given by eq.(\[Frinf\]); [*they explicitly verify*]{} eq.(\[frinf\]), and this provides a positive test of regular $N \to +\infty$ behaviour for the main functional relation in the form (\[Ffr\]).

The $N=\infty$ coboundary and cocycle identities
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For the homogeneous finite-$N$ problem, a closed functional equation for the complete determinant $D(\lambda)$ is supplied in Appendix as “the cocycle identity" (\[COC\]), a sum of $L =O(N)$ terms. Its naive $N \to +\infty$ limit will be an integral relation for $D(\lambda)$, further reducible by the residue calculus. It is however simpler to work out the $N \to +\infty$ limit directly upon the (logarithm of the) underlying “coboundary identity" (\[Bfr\]) equivalent to eq.(\[Ffr\]): this limit is manifestly equivalent to eq.(\[frinf\]) and has the (additive) coboundary form, $$\label{COBI}
-2\lambda {\d \over \d\lambda} \,\log \Delta^{\rm P}(\lambda) \equiv 
{1 \over \Delta(\lambda)} - 1 .$$ $\Delta^{\rm P}$ being meromorphic and $\Delta$ entire, the main solvability condition for eq.(\[COBI\]) is that the residues at the poles of $1/\Delta$ must match the explicit residues of the left-hand side, resulting in a curious constraint upon $\Delta$ alone at its zeros, $$\label{COCI}
-2 \Bigl[\lambda {\d \over \d\lambda} \Delta(\lambda) \Bigr] _{\lambda=-E_k} =
(-1)^k
\qquad \mbox{for all } k \in \Bbb N ,$$ which stands 