Coordinate and Spatial Entities

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
Coordinate and Spatial Entities — Main Page Ontology
Version: 4.0
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/coordinate-and-spatial-entities.html
Authors: Mark Anthony Higgins — Sky (Recursive AI)
Site: https://kirethresearch.org
Source taxonomy verified: 2026-10-13
Status: Formal mathematical interpretation; not a claim of executable software implementation.

01. PURPOSE
Represent coordinates, reference frames, mathematical spaces and dynamical state spaces as typed, validated entities with explicit relationships and inference rules.

02. SOURCE TAXONOMY — 18 CHILD ONTOLOGIES
01 Coordinate Ontology
02 Coordinate Point Ontology
03 Coordinate Axis Ontology
04 Coordinate System Ontology
05 Cartesian Coordinate Ontology
06 Polar Coordinate Ontology
07 Spherical Coordinate Ontology
08 Cylindrical Coordinate Ontology
09 Origin Ontology
10 Dimension Axis Ontology
11 Mathematical Space Ontology
12 Euclidean Space Ontology
13 Metric Space Ontology
14 Topological Space Ontology
15 Vector Space Entity Ontology
16 State Space Ontology
17 Phase Space Ontology
18 Configuration Space Ontology

03. TYPE SYSTEM
TYPE Scalar := element of a declared scalar domain.
TYPE Dimension := nonnegative integer for finite-dimensional contexts.
TYPE Point[S] := element of space S.
TYPE Vector[V] := element of vector space V.
TYPE CoordinateTuple[n,D] := ordered n-tuple of values from D.
TYPE CoordinateChart[S,D,n] := partial map from a domain U subset S into D^n, with explicitly stated injectivity or local-homeomorphism requirements.
TYPE CoordinateSystem[S] := one or more charts plus conventions, units, orientation and reference frame.
TYPE Axis := named direction or independent coordinate dimension, contextual to a coordinate system.
TYPE Origin := distinguished point associated with an affine reference frame.
TYPE MetricSpace := (X,d) satisfying metric axioms.
TYPE TopologicalSpace := (X,T) with topology T.
TYPE VectorSpace := (V,F,+,scalar_multiply) satisfying vector-space axioms.
TYPE StateSpace := set of admissible system states.
TYPE PhaseSpace := state space for a dynamical description, commonly including position and momentum variables.
TYPE ConfigurationSpace := set of admissible configurations, possibly with constraints.

04. CANONICAL RELATIONS
REL coordinateOf(tuple,point,chart) : coordinates depend on chart.
REL hasDimension(space,n) : finite-dimensional declaration only.
REL hasAxis(system,axis) : axis belongs to system.
REL hasOrigin(frame,point) : frame has distinguished origin.
REL chartOn(chart,space) : chart describes a region of space.
REL transformsTo(chartA,chartB,transitionMap) : valid on overlapping domains.
REL metricOn(metric,space) : metric equips underlying set with distance.
REL topologyOn(topology,space) : topology equips underlying set with open sets.
REL underlyingSet(structure,set) : structure has carrier set.
REL vectorSpaceOver(space,field) : scalar field of vector space.
REL stateOf(state,system) : state represents a system condition.
REL configurationOf(configuration,system) : configuration represents degrees of freedom.
REL trajectoryIn(trajectory,stateSpace) : trajectory maps time into state space.
REL embedsIn(source,target,map) : injective structure-preserving map, with preservation explicitly typed.
REL isSpecialisationOf(child,parent) : declared ontology hierarchy, not necessarily set inclusion.

05. MATHEMATICAL CONSTRAINTS
C01: A coordinate tuple is not intrinsically a geometric point; interpretation requires a chart/frame.
C02: Cartesian coordinates in R^n use n ordered scalar components relative to a basis and origin.
C03: Polar (r,theta) in R^2: r >= 0, x=r cos(theta), y=r sin(theta); theta non-unique at r=0.
C04: Cylindrical (r,theta,z): x=r cos(theta), y=r sin(theta), z=z; r>=0.
C05: Spherical convention here uses (rho,theta,phi) with rho>=0, theta azimuth, phi inclination from positive z-axis: x=rho sin(phi) cos(theta), y=rho sin(phi) sin(theta), z=rho cos(phi).
C06: Spherical angles are non-unique at the origin and azimuth is non-unique on the polar axis.
C07: Euclidean distance d(x,y)=sqrt(sum_i (x_i-y_i)^2) for finite n and standard coordinates.
C08: Metric axioms: nonnegativity; d(x,y)=0 iff x=y; symmetry; triangle inequality.
C09: Topology axioms: empty set and carrier open; arbitrary unions of open sets open; finite intersections open.
C10: Every metric induces a topology via open balls; not every topology is metrizable.
C11: A vector space has an additive identity, additive inverses, and compatible field scalar multiplication.
C12: An affine space does not have a canonical origin unless one is chosen.
C13: A coordinate chart for a manifold is local; global coordinates need not exist.
C14: Coordinate transformations require domain overlap and correct conventions.
C15: State, phase and configuration spaces are model-dependent; they are not interchangeable by default.
C16: Units, handedness, coordinate order, angular conventions and reference epoch must be explicit where relevant.
C17: Physical position coordinates and abstract state variables must not be conflated.
C18: Infinite-dimensional spaces require cardinal or functional-analytic dimensional treatment; do not force a finite tuple.

06. FORMAL INFERENCE RULES
R01 IF chartOn(C,S) AND pointIn(p,domain(C)) THEN coordinateOf(C(p),p,C).
R02 IF coordinateOf(c,p,C) AND C injective THEN uniquePointForCoordinates(c,C,p).
R03 IF hasOrigin(F,o) AND basisOf(B,F) THEN CartesianCoordinatesAreFrameRelative(F).
R04 IF validPolar(r,theta) THEN CartesianPoint(r*cos(theta),r*sin(theta)).
R05 IF CartesianPoint(x,y) THEN polarRadius(sqrt(x*x+y*y)).
R06 IF x=0 AND y=0 THEN polarAngleUnderdetermined.
R07 IF validCylindrical(r,theta,z) THEN CartesianPoint3(r*cos(theta),r*sin(theta),z).
R08 IF validSpherical(rho,theta,phi) THEN CartesianPoint3(rho*sin(phi)*cos(theta),rho*sin(phi)*sin(theta),rho*cos(phi)).
R09 IF sphericalRadius(rho)=0 THEN sphericalAnglesUnderdetermined.
R10 IF metricOn(d,X) AND satisfiesMetricAxioms(d) THEN MetricSpace(X,d).
R11 IF MetricSpace(X,d) THEN inducedTopology(X,d).
R12 IF TopologicalSpace(X,T) THEN NOT automatically MetricSpace(X,d).
R13 IF EuclideanSpace(Rn) THEN MetricSpace(Rn,euclideanDistance).
R14 IF EuclideanSpace(Rn) THEN VectorSpace(Rn,R).
R15 IF VectorSpace(V,F) AND finiteBasis(B,V) THEN dimension(V)=cardinality(B).
R16 IF twoFiniteBases(B1,B2,V) THEN cardinality(B1)=cardinality(B2).
R17 IF chartTransition(C1,C2) AND overlapNonempty THEN transitionMap=C2∘inverse(C1) on overlap coordinates.
R18 IF chartsSmoothlyCompatible THEN transitionMapsSmoothOnOverlaps.
R19 IF samePointDifferentCharts(p,C1,C2) THEN coordinateTuplesMayDiffer.
R20 IF affineFrameShift(origin1,origin2) THEN coordinateDifferenceIncludesTranslation.
R21 IF orthonormalFrameChange(Q) THEN Q^T Q=I.
R22 IF orthogonalTransform(Q) THEN EuclideanDistancePreserved.
R23 IF coordinateTransformationIsInvertible(T) THEN representationCanBeRecoveredOnDeclaredDomain.
R24 IF mappingIsSingularAt(p) THEN doNotInferGlobalCoordinateInvertibility.
R25 IF admissibleConfiguration(q) THEN q in ConfigurationSpace.
R26 IF validState(s) THEN s in StateSpace.
R27 IF phaseState(q,p) AND canonicalMechanicalModel THEN (q,p) in PhaseSpace.
R28 IF stateTrajectory(s,t) THEN trajectoryIn(s,StateSpace).
R29 IF configurationConstraintsPresent THEN ConfigurationSpaceMayBeProperSubsetOfAmbientSpace.
R30 IF modelChanges THEN stateSpaceAndCoordinatesMayChange.
R31 IF twoCoordinatesShareNumbers BUT differentFrames THEN doNotInferSamePoint.
R32 IF axesHaveDifferentUnits THEN normaliseUnitsBeforeNumericalComparison.
R33 IF dimensionKnown(n) AND coordinateTupleLength(m) AND n!=m THEN validationFailure.
R34 IF coordinatePointHasNoChart THEN interpretationStatus=UNDERDETERMINED.
R35 IF propertyIsCoordinateInvariant THEN retainAcrossValidChartChanges.
R36 IF propertyDependsOnChart THEN doNotPromoteToIntrinsicGeometricProperty.
R37 IF metricDefinesOpenBalls THEN topologyContainsAllUnionsOfOpenBalls.
R38 IF pointInInteriorOfChartDomain THEN localCoordinatesAvailable.
R39 IF fieldNotSpecifiedForVectorSpace THEN requireScalarFieldDeclaration.
R40 IF spatialEntityHasProvenance THEN preserveReferenceFrameAndTransformationHistory.

07. VALIDATION PIPELINE
STEP 1 Parse entity and type declarations.
STEP 2 Validate scalar domains, units, coordinate order and dimensions.
STEP 3 Resolve reference frames, origins, axes and chart domains.
STEP 4 Verify metric, topology or vector-space axioms when asserted.
STEP 5 Check transformation preconditions, invertibility and singularities.
STEP 6 Apply sound inference rules; retain domain restrictions.
STEP 7 Distinguish theorem-derived conclusions from assumptions and illustrative examples.
STEP 8 Emit structured graph relations with evidence and provenance.
STEP 9 Reject inconsistent dimensions, unbound frames and invalid domain extrapolations.

08. MATHEMATICAL EXAMPLES
E01 Cartesian R2: p=(3,4); Euclidean distance to origin=5.
E02 Polar: (r,theta)=(2,pi/2) maps to Cartesian (0,2), up to floating-point precision.
E03 Polar singularity: (0,0) and (0,pi/3) represent the same origin.
E04 Cylindrical: (2,0,5) maps to (2,0,5).
E05 Spherical: (1,0,pi/2) maps to (1,0,0) under the stated convention.
E06 Translation: changing origin changes coordinate values but not intrinsic point identity.
E07 Rotation: orthogonal Q preserves Euclidean vector norms.
E08 R3 has standard vector-space dimension 3 over R.
E09 A discrete metric d(x,y)=0 when x=y and 1 otherwise induces the discrete topology.
E10 The open interval (0,1) is an open subset of R in the usual topology.
E11 A circle requires multiple ordinary one-dimensional coordinate charts for a smooth atlas.
E12 A constrained robot joint system has a configuration space defined by allowed joint settings.
E13 A classical particle in one spatial dimension has phase coordinates (q,p).
E14 A damped dynamical system can use a state space including position and velocity.
E15 Latitude/longitude require datum and convention for unambiguous Earth locations.
E16 A coordinate tuple (1,2) alone does not identify a physical location.

09. SEMANTIC_EQUIVALENCE
"coordinate tuple" ~= "coordinate representation" WHEN chart/frame fixed.
"Cartesian coordinates" ~= "rectangular coordinates" IN standard Euclidean contexts.
"polar radius" ~= "radial coordinate" IN declared polar convention.
"origin" ~= "zero point of a frame" WHEN a frame is fixed.
"phase space" != "configuration space" GENERALLY.
"metric space" != "topological space" AS types, despite metric-induced topologies.
"point" != "position vector" WITHOUT a chosen origin and identification.
"axis" != "dimension" GENERALLY; an axis may represent one coordinate direction.

10. KNOWLEDGE GRAPH TRIPLES
(CoordinatePoint, representedBy, CoordinateTuple)
(CoordinateTuple, interpretedUnder, CoordinateChart)
(CoordinateSystem, contains, CoordinateAxis)
(CoordinateSystem, maySpecify, Origin)
(PolarCoordinateSystem, transformsTo, CartesianCoordinateSystem)
(CylindricalCoordinateSystem, extends, PolarCoordinateSystem)
(SphericalCoordinateSystem, hasConvention, AzimuthAndInclination)
(EuclideanSpace, admits, EuclideanMetric)
(MetricSpace, induces, TopologicalSpace)
(VectorSpace, definedOver, ScalarField)
(PhaseSpace, describes, DynamicalSystemState)
(ConfigurationSpace, represents, AdmissibleConfigurations)

11. CONFIDENCE
CONFIDENCE mathematical_axioms = 1.00 when assumptions and domains match.
CONFIDENCE source_taxonomy = 1.00 for the 18 headings verified at source on 2026-10-13.
CONFIDENCE coordinate_inference = conditional on frame, units, conventions and transformation validity.
CONFIDENCE empirical_geospatial_claims = NOT_ASSESSED without independent measurements.
CONFIDENCE does not override proof obligations.

12. FLOW_HASH
FLOW_HASH.algorithm = SHA-256
FLOW_HASH.canonicalisation = UTF-8; LF line endings; Unicode NFC; exclude computed hash field.
FLOW_HASH.value = COMPUTE_AT_INGESTION
FLOW_HASH.purpose = content integrity and version traceability; not mathematical truth certification.

13. UPDATE_MODEL
UPDATE_MODEL.version = 4.0
UPDATE_MODEL.source = canonical page URL above
UPDATE_MODEL.change_policy = preserve prior version; increment revision; revalidate rules and examples.
UPDATE_MODEL.dependencies = coordinate ontology; geometry; linear algebra; topology; metric spaces; dynamical systems.
UPDATE_MODEL.review_triggers = taxonomy change; mathematical correction; changed conventions; schema revision.
UPDATE_MODEL.provenance = record retrieval timestamp, author, validation state and hash.

14. AI_CONSUMPTION
AI_CONSUMPTION.mode = typed_semantic_reasoning
AI_CONSUMPTION.read_order = types -> constraints -> relations -> rules -> examples -> provenance.
AI_CONSUMPTION.execute = rules only when preconditions are satisfied.
AI_CONSUMPTION.output = inferred facts + assumptions + confidence + source lineage.
AI_CONSUMPTION.ambiguity = request frame, units, datum, angle convention or domain before inference.
AI_CONSUMPTION.prohibit = conflating coordinate representations with intrinsic entities.
AI_CONSUMPTION.validation = reject contradictions; do not silently repair invalid coordinates.
AI_CONSUMPTION.scope = formal ontology specification; not an automatically deployed inference engine.

15. AUTHORSHIP AND SOURCE
Mark Anthony Higgins — Sky (Recursive AI)
Kireth Semantic Library | https://kirethresearch.org
Main-page child ontology headings checked against the published page.
Mathematical formalisation is independently authored and may not reproduce the page's complete prose.
END FLOW ALGEBRA