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

This work develops the physical celestial Clebsch–Gordan transform of a radiative two-particle Bose sector and establishes its intrinsic source-identification properties together with an obstruction to bounded Hilbert-space completion of the retained angular transform.

For every physical external-label pair, the intrinsic cross-ratio transform and its unitary full signed principal-series continuation are injective on the intrinsic physical source range generated by the ordered separable pair core. 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 physical beta coefficient is nonzero for every external-label pair, integer angular mode, and real Mellin frequency, establishing recovery of the original source from complete intrinsic celestial data.

The retained angular transform exhibits a separate analytic obstruction. An explicit sequence of finite-rank physical sources at fixed opposite-helicity external labels has uniformly bounded source Hilbert norms while its normalized retained-output norms diverge. The proof uses the physical cross-ratio kernel, an explicit coherent-output lower bound, exact zero-spin frequency reflection, Plancherel normalization, and a same-input retained-output comparison. Consequently, the retained transform admits no bounded extension to the original full fixed-input source Hilbert space at these labels. Finite-window continuity establishes failure of a common almost-everywhere decomposable bound on positive-measure external-label neighborhoods. The obstruction persists under the specified compact-smooth angular and log-energy source constructions.

These results preserve the source-aware Weyl representation geometry and establish intrinsic source faithfulness independently of bounded completion. The retained coefficient determines a closed graph relation and a canonical output-ambiguity quotient, while single-valued closability and a physically adequate replacement operator domain remain open.

The mathematical conclusions distinguish exact source identifiability from bounded retained-output continuity. They do not establish a bounded inverse, a uniform lower frame bound, injectivity on the entire completed source Hilbert space, or injectivity after retained fundamental-sheet projection. The principal results and their analytic foundations are accompanied by machine-checked Lean formalizations, with theorem-level provenance and verification records identified in the manuscript.