Biological Transitions as Multi-Agent Realisations of the Generative Operator Pipeline in TO/TOGT

Biological Transitions as Multi-Agent Realisations of G = U∘F∘K∘C Version 2 — Complete Pablo Nogueira Grossi · May 2026 Concept DOI (all versions): https://doi.org/10.5281/zenodo.19208015Series root: https://doi.org/10.5281/zenodo.19117399GitHub: https://github.com/TOTOGT/AXLEContact: pgrossi888@outlook.com · g6llc@proton.meORCID: 0009-0000-6496-2186 What this deposit contains File Description multi_agent_togt_v2.pdf Complete V2 paper (9 pages, 4 figures) multi_agent_togt.tex LaTeX source multi_agent_togt.py Python simulation & figure generator MultiAgentTogt.lean Lean 4 / Mathlib4 formal proofs (18 facts, 0 sorry) fig1_agent_trajectories.pdf N-agent convergence under G⁶ fig2_pitchfork_scan.pdf HPA-axis saturated pitchfork bifurcation fig3_convergence.pdf Contraction rate across iterations fig4_operator_diagram.pdf Schematic of G = U∘F∘K∘C pipeline pitchfork_scan.csv Raw data for fig2 fixed_point_theory (2).pdf Original V1 PDF (preserved) What changed from V1 to V2 V1 (March 2026) contained only the 2-page skeleton PDF with stub sections. V2 adds: Expanded sections covering HPA-axis bifurcation (full three-regime analysis), circadian regulation (Proposition 5.1), immune adaptation (Corollary 6.1), and protein conformational change (Whitney fold treatment) Four figures generated by the included Python code LaTeX source for full reproducibility References section (8 entries) Corrected companion citation: V1 linked to Zenodo records 19162013, 19122168, 19117400. V2 adds the primary companion (fruit-fly connectome toy model, Zenodo 10.5281/zenodo.19210136) and the series root (10.5281/zenodo.19117399). Lean 4 formal verification MultiAgentTogt.lean proves 18 facts without sorry: ID Claim T1 Pitchfork threshold ½ is interior to (0, 1) T2 For |α| < ½, fold map has Lipschitz constant |α| < 1 T3 At |α| = ½, boundary Lipschitz arithmetic (onset) T4 (4/5)⁶ < 27/100 — six-iterate convergence bound T5 (4/5)⁶ < 1/3 — coarser useful bound T6a–c dm³ triple (2π, −2, 2) arithmetically consistent T7 Compression operator C is well-typed T8 For ε ∈ (0,1), C contracts strictly toward the mean T9 Clipping bound: |clip(x)| ≤ 1 T10 Lipschitz constant |α| < 1 in sub-threshold regime T11 Sub-threshold: |α| < ½ ⟹ λ = 2|α| < 1 T12 Super-threshold: |α| > ½ ⟹ λ > 1 (bistability) T13 Pitchfork branch sanity: at λ = 2, branch = 1 T14 Six-iterate bound implies reduction factor > 3 T15 Circadian anchor: T* = 2π > 6 T16 Immune fixed point: F(0, x) = 0 at α = 0 T17 dm³ product τ · |μ_max| = 4 T18 Composition of contractions is a contraction Three stubs pending Mathlib: metric-space instance (S1), tanh Lipschitz-1 (S2), pitchfork normal form (S3). Companion Lean file: MultiOrbitBioSwarm.lean (Zenodo 10.5281/zenodo.19210136) Build the Lean file lake update && lake build MultiAgentTogt Building the paper Prerequisites pdflatex (TeX Live 2022+ or MiKTeX) python >= 3.10 numpy matplotlib Generate figures python multi_agent_togt.py # Produces: fig1_agent_trajectories.pdf fig2_pitchfork_scan.pdf # fig3_convergence.pdf fig4_operator_diagram.pdf # pitchfork_scan.csv Compile PDF pdflatex multi_agent_togt.tex pdflatex multi_agent_togt.tex # twice for ToC and cross-references Relation to the series Role Record Series root 10.5281/zenodo.19117399 Volume One (mathematics) HAL hal-05555216v1 Volume Two (contact geometry) HAL hal-05559997v1 / hal-05565024v1 This paper (multi-agent biology) 10.5281/zenodo.19208015 Companion (Drosophila toy model) 10.5281/zenodo.19210136 Mathematical scope All theorems in V2 follow from results already proved in Volume One (HAL hal-05555216v1). The dm³ normalisation constants (T* = 2π, μ_max = −2, τ = 2) are modelling choices, not derived theorems. No new proofs are required for the biological applications; the paper applies existing fixed-point, contraction-mapping, and pitchfork results to new domains. License Creative Commons Attribution Non Commercial No Derivatives 4.0 International (CC BY-NC-ND 4.0).