41. The object to be certified
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Fix a real predictive space \(E=\mathbb R^r\), a nonzero row \(\lambda\), and a finite family \(B_s\), indexed by \(s=(a,o)\). The action and observation alphabets are finite and nonempty. Every declared observation branch is present, including branches represented by the zero operator. Temporal words act from right to left:
The mass identity is an exact constraint,
An initial state has \(\lambda z_0=1\). The proposed probability of a word is \(\lambda B_wz_0\). An invertible coordinate change must transform \(z_0\), \(B_s\), \(\lambda\) and the admissible cone together, as in Proposition 37.
A probability-domain certificate specifies a closed, pointed, generating cone \(C\subset E\), with \(\lambda\) strictly positive on \(C\setminus\{0\}\), such that
Its normalized base is \(K=\{z\in C:\lambda z=1\}\). These conditions imply nonnegative word probabilities and their per-action normalization at every depth. Positive-probability updates remain in \(K\). Zero-probability branches have no conditional posterior and must not be assigned one by denominator flooring.
There are three separate claims. Internal validity says the fitted model is a stochastic model. Data compatibility says it agrees with the observations within a specified error budget. Coverage of the generating process says the assumptions and uncertainty set actually include the process that generated those observations. The first can have a deterministic certificate even when the other two remain conditional.