From Matrix Exponentials to Nonlinear Functions: A Common Structural View of Fourier Analysis, Hyperbolic Transformations, and Neural Learning (v1.0-en)

This research note presents a cross-disciplinary exposition of familiar mathematical structures, tracing a route from the natural exponential and matrix generators to projective nonlinear functions, tensor actions, and functional approximation. The two-dimensional generator family satisfying Aε2 = εI (ε = −1, 0, +1) yields elliptic, nilpotent, and hyperbolic linear flows. Projective component ratios turn these flows into the elementary Riccati equations underlying tan t, t, and tanh t. A brief Lorentz-boost example distinguishes a quadratic-form-preserving linear action from a nonlinear projective readout; a tensor-product example illustrates the induced generator through the Leibniz rule. Fourier modes, logistic/tanh functions, and simple neural-network representations are used as explanatory comparisons rather than claims of physical or algorithmic equivalence. The note also addresses higher-order derivative outer products, gradient propagation, and frequency-domain representations, with explicit qualifications and links to established literature. Its contribution is pedagogical and structural: it organizes known identities into a coherent, checkable line of reasoning and identifies questions for subsequent research notes. No new theorem, new neural-network architecture, or Lorentz symmetry of ordinary neural units is claimed. Files: the English PDF edition, with a ZIP containing the LaTeX source, the unmodified Markdown manuscript, PNG/SVG figures, and the reconstructed build settings. Version v1.0-en is the Zenodo release of manuscript revision v0.1-r3 (dated 2026-11-02); the text of the note is unchanged. The note has not been peer reviewed.