M20c Operational Engineering: I. Musical and Audio Engineering Lexemic Synthesis, Coding, Neural Models & II. Hyperderivative Engineering Rotaderivative Filters, Logaderivative Optimisation, and Approximation Methods

This monograph develops Operational Engineering as a constructive discipline grounded primarily in Hyperderivatives, Rotaderivatives, and Lexemic Calculus. Its aim is to demonstrate that hyperderivative structures constitute a stand‑alone engineering calculus, capable of generating concrete algorithms for signal processing, neural computation, numerical approximation, and optimisation. The central objects are the logaderivative, rotaderivative, and their lexemic representations. These operators define a differential–algebraic language that replaces classical polynomial, Fourier, and finite‑difference methods by rank‑adaptive, exponentially efficient alternatives. All engineering constructions in the monograph are derived directly from the analytic and algebraic properties of these operators, without reliance on symbolic manipulation or heuristic tuning. Part I develops lexemic engineering architectures. A complete lexemic synthesizer is specified, including encoding, transformation, and reconstruction, where musical consonance, harmonic motion, and the circle of fifths arise as consequences of lexemic rotation and irrational crate paths. A lexemic codec is formulated algorithmically, with explicit complexity bounds and approximation guarantees. In parallel, a lexemic neural architecture is introduced, defining tokenisation, embedding, and training objectives natively in lexem space, rather than as discretisations of Euclidean data. Part II develops hyperderivative engineering proper. The logaderivative is shown to converge optimally to logarithmic behaviour and to act as an attractor for multiplicative dynamics. This property is exploited to construct adaptive filters with provably optimal sensitivity profiles. The rotaderivative, when expressed in the lexem basis, yields fractional‑order filters requiring only linear computational complexity, while remaining strictly optimal for exponential‑type signals. These constructions provide deterministic alternatives to polynomial approximation and spectral decomposition.