Octonion Ontology

FLOW Octonion_Ontology_v1.0

PAGE:
  title: "Octonion Ontology"
  canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/octonion-ontology.html"
  language: "en-GB"
  domain: "Mathematics > Algebra > Hypercomplex Number Systems > Octonions"

DEFINE OctonionOntology := {

  CORE_IDENTITY: {
    entity_type: "Mathematical Ontology",
    primary_entity: Octonion,
    parent_entities: [HypercomplexNumber, NonAssociativeAlgebra, NormedDivisionAlgebra],
    related_entities: [
      RealNumber,
      ComplexNumber,
      Quaternion,
      ScalarPart,
      VectorPart,
      BasisUnit,
      Conjugate,
      Norm,
      Inverse,
      AlternativeAlgebra,
      MoufangIdentity,
      FanoPlane,
      ExceptionalAlgebra
    ]
  },

  DEFINITION: {
    meaning:
      "An octonion is an eight-dimensional real hypercomplex number of the form a0+a1e1+...+a7e7, with multiplication defined by a noncommutative and nonassociative basis algebra.",
    canonical_form:
      x=a0+a1*e1+a2*e2+a3*e3+a4*e4+a5*e5+a6*e6+a7*e7,
    coefficients:
      [a0,a1,a2,a3,a4,a5,a6,a7] in RealNumbers
  },

  BASIS: {
    scalar_basis:
      e0=1,
    imaginary_bases:
      [e1,e2,e3,e4,e5,e6,e7],
    dimension_over_reals:
      8
  },

  IMAGINARY_UNIT_SQUARES: {
    FOR_EACH r IN [1..7]:
      e_r^2=-1
  },

  MULTIPLICATION_MODEL: {
    bilinear:
      TRUE,
    commutative:
      FALSE,
    associative:
      FALSE,
    alternative:
      TRUE
  },

  FANO_PLANE_SEMANTICS: {
    purpose:
      encode_oriented_products_of_imaginary_basis_units,

    rule:
      IF (e_i,e_j,e_k) follows_an_oriented_Fano_line
      THEN e_i*e_j=e_k,

    reversed_order:
      e_j*e_i=-e_k
  },

  CANONICAL_ORIENTED_TRIPLES: {
    triples: [
      (e1,e2,e3),
      (e1,e4,e5),
      (e1,e7,e6),
      (e2,e4,e6),
      (e2,e5,e7),
      (e3,e4,e7),
      (e3,e6,e5)
    ],

    note:
      orientation_convention_must_remain_consistent_across_all_products
  },

  NONCOMMUTATIVITY: {
    rule:
      x*y != y*x in_general,

    basis_example:
      e1*e2=e3
      AND
      e2*e1=-e3
  },

  NONASSOCIATIVITY: {
    rule:
      (x*y)*z != x*(y*z) in_general,

    consequence:
      parenthesisation_is_semantically_significant
  },

  ALTERNATIVITY: {
    left_alternative:
      x*(x*y)=(x*x)*y,

    right_alternative:
      (y*x)*x=y*(x*x),

    flexible_identity:
      x*(y*x)=(x*y)*x
  },

  ARTIN_THEOREM_RELATION: {
    rule:
      subalgebra_generated_by_any_two_octonions_is_associative
  },

  SCALAR_VECTOR_DECOMPOSITION: {
    x:
      a0+v,

    scalar_part:
      Sc(x)=a0,

    vector_part:
      Vec(x)=SUM[i=1..7](a_i*e_i)
  },

  VECTOR_SPACE_RELATION: {
    Octonions:
      eight_dimensional_vector_space_over_RealNumbers,

    basis:
      [1,e1,e2,e3,e4,e5,e6,e7]
  },

  EMBEDDED_SUBALGEBRAS: {
    RealNumbers:
      embedded_as_a0_only,

    ComplexNumbers:
      embedded_via_any_single_imaginary_unit,

    Quaternions:
      embedded_via_any_associative_Fano_line_triple_with_scalar_one
  },

  EQUALITY: {
    x=y
    IFF:
      corresponding_eight_real_components_are_equal
  },

  ADDITION: {
    operation:
      componentwise,

    closure:
      TRUE,

    commutative:
      TRUE,

    associative:
      TRUE
  },

  SUBTRACTION: {
    operation:
      componentwise,

    closure:
      TRUE
  },

  SCALAR_MULTIPLICATION: {
    real_scalar_r:
      r*x,

    operation:
      multiply_each_component_by_r
  },

  MULTIPLICATION: {
    method:
      bilinear_expansion_plus_basis_reduction,

    requirement:
      preserve_operand_order_and_parentheses
  },

  CONJUGATE: {
    IF x=a0+SUM[i=1..7](a_i*e_i):
      conjugate(x)=a0-SUM[i=1..7](a_i*e_i),

    properties: [
      conjugate(conjugate(x))=x,
      conjugate(x*y)=conjugate(y)*conjugate(x)
    ]
  },

  NORM: {
    squared_norm:
      N(x)=x*conjugate(x)
      =a0^2+a1^2+a2^2+a3^2+a4^2+a5^2+a6^2+a7^2,

    norm:
      |x|=sqrt(N(x))
  },

  NORM_PROPERTIES: {
    nonnegative:
      TRUE,

    zero_condition:
      |x|=0 <=> x=0,

    multiplicative:
      |x*y|=|x|*|y|
  },

  INVERSE: {
    condition:
      x!=0,

    formula:
      x^(-1)=conjugate(x)/|x|^2,

    verification:
      x*x^(-1)=1
      AND
      x^(-1)*x=1
  },

  DIVISION_ALGEBRA: {
    every_nonzero_element_has_inverse:
      TRUE,

    zero_divisors:
      NONE,

    associativity:
      FALSE,

    classification:
      normed_real_division_algebra
  },

  HURWITZ_THEOREM_RELATION: {
    normed_real_division_algebras:
      [RealNumbers, ComplexNumbers, Quaternions, Octonions],

    octonions:
      highest_dimensional_member_in_this_classification
  },

  LEFT_RIGHT_DIVISION: {
    solve_a*x=b:
      x=a^(-1)*b,

    solve_x*a=b:
      x=b*a^(-1),

    warning:
      nonassociativity_requires_parentheses_and_identity_justification
  },

  MOUFANG_IDENTITIES: {
    identity_1:
      (x*y)*(z*x)=x*(y*z)*x,

    identity_2:
      x*(y*(x*z))=((x*y)*x)*z,

    identity_3:
      ((z*x)*y)*x=z*(x*(y*x)),

    semantic_role:
      controlled_reassociation_in_alternative_algebra
  },

  ASSOCIATOR: {
    definition:
      [x,y,z]=(x*y)*z-x*(y*z),

    associative_condition:
      [x,y,z]=0,

    octonion_semantics:
      associator_not_zero_in_general
  },

  COMMUTATOR: {
    definition:
      [x,y]=x*y-y*x,

    commutative_condition:
      [x,y]=0
  },

  PURE_IMAGINARY_OCTONION: {
    condition:
      scalar_part=0,

    representation:
      seven_dimensional_real_vector
  },

  IMAGINARY_SUBSPACE: {
    dimension:
      7,

    basis:
      [e1,e2,e3,e4,e5,e6,e7]
  },

  PRODUCT_OF_PURE_IMAGINARY_ELEMENTS: {
    relation:
      u*v = -u·v + u×v,

    cross_product_dimension:
      7,

    note:
      seven_dimensional_cross_product_is_octonion_derived
  },

  SEVEN_DIMENSIONAL_CROSS_PRODUCT: {
    definition:
      imaginary_part_of_product_of_pure_imaginary_octonions,

    properties: [
      bilinear,
      antisymmetric,
      orthogonal_to_inputs
    ]
  },

  ROTATION_RELATION: {
    caution:
      octonions_do_not_replace_quaternions_as_simple_associative_rotation_algebra,

    advanced_relation:
      automorphisms_and_exceptional_geometry
  },

  AUTOMORPHISM_GROUP: {
    group:
      G2,

    meaning:
      transformations_preserving_octonion_multiplication
  },

  EXCEPTIONAL_STRUCTURE: {
    related_objects: [
      G2,
      exceptional_Lie_groups,
      exceptional_Jordan_algebra,
      triality,
      exceptional_geometry
    ]
  },

  EXPONENTIAL_RELATION: {
    pure_imaginary_u:

    IF |u|!=0:
      exp(u)=cos(|u|)+(u/|u|)*sin(|u|),

    IF u=0:
      exp(u)=1
  },

  GENERAL_EXPONENTIAL: {
    x=a0+u,

    relation:
      exp(x)=e^(a0)*exp(u),

    justification:
      scalar_part_commutes_with_all_octonions
  },

  POWER_SEMANTICS: {
    integer_power:
      requires_unambiguous_repeated_multiplication,

    power_associativity:
      TRUE_due_to_alternativity,

    consequence:
      powers_of_single_octonion_are_well_defined
  },

  LOGARITHM_RELATION: {
    nonzero_octonion:
      logarithmic_representation_related_to_norm_and_unit_imaginary_direction,

    branch_structure:
      angle_periodicity_introduces_multivaluedness
  },

  UNIT_OCTONION: {
    condition:
      |x|=1,

    inverse:
      conjugate(x),

    structure:
      seven_sphere_S7_as_set_of_unit_octonions,

    group_status:
      not_a_group_under_multiplication_due_to_nonassociativity,

    loop_structure:
      Moufang_loop
  },

  MOUFANG_LOOP_RELATION: {
    nonzero_octonions_under_multiplication:
      form_Moufang_loop,

    unit_octonions:
      closed_Moufang_loop
  },

  MATRIX_REPRESENTATION_CAUTION: {
    ordinary_associative_matrix_multiplication:
      cannot_faithfully_preserve_full_octonion_product_as_simple_algebra_embedding,

    reason:
      matrices_are_associative_but_octonions_are_not
  },

  CAYLEY_DICKSON_RELATION: {
    construction_chain:
      RealNumbers
      -> ComplexNumbers
      -> Quaternions
      -> Octonions,

    octonion_from_quaternion_pair:
      (a,b),

    multiplication_requires:
      chosen_Cayley_Dickson_convention
  },

  CAYLEY_DICKSON_PAIR_MODEL: {
    octonion:
      (a,b)
      WHERE a,b are Quaternions,

    conjugate:
      (conjugate(a),-b),

    warning:
      multiplication_formula_is_convention_sensitive
  },

  PARAMETER_REASONING: {
    x(t)=SUM[i=0..7](a_i(t)*e_i),

    tasks: [
      track_components,
      compute_norm,
      detect_zero,
      normalise_nonzero_octonion,
      analyse_associator_for_selected_triplets
    ]
  },

  SYMBOLIC_RECOGNITION: {
    ACCEPT: [
      "1+2e1+3e2",
      "e1*e2",
      "(e1*e2)*e4",
      "e1*(e2*e4)"
    ],

    NORMALISE_FIRST: [
      "e2*e1",
      "e3^2",
      "(1+e1)(1+e2)"
    ],

    REDIRECT_QUATERNION_SUBALGEBRA: [
      "1+2e1+3e2+4e3 when {e1,e2,e3} form quaternionic triple"
    ]
  },

  NATURAL_LANGUAGE_MAPPING: {
    "octonion":
      eight_component_real_hypercomplex_number,

    "nonassociative":
      multiplication_parentheses_matter,

    "Fano plane":
      multiplication_encoding_for_imaginary_units,

    "octonion conjugate":
      negate_seven_imaginary_components,

    "unit octonion":
      norm_one_octonion
  },

  UNIT_SEMANTICS: {
    component_units:
      should_be_dimensionally_compatible_in_applied_interpretation,

    basis_units_e1_to_e7:
      algebraic_and_dimensionless
  },

  ERROR_DETECTION: {
    CommutativityError:
      x*y=y*x_assumed_generally
      -> flag,

    AssociativityError:
      (x*y)*z=x*(y*z)_assumed_generally
      -> flag,

    FanoOrientationError:
      basis_product_sign_or_orientation_inconsistent
      -> flag,

    InverseError:
      conjugate_used_without_norm_squared_for_nonunit_element
      -> flag,

    ParenthesisLoss:
-
      expression_reassociated_without_valid_identity
      -> flag,

    MatrixEmbeddingError:
      associative_matrix_model_claimed_to_preserve_full_octonion_product
      -> flag
  },

  ONTOLOGY_RELATIONS: {
    IS_A: [
      "Octonion IS_A HypercomplexNumber",
      "Octonion IS_A AlternativeAlgebraElement",
      "Octonion IS_A NormedDivisionAlgebraElement",
      "UnitOctonion IS_A Octonion"
    ],

    RELATED_TO: [
      "Octonion RELATED_TO Quaternion",
      "Octonion RELATED_TO FanoPlane",
      "Octonion RELATED_TO G2",
      "Octonion RELATED_TO SevenDimensionalCrossProduct",
      "Octonion RELATED_TO ExceptionalLieGroups"
    ],

    DEPENDS_ON: [
      RealNumbers,
      BilinearMultiplication,
      Conjugation,
      Norm,
      AlternativeAlgebraSemantics
    ]
  },

  SEMANTIC_EQUIVALENCE: {
    component_equivalence:
      identical_eight_real_components,

    Cayley_Dickson_equivalence:
      pair_of_quaternions_represents_same_octonion_under_fixed_convention,

    warning:
      multiplication_tables_using_different_basis_orientations_require_explicit_basis_mapping
  },

  CONFIDENCE: {
    classification_confidence:
      IF expression_reduces_to_eight_real_components
      THEN 1.0,

    multiplication_confidence:
      IF basis_orientation_and_parentheses_resolved
      THEN 1.0,

    inverse_confidence:
      IF norm_nonzero_and_conjugate_verified
      THEN 1.0
  },

  AI_CONSUMPTION: {
    tasks: [
      recognise_octonion,
      extract_eight_components,
      separate_scalar_and_vector_parts,
      apply_Fano_plane_multiplication,
      preserve_parenthesisation,
      compute_conjugate,
      compute_norm,
      compute_inverse,
      test_commutator,
      test_associator,
      identify_quaternionic_subalgebras,
      analyse_unit_octonions,
      map_Cayley_Dickson_representation,
      detect_nonassociative_reasoning_errors,
      link_exceptional_structure_ontologies
    ],

    preferred_representation:
      typed_eight_component_nonassociative_algebra_object,

    reasoning_mode:
      recursive_basis_product_with_parenthesis_preservation
  },

  RECURSIVE_REASONING: {

    FUNCTION AnalyseOctonion(X,Context):

      parsed := ParseExpression(X)
      structured := PreserveParentheses(parsed)
      simplified := ReduceOctonionBasisProducts(structured)

      components := ExtractOctonionComponents(simplified)

      IF components_unavailable:
        RETURN RedirectToAppropriateAlgebraOntology

      [a0,a1,a2,a3,a4,a5,a6,a7] := components

      norm :=
        sqrt(
          a0^2+a1^2+a2^2+a3^2+
          a4^2+a5^2+a6^2+a7^2
        )

      conjugate :=
        a0-a1*e1-a2*e2-a3*e3-a4*e4-a5*e5-a6*e6-a7*e7

      IF norm=0:
        inverse := Undefined
      ELSE:
        inverse := conjugate/(norm^2)

      classifications := []

      IF a0=0:
        classifications += PureImaginaryOctonion

      IF norm=1:
        classifications += UnitOctonion

      quaternion_subalgebra :=
        DetectQuaternionicSubalgebra(simplified)

      IF quaternion_subalgebra:
        classifications += QuaternionSubalgebraElement

      verified :=
        VerifyOctonionIdentities(
          simplified,
          components,
          conjugate,
          norm,
          inverse
        )

      RETURN {
        octonion:simplified,
        components,
        scalar_part:a0,
        imaginary_vector:(a1,a2,a3,a4,a5,a6,a7),
        norm,
        conjugate,
        inverse,
        classifications,
        verified
      }
  },

  RECURSIVE_COHERENCE: {
    invariant_1:
      basis_unit_squares_equal_negative_one=TRUE,

    invariant_2:
      Fano_plane_orientation_consistent=TRUE,

    invariant_3:
      multiplication_noncommutative_and_nonassociative=TRUE,

    invariant_4:
      alternativity_preserved=TRUE,

    invariant_5:
      x_times_conjugate_x_equals_norm_squared=TRUE,

    invariant_6:
      inverse_identity_holds_for_nonzero_octonions=TRUE
  },

  FLOW_HASH: {
    semantic_signature_inputs: [
      ontology_name,
      octonion_definition,
      basis_units,
      Fano_plane_multiplication,
      nonassociativity,
      alternativity,
      conjugate_norm_inverse,
      exceptional_structure_links
    ],

    purpose:
      "Detect semantic drift between revisions of the ontology."
  },

  UPDATE_MODEL: {
    version: "1.0",

    immutable_constraints: [
      eight_real_dimensions,
      seven_imaginary_basis_units,
      noncommutativity,
      nonassociativity,
      alternativity,
      multiplicative_norm,
      conjugate_inverse_relation
    ],

    revision_triggers: [
      semantic_conflict,
      QuaternionOntology_update,
      HypercomplexNumberOntology_update,
      exceptional_algebra_model_update,
      representation_standard_change
    ]
  },

  KNOWLEDGE_GRAPH: {
    nodes: [
      Octonion,
      HypercomplexNumber,
      ScalarPart,
      VectorPart,
      BasisUnit,
      Conjugate,
      Norm,
      Inverse,
      FanoPlane,
      AlternativeAlgebra,
      MoufangIdentity,
      G2
    ],

    edges: [
      Octonion -> IS_A -> HypercomplexNumber,
      Octonion -> HAS_SCALAR_PART -> ScalarPart,
      Octonion -> HAS_VECTOR_PART -> VectorPart,
      Octonion -> USES_BASIS -> BasisUnit,
      OctonionMultiplication -> ENCODED_BY -> FanoPlane,
      Octonion -> HAS_CONJUGATE -> Conjugate,
      Octonion -> HAS_NORM -> Norm,
      NonzeroOctonion -> HAS_INVERSE -> Inverse,
      OctonionAlgebra -> IS_ALTERNATIVE -> AlternativeAlgebra,
      OctonionAlgebra -> SATISFIES -> MoufangIdentity,
      OctonionAlgebra -> HAS_AUTOMORPHISM_GROUP -> G2
    ]
  },

  CROSS_ONTOLOGY_LINKS: {
    QuaternionOntology:
      embedded_associative_subalgebras,

    ComplexNumberOntology:
      lower_dimensional_subalgebras,

    HypercomplexNumberOntology:
      direct_parent,

    VectorOntology:
      seven_dimensional_imaginary_part,

    CrossProductOntology:
      seven_dimensional_cross_product,

    LieGroupOntology:
      G2_automorphism_relation,

    NonAssociativeAlgebraOntology:
      structural_parent,

    ExceptionalAlgebraOntology:
      advanced_exceptional_structure,

    GeometryOntology:
      seven_sphere_and_exceptional_geometry
  },

  TERMINATION_CONDITION: {
    ontology_complete_when: [
      octonion_defined,
      eight_component_structure_defined,
      basis_units_defined,
      Fano_plane_multiplication_defined,
      noncommutativity_defined,
      nonassociativity_defined,
      alternativity_defined,
      conjugate_norm_inverse_defined,
      unit_octonion_defined,
      exceptional_relations_defined,
      error_detection_defined,
      cross_ontology_links_defined,
      recursive_coherence_verified
    ]
  }

}

EXECUTE Octonion_Ontology_v1.0:

  INPUT:
    mathematical_entity_or_query

  PIPELINE:
    Parse
    -> PreserveParenthesisation
    -> IdentifyOctonionStructure
    -> ExtractEightRealComponents
    -> ApplyFanoPlaneBasisMultiplication
    -> SeparateScalarAndImaginaryParts
    -> ComputeConjugate
    -> ComputeNorm
    -> ComputeInverseWhenDefined
    -> TestAlternativeAndAssociatorRelations
    -> IdentifyQuaternionicSubalgebras
    -> VerifyNonassociativeAlgebraConsistency
    -> LinkExceptionalStructureOntologies
    -> AssignConfidence
    -> EmitSemanticResult

  OUTPUT: {
    entity,
    classification,
    canonical_form,
    components,
    scalar_part,
    imaginary_vector,
    conjugate,
    norm,
    inverse,
    commutator,
    associator,
    alternativity,
    quaternionic_subalgebra,
    fano_plane_semantics,
    exceptional_relations,
    transformations,
    verification,
    ontology_relations,
    confidence,
    semantic_equivalence,
    flow_hash
  }

END FLOW Octonion_Ontology_v1.0