3.py (measurements, figures, checks) CHANGES_BioSwarm_V3.md, OPEN_QUESTIONS_BioSwarm.md Carried over from version 2 (not re-checked in version 3) These files were in the version 2 record and remain available there. If they are kept in this version, they are unchanged and have not been re-examined: multi_agent_togt_v2.pdf and multi_agent_togt.tex: the companion paper (broader biological scope; Zenodo 10.5281/zenodo.19208015) AutophagyDm3.lean, AutophagyDm3_v2.lean: Lean files for the Book 3 autophagy chapter Multi_Orbit_Theory_Grossi2026 (3).pdf, fixed_point_theory (2).pdf: the original version 1 PDFs Withdrawn in version 3 These version 2 files are superseded and stay in the version 2 record (10.5281/zenodo.20128568): multi_orbit_bioswarm_v2.pdf and its source; MultiOrbitBioSwarm.lean (its contraction constant is a definition not derived from G, and its three obligations are True or an identity placeholder); fig2_pitchfork_scan.pdf and pitchfork_scan.csv (the plotted quantity is 1 by construction); fig3_convergence.pdf (it drew the withdrawn bound); fig1_swarm_trajectories.pdf. The version 1 citation correction made in version 2 stands. Repository: github.com/TOTOGT/AXLE. Concept DOI (all versions): 10.5281/zenodo.19210136. MSC: 37C25, 37N25, 92B20. License: CC BY 4.0 (paper, figures), MIT (code, Lean). © 2026 Pablo Nogueira Grossi, G6 LLC.