66. A complete continuous-observation example
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Use the same three-coordinate matrix embedding as Section 57. Set
The direct action uses \(X\mapsto K(u)XK(u)^{\mathsf T}\). The rotated action first applies the unchanged rotation congruence from Section 57, giving
Both are exactly normalized because \(\int K(u)^{\mathsf T}K(u)\,du/2=I\) and \(W^{\mathsf T}W=5I\). In predictive coordinates, write
Although \(c(u)\) is negative for part of the interval, both actions preserve the disk cone. Entrywise sign is not the positivity criterion in these coordinates. The initial state is \(z_0=(1,0,0)^{\mathsf T}\), or the matrix \(I/2\).
66.1 Event tables and moment reconstruction
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Let the three future tests be the empty test, a nonnegative observation after action zero, and a nonnegative observation after action one. Their rows form
Use histories consisting of the empty history, a nonnegative observation under action zero, and that history followed by a nonnegative observation under action one. Their unnormalized columns are
The learner is supplied only the resulting integral tables and the declared class. Its core is
The core is not required to be symmetric. Its rows are tests and its columns are histories, which are different experiments. The three moment orders use
Proposition 54 reconstructs \(QB_a(u)Q^{-1}\) from \(H\) and the six inserted-moment tables. The code that performs this reconstruction imports no generating matrices. The separate generator produces the tables and is used only in the verification suite. These are population values from an exact example, not learned sample estimates.
66.2 Two densities and a genuine event probability
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For a direct observation value \(u=1/2\), the density and updated state are
If the rotated action next gives \(u=-1/2\), they are
Their product \(413/1280\) is a joint density with respect to \((du/2)^{\otimes2}\), not the probability of those two exact values. The two point values each have zero probability in this nonatomic model.
By contrast, the event \(u_1\geq0\) under action zero followed by \(u_2\geq0\) under action one has actual probability
and its conditional state is \((1,96/107,-24/107)^{\mathsf T}\). Exact bin operators produce this value without quadrature.
The final point-conditioned state remains strictly inside the disk:
A pure boundary state can instead have zero density at isolated observations. The checker rejects conditioning on such a zero-density branch. It does not manufacture a posterior by flooring its denominator.
66.3 The missing covariance is exactly calculable
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On an interval with midpoint \(v\) and width \(h\), the average matrix instrument is
The correction is itself a positive congruence. The cell operator multiplies this mean by the cell mass \(h/2\). Equal bins have width \(2/n\), so omitting the covariance loses trace \(3/(4n^2)\) over the full partition. The reference calculation checks both the exact matrix identity and this mass defect for several rational partitions.