The remaining finite-capacity question and partial slack

For a fixed $N\ge 3$, do the requirements defining ${\mathcal{A}}_N$, without diagonal faithfulness, force every capacity-$N$ outcome law and every positive residual ray to be Born and destructive projective, respectively?

A witness-specific version asks whether an admissible strategy can deviate at a specified input not already anchored by a certificate. For capacity three, one possible diagnostic is

No claim is made that this diagnostic captures every possible deviation.

The diagnostic at ${\chi }_{1/3}$ is identically zero on ${\mathcal{A}}_3$ by Corollary cor:third. It is therefore unsuitable as an open capacity-three witness.