Foundational Theory

Hierarchical Calculus is a framework designed to unify descriptions of change by moving across measurement scales. Each step in rank changes the “reference operation” used to measure variation.

Concept DOI: 10.5281/zenodo.17917302

1) The hierarchical principle

Core idea

Rank increases when we measure variation using a more abstract operation (from additive to relative, then to log-relative, etc.).

\[ \text{Reference operation}: \begin{cases} \text{Addition} &\Rightarrow \text{rank }0 \ (\;D_0^1\;)\\ \text{Division (relative)} &\Rightarrow \text{rank }1 \ (\;D_1^1\;)\\ \text{Logarithm of logarithms} &\Rightarrow \text{rank }2 \ (\;D_2^1\;)\\ \text{Iterated logs (rank }3\text{ and above)} &\Rightarrow D_r^1,\ r\ge 3 \end{cases} \]

Notation rule: rank is \(r\), degree is \(n\). In this theory page we focus mainly on degree 1 (first-degree derivatives): \(D_r^1\).

2) General definition (rank \(r\), degree 1)

Let \(f(x)>0\) on a suitable domain. Define \(\ln^{(r)}\) as the \(r\)-fold iterated logarithm: \(\ln^{(0)}(x)=x\), \(\ln^{(1)}(x)=\ln x\), \(\ln^{(2)}(x)=\ln(\ln x)\), etc.

\[ \boxed{ D_{r}^{1} f(x) = \frac{d\,\ln^{(r)}\!\big(f(x)\big)}{d\,\ln^{(r)}(x)} } \]

This definition makes rank a change of measurement scale: we take the classical derivative after transforming both input and output by the same iterated logarithm level.

3) Special cases (degree 1)

Rank 0, 1, 2, 3

\[ D_0^{1} f(x)=\frac{df(x)}{dx} \]

\[ D_1^{1} f(x)=\frac{d\ln f(x)}{d\ln x} \quad\text{(relative derivative)} \]

\[ D_2^{1} f(x)=\frac{d\ln(\ln f(x))}{d\ln(\ln x)} \]

\[ D_3^{1} f(x)=\frac{d\ln^{(3)}(f(x))}{d\ln^{(3)}(x)} \]

Domain reminder: when rank \(\ge 1\), you must ensure the corresponding iterated logarithms are defined (e.g., for rank 2 typically \(x>1\) and \(f(x)>1\)).

4) Hierarchical consistency

Principle

If a function admits a rank-\(r\) derivative on a domain where the required iterated logarithms are valid, then the lower ranks are also valid on the same domain (or a superset).

\[ D_{r}^{1} f(x)\ \Rightarrow\ D_{r-1}^{1} f(x)\ \Rightarrow\ \cdots\ \Rightarrow\ D_{0}^{1} f(x) \]

5) Golden relations (rank-to-rank links)

These identities connect the ranks by explicit rescaling factors. For degree 1:

\[ D_{0}^{1} f=\frac{df}{dx} \] \[ D_{1}^{1} f=\frac{x}{f}\,D_{0}^{1} f \] \[ D_{2}^{1} f=D_{1}^{1} f\;\frac{\ln x}{\ln f} \] \[ D_{3}^{1} f=D_{2}^{1} f\;\frac{\ln(\ln x)}{\ln(\ln f)} \]

\[ \boxed{ D_{r+1}^{1} f = D_{r}^{1} f\;\frac{\ln^{(r)}(x)}{\ln^{(r)}(f)} } \]

Interpretation: moving from rank \(r\) to rank \(r+1\) is a controlled rescaling governed by the ratio of the rank-\(r\) logarithmic coordinates.

6) Interpretation of ranks

Rank 0: absolute (additive) change in \(x\).
Rank 1: relative change (scale-invariant viewpoint).
Rank 2: log-relative structural change (log-of-log scale).
Rank \(\ge 3\): higher-order hierarchical scales via iterated logarithms.

7) Citation

\[ \texttt{GOSSA AHMED. Hierarchical Calculus — Theory (definition of D\_r^n and golden relations). Official Website. Zenodo. https://doi.org/10.5281/zenodo.17917302 (2025).} \]