60. Learning from integral observations rather than point probabilities
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A finite polynomial family can be identified through moment operators. This gives a continuous-observation analogue of the core-table construction without pretending that a point has positive probability.
Assume the true operator density belongs to the declared class
Define the moment Gram matrix \(W_D\) by
A history is now a prescribed action sequence with observations constrained to measurable events of positive or zero probability. It produces an unnormalized predictive vector \(h_j\). A future event experiment produces a row functional \(t_i\). Choose \(r\) of each and form
The index \(j\) in \(G_{a,j}\) is a moment order. Odd-order entries can be signed expectations. They are not branch probabilities and are never normalized as if they were.
Proposition 54: Conditional finite-moment identification
Suppose the true predictive dimension is \(r\), the degree bound \(D\) is correct, and the selected \(r\times r\) core \(H\) is invertible. Exact population values of \(H\), every \(G_{a,j}\) and the initial test vector determine the complete polynomial operator family in test coordinates:
If every moment-operator error is at most \(\eta\) in one fixed induced norm, then coefficient reconstruction alone obeys
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Proof
A nonzero polynomial has strictly positive squared integral, so its coefficient vector \(c\) satisfies \(c^{\mathsf T}W_Dc>0\). Thus \(W_D\) is invertible. Let \(T\) have rows \(t_i\) and \(S\) have columns \(h_j\). Since \(H=TS\) is an invertible square core in an \(r\)-dimensional space, both \(T\) and \(S\) are invertible. Therefore \(G_{a,j}H^{-1}=TM_{a,j}T^{-1}\). The relation \(M_{a,j}=\sum_k(W_D)_{jk}C_{a,k}\) is exact integration of the polynomial expansion. Invert it and conjugate by \(T\) to obtain the displayed reconstruction. The error bound is the triangle inequality for that same fixed linear inversion. \(\square\)
The rank and degree assumptions are not learned from the determinant of \(H\). The determinant establishes coverage only after the ambient predictive dimension has been justified. The moment Gram can become poorly conditioned at higher degree. An orthogonal polynomial basis can improve this representation without removing an incorrect degree assumption.
The entries have an operational population meaning. A reset experiment for an event word, with one observation weighted by \(u^j\), returns a bounded random variable in \([-1,1]\). Its expectation is the corresponding entry of \(G_{a,j}\). With \(N\) independent trials for each of \(M\) predeclared entries, the same exponential-moment argument as Proposition 35 gives the simultaneous bound
with failure probability at most \(\delta\). The factor differs from the Bernoulli bound because the value range is two. Dependent trajectories, unknown resets and post-selection are not covered by that observation protocol. This edition uses exact constructed population tables, not a new sampling experiment.