Geometric Number Scale (GeoNum)

We introduce GeoNum, a logarithmic-domain numerical representation that augments floating-point arithmetic with explicit tracking of quantization and conditioning-induced uncertainty ("drift"). GeoNum partitions logarithmic magnitude space into zones and represents values via normalized intensity within each zone. Arithmetic is performed in log-space, enabling closed-form multiplication and division and stable evaluation of addition via log-sum-exp and log-diff-exp formulations. We define a conditional stability theorem: for operations outside the cancellation regime, the accumulated drift provides an upper bound on logarithmic error proportional to zone width. We prove this bound for encoding and multiplicative operations under a zone-normalized propagation rule, and show that subtraction near cancellation is inherently ill-conditioned and must be excluded from bounded-error guarantees. We introduce a trust classification system that encodes these conditions explicitly, enabling downstream solvers to reason about numerical validity.