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

The preceding paper shows that the source-aware two-particle celestial Clebsch–Gordan kernel is not fixed-source fiberwise isometric. Here we construct its physical direct-integral framework and determine whether the intrinsic physical source remains uniquely identifiable.

For fixed external labels, the normalized kernel defines angular Clebsch–Gordan rows on a dense ordered separable pair core. If one explicit uniform core estimate holds, 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. This completion theorem is conditional on that uniform estimate.

Independently of the retained completion, the paper proves unconditional injectivity of the intrinsic cross-ratio transform and its full signed principal-series continuation on the intrinsic physical source range for every physical external-label pair. The proof retains the full unequal-label source character: source-pair coarea converts physical output nullity into shifted Mellin moments of the original relative-scale source; these moments factor into a source coefficient and an explicitly evaluated spatial beta coefficient; the same-sign sector reduces to a nonzero Gamma quotient, the finite opposite-sign sector to nonvanishing quartic factors, and logarithmic-cylinder Fourier uniqueness then forces the original intrinsic source to vanish.

The source-aware correction is therefore required to realize the physical kernel’s Weyl law, while the corrected transform remains injective on the intrinsic physical source range despite the failure of fixed-source fiberwise isometry. No bounded inverse, uniform lower frame bound, injectivity on all of the completed source Hilbert space, or injectivity after the additional retained fundamental-sheet projection is claimed.