A.3.3:4.4 - State-space and transition-law fields

The following optional view groups the claim content of one C.2.1 episteme:

U.Dynamics membership view {
  candidateEpisteme: U.Episteme
  entityOfConcern: EntityOfConcern
  effectiveReferenceScheme: U.ReferenceScheme
  claimGraph: {
    stateSpace: reference to an A.19 CharacteristicSpace with the admitted-state constraints
    transitionLaw: state-transition claim
    timeReference: continuous | discrete | hybrid
    transitionChoices: permitted continuations and conditions selecting among them
    probabilityLawIfSpecified?: conditional probability law over continuations
    inputsOrDisturbances?: CharacteristicSet
    observationRelation?: claim or exact relation reference
    constraintsOrInvariants?: claim content
    claimScopeIfReliedOn?: U.ClaimScope
    operatingRegionAndApplicabilityWindow?: ConditionSet
    calibrationOrParameterSourceIfReliedOn?: exact source or calibration-episteme reference
  }
}

stateSpace is claim content of this U.Dynamics episteme. It refers to an A.19 CharacteristicSpace: the product of declared characteristic value sets. The model states which combinations are admitted, through constraints or a predicate over that product. A.3.3.CC constructs such a configuration description; a predictive state can require additional information under :4.4.1. Characteristics retain their local meanings, units, Scales and comparability rules; C.16 supplies measurement construction when needed. A receiving evaluation may reuse this CharacteristicSpace; its scoring and judgement belong to that evaluation. Topology, geometry, aggregation or coordinate transformations are supplied over the domain where trajectories or comparisons use them; an independently selected organization of constituents and obtaining relations remains A.22 U.Structure.

transitionLaw is paradigm-agnostic. It can be an equation, relation, kernel, finite-state transition, queueing model, Bayesian update, Petri-net firing relation, simulation rule, learned predictor, or hybrid model, provided the state space, semantic basis, and applicability boundary are declared.

transitionLaw, observationRelation, constraintsOrInvariants, and calibrationOrParameterSourceIfReliedOn are ClaimGraph content or exact references inside the U.Dynamics episteme unless another governing pattern independently identifies one as an episteme, source, relation, or structure.

observationRelation specifies how the model connects its state to the observed quantity. For a deterministic observation, give the map y = h(x), where x is the model state and y the observed quantity. Identity observation (h(x) = x) is allowed only when the claim says the state coordinate is directly observed.

When proposing an exact deterministic one-step law on measured or aggregated coordinates, check whether two admitted states with the same current values of those coordinates, time and inputs can give different next coordinate values. Such a pair disproves that proposed law. Section 5.6 shows how to recover the missing predictive information or give a bounded answer.

For a proposed stochastic one-step law on aggregated coordinates, compare the next-observation distributions from the states it merges under the same time and inputs. If those distributions differ, the current aggregate omits predictive information. Retain a more informative state, condition a distribution over hidden states on the available history, or use a bound sufficient for the question. Equality for every merged group supports the aggregated one-step law under those conditions; longer use must preserve the later outputs and conditions it needs. Section :5.8 separates this question from long-run averaging.

A.3.3:4.4.1 - Construct the state and allowed continuations

1. Start with the question and participants. Name what can change, which result is needed and the conditions being considered. From the relevant subject account, identify the interacting participants and which of their differences can affect that result.
2. Describe allowed configurations. Use A.3.3.CC to select the participants’ retained differences, express compatible combinations, and choose independent coordinates, implicit constraints or finite enumeration for the receiving operation. Keep constraints on configurations separate from restrictions on rates or transitions. Sections :5.7 and A.3.3.CC:5 show how changed lengths, stock and job-order questions alter the description.
3. Recover the information needed for continuation. Separate changing state from parameters held fixed by the model and externally supplied inputs. Determine the initial data required by the proposed law. A position may also need its velocity; a computation may need its instruction position and saved local values. For a field, name its argument domain and value quantities, then obtain the needed initial and boundary data from its law and the modeled arrangement.
4. Construct the transition. Use the subject’s laws or operation rules to relate admitted states under the inputs. A.3.3.TR constructs that rule: state local effects, combine jointly active relations separately from alternative actions, and derive a small case. Check that the proposed continuation respects the constraints. If a constraint leaves the next state unresolved, supply the missing interaction or operation rule, or retain the alternatives it permits.
5. Interpret the alternatives. State who or what can select a continuation and under which conditions. Use a probability law when one is supplied or supported for that use. Counting possible continuations establishes their number; probabilities require a rule assigning them weights. The distinction changes the result in :5.9.
6. Test the description and choose the return. Apply the state-sufficiency comparison above to the prediction or observation needed now. A.3.3.PI constructs the retained state, usable history, predictive distribution or bound for that question and horizon. Return the state and transition account, a sufficient range or conditional conclusion, or the state distinction, law, input or observation still needed. Use C.11.DUA when choosing whether further information is worth obtaining.

The construction can finish before a complete dynamics model exists: a useful result may identify the missing physical interaction or computational rule. Section :4.1 admits U.Dynamics only when the episteme substantively states both the state space and the transition law.