UDCT v7.13 Preregistration: Refined Jeans Sign-Boundary and Hessian Diagnostics in Hydrus I

UDCT v7.13 — Preregistration Refined Jeans Sign-Boundary and Hessian Diagnostics in Hydrus I This preregistration specifies a prospective follow-up to UDCT v7.12, which mapped signed Jeans solutions in Hydrus I across velocity anisotropy and screening-length parameters. The previous study identified 274 negative cases, 48 positive sampled candidates, and 41 indeterminate cases among 363 primary decisions. Although 24 adjacent intervals exhibited strict raw sign changes, their transition locations remained numerically unresolved. UDCT v7.13 aims to investigate these unresolved boundaries through predefined local refinement, numerical convergence checks, and separate Hessian diagnostics. The objective is to distinguish robust sign transitions from numerical uncertainty and examine whether any relationship exists between the Jeans admissibility boundaries and the local properties of the gravitational field. The investigation will retain the established model definitions and inherited physical assumptions. Numerical tolerances, parameter grids, stopping criteria, and outcome classifications will be fixed before confirmatory execution. This is a preregistration, not a report of completed v7.13 simulations. Positive sampled Jeans solutions do not by themselves establish physical equilibrium, dynamical stability, or observational agreement. No universal Hessian threshold or validation of modified gravity is assumed. Reproducibility and registration GitHub: https://github.com/wspaik5/UDCT_v7.13_Preregistration_Jeans_Boundary_Refinement Reserved DOI: https://doi.org/10.5281/zenodo.23342971 Previous results (UDCT v7.12): https://doi.org/10.5281/zenodo.23296444 Author: Won Shik PaikIndependent Researcher, Auckland, New Zealand Preregistration Protocol Revision — 14 October 2026 This revision incorporates additional prespecified numerical validation criteria into the v7.13 preregistration, including convergence checks, numerical uncertainty tolerances, sign-boundary classification and stopping rules, and a defined procedure for three-dimensional Hessian diagnostics. The original scientific scope, inherited v7.12 results, and all 24 predefined unresolved boundary intervals remain unchanged. This amendment was made after the initial publication of the preregistration and before the proposed confirmatory v7.13 calculations. No new scientific results are reported. Implementation and independent validation of the additional numerical procedures remain required before confirmatory execution.