The Geometry of the Ternary Path
Abstract
The continuous ternary path stays in a rational-axis cylinder and revisits every tangential line with bounded gaps.
Definition 1.1 (The diagonal axis).
Formalization. D5/S1/Words/AbelianBorders/AbelianBorderQuestionGeometry.axis (✓ std3).
Source. Repository-derived.
Acknowledgement. Émilie Charlier, Tero Harju, Svetlana Puzynina, Luca Q. Zamboni (2015). Abelian bordered factors and periodicity. DOI: 10.48550/arXiv.1501.07464. URL: https://arxiv.org/abs/1501.07464v1.
Commentary.
The axis direction is (1,1,1), a nonzero vector with rational coordinates.
Definition 1.2 (Four vertices modulo the axis).
Formalization. D5/S1/Words/AbelianBorders/AbelianBorderQuestionGeometry.vertex (✓ std3).
Source. Repository-derived.
Acknowledgement. Émilie Charlier, Tero Harju, Svetlana Puzynina, Luca Q. Zamboni (2015). Abelian bordered factors and periodicity. DOI: 10.48550/arXiv.1501.07464. URL: https://arxiv.org/abs/1501.07464v1.
Commentary.
The representatives are (-3,-3,0), (0,-3,0), (2,2,0), and (0,2,0). Projection by T(p)=(p0-p2,p1-p2) sends them to the four vertices of a quadrilateral containing the projected path.
Theorem 1.3 (Cylinder containment and recurrent tangential lines).
Proof. Machine-checked in Lean as D5/S1/Words/AbelianBorders/AbelianBorderQuestionGeometry.geometric (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. Émilie Charlier, Tero Harju, Svetlana Puzynina, Luca Q. Zamboni (2015). Abelian bordered factors and periodicity. DOI: 10.48550/arXiv.1501.07464. URL: https://arxiv.org/abs/1501.07464v1.
Commentary.
Every point of the continuous graph is a convex combination of the four representatives plus a real multiple of the diagonal axis. The four exposed vertex lines each contain a cut point in every window from n through n+60. A supporting hyperplane whose contacts lie on a single axis-parallel line forces that line to be one of these exposed vertex lines. Thus cylinder containment and the required bounded tangential gaps both hold.