FORMAL APPENDIX: Mathematical Foundations of the Silent Radix
## A.1 Positional Notation as a Rooted Tree **Definition A.1 (Positional Tree).** Let $b \geq 2$ be an integer (the base or radix). The positional tree $\mathcal{T}_b$ is the infinite rooted tree where: - The root is the empty string $\varepsilon$, representing zero. - Each node at depth $k$ has exactly $b$ children, labeled $0, 1, \dots, b-1$ (the digits). - A node is identified by the string of edge labels from the root to that node. - The integer represented by a node with label $d_k d_{k-1
FORMAL APPENDIX: Mathematical Foundations of the Silent Radix
Authors: Rowan Brad Quni-Gudzinas
Published: 2026-07
Abstract
A.1 Positional Notation as a Rooted Tree Definition A.1 (Positional Tree). Let $b \geq 2$ be an integer (the base or radix). The positional tree $\mathcal{T}_b$ is the infinite rooted tree where: - The root is the empty string $\varepsilon$, representing zero. - Each node at depth $k$ has exactly $b$ children, labeled $0, 1, \dots, b-1$ (the digits). - A node is identified by the string of edge labels from the root to that node. - The integer represented by a node with label $d_k d_{k-1
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