Lemma: Compactness of the selective-output family

The space ${\mathcal{K}}_{N,\nu }$, with the uniform trace-norm topology on all branch maps, is compact.

Proof. There are finitely many canonical register sizes, subsets, and outcomes at capacity $N$. Each ray space is compact, and each finite-dimensional output set is closed and bounded. The common modulus gives equicontinuity.

For completeness, choose a countable dense subset of each input ray space. Repeated subsequence extraction makes a sequence converge on every selected point. The common modulus then makes that subsequence uniformly Cauchy: cover each input space by finitely many sufficiently small balls centered at selected points. The resulting uniform limits satisfy the same modulus. Positivity, rank at most one, trace normalization, and equivariance are closed under uniform limits. Thus every sequence has a convergent subsequence in ${\mathcal{K}}_{N,\nu }$.