M10 Hyperderivatives - The Algebra of Derivative Cycles, Logaderivatives, and the Spectral Theory of Differentiation

This monograph develops the theory of hyperderivatives, defined as iterated logarithmic derivatives that act as operators on functions rather than on values. Starting from the basic log‑derivative f↦ln⁡∣f′∣, the work studies repeated application of this operator and classifies functions according to the number of steps required to reach the universal attractor ln⁡x xlnx up to an additive constant. This yields a finite derivative‑cycle stratification of functions into classes characterised by convergence depth, distinguishing power functions, logarithmic families, and degenerate affine cases. The monograph introduces logaderivative operators and establishes exact results for polynomial, monomial, and mixed families, including critical cases where iteration terminates or becomes undefined. A general basin stratification theorem is proved, providing necessary and sufficient conditions for the convergence depth of a function under hyperderivative iteration. Beyond classification, hyperderivatives provide a compact language for ordinary differential equations, recasting qualitative behaviour of ODEs—such as scaling, damping, and attractor structure—in terms of derivative cycles rather than explicit solutions. This viewpoint allows families of evolution equations to be compared structurally and supports applications where logarithmic perception and scaling invariance play a central role, including models arising in the theory of music. Hyperderivatives supply a spectral view of differentiation that complements the rank‑based analysis of hyperoperations, offering a systematic framework for analysing functional dynamics across mathematics. The purpose of this deposit is to document the definitions, results, and conceptual priority of hyperderivatives within the hyperoperation theory series.