Dipole energy and local admissibility boundaries in matter-supported GFF neutron stars

This paper investigates where static neutron-star branches admit a regular physical perturbation problem and how stability can be assessed within the locally admissible region. The analysis considers even-parity dipole (\(l=1\)) perturbations in a conformally matter-coupled scalar–Gauss–Bonnet realization of the G-Field Framework (GFF). A common-action baseline at \(\beta=0\) is compared with an extension containing the cubic scalar derivative interaction \(\beta X\Box\chi\). A source-code analysis identifies a relative normalization inconsistency between the metric and scalar Gauss–Bonnet terms in a historical implementation, motivating background calculations consistent with the action used for the perturbations. The study distinguishes three local admissibility mechanisms on the analyzed equilibrium sequences: obstruction of a regular central expansion, degeneration of an interior scalar spatial principal coefficient, and reversal of the physical kinetic inertia. It derives algebraic relations connecting central compatibility and background highest-derivative degeneracy to the perturbation principal coefficients. By retaining an angular-fluid variable eliminated in a frequency-dependent reduction, the analysis reconstructs an action-compatible dipole energy and clarifies the complementary zero-frequency obstructions of different reduced variable systems. A two-resolution, three-radius static dipole matching calculation on an action-consistent baseline finds no resolved kernel signal in the examined time-even quotient problem. Exact rational calculations certify positivity for 413 stored polynomial cells and zero winding for two normalized Evans polygons. These are certificates for finite numerical objects, not a global stability theorem for the continuous stellar perturbation operator. The paper states the additional coefficient-error, boundary-domain, exterior, and spectral-enclosure conditions required to connect the finite certificates to an exclusion of negative eigenvalues. Its central contribution is a framework that separates equilibrium existence, local admissibility, energy structure, and spectral evidence, without identifying any one of these with complete stellar stability. Monopole dynamics, stability across all multipole sectors, and nonlinear stability remain outside the conclusions established here.