The ODE lives on (M, ξ) = (ℝ³, ker α) in cylindrical coordinates, with ε = 2 throughout: ṙ = r(1−r²) + ε(r−1)e−z, θ̇ = 1, ż = r² − ε(r−1)²e−z The limit set is a helix — the Reeb orbit Γ on the unit cylinder r = 1. Non-integrability of ξ forces any planar periodic limit to become helical in 3D.
Volume IV is published as a living document: the book is the anchor, and everything around it is open and reproducible. Each tile below opens a distinct surface of the same mathematical object.
Chain.lean — the C→K→F→U structures on metric spaces, the Spiral Return statement, and the open kappa_lipschitz proof obligation (AXLE Issue #12).G = U∘F∘K∘C formalised over metric/normed spaces, the Spiral Return theorem, the open kappa_lipschitz obligation, and all Mathlib 4 dependencies.Three sessions. Three surfaces of one system. Pre-requisites: undergraduate ODE. Lean 4 is not assumed. Every artefact below is open, linkable, and reproducible; together they constitute the full handout set distributed to enrolled students.
Chain.lean, the operator structures over metric spaces, the Spiral Return statement, and the kappa_lipschitz obligation still open as AXLE Issue #12.Volume IV is the Brazil Edition — the slim, formal, numerically rigorous companion to the full Book 3. It is designed to be read in a week, taught in three hours, and formally verified line-by-line.
| Vol | Title | ISBN / DOI |
|---|---|---|
| G¹ | The Orthogonal Operator Framework | — |
| G² | TOGT: Applications Across Domains | — |
| G³ | The Mini-Beast: Biological Instantiations | — |
| G⁴ | Helical Attractors on Contact 3-Manifolds · GTCT T1 — The Brazil Edition (this volume) | 10.5281/zenodo.19117400 |
| G⁵ | The Seed — Complete Completeness | — |
Only registered allocations are listed. Volumes shown with — have no ISBN of their own; the reserve numbers previously printed here are unallocated (no group, no format) per the Bowker registry, and are not valid fallbacks. Cite the Principia Orthogona community instead.