UDCT v7.14-TN01: Phase-Space Admissibility Audit (p=1.5, r_a/a≈0.4665)

Overview UDCT v7.14-TN01: Phase-Space Admissibility Audit (p=1.5, r_a/a≈0.4665) This independent technical note provides a retrospective, source-grounded examination of exploratory results developed during UDCT v7.14, focusing on phase-space consistency, gravitational-field diagnostics, and the physical admissibility of specified equilibrium models. The note consolidates mathematical interpretations, numerical boundary conditions, reproducibility considerations, and limitations arising from the earlier 50-step exploratory program and subsequent audits. Principal Findings 1. Central asymptotic cancellation at p = 1.5 Within a specified isolated spherical model, the exploratory asymptotic analysis identifies a cancellation of the leading distribution-function inversion coefficient at p = 1.5. This is a conditional mathematical result. The sign of the complete distribution function at the equality case remains unresolved without higher-order analysis and full-domain verification. Physical and mathematical interpretation: Within the specified isolated spherical QUMOND-inspired Plummer model, the generalized tracer-density family is ρₚ(r) ∝ [1 + (r/a)ᵖ]⁻⁵/ᵖ. Under the adopted central deep-MOND asymptotics, the potential difference scales as r³ᐟ², giving a tracer-density expansion exponent s = 2p/3. At p = 1.5, s = 1, causing the leading singular distribution-function inversion coefficient to vanish. This explains the mathematical origin of the identified cancellation without establishing the sign of the complete distribution function. 2. Local anisotropy boundary near r_a/a = 0.4665 Under the specified Osipkov–Merritt anisotropy prescription, the numerical analysis identifies a local finite-window tangency near r_a/a = 0.466466787396 for p = 1.5. This value is model-dependent and does not establish a universal anisotropy threshold or global phase-space positivity. Physical and mathematical interpretation: The dimensionless quantity r_a/a represents the Osipkov–Merritt anisotropy radius relative to the model scale radius. At the reported critical value, a sampled negative distribution-function region reaches a local zero-contact condition within the prescribed finite numerical window. The corresponding tangency coordinate is separately reported as r/a ≈ 0.47409755. Thus, r_a/a ≈ 0.4665 is an anisotropy-parameter boundary, not the physical radius of the tangency point or a globally established positivity threshold. 3. Physical admissibility and model consistency The technical note distinguishes gravitational Hessian diagnostics, Jeans-equation consistency, distribution-function non-negativity, and dynamical stability. A negative distribution function indicates inconsistency under the specified equilibrium assumptions; it does not independently falsify MOND, QUMOND, GQUMOND, or modified gravity generally. Scientific Scope and Limitations The early three-dimensional Hessian investigations and later spherical distribution-function calculations are separate computational branches. Hydrus I provides observational motivation and selected model inputs, but the present analysis is not a direct likelihood fit to the complete individual-star line-of-sight velocity dataset. Previously archived reproducibility checks do not constitute independent verification of every computational step. This note should therefore be interpreted as a retrospective exploratory technical audit rather than a peer-reviewed observational validation. Future Research Directions Future work will investigate whether the mathematical mechanisms associated with p = 1.5 and r_a/a ≈ 0.4665 persist, shift, or disappear under alternative physical assumptions and observationally constrained models. Hydrus I: Extend the analysis toward observationally constrained stellar-density profiles, anisotropy models, external-field effects, and direct comparisons with individual stellar velocities. Other dwarf galaxies: Test the sensitivity of corresponding phase-space admissibility boundaries to galaxy structure, tracer profiles, gravitational potentials, and environmental conditions. Bullet Cluster: Investigate whether analogous consistency constraints can be formulated in three-dimensional, non-equilibrium cluster dynamics, including evolving mass distributions and gravitational-lensing comparisons. The spherical equilibrium values p = 1.5 and r_a/a ≈ 0.4665 will not be assumed to transfer directly to a colliding galaxy cluster. Further work will also address higher-order asymptotic terms, full-energy-domain distribution-function positivity, numerical convergence, and independently reproducible confirmatory tests. Research Status Version: UDCT v7.14-TN01 Document type: Independent Technical Note Status: Retrospective exploratory audit; not a preregistered confirmatory result or a claim of a new universal gravitational theory. Author: Won Shik Paik — Independent Researcher, Auckland, New Zealand Zenodo DOI: https://doi.org/10.5281/zenodo.24107687 GitHub repository: https://github.com/wspaik5/UDCT_v7.14_TN01_Phase_Space_Admissibility_Audit