THE SILENT RADIX: Positional Notation as Ultrametric Tree and the Calculus of Indications as Remedy

Positional notation is not merely a convention for writing numbers — it is an ultrametric tree whose place-value columns encode a hierarchical distance function. The base (radix) is the structural parameter that determines the tree topology, yet in everyday practice it goes unstated and unexamined: a silent radix. This paper argues that the distinction between a number and its representation collapses when the radix is treated as arbitrary convention rather than structural geometry.