Unit Vector Ontology

FLOW ALGEBRA
ONTOLOGY: Unit Vector Ontology
VERSION: 4.0
SOURCE: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/linear-algebra-entities/unit-vector-ontology.html
STATUS: Independently constructed mathematical ontology; source-page alignment not verified.
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library

1. PURPOSE
Represent unit vectors as norm-one vectors in normed vector spaces, with explicit dependencies, computational rules, inference constraints and semantic interoperability.

2. TYPE SIGNATURE
TYPE ScalarField := R | C | other_norm_compatible_field
TYPE VectorSpace(V,F) := vector space V over F
TYPE Norm := map V -> R_nonnegative satisfying norm axioms
TYPE NormedVectorSpace := (V,F,norm)
TYPE Vector := element of V
TYPE UnitVector := {v in V | norm(v)=1}
TYPE InnerProductSpace := (V,F,inner_product)
TYPE OrthonormalSet := set of pairwise orthogonal unit vectors
TYPE UnitSphere := {v in V | norm(v)=1}
TYPE StandardBasisVector := e_i in F^n with component i=1 and other components=0, under standard Euclidean norm

3. CORE AXIOMS
AXIOM N1: norm(v)>=0
AXIOM N2: norm(v)=0 iff v=0
AXIOM N3: norm(a*v)=abs(a)*norm(v)
AXIOM N4: norm(u+v)<=norm(u)+norm(v)
AXIOM U1: UnitVector(v) iff norm(v)=1
AXIOM U2: UnitVector(v) implies v != 0
AXIOM U3: UnitVector(v) implies v belongs to UnitSphere(V,norm)
AXIOM U4: norm(-v)=norm(v)
AXIOM IP1: In an inner-product space, norm(v)=sqrt(inner_product(v,v))

4. DEFINITIONS
DEFINE unit(v): norm(v)=1
DEFINE normalise(v): v/norm(v), with precondition norm(v)>0
DEFINE direction_equivalence(u,v): exists positive real c such that u=c*v
DEFINE opposite_direction(u,v): exists negative real c such that u=c*v
DEFINE unit_sphere(V): {v | norm(v)=1}
DEFINE EuclideanNorm(x_1,...,x_n): sqrt(sum_i abs(x_i)^2)
DEFINE dot_unit(u,v): inner_product(u,v), when the relevant inner product exists
DEFINE orthogonal(u,v): inner_product(u,v)=0
DEFINE orthonormal(S): all elements unit and distinct elements pairwise orthogonal

5. TYPED RELATIONSHIPS
REL UnitVector --is_a--> Vector
REL UnitVector --belongs_to--> NormedVectorSpace
REL UnitVector --has_norm--> One
REL UnitVector --lies_on--> UnitSphere
REL Normalisation --maps_nonzero_vector_to--> UnitVector
REL UnitVector --may_be_element_of--> OrthonormalSet
REL StandardBasisVector --is_a--> UnitVector [standard Euclidean norm]
REL UnitVector --depends_on--> NormChoice
REL UnitVector --is_not_equivalent_to--> BasisVector
REL UnitVector --is_not_equivalent_to--> ZeroVector
REL OrthonormalBasis --consists_of--> UnitVector

6. FORMAL INFERENCE RULES
RULE 01: norm(v)=1 => UnitVector(v)
RULE 02: UnitVector(v) => norm(v)=1
RULE 03: UnitVector(v) => v!=0
RULE 04: UnitVector(v) => UnitVector(-v)
RULE 05: norm(v)>0 => UnitVector(v/norm(v))
RULE 06: v=0 => normalise(v) undefined
RULE 07: UnitVector(v) and abs(a)=1 => UnitVector(a*v)
RULE 08: UnitVector(v) and UnitVector(a*v) => abs(a)=1
RULE 09: UnitVector(u) and UnitVector(v) and u=c*v with c>0 => u=v
RULE 10: UnitVector(u) and UnitVector(v) and u=c*v with c<0 => u=-v [real scalars]
RULE 11: In Euclidean R^n, sum_i v_i^2=1 => UnitVector(v)
RULE 12: In Euclidean C^n, sum_i abs(v_i)^2=1 => UnitVector(v)
RULE 13: In standard Euclidean F^n, e_i => UnitVector(e_i)
RULE 14: Orthonormal(S) => every v in S is UnitVector
RULE 15: Orthonormal(S) => distinct u,v in S satisfy inner_product(u,v)=0
RULE 16: UnitVector(v) does not imply v is a basis vector
RULE 17: UnitVector(v) does not imply orthogonality to any other vector
RULE 18: In an inner-product space, UnitVector(v) => inner_product(v,v)=1
RULE 19: In an inner-product space, abs(inner_product(u,v))<=norm(u)*norm(v)
RULE 20: UnitVector(u) and UnitVector(v) => abs(inner_product(u,v))<=1 [inner-product space]
RULE 21: A norm choice must be supplied before unit status is evaluated
RULE 22: If two norms differ, unit status may differ
RULE 23: UnitVector(v) => v belongs to unit sphere of its normed space
RULE 24: Every nonzero vector in a real or complex normed vector space admits normalisation
RULE 25: A normed vector space of dimension zero has no unit vectors
RULE 26: If v is a unit vector and t is real, norm(t*v)=abs(t)
RULE 27: If u,v are unit and orthogonal in a real inner-product space, norm(u+v)=sqrt(2)
RULE 28: If u,v are unit and orthogonal in an inner-product space, norm(u-v)=sqrt(2)
RULE 29: If v is unit, projection of x onto span(v) equals inner_product(x,v)*v [inner product linear in first argument]
RULE 30: UnitVector(v) is invariant under norm-preserving linear transformations
RULE 31: If T is an isometry and UnitVector(v), then UnitVector(T(v))
RULE 32: ZeroVector(v) => not UnitVector(v)

7. COMPUTATION
INPUT: vector components, scalar field, chosen norm, coordinate system
VALIDATE: vector belongs to domain of norm
COMPUTE: n = norm(v)
BRANCH: n=0 => normalisation undefined
BRANCH: n=1 => already a unit vector
BRANCH: n>0 => return v/n
VERIFY: norm(v/n)=1, within declared numerical tolerance for approximate calculations
OUTPUT: normalised vector, norm, proof obligations, uncertainty metadata

8. EXAMPLES
EXAMPLE 01: v=(3,4) in Euclidean R^2; norm(v)=5; normalise(v)=(3/5,4/5)
EXAMPLE 02: v=(1,0,0) in Euclidean R^3; norm(v)=1; UnitVector(v)=true
EXAMPLE 03: v=(0,0) in Euclidean R^2; norm(v)=0; normalisation undefined
EXAMPLE 04: v=(-1,0) in Euclidean R^2; norm(v)=1; UnitVector(v)=true
EXAMPLE 05: v=(1/sqrt(2),1/sqrt(2)); Euclidean norm=1
EXAMPLE 06: v=(1,1) under L1 norm; norm(v)=2; not unit
EXAMPLE 07: v=(1,1) under L-infinity norm; norm(v)=1; unit
EXAMPLE 08: v=(i,0) in Euclidean C^2; norm(v)=1
EXAMPLE 09: Standard basis vectors e1,e2 in R^2 are orthonormal
EXAMPLE 10: (cos(theta),sin(theta)) in Euclidean R^2 is unit for real theta
EXAMPLE 11: For unit v, 2*v has norm 2 and is not unit
EXAMPLE 12: Rotation matrix R in R^2 preserves unit vectors
EXAMPLE 13: In R^3, v=(1,1,1)/sqrt(3) is unit
EXAMPLE 14: A nonzero vector can be normalised even if it is not a basis vector
EXAMPLE 15: In an inner-product space, unit u,v with inner_product(u,v)=0 form an orthonormal pair
EXAMPLE 16: In a complex inner-product space, multiplication by exp(i*theta) preserves unit norm

9. VALIDATION CONSTRAINTS
CHECK V01: norm must be explicitly identified
CHECK V02: zero vector must not be labelled unit
CHECK V03: numerical tolerance must be recorded when using floats
CHECK V04: component coordinates must match declared basis and dimension
CHECK V05: norm axioms must hold for supplied norm
CHECK V06: normalisation requires a nonzero input
CHECK V07: complex norms use absolute-square components, not plain squares
CHECK V08: orthogonality requires an inner product, not merely a norm
CHECK V09: orthonormality requires both orthogonality and unit norms
CHECK V10: basis status requires spanning and linear independence
CHECK V11: transformations preserving norm may preserve unit status
CHECK V12: avoid conflating unit vector with vector of all ones

10. SEMANTIC EQUIVALENCE
EQUIVALENT: unit vector <=> norm-one vector [same normed space]
EQUIVALENT: unit vector <=> point on unit sphere [same norm]
CONTEXTUAL: direction vector normalised to unit length [nonzero vector]
NOT_EQUIVALENT: unit vector != zero vector
NOT_EQUIVALENT: unit vector != arbitrary basis vector
NOT_EQUIVALENT: unit vector != orthogonal vector
NOT_EQUIVALENT: unit vector != scalar one
NOT_EQUIVALENT: unit vector != unit matrix

11. KNOWLEDGE GRAPH TRIPLES
(UnitVector, rdf:type, MathematicalEntity)
(UnitVector, rdfs:subClassOf, Vector)
(UnitVector, definedBy, NormEqualsOne)
(UnitVector, existsIn, NormedVectorSpace)
(UnitVector, belongsTo, UnitSphere)
(Normalisation, requires, NonzeroVector)
(Normalisation, produces, UnitVector)
(OrthonormalSet, contains, UnitVector)
(OrthonormalSet, requires, PairwiseOrthogonality)
(StandardBasisVector, isA, UnitVector) [Euclidean norm]
(ZeroVector, incompatibleWith, UnitVector)
(UnitVector, dependentOn, NormChoice)

12. CONFIDENCE
CONFIDENCE mathematical_axioms: 1.00 [definition-based]
CONFIDENCE derived_rules: 1.00 [subject to stated assumptions]
CONFIDENCE computed_approximation: conditional_on_precision_and_tolerance
CONFIDENCE source_page_alignment: unverified
REQUIRE evidence_trace for inferred classifications

13. FLOW HASH
FLOW_HASH algorithm: SHA-256
FLOW_HASH input: canonical UTF-8 serialisation of ontology content excluding the hash value
FLOW_HASH value: compute upon ingestion; not precomputed
FLOW_HASH purpose: content integrity and version comparison

14. UPDATE MODEL
UPDATE_MODEL version: 4.0
UPDATE_MODEL policy: preserve identifiers and mathematical invariants
UPDATE_MODEL trigger: corrected definition, revised inference, new example, new norm family, or new source evidence
UPDATE_MODEL method: semantic diff, consistency checks, provenance record, recompute Flow Hash
UPDATE_MODEL breaking_change: flag changes to type signatures or axiom semantics

15. AI CONSUMPTION
AI_CONSUMPTION mode: typed semantic graph + deterministic inference
AI_CONSUMPTION steps: parse -> type-check -> validate assumptions -> apply rules -> verify result -> attach provenance
AI_CONSUMPTION do_not: infer unit status without a norm
AI_CONSUMPTION do_not: normalise a zero vector
AI_CONSUMPTION do_not: infer basis or orthogonality from unit status alone
AI_CONSUMPTION prefer: exact arithmetic where possible
AI_CONSUMPTION report: norm, field, dimension, assumptions, proof path, numerical tolerance
AI_CONSUMPTION link: Scalar Ontology, Vector Space Ontology, Zero Vector Ontology, Vector Ontology, Basis Ontology, Norm Ontology, Inner Product Ontology

END FLOW ALGEBRA