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

In this research paper I introduce the Growth Depth Index (GDI), a coordinate-based system that provides explicit, computable invariants for asymptotic growthclassification of logarithmico-exponential functions. Rosenlicht’s canonical valu-ation provides an abstract framework for comparing asymptotic growth rates inHardy fields, but it remains non-constructive, with values defined through equiva-lence classes.We define GDI as a partial invariant, represented by quadruple coordinatesGDI(f ) = (kf , rf , af , cf ) ∈ R × R3+, for functions f in a class of logarithmico-exponential functions where the relevant asymptotic limits exist. The coordinatekf ∈ R provides a continuous parameter interpolating between discrete logarithmic-exponential levels (what I call ”between floors” functions), while rf , af , and cf offerdetailed classification within growth classes.We establish that GDI serves as a complete asymptotic invariant for functionswithin its domain: GDI(f ) = GDI(g) if and only if f ∼ Cg for some constant C > 0,and the lexicographic order on GDI coordinates corresponds exactly to asymptoticdomination. While GDI captures important aspects of growth structure, it is notequivalent to Rosenlicht’s full valuation. The main result establishes an order-preserving representation of growth comparison that is computationally accessible.