Point Ontology

FLOW ALGEBRA
POINT ONTOLOGY — v4.0
Kireth Semantic Library
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/geometric-entities/point-ontology-2.html
Status: Independently formalised; source-page wording not verified.

01. ONTOLOGY IDENTITY
ONTOLOGY_ID: geometric.point
ENTITY_TYPE: Point
PARENT: GeometricEntity
DOMAIN: Geometry
VERSION: 4.0
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org

02. SEMANTIC DEFINITION
A point is a primitive geometric entity representing a location without intrinsic spatial extent in a specified geometric space. Coordinates represent a point relative to a chosen reference frame; the point itself is not identical to its coordinate tuple.

03. TYPE SIGNATURES
Space := EuclideanSpace | AffineSpace | MetricSpace | TopologicalSpace | ProjectiveSpace | Manifold
Point[S:Space] := element_of(S)
CoordinateFrame[S] := reference_system(S)
Coordinates[S,F] := representation(Point[S], F)
Distance[S] := optional_metric(S)
Vector[S] := displacement_or_tangent_entity(S)

04. CORE AXIOMS
P01: forall p:Point[S], p in S.
P02: points have no intrinsic length, area, or volume.
P03: p = q is a geometric identity assertion independent of coordinate representation.
P04: coordinate(p,F) is defined only if frame F supports p.
P05: coordinate(p,F1) and coordinate(p,F2) may differ for the same p.
P06: in Euclidean R^n, each p has an n-tuple of coordinates in a Cartesian frame.
P07: in affine geometry, differences of points define displacement vectors; addition of two points is not intrinsically defined.
P08: distance(p,q) requires an appropriate metric, not merely a topological space.
P09: incidence relations are defined relative to lines, curves, planes, or other subsets.
P10: the same point may belong to multiple geometric objects.

05. RELATIONSHIP SIGNATURES
IS_A(Point,GeometricEntity)
LOCATED_IN(Point,Space)
HAS_COORDINATES(Point,CoordinateTuple,CoordinateFrame)
LIES_ON(Point,Line|Curve|Surface)
BELONGS_TO(Point,GeometricSubset)
DISTINCT_FROM(Point,Point)
COINCIDES_WITH(Point,Point)
CONNECTED_BY(Point,Point,Segment)
TRANSFORMED_TO(Point,Point,Transformation)
DISTANCE_BETWEEN(Point,Point,NonnegativeReal) [metric required]
MIDPOINT_OF(Point,Point,Point) [affine midpoint structure required]

06. EUCLIDEAN AND AFFINE OPERATIONS
E01: p=(p1,...,pn) in R^n.
E02: displacement(p,q)=q-p in R^n.
E03: d(p,q)=sqrt(sum_i (qi-pi)^2).
E04: midpoint(p,q)=(p+q)/2 in affine coordinates over R.
E05: segment(p,q)={(1-t)p+tq | 0<=t<=1}.
E06: affine_combination(p_i,a_i)=sum_i a_i p_i provided sum_i a_i=1.
E07: translation_Tv(p)=p+v.
E08: linear_map_A_on_points(p)=Ap only after an origin is chosen.
E09: affine_map(p)=Ap+b.
E10: coordinate_change(p,F1,F2) preserves geometric identity.

07. TOPOLOGICAL AND PROJECTIVE DISTINCTIONS
T01: a topological point is an element of a topological space; distance need not exist.
T02: a manifold point may admit local chart coordinates, without a single global coordinate chart.
T03: a projective point is an equivalence class of nonzero homogeneous coordinate vectors under nonzero scalar multiplication.
T04: homogeneous vectors x and lambda*x, lambda!=0, denote the same projective point.
T05: points at infinity are valid in projective completion but not ordinary finite Euclidean points.
T06: an isolated point is defined relative to a subset and its topology.
T07: a limit point is defined by neighbourhood intersection conditions, not by geometric distance alone.

08. FORMAL INFERENCE RULES
R01: Point(p,S) -> Member(p,S).
R02: HasCoordinates(p,c,F) -> Point(p,space(F)).
R03: HasCoordinates(p,c,F) and HasCoordinates(q,c,F) and FrameInjective(F) -> p=q.
R04: p=q -> Coordinate(p,F)=Coordinate(q,F) where defined.
R05: LiesOn(p,L) -> Member(p,L).
R06: LiesOn(p,L) and Subset(L,S) -> Member(p,S).
R07: MetricSpace(S) and Point(p,S) and Point(q,S) -> DistanceDefined(p,q).
R08: Distance(p,q)=0 and MetricSpace(S) -> p=q.
R09: p=q -> Distance(p,q)=0 when metric defined.
R10: Distance(p,q)=Distance(q,p) when metric defined.
R11: Distance(p,r)<=Distance(p,q)+Distance(q,r) when metric defined.
R12: EuclideanSpace(S) and p,q in S -> SegmentDefined(p,q).
R13: p=q -> Segment(p,q) is singleton {p}.
R14: AffineSpace(S) and p,q in S -> DisplacementVector(q-p).
R15: AffineSpace(S) and v a translation vector -> Translate(p,v) in S.
R16: AffineMap(T) and p in Domain(T) -> Point(T(p),Codomain(T)).
R17: BijectiveTransformation(T) and p!=q -> T(p)!=T(q).
R18: Isometry(T) -> Distance(T(p),T(q))=Distance(p,q).
R19: p in Intersection(A,B) -> p in A and p in B.
R20: p in A and p in B -> p in Intersection(A,B).
R21: p in Boundary(A) -> p in Closure(A).
R22: p in Interior(A) -> p in A.
R23: p in Closure(A) -> every neighbourhood of p intersects A.
R24: p is AccumulationPoint(A) -> every neighbourhood of p intersects A\{p}.
R25: FiniteEuclideanPoint(p) -> CoordinatesFinite(p) in a finite Cartesian frame.
R26: ProjectiveEquivalent(x,y) -> SameProjectivePoint([x],[y]).
R27: AffineCombinationWeightsSumOne -> ResultIsPoint.
R28: VectorCombinationWeightsArbitrary -> ResultIsVector, not automatically a point.
R29: p in R^n -> CoordinateDimension(p)=n in standard coordinates.
R30: Coincident(p,q) -> GeometricIdentity(p,q).
R31: Distinct(p,q) -> not GeometricIdentity(p,q).
R32: CoordinateChange(F,G) and valid chart overlap -> Coordinate_G(p)=Transition_F_to_G(Coordinate_F(p)).
R33: Midpoint(m,p,q) -> displacement(p,m)=displacement(m,q) in affine real space.
R34: Collinear(p,q,r) -> points share an affine line (allow repeated points).
R35: Noncollinear(p,q,r) in Euclidean plane -> unique line through p,q and unique plane containing all three in R^3.
R36: Point(p,S) and Subset(A,S) does not imply p in A without membership evidence.

09. VALIDATION CONSTRAINTS
V01: Coordinate tuple dimension must match chosen chart or frame.
V02: A point must identify its ambient space or context.
V03: Distances must not be inferred absent a metric.
V04: Coordinate equality across different frames must not imply point equality without conversion.
V05: Reject addition of affine points unless an origin/affine combination is specified.
V06: Reject coordinate-free claims of point magnitude.
V07: Reject treating homogeneous coordinate representatives as unique.
V08: Require chart-domain membership before applying manifold coordinate transformations.
V09: Keep geometric identity distinct from numeric approximate equality.
V10: Require units and compatible coordinate systems for physical locations.
V11: Zero-dimensional geometric entity does not imply singleton ambient space.
V12: Euclidean point claims should not be automatically transferred to all geometries.

10. EXAMPLES
X01: (0,0) is the origin in the standard R^2 Cartesian frame.
X02: (3,4) and (3,4) in the same injective frame denote the same point.
X03: p=(1,2), q=(4,6): distance(p,q)=5.
X04: p=(1,2), q=(5,6): midpoint=(3,4).
X05: translation by (2,-1) maps (1,3) to (3,2).
X06: the point (0,0) lies on x-axis and y-axis.
X07: point (1,1) lies on line y=x.
X08: p=(1,0) and q=(0,1) define a segment.
X09: point p may be interior to a disk while lying on no boundary circle.
X10: a point on a sphere surface belongs to the surface and ambient R^3.
X11: latitude/longitude requires a geographic reference model; raw pairs are not universal Cartesian coordinates.
X12: in projective plane [1:2:1]=[2:4:2].
X13: a manifold point may have coordinates in overlapping charts.
X14: two different coordinate frames may assign different tuples to the same point.
X15: the zero vector is not a point intrinsically, though an origin may be represented by zero coordinates.
X16: in a discrete topological space, every singleton is open and every point is isolated.

11. SEMANTIC EQUIVALENCE
SE01: Point ~= GeometricLocation [context-dependent; not universal identity].
SE02: CoordinatePoint ~= PointWithSpecifiedFrame [representation-specific].
SE03: OrderedTuple != IntrinsicPoint [distinguish representation].
SE04: PositionVector != AffinePoint [origin-dependent association].
SE05: EuclideanPoint IS_A Point.
SE06: ProjectivePoint IS_A Point [in projective geometry].
SE07: ManifoldPoint IS_A Point [in manifold context].
SE08: Origin IS_A Point [in a chosen affine frame].

12. KNOWLEDGE GRAPH TRIPLES
(Point, rdf:type, GeometricEntity)
(Point, belongsToDomain, Geometry)
(Point, memberOf, GeometricSpace)
(Point, mayHaveRepresentation, CoordinateTuple)
(CoordinateTuple, interpretedUnder, CoordinateFrame)
(Point, participatesIn, IncidenceRelation)
(Point, participatesIn, GeometricTransformation)
(EuclideanPoint, subclassOf, Point)
(ProjectivePoint, subclassOf, Point)
(ManifoldPoint, subclassOf, Point)
(GeometricPointIdentity, distinctFrom, CoordinateTupleEqualityAcrossFrames)

13. CONFIDENCE
CONFIDENCE:
  mathematical_axioms: 0.99
  euclidean_examples: 0.99
  contextual_semantic_mappings: 0.90
  exact_source_page_alignment: unverified
  policy: "Scores express editorial confidence, not calibrated probabilities."

14. FLOW_HASH
FLOW_HASH:
  algorithm: SHA-256
  canonicalisation: UTF-8, LF newlines, exclude FLOW_HASH value from hash input
  value: COMPUTE_ON_PUBLICATION
  purpose: version integrity and change detection

15. UPDATE_MODEL
UPDATE_MODEL:
  version: 4.0
  change_tracking: semantic-versioned
  triggers: [axiom_correction, relationship_change, new_geometry_context, inference_revision, source_page_revision]
  review: verify mathematical typing, domain assumptions and counterexamples
  backwards_compatibility: preserve stable entity and predicate identifiers where possible

16. AI_CONSUMPTION
AI_CONSUMPTION:
  mode: typed_semantic_reasoning
  required_context: [ambient_space, geometry_type, coordinate_frame_if_used]
  operations: [classify, validate, infer, map, transform, compare]
  safeguards: [do_not_conflate_point_and_vector, do_not_assume_metric, do_not_equate_cross_frame_coordinates]
  output: [entity_type, inferred_relations, assumptions, evidence, confidence, validation_status]
  provenance: "Independent mathematical formalisation for Kireth Semantic Library; page not verified."

END FLOW ALGEBRA