The Physical Celestial Clebsch–Gordan Transform of a Radiative Bose Pair

This work develops the physical celestial Clebsch–Gordan transform for a radiative two-particle Bose sector and determines whether the intrinsic physical source remains uniquely identifiable after the source-aware Weyl correction required by the kernel is retained. For fixed external labels, the normalized physical kernel defines angular Clebsch–Gordan rows on a dense ordered separable pair core. Under one explicit uniform core estimate, these rows extend uniquely to a bounded retained direct-integral transform with strongly measurable external-label dependence and global assembly over the physical two-particle carrier. Independently of that conditional completion theorem, the paper proves unconditional all-label injectivity of both the intrinsic cross-ratio transform and its full signed principal-series continuation on the intrinsic physical source range. The proof retains the complete unequal-label source character and combines source-pair coarea, Gaussian and harmonic testing, Abel–Fresnel reduction, shifted Mellin factorization, explicit beta-function evaluation, and logarithmic-cylinder Fourier uniqueness. The relevant beta coefficient is shown to be nonzero for every physical external-label pair, every integer angular mode, and every real Mellin frequency, forcing recovery of the original source. The result clarifies the relationship with the preceding source-aware representation theory: the source correction remains required by the physical kernel’s Weyl law, fixed-source fiberwise isometry fails, and the corrected source-aware transform is nevertheless injective on the intrinsic physical source range. The theorem does not assert a bounded inverse, a uniform lower frame bound, injectivity on the entire completed source Hilbert space, or injectivity after the additional retained fundamental-sheet projection. The principal theorem and its supporting analytic chain were developed and verified within a machine-checked Lean formalization.