The equal-time canonical commutation relation for the transverse-traceless graviton and its canonical momentum is the transverse-traceless projector. At zero momentum the supplied polarization completeness relation gives
(1)
where symmetry and tracelessness of remove the trace term. Thus the charge generates precisely the transformation in part i:
(2)
This is the soft, field-independent part of the Noether charge associated with the large diffeomorphism.

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