30. Probability validity is not a rank condition
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A real matrix product can yield a negative number. Therefore a fitted factorization is not a stochastic model merely because it is low rank or exactly interpolates a probability table. Spectral estimators can encounter negative-probability problems in practice [R10]. We address the structural issue directly.
In the canonical behavior space \(V\), define
Proposition 29: Canonical probability cone
The set \(\mathcal C\) is a closed convex pointed cone that spans \(V\). Every \(B_s\) preserves it. The mass functional is strictly positive on its nonzero elements, and
Consequently the normalized base
is a compact convex probability-valid state space. Every positive-probability branch maps \(\mathcal K\) into itself by normalized updating. In the core-test coordinates of Proposition 28, \(\mathcal K\subseteq\{1\}\times[0,1]^{r-1}\).
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Proof
Each test evaluation is a continuous linear functional on finite-dimensional \(V\), so the cone is closed and convex. If both \(v\) and \(-v\) belong to it, every test value is zero and \(v=0\). It spans \(V\) because it contains every column \(v_h\). Also \((B_sv)(t)=v(st)\), proving invariance.
The consistency law \(\sum_ov(t(a,o))=v(t)\) holds for each history column and hence throughout \(V\). Iterating it along any fixed action schedule expresses \(v(\epsilon)\) as a sum of nonnegative values for all observation words on that schedule. If \(v(\epsilon)=0\), each summand is zero, for every schedule and length, so \(v=0\). The same sums show \(0\leq v(t)\leq1\) when \(\lambda v=1\). They also give mass conservation. The base is closed and bounded in the finite core coordinates and thus compact. Invariance and positive normalization prove the posterior assertion. \(\square\)
This cone is defined by all finite tests. Its existence is not an algorithm for finding a finite list of inequalities. A finite list gives an outer approximation and can miss a longer-test violation. The cone also need not equal the convex hull of the states reachable from the fixed initial distribution.
A practical sufficient certificate can instead use a known closed cone \(C_0\) with nonempty normalized base, a strictly positive mass functional, \(b_0\in C_0\), \(B_sC_0\subseteq C_0\), and mass conservation. A nonnegative hidden-state realization supplies one after a coordinate change. Such a certificate may use information unavailable to an observable-only learner and must be labeled accordingly.
Negative entries of \(B_s\) do not, by themselves, prove invalidity. Coordinates can be signed while the represented state remains in the correct cone. Conversely, clipping negative coefficients can destroy the operator identities. Validity belongs to the pair consisting of the operators and their admissible cone, not to a coordinatewise heuristic.