UDCT v7.14: Exploratory QUMOND Hessian and Phase-Space Consistency Tests (Steps 01–50)

UDCT v7.14 — Exploratory Working Paper (Steps 01–50) Overview This pre-preregistration working paper consolidates 50 stages of exploratory computational research investigating gravitational-field structure, Hessian diagnostics, Jeans modeling, and phase-space distribution-function (DF) consistency in dwarf-galaxy-motivated modified-gravity models. The study builds on the QUMOND framework and discusses its relationship to Milgrom's 2023 generalized QUMOND (GQUMOND) formulation, in which higher spatial derivatives can enter the gravitational theory. Computing Hessian diagnostics is not, by itself, evidence that the full GQUMOND field equations have been implemented. Principal Findings The later computational branch examines an isolated spherical Plummer gravitational source with a family of tracer-density profiles. Conditional central asymptotic analysis identifies a transition in the leading DF sign at the tracer-shape exponent p = 1.5. At exact equality, the leading term vanishes; the sign of the complete DF requires additional higher-order analysis. Under Osipkov–Merritt anisotropy, Step 50 identifies a local finite-window tangency at r_a/a = 0.466466787396 for p = 1.5. This result characterizes the explored model and numerical interval, not a universal or globally established phase-space positivity threshold. Scientific Scope and Limitations Negative DF values indicate phase-space inconsistency under the specified equilibrium assumptions. They do not establish dynamical instability or falsify MOND, QUMOND, or GQUMOND in general. Hydrus I provides observational motivation and selected model inputs, but this study does not claim a direct likelihood fit to the complete individual stellar line-of-sight velocity dataset. The early three-dimensional Hessian investigations and the later isolated spherical DF calculations are distinct computational branches. Reproducibility and Research Status This publication documents exploratory research Steps 01–50, including mathematical interpretation, computational provenance, numerical results, and identified limitations. Earlier archived audits reported successful reexecution of five selected scripts and agreement of 13 generated output files. Independent replication of all 50 stages has not been established. This work is an exploratory research record, not a completed confirmatory preregistration, peer-reviewed validation, or claim of a new universally applicable gravitational theory. Repository: https://github.com/wspaik5/UDCT_v7.14_PrePreregistration_Exploratory Reserved DOI: https://doi.org/10.5281/zenodo.23663161 Future Work Further investigations will examine higher-order asymptotics at p = 1.5, full-energy-domain DF positivity, alternative anisotropy prescriptions, external-field effects, and direct observational comparisons with individual stellar velocities.