Lemma: Common probability modulus

For a fixed admissible strategy at capacity $N$, there is a nondecreasing function ${\omega }_N$, tending to zero at zero, such that

for every same-size pair with at most $N$ qubits.

Proof. Each fixed ray space is compact, so continuity implies uniform continuity. Up to name relabeling there are finitely many register sizes and measured subsets at capacity $N$. Take the maximum of their uniform moduli.