Helical Attractors on Contact 3-Manifolds — book cover
Principia Orthogona · Vol. IV · GTCT T1
PRINCIPIA ORTHOGONA · VOL. IV · IMPA EDITION

Helical Attractors on
Contact 3-Manifolds

A numerical and formal study of the dm³ system — contact geometry, exponential convergence, and the asymmetric basin the Lyapunov bound cannot see.
Pablo Nogueira Grossi · G6 LLC · Newark, NJ · 2026
C→ K→ F→ U→ T→ Γ*
DOI · Zenodo Open Access10.5281/zenodo.19117400 Mini-curso · XII Bienal UFRN3-page preview (PDF)
▶ Interactive 3D simulator Buy Book 3
Series: Principia Orthogona · Volume: IV (GTCT T1) · Edition: IMPA · License: MIT · ORCID: 0009-0000-6496-2186
"The internal basin boundary is asymmetric: real r* ≈ 0.776, not the r = 2/3 that the symmetric Lyapunov bound predicts. The study of this correction remains open for scientific initiation. Its study is yours. No one can take it from you."
"A fronteira interna real da bacia é assimétrica: r* ≈ 0,80, não r = 2/3. Seu estudo é seu. Ninguém pode tirá-lo de você." — § 5 do mini-curso
Canonical Invariants · Contact Form α = dz − r² dθ

The dm³ System at a Glance

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.

−2
μ · Lyapunov
2π
T* · period
2
τ · embodiment
1/3
ε₀ · Lyapunov
0.80
r* · real basin
Interactive Visualisation

The Helical Attractor — 3D Simulator

▶ dm³ Contact Flow · Reeb Helix Γ on r = 1

Blue trajectory spiraling in from r(0) > 1 · gold Γ helix on the unit cylinder · red cautionary curve from r(0) < r*. Drag to rotate · scroll to zoom · adjust ε, r₀, and N.
Engine: Three.js r128  ·  Integrator: RK4 dt = 0.01  ·  Opens standalone: sims/helical-attractor.html
Companion Materials · All MIT · All Linkable

Read, Simulate, Prove, Teach

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.

Bonus Chapter · E

GTCT for Everyone

Nine axioms, twelve operators, four theorems, one fixed point — made accessible to ESL students and STEM teachers. Bilingual EN/PT. Embedded G-orbit machine. Seven CEFR prompt levels A1 → D1.
Read Chapter E
Mini-Curso · XII Bienal 2026

3-page Preview · Natal, UFRN

The accepted SBM mini-course abstract: sistema dm³, teorema de atrator helicoidal, correção assimétrica da bacia de atração, e ligação com a formalização em Lean 4. Três sessões × 60 min.
Download preview PDF
SBM Submission · Chapter 10

3-page Bilingual PT / EN

The polished Sociedade Brasileira de Matemática submission: Theorem 2.1 (exponential convergence µ → −2), the Lyapunov stability radius[Ch 10] ε₀ = 1/3, and the numerical correction to r* ≈ 0.776.
Read the submission
Mini-Curso · Session S1

Contact Geometry & the dm³ System

The arena: contact manifolds, Reeb vector fields, and the helical ODE. Interactive embed of the simulator. Ends with the ε₀ = 1/3 Lyapunov basin derivation ready for S2.
Open Session 1 handout
Mini-Curso · Session S2

Theorem 2.1 & the Asymmetric Basin

The proof sketch for exponential convergence and the pedagogical correction: symmetric r-balls misclassify the basin. Numerics confirm r* ≈ 0.776. Reproduce Table 1 yourself.
Open Session 2 handout
Mini-Curso · Session S3

Lean 4 Skeleton & the Open Sorry

Walk through 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).
Open Session 3 handout
Numeric Lab · Python

DOP853 High-Precision Reproducer

Runnable script that integrates (1) with rtol = 10⁻¹⁰, fits log|r − 1|, and reproduces the 5-row Table 1 — including the basin-boundary scan that corrects the Lyapunov bound.
View the Python script
Formal Verification · Lean 4

AXLE Repository

The axis of the operator chain: G = U∘F∘K∘C formalised over metric/normed spaces, the Spiral Return theorem, the open kappa_lipschitz obligation, and all Mathlib 4 dependencies.
github.com/TOTOGT/AXLE
Simulation Source · Python + Lean

GTCT Repository

The wider GTCT workshop: Python integrators, FINDINGS.md with the full 24-orbit table, LaTeX sources for the preprint, and the Lean stubs mirrored into AXLE.
github.com/TOTOGT/GTCT
Open Access · DOI

Zenodo Canonical Record

Citable archival record of Volume IV, including the preprint PDF, Python sources, Lean proofs, and figure assets. Versioned; each revision gets its own DOI suffix.
doi.org/10.5281/zenodo.19117400
IMPA Portal

For Reviewers & Visitors

The reviewer-facing landing: the four Theorems of GTCT, how the Lean proofs relate to the preprint, and a directed path through the Principia Orthogona series from G¹ to G⁵.
Open the IMPA portal
Book 6 · Ch 22 · Live

The Stone Fold

Seismically stable forms as dm³ fixed points. Roman amphitheaters · Borobudur · Bagan · Hindu stepwells · Sierpiński phononic crystals · orthogenesis theorem: bee hive, NASA O'Neill, amphitheater — same attractor.
ch-seismic.html
Book 6 · WP-18 · Preprint

Topographical Orthogenetic Architecture

n-bonacci growth law for seismically stable construction. g = η derived from first principles. Mars/Moon spectrum adaptation via InSight data. Autonomous robotic fabrication algorithm.
toa-preprint.html
Book 3 · Book 6 · Dual-Registered · Live

HVEH — HALO

Helical Vortex Energy Harvester. Flood attenuation + storm-peak energy revenue. 25 sites, Passaic River corridor. Physics origin: El Ojo (Book 3, Vol III). Industrial deployment: Book 6.
HVEH/chHALO.html
Series Home

Principia Orthogona

The full homepage for the series: five volumes from the operator framework (G¹) through The Seed (G⁵). Sister site to this one, sharing the formal backbone but with different emphasis.
totogt.github.io/AXLE
3-Hour Course Program · XII Bienal de Matemática 2026 · UFRN, Natal

The Mini-Curso in Three 60-Minute Sessions

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.

60 min
SESSION S1

Contact Geometry & the dm³ System

The contact 3-manifold (M, ξ), Reeb field, the helical ODE with ε = 2, intuition for why the attractor must be a helix, and an embedded run of the simulator.
Open S1
 60 min
SESSION S2

Theorem 2.1 & the Asymmetric Basin

Linearisation at r = 1 giving µ → −2, the Lyapunov basin derivation for the symmetric ε₀ = 1/3 ball, the numerical correction r* ≈ 0.776, and a live reproduction of Table 1.
Open S2
 60 min
SESSION S3

Lean 4 Skeleton & the Open Sorry

Reading Chain.lean, the operator structures over metric spaces, the Spiral Return statement, and the kappa_lipschitz obligation still open as AXLE Issue #12.
Open S3
Every session leaves one open problem in the student's hands. The asymmetric basin boundary r* is open for iniciação científica. The kappa_lipschitz obligation is open for Lean contributors. This is Axiom 9 (Honest Incompleteness) made pedagogical.
The Principia Orthogona Series

Where Vol. IV Sits

Volume IV is the IMPA 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
G¹The Orthogonal Operator Framework979-8-9954416-2-5
G²TOGT: Applications Across Domains979-8-9954416-4-9
G³The Mini-Beast: Biological Instantiations979-8-9954416-6-3
G⁴Helical Attractors on Contact 3-Manifolds · GTCT T1 — The IMPA Edition (this volume)10.5281/zenodo.19117400
G⁵The Seed — Complete Completeness979-8-9954416-5-6
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