Geometry Domain Ontology

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

flow "GeometryDomainOntology" v1.0 {
  ontology:
    id: "geometry-domain-ontology"
    parent: "mathematical-domains"
    purpose: "Represent geometric entities, relationships, transformations, measurements and justified geometric inferences."

  types:
    GeometrySpace: {id: string, dimension: integer, metric: string?, coordinate_system: string?, axioms: string[]}
    Point: {id: string, space: string, coordinates: number[]?}
    Line: {id: string, space: string, representation: string}
    Plane: {id: string, space: string, representation: string}
    Shape: {id: string, space: string, kind: string, boundary: string?, properties: string[]}
    Transformation: {id: string, source_space: string, target_space: string, kind: enum[translation, rotation, reflection, scaling, affine, projective, other], parameters: string}
    Measurement: {id: string, target: string, kind: enum[length, angle, area, volume, curvature, other], value: number, unit: string, uncertainty: number?}
    Assertion: {id: string, subject: string, predicate: string, object: string, assumptions: string[], evidence: string[], confidence: number}

  predicates:
    lies_on(point: Point, line: Line)
    lies_in(point: Point, plane: Plane)
    intersects(a: string, b: string)
    parallel_to(a: string, b: string)
    perpendicular_to(a: string, b: string)
    congruent_to(a: Shape, b: Shape)
    similar_to(a: Shape, b: Shape)
    contains(a: Shape, b: string)
    transformed_to(a: string, b: string, by: Transformation)
    measures(measurement: Measurement, entity: string)

  constraints:
    - "Every relationship endpoint must resolve to a typed entity."
    - "Geometric claims must specify the applicable space and axiomatic assumptions."
    - "Coordinates must match the declared space dimension and coordinate system."
    - "Measurements require units; uncertainty must be non-negative when provided."
    - "Parallelism, perpendicularity and distance require appropriate geometric definitions."
    - "Congruence and similarity must not be conflated."
    - "Do not infer Euclidean properties in non-Euclidean spaces without justification."
    - "Confidence must lie within [0, 1]."

  inference:
    rule congruence_symmetry:
      if congruent_to(a,b)
      then congruent_to(b,a)
      confidence: "retain proof context"
    rule congruence_transitivity:
      if congruent_to(a,b) and congruent_to(b,c) and compatible_spaces(a,b,c)
      then congruent_to(a,c)
      confidence: "requires valid premises"
    rule transformation_composition:
      if transformed_to(a,b,t1) and transformed_to(b,c,t2) and composable(t1,t2)
      then transformed_to(a,c,compose(t2,t1))
      confidence: "retain transformation provenance"
    rule similarity_symmetry:
      if similar_to(a,b)
      then similar_to(b,a)
      confidence: "retain scale and assumptions"

  SEMANTIC_EQUIVALENCE:
    - "Normalise accepted geometric terminology without merging distinct mathematical concepts."
    - "Treat equivalent coordinate representations as equal only after validated transformation."
    - "Distinguish exact equality, congruence, similarity and approximate numerical agreement."

  CONFIDENCE:
    range: [0,1]
    factors: [proof_validity, axiom_compatibility, measurement_uncertainty, numerical_precision, provenance]
    policy: "Deductive conclusions require valid premises; numerical approximation does not establish exact proof."

  VALIDATION:
    - "Check types, dimensions, coordinate systems, units and axioms."
    - "Verify transformation domains, codomains and composition order."
    - "Flag inconsistent geometric constraints and unsupported claims."
    - "Keep exact symbolic results distinct from approximate measurements."

  FLOW_HASH:
    algorithm: "SHA-256"
    input: "canonicalised ontology version, typed assertions, axioms, 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 geometric knowledge graph with proof and measurement annotations"
    outputs: [geometric_entities, relation_triples, transformation_chains, validated_inferences, uncertainty_annotations, provenance_records]
    safeguards: [preserve_axioms, preserve_coordinate_frames, distinguish_exact_from_approximate, reject_unsupported_inference]
}

END FLOW ALGEBRA