Dynamical Ill-Conditioning of a Faithful Celestial Representation

A faithful celestial representation can uniquely determine physical radiative states while admitting gravitational observables that are discontinuous in the inherited celestial Hilbert norm. We establish this phenomenon for a completed celestial Clebsch–Gordan representation of radiative Bose pairs. Under an explicit uniform completion estimate, the retained celestial transform is injective, while the Riesz vector of a distinguished action-generated gravitational cubic lies outside the range of its Hilbert adjoint. The cubic is constructed from ordered third tetrad-coordinate variations of the complete finite-regulator six-sector action, followed by physical radiative readout, incoming Bose symmetrization, and outgoing Hilbert duality. An exact rational evaluation on a physical mean-helicity packet establishes a nonzero gravitational response of magnitude at least 1/17280. The adjoint-range obstruction follows from a source-regularity mismatch: retained-adjoint fibers satisfy finite logarithmic translation-energy estimates on fixed Fourier annuli, while the physical cubic exhibits a nonzero momentum-cell face jump producing an incompatible translation tail. The resulting representation determines physical states exactly and permits arbitrarily accurate approximation of the gravitational observable by retained-data functionals, uniformly on the physical Hilbert unit ball. Every such convergent approximation requires reconstruction norms that diverge. The obstruction persists under equivalent retained Hilbert norms and boundedly invertible reparameterizations of the celestial data. We also establish that no nonzero continuous observable on the completed physical Bose space descends through a single fixed-label source extraction. These results identify an interaction-sensitive topological constraint on faithful celestial state representations and distinguish exact state determination, approximate observable recovery, and continuous gravitational reconstruction. The conclusions concern the specified finite-regulator radiative construction, conditional on the stated completion estimate. Stability under momentum-cell smoothing and regulator refinement, as well as continuum scattering, celestial operator products, and interacting Ward identities, remain open directions. This is Paper 7 in the series Gaussian Physical States and Celestial Representation. An associated private Lean development provides machine-checked verification of the principal formal constructions and mathematical dependencies.