Scalar Ontology

FLOW ALGEBRA
TITLE: Scalar Ontology
VERSION: 4.0
DOMAIN: Mathematical Entities / Linear Algebra Entities
SOURCE_URL: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/linear-algebra-entities/scalar-ontology-2.html
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library
SOURCE_ALIGNMENT: Independently constructed; source page not verified.

[ONTOLOGY]
ENTITY: Scalar
TYPE: MathematicalEntity
PARENT: LinearAlgebraEntity
DEFINITION: An element of a specified field K acting on a vector space V through scalar multiplication.
SCOPE: Field-based linear algebra; distinguish scalars from vectors, matrices, and dimensioned physical quantities.

[TYPE SYSTEM]
Field := (K, +, *, 0, 1)
VectorSpace := (V, +_V, scalar_mul, K)
Scalar := element(K)
Vector := element(V)
ScalarMultiplication := K x V -> V
ScalarAddition := K x K -> K
ScalarProduct := K x K -> K
ScalarInverse := K\{0} -> K

[CORE AXIOMS]
A01 | ScalarClosureAddition | a,b in K => a+b in K
A02 | ScalarClosureMultiplication | a,b in K => a*b in K
A03 | ScalarAddAssociative | (a+b)+c = a+(b+c)
A04 | ScalarAddCommutative | a+b = b+a
A05 | ScalarZero | a+0 = a
A06 | ScalarAddInverse | a+(-a)=0
A07 | ScalarMulAssociative | (a*b)*c=a*(b*c)
A08 | ScalarMulCommutative | a*b=b*a
A09 | ScalarOne | a*1=a
A10 | ScalarMulInverse | a!=0 => a*(1/a)=1
A11 | Distributive | a*(b+c)=a*b+a*c
A12 | NontrivialField | 0!=1
A13 | VectorDistributive | a*(u+v)=a*u+a*v
A14 | ScalarDistributive | (a+b)*v=a*v+b*v
A15 | ScalarActionAssociative | (a*b)*v=a*(b*v)
A16 | ScalarIdentityAction | 1*v=v

[ENTITY RELATIONSHIPS]
R01 | Scalar | IS_ELEMENT_OF | Field
R02 | Field | DEFINES_SCALARS_FOR | VectorSpace
R03 | Scalar | PARTICIPATES_IN | ScalarMultiplication
R04 | ScalarMultiplication | MAPS | Field x VectorSpace -> VectorSpace
R05 | Scalar | OPERAND_OF | ScalarAddition
R06 | Scalar | OPERAND_OF | ScalarProduct
R07 | NonzeroScalar | HAS | MultiplicativeInverse
R08 | ZeroScalar | IS | AdditiveIdentity
R09 | UnitScalar | IS | MultiplicativeIdentity
R10 | ScalarMultiplication | PRESERVES | VectorSpaceMembership
R11 | Scalar | MAY_BE | RealNumber
R12 | Scalar | MAY_BE | ComplexNumber
R13 | Scalar | MAY_BE | RationalNumber
R14 | Scalar | MAY_BE | FiniteFieldElement
R15 | Scalar | IS_DISTINCT_FROM | Vector
R16 | ScalarMatrix | IS_NOT_EQUIVALENT_TO | ScalarWithoutContext
R17 | Scalar | PARAMETERISES | LinearCombination
R18 | Scalar | SCALES | Vector
R19 | Scalar | SCALES | LinearTransformation
R20 | Scalar | PARTICIPATES_IN | EigenvalueEquation

[FORMAL INFERENCE RULES]
I01 | a in K AND b in K => a+b in K
I02 | a in K AND b in K => a*b in K
I03 | a in K => -a in K
I04 | a in K AND a!=0 => inverse(a) in K
I05 | a in K AND v in V => a*v in V
I06 | a=0 AND v in V => a*v=zero_vector(V)
I07 | a in K => a*zero_vector(V)=zero_vector(V)
I08 | a=1 AND v in V => a*v=v
I09 | a=-1 AND v in V => a*v=-v
I10 | a,b in K AND v in V => (a+b)*v=a*v+b*v
I11 | a in K AND u,v in V => a*(u+v)=a*u+a*v
I12 | a,b in K AND v in V => (a*b)*v=a*(b*v)
I13 | a!=0 AND a*v=zero_vector(V) => v=zero_vector(V)
I14 | v!=zero_vector(V) AND a*v=zero_vector(V) => a=0
I15 | a*v=b*v AND v!=zero_vector(V) => a=b
I16 | a*v=a*w AND a!=0 => v=w
I17 | a,b in K AND a=b => a*v=b*v
I18 | a in K AND v=w => a*v=a*w
I19 | a in K AND v in V => (-a)*v=-(a*v)
I20 | a in K AND v in V => a*(-v)=-(a*v)
I21 | a,b in K AND v in V => (a-b)*v=a*v-b*v
I22 | a in K AND u,v in V => a*(u-v)=a*u-a*v
I23 | a in K AND a!=0 AND v in V => (1/a)*(a*v)=v
I24 | a in K AND a!=0 AND v in V => a*((1/a)*v)=v
I25 | finiteLinearCombination(sum_i a_i*v_i) => each a_i in K AND each v_i in V
I26 | lambda in K AND T(v)=lambda*v AND v!=0 => lambda IS_EIGENVALUE_OF T
I27 | a in K AND T K-linear => T(a*v)=a*T(v)
I28 | K=R AND a in K => a is real scalar
I29 | K=C AND a in K => a is complex scalar
I30 | K=F_p AND p prime AND a in K => a is finite-field scalar
I31 | v!=0 AND a*v=v => a=1
I32 | a*v=0 AND a!=0 => v=0

[VALIDATION CONSTRAINTS]
V01 | Every scalar has a declared coefficient field or a resolvable field context.
V02 | Scalar multiplication operands must be (scalar, vector) in that order or explicitly typed equivalent.
V03 | Do not infer real-valued scalars from a generic vector-space declaration.
V04 | Multiplicative inverse exists only for nonzero field elements.
V05 | The scalar 0 and the zero vector belong to different types even when notation coincides.
V06 | Physical scalar quantities can carry units; their physical dimensions are separate from field-element typing.
V07 | A one-by-one matrix may be isomorphic to a scalar representation, but is not automatically the same typed entity.
V08 | Eigenvalues are scalars over the declared field only when the eigenvalue exists in that field.
V09 | Linear combinations require compatible vectors from the same vector space.
V10 | In modules over rings, coefficients are ring elements and need not form a field.
V11 | Avoid division by zero and cross-field arithmetic without explicit coercion.
V12 | Zero-vector cancellation is invalid: a*0=b*0 does not imply a=b.

[MATHEMATICAL EXAMPLES]
E01 | K=R, V=R^2, a=3, v=(2,-1) => a*v=(6,-3)
E02 | K=R, V=R^2, a=0, v=(4,7) => a*v=(0,0)
E03 | K=R, V=R^2, a=-2, v=(1,3) => a*v=(-2,-6)
E04 | K=C, V=C, a=i, v=1+i => a*v=-1+i
E05 | K=Q, V=Q^2, a=1/2, v=(2,6) => a*v=(1,3)
E06 | K=F_5, V=F_5, a=3, v=4 => a*v=2 (mod 5)
E07 | K=R, V=R^2, 2*(1,0)+3*(0,1)=(2,3)
E08 | K=R, V=R^2, a=1 => a*(5,-2)=(5,-2)
E09 | K=R, V=R^2, a=-1 => a*(5,-2)=(-5,2)
E10 | K=R, V=R^2, a=2, b=3, v=(1,4) => (a+b)*v=(5,20)
E11 | K=R, T=diag(2,3), v=(1,0) => T(v)=2*v; eigenvalue=2
E12 | K=R, V=R^3, a=4, v=(0,0,0) => a*v=(0,0,0)
E13 | K=R, V=R^2, a=2, u=(1,1), v=(2,-1) => a*(u+v)=(6,0)
E14 | K=C, V=C^2, a=1+i, v=(1,0) => a*v=(1+i,0)
E15 | K=F_2, V=F_2^2, a=1, v=(1,0) => a*v=(1,0)
E16 | K=R, V=R, a=1/4, v=8 => a*v=2

[SEMANTIC EQUIVALENCE]
Scalar <=> FieldElement [context: linear algebra, declared field]
ScalarCoefficient <=> Scalar [context: linear combination]
ScalarFactor <=> Scalar [context: scalar multiplication]
RealScalar <=> RealNumber [context: K=R]
ComplexScalar <=> ComplexNumber [context: K=C]
NOT_EQUIVALENT: Scalar != Vector
NOT_EQUIVALENT: Scalar != ZeroVector
NOT_EQUIVALENT: Scalar != ArbitraryRingElement without module context

[KNOWLEDGE GRAPH TRIPLES]
(Scalar, rdf:type, MathematicalEntity)
(Scalar, rdfs:subClassOf, LinearAlgebraEntity)
(Scalar, belongsTo, Field)
(ScalarMultiplication, hasOperandType, Scalar)
(ScalarMultiplication, hasOperandType, Vector)
(ScalarMultiplication, hasResultType, Vector)
(LinearCombination, hasCoefficientType, Scalar)
(Eigenvalue, rdfs:subClassOf, Scalar)
(RealScalar, rdfs:subClassOf, Scalar)
(ComplexScalar, rdfs:subClassOf, Scalar)

[CONFIDENCE]
MathematicalAxioms: 1.00
FieldContextDependentAssertions: 0.99
SemanticMappings: 0.95
SourcePageAlignment: UNVERIFIED
EvidencePolicy: Distinguish proved algebraic statements from inferred semantic labels.

[FLOW_HASH]
Algorithm: SHA-256
Input: Canonical UTF-8 normalised ontology content excluding the FLOW_HASH value itself.
Value: COMPUTE_ON_PUBLICATION

[UPDATE_MODEL]
Version: 4.0
ChangeDetection: Compare definitions, relationships, rules, examples and source-page metadata.
OnUpdate: Revalidate field assumptions, type constraints, all inference rules and knowledge graph triples.
BackwardCompatibility: Preserve stable entity IDs; version changed assertions explicitly.

[AI_CONSUMPTION]
Mode: TypedSemanticReasoning
Input: Field declaration, vector-space declaration, candidate scalar, operation context.
Resolve: Determine field membership before applying scalar rules.
Validate: Check type compatibility, nonzero preconditions, and vector-space axioms.
Infer: Apply sound rules only when premises are established.
Explain: Return the governing axiom or inference rule with each conclusion.
Uncertainty: Report absent field context rather than guessing.
Output: Typed entities, relations, proofs or counterexamples, provenance, confidence.

[END FLOW ALGEBRA]