The Growth Depth Index: Explicit Coordinates for Growth Classification in Hardy Fields

We introduce the Growth Depth Index (GDI), a coordinate system that provides explicit, computable invariants for asymptotic growth classification of functions with regular asymptotic behavior. Whereas Rosenlicht’s canonical valuationfor Hardy fields is abstract and non-constructive, GDI assigns to each function aquadruple of real parameters that capture its growth in a computationally accessible way.The GDI of a function f is defined as GDI(f) = (kf , rf , af , cf ) ∈ R×(0, ∞)2×R,provided f is GDI-admissible—meaning it admits a stable asymptotic expansion.This class includes all logarithmico-exponential functions and many special functions with known asymptotics. The parameter kf interpolates continuously betweendiscrete growth levels (“between floors” functions), while rf , af , and cf refine theclassification within each level.We prove that GDI is a complete asymptotic invariant for its domain: (kf , rf , af ) =(kg, rg, ag) if and only if f and g are asymptotically equivalent up to a constant factor (i.e., f ∼ (cf /cg)g). Moreover, the lexicographic order on the triple (kf , rf , af )exactly reflects asymptotic domination: (kf , rf , af ) <lex (kg, rg, ag) if and only iff = o(g). Full equality GDI(f) = GDI(g) holds if and only if f ∼ g. Thus, GDIprovides an order-preserving, explicit representation of growth rates for a broadclass of functions, bridging the gap between abstract valuation theory and practical computation.