We present GeoNum (Geometric Number Scale), a novel number representation system that achieves high-precision scientific computing within standard browser environments constrained to IEEE 754 double-precision floating point. GeoNum encodes numerical values as field intensities within a zone-tessellated geometric structure, using sequence-derived zone boundaries and 2048 discrete shade levels aligned with the IEEE 754 exponent field width. The system introduces drift compartments as first-class quantities, providing transparent uncertainty quantification absent from conventional floating-point arithmetic. Validated on Hawking radiation temperature calculations spanning 60 orders of magnitude, GeoNum achieves 0.27% relative error with sub-shade drift accumulation of 0.345, while simultaneously reducing storage footprint compared to raw 64-bit doubles. The architecture supports domain-configurable zone boundaries, enabling a single precision framework to serve quantum mechanics, fluid dynamics, plasma physics, materials science, geophysics, and ballistics applications. GeoNum represents a fundamental departure from format-level solutions (IEEE 754 extensions, posits, takums) by relocating precision from bit-width to geometric structure.