Non-Euclidean Geometry Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of non-Euclidean geometry within the Kireth Research Semantic Library. Non-Euclidean geometry studies geometries that do not satisfy the Euclidean parallel postulate in its standard form. The two principal classical families are hyperbolic geometry and elliptic geometry. This record is designed for semantic retrieval, recursive geometric reasoning, ontology linking, proof support, knowledge-graph construction and AI consumption.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Non-Euclidean Geometry Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Non-Euclidean Geometry"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/geometry-ontology/non-euclidean-geometry-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = NonEuclideanGeometry
 ENTITY_CLASS = GeometricTheoryFamily
 PRIMARY_CONTEXT = Geometry
 DEFINITION = "Non-Euclidean geometry is the study of geometric systems in which Euclid's parallel postulate is replaced or modified, producing geometries such as hyperbolic and elliptic geometry."
 SEMANTIC_ROLE = "Provides a formal framework for geometric spaces whose line, distance, angle and triangle behaviour differs from Euclidean geometry."
}

SCOPE {
 INCLUDE = [
  NonEuclideanGeometry,
  HyperbolicGeometry,
  EllipticGeometry,
  SphericalGeometry,
  ParallelPostulate,
  Geodesic,
  Curvature,
  TriangleAngleSum,
  Distance,
  Angle,
  Model,
  AxiomSystem,
  Consistency,
  Isometry,
  Congruence,
  Transformation,
  ManifoldContext
 ]

 EXCLUDE_AS_PRIMARY = [
  EuclideanGeometry,
  ArbitraryMetricSpaceWithoutGeometricContext,
  ProjectiveGeometryUnlessExplicitlyLinked,
  PureTopologyWithoutMetricOrGeodesicStructure
 ]

 DISAMBIGUATION_RULE =
  "When geometry is described by alternative parallel behaviour, constant positive or negative curvature, geodesics, or triangle angle sums different from pi, interpret it as non-Euclidean geometry."
}

CORE_CONCEPTS {

 NonEuclideanGeometry {
  definition = "A family of geometries differing from Euclidean geometry through alternative axiomatic treatment of parallel lines."
  principal_branches = [
   HyperbolicGeometry,
   EllipticGeometry
  ]
 }

 HyperbolicGeometry {
  definition = "A non-Euclidean geometry in which through a point not on a given line there are multiple distinct lines not intersecting the given line."
  curvature = "negative in standard constant-curvature models"
  triangle_angle_sum = "less than pi"
 }

 EllipticGeometry {
  definition = "A non-Euclidean geometry in which no line through a point is parallel to a given line."
  curvature = "positive in standard constant-curvature models"
  triangle_angle_sum = "greater than pi"
 }

 SphericalGeometry {
  definition = "A geometry on the sphere where great circles act as geodesics; closely related to elliptic geometry but with antipodal-point distinctions depending on the model."
 }

 ParallelPostulate {
  euclidean_form = "Through a point not on a line, exactly one parallel line exists."
  hyperbolic_alternative = "More than one nonintersecting line may pass through the external point."
  elliptic_alternative = "No parallel line exists."
 }

 Geodesic {
  definition = "A locally distance-minimising or intrinsically straight path in the geometry."
 }

 Curvature {
  role = "Characterises geometric deviation from Euclidean flatness."
  euclidean = "zero"
  hyperbolic = "negative in standard models"
  elliptic = "positive in standard models"
 }

 Model {
  definition = "A mathematical structure realising the axioms of a non-Euclidean geometry."
  examples = [
   PoincareDiskModel,
   UpperHalfPlaneModel,
   HyperboloidModel,
   SphereModel
  ]
 }
}

RELATIONS {
 NonEuclideanGeometry --is_subdomain_of--> Geometry
 NonEuclideanGeometry --is_formalised_by--> AxiomSystem
 HyperbolicGeometry --is_special_case_of--> NonEuclideanGeometry
 EllipticGeometry --is_special_case_of--> NonEuclideanGeometry
 SphericalGeometry --is_related_to--> EllipticGeometry
 ParallelPostulate --distinguishes--> EuclideanHyperbolicEllipticGeometry
 HyperbolicGeometry --typically_has--> NegativeCurvature
 EllipticGeometry --typically_has--> PositiveCurvature
 Geodesic --generalises--> StraightLine
 NonEuclideanGeometry --can_have--> Model
 Model --satisfies--> AxiomSystem
 NonEuclideanGeometry --supports--> Isometry
 NonEuclideanGeometry --supports--> Congruence
}

OPERATORS {
 CLASSIFY_GEOMETRY(PARALLEL_BEHAVIOUR) :=
  IF exactly_one_parallel THEN EuclideanGeometry
  ELSE IF multiple_parallels THEN HyperbolicGeometry
  ELSE IF no_parallels THEN EllipticGeometry

 TRIANGLE_ANGLE_SUM(TRIANGLE,GEOMETRY) :=
  compute_intrinsic_angle_sum(TRIANGLE,GEOMETRY)

 GEODESIC_BETWEEN(P,Q,GEOMETRY) :=
  determine_geodesic(P,Q,GEOMETRY)

 DISTANCE(P,Q,MODEL) :=
  compute_model_specific_intrinsic_distance(P,Q,MODEL)

 IS_ISOMETRY(T,GEOMETRY) :=
  preserves_intrinsic_distance(T)

 IS_MODEL(M,SYSTEM) :=
  M satisfies all axioms of SYSTEM

 CURVATURE_AT(P,GEOMETRY) :=
  determine_intrinsic_curvature(P,GEOMETRY)

 VERIFY_THEOREM(statement,system) :=
  derive_or_validate_using_non_euclidean_axioms(statement,system)
}

DEFINITIONS {
 D1 = "Non-Euclidean geometry modifies the Euclidean parallel postulate."
 D2 = "Hyperbolic geometry admits multiple parallels through an external point."
 D3 = "Elliptic geometry admits no parallels."
 D4 = "Hyperbolic triangle angle sums are less than pi in standard constant-curvature geometry."
 D5 = "Elliptic triangle angle sums are greater than pi in standard constant-curvature geometry."
 D6 = "Geodesics replace ordinary Euclidean straight lines as intrinsic straight paths."
 D7 = "Models may represent the same intrinsic geometry in different coordinate forms."
 D8 = "Curvature helps classify local geometric structure."
 D9 = "Non-Euclidean geometry is logically consistent relative to standard mathematical foundations through appropriate models."
}

AXIOMS {
 A1_AXIOM_SYSTEM:
  Every non-Euclidean geometry is interpreted relative to an explicit or implicit axiom system

 A2_PARALLEL_VARIATION:
  The Euclidean parallel postulate is not retained in its standard unique-parallel form

 A3_MODEL_INVARIANCE:
  Different models of the same geometry represent identical intrinsic relations through appropriate isometries

 A4_GEODESIC_ROLE:
  Geodesics serve as intrinsic analogues of straight lines

 A5_METRIC_CONTEXT:
  Distance and angle must be interpreted using the geometry's intrinsic metric

 A6_PROOF_VALIDITY:
  Theorems must be derived using axioms valid in the active geometry

 A7_EUCLIDEAN_NONTRANSFER:
  Euclidean theorems are not assumed valid unless proved in the non-Euclidean system
}

REASONING {

 R1_GEOMETRY_CLASSIFICATION {
  INPUT = geometric_axiom_or_parallel_behaviour
  IF unique_parallel
   RETURN EuclideanGeometry
  ELSE IF multiple_parallels
   RETURN HyperbolicGeometry
  ELSE IF no_parallels
   RETURN EllipticGeometry
  ELSE
   RETURN FurtherSpecificationRequired
 }

 R2_MODEL_REASONING {
  INPUT = [candidate_model,axiom_system]
  FOR each axiom
   TEST = model_satisfaction
  IF all_satisfied
   RETURN ValidModel
  ELSE
   RETURN InvalidModel
 }

 R3_TRIANGLE_REASONING {
  INPUT = [triangle,geometry]
  IDENTIFY = geometry_type
  COMPUTE_OR_INFER = angle_sum
  RETURN = relation_to_pi
 }

 R4_GEODESIC_REASONING {
  INPUT = [points P,Q,model]
  IDENTIFY = model_specific_geodesic_family
  CONSTRUCT = geodesic_through(P,Q)
  RETURN = geodesic
 }

 R5_DISTANCE_REASONING {
  INPUT = [P,Q,model]
  APPLY = intrinsic_metric_formula
  RETURN = distance
 }

 R6_ISOMETRY_REASONING {
  INPUT = transformation T
  TEST = intrinsic_distance_preservation
  IF true
   RETURN Isometry
  ELSE
   RETURN NotIsometry
 }

 R7_THEOREM_TRANSFER {
  INPUT = Euclidean_theorem
  CHECK = proof_dependencies
  IF proof_uses_parallel_postulate_or_equivalent
   RETURN "Cannot transfer automatically"
  ELSE
   TEST = derivability_in_target_geometry
  RETURN = status
 }

 R8_CONSISTENCY_REASONING {
  INPUT = non_euclidean_axiom_system
  IDENTIFY = known_model
  IF valid_model_exists
   RETURN "System satisfiable and consistent relative to ambient foundational theory under standard soundness assumptions."
 }
}

CONSTRAINTS {
 C1 = "Euclidean parallel-line intuition must not be imported into non-Euclidean contexts."
 C2 = "Triangle angle-sum behaviour depends on the active geometry."
 C3 = "Coordinates of a model are representational and not the intrinsic geometry itself."
 C4 = "Curvature statements require geometric and metric context."
 C5 = "Spherical geometry and elliptic geometry must not be conflated without noting antipodal identification conventions."
 C6 = "Model-specific formulas must preserve intrinsic invariants."
 C7 = "Geodesics depend on the geometry and model."
 C8 = "Theorems require proof within the active axiom system."
 C9 = "Do not interpret non-Euclidean as mathematically inconsistent or merely approximate."
}

SEMANTIC_EQUIVALENCE {
 "NonEuclideanGeometry" <=> "GeometryWithAlternativeParallelPostulate"
 "HyperbolicGeometry" <=> "NegativeCurvatureClassicalNonEuclideanGeometry" WHEN constant_curvature_context
 "EllipticGeometry" <=> "PositiveCurvatureClassicalNonEuclideanGeometry" WHEN constant_curvature_context
 "Geodesic" <=> "IntrinsicStraightPath"
 "Model" <=> "RepresentationSatisfyingGeometricAxioms"
}

VALIDATION {
 V1 = "Hyperbolic and elliptic geometries are distinguished by parallel behaviour."
 V2 = "Euclidean theorems are not assumed automatically."
 V3 = "Triangle angle sums match the active geometry."
 V4 = "Model representations preserve intrinsic geometry."
 V5 = "Geodesics are interpreted model-specifically."
 V6 = "Curvature classification is consistent with the chosen geometry."
 V7 = "Spherical and elliptic distinctions are made when relevant."
 V8 = "Proofs use the correct axiom system."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 NON_EUCLIDEAN_DEFINITION = 0.99
 HYPERBOLIC_CLASSIFICATION = 0.99
 ELLIPTIC_CLASSIFICATION = 0.99
 TRIANGLE_ANGLE_SUM_RELATION = 0.99
 MODEL_RELATION = 0.99
 GEODESIC_RELATION = 0.99
 CURVATURE_RELATION = 0.98

 POLICY =
  "Reduce confidence when the geometry type, curvature convention, model, metric or parallel axiom is underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  NonEuclideanReasoning,
  HyperbolicGeometryReasoning,
  EllipticGeometryReasoning,
  GeodesicReasoning,
  CurvatureReasoning,
  ModelReasoning,
  ProofReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "non-Euclidean geometry" -> NonEuclideanGeometry
  "hyperbolic geometry" -> HyperbolicGeometry
  "elliptic geometry" -> EllipticGeometry
  "spherical geometry" -> SphericalGeometry
  "parallel postulate" -> ParallelPostulate
  "geodesic" -> Geodesic
  "Poincare disk" -> PoincareDiskModel
 }

 INTERPRETATION_ORDER = [
  detect_non_euclidean_context,
  identify_geometry_type,
  identify_parallel_axiom,
  identify_model_if_present,
  identify_metric_and_curvature_context,
  identify_geodesic_structure,
  route_to_hyperbolic_or_elliptic_reasoning,
  perform_recursive_reasoning,
  verify_theorem_under_active_axioms,
  return_semantic_output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("non-Euclidean geometry")
  THEN activate(NonEuclideanGeometryOntology)

 E2:
  IF query_mentions("hyperbolic geometry")
  THEN classify(HyperbolicGeometry)

 E3:
  IF query_mentions("elliptic geometry")
  THEN classify(EllipticGeometry)

 E4:
  IF query_mentions("spherical geometry")
  THEN classify(SphericalGeometry)
  AND inspect_antipodal_identification_context

 E5:
  IF query_mentions("parallel postulate")
  THEN classify_geometry_by_parallel_behaviour

 E6:
  IF model_name_present
  THEN activate_model_specific_reasoning

 E7:
  IF Euclidean_theorem_is_requested_in_non_euclidean_context
  THEN rederive_under_active_axioms

 E8:
  IF triangle_context
  THEN inspect_angle_sum_relation

 E9:
  IF geodesic_context
  THEN route_to_intrinsic_straight_path_reasoning
}

RECURSION {
 ENTRY = NonEuclideanGeometry

 EXPAND(NonEuclideanGeometry) -> [
  HyperbolicGeometry,
  EllipticGeometry,
  SphericalGeometry,
  ParallelPostulate,
  Geodesic,
  Curvature,
  Model,
  Isometry,
  TriangleAngleSum
 ]

 EXPAND(HyperbolicGeometry) -> [
  PoincareDiskModel,
  UpperHalfPlaneModel,
  HyperboloidModel,
  NegativeCurvature,
  HyperbolicDistance,
  HyperbolicGeodesic
 ]

 EXPAND(EllipticGeometry) -> [
  SphereModel,
  PositiveCurvature,
  GreatCircle,
  EllipticDistance,
  NoParallelLines
 ]

 EXPAND(Model) -> [
  Coordinates,
  Metric,
  GeodesicRepresentation,
  Isometry,
  AxiomSatisfaction
 ]

 STOP_CONDITION =
  "Stop when the geometry type, model-dependent relations, theorem or metric question is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Geometry Ontology"

 IMMEDIATE_CONTEXTS = [
  "Axiomatic Geometry Ontology",
  "Synthetic Geometry Ontology",
  "Euclidean Geometry Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Geometry Ontology",
  "Axiomatic Geometry Ontology",
  "Synthetic Geometry Ontology",
  "Transformation Ontology",
  "Dimension Ontology",
  "Mathematical Logic Ontologies"
 ]

 ROUTE_RULES = [
  "Non-Euclidean Geometry -> Axiomatic Geometry for parallel-postulate foundations",
  "Non-Euclidean Geometry -> Hyperbolic Geometry for multiple-parallel systems",
  "Non-Euclidean Geometry -> Elliptic Geometry for no-parallel systems",
  "Non-Euclidean Geometry -> Synthetic Geometry for coordinate-free proofs",
  "Non-Euclidean Geometry -> Geometry for core entity semantics",
  "Non-Euclidean Geometry -> Mathematical Logic for consistency and model reasoning"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible models, geometry classes, relations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve alternative-parallel-postulate semantics.",
  "Preserve distinction between hyperbolic and elliptic geometry.",
  "Preserve model-versus-intrinsic-geometry distinction.",
  "Preserve geometry-specific triangle and geodesic behaviour.",
  "Preserve curvature context.",
  "Preserve theorem validity under the active axiom system.",
  "Preserve machine-readable non-Euclidean semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Non-Euclidean geometry studies geometric systems in which the Euclidean parallel postulate is replaced or modified."

 HYPERBOLIC_SUMMARY =
  "Hyperbolic geometry admits multiple parallels through an external point and has triangle angle sums less than pi in standard constant-curvature models."

 ELLIPTIC_SUMMARY =
  "Elliptic geometry admits no parallels and has triangle angle sums greater than pi in standard constant-curvature models."

 GEODESIC_SUMMARY =
  "Geodesics are intrinsic analogues of straight lines."

 MODEL_SUMMARY =
  "Different coordinate models can represent the same intrinsic non-Euclidean geometry."

 PROOF_SUMMARY =
  "Theorems must be derived using the axioms of the active geometry."
}

FLOW {
 INPUT
 -> detect_non_euclidean_geometry_context
 -> identify_parallel_axiom
 -> classify_hyperbolic_elliptic_or_related_geometry
 -> identify_model_if_present
 -> identify_metric_curvature_and_geodesic_context
 -> identify_target_relation_or_theorem
 -> perform_geometry_specific_reasoning
 -> prevent_invalid_euclidean_transfer
 -> verify_axiom_and_model_consistency
 -> link_related_ontologies
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "Non-Euclidean geometry encompasses geometric systems that replace the Euclidean parallel postulate. Hyperbolic geometry permits multiple parallels through an external point, while elliptic geometry permits none. These geometries possess their own intrinsic metrics, geodesics, curvature and triangle behaviour, and their theorems must be interpreted and proved within the appropriate axiomatic and model context."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-GEOMETRY-NON-EUCLIDEAN-GEOMETRY-ONTOLOGY-V1"
}

}

END_FLOW