A.3.3:4 - Solution

A.3.3:4.1 - Definition

U.Dynamics is a same-individual dependent kind of U.Episteme. Membership holds when one already identified episteme has the changing subject as its exact C.2.1 EntityOfConcern and its ClaimGraph, interpreted under the effective U.ReferenceScheme, substantively declares both a state space and a state-transition law for that subject. The law may include exogenous inputs, constraints, disturbances, and an observation relation.

The C.2.1 ClaimGraph, exact EntityOfConcern, and effective U.ReferenceScheme remain the episteme’s identity discriminators. A.3.3 adds no context field or second dynamics identity. A U.ClaimScope, operating region, applicability window, qualification interval, parameter regime, or scale band enters only through the exact claim that uses it and its subject pattern; changing one can change claim content without becoming an ambient container.

U.Dynamics can declare a deterministic transition, permitted alternatives, a probability law over continuations, or a combination of these. Its time description can be continuous, discrete or hybrid. It can make state-change claims about physical systems, software services, organizations, epistemes, claim portfolios, resource states, architecture characteristics, or another exact EntityOfConcern. If several subjects are jointly modelled, the exact C.2.1 EntityOfConcern must itself be an independently identified collection, system, or other admitted subject.

If empirical grounding is claimed, state the exact C.2.1 EpistemeEmpiricalGroundingRelation.

A.3.3:4.2 - Dynamics statement

Use this compact aid only when the ordinary sentence is insufficient for the current decision:

Dynamics statement:
  CandidateEpisteme:
  EntityOfConcern:
  EffectiveReferenceScheme:
  StateSpace:
  TransitionLaw:
  TimeReference:
  TransitionChoices:
  ProbabilityLawIfSpecified:
  InputsOrDisturbances:
  ObservationRelation:
  ConstraintsOrInvariants:
  ClaimScopeIfReliedOn:
  OperatingRegionAndApplicabilityWindow:
  CalibrationOrParameterSourceIfReliedOn:
  PredictionUse:
  EvidenceOrAssurancePathIfReliedOn:
  StopCondition:

These rows are an optional aid for the minimum claim content and separately governed references needed by the current use. C.2.1 identifies the candidate episteme.

A.3.3:4.3 - Conditional contributions to the work

Use a contribution below when its question arises while constructing or using the state-law account.

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.

A.3.3:4.5 - Evidence, prediction, conformance, drift, and calibration

Let D be a U.Dynamics about exact EntityOfConcern E. Let W denote only exact dated U.Work occurrences when Work is current, and let O denote separately identified observation, telemetry, source, or measurement records.

These expressions name claim-side calculations or questions. When an observation, conformance, drift, measurement, evaluation, gate, or assurance result is claimed, the applicable evaluation or measurement declaration states the criterion and result semantics, and the actual application and result are identified separately; C.2.1 identifies any persisted result episteme, and use A.10 or B.3 only for the separately claimed reliance or assurance use.

Calibration Work and its domain result may support a later dynamics episteme whose changed ClaimGraph receives its own C.2.1 identity; an EpistemeEditionRelation obtains only when C.2.1’s exact continuation predicate is separately established.

A.3.3:4.6 - Prediction use in comparison or gating

A prediction used for comparison, release, gate, assurance, or work preparation states the exact dynamics edition, predicted Coordinates, operating region, horizon, and relevant error or uncertainty. State any time step, parameter regime or source-currentness condition on which the prediction or receiving use depends. The direct consumer’s policy then states which observation, validation, sensitivity, robustness, stability, or normalization-composition conditions that use requires.

A fresh observation may replace or check the prediction when the policy calls for it. A non-expansive bound, another sensitivity bound, or commutation with a normalization step is required only when the named use relies on that property. If the required conditions are absent or fail, the prediction cannot carry that use. State a needed currentness claim through C.27.TA. Use C.27 when a separate authored temporal-claim adequacy question remains. Obtain a needed internal-constraint result through A.20; use A.21 for a gate decision or the applicable decision pattern for another decision.

A.3.3:4.7 - Apply or transfer a dynamics model

Stay in A.3.3 when the transition law or observation relation uses accepted local dynamics under one explicit semantic basis and applicability boundary.

Use C.29 when the law’s use depends on a contested transfer, cross-domain analogy, learned or speculative mathematical lens, scale change, abstraction, quotienting or reusable explanation across contexts. Establish the preserved and lost structure, operating region or scale window, applicable rival, lens-use boundary and stop condition. Then state the resulting dynamics law and its observation, constraint and calibration conditions here. The direct prediction consumer still determines reliance under :4.6.

A.3.3:4.8 - Recover an ambiguous source claim

When a source label such as “process” or “model” leaves the asserted relation unclear, recover one concrete claim before assigning a kind. Establish U.Dynamics membership on the episteme under :4.1 and a Method claim on the semantic way of doing under A.3.1. When an episteme is also claimed to describe that Method, apply A.3.2’s membership rule independently. Section 4.3 supplies the other conditional contributions; use E.10.ARCH if the represented relation remains unresolved.