Dyadic Boundary Points and Scale-Invariant Coordinate Decomposition on Logarithmic Phase Grids

This paper investigates the geometric properties of positive scalars through the lens of dyadic scale decomposition. By mapping real numbers into a two-dimensional logarithmic phase space consisting of a macroscopic scale exponent and a fractional offset, we isolate specific boundary points that emerge naturally at the limits of dyadic intervals. We demonstrate that under continuous logarithmic polar embedding, these boundary points align on a single invariant axis. Furthermore, we formalize the scale invariance of the fractional offset under dyadic translation. This framework provides an alternative, geometrically intuitive visualization of numerical phase tracking, with potential didactic and structural applications in discrete signal analysis.