Interaction Topologies of Celestial Radiative States

This eighth paper in the Gaussian Physical States and Celestial Representation series establishes the successive interaction-dependent topologies required for continuous gravitational observables on a faithful physical celestial representation. Building on the physical Clebsch–Gordan transform of Paper VI and the cubic gravitational instability established in Paper VII, the analysis incorporates the complete finite-regulator cubic and fixed-action-leg quartic observables of the radiative Bose-pair sector. The original retained celestial coefficient is densely defined, injective, unbounded, and closable, with a canonical closed realization on a proper dense source domain. On its exact retained image, the complete cubic and quartic interaction families generate the strict hierarchy [τᵣₑₜτ_3τ3,4.] The first strict inclusion follows from the gravitational cubic adjoint-range obstruction. The second establishes that a physical quartic Riesz vector lies outside the sum of the genuine partial-adjoint range and the span of the full admissible cubic family. Its proof combines annular logarithmic source regularity, one-sided physical cell-face traces, two-log Mellin reconstruction, finite-regulator support reduction, and a seven-row augmented minor of complete-action gravitational coefficients. The determinant is certified through exact rational estimates involving five physical polarization moments. Further results distinguish compatible native null-face quartic refinements from occupied-to-null-face comparisons with persistent response defects. Finite-action cubic mesh and radius-child transport calculations provide additional tests of regulator compatibility. Together, these results characterize the continuity requirements imposed by successive gravitational interaction orders on the retained celestial state space. They provide a mathematical basis for investigating compatible asymptotic domains, physical evolution, and regulator transport in subsequent celestial scattering constructions. The present analysis concerns physical Hilbert states and complete finite-regulator action observables; it does not assume or construct a continuum scattering operator, connected scattering amplitudes, or interacting celestial operator products.