Biological Transitions: Fruit-Fly Connectome Toy Model (Drosophila) — V2 Complete

**Version 2** of the fruit-fly connectome toy model paper. V1 (Zenodo 10. 5281/zenodo. 19210137, March 2026) contained only the 3-page skeleton PDF. V2 adds the complete LaTeX source, four figures, Python simulation code, Lean 4 formal verification, a full references section, and corrects a circular self-citation in the V1 abstract. **Correction from V1: ** The V1 abstract cited "companion paper (Zenodo 10. 5281/zenodo. 19210137) " — that DOI is this paper's own V1 record, a circular self-citation. The correct companion (broader biological scope) is Zenodo 10. 5281/zenodo. 19208015. Corrected throughout V2. --- We model biological transitions as multi-agent realisations of the composite operator pipeline G = U ∘ F ∘ K ∘ C from Topographical Orthogonal Generative Theory (TO/TOGT). Using the fixed-point, contraction-mapping, and saturated-pitchfork theorems proved in the companion paper (Zenodo 10. 5281/zenodo. 19208015), we construct a minimal toy model of the Drosophila connectome. Neural clusters are treated as coupled agents whose trajectories stabilise under six applications of G. The model reproduces qualitative features of neural coherence, oscillatory entrainment, and stress-response switching (HPA-axis analogue) while remaining strictly within what is rigorously established. The dm³ canonical invariants (T*, μₘax, τ) = (2π, −2, 2) are used solely as convenient normalisation choices. The Lean 4/Mathlib4 file MultiOrbitBioSwarm. lean proves 16 facts without sorry: the pitchfork threshold |α| = ½ is interior to (0, 1) ; for |α| < ½ the swarm map is a strict contraction (L < 1) ; at |α| = ½ the map is non-expansive (L = 1, pitchfork onset) ; (4/5) ⁶ < 27/100 (six-iterate convergence bound) ; and the dm³ normalisation triple (2π, −2, 2) is arithmetically consistent. Three obligations carry trivial stubs pending Mathlib infrastructure (BioSwarm metric instance, bifurcation library). Also included: AutophagyDm3. lean, the Lean 4 verification file for the autophagy/triple-alpha chapter of Principia Orthogona Book 3 (18 theorems, zero actual sorry) ; and the companion paper multiₐgentₜogtᵥ2. pdf + source (broader biological scope: neural oscillations, HPA axis, circadian regulation, immune adaptation, protein conformational change). --- **Files in this deposit: **- multiₒrbitbioswarmᵥ2. pdf — complete V2 paper (9 pages, with figures) - multiₒrbitbioswarm. tex — LaTeX source- MultiOrbitBioSwarm. lean — Lean 4/Mathlib4 proofs- multiₒrbitbioswarm. py — Python simulation and figure generator- fig1ₛwarmₜrajectories. pdf, fig2ₚitchforkₛcan. pdf, fig3convergence. pdf, fig4ₒperatordiagram. pdf- pitchforkₛcan. csv — raw data for fig2- README. md — manifest and build instructions- multiₐgentₜogtᵥ2. pdf + multiₐgentₜogt. tex — companion paper V2- AutophagyDm3. lean — Lean 4 proofs for Book 3 autophagy chapter- MultiOrbitTheoryGrossi2026 (3). pdf + fixedₚointₜheory (2). pdf — original V1 PDFs preserved - AutophagyDm3ᵥ2. lean — Lean 4 proofs for Book 3 autophagy chapter **Repository: ** https: //github. com/TOTOGT/AXLE**Concept DOI (all versions): ** 10. 5281/zenodo. 19210136**This version (V2): ** 10. 5281/zenodo. 20128568