Part V. Curved domains and continuous observations
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Part IV separated a valid predictive model from a successful fit. Its finite positive lift gives a useful certificate, but its rank-three rotation example has no finite polyhedral probability domain. The same block left continuous observations outside the finite-alphabet construction. This part addresses both boundaries in a declared finite sector.
A curved domain can have a short matrix certificate even when it has infinitely many extreme states. A continuous observation can have a finite operator description even when its possible values are uncountable. Neither statement follows from finite predictive dimension alone. The constructions below require an explicit domain, a reference measure and a restricted observation-kernel family.
The block connects matrix positivity, integral observation semantics, population-moment identification, exact coarsening and approximation. It also identifies a defect that is easy to overlook: averaging a matrix transformation before applying its quadratic action does not equal averaging that action. The difference is a positive covariance contribution. Omitting it can destroy normalization even in a very small example.
There are established antecedents. Congruence representations and their positive semidefinite Gram matrices belong to the theory of completely positive maps [R16]. Polynomial Gram certificates are standard sum-of-squares machinery [R17]. Continuous-valued observable operator models already occur in Jaeger's work [R18]. The proofs here develop their specific connection to the realization and defect contracts of the large signal model. They do not claim to originate those ingredients.
This is a discrete-time theory with continuous-valued observations. It is not a continuous-time stochastic dynamics theorem. Production PR-LASM 2.3.0 is unchanged.