44. Robustness is a quantified invariance statement
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A certificate for one fitted operator does not automatically cover neighboring operators. For uncertainty, the quantifier must range over the whole proposed model set, including correlations required by mass conservation.
Let \(\mathcal U\) be the convex hull of finitely many complete operator families \(\{B_s^{(j)}\}_s\). Every family must satisfy the exact per-action mass identities. The convex coefficient is shared across that family's symbols. A product of independent uncertainty sets can instead be handled by its joint vertices.
Proposition 40: Robust finite-domain certificate
For a fixed normalized generator matrix \(V\), suppose every vertex family has witnesses
Then every family in \(\mathcal U\) preserves \(C=V\mathbb R_+^m\) and is mass conserving. If its initial state lies in \(K\), it defines valid finite observation-tree probabilities at every horizon. This remains true when a family in \(\mathcal U\) is selected separately at each history.
For a complete facet description \(C=\{z:Gz\geq0\}\), let \(g_i^\top\) be its rows. A sufficient norm-ball certificate around \(\widehat B_s\) is
It covers all perturbations with \(\|B_s-\widehat B_s\|_2\leq\epsilon_s\) that also satisfy the exact mass constraints.
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Proof
For convex weights \(\theta_j\), take \(A_s=\sum_j\theta_jA_s^{(j)}\). The equality and nonnegativity survive convex combination, as does mass conservation. At each history, every selected family maps the current cone state back into the cone. Applying the mass identity at that node proves recursive normalization, regardless of how the family was selected. For the norm ball,
The margin makes every perturbed generator image satisfy all facets. Proposition 39, or the direct generator test, then supplies invariance. \(\square\)
The switching conclusion is structural validity, not an assertion that time-varying selection represents one fixed unknown parameter. A likelihood for a fixed mechanism and a likelihood for a mechanism selected anew at each history are different statistical models.
The norm-ball inequality is sufficient and can be conservative, especially at boundary images with zero slack. Failure of this margin is not a proof that no robust certificate exists. The exact vertex test retains uncertainty correlations that a separate bound for each entry may discard.