UDCT v7.13 Post-registration: Hessian Sign Boundaries and Jeans Admissibility in Dwarf Galaxies

UDCT v7.13 Post-registration: Hessian Sign Boundaries and Jeans Admissibility Author: Won Shik PaikIndependent Researcher, Auckland, New Zealand DOI: https://doi.org/10.5281/zenodo.23539127 GitHub: https://github.com/wspaik5/UDCT_v7.13_Postregistration_Hessian_Archive Abstract This post-registration research archive documents a numerical investigation of Hessian-related sign boundaries and physically admissible solutions in modified-gravity models under the stationary Jeans framework for dwarf galaxies. The central objective is not to establish a new gravitational theory, but to examine the mathematical and physical conditions under which modified-gravity formulations remain viable, develop negative or otherwise inadmissible solutions, or become sensitive to higher-order spatial derivatives. The study consolidates Hessian-related numerical investigations, sign-boundary assessments, Jeans admissibility considerations, and Hydrus I-informed model calculations. Both successful and unsuccessful numerical outcomes are retained as part of the research record. Research Scope and Main Contributions Hessian sign-boundary analysis: Investigation of the influence of higher-order spatial derivatives on the sign structure and admissibility of candidate modified-gravity solutions. Jeans admissibility: Examination of physically acceptable and unacceptable regimes under specified stationary Jeans assumptions, including attention to negative velocity-dispersion-related solutions. Hydrus I-informed calculations: Use of observationally motivated properties of the ultrafaint dwarf galaxy Hydrus I as inputs for model-based numerical investigations. Numerical research archive: Preservation of computational investigations, including unsuccessful cases, limitations, and reproducibility materials. Post-registration assessment: Consolidation of the research record while distinguishing exploratory calculations from independently validated observational results. Observational Limitations An essential distinction is maintained between simulations using observationally informed Hydrus I model inputs and direct observational validation using the complete individual stellar line-of-sight velocity dataset. The former does not constitute the latter. Hydrus I-related model investigations also predate v7.13 within the UDCT research sequence. Therefore, the present work does not claim that Hydrus I was first introduced in v7.13 or that direct stellar-kinematic validation has been completed. The results should be interpreted within their stated model assumptions and numerical limitations. Conclusions This archive provides a documented framework for investigating Hessian-related sign boundaries and the physical admissibility of stationary Jeans solutions in dwarf-galaxy modified-gravity calculations. It does not establish observational confirmation of UDCT, demonstrate the general failure of MOND, or resolve the dark-matter problem. Its contribution is the preservation and critical examination of numerical boundary tests, including the conditions under which particular model assumptions lead to inadmissible solutions. Future Research Future investigations may include finer numerical resolution near sign-transition boundaries, systematic sensitivity and convergence tests, broader dwarf-galaxy model comparisons, and direct statistical confrontation with individual stellar velocity observations. Independent reproduction and comparison with established gravitational formulations remain important priorities. Archive Contents The deposited research materials include: Consolidated post-registration research report (PDF), including the principal analysis, figures, conclusions, and limitations. Original Hessian research archive (ZIP). README.md documenting the research scope and repository contents. requirements.txt describing the Python software dependencies. The accompanying PDF is the primary reference for detailed numerical results and their interpretation. Research status: Independent post-registration numerical research archive; not an observationally confirmed alternative gravitational theory. Keywords: Modified Gravity; Hessian; Jeans Equation; Dwarf Galaxies; Hydrus I; MOND; Numerical Analysis; Sign Boundaries; Reproducibility; Post-registration.