Euclidean Space Ontology

FLOW ALGEBRA
Ontology: Euclidean Space Ontology
Version: 4.0
Date: 2026-10-13
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/coordinate-and-spatial-entities/euclidean-space-ontology.html
Status: Independent mathematical formalisation; source-page alignment not verified.
Authors: Mark Higgins — Sky (Recursive AI)

1. PURPOSE
Represent Euclidean spaces, points, displacement vectors, inner products, distances, affine structure, orthogonality and isometries with explicit type constraints and inference rules.

2. TYPE SYSTEM
TYPE Real := ordered complete field R.
TYPE NaturalDimension := n in N with n >= 0.
TYPE EuclideanVectorSpace(n) := (R^n, +, scalar_mul, dot).
TYPE EuclideanAffineSpace(n) := nonempty affine space E modelled on R^n with positive-definite inner product on its translation space.
TYPE Point(E) := p in E.
TYPE Vector(E) := v in translation_space(E).
TYPE Distance(E) := map E x E -> R_nonnegative.
TYPE LinearIsometry(n) := Q in R^(n x n) with Q^T Q = I.
TYPE AffineIsometry(E) := map f:E->E preserving distances.
TYPE OrthogonalFrame(E) := (origin, ordered orthonormal basis).
TYPE Subspace(V) := subset closed under vector addition and scalar multiplication.
TYPE AffineSubspace(E) := p + W for p in E and linear subspace W.

3. FOUNDATIONAL DEFINITIONS
DEFINE EuclideanSpace(E,n): affine_space(E,R^n) AND inner_product(translation_space(E)) AND positive_definite(inner_product).
DEFINE Displacement(p,q): q-p in translation_space(E).
DEFINE Dot(x,y): sum(i=1..n, x_i*y_i) in orthonormal Cartesian coordinates.
DEFINE Norm(v): sqrt(Dot(v,v)).
DEFINE Distance(p,q): Norm(q-p).
DEFINE Orthogonal(u,v): Dot(u,v)=0.
DEFINE UnitVector(v): Norm(v)=1.
DEFINE UnitDirection(v): v/Norm(v), requiring v != 0.
DEFINE Sphere(c,r): {p in E : Distance(p,c)=r}, r>=0.
DEFINE ClosedBall(c,r): {p in E : Distance(p,c)<=r}, r>=0.
DEFINE OpenBall(c,r): {p in E : Distance(p,c)<r}, r>0.
DEFINE Segment(p,q): {p+t(q-p): 0<=t<=1}.
DEFINE Line(p,v): {p+t*v: t in R}, v!=0.
DEFINE Hyperplane(p,n): {x in E: Dot(x-p,n)=0}, n!=0.
DEFINE AffineDimension(p+W): dim(W).
DEFINE EuclideanDimension(E): dim(translation_space(E)).
DEFINE Angle(u,v): arccos(Dot(u,v)/(Norm(u)*Norm(v))), u!=0, v!=0.
DEFINE ProjectionOnto(v,u): (Dot(v,u)/Dot(u,u))*u, u!=0.
DEFINE Midpoint(p,q): p+(q-p)/2.
DEFINE Convex(C): forall p,q in C, forall t in [0,1], p+t(q-p) in C.
DEFINE Isometry(f): forall p,q, Distance(f(p),f(q))=Distance(p,q).

4. AXIOMS AND INVARIANTS
AXIOM E01: forall v, Dot(v,v)>=0.
AXIOM E02: Dot(v,v)=0 iff v=0.
AXIOM E03: Dot(u,v)=Dot(v,u).
AXIOM E04: Dot(a*u+b*v,w)=a*Dot(u,w)+b*Dot(v,w).
AXIOM E05: forall p,q, Distance(p,q)>=0.
AXIOM E06: Distance(p,q)=0 iff p=q.
AXIOM E07: Distance(p,q)=Distance(q,p).
AXIOM E08: Distance(p,r)<=Distance(p,q)+Distance(q,r).
AXIOM E09: Displacement(p,q)+Displacement(q,r)=Displacement(p,r).
AXIOM E10: Every Euclidean affine space admits an orthonormal coordinate frame when finite-dimensional.
AXIOM E11: Coordinate representations depend on frame; distances and angles do not.
AXIOM E12: The zero-dimensional Euclidean space consists of one point up to isometry.

5. ENTITY RELATIONSHIPS
REL EuclideanSpace --is_a--> MetricSpace.
REL EuclideanSpace --has_underlying--> AffineSpace.
REL EuclideanSpace --modelled_on--> InnerProductSpace.
REL EuclideanSpace --has_dimension--> NaturalDimension.
REL Point --belongs_to--> EuclideanSpace.
REL Vector --translates--> Point.
REL PointPair --determines--> DisplacementVector.
REL InnerProduct --induces--> Norm.
REL Norm --induces--> EuclideanDistance.
REL OrthogonalFrame --coordinates--> EuclideanSpace.
REL EuclideanSpace --contains--> AffineSubspace.
REL AffineSubspace --has_direction--> LinearSubspace.
REL EuclideanDistance --invariant_under--> Isometry.
REL OrthogonalTransformation --is_a--> LinearIsometry.
REL Translation --is_a--> AffineIsometry.
REL Rotation --is_a--> OrientationPreservingIsometry.
REL Reflection --is_a--> OrientationReversingIsometry.
REL EuclideanSpace --supports--> ConvexGeometry.
REL EuclideanSpace --supports--> AnalyticGeometry.

6. INFERENCE RULES
RULE R01: EuclideanSpace(E,n) => MetricSpace(E,Distance).
RULE R02: InnerProduct(V) => Norm(v)=sqrt(Dot(v,v)).
RULE R03: Norm(v) defined => Distance(p,q)=Norm(q-p).
RULE R04: Distance(p,q)=0 => p=q.
RULE R05: p=q => Distance(p,q)=0.
RULE R06: Orthogonal(u,v) AND u!=0 AND v!=0 => Angle(u,v)=pi/2.
RULE R07: Dot(u,v)=0 => Norm(u+v)^2=Norm(u)^2+Norm(v)^2.
RULE R08: Norm(v)=1 => UnitVector(v).
RULE R09: v!=0 => Norm(UnitDirection(v))=1.
RULE R10: Q^T*Q=I => Norm(Q*v)=Norm(v).
RULE R11: Q^T*Q=I => Dot(Q*u,Q*v)=Dot(u,v).
RULE R12: Q^T*Q=I => abs(det(Q))=1.
RULE R13: Q^T*Q=I AND det(Q)=1 => ProperOrthogonalTransformation(Q).
RULE R14: Q^T*Q=I AND det(Q)=-1 => ImproperOrthogonalTransformation(Q).
RULE R15: f(p)=Q*p+b AND Q^T*Q=I => Isometry(f).
RULE R16: Isometry(f) => preserves_pairwise_distances(f).
RULE R17: Isometry(f) => preserves_collinearity(f).
RULE R18: Isometry(f) => preserves_angles_between_nonzero_displacements(f).
RULE R19: W linear_subspace_of V AND p in E => p+W affine_subspace_of E.
RULE R20: W linear_subspace_of V => 0 in W.
RULE R21: C convex AND p,q in C => Segment(p,q) subset_of C.
RULE R22: p,q in E => Midpoint(p,q) lies_on Segment(p,q).
RULE R23: u!=0 => Dot(v-ProjectionOnto(v,u),u)=0.
RULE R24: Orthogonal(u,v) => Dot(u,v)=0.
RULE R25: Norm(u+v)<=Norm(u)+Norm(v).
RULE R26: abs(Dot(u,v))<=Norm(u)*Norm(v).
RULE R27: EuclideanSpace(E,n) AND n>=1 => complete_metric_space(E).
RULE R28: EuclideanSpace(E,n) AND n>=1 => locally_compact(E).
RULE R29: ClosedAndBounded(S) AND S subset_of R^n => Compact(S).
RULE R30: Continuous(f) AND Compact(S) => Compact(f(S)).
RULE R31: OrthonormalBasis(B,V) => cardinality(B)=dim(V).
RULE R32: OrthogonalNonzeroSet(S) => LinearlyIndependent(S).
RULE R33: EuclideanSpace(E,n) => HomeomorphicToRn(E,n) after choosing affine frame.
RULE R34: n>=1 AND OpenBall(c,r) => Convex(OpenBall(c,r)).
RULE R35: n>=1 AND ClosedBall(c,r) => Convex(ClosedBall(c,r)).
RULE R36: n>=1 AND Sphere(c,r) AND r>0 => boundary_of(ClosedBall(c,r)).
RULE R37: p,q in E AND p!=q => unique_affine_line_through(p,q).
RULE R38: Distance(p,q)=d AND d>=0 => symmetric_distance(q,p,d).
RULE R39: CoordinateFrameChangeOrthogonal(Q,b) => EuclideanMetricInvariant.
RULE R40: AffineIsometry(E) AND EuclideanSpace(E,n) => distance_preserving_bijection(E,E) when isometry is defined as global Euclidean motion.

7. VALIDATION CONSTRAINTS
CHECK C01: dimension must be a nonnegative integer.
CHECK C02: point and vector coordinates must share dimension and frame when used together.
CHECK C03: distances must be nonnegative finite real values for finite points.
CHECK C04: inner product must be symmetric, bilinear and positive definite over R.
CHECK C05: unit-vector classification requires norm exactly one (or stated numerical tolerance).
CHECK C06: angle calculation requires two nonzero vectors.
CHECK C07: projection requires a nonzero target vector.
CHECK C08: orthogonal transformation requires square Q and Q^T Q=I.
CHECK C09: open-ball radius must be positive; closed-ball radius may be zero.
CHECK C10: distinguish a sphere (boundary) from a ball (region).
CHECK C11: distinguish points from free vectors; a chosen origin is required for coordinate identification.
CHECK C12: distinguish affine dimension from ambient dimension.
CHECK C13: do not infer Euclidean structure from arbitrary metric-space structure.
CHECK C14: do not infer finite dimensionality from an arbitrary inner-product space.
CHECK C15: specify whether a mapping is linear, affine, or general before classifying isometry.
CHECK C16: numerical equality tests must declare tolerance and precision.

8. MATHEMATICAL EXAMPLES
EXAMPLE X01: R^2 with standard dot product is a 2D Euclidean vector space.
EXAMPLE X02: R^3 with standard dot product is a 3D Euclidean vector space.
EXAMPLE X03: p=(1,2), q=(4,6) => displacement=(3,4), distance=5.
EXAMPLE X04: u=(1,0), v=(0,1) => Dot(u,v)=0.
EXAMPLE X05: v=(3,4) => Norm(v)=5, UnitDirection(v)=(3/5,4/5).
EXAMPLE X06: p=(0,0), q=(2,4) => Midpoint=(1,2).
EXAMPLE X07: Segment((0,0),(2,0))={(t,0):0<=t<=2}.
EXAMPLE X08: Sphere((0,0,0),2)={x: x1^2+x2^2+x3^2=4}.
EXAMPLE X09: ClosedBall((0,0),1)={x:x1^2+x2^2<=1}.
EXAMPLE X10: Q=[[0,-1],[1,0]] is orthogonal with determinant 1.
EXAMPLE X11: Reflection Q=[[-1,0],[0,1]] has determinant -1.
EXAMPLE X12: Translation f(x)=x+(3,2) preserves Euclidean distances.
EXAMPLE X13: W={(x,0):x in R} is a linear subspace of R^2.
EXAMPLE X14: (0,1)+W={(x,1):x in R} is an affine subspace, not a linear subspace.
EXAMPLE X15: ProjectionOnto((3,4),(1,0))=(3,0).
EXAMPLE X16: The Euclidean distance between (0,0,0) and (1,2,2) is 3.

9. SEMANTIC EQUIVALENCE AND DISAMBIGUATION
EQUIV Euclidean_n_space <-> n_dimensional_Euclidean_space [context: standard finite-dimensional geometry].
EQUIV Euclidean_distance <-> L2_distance [context: standard orthonormal coordinates].
EQUIV orthogonal_matrix <-> real_square_matrix_Q_with_QtQ_equals_I.
RELATED EuclideanSpace ~ InnerProductSpace [not universally equivalent: affine versus vector structure].
RELATED EuclideanSpace ~ MetricSpace [not equivalent: Euclidean is specialised].
DISTINCT Sphere != ClosedBall.
DISTINCT AffinePoint != FreeVector.
DISTINCT EuclideanDimension != CoordinateIndex.
DISTINCT Orthogonal != Independent [orthogonal nonzero vectors imply independence; converse generally false].
DISTINCT CoordinateChange != GeometricMotion [passive vs active interpretation].

10. KNOWLEDGE GRAPH TRIPLES
(EuclideanSpace, subClassOf, MetricSpace)
(EuclideanSpace, hasStructure, EuclideanMetric)
(EuclideanMetric, inducedBy, InnerProduct)
(InnerProduct, defines, Orthogonality)
(EuclideanSpace, admits, OrthonormalFrame)
(EuclideanSpace, contains, Point)
(PointPair, determines, DisplacementVector)
(DisplacementVector, hasMeasure, Norm)
(OrthogonalMatrix, represents, LinearIsometry)
(Translation, instanceOf, AffineIsometry)
(AffineSubspace, hasDirectionSpace, LinearSubspace)
(Sphere, isBoundaryOf, ClosedBall)

11. CONFIDENCE
CONFIDENCE mathematical_axioms: 1.00 [standard definitions, conditional on stated model].
CONFIDENCE exact_theorems: 1.00 [valid under explicit hypotheses].
CONFIDENCE illustrative_examples: 1.00 [symbolically verifiable].
CONFIDENCE source_page_alignment: unverified.
CONFIDENCE generated_semantic_mappings: 0.95 [review recommended before canonical ingestion].

12. FLOW HASH
FLOW_HASH.algorithm: SHA-256.
FLOW_HASH.canonicalisation: UTF-8; LF line endings; normalised Unicode NFC; exclude FLOW_HASH.digest field.
FLOW_HASH.digest: computed on ingestion, not fabricated in this file.
FLOW_HASH.scope: complete canonical flow content.

13. UPDATE MODEL
UPDATE_MODEL.version: 4.0.
UPDATE_MODEL.mode: semantic_versioning.
UPDATE_MODEL.trigger: corrected theorem, new formal relationship, revised source page, or ontology taxonomy change.
UPDATE_MODEL.procedure: validate definitions -> validate inference rules -> rerun examples -> recompute hash -> record provenance.
UPDATE_MODEL.compatibility: preserve stable entity identifiers; declare breaking changes explicitly.
UPDATE_MODEL.provenance: distinguish sourced claims from independently formalised claims.

14. AI CONSUMPTION
AI_CONSUMPTION.read_mode: typed_semantic_graph.
AI_CONSUMPTION.primary_entity: EuclideanSpace.
AI_CONSUMPTION.reasoning: deterministic symbolic inference with precondition checks.
AI_CONSUMPTION.units: distances inherit coordinate units; angles in radians unless explicitly converted.
AI_CONSUMPTION.frame_awareness: required.
AI_CONSUMPTION.ambiguity_policy: request dimension, frame, metric and point/vector type when unspecified.
AI_CONSUMPTION.validation: reject invalid dimensions, inconsistent coordinate frames and violated theorem hypotheses.
AI_CONSUMPTION.uncertainty: separate numerical tolerance from logical confidence.
AI_CONSUMPTION.output: typed entities, justified relations, rule IDs, assumptions, provenance and confidence.
AI_CONSUMPTION.no_hallucination: never invent verified source alignment or hash digests.

END FLOW ALGEBRA