Standard positional numeral systems carry geometric boundary distortion across interval steps $(B^{x-1}, B^x]$[span_0](start_span)[span_0](end_span). When modeled by the universal divisor $\Omega(B) = \sqrt{(B^x / B^{x-1}) + 1} = \sqrt{B + 1}$, standard decimal ($\Omega(10) \approx 3.316$) and binary ($\Omega(2) \approx 1.732$) systems introduce trailing fractional radical noise across scaling boundaries[span_1](start_span)[span_1](end_span).
Solving the boundary integer condition $\sqrt{B + 1} = k$ ($k \in \mathbb{Z}^+$) yields the Diophantine identity $B = k^2 - 1$[span_2](start_span)[span_2](end_span). For minimal non-trivial integer bases ($k=2$), Base-3 emerges as the unique minimal base where the boundary metric collapses completely into a rational integer ($\Omega(3) = 2$)[span_3](start_span)[span_3](end_span).
This interactive plot compares the geometric divisor $\Omega(B)$ across integer bases $B \in [2, 16]$[span_4](start_span)[span_4](end_span). Discrete highlighted nodes illustrate the rational collapse occurring at Base-3 and Base-8, contrasted against the continuous irrational noise floor of non-optimized baselines[span_5](start_span)[span_5](end_span).