Source-Aware Weyl-Reduced Celestial Representation

The preceding paper constructs a qualified physical radiative Bose pair and hands forward its generic completed two-particle Hilbert space. This paper studies the Weyl/shadow structure of the normalized celestial Clebsch–Gordan kernel at fixed external representation labels and tests whether its contact-normalized transition can be realized fiberwise by isometries.

For fixed external labels, the normalized kernel determines a source unitary, a unit-modulus channel phase, and a completed angular shadow. The source unitaries form a strongly continuous representation of Z × R and become explicit translations in source spectral coordinates. Combined with principal-series Weyl reflection, they define a measure-preserving involution on the joint source/output spectral space. The Weyl-fixed subspace is unitarily equivalent, by normalized restriction, to the L² space over a fundamental output domain while retaining the full source Hilbert space.

The contact-normalized transition obeys a stronger metric constraint. An independent reconstruction gives the exact spin-one value T₁,₀ = 1 − 4 log 2. Joint continuity in the two incoming Mellin frequencies and the output principal-series frequency extends |T|² > 1 to an open positive-measure subset of the joint spectral parameter space. Since fiberwise isometry requires unit Gram, no almost-everywhere fixed-source isometric realization agrees with the normalized physical kernel.

The paper also classifies the remaining completion and source-frame freedoms. Every tempered extension of the audited punctured kernel differs by a unique Dirac contact term; the homogeneous/Fourier-normalized completion selects the canonical zero-contact coordinate. Global source-basis changes preserve the nontrivial source action up to conjugacy, while a label-dependent measurable source-frame change trivializes the bare two-sheet cocycle. These results provide the representation-theoretic input for the direct-integral completion and range analysis developed in Paper VI.