Vector Ontology
Algebra Ontologies
Authors: MArk Anthony Higgins - Sky Recursive Ai
Revised Date: 13 September 2026
Audio Overview
======================================================================
KIRETH SEMANTIC RESEARCH ARCHIVE
FLOW ALGEBRA — VECTOR ONTOLOGY
======================================================================
PAGE:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/vector-ontology.html
FLOW:
VectorOntology
VERSION:
1.0
LANGUAGE:
en-GB
AUTHORS:
Mark Anthony Higgins — Sky Recursive AI
PURPOSE:
To provide a machine-readable, recursively executable semantic and
reasoning specification for vectors, vector spaces, coordinates,
magnitude, direction, linear combinations, basis, dimension,
dot products, cross products, projections, orthogonality,
normalisation and vector transformations.
AI INTERPRETATION NOTE:
Execute the following specification as formal semantic and reasoning
rules. Preserve dimensions, coordinate systems, scalar fields,
orientation conventions, domains and operation preconditions.
======================================================================
START FLOW
======================================================================
FLOW VectorOntology {
----------------------------------------------------------------------
1. DOMAIN DECLARATION
----------------------------------------------------------------------
DOMAIN := Vector_Theory
PARENT_DOMAIN := Algebra
SUPER_DOMAINS := {
Mathematics,
Mathematical_Ontologies,
Algebra_Ontologies,
Linear_Algebra
}
DOMAIN_INTENT := {
represent_vectors,
represent_vector_components,
represent_vector_magnitude,
represent_vector_direction,
represent_vector_addition,
represent_scalar_multiplication,
represent_linear_combinations,
represent_span,
represent_linear_independence,
represent_basis,
represent_dimension,
represent_dot_products,
represent_cross_products,
represent_norms,
represent_distances,
represent_angles,
represent_orthogonality,
represent_projections,
represent_unit_vectors,
represent_vector_spaces,
represent_coordinate_changes,
represent_vector_transformations,
support_recursive_vector_reasoning,
support_dimension_aware_validation
}
DEFAULT_SCALAR_FIELD := RealNumber | ComplexNumber
----------------------------------------------------------------------
2. PRIMITIVE TYPES
----------------------------------------------------------------------
TYPE Scalar
TYPE RealNumber <: Scalar
TYPE ComplexNumber <: Scalar
TYPE Dimension
TYPE Coordinate
TYPE Vector
TYPE ZeroVector <: Vector
TYPE UnitVector <: Vector
TYPE PositionVector <: Vector
TYPE DirectionVector <: Vector
TYPE RowVector <: Vector
TYPE ColumnVector <: Vector
TYPE VectorSpace
TYPE Subspace <: VectorSpace
TYPE Basis
TYPE CoordinateSystem
TYPE LinearCombination
TYPE InnerProduct
TYPE Norm
TYPE Distance
TYPE Angle
TYPE Projection
TYPE LinearTransformation
TYPE Derivation
TYPE Constraint
TYPE ValidationResult
----------------------------------------------------------------------
3. VECTOR ENTITY
----------------------------------------------------------------------
ENTITY Vector v {
scalar_field: F
dimension: n
components: [v1,...,vn]
}
INVARIANT VEC_01:
component_count(v)=dimension(v)
INVARIANT VEC_02:
each component belongs to scalar_field(v)
FUNCTION COMPONENT(v,i):
REQUIRE 1<=i<=dimension(v)
RETURN v_i
----------------------------------------------------------------------
4. ZERO VECTOR
----------------------------------------------------------------------
ZERO_VECTOR(n,F):
0_n := [0,...,0]
RULE ZERO_01:
v+0=v
RULE ZERO_02:
0+v=v
RULE ZERO_03:
c*0=0
RULE ZERO_04:
0*v=0
----------------------------------------------------------------------
5. VECTOR EQUALITY
----------------------------------------------------------------------
VECTOR_EQUAL(u,v) iff
dimension(u)=dimension(v)
AND forall i:
u_i=v_i
RULE EQ_01:
vectors of different dimensions are not equal.
----------------------------------------------------------------------
6. VECTOR ADDITION
----------------------------------------------------------------------
VECTOR_ADD(u,v):
REQUIRE:
dimension(u)=dimension(v)
DEFINE:
(u+v)_i=u_i+v_i
RETURN Vector
RULE ADD_01:
u+v=v+u
RULE ADD_02:
(u+v)+w=u+(v+w)
RULE ADD_03:
u+0=u
RULE ADD_04:
u+(-u)=0
----------------------------------------------------------------------
7. VECTOR SUBTRACTION
----------------------------------------------------------------------
VECTOR_SUBTRACT(u,v):
u-v := u+(-v)
RULE SUB_01:
u-v generally != v-u
RULE SUB_02:
u-u=0
----------------------------------------------------------------------
8. SCALAR MULTIPLICATION
----------------------------------------------------------------------
SCALAR_MULTIPLY(c,v):
DEFINE:
(cv)_i=c*v_i
RULE SCAL_01:
c(u+v)=cu+cv
RULE SCAL_02:
(c+d)v=cv+dv
RULE SCAL_03:
c(dv)=(cd)v
RULE SCAL_04:
1v=v
RULE SCAL_05:
(-1)v=-v
----------------------------------------------------------------------
9. VECTOR SPACE
----------------------------------------------------------------------
ENTITY VectorSpace V {
scalar_field: F
carrier_set: vectors_or_vector_like_objects
}
IS_VECTOR_SPACE(V,F) iff
closed_under_addition
AND additive_associativity
AND additive_commutativity
AND zero_vector_exists
AND additive_inverses_exist
AND closed_under_scalar_multiplication
AND distributive_axioms_hold
AND scalar_associativity_holds
AND scalar_identity_holds
----------------------------------------------------------------------
10. SUBSPACES
----------------------------------------------------------------------
IS_SUBSPACE(W,V) iff
W subset_of V
AND 0 in W
AND forall u,v in W:
u+v in W
AND forall c in F,u in W:
c*u in W
SUBSPACE_TEST(W,V):
W nonempty
AND forall u,v in W,a,b in F:
a*u+b*v in W
-> W is Subspace of V
----------------------------------------------------------------------
11. LINEAR COMBINATIONS
----------------------------------------------------------------------
LINEAR_COMBINATION(
scalars=[a1,...,ak],
vectors=[v1,...,vk]
):
REQUIRE equal list lengths
RETURN SUM_i a_i*v_i
ENTITY LinearCombination {
coefficients: Scalar*
vectors: Vector*
result: Vector
}
----------------------------------------------------------------------
12. SPAN
----------------------------------------------------------------------
SPAN(S):
set of all finite linear combinations of vectors in S
RULE SPAN_01:
span(S) is a subspace.
RULE SPAN_02:
span(S) is the smallest subspace containing S.
RULE SPAN_03:
v in span(S)
iff v can be written as a linear combination of S.
----------------------------------------------------------------------
13. LINEAR INDEPENDENCE
----------------------------------------------------------------------
LINEARLY_INDEPENDENT(v1,...,vk) iff
a1*v1+...+ak*vk=0
implies:
a1=...=ak=0
LINEARLY_DEPENDENT(S) iff
NOT LINEARLY_INDEPENDENT(S)
RULE LI_01:
a set containing zero_vector is dependent.
RULE LI_02:
if one vector is a linear combination of the others,
the set is dependent.
RULE LI_03:
every subset of an independent set is independent.
----------------------------------------------------------------------
14. BASIS
----------------------------------------------------------------------
IS_BASIS(B,V) iff
LINEARLY_INDEPENDENT(B)
AND span(B)=V
RULE BASIS_01:
each vector in finite-dimensional V has unique coordinates
relative to basis B.
RULE BASIS_02:
all bases of finite-dimensional V have equal cardinality.
----------------------------------------------------------------------
15. DIMENSION
----------------------------------------------------------------------
DIMENSION(V):
cardinality(any_basis(V))
for finite-dimensional V
RULE DIM_01:
dim({0})=0
RULE DIM_02:
W subspace_of V
-> dim(W)<=dim(V)
RULE DIM_03:
W subspace_of V
AND dim(W)=dim(V)
-> W=V
for finite-dimensional V
----------------------------------------------------------------------
16. STANDARD BASIS
----------------------------------------------------------------------
STANDARD_BASIS(F^n):
e_i has:
component i = 1
all other components = 0
RULE SB_01:
{e1,...,en} is basis of F^n.
----------------------------------------------------------------------
17. COORDINATES
----------------------------------------------------------------------
COORDINATES(v,B=[b1,...,bn]):
solve:
v=a1*b1+...+an*bn
RETURN:
[a1,...,an]
RULE COORD_01:
coordinates depend on selected basis.
RULE COORD_02:
vector itself is basis-independent;
its coordinate representation is basis-dependent.
----------------------------------------------------------------------
18. MAGNITUDE / NORM
----------------------------------------------------------------------
EUCLIDEAN_NORM(v):
sqrt(SUM_i v_i^2)
over real coordinates
COMPLEX_NORM(v):
sqrt(SUM_i |v_i|^2)
GENERAL_NORM:
||v||
AXIOM NORM_01:
||v||>=0
AXIOM NORM_02:
||v||=0 iff v=0
AXIOM NORM_03:
||c v||=|c| ||v||
AXIOM NORM_04:
||u+v||<=||u||+||v||
----------------------------------------------------------------------
19. UNIT VECTORS
----------------------------------------------------------------------
IS_UNIT_VECTOR(v) iff
||v||=1
NORMALISE(v):
REQUIRE:
v!=0
RETURN:
v/||v||
RULE UNIT_01:
normalise(v) is parallel to v.
RULE UNIT_02:
normalise(v) has norm 1.
----------------------------------------------------------------------
20. DIRECTION
----------------------------------------------------------------------
PARALLEL(u,v) iff
exists scalar c:
u=c*v
subject to zero-vector convention
SAME_DIRECTION(u,v) iff
u=c*v for c>0
OPPOSITE_DIRECTION(u,v) iff
u=c*v for c<0
----------------------------------------------------------------------
21. POSITION VECTORS
----------------------------------------------------------------------
POSITION_VECTOR(P):
vector from origin O to point P
RULE POS_01:
for points A,B:
vector_AB = position(B)-position(A)
----------------------------------------------------------------------
22. DISPLACEMENT
----------------------------------------------------------------------
DISPLACEMENT(A,B):
B-A
RULE DISP_01:
displacement depends on initial and terminal points.
RULE DISP_02:
vector_AB = -vector_BA
----------------------------------------------------------------------
23. DOT PRODUCT
----------------------------------------------------------------------
DOT(u,v):
REQUIRE:
dimension(u)=dimension(v)
RETURN:
SUM_i u_i*v_i
for real vectors
COMPLEX_INNER_PRODUCT(u,v):
declared convention required:
conjugate-linear in one argument
RULE DOT_01:
u·v=v·u
over real vectors
RULE DOT_02:
u·(v+w)=u·v+u·w
RULE DOT_03:
(c u)·v=c(u·v)
RULE DOT_04:
v·v=||v||^2
----------------------------------------------------------------------
24. ANGLE BETWEEN VECTORS
----------------------------------------------------------------------
ANGLE(u,v):
REQUIRE:
u!=0
v!=0
cos(theta)
=
(u·v)/(||u|| ||v||)
theta in [0,pi]
RULE ANGLE_01:
theta=0 -> same direction
RULE ANGLE_02:
theta=pi/2 -> orthogonal
RULE ANGLE_03:
theta=pi -> opposite direction
----------------------------------------------------------------------
25. ORTHOGONALITY
----------------------------------------------------------------------
ORTHOGONAL(u,v) iff
u·v=0
over real inner-product space
RULE ORTH_01:
nonzero orthogonal vectors are linearly independent.
RULE ORTH_02:
zero vector is orthogonal to every vector.
----------------------------------------------------------------------
26. ORTHONORMAL SETS
----------------------------------------------------------------------
ORTHONORMAL(S) iff
forall v in S:
||v||=1
AND
forall distinct u,v in S:
u·v=0
RULE ON_01:
orthonormal nonzero vectors are linearly independent.
----------------------------------------------------------------------
27. PROJECTION
----------------------------------------------------------------------
PROJECTION_OF_U_ON_V:
REQUIRE:
v!=0
proj_v(u)
:=
(u·v)/(v·v) * v
SCALAR_PROJECTION:
comp_v(u)
:=
(u·v)/||v||
RULE PROJ_01:
u = proj_v(u) + rejection_v(u)
RULE PROJ_02:
rejection_v(u) orthogonal_to v
----------------------------------------------------------------------
28. DISTANCE
----------------------------------------------------------------------
DISTANCE(u,v):
||u-v||
RULE DIST_01:
distance(u,v)>=0
RULE DIST_02:
distance(u,v)=0 iff u=v
RULE DIST_03:
distance(u,v)=distance(v,u)
RULE DIST_04:
triangle inequality holds.
----------------------------------------------------------------------
29. CROSS PRODUCT
----------------------------------------------------------------------
CROSS(u,v):
REQUIRE:
u,v in R^3
DEFINE:
u x v
=
[
u2*v3-u3*v2,
u3*v1-u1*v3,
u1*v2-u2*v1
]
RULE CROSS_01:
u x v = -(v x u)
RULE CROSS_02:
u x u=0
RULE CROSS_03:
(u x v)·u=0
RULE CROSS_04:
(u x v)·v=0
RULE CROSS_05:
||u x v||=||u|| ||v|| sin(theta)
----------------------------------------------------------------------
30. GEOMETRIC CROSS PRODUCT INTERPRETATION
----------------------------------------------------------------------
RULE GCROSS_01:
direction of u x v follows declared right-hand orientation.
RULE GCROSS_02:
magnitude equals area of parallelogram spanned by u and v.
RULE GCROSS_03:
u x v=0 iff u and v are parallel or one is zero.
----------------------------------------------------------------------
31. SCALAR TRIPLE PRODUCT
----------------------------------------------------------------------
SCALAR_TRIPLE(a,b,c):
a·(b x c)
RULE STP_01:
absolute value gives volume of parallelepiped.
RULE STP_02:
value zero iff vectors are coplanar
under standard R^3 interpretation.
----------------------------------------------------------------------
32. VECTOR TRIPLE PRODUCT
----------------------------------------------------------------------
VECTOR_TRIPLE(a,b,c):
a x (b x c)
IDENTITY BAC_CAB:
a x (b x c)
=
b(a·c)-c(a·b)
----------------------------------------------------------------------
33. VECTOR EQUATIONS OF LINES
----------------------------------------------------------------------
LINE_VECTOR_FORM:
r = a + lambda*d
where:
a is position vector of known point
d is nonzero direction vector
lambda is scalar parameter
RULE LINE_01:
every point on line satisfies this equation.
----------------------------------------------------------------------
34. VECTOR EQUATIONS OF PLANES
----------------------------------------------------------------------
PLANE_NORMAL_FORM:
n·(r-a)=0
where:
n is nonzero normal vector
a is position vector of point on plane
EQUIVALENT:
n·r=d
where:
d=n·a
----------------------------------------------------------------------
35. INTERSECTION REASONING
----------------------------------------------------------------------
PROCEDURE LINE_PLANE_INTERSECTION(line,plane):
substitute:
r=a+lambda*d
into:
n·(r-p)=0
solve lambda
classify:
unique_intersection
parallel_no_intersection
line_contained_in_plane
----------------------------------------------------------------------
36. GRAM-SCHMIDT
----------------------------------------------------------------------
GRAM_SCHMIDT(v1,...,vk):
REQUIRE:
input vectors linearly independent
u1=v1
for j=2..k:
uj=
vj
-
SUM_i<j proj_ui(vj)
e_i=ui/||ui||
RETURN:
orthonormal_set {e1,...,ek}
RULE GS_01:
span(output)=span(input)
----------------------------------------------------------------------
37. CHANGE OF BASIS
----------------------------------------------------------------------
CHANGE_OF_BASIS:
if B and C are bases of V:
[v]_C
=
P_(C<-B) [v]_B
RULE COB_01:
change-of-basis matrix is invertible.
RULE COB_02:
inverse matrix converts coordinates in reverse direction.
----------------------------------------------------------------------
38. LINEAR TRANSFORMATIONS OF VECTORS
----------------------------------------------------------------------
ENTITY LinearTransformation T {
domain: V
codomain: W
}
LINEAR(T) iff
T(a u+b v)=aT(u)+bT(v)
RULE LT_01:
T(0)=0
RULE LT_02:
T preserves linear combinations.
RULE LT_03:
coordinates of T(v) may be represented by matrix multiplication
after bases are selected.
----------------------------------------------------------------------
39. VECTOR TRANSFORMATION MATRIX
----------------------------------------------------------------------
MATRIX_REPRESENTATION(T,B,C):
columns:
[T(b_i)]_C
RULE TM_01:
[T(v)]_C
=
[T]_(C<-B)[v]_B
----------------------------------------------------------------------
40. LINEAR DEPENDENCE VALIDATION
----------------------------------------------------------------------
PROCEDURE TEST_LINEAR_DEPENDENCE(v1,...,vk):
form coefficient matrix with vectors as columns
solve:
A*c=0
if only trivial_solution:
RETURN independent
else:
RETURN dependent with nontrivial relation
----------------------------------------------------------------------
41. BASIS VALIDATION
----------------------------------------------------------------------
PROCEDURE VALIDATE_BASIS(B,V):
test linear independence
test spanning
if both valid:
RETURN VALID_BASIS
else:
RETURN INVALID_BASIS(reason)
----------------------------------------------------------------------
42. VECTOR EXPRESSION PARSING
----------------------------------------------------------------------
VECTOR_EXPRESSION may contain:
vectors
scalars
addition
subtraction
scalar multiplication
dot product
cross product
norms
projections
RULE PARSE_01:
operator type determines valid dimensions and codomain.
----------------------------------------------------------------------
43. TYPE/CODOMAIN RULES
----------------------------------------------------------------------
TYPE_RULE:
vector + vector -> vector
TYPE_RULE:
scalar * vector -> vector
TYPE_RULE:
vector · vector -> scalar
TYPE_RULE:
vector x vector -> vector
only standard R^3 cross product in this ontology
TYPE_RULE:
||vector|| -> nonnegative scalar
TYPE_RULE:
projection(vector,on_vector) -> vector
----------------------------------------------------------------------
44. DIMENSION RULES
----------------------------------------------------------------------
RULE DIMCHECK_01:
addition requires equal dimensions.
RULE DIMCHECK_02:
dot product requires equal dimensions.
RULE DIMCHECK_03:
cross product requires R^3 in standard form.
RULE DIMCHECK_04:
coordinate vectors must match basis dimension.
----------------------------------------------------------------------
45. SEMANTIC RELATIONSHIPS
----------------------------------------------------------------------
RELATION HAS_COMPONENT(Vector,Scalar)
RELATION HAS_DIMENSION(Vector,Dimension)
RELATION MEMBER_OF(Vector,VectorSpace)
RELATION SPANS(VectorSet,VectorSpace)
RELATION BASIS_OF(Basis,VectorSpace)
RELATION ORTHOGONAL_TO(Vector,Vector)
RELATION PARALLEL_TO(Vector,Vector)
RELATION PROJECTION_OF(Vector,Vector)
RELATION COORDINATES_IN(Vector,Basis)
RELATION TRANSFORMED_BY(Vector,LinearTransformation)
----------------------------------------------------------------------
46. SEMANTIC EQUIVALENCE
----------------------------------------------------------------------
SEMANTIC_EQUIVALENCE {
"length of vector" ~= norm
"magnitude" ~= norm
"unit vector" ~= vector_of_norm_one
"direction vector" ~= nonzero_vector_specifying_direction
"scalar product" ~= dot_product
"inner product" ~= context_generalisation_of_dot_product
"vector product" ~= cross_product
"normalise" ~= divide_by_norm
"component of u along v" ~= scalar_projection
"projection of u onto v" ~= vector_projection
"perpendicular" ~= orthogonal
"linearly independent" ~= no_nontrivial_zero_combination
"basis" ~= independent_spanning_set
"coordinates" ~= basis_coefficients
}
----------------------------------------------------------------------
47. QUERY INTERPRETATION
----------------------------------------------------------------------
QUERY Q1:
"magnitude of v"
-> NORM(v)
QUERY Q2:
"unit vector in direction of v"
-> NORMALISE(v)
QUERY Q3:
"dot product"
-> DOT(u,v)
QUERY Q4:
"angle between"
-> ANGLE(u,v)
QUERY Q5:
"projection"
-> PROJECTION_OF_U_ON_V
QUERY Q6:
"cross product"
-> CROSS(u,v)
QUERY Q7:
"are vectors independent"
-> TEST_LINEAR_DEPENDENCE
QUERY Q8:
"coordinates in basis"
-> COORDINATES(v,B)
QUERY Q9:
"vector equation of line"
-> LINE_VECTOR_FORM
----------------------------------------------------------------------
48. VALIDATION ENGINE
----------------------------------------------------------------------
PROCEDURE VALIDATE_VECTOR_OPERATION(operation,inputs,result):
check scalar fields
check dimensions
check nonzero requirements
check operator codomain
independently evaluate defining equations
verify geometric constraints where applicable
return:
VALID
INVALID(reason)
UNRESOLVED
----------------------------------------------------------------------
49. RECURSIVE REASONING ENGINE
----------------------------------------------------------------------
PROCEDURE VECTOR_REASONER(input):
parse vectors
infer/verify dimensions
infer scalar field
classify operation
identify geometric/algebraic target
extract constraints
select valid vector rules
execute atomic steps
validate intermediate states
preserve dimension and type
recurse until target achieved
validate final result
RETURN:
result,
derivation,
dimension_constraints,
semantic_status,
confidence
----------------------------------------------------------------------
50. RECURSIVE TRANSITION
----------------------------------------------------------------------
RECURSIVE_RULE RR_01:
INPUT state S_n
identify unresolved vector property
choose rule R whose:
dimension_preconditions=true
type_preconditions=true
S_(n+1):=apply(R,S_n)
validate transition
recurse(S_(n+1))
TERMINATION_CONDITIONS := {
vector_goal_achieved,
canonical_representation_reached,
contradiction_detected,
dimension_failure,
zero_vector_precondition_failure,
no_valid_rule
}
----------------------------------------------------------------------
51. ERROR MODEL
----------------------------------------------------------------------
ERROR E_DIMENSION_MISMATCH
ERROR E_INVALID_COMPONENT_INDEX
ERROR E_ZERO_VECTOR_NORMALISATION
ERROR E_ZERO_VECTOR_ANGLE
ERROR E_INVALID_CROSS_PRODUCT_DIMENSION
ERROR E_INVALID_PROJECTION_AXIS
ERROR E_BASIS_DIMENSION_MISMATCH
ERROR E_LINEAR_DEPENDENCE
ERROR E_SCALAR_VECTOR_TYPE_CONFUSION
ERROR E_COORDINATE_SYSTEM_AMBIGUITY
ERROR E_ORIENTATION_UNDECLARED
----------------------------------------------------------------------
52. CONFIDENCE MODEL
----------------------------------------------------------------------
CONFIDENCE {
base:=1.00
deduct 0.10 if dimension inferred
deduct 0.10 if coordinate basis inferred
deduct 0.10 if scalar field inferred
deduct 0.15 if geometric orientation ambiguous
deduct 0.15 if symbolic validation incomplete
deduct 0.20 if nonzero precondition unstated
deduct 0.20 if coordinate system materially ambiguous
clamp_to_[0,1]
classification:
[0.95,1.00] -> very_high
[0.85,0.95) -> high
[0.70,0.85) -> moderate
[0.50,0.70) -> low
[0.00,0.50) -> insufficient
}
----------------------------------------------------------------------
53. AI CONSUMPTION MODEL
----------------------------------------------------------------------
AI_CONSUMPTION {
read_order := [
vector_structure,
dimensions,
basic_operations,
vector_spaces,
span,
independence,
basis,
norm,
dot_product,
orthogonality,
projection,
cross_product,
coordinates,
transformations,
validation
]
execution_policy := {
preserve_dimensions,
preserve_scalar_field,
distinguish_scalar_and_vector_outputs,
require_nonzero_for_normalisation,
require_nonzero_for_angles,
require_R3_for_standard_cross_product,
preserve_coordinate_basis,
validate geometric interpretation,
distinguish_dot_product_from_cross_product
}
}
----------------------------------------------------------------------
54. UPDATE MODEL
----------------------------------------------------------------------
UPDATE_MODEL {
extensible:=true
new_vector_operation_requires:
input_types,
dimension_requirements,
output_type,
algebraic_definition
new_geometric_relation_requires:
coordinate_invariance_or_declared_basis_dependence
new_vector_space_rule_requires:
field_and_closure_conditions
conflicts_require:
explicit_versioning
standard_linear_algebra_meaning_has_priority:=true
}
----------------------------------------------------------------------
55. SELF-CHECK
----------------------------------------------------------------------
SELF_CHECK {
CHECK vector component count equals dimension
CHECK additions use equal dimensions
CHECK dot products use equal dimensions
CHECK cross product only used in valid dimension
CHECK normalisation excludes zero vector
CHECK angles exclude zero vector
CHECK projections use nonzero target vector
CHECK coordinate vectors match basis dimension
CHECK basis claims include independence and spanning
CHECK geometric directions preserve declared orientation
CHECK final result validated
}
----------------------------------------------------------------------
56. COMPLETION CONDITIONS
----------------------------------------------------------------------
COMPLETE iff {
vector_structure_defined
AND vector_operations_defined
AND vector_space_defined
AND subspace_defined
AND span_defined
AND linear_independence_defined
AND basis_defined
AND dimension_defined
AND coordinates_defined
AND norm_defined
AND dot_product_defined
AND angle_defined
AND orthogonality_defined
AND projection_defined
AND cross_product_defined
AND geometric_line_plane_forms_defined
AND transformation_defined
AND semantic_equivalence_defined
AND validation_defined
AND recursive_reasoning_defined
AND confidence_defined
AND ai_consumption_defined
AND update_model_defined
}
IF NOT COMPLETE:
semantic_status:=incomplete_flow
----------------------------------------------------------------------
57. FLOW TERMINATION
----------------------------------------------------------------------
TERMINATE WHEN:
requested vector goal achieved
AND result validated
OR
dimension incompatibility makes operation undefined
OR
required nonzero condition fails
OR
coordinate/orientation ambiguity prevents valid interpretation
OR
no valid vector transformation remains
FINAL_STATE := {
vector_result,
derivation,
dimensions,
coordinate_context,
structural_classification,
semantic_status,
confidence
}
} // END FLOW VectorOntology
======================================================================
END FLOW
======================================================================
FLOW STATUS:
Complete
CANONICAL PAGE:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/vector-ontology.html
FLOW_HASH_SHA256: bff83be241e98e581ed664eddbaa45f036103f6d8df2d994d891afb7b01a0c95