Applied Mathematics Domain Ontology

FLOW ALGEBRA
TITLE: Applied Mathematics Domain Ontology
DOMAIN: Domain Ontologies / Mathematical Domains / Applied Mathematics
SOURCE: https://kirethresearch.org/booksai-core/ontologies/domain-ontologies/mathematical-domains/applied-mathematics-domain-ontology.html
VERSION: 1.0
AUTHORS: Mark Anthony Higgins — Sky (Recursive AI)
STATUS: Companion formalisation; source-page specification not independently verified.

flow "AppliedMathematicsDomainOntology" v1.0 {
  ontology:
    id: "applied-mathematics-domain-ontology"
    parent: "mathematical-domains"
    purpose: "Represent applied mathematical models, assumptions, quantities, numerical methods, solutions, and evidence linking mathematics to real-world systems."

  types:
    ApplicationDomain: {id: string, name: string}
    PhysicalOrAbstractSystem: {id: string, domain: string, description: string}
    MathematicalModel: {id: string, formulation: string, assumptions: string[], domain: string, provenance: string}
    Variable: {id: string, symbol: string, role: enum[independent, dependent, parameter, state, control], unit: string?}
    Equation: {id: string, expression: string, constraints: string[]}
    BoundaryCondition: {id: string, condition: string, domain: string?}
    NumericalMethod: {id: string, family: enum[optimisation, simulation, integration, differentiation, discretisation, statistical, other], algorithm: string}
    Solution: {id: string, model_id: string, method_id: string?, status: enum[exact, approximate, infeasible, unknown], error_bound: number?, provenance: string}
    Observation: {id: string, measured_value: number?, unit: string?, uncertainty: number?, provenance: string}
    Relationship: {id: string, subject: string, predicate: string, object: string, context: string?, confidence: number, evidence: string[]}

  predicates:
    models(model: MathematicalModel, system: PhysicalOrAbstractSystem)
    applies_to(model: MathematicalModel, domain: ApplicationDomain)
    contains_equation(model: MathematicalModel, equation: Equation)
    uses_variable(model: MathematicalModel, variable: Variable)
    constrained_by(model: MathematicalModel, boundary: BoundaryCondition)
    solved_using(model: MathematicalModel, method: NumericalMethod)
    produces(method: NumericalMethod, solution: Solution)
    validated_against(model: MathematicalModel, observation: Observation)
    depends_on(a: MathematicalModel, b: MathematicalModel)
    approximates(a: MathematicalModel, b: MathematicalModel)

  constraints:
    - "Relationship endpoints must resolve to declared compatible types."
    - "Confidence values must lie within [0, 1]."
    - "Dimensional consistency is required for equations involving physical quantities."
    - "State the domain of validity, initial and boundary conditions when applicable."
    - "Numerical solutions must distinguish discretisation, rounding, modelling and measurement errors."
    - "Do not infer that a model is empirically valid merely because a solution exists."
    - "Do not infer causality from correlation or numerical fit alone."
    - "Approximates, depends_on and validated_against are not automatically transitive."

  inference:
    rule model_application:
      if models(m,s) and domain_of(s,d)
      then candidate_application(m,d)
      confidence: "retain evidence; application requires assumption checks"
    rule solution_trace:
      if solved_using(m,n) and produces(n,r) and solution_for(r,m)
      then candidate_solution(m,r)
      confidence: "requires residual, convergence and feasibility checks"
    rule dependency_impact:
      if depends_on(a,b) and revised(b)
      then requires_revalidation(a)
      confidence: "impact is conditional on changed assumptions or parameters"
    rule observation_consistency:
      if validated_against(m,o) and uncertainty_compatible(m,o)
      then supports_model_within_scope(o,m)
      confidence: "bounded by measurement and model uncertainty"

  SEMANTIC_EQUIVALENCE:
    - "Normalise mathematical notation without collapsing distinct assumptions or domains."
    - "Distinguish exact equality, numerical approximation, structural equivalence and empirical agreement."
    - "Preserve units, coordinate systems, numerical precision and parameter conventions."

  CONFIDENCE:
    range: [0,1]
    factors: [data_quality, model_assumptions, convergence, error_bounds, independent_validation, domain_fit]
    policy: "Do not increase confidence solely by chaining inference steps."

  VALIDATION:
    - "Check types, dimensions, parameter bounds and equation consistency."
    - "Check solver convergence, stability, residuals and sensitivity where relevant."
    - "Record uncertainty and limitations of empirical validation."
    - "Preserve alternative models when evidence is inconclusive."

  FLOW_HASH:
    algorithm: "SHA-256"
    input: "canonicalised ontology version, model assertions, assumptions, rules and provenance"
    note: "Compute hash at execution; no hash value is claimed here."

  UPDATE_MODEL:
    strategy: "versioned, provenance-preserving incremental update"
    steps: [ingest, normalise, validate, compare, reconcile, recompute_inferences, publish]
    rollback: true

  AI_CONSUMPTION:
    representation: "typed applied-mathematics knowledge graph with model and evidence links"
    outputs: [model_links, equation_relations, solution_records, uncertainty_annotations, inferred_edges, provenance_records]
    safeguards: [preserve_assumptions, preserve_units, distinguish_model_from_reality, no_unsupported_causal_claims]
}

END FLOW ALGEBRA