A Unified Numerically Stable Framework for Cubic Equations: Classification by Injectivity (One-to-One) and Elimination of Catastrophic Cancellation

This work presents a unified numerically stable framework for solving cubic equations with real coefficients. It is a direct improvement and consolidation of our two previous publications: - Previous Work I (Zenodo. 19376225): Covered the special cases b² = 3ac and b² 3ac. While those works treated each case independently, the present paper provides a complete unification through a novel classification based on the injectivity (one-to-one property) of the cubic function. This new perspective eliminates case-by-case fragmentation and offers a coherent algorithmic framework. Key improvements over the previous works: 1. Unified classification theorem based on the one-to-one property2. Single unified algorithm covering all regimes3. Comprehensive decision table and decision tree4. Consolidated numerical examples (20 examples) 5. Detailed practical applications (17 applications) 6. Complete numerical stability analysis with backward stability proof The proposed method is scale-invariant, backward stable under IEEE 754 double-precision arithmetic, and achieves relative errors below 10^-14 in all test cases, including near-degenerate roots and extreme coefficient scaling where classical Cardano loses up to 10^-4 accuracy or overflows. This work does not replace the previous publications; rather, it builds upon them to provide a higher-level unified perspective. All three works can be cited independently according to the reader's needs. References: 1 Moqadem, W. M. K. (2026). Numerically Stable Solutions for Cubic Equations: Eliminating Catastrophic Cancellation in Cardano's Formula. Zenodo. https: //doi. org/10. 5281/zenodo. 193762252 Moqadem, W. M. K. (2026). A Numerically Stable Reformulation of Cardano's Method for Cubic Equations with b² > 3ac. Zenodo. https: //doi. org/10. 5281/zenodo. 19412534