Finite-rank gravitational kernels

Let ${\mathsf{P}}_N$ be finite-rank orthogonal projections on ${\mathfrak{h}}_g$, increasing strongly to $I$. Project the gravitational legs:

using adjoint or conjugate projections on the other two legs. The induced $B_{R,N,M}$ and ${A^{(2)}}_{R,N,M}$ use the same number projections and the same free scalar term.

At fixed $R,M$, Hilbert–Schmidt approximation of the cutoff kernels and Lemma lem:quadratic2 give operator-norm convergence as $N\to \infty$. There is also a joint estimate at fixed $M$:

where

The first convergence uses compactness of $L$, and the second uses Hilbert tensor approximation. Thus any claimed interchange of $R,N$ at fixed $M$ is supported by an explicit bound. No interchange with measured $M\to \infty$ dynamics is claimed.

Figure fig:fixed-M-convergence summarizes the spectral and finite-rank limits supported by the fixed-$M$ estimates.