H9T

H9T: A Hierarchical Nine-Point Orthogonal Transform — Mathematical Foundations, Numerical Stability, and Comparative Performance H9T (Hierarchical Nine-Point Transform) is a computational framework investigating the application of nine-point orthogonal basis transformations to hierarchical data representation, numerical analysis, image and signal processing, compression, and anomaly detection. The framework employs a normalized nine-point Helmert basis and extends its application through structured multilevel transformations, allowing reversible analysis of multidimensional data. Its mathematical formulation provides an orthogonal change of basis, while its hierarchical organization supports experimentation with localized signal characteristics, coefficient selection, and multiscale representation. This publication presents the mathematical foundations, Python implementation, numerical verification, and preliminary comparative evaluation of H9T. Numerical experiments demonstrate near-machine-precision reconstruction using IEEE 754 double-precision arithmetic. A seven-level verification involving 4,782,969 data points produced a relative reconstruction error of approximately 5.61 × 10⁻¹⁶. Additional experiments using 50-digit and 100-digit arithmetic support the conclusion that the observed residuals arise primarily from finite-precision arithmetic rather than a defect in the underlying orthogonal transformation. The framework was also evaluated in preliminary image compression, signal processing, and anomaly-detection experiments against established alternatives, including the discrete cosine transform (DCT-II), Haar wavelets, and Fourier-derived features. Results establish computational feasibility and numerical stability while identifying limitations in compression efficiency and predictive performance. In the present experiments, DCT-II generally outperformed H9T in image reconstruction and synthetic anomaly classification, although H9T demonstrated measurable improvements over raw signal fea