Synthetic Geometry Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of synthetic geometry within the Kireth Research Semantic Library. Synthetic geometry studies geometric objects and relations directly through axioms, constructions, incidence, congruence, similarity and deductive reasoning, rather than primarily through coordinates or algebraic equations. 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 = "Synthetic Geometry Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Geometry"
 SPECIALISATION = "Synthetic Geometry"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/geometry-ontologies/geometry-ontology/synthetic-geometry-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = SyntheticGeometry
 ENTITY_CLASS = GeometricReasoningFramework
 PRIMARY_CONTEXT = Geometry
 DEFINITION = "Synthetic geometry is the study of geometric entities and relations through axioms, constructions and direct geometric reasoning without relying primarily on coordinate or algebraic representation."
 SEMANTIC_ROLE = "Provides a coordinate-free framework for proving geometric properties from incidence, congruence, similarity, order and construction relations."
}

SCOPE {
 INCLUDE = [
  SyntheticGeometry,
  EuclideanSyntheticGeometry,
  IncidenceGeometry,
  Congruence,
  Similarity,
  Betweenness,
  Parallelism,
  Perpendicularity,
  GeometricConstruction,
  AxiomaticGeometry,
  TriangleGeometry,
  CircleGeometry,
  Locus,
  Symmetry,
  ClassicalTheorem,
  SyntheticProof
 ]

 EXCLUDE_AS_PRIMARY = [
  CoordinateGeometry,
  AnalyticGeometry,
  PureAlgebraicGeometry,
  NumericalGeometryWithoutSyntheticReasoning
 ]

 DISAMBIGUATION_RULE =
  "When geometry is reasoned about through points, lines, circles, incidence, congruence, similarity, constructions and theorems without coordinates, interpret it as synthetic geometry."
}

CORE_CONCEPTS {

 SyntheticGeometry {
  definition = "Geometric reasoning conducted directly in terms of geometric objects and their intrinsic relations."
  methodological_character = "Coordinate-free or coordinate-secondary."
 }

 Incidence {
  definition = "Relation expressing that a geometric entity lies on or belongs to another geometric entity."
  examples = [
   "Point lies on line",
   "Line lies in plane"
  ]
 }

 Betweenness {
  definition = "Order relation identifying when one point lies between two others on a line."
 }

 Congruence {
  definition = "Equality of geometric size and shape under rigid motion."
 }

 Similarity {
  definition = "Equality of shape under scaling and rigid motion."
 }

 Parallelism {
  definition = "Relation between coplanar lines that do not meet in Euclidean geometry."
 }

 Perpendicularity {
  definition = "Relation between lines or segments meeting at a right angle."
 }

 GeometricConstruction {
  definition = "Creation of geometric objects by exact rule-governed procedures."
 }

 SyntheticProof {
  definition = "A proof relying primarily on geometric definitions, axioms, diagrams, constructions and previously established theorems."
 }

 Locus {
  definition = "The set of all points satisfying a stated geometric condition."
 }
}

RELATIONS {
 SyntheticGeometry --is_subdomain_of--> Geometry
 SyntheticGeometry --uses--> AxiomaticGeometry
 SyntheticGeometry --uses--> GeometricConstruction
 SyntheticGeometry --uses--> SyntheticProof
 SyntheticGeometry --reasons_about--> Incidence
 SyntheticGeometry --reasons_about--> Congruence
 SyntheticGeometry --reasons_about--> Similarity
 SyntheticGeometry --reasons_about--> Parallelism
 SyntheticGeometry --reasons_about--> Perpendicularity
 Point --can_lie_on--> Line
 Point --can_lie_in--> Plane
 Shape --can_be_related_by--> Congruence
 Shape --can_be_related_by--> Similarity
 SyntheticProof --can_use--> Construction
 SyntheticProof --can_use--> ClassicalTheorem
}

OPERATORS {
 IS_INCIDENT(P,L) :=
  P lies_on L

 IS_BETWEEN(A,B,C) :=
  B lies_between A and C

 IS_CONGRUENT(A,B) :=
  establish_congruence_by_valid_geometric_criteria(A,B)

 IS_SIMILAR(A,B) :=
  establish_similarity_by_valid_geometric_criteria(A,B)

 IS_PARALLEL(L1,L2) :=
  establish_parallel_relation(L1,L2)

 IS_PERPENDICULAR(L1,L2) :=
  establish_right_angle_relation(L1,L2)

 CONSTRUCT(target,givens) :=
  produce_exact_geometric_construction(target,givens)

 PROVE(statement) :=
  construct_synthetic_geometric_proof(statement)

 LOCUS(condition) :=
  set_of_points_satisfying(condition)
}

DEFINITIONS {
 D1 = "Synthetic geometry reasons directly with geometric entities rather than relying primarily on coordinates."
 D2 = "Incidence expresses containment or membership relationships among geometric entities."
 D3 = "Congruence preserves size and shape."
 D4 = "Similarity preserves shape and proportional structure."
 D5 = "Parallelism and perpendicularity are fundamental line relations."
 D6 = "Synthetic construction is exact and rule-governed."
 D7 = "A synthetic proof derives conclusions using geometric axioms, definitions and theorems."
 D8 = "A locus is defined by a geometric property."
 D9 = "Coordinate methods may verify a synthetic result but are not required for the synthetic argument."
}

AXIOMS {
 A1_GEOMETRIC_PRIMITIVES:
  Synthetic reasoning begins from primitive geometric entities and relations

 A2_INCIDENCE:
  Incidence relations are governed by the chosen geometric axiom system

 A3_CONGRUENCE:
  Congruence relations preserve corresponding lengths and angles

 A4_SIMILARITY:
  Similarity preserves corresponding angle equality and proportional lengths

 A5_CONSTRUCTION_VALIDITY:
  Constructed entities must follow permitted geometric procedures

 A6_PROOF_VALIDITY:
  Every synthetic proof step must follow from accepted geometric axioms, definitions or theorems

 A7_COORDINATE_INDEPENDENCE:
  Synthetic validity does not depend on a particular coordinate representation
}

REASONING {

 R1_SYNTHETIC_CLASSIFICATION {
  INPUT = geometric_problem
  IF reasoning_uses_coordinates_or_equations_as_primary_method
   RETURN "Analytic or coordinate geometry context"
  ELSE IF reasoning_uses_direct_geometric_relations
   RETURN SyntheticGeometry
 }

 R2_TRIANGLE_PROOF {
  INPUT = triangle_configuration_and_goal
  IDENTIFY = congruence_or_similarity_relations
  APPLY = triangle_theorems
  DERIVE = target_relation
  RETURN = proof_status
 }

 R3_CIRCLE_PROOF {
  INPUT = circle_configuration_and_goal
  IDENTIFY = radii_chords_arcs_angles_tangents
  APPLY = circle_theorems
  RETURN = proof_status
 }

 R4_PARALLELISM_PROOF {
  INPUT = line_configuration
  APPLY = angle_relations_or_axiomatic_parallel_rules
  RETURN = parallelism_status
 }

 R5_PERPENDICULARITY_PROOF {
  INPUT = geometric_configuration
  IDENTIFY = right_angle_or_congruence_conditions
  DERIVE = perpendicular_relation
  RETURN = perpendicularity_status
 }

 R6_LOCUS_REASONING {
  INPUT = condition
  PROVE_NECESSITY = every_point_on_candidate_locus_satisfies_condition
  PROVE_SUFFICIENCY = every_point_satisfying_condition_lies_on_candidate_locus
  RETURN = locus_characterisation
 }

 R7_CONSTRUCTION_PROOF {
  INPUT = construction_sequence
  VERIFY = every_step
  PROVE = final_geometric_conditions
  RETURN = ValidConstructionOrInvalid
 }

 R8_SYNTHETIC_THEOREM_PROOF {
  INPUT = proposition
  IDENTIFY = givens
  IDENTIFY = relevant_classical_theorems
  CONSTRUCT = deductive_chain
  VERIFY = conclusion
  RETURN = SyntheticProofStatus
 }
}

CONSTRAINTS {
 C1 = "Synthetic arguments must preserve the geometry's axioms and definitions."
 C2 = "Diagrams may aid reasoning but do not alone establish proof."
 C3 = "Visual appearance cannot replace geometric justification."
 C4 = "Coordinate assumptions must not be silently imported into a purely synthetic proof."
 C5 = "Congruence and similarity criteria must be applied correctly."
 C6 = "Locus proofs require both necessity and sufficiency when full characterisation is claimed."
 C7 = "Construction reasoning must remain exact."
 C8 = "Euclidean conclusions require Euclidean assumptions unless otherwise justified."
}

SEMANTIC_EQUIVALENCE {
 "SyntheticGeometry" <=> "CoordinateFreeGeometricReasoning"
 "SyntheticProof" <=> "DirectGeometricDeduction"
 "CongruenceReasoning" <=> "RigidShapeEquivalenceReasoning"
 "SimilarityReasoning" <=> "ScaleInvariantShapeReasoning"
 "Locus" <=> "SetOfPointsSatisfyingGeometricCondition"
}

VALIDATION {
 V1 = "Synthetic proof steps are geometrically justified."
 V2 = "Diagrams are not treated as proof by themselves."
 V3 = "Congruence and similarity are distinguished."
 V4 = "Coordinate-free reasoning remains valid under coordinate changes."
 V5 = "Locus proofs establish necessity and sufficiency."
 V6 = "Construction steps are exact and admissible."
 V7 = "Euclidean assumptions are not silently applied in non-Euclidean settings."
 V8 = "Final conclusion follows from stated givens."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SYNTHETIC_GEOMETRY_DEFINITION = 0.99
 CONGRUENCE_REASONING = 0.99
 SIMILARITY_REASONING = 0.99
 CONSTRUCTION_REASONING = 0.99
 LOCUS_REASONING = 0.99
 SYNTHETIC_PROOF = 0.99

 POLICY =
  "Reduce confidence when the geometric axiom system, diagram configuration or theorem assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  SyntheticGeometricReasoning,
  TriangleReasoning,
  CircleReasoning,
  ConstructionReasoning,
  CongruenceReasoning,
  SimilarityReasoning,
  ProofReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "synthetic geometry" -> SyntheticGeometry
  "synthetic proof" -> SyntheticProof
  "pure geometry proof" -> SyntheticProof
  "geometric locus" -> Locus
  "congruent triangles" -> Congruence
  "similar triangles" -> Similarity
 }

 INTERPRETATION_ORDER = [
  identify_synthetic_geometry_context,
  identify_givens,
  identify_target_relation,
  identify_incidence_and_metric_relations,
  retrieve_relevant_geometric_theorems,
  construct_direct_geometric_reasoning_chain,
  verify_each_step,
  verify_target_conclusion,
  return_semantic_output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("synthetic geometry")
  THEN activate(SyntheticGeometryOntology)

 E2:
  IF geometric_proof_uses_no_primary_coordinate_method
  THEN prefer_synthetic_reasoning

 E3:
  IF triangle_context
  THEN activate_triangle_congruence_similarity_reasoning

 E4:
  IF circle_context
  THEN activate_circle_theorem_reasoning

 E5:
  IF request=="prove parallel"
  THEN use_valid_synthetic_parallelism_criteria

 E6:
  IF request=="prove perpendicular"
  THEN use_valid_synthetic_perpendicularity_criteria

 E7:
  IF request=="find locus"
  THEN prove_necessity_and_sufficiency

 E8:
  IF construction_context
  THEN route_to(GeometricConstructionOntology)
}

RECURSION {
 ENTRY = SyntheticGeometry

 EXPAND(SyntheticGeometry) -> [
  Point,
  Line,
  Circle,
  Triangle,
  Incidence,
  Betweenness,
  Congruence,
  Similarity,
  Parallelism,
  Perpendicularity,
  GeometricConstruction,
  SyntheticProof,
  Locus
 ]

 EXPAND(Congruence) -> [
  SegmentCongruence,
  AngleCongruence,
  TriangleCongruence,
  RigidTransformation
 ]

 EXPAND(Similarity) -> [
  Proportionality,
  EqualAngles,
  TriangleSimilarity,
  ScaleTransformation
 ]

 EXPAND(SyntheticProof) -> [
  Given,
  Construction,
  Theorem,
  Deduction,
  Conclusion
 ]

 STOP_CONDITION =
  "Stop when the requested geometric relation, construction, locus or theorem is established through valid synthetic reasoning."
}

ROUTING {
 PARENT_ONTOLOGY = "Geometry Ontology"

 IMMEDIATE_CONTEXTS = [
  "Axiomatic Geometry Ontology",
  "Geometric Construction Ontology",
  "Euclidean Geometry Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Geometry Ontology",
  "Axiomatic Geometry Ontology",
  "Geometric Construction Ontology",
  "Algebraic Proof Ontology",
  "Transformation Ontology",
  "Affine Transformation Ontology"
 ]

 ROUTE_RULES = [
  "Synthetic Geometry -> Axiomatic Geometry for foundational assumptions",
  "Synthetic Geometry -> Geometric Construction for exact construction procedures",
  "Synthetic Geometry -> Euclidean Geometry for classical metric theorems",
  "Synthetic Geometry -> Geometry for core entity semantics",
  "Synthetic Geometry -> Algebraic Proof for formal proof structure when needed"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible synthetic methods, relations or theorem pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve direct geometric reasoning semantics.",
  "Preserve distinction from coordinate-first methods.",
  "Preserve congruence and similarity criteria.",
  "Preserve exact construction reasoning.",
  "Preserve full locus characterisation requirements.",
  "Preserve proof validity under the active geometry.",
  "Preserve machine-readable synthetic-geometric semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Synthetic geometry studies geometric entities and relations directly through axioms, constructions and geometric theorems rather than primarily through coordinates."

 PROOF_SUMMARY =
  "givens -> geometric theorems and constructions -> justified deductions -> conclusion"

 CONGRUENCE_SUMMARY =
  "Congruence establishes equality of size and shape."

 SIMILARITY_SUMMARY =
  "Similarity establishes equality of shape up to scale."

 LOCUS_SUMMARY =
  "A full locus proof establishes both necessity and sufficiency."

 CONSTRUCTION_SUMMARY =
  "Synthetic constructions produce exact geometric objects through permitted steps."
}

FLOW {
 INPUT
 -> detect_synthetic_geometry_context
 -> identify_givens
 -> identify_target_geometric_relation
 -> identify_incidence_congruence_similarity_and_order_relations
 -> retrieve_relevant_axioms_definitions_and_theorems
 -> construct_synthetic_reasoning_chain
 -> invoke_construction_or_locus_reasoning_when_required
 -> verify_every_geometric_step
 -> verify_final_conclusion
 -> link_related_ontologies
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "Synthetic geometry reasons directly with geometric objects, constructions and intrinsic relations rather than relying primarily on coordinates. It uses incidence, congruence, similarity, parallelism, perpendicularity, loci and classical theorems to construct valid geometric proofs. Its strength is structural reasoning that remains independent of a particular coordinate representation."
}

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

}

END_FLOW