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UOR Foundation - Foundation Model - Categorical X
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| { | |
| "@context": { | |
| "@base": "https://uor.foundation/categorical-x/v1/", | |
| "@vocab": "https://uor.foundation/categorical-x/v1#", | |
| "AlgebraicProperty": "cx:AlgebraicProperty", | |
| "ApproximateNumber": "cx:ApproximateNumber", | |
| "ArbitraryPrecisionContext": "cx:ArbitraryPrecisionContext", | |
| "AssertionStatus": "cx:AssertionStatus", | |
| "AutomorphismGroup": "cx:AutomorphismGroup", | |
| "Axiom": "cx:Axiom", | |
| "AxiomsContainer": "cx:AxiomsContainer", | |
| "Ball": "cx:Ball", | |
| "BaseCaseValue": "cx:BaseCaseValue", | |
| "CategoricalStructure": "cxm:CategoricalStructure", | |
| "CategoricalStructureKind": "cxm:CategoricalStructureKind", | |
| "Category": "cx:Category", | |
| "CocycleClass": "cx:CocycleClass", | |
| "CocycleRelation": "cx:CocycleRelation", | |
| "CompositionLaw": "cxm:CompositionLaw", | |
| "ComputationArtifact": "cx:ComputationArtifact", | |
| "ConstantsContainer": "cx:ConstantsContainer", | |
| "ConstructionRule": "cxm:ConstructionRule", | |
| "ConstructorFunctor": "cxm:ConstructorFunctor", | |
| "ConstructorFunctorsContainer": "cxm:ConstructorFunctorsContainer", | |
| "ConversionMorphism": "cx:ConversionMorphism", | |
| "Correspondence": "cx:Correspondence", | |
| "CorrespondencesContainer": "cx:CorrespondencesContainer", | |
| "CountingFunctionInstance": "cx:CountingFunctionInstance", | |
| "CountingSequence": "cx:CountingSequence", | |
| "DedekindNumber": "cx:DedekindNumber", | |
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| "DerivationConstraints": "cxm:DerivationConstraints", | |
| "DerivationLevel": "cxm:DerivationLevel", | |
| "DerivationRule": "cx:DerivationRule", | |
| "DerivationRuleValue": "cx:DerivationRuleValue", | |
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| "cx": "https://uor.foundation/categorical-x/v1#", | |
| "cxm": "https://uor.foundation/categorical-x/v1/meta#", | |
| "cxs": "https://uor.foundation/categorical-x/v1/schema#", | |
| "dcterms": "http://purl.org/dc/terms/", | |
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| "losesProperty": { | |
| "@id": "cx:losesProperty", | |
| "@type": "@id" | |
| }, | |
| "lostAtLevel": { | |
| "@id": "cx:lostAtLevel", | |
| "@type": "@id" | |
| }, | |
| "lostPropertyRule": { | |
| "@id": "cx:lostPropertyRule" | |
| }, | |
| "lowerBound": { | |
| "@id": "cx:lowerBound" | |
| }, | |
| "map": { | |
| "@id": "cx:map" | |
| }, | |
| "mapsPhase": { | |
| "@container": "@set", | |
| "@id": "cx:mapsPhase", | |
| "@type": "@id" | |
| }, | |
| "maxDerivationLevel": { | |
| "@id": "cxm:maxDerivationLevel", | |
| "@type": "xsd:integer" | |
| }, | |
| "modifiedAtLevel": { | |
| "@id": "cx:modifiedAtLevel", | |
| "@type": "@id" | |
| }, | |
| "monsterAntichain": { | |
| "@id": "cx:monsterAntichain", | |
| "@type": "@id" | |
| }, | |
| "morphismMapping": { | |
| "@id": "cx:morphismMapping" | |
| }, | |
| "multiplicityRule": { | |
| "@id": "cx:multiplicityRule" | |
| }, | |
| "n": { | |
| "@id": "cx:n", | |
| "@type": "xsd:integer" | |
| }, | |
| "name": "rdfs:label", | |
| "naturalComponents": { | |
| "@id": "cx:naturalComponents" | |
| }, | |
| "note": { | |
| "@id": "cx:note" | |
| }, | |
| "numericalOrder": { | |
| "@container": "@list", | |
| "@id": "cx:numericalOrder" | |
| }, | |
| "objectMapping": { | |
| "@id": "cx:objectMapping" | |
| }, | |
| "octaveCorrection": { | |
| "@id": "cx:octaveCorrection", | |
| "@type": "@id" | |
| }, | |
| "oeisId": { | |
| "@id": "cx:oeisId" | |
| }, | |
| "operatesOn": { | |
| "@container": "@set", | |
| "@id": "cx:operatesOn", | |
| "@type": "@id" | |
| }, | |
| "order": { | |
| "@container": "@list", | |
| "@id": "cx:order", | |
| "@type": "@id" | |
| }, | |
| "owl": "http://www.w3.org/2002/07/owl#", | |
| "parameterCondition": { | |
| "@id": "cx:parameterCondition" | |
| }, | |
| "parameterDomain": { | |
| "@id": "cx:parameterDomain" | |
| }, | |
| "parameterVariable": { | |
| "@id": "cx:parameterVariable" | |
| }, | |
| "partOf": { | |
| "@id": "cx:partOf", | |
| "@type": "@id" | |
| }, | |
| "pattern": { | |
| "@id": "cx:pattern" | |
| }, | |
| "period": { | |
| "@id": "cx:period", | |
| "@type": "@id" | |
| }, | |
| "periodicity": { | |
| "@id": "cx:periodicity", | |
| "@type": "@id" | |
| }, | |
| "phase": { | |
| "@id": "cx:phase", | |
| "@type": "@id" | |
| }, | |
| "phaseAlignment": { | |
| "@container": "@set", | |
| "@id": "cx:phaseAlignment", | |
| "@type": "@id" | |
| }, | |
| "phaseBoundary": { | |
| "@id": "cx:phaseBoundary", | |
| "@type": "@id" | |
| }, | |
| "phaseFormula": { | |
| "@id": "cx:phaseFormula" | |
| }, | |
| "phaseII": { | |
| "@id": "cx:phaseII", | |
| "@type": "@id" | |
| }, | |
| "phaseIV": { | |
| "@id": "cx:phaseIV", | |
| "@type": "@id" | |
| }, | |
| "phaseRange": { | |
| "@id": "cx:phaseRange" | |
| }, | |
| "preTrialityRegime": { | |
| "@id": "cx:preTrialityRegime", | |
| "@type": "@id" | |
| }, | |
| "precedesPhase": { | |
| "@id": "cx:precedesPhase", | |
| "@type": "@id" | |
| }, | |
| "precisionBits": { | |
| "@id": "cx:precisionBits", | |
| "@type": "xsd:integer" | |
| }, | |
| "precisionDigits": { | |
| "@id": "cx:precisionDigits", | |
| "@type": "xsd:integer" | |
| }, | |
| "preserves": { | |
| "@container": "@set", | |
| "@id": "cx:preserves", | |
| "@type": "@id" | |
| }, | |
| "prime": { | |
| "@id": "cx:prime" | |
| }, | |
| "product": { | |
| "@id": "cx:product", | |
| "@type": "xsd:integer" | |
| }, | |
| "projectsTo": { | |
| "@id": "cx:projectsTo", | |
| "@type": "@id" | |
| }, | |
| "proof": { | |
| "@id": "cx:proof" | |
| }, | |
| "proofStep": { | |
| "@container": "@list", | |
| "@id": "cx:proofStep", | |
| "@type": "@id" | |
| }, | |
| "propertyClosureDimension": { | |
| "@id": "cx:propertyClosureDimension" | |
| }, | |
| "propertyClosureLevel": { | |
| "@id": "cx:propertyClosureLevel", | |
| "@type": "xsd:integer" | |
| }, | |
| "proves": { | |
| "@id": "cx:proves", | |
| "@type": "@id" | |
| }, | |
| "providesConstructor": { | |
| "@id": "cxm:providesConstructor", | |
| "@type": "@id" | |
| }, | |
| "publisher": { | |
| "@id": "dcterms:publisher" | |
| }, | |
| "range": { | |
| "@id": "cx:range" | |
| }, | |
| "rdfs": "http://www.w3.org/2000/01/rdf-schema#", | |
| "relation": { | |
| "@id": "cx:relation", | |
| "@type": "@id" | |
| }, | |
| "relationToDedekind": { | |
| "@id": "cx:relationToDedekind" | |
| }, | |
| "relations": { | |
| "@container": "@list", | |
| "@id": "cx:relations" | |
| }, | |
| "representative": { | |
| "@id": "cx:representative" | |
| }, | |
| "representativeRule": { | |
| "@id": "cx:representativeRule", | |
| "@type": "@id" | |
| }, | |
| "requiresStructure": { | |
| "@id": "cxm:requiresStructure", | |
| "@type": "@id" | |
| }, | |
| "resolutionComplexity": { | |
| "@id": "cx:resolutionComplexity" | |
| }, | |
| "resolutionMethod": { | |
| "@id": "cx:resolutionMethod" | |
| }, | |
| "resolutionRules": { | |
| "@id": "cx:resolutionRules", | |
| "@type": "@id" | |
| }, | |
| "result": { | |
| "@id": "cx:result" | |
| }, | |
| "rhs": { | |
| "@id": "cx:rhs" | |
| }, | |
| "roundingMode": { | |
| "@id": "cx:roundingMode", | |
| "@type": "@id" | |
| }, | |
| "rule": { | |
| "@id": "cx:rule" | |
| }, | |
| "rules": { | |
| "@container": "@set", | |
| "@id": "cx:rules", | |
| "@type": "@id" | |
| }, | |
| "sameAs": { | |
| "@id": "owl:sameAs", | |
| "@type": "@id" | |
| }, | |
| "satisfiesUniversalProperty": { | |
| "@id": "cxm:satisfiesUniversalProperty", | |
| "@type": "@id" | |
| }, | |
| "schemaOrg": "https://schema.org/", | |
| "sectorProof": { | |
| "@id": "cx:sectorProof", | |
| "@type": "@id" | |
| }, | |
| "sequenceIndex": { | |
| "@id": "cx:sequenceIndex", | |
| "@type": "xsd:integer" | |
| }, | |
| "sharedStructure": { | |
| "@id": "cx:sharedStructure", | |
| "@type": "@id" | |
| }, | |
| "sharedType": { | |
| "@id": "cx:sharedType" | |
| }, | |
| "signature": { | |
| "@id": "cx:signature" | |
| }, | |
| "significance": { | |
| "@id": "cx:significance" | |
| }, | |
| "skos": "http://www.w3.org/2004/02/skos/core#", | |
| "source": { | |
| "@id": "cx:source", | |
| "@type": "@id" | |
| }, | |
| "sourceAxiom": { | |
| "@id": "cx:sourceAxiom" | |
| }, | |
| "sourceLevel": { | |
| "@id": "cx:sourceLevel", | |
| "@type": "@id" | |
| }, | |
| "sourcePhase": { | |
| "@id": "cx:sourcePhase" | |
| }, | |
| "stabilityCondition": { | |
| "@id": "cx:stabilityCondition" | |
| }, | |
| "statement": { | |
| "@id": "cx:statement" | |
| }, | |
| "status": { | |
| "@id": "cx:status" | |
| }, | |
| "stepDependsOn": { | |
| "@id": "cxm:stepDependsOn", | |
| "@type": "@id" | |
| }, | |
| "stepProduces": { | |
| "@id": "cxm:stepProduces", | |
| "@type": "@id" | |
| }, | |
| "stepVia": { | |
| "@id": "cxm:stepVia", | |
| "@type": "@id" | |
| }, | |
| "structuralFormula": { | |
| "@id": "cx:structuralFormula" | |
| }, | |
| "supportedBy": { | |
| "@id": "cx:supportedBy", | |
| "@type": "@id" | |
| }, | |
| "symbol": { | |
| "@id": "cx:symbol" | |
| }, | |
| "targetCategory": { | |
| "@id": "cx:targetCategory", | |
| "@type": "@id" | |
| }, | |
| "targetField": { | |
| "@id": "cx:targetField" | |
| }, | |
| "targetLevel": { | |
| "@id": "cx:targetLevel", | |
| "@type": "@id" | |
| }, | |
| "targetPhase": { | |
| "@id": "cx:targetPhase" | |
| }, | |
| "targets": { | |
| "@container": "@set", | |
| "@id": "cx:targets", | |
| "@type": "@id" | |
| }, | |
| "terminalAxiom": { | |
| "@id": "cxm:terminalAxiom", | |
| "@type": "@id" | |
| }, | |
| "termination": { | |
| "@id": "cx:termination" | |
| }, | |
| "tracesToAxiom": { | |
| "@id": "cx:tracesToAxiom", | |
| "@type": "@id" | |
| }, | |
| "transitionLevel": { | |
| "@id": "cx:transitionLevel", | |
| "@type": "xsd:integer" | |
| }, | |
| "trialityInvariant": { | |
| "@id": "cx:trialityInvariant", | |
| "@type": "xsd:boolean" | |
| }, | |
| "trialityRegime": { | |
| "@id": "cx:trialityRegime", | |
| "@type": "@id" | |
| }, | |
| "trialityTheorem": { | |
| "@id": "cx:trialityTheorem", | |
| "@type": "@id" | |
| }, | |
| "type": { | |
| "@id": "cx:type" | |
| }, | |
| "typeParameterDomain": { | |
| "@id": "cx:typeParameterDomain", | |
| "@type": "@id" | |
| }, | |
| "uniquenessCondition": { | |
| "@id": "cx:uniquenessCondition" | |
| }, | |
| "universalArrow": { | |
| "@id": "cx:universalArrow" | |
| }, | |
| "upperBound": { | |
| "@id": "cx:upperBound" | |
| }, | |
| "usesDigitSet": { | |
| "@id": "cx:usesDigitSet", | |
| "@type": "@id" | |
| }, | |
| "usesRuntimeConstruct": { | |
| "@id": "cx:usesRuntimeConstruct", | |
| "@type": "@id" | |
| }, | |
| "validationRule": { | |
| "@id": "cx:validationRule" | |
| }, | |
| "value": { | |
| "@id": "cx:value" | |
| }, | |
| "values": { | |
| "@container": "@set", | |
| "@id": "cx:values", | |
| "@type": "@id" | |
| }, | |
| "verification": { | |
| "@id": "cx:verification", | |
| "@type": "@id" | |
| }, | |
| "verificationStatus": { | |
| "@id": "cx:verificationStatus" | |
| }, | |
| "version": { | |
| "@id": "owl:versionInfo" | |
| }, | |
| "viaIsomorphism": { | |
| "@id": "cx:viaIsomorphism" | |
| }, | |
| "when": { | |
| "@id": "cx:when" | |
| }, | |
| "whyPentality": { | |
| "@id": "cx:whyPentality" | |
| }, | |
| "whySeptality": { | |
| "@id": "cx:whySeptality" | |
| }, | |
| "whyT": { | |
| "@id": "cx:whyT" | |
| }, | |
| "xsd": "http://www.w3.org/2001/XMLSchema#", | |
| "yieldsValue": { | |
| "@id": "cx:yieldsValue" | |
| } | |
| }, | |
| "@graph": [ | |
| { | |
| "@id": "owl:Class", | |
| "@type": "rdfs:Class", | |
| "label": "Class" | |
| }, | |
| { | |
| "@id": "owl:ObjectProperty", | |
| "@type": "rdfs:Class", | |
| "label": "Object Property" | |
| }, | |
| { | |
| "@id": "owl:DatatypeProperty", | |
| "@type": "rdfs:Class", | |
| "label": "Datatype Property" | |
| }, | |
| { | |
| "@id": "cx:SingletonInstance", | |
| "@type": "owl:Class", | |
| "comment": "The unique canonical instance of the Categorical X structure", | |
| "label": "Singleton Instance" | |
| }, | |
| { | |
| "@id": "cx:Primitive", | |
| "@type": "owl:Class", | |
| "comment": "Atomic element that cannot be decomposed further", | |
| "label": "Primitive" | |
| }, | |
| { | |
| "@id": "cx:Axiom", | |
| "@type": "owl:Class", | |
| "comment": "Foundational assertion", | |
| "label": "Axiom" | |
| }, | |
| { | |
| "@id": "cx:DerivedConstant", | |
| "@type": "owl:Class", | |
| "comment": "Constant derived from primitives via operations", | |
| "label": "Derived Constant" | |
| }, | |
| { | |
| "@id": "cx:TypeLevel", | |
| "@type": "owl:Class", | |
| "comment": "A level in the type hierarchy", | |
| "label": "Type Level" | |
| }, | |
| { | |
| "@id": "cx:TowerLevel", | |
| "@type": "owl:Class", | |
| "comment": "A level in the Cayley-Dickson tower", | |
| "label": "Tower Level" | |
| }, | |
| { | |
| "@id": "cx:TowerTransition", | |
| "@type": "owl:Class", | |
| "comment": "Cayley-Dickson doubling functor between levels", | |
| "label": "Tower Transition" | |
| }, | |
| { | |
| "@id": "cx:Operator", | |
| "@type": "owl:Class", | |
| "comment": "Closure operation on the categorical structure", | |
| "label": "Operator" | |
| }, | |
| { | |
| "@id": "cx:Projection", | |
| "@type": "owl:Class", | |
| "comment": "Domain-specific functor from the categorical structure", | |
| "label": "Projection" | |
| }, | |
| { | |
| "@id": "cx:Correspondence", | |
| "@type": "owl:Class", | |
| "comment": "Cross-projection relationship", | |
| "label": "Correspondence" | |
| }, | |
| { | |
| "@id": "cx:PhaseTransition", | |
| "@type": "owl:Class", | |
| "comment": "Boundary point in phase behavior", | |
| "label": "Phase Transition" | |
| }, | |
| { | |
| "@id": "cx:PhaseBehavior", | |
| "@type": "owl:Class", | |
| "comment": "Operational formula for a phase", | |
| "label": "Phase Behavior" | |
| }, | |
| { | |
| "@id": "cx:PhaseModification", | |
| "@type": "owl:Class", | |
| "comment": "Operator behavior modification at a phase boundary", | |
| "label": "Phase Modification" | |
| }, | |
| { | |
| "@id": "cx:AutomorphismGroup", | |
| "@type": "owl:Class", | |
| "comment": "Symmetry group of an algebra", | |
| "label": "Automorphism Group" | |
| }, | |
| { | |
| "@id": "cx:CocycleClass", | |
| "@type": "owl:Class", | |
| "comment": "Cohomological obstruction class", | |
| "label": "Cocycle Class" | |
| }, | |
| { | |
| "@id": "cx:AlgebraicProperty", | |
| "@type": "owl:Class", | |
| "comment": "Property that may be lost in the tower", | |
| "label": "Algebraic Property" | |
| }, | |
| { | |
| "@id": "cx:Category", | |
| "@type": "owl:Class", | |
| "comment": "Target category for projections", | |
| "label": "Mathematical Category" | |
| }, | |
| { | |
| "@id": "cx:Stratum", | |
| "@type": "owl:Class", | |
| "comment": "Stratification of tower levels", | |
| "label": "Stratum" | |
| }, | |
| { | |
| "@id": "cx:DimensionSpec", | |
| "@type": "owl:Class", | |
| "comment": "Dimension specification", | |
| "label": "Dimension Spec" | |
| }, | |
| { | |
| "@id": "cx:DerivationRule", | |
| "@type": "owl:Class", | |
| "comment": "Rule for deriving values", | |
| "label": "Derivation Rule" | |
| }, | |
| { | |
| "@id": "cx:Theorem", | |
| "@type": "owl:Class", | |
| "comment": "Mathematical theorem", | |
| "label": "Theorem" | |
| }, | |
| { | |
| "@id": "cx:Proof", | |
| "@type": "owl:Class", | |
| "comment": "Mathematical proof", | |
| "label": "Proof" | |
| }, | |
| { | |
| "@id": "cx:ProofStep", | |
| "@type": "owl:Class", | |
| "comment": "Step in a proof", | |
| "label": "Proof Step" | |
| }, | |
| { | |
| "@id": "cx:CriticalBoundary", | |
| "@type": "owl:Class", | |
| "comment": "Axiom-derived boundary constraint (T=3 division, O=8 tower)", | |
| "label": "Critical Boundary" | |
| }, | |
| { | |
| "@id": "cx:CompletenessCertificate", | |
| "@type": "owl:Class", | |
| "comment": "Formal guarantee of lossless coverage for arbitrary scale", | |
| "label": "Completeness Certificate" | |
| }, | |
| { | |
| "@id": "cx:PrimitiveOperation", | |
| "@type": "owl:Class", | |
| "comment": "Basic arithmetic operation", | |
| "label": "Primitive Operation" | |
| }, | |
| { | |
| "@id": "cx:PrimitiveRelation", | |
| "@type": "owl:Class", | |
| "comment": "Basic mathematical relation", | |
| "label": "Primitive Relation" | |
| }, | |
| { | |
| "@id": "cx:DerivedOperation", | |
| "@type": "owl:Class", | |
| "comment": "Operation derived from primitives", | |
| "label": "Derived Operation" | |
| }, | |
| { | |
| "@id": "cx:FirstOrderType", | |
| "@type": "owl:Class", | |
| "comment": "Type from binary operations on primitives", | |
| "label": "First-Order Type" | |
| }, | |
| { | |
| "@id": "cx:SecondOrderType", | |
| "@type": "owl:Class", | |
| "comment": "Type from first-order types", | |
| "label": "Second-Order Type" | |
| }, | |
| { | |
| "@id": "cx:StructuralType", | |
| "@type": "owl:Class", | |
| "comment": "Abstract structural type", | |
| "label": "Structural Type" | |
| }, | |
| { | |
| "@id": "cx:OctaveCocycle", | |
| "@type": "owl:Class", | |
| "comment": "Fundamental octave cocycle", | |
| "label": "Octave Cocycle" | |
| }, | |
| { | |
| "@id": "cx:OctaveConstants", | |
| "@type": "owl:Class", | |
| "comment": "Octave correction constants", | |
| "label": "Octave Constants" | |
| }, | |
| { | |
| "@id": "cx:NamedOctave", | |
| "@type": "owl:Class", | |
| "comment": "Named octave in periodicity", | |
| "label": "Named Octave" | |
| }, | |
| { | |
| "@id": "cx:PhaseAlignment", | |
| "@type": "owl:Class", | |
| "comment": "Alignment of phases across projections", | |
| "label": "Phase Alignment" | |
| }, | |
| { | |
| "@id": "cx:PhaseMapEntry", | |
| "@type": "owl:Class", | |
| "comment": "Entry in phase mapping", | |
| "label": "Phase Map Entry" | |
| }, | |
| { | |
| "@id": "cx:BaseCaseValue", | |
| "@type": "owl:Class", | |
| "comment": "Base case value for resolution", | |
| "label": "Base Case Value" | |
| }, | |
| { | |
| "@id": "cx:LatticeInstance", | |
| "@type": "owl:Class", | |
| "comment": "Instance of a lattice structure", | |
| "label": "Lattice Instance" | |
| }, | |
| { | |
| "@id": "cx:FilterInstance", | |
| "@type": "owl:Class", | |
| "comment": "Instance of a filter structure", | |
| "label": "Filter Instance" | |
| }, | |
| { | |
| "@id": "cx:SieveInstance", | |
| "@type": "owl:Class", | |
| "comment": "Instance of a sieve structure", | |
| "label": "Sieve Instance" | |
| }, | |
| { | |
| "@id": "cx:FiniteGroup", | |
| "@type": "owl:Class", | |
| "comment": "Finite automorphism group", | |
| "label": "Finite Group" | |
| }, | |
| { | |
| "@id": "cx:LieGroup", | |
| "@type": "owl:Class", | |
| "comment": "Continuous automorphism group", | |
| "label": "Lie Group" | |
| }, | |
| { | |
| "@id": "cx:PrimitivesContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for primitive elements", | |
| "label": "Primitives Container" | |
| }, | |
| { | |
| "@id": "cx:IntegersContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for primitive integers", | |
| "label": "Integers Container" | |
| }, | |
| { | |
| "@id": "cx:OperationsContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for primitive operations", | |
| "label": "Operations Container" | |
| }, | |
| { | |
| "@id": "cx:RelationsContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for primitive relations", | |
| "label": "Relations Container" | |
| }, | |
| { | |
| "@id": "cx:TowerContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for tower structure", | |
| "label": "Tower Container" | |
| }, | |
| { | |
| "@id": "cx:TowerLevelsContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for tower levels", | |
| "label": "Tower Levels Container" | |
| }, | |
| { | |
| "@id": "cx:TransitionsContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for tower transitions", | |
| "label": "Transitions Container" | |
| }, | |
| { | |
| "@id": "cx:StrataContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for tower strata", | |
| "label": "Strata Container" | |
| }, | |
| { | |
| "@id": "cx:OperatorsContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for categorical operators", | |
| "label": "Operators Container" | |
| }, | |
| { | |
| "@id": "cx:ProjectionsContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for projections", | |
| "label": "Projections Container" | |
| }, | |
| { | |
| "@id": "cx:CorrespondencesContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for correspondences", | |
| "label": "Correspondences Container" | |
| }, | |
| { | |
| "@id": "cx:TypesContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for type system", | |
| "label": "Types Container" | |
| }, | |
| { | |
| "@id": "cx:AxiomsContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for axioms", | |
| "label": "Axioms Container" | |
| }, | |
| { | |
| "@id": "cx:ConstantsContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for constants", | |
| "label": "Constants Container" | |
| }, | |
| { | |
| "@id": "cx:PhasesContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for phase system", | |
| "label": "Phases Container" | |
| }, | |
| { | |
| "@id": "cx:DedekindNumber", | |
| "@type": "owl:Class", | |
| "comment": "Dedekind number D(n) counting antichains", | |
| "label": "Dedekind Number" | |
| }, | |
| { | |
| "@id": "cx:Lattice", | |
| "@type": "owl:Class", | |
| "comment": "Mathematical lattice structure", | |
| "label": "Lattice" | |
| }, | |
| { | |
| "@id": "cx:Filter", | |
| "@type": "owl:Class", | |
| "comment": "Mathematical filter structure", | |
| "label": "Filter" | |
| }, | |
| { | |
| "@id": "cx:Sieve", | |
| "@type": "owl:Class", | |
| "comment": "Mathematical sieve structure", | |
| "label": "Sieve" | |
| }, | |
| { | |
| "@id": "cx:CountingFunction", | |
| "@type": "owl:Class", | |
| "comment": "Function counting combinatorial objects", | |
| "label": "Counting Function" | |
| }, | |
| { | |
| "@id": "cx:ArithmeticObject", | |
| "@type": "owl:Class", | |
| "comment": "Object in arithmetic projection", | |
| "label": "Arithmetic Object" | |
| }, | |
| { | |
| "@id": "cx:CombinatorialObject", | |
| "@type": "owl:Class", | |
| "comment": "Object in combinatorial projection", | |
| "label": "Combinatorial Object" | |
| }, | |
| { | |
| "@id": "cx:SpectralObject", | |
| "@type": "owl:Class", | |
| "comment": "Object in spectral projection", | |
| "label": "Spectral Object" | |
| }, | |
| { | |
| "@id": "cx:ModularObject", | |
| "@type": "owl:Class", | |
| "comment": "Object in modular projection", | |
| "label": "Modular Object" | |
| }, | |
| { | |
| "@id": "cx:PrimeSystem", | |
| "@type": "owl:Class", | |
| "comment": "Prime number system in arithmetic projection", | |
| "label": "Prime System" | |
| }, | |
| { | |
| "@id": "cx:ResidueClass", | |
| "@type": "owl:Class", | |
| "comment": "Residue class modular arithmetic", | |
| "label": "Residue Class" | |
| }, | |
| { | |
| "@id": "cx:SporadicGroup", | |
| "@type": "owl:Class", | |
| "comment": "Sporadic simple group", | |
| "label": "Sporadic Group" | |
| }, | |
| { | |
| "@id": "cx:MathieuGroup", | |
| "@type": "owl:Class", | |
| "comment": "Mathieu sporadic group", | |
| "label": "Mathieu Group" | |
| }, | |
| { | |
| "@id": "cx:ConwayGroup", | |
| "@type": "owl:Class", | |
| "comment": "Conway sporadic group", | |
| "label": "Conway Group" | |
| }, | |
| { | |
| "@id": "cx:MonsterGroup", | |
| "@type": "owl:Class", | |
| "comment": "Monster simple group", | |
| "label": "Monster Group" | |
| }, | |
| { | |
| "@id": "cx:BabyMonster", | |
| "@type": "owl:Class", | |
| "comment": "Baby Monster sporadic group", | |
| "label": "Baby Monster" | |
| }, | |
| { | |
| "@id": "cx:LeechLattice", | |
| "@type": "owl:Class", | |
| "comment": "24-dimensional even unimodular lattice", | |
| "label": "Leech Lattice" | |
| }, | |
| { | |
| "@id": "cx:E8Lattice", | |
| "@type": "owl:Class", | |
| "comment": "E8 root lattice", | |
| "label": "E8 Lattice" | |
| }, | |
| { | |
| "@id": "cx:ModularForm", | |
| "@type": "owl:Class", | |
| "comment": "Holomorphic function on upper half-plane", | |
| "label": "Modular Form" | |
| }, | |
| { | |
| "@id": "cx:JFunction", | |
| "@type": "owl:Class", | |
| "comment": "Klein j-invariant", | |
| "label": "J Function" | |
| }, | |
| { | |
| "@id": "cx:MoonshineModule", | |
| "@type": "owl:Class", | |
| "comment": "Graded vertex algebra for moonshine", | |
| "label": "Moonshine Module" | |
| }, | |
| { | |
| "@id": "cx:GolayCode", | |
| "@type": "owl:Class", | |
| "comment": "Extended binary Golay code", | |
| "label": "Golay Code" | |
| }, | |
| { | |
| "@id": "cx:HammingCode", | |
| "@type": "owl:Class", | |
| "comment": "Error-correcting Hamming code", | |
| "label": "Hamming Code" | |
| }, | |
| { | |
| "@id": "cx:ReedMullerCode", | |
| "@type": "owl:Class", | |
| "comment": "Reed-Muller error-correcting code", | |
| "label": "Reed-Muller Code" | |
| }, | |
| { | |
| "@id": "cx:Spinor", | |
| "@type": "owl:Class", | |
| "comment": "Spinor representation", | |
| "label": "Spinor" | |
| }, | |
| { | |
| "@id": "cx:CliffordAlgebra", | |
| "@type": "owl:Class", | |
| "comment": "Clifford algebra structure", | |
| "label": "Clifford Algebra" | |
| }, | |
| { | |
| "@id": "cx:ExceptionalGroup", | |
| "@type": "owl:Class", | |
| "comment": "Exceptional Lie group", | |
| "label": "Exceptional Group" | |
| }, | |
| { | |
| "@id": "cx:TrialityAutomorphism", | |
| "@type": "owl:Class", | |
| "comment": "Triality outer automorphism of D4", | |
| "label": "Triality Automorphism" | |
| }, | |
| { | |
| "@id": "cx:OctonionAlgebra", | |
| "@type": "owl:Class", | |
| "comment": "8-dimensional normed division algebra", | |
| "label": "Octonion Algebra" | |
| }, | |
| { | |
| "@id": "cx:QuaternionAlgebra", | |
| "@type": "owl:Class", | |
| "comment": "4-dimensional normed division algebra", | |
| "label": "Quaternion Algebra" | |
| }, | |
| { | |
| "@id": "cx:ComplexNumbers", | |
| "@type": "owl:Class", | |
| "comment": "2-dimensional normed division algebra", | |
| "label": "Complex Numbers" | |
| }, | |
| { | |
| "@id": "cx:RealNumbers", | |
| "@type": "owl:Class", | |
| "comment": "1-dimensional normed division algebra", | |
| "label": "Real Numbers" | |
| }, | |
| { | |
| "@id": "cx:Sedenions", | |
| "@type": "owl:Class", | |
| "comment": "16-dimensional Cayley-Dickson algebra", | |
| "label": "Sedenions" | |
| }, | |
| { | |
| "@id": "cx:CayleyDicksonConstruction", | |
| "@type": "owl:Class", | |
| "comment": "Procedure for doubling algebras", | |
| "label": "Cayley-Dickson Construction" | |
| }, | |
| { | |
| "@id": "cx:NormedDivisionAlgebra", | |
| "@type": "owl:Class", | |
| "comment": "Algebra with multiplicative norm", | |
| "label": "Normed Division Algebra" | |
| }, | |
| { | |
| "@id": "cx:CompositionAlgebra", | |
| "@type": "owl:Class", | |
| "comment": "Algebra satisfying composition property", | |
| "label": "Composition Algebra" | |
| }, | |
| { | |
| "@id": "cx:AlternativeAlgebra", | |
| "@type": "owl:Class", | |
| "comment": "Algebra satisfying alternative law", | |
| "label": "Alternative Algebra" | |
| }, | |
| { | |
| "@id": "cx:FlexibleAlgebra", | |
| "@type": "owl:Class", | |
| "comment": "Algebra satisfying flexible identity", | |
| "label": "Flexible Algebra" | |
| }, | |
| { | |
| "@id": "cx:PowerAssociativeAlgebra", | |
| "@type": "owl:Class", | |
| "comment": "Algebra where powers associate", | |
| "label": "Power Associative Algebra" | |
| }, | |
| { | |
| "@id": "cx:JordanAlgebra", | |
| "@type": "owl:Class", | |
| "comment": "Commutative non-associative algebra", | |
| "label": "Jordan Algebra" | |
| }, | |
| { | |
| "@id": "cx:ExceptionalJordan", | |
| "@type": "owl:Class", | |
| "comment": "27-dimensional exceptional Jordan algebra", | |
| "label": "Exceptional Jordan" | |
| }, | |
| { | |
| "@id": "cx:FreudenthalMagicSquare", | |
| "@type": "owl:Class", | |
| "comment": "Magic square of Lie algebras", | |
| "label": "Freudenthal Magic Square" | |
| }, | |
| { | |
| "@id": "cx:G2Group", | |
| "@type": "owl:Class", | |
| "comment": "Exceptional G2 automorphism group of octonions", | |
| "label": "G2 Group" | |
| }, | |
| { | |
| "@id": "cx:F4Group", | |
| "@type": "owl:Class", | |
| "comment": "Exceptional F4 Lie group", | |
| "label": "F4 Group" | |
| }, | |
| { | |
| "@id": "cx:E6Group", | |
| "@type": "owl:Class", | |
| "comment": "Exceptional E6 Lie group", | |
| "label": "E6 Group" | |
| }, | |
| { | |
| "@id": "cx:E7Group", | |
| "@type": "owl:Class", | |
| "comment": "Exceptional E7 Lie group", | |
| "label": "E7 Group" | |
| }, | |
| { | |
| "@id": "cx:E8Group", | |
| "@type": "owl:Class", | |
| "comment": "Exceptional E8 Lie group", | |
| "label": "E8 Group" | |
| }, | |
| { | |
| "@id": "cx:SubobjectClassifier", | |
| "@type": "owl:Class", | |
| "comment": "Object Ω classifying subobjects in categorical structure", | |
| "label": "Subobject Classifier" | |
| }, | |
| { | |
| "@id": "cx:CardinalityFunctor", | |
| "@type": "owl:Class", | |
| "comment": "Functor |·| mapping objects to their cardinality", | |
| "label": "Cardinality Functor" | |
| }, | |
| { | |
| "@id": "cx:MetamodelEntity", | |
| "@type": "owl:Class", | |
| "comment": "Superclass for all specification/generation entities (non-domain). Uses cxm: namespace.", | |
| "label": "Metamodel Entity" | |
| }, | |
| { | |
| "@id": "cx:TypeConstructor", | |
| "@type": "owl:Class", | |
| "comment": "Generative mechanism that produces RDF instances from parameters. Subclass of MetamodelEntity.", | |
| "label": "Type Constructor" | |
| }, | |
| { | |
| "@id": "cx:ConstructorFunctor", | |
| "@type": "owl:Class", | |
| "comment": "Morphism between TypeConstructors, e.g., Cayley-Dickson doubling functor. Subclass of MetamodelEntity.", | |
| "label": "Constructor Functor" | |
| }, | |
| { | |
| "@id": "cx:ConstructionRule", | |
| "@type": "owl:Class", | |
| "comment": "Rule mapping a parameter to a field value via formula. Subclass of MetamodelEntity.", | |
| "label": "Construction Rule" | |
| }, | |
| { | |
| "@id": "cx:ParameterSpec", | |
| "@type": "owl:Class", | |
| "comment": "Specification of parameters for a TypeConstructor (variable, domain, condition). Subclass of MetamodelEntity.", | |
| "label": "Parameter Specification" | |
| }, | |
| { | |
| "@id": "cx:DerivationSource", | |
| "@type": "owl:Class", | |
| "comment": "Source tracing a derived value back to axioms T=3 or O=8. Subclass of MetamodelEntity.", | |
| "label": "Derivation Source" | |
| }, | |
| { | |
| "@id": "cx:MetaContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for meta-level generative constructs (TypeConstructors, Functors). Subclass of MetamodelEntity.", | |
| "label": "Meta Container" | |
| }, | |
| { | |
| "@id": "cx:TypeConstructorsContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for TypeConstructor definitions. Subclass of MetamodelEntity.", | |
| "label": "TypeConstructors Container" | |
| }, | |
| { | |
| "@id": "cx:ConstructorFunctorsContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for ConstructorFunctor definitions. Subclass of MetamodelEntity.", | |
| "label": "ConstructorFunctors Container" | |
| }, | |
| { | |
| "@id": "cx:DerivationLevel", | |
| "@type": "owl:Class", | |
| "comment": "Level in the stratified derivation hierarchy (Axiom -> DerivedConstant -> Formula -> Construction -> Extension). Subclass of MetamodelEntity.", | |
| "label": "Derivation Level" | |
| }, | |
| { | |
| "@id": "cx:ExtensionPoint", | |
| "@type": "owl:Class", | |
| "comment": "Declares what can be extended and the categorical structure required. Subclass of MetamodelEntity.", | |
| "label": "Extension Point" | |
| }, | |
| { | |
| "@id": "cx:UniversalProperty", | |
| "@type": "owl:Class", | |
| "comment": "Defines objects by their mapping properties (Initial, Terminal, Limit, Colimit, Adjoint, Free). Subclass of MetamodelEntity.", | |
| "label": "Universal Property" | |
| }, | |
| { | |
| "@id": "cx:UniversalPropertyKind", | |
| "@type": "owl:Class", | |
| "comment": "The kind of universal property (Initial, Terminal, Limit, Colimit, LeftAdjoint, RightAdjoint, Free, Cofree).", | |
| "label": "Universal Property Kind" | |
| }, | |
| { | |
| "@id": "cx:FreeConstruction", | |
| "@type": "owl:Class", | |
| "comment": "TypeConstructor that generates minimal structures via universal property. Subclass of TypeConstructor.", | |
| "label": "Free Construction" | |
| }, | |
| { | |
| "@id": "cx:Extension", | |
| "@type": "owl:Class", | |
| "comment": "Concrete implementation of an ExtensionPoint with complete derivation chain. Subclass of MetamodelEntity.", | |
| "label": "Extension" | |
| }, | |
| { | |
| "@id": "cx:DerivationChain", | |
| "@type": "owl:Class", | |
| "comment": "Complete chain of derivation steps tracing back to axioms T=3 and O=8.", | |
| "label": "Derivation Chain" | |
| }, | |
| { | |
| "@id": "cx:DerivationStep", | |
| "@type": "owl:Class", | |
| "comment": "Single step in a derivation chain (produces, depends_on, via).", | |
| "label": "Derivation Step" | |
| }, | |
| { | |
| "@id": "cx:CompositionLaw", | |
| "@type": "owl:Class", | |
| "comment": "Law that must be preserved by extensions (associativity, identity, functoriality, naturality).", | |
| "label": "Composition Law" | |
| }, | |
| { | |
| "@id": "cx:CategoricalStructureKind", | |
| "@type": "owl:Class", | |
| "comment": "Kind of categorical structure (Functor, NaturalTransformation, Adjunction, Monad, Endofunctor).", | |
| "label": "Categorical Structure Kind" | |
| }, | |
| { | |
| "@id": "cx:CategoricalStructure", | |
| "@type": "owl:Class", | |
| "comment": "Concrete categorical structure with object/morphism mappings, components, unit/counit.", | |
| "label": "Categorical Structure" | |
| }, | |
| { | |
| "@id": "cx:DerivationConstraints", | |
| "@type": "owl:Class", | |
| "comment": "Constraints on derivation chains (required axioms, max derivation level).", | |
| "label": "Derivation Constraints" | |
| }, | |
| { | |
| "@id": "cx:VerificationStatus", | |
| "@type": "owl:Class", | |
| "comment": "Status of extension verification (Verified, Pending, Failed, Unchecked).", | |
| "label": "Verification Status" | |
| }, | |
| { | |
| "@id": "cx:Number", | |
| "@type": "owl:Class", | |
| "comment": "Abstract numeric value (domain element).", | |
| "label": "Number" | |
| }, | |
| { | |
| "@id": "cx:NumericDomain", | |
| "@type": "owl:Class", | |
| "comment": "A set/structure in which numbers live (ℕ, ℤ, ℚ, ℝ, ℂ).", | |
| "label": "Numeric Domain" | |
| }, | |
| { | |
| "@id": "cx:NaturalNumbers", | |
| "@type": "owl:Class", | |
| "comment": "Domain ℕ with 0, successor; canonical counting domain.", | |
| "label": "Natural Numbers" | |
| }, | |
| { | |
| "@id": "cx:Integers", | |
| "@type": "owl:Class", | |
| "comment": "Domain ℤ.", | |
| "label": "Integers" | |
| }, | |
| { | |
| "@id": "cx:Rationals", | |
| "@type": "owl:Class", | |
| "comment": "Domain ℚ.", | |
| "label": "Rationals" | |
| }, | |
| { | |
| "@id": "cx:ExactNumber", | |
| "@type": "owl:Class", | |
| "comment": "A number represented exactly (BigInt, rational).", | |
| "label": "Exact Number" | |
| }, | |
| { | |
| "@id": "cx:ApproximateNumber", | |
| "@type": "owl:Class", | |
| "comment": "A number with finite precision (float, interval, ball).", | |
| "label": "Approximate Number" | |
| }, | |
| { | |
| "@id": "cx:Numeral", | |
| "@type": "owl:Class", | |
| "comment": "A syntactic representation of a number in some numeral system.", | |
| "label": "Numeral" | |
| }, | |
| { | |
| "@id": "cx:NumeralSystem", | |
| "@type": "owl:Class", | |
| "comment": "A scheme for representing numbers (positional, mixed-radix, etc.).", | |
| "label": "Numeral System" | |
| }, | |
| { | |
| "@id": "cx:PositionalSystem", | |
| "@type": "owl:Class", | |
| "comment": "Fixed-radix positional representation (e.g., decimal, binary).", | |
| "label": "Positional System" | |
| }, | |
| { | |
| "@id": "cx:MixedRadixSystem", | |
| "@type": "owl:Class", | |
| "comment": "Variable radices per position (e.g., mod-96 CRT, factorial).", | |
| "label": "Mixed-Radix System" | |
| }, | |
| { | |
| "@id": "cx:NonPositionalSystem", | |
| "@type": "owl:Class", | |
| "comment": "Non-positional representation (e.g., continued fractions, Roman).", | |
| "label": "Non-Positional System" | |
| }, | |
| { | |
| "@id": "cx:DigitSet", | |
| "@type": "owl:Class", | |
| "comment": "Alphabet of digit symbols used by a numeral system.", | |
| "label": "Digit Set" | |
| }, | |
| { | |
| "@id": "cx:DigitSymbol", | |
| "@type": "owl:Class", | |
| "comment": "A symbol used as a digit (character, token, glyph).", | |
| "label": "Digit Symbol" | |
| }, | |
| { | |
| "@id": "cx:ConversionMorphism", | |
| "@type": "owl:Class", | |
| "comment": "A morphism between numeral systems (parse/print + correctness).", | |
| "label": "Conversion Morphism" | |
| }, | |
| { | |
| "@id": "cx:PrecisionContext", | |
| "@type": "owl:Class", | |
| "comment": "Defines precision/rounding/exponent constraints for numerics.", | |
| "label": "Precision Context" | |
| }, | |
| { | |
| "@id": "cx:ArbitraryPrecisionContext", | |
| "@type": "owl:Class", | |
| "comment": "Unbounded precision (BigInt, exact rational).", | |
| "label": "Arbitrary Precision Context" | |
| }, | |
| { | |
| "@id": "cx:FixedPrecisionContext", | |
| "@type": "owl:Class", | |
| "comment": "Fixed precision and exponent range (IEEE-like, mod-96).", | |
| "label": "Fixed Precision Context" | |
| }, | |
| { | |
| "@id": "cx:RoundingMode", | |
| "@type": "owl:Class", | |
| "comment": "Rounding policy (ties-to-even, toward-zero, etc.).", | |
| "label": "Rounding Mode" | |
| }, | |
| { | |
| "@id": "cx:Interval", | |
| "@type": "owl:Class", | |
| "comment": "[lower, upper] bound for approximate values.", | |
| "label": "Interval" | |
| }, | |
| { | |
| "@id": "cx:Ball", | |
| "@type": "owl:Class", | |
| "comment": "Midpoint + radius bound (ball arithmetic).", | |
| "label": "Ball" | |
| }, | |
| { | |
| "@id": "cx:NumericAssertion", | |
| "@type": "owl:Class", | |
| "comment": "Epistemic wrapper: computed/predicted/bounded with provenance.", | |
| "label": "Numeric Assertion" | |
| }, | |
| { | |
| "@id": "cx:AssertionStatus", | |
| "@type": "owl:Class", | |
| "comment": "ComputedExact, ComputedApprox, Predicted, Conjectured, Bounded, Unknown.", | |
| "label": "Assertion Status" | |
| }, | |
| { | |
| "@id": "cx:ComputationArtifact", | |
| "@type": "owl:Class", | |
| "comment": "Program run, proof certificate, or derivation trace.", | |
| "label": "Computation Artifact" | |
| }, | |
| { | |
| "@id": "cx:CountingSequence", | |
| "@type": "owl:Class", | |
| "comment": "A counting function ℕ → NumericDomain.", | |
| "label": "Counting Sequence" | |
| }, | |
| { | |
| "@id": "cx:SequenceEntry", | |
| "@type": "owl:Class", | |
| "comment": "An indexed entry in a counting sequence with epistemic status.", | |
| "label": "Sequence Entry" | |
| }, | |
| { | |
| "@id": "cx:TypeVariable", | |
| "@type": "owl:Class", | |
| "comment": "A metavariable for polymorphic signatures (distinct from T=3 axiom).", | |
| "label": "Type Variable" | |
| }, | |
| { | |
| "@id": "cx:UniversalOperator", | |
| "@type": "owl:Class", | |
| "comment": "The categorical limit M_∞ of gauge extensions.", | |
| "label": "Universal Operator" | |
| }, | |
| { | |
| "@id": "cx:GaugeTower", | |
| "@type": "owl:Class", | |
| "comment": "Directed system of gauge extensions {2,3}→{2,3,5}→...", | |
| "label": "Gauge Tower" | |
| }, | |
| { | |
| "@id": "cx:GaugeLevel", | |
| "@type": "owl:Class", | |
| "comment": "Single level in the gauge tower with prime set.", | |
| "label": "Gauge Level" | |
| }, | |
| { | |
| "@id": "cx:SpectralTriple", | |
| "@type": "owl:Class", | |
| "comment": "AltSpec(480) × ZDSpec(15) × JordSpec(27) = 522 dimensions.", | |
| "label": "Spectral Triple" | |
| }, | |
| { | |
| "@id": "cx:SpectralPrime", | |
| "@type": "owl:Class", | |
| "comment": "Prime arising from transfer eigenvalue structure.", | |
| "label": "Spectral Prime" | |
| }, | |
| { | |
| "@id": "cx:PhaseBridge", | |
| "@type": "owl:Class", | |
| "comment": "Linking factor between triality phases (401, 433).", | |
| "label": "Phase Bridge" | |
| }, | |
| { | |
| "@id": "cx:Diagram", | |
| "@type": "owl:Class", | |
| "comment": "Functor from index category to target category.", | |
| "label": "Diagram" | |
| }, | |
| { | |
| "@id": "cx:Cone", | |
| "@type": "owl:Class", | |
| "comment": "Natural transformation from constant diagram to F.", | |
| "label": "Cone" | |
| }, | |
| { | |
| "@id": "cx:Limit", | |
| "@type": "owl:Class", | |
| "comment": "Universal cone over a diagram.", | |
| "label": "Limit" | |
| }, | |
| { | |
| "@id": "cx:IndexCategory", | |
| "@type": "owl:Class", | |
| "comment": "Shape category indexing a diagram.", | |
| "label": "Index Category" | |
| }, | |
| { | |
| "@id": "cx:Gauge", | |
| "@type": "owl:Class", | |
| "comment": "Finite prime-indexed approximation to the universal operator.", | |
| "label": "Gauge" | |
| }, | |
| { | |
| "@id": "cx:GaugesContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for gauge structures.", | |
| "label": "Gauges Container" | |
| }, | |
| { | |
| "@id": "cx:Hub", | |
| "@type": "owl:Class", | |
| "comment": "Categorical boundary for hierarchical state management.", | |
| "label": "Hub" | |
| }, | |
| { | |
| "@id": "cx:HubsContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for hub boundaries.", | |
| "label": "Hubs Container" | |
| }, | |
| { | |
| "@id": "cx:ExtensionFunctor", | |
| "@type": "owl:Class", | |
| "comment": "Gauge extension E_p: Ω_P → Ω_{P∪{p}} for prime p.", | |
| "label": "Extension Functor" | |
| }, | |
| { | |
| "@id": "cx:LimitProjection", | |
| "@type": "owl:Class", | |
| "comment": "Projection πᵢ: M_∞ → Ωᵢ from universal limit to gauge level.", | |
| "label": "Limit Projection" | |
| }, | |
| { | |
| "@id": "cx:FunctorsContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for extension functors.", | |
| "label": "Functors Container" | |
| }, | |
| { | |
| "@id": "cx:LimitProjectionsContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for limit projections.", | |
| "label": "Limit Projections Container" | |
| }, | |
| { | |
| "@id": "cx:IndexCategoriesContainer", | |
| "@type": "owl:Class", | |
| "comment": "Container for index category instances.", | |
| "label": "Index Categories Container" | |
| }, | |
| { | |
| "@id": "cx:TrialityPhase", | |
| "@type": "owl:Class", | |
| "comment": "Phase in the T=3 triality structure (Tonic, LeadingTone, Overtone).", | |
| "label": "Triality Phase" | |
| }, | |
| { | |
| "@id": "cx:Tonic", | |
| "@type": "owl:Class", | |
| "comment": "Phase for n ≡ 0 mod 3; aligned with Jordan spectrum.", | |
| "label": "Tonic Phase" | |
| }, | |
| { | |
| "@id": "cx:LeadingTone", | |
| "@type": "owl:Class", | |
| "comment": "Phase for n ≡ 1 mod 3; aligned with zero-divisor spectrum.", | |
| "label": "Leading Tone Phase" | |
| }, | |
| { | |
| "@id": "cx:Overtone", | |
| "@type": "owl:Class", | |
| "comment": "Phase for n ≡ 2 mod 3; aligned with alternating spectrum.", | |
| "label": "Overtone Phase" | |
| }, | |
| { | |
| "@id": "cx:GaugeExtension", | |
| "@type": "owl:Class", | |
| "comment": "Abstract adjunction E_p ⊣ R_p for gauge extensions.", | |
| "label": "Gauge Extension" | |
| }, | |
| { | |
| "@id": "cx:RestrictionFunctor", | |
| "@type": "owl:Class", | |
| "comment": "Gauge restriction R_p: Ω_{P∪{p}} → Ω_P for prime p.", | |
| "label": "Restriction Functor" | |
| }, | |
| { | |
| "@id": "cx:AdjunctionUnit", | |
| "@type": "owl:Class", | |
| "comment": "Unit natural transformation η: 1 → R_p ∘ E_p.", | |
| "label": "Adjunction Unit" | |
| }, | |
| { | |
| "@id": "cx:AdjunctionCounit", | |
| "@type": "owl:Class", | |
| "comment": "Counit natural transformation ε: E_p ∘ R_p → 1.", | |
| "label": "Adjunction Counit" | |
| }, | |
| { | |
| "@id": "cx:TransferEigenvalue", | |
| "@type": "owl:Class", | |
| "comment": "Eigenvalue of the transfer matrix (λ₁=10, λ₂=2, λ₃=7, λ₄=-1).", | |
| "label": "Transfer Eigenvalue" | |
| }, | |
| { | |
| "@id": "cx:DominantEigenvalue", | |
| "@type": "owl:Class", | |
| "comment": "Tonic dominant eigenvalue λ₁ = O + 2 = 10.", | |
| "label": "Dominant Eigenvalue" | |
| }, | |
| { | |
| "@id": "cx:SubdominantEigenvalue", | |
| "@type": "owl:Class", | |
| "comment": "Subdominant eigenvalue for phase coupling.", | |
| "label": "Subdominant Eigenvalue" | |
| }, | |
| { | |
| "@id": "cx:hasPart", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has part", | |
| "owl:inverseOf": { | |
| "@id": "cx:partOf" | |
| }, | |
| "rdfs:domain": { | |
| "@id": "cx:SingletonInstance" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Thing" | |
| } | |
| }, | |
| { | |
| "@id": "cx:partOf", | |
| "@type": "owl:ObjectProperty", | |
| "label": "part of", | |
| "owl:inverseOf": { | |
| "@id": "cx:hasPart" | |
| }, | |
| "rdfs:domain": { | |
| "@id": "cx:Thing" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:SingletonInstance" | |
| } | |
| }, | |
| { | |
| "@id": "cx:sourceLevel", | |
| "@type": "owl:ObjectProperty", | |
| "label": "source level", | |
| "rdfs:domain": { | |
| "@id": "cx:TowerTransition" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:TowerLevel" | |
| } | |
| }, | |
| { | |
| "@id": "cx:targetLevel", | |
| "@type": "owl:ObjectProperty", | |
| "label": "target level", | |
| "rdfs:domain": { | |
| "@id": "cx:TowerTransition" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:TowerLevel" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasDimension", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has dimension", | |
| "rdfs:domain": { | |
| "@id": "cx:TowerLevel" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:DimensionSpec" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasAutomorphismGroup", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has automorphism group", | |
| "rdfs:domain": { | |
| "@id": "cx:TowerLevel" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:AutomorphismGroup" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasCocycle", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has cocycle", | |
| "rdfs:domain": { | |
| "@id": "cx:TowerLevel" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:CocycleClass" | |
| } | |
| }, | |
| { | |
| "@id": "cx:losesProperty", | |
| "@type": "owl:ObjectProperty", | |
| "label": "loses property", | |
| "rdfs:domain": { | |
| "@id": "cx:TowerLevel" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:AlgebraicProperty" | |
| } | |
| }, | |
| { | |
| "@id": "cx:targetCategory", | |
| "@type": "owl:ObjectProperty", | |
| "label": "target category", | |
| "rdfs:domain": { | |
| "@id": "cx:Thing" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Category" | |
| } | |
| }, | |
| { | |
| "@id": "cx:correspondsTo", | |
| "@type": "owl:ObjectProperty", | |
| "label": "corresponds to", | |
| "rdfs:domain": { | |
| "@id": "cx:Correspondence" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Projection" | |
| } | |
| }, | |
| { | |
| "@id": "cx:phaseAlignment", | |
| "@type": "owl:ObjectProperty", | |
| "label": "phase alignment", | |
| "rdfs:domain": { | |
| "@id": "cx:Correspondence" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:PhaseAlignment" | |
| } | |
| }, | |
| { | |
| "@id": "cx:governsPhase", | |
| "@type": "owl:ObjectProperty", | |
| "label": "governs phase", | |
| "rdfs:domain": { | |
| "@id": "cx:Operator" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:PhaseModification" | |
| } | |
| }, | |
| { | |
| "@id": "cx:activatesAt", | |
| "@type": "owl:ObjectProperty", | |
| "label": "activates at", | |
| "rdfs:domain": { | |
| "@id": "cx:PhaseBehavior" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:TowerLevel" | |
| } | |
| }, | |
| { | |
| "@id": "cx:derivedFrom", | |
| "@type": "owl:ObjectProperty", | |
| "label": "derived from", | |
| "rdfs:domain": { | |
| "@id": "cx:DerivedConstant" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Primitive" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasProof", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has proof", | |
| "rdfs:domain": { | |
| "@id": "cx:Theorem" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Proof" | |
| } | |
| }, | |
| { | |
| "@id": "cx:proves", | |
| "@type": "owl:ObjectProperty", | |
| "label": "proves", | |
| "rdfs:domain": { | |
| "@id": "cx:Proof" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Theorem" | |
| } | |
| }, | |
| { | |
| "@id": "cx:contains", | |
| "@type": "owl:ObjectProperty", | |
| "label": "contains", | |
| "owl:inverseOf": { | |
| "@id": "cx:containedIn" | |
| }, | |
| "rdfs:domain": { | |
| "@id": "cx:Stratum" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:TowerLevel" | |
| } | |
| }, | |
| { | |
| "@id": "cx:containedIn", | |
| "@type": "owl:ObjectProperty", | |
| "label": "contained in", | |
| "owl:inverseOf": { | |
| "@id": "cx:contains" | |
| }, | |
| "rdfs:domain": { | |
| "@id": "cx:TowerLevel" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Stratum" | |
| } | |
| }, | |
| { | |
| "@id": "cx:mapsPhase", | |
| "@type": "owl:ObjectProperty", | |
| "label": "maps phase", | |
| "rdfs:domain": { | |
| "@id": "cx:Projection" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:PhaseMapEntry" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasInstance", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has instance", | |
| "owl:inverseOf": { | |
| "@id": "cx:instanceOf" | |
| }, | |
| "rdfs:domain": { | |
| "@id": "cx:StructuralType" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Thing" | |
| } | |
| }, | |
| { | |
| "@id": "cx:instanceOf", | |
| "@type": "owl:ObjectProperty", | |
| "label": "instance of", | |
| "owl:inverseOf": { | |
| "@id": "cx:hasInstance" | |
| }, | |
| "rdfs:domain": { | |
| "@id": "cx:Thing" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:StructuralType" | |
| } | |
| }, | |
| { | |
| "@id": "cx:generatesType", | |
| "@type": "owl:ObjectProperty", | |
| "label": "generates type", | |
| "rdfs:domain": { | |
| "@id": "cx:TypeConstructor" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:owl:Class" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasConstructionRule", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has construction rule", | |
| "rdfs:domain": { | |
| "@id": "cx:TypeConstructor" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:ConstructionRule" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasParameter", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has parameter", | |
| "rdfs:domain": { | |
| "@id": "cx:TypeConstructor" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:ParameterSpec" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasDerivationSource", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has derivation source", | |
| "rdfs:domain": { | |
| "@id": "cx:Thing" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:DerivationSource" | |
| } | |
| }, | |
| { | |
| "@id": "cx:functorSource", | |
| "@type": "owl:ObjectProperty", | |
| "label": "functor source", | |
| "rdfs:domain": { | |
| "@id": "cx:ConstructorFunctor" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:TypeConstructor" | |
| } | |
| }, | |
| { | |
| "@id": "cx:functorTarget", | |
| "@type": "owl:ObjectProperty", | |
| "label": "functor target", | |
| "rdfs:domain": { | |
| "@id": "cx:ConstructorFunctor" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:TypeConstructor" | |
| } | |
| }, | |
| { | |
| "@id": "cx:tracesToAxiom", | |
| "@type": "owl:ObjectProperty", | |
| "label": "traces to axiom", | |
| "rdfs:domain": { | |
| "@id": "cx:DerivationSource" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Axiom" | |
| } | |
| }, | |
| { | |
| "@id": "cx:canDeriveFrom", | |
| "@type": "owl:ObjectProperty", | |
| "label": "can derive from", | |
| "rdfs:domain": { | |
| "@id": "cx:DerivationLevel" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:DerivationLevel" | |
| } | |
| }, | |
| { | |
| "@id": "cx:atDerivationLevel", | |
| "@type": "owl:ObjectProperty", | |
| "label": "at derivation level", | |
| "rdfs:domain": { | |
| "@id": "cx:Thing" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:DerivationLevel" | |
| } | |
| }, | |
| { | |
| "@id": "cx:requiresStructure", | |
| "@type": "owl:ObjectProperty", | |
| "label": "requires structure", | |
| "rdfs:domain": { | |
| "@id": "cx:ExtensionPoint" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:CategoricalStructureKind" | |
| } | |
| }, | |
| { | |
| "@id": "cx:extendsConstructor", | |
| "@type": "owl:ObjectProperty", | |
| "label": "extends constructor", | |
| "rdfs:domain": { | |
| "@id": "cx:ExtensionPoint" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:TypeConstructor" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasCompositionLaw", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has composition law", | |
| "rdfs:domain": { | |
| "@id": "cx:ExtensionPoint" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:CompositionLaw" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasDerivationConstraints", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has derivation constraints", | |
| "rdfs:domain": { | |
| "@id": "cx:ExtensionPoint" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:DerivationConstraints" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasUniversalPropertyKind", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has universal property kind", | |
| "rdfs:domain": { | |
| "@id": "cx:UniversalProperty" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:UniversalPropertyKind" | |
| } | |
| }, | |
| { | |
| "@id": "cx:definesObject", | |
| "@type": "owl:ObjectProperty", | |
| "label": "defines object", | |
| "rdfs:domain": { | |
| "@id": "cx:UniversalProperty" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:owl:Thing" | |
| } | |
| }, | |
| { | |
| "@id": "cx:inCategory", | |
| "@type": "owl:ObjectProperty", | |
| "label": "in category", | |
| "rdfs:domain": { | |
| "@id": "cx:UniversalProperty" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Thing" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasBaseStructure", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has base structure", | |
| "rdfs:domain": { | |
| "@id": "cx:FreeConstruction" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:owl:Class" | |
| } | |
| }, | |
| { | |
| "@id": "cx:adjointTo", | |
| "@type": "owl:ObjectProperty", | |
| "label": "adjoint to", | |
| "rdfs:domain": { | |
| "@id": "cx:FreeConstruction" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:ConstructorFunctor" | |
| } | |
| }, | |
| { | |
| "@id": "cx:satisfiesUniversalProperty", | |
| "@type": "owl:ObjectProperty", | |
| "label": "satisfies universal property", | |
| "rdfs:domain": { | |
| "@id": "cx:FreeConstruction" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:UniversalProperty" | |
| } | |
| }, | |
| { | |
| "@id": "cx:implementsExtensionPoint", | |
| "@type": "owl:ObjectProperty", | |
| "label": "implements extension point", | |
| "rdfs:domain": { | |
| "@id": "cx:Extension" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:ExtensionPoint" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasDerivationChain", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has derivation chain", | |
| "rdfs:domain": { | |
| "@id": "cx:Extension" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:DerivationChain" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasDerivationStep", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has derivation step", | |
| "rdfs:domain": { | |
| "@id": "cx:DerivationChain" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:DerivationStep" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasCategoricalStructure", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has categorical structure", | |
| "rdfs:domain": { | |
| "@id": "cx:Extension" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:CategoricalStructure" | |
| } | |
| }, | |
| { | |
| "@id": "cx:providesConstructor", | |
| "@type": "owl:ObjectProperty", | |
| "label": "provides constructor", | |
| "rdfs:domain": { | |
| "@id": "cx:Extension" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:TypeConstructor" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasVerificationStatus", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has verification status", | |
| "rdfs:domain": { | |
| "@id": "cx:Extension" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:VerificationStatus" | |
| } | |
| }, | |
| { | |
| "@id": "cx:terminalAxiom", | |
| "@type": "owl:ObjectProperty", | |
| "label": "terminal axiom", | |
| "rdfs:domain": { | |
| "@id": "cx:DerivationChain" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Axiom" | |
| } | |
| }, | |
| { | |
| "@id": "cx:stepProduces", | |
| "@type": "owl:ObjectProperty", | |
| "label": "step produces", | |
| "rdfs:domain": { | |
| "@id": "cx:DerivationStep" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Thing" | |
| } | |
| }, | |
| { | |
| "@id": "cx:stepDependsOn", | |
| "@type": "owl:ObjectProperty", | |
| "label": "step depends on", | |
| "rdfs:domain": { | |
| "@id": "cx:DerivationStep" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Thing" | |
| } | |
| }, | |
| { | |
| "@id": "cx:stepVia", | |
| "@type": "owl:ObjectProperty", | |
| "label": "step via", | |
| "rdfs:domain": { | |
| "@id": "cx:DerivationStep" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:DerivationSource" | |
| } | |
| }, | |
| { | |
| "@id": "cx:appliesTo", | |
| "@type": "owl:ObjectProperty", | |
| "label": "applies to", | |
| "rdfs:domain": { | |
| "@id": "cx:CompositionLaw" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:CategoricalStructureKind" | |
| } | |
| }, | |
| { | |
| "@id": "cx:inDomain", | |
| "@type": "owl:ObjectProperty", | |
| "label": "in domain", | |
| "rdfs:domain": { | |
| "@id": "cx:Number" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:NumericDomain" | |
| } | |
| }, | |
| { | |
| "@id": "cx:embedsInto", | |
| "@type": "owl:ObjectProperty", | |
| "label": "embeds into", | |
| "rdfs:domain": { | |
| "@id": "cx:NumericDomain" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:NumericDomain" | |
| } | |
| }, | |
| { | |
| "@id": "cx:usesDigitSet", | |
| "@type": "owl:ObjectProperty", | |
| "label": "uses digit set", | |
| "rdfs:domain": { | |
| "@id": "cx:NumeralSystem" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:DigitSet" | |
| } | |
| }, | |
| { | |
| "@id": "cx:denotes", | |
| "@type": "owl:ObjectProperty", | |
| "label": "denotes", | |
| "rdfs:domain": { | |
| "@id": "cx:Numeral" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Number" | |
| } | |
| }, | |
| { | |
| "@id": "cx:inSystem", | |
| "@type": "owl:ObjectProperty", | |
| "label": "in system", | |
| "rdfs:domain": { | |
| "@id": "cx:Numeral" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:NumeralSystem" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasConversion", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has conversion", | |
| "rdfs:domain": { | |
| "@id": "cx:NumeralSystem" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:ConversionMorphism" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasPrecisionContext", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has precision context", | |
| "rdfs:domain": { | |
| "@id": "cx:ApproximateNumber" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:PrecisionContext" | |
| } | |
| }, | |
| { | |
| "@id": "cx:roundingMode", | |
| "@type": "owl:ObjectProperty", | |
| "label": "rounding mode", | |
| "rdfs:domain": { | |
| "@id": "cx:PrecisionContext" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:RoundingMode" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasInterval", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has interval", | |
| "rdfs:domain": { | |
| "@id": "cx:ApproximateNumber" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Interval" | |
| } | |
| }, | |
| { | |
| "@id": "cx:assertsValue", | |
| "@type": "owl:ObjectProperty", | |
| "label": "asserts value", | |
| "rdfs:domain": { | |
| "@id": "cx:NumericAssertion" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:Number" | |
| } | |
| }, | |
| { | |
| "@id": "cx:status", | |
| "@type": "owl:ObjectProperty", | |
| "label": "status", | |
| "rdfs:domain": { | |
| "@id": "cx:NumericAssertion" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:AssertionStatus" | |
| } | |
| }, | |
| { | |
| "@id": "cx:supportedBy", | |
| "@type": "owl:ObjectProperty", | |
| "label": "supported by", | |
| "rdfs:domain": { | |
| "@id": "cx:NumericAssertion" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:ComputationArtifact" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasProofStep", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has proof step", | |
| "rdfs:domain": { | |
| "@id": "cx:NumericAssertion" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:ProofStep" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasEntry", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has entry", | |
| "rdfs:domain": { | |
| "@id": "cx:CountingSequence" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:SequenceEntry" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasAssertion", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has assertion", | |
| "rdfs:domain": { | |
| "@id": "cx:SequenceEntry" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:NumericAssertion" | |
| } | |
| }, | |
| { | |
| "@id": "cx:codomain", | |
| "@type": "owl:ObjectProperty", | |
| "label": "codomain", | |
| "rdfs:domain": { | |
| "@id": "cx:CountingSequence" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:NumericDomain" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasTypeParameter", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has type parameter", | |
| "rdfs:domain": { | |
| "@id": "cx:PrimitiveOperation" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:TypeVariable" | |
| } | |
| }, | |
| { | |
| "@id": "cx:typeParameterDomain", | |
| "@type": "owl:ObjectProperty", | |
| "label": "type parameter domain", | |
| "rdfs:domain": { | |
| "@id": "cx:TypeVariable" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:owl:Class" | |
| } | |
| }, | |
| { | |
| "@id": "cx:extendsVia", | |
| "@type": "owl:ObjectProperty", | |
| "label": "extends via", | |
| "owl:inverseOf": { | |
| "@id": "cx:extendedBy" | |
| }, | |
| "rdfs:domain": { | |
| "@id": "cx:GaugeLevel" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:GaugeLevel" | |
| } | |
| }, | |
| { | |
| "@id": "cx:extendedBy", | |
| "@type": "owl:ObjectProperty", | |
| "label": "extended by", | |
| "owl:inverseOf": { | |
| "@id": "cx:extendsVia" | |
| }, | |
| "rdfs:domain": { | |
| "@id": "cx:GaugeLevel" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:GaugeLevel" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasSpectralPrime", | |
| "@type": "owl:ObjectProperty", | |
| "label": "has spectral prime", | |
| "rdfs:domain": { | |
| "@id": "cx:SpectralTriple" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:SpectralPrime" | |
| } | |
| }, | |
| { | |
| "@id": "cx:coherentWith", | |
| "@type": "owl:ObjectProperty", | |
| "label": "coherent with", | |
| "owl:inverseOf": { | |
| "@id": "cx:coherentWith" | |
| }, | |
| "rdfs:domain": { | |
| "@id": "cx:GaugeLevel" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:GaugeLevel" | |
| } | |
| }, | |
| { | |
| "@id": "cx:extensionSource", | |
| "@type": "owl:ObjectProperty", | |
| "label": "extension source", | |
| "rdfs:domain": { | |
| "@id": "cx:ExtensionFunctor" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:GaugeLevel" | |
| } | |
| }, | |
| { | |
| "@id": "cx:extensionTarget", | |
| "@type": "owl:ObjectProperty", | |
| "label": "extension target", | |
| "rdfs:domain": { | |
| "@id": "cx:ExtensionFunctor" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:GaugeLevel" | |
| } | |
| }, | |
| { | |
| "@id": "cx:projectionSource", | |
| "@type": "owl:ObjectProperty", | |
| "label": "projection source", | |
| "rdfs:domain": { | |
| "@id": "cx:LimitProjection" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:UniversalOperator" | |
| } | |
| }, | |
| { | |
| "@id": "cx:projectionTarget", | |
| "@type": "owl:ObjectProperty", | |
| "label": "projection target", | |
| "rdfs:domain": { | |
| "@id": "cx:LimitProjection" | |
| }, | |
| "rdfs:range": { | |
| "@id": "cx:GaugeLevel" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasValue", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "has value", | |
| "rdfs:domain": { | |
| "@id": "cx:DerivedConstant" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:dimension", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "dimension", | |
| "rdfs:domain": { | |
| "@id": "cx:DimensionSpec" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:symbol", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "symbol", | |
| "rdfs:domain": { | |
| "@id": "cx:Primitive" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:formula", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "formula", | |
| "rdfs:domain": { | |
| "@id": "cx:DerivedConstant" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:index", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "index", | |
| "rdfs:domain": { | |
| "@id": "cx:TowerLevel" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:levelIndex", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "level index", | |
| "rdfs:domain": { | |
| "@id": "cx:TowerLevel" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:activatesAtIndex", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "activates at index", | |
| "rdfs:domain": { | |
| "@id": "cx:PhaseBehavior" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:sourcePhase", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "source phase", | |
| "rdfs:domain": { | |
| "@id": "cx:PhaseTransition" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:targetPhase", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "target phase", | |
| "rdfs:domain": { | |
| "@id": "cx:PhaseTransition" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:transitionLevel", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "transition level", | |
| "rdfs:domain": { | |
| "@id": "cx:PhaseTransition" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:criticalConstant", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "critical constant", | |
| "rdfs:domain": { | |
| "@id": "cx:PhaseTransition" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:targetField", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "target field", | |
| "rdfs:domain": { | |
| "@id": "cx:ConstructionRule" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:constructionFormula", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "construction formula", | |
| "rdfs:domain": { | |
| "@id": "cx:ConstructionRule" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:parameterVariable", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "parameter variable", | |
| "rdfs:domain": { | |
| "@id": "cx:ParameterSpec" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:parameterDomain", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "parameter domain", | |
| "rdfs:domain": { | |
| "@id": "cx:ParameterSpec" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:parameterCondition", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "parameter condition", | |
| "rdfs:domain": { | |
| "@id": "cx:ParameterSpec" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:functorFormula", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "functor formula", | |
| "rdfs:domain": { | |
| "@id": "cx:ConstructorFunctor" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:sourceAxiom", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "source axiom", | |
| "rdfs:domain": { | |
| "@id": "cx:DerivationSource" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:derivationDepth", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "derivation depth", | |
| "rdfs:domain": { | |
| "@id": "cx:DerivationLevel" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:validationRule", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "validation rule", | |
| "rdfs:domain": { | |
| "@id": "cx:DerivationLevel" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:domainCategory", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "domain category", | |
| "rdfs:domain": { | |
| "@id": "cx:ExtensionPoint" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:codomainCategory", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "codomain category", | |
| "rdfs:domain": { | |
| "@id": "cx:ExtensionPoint" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:maxDerivationLevel", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "max derivation level", | |
| "rdfs:domain": { | |
| "@id": "cx:DerivationConstraints" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:diagramShape", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "diagram shape", | |
| "rdfs:domain": { | |
| "@id": "cx:UniversalProperty" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:universalArrow", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "universal arrow", | |
| "rdfs:domain": { | |
| "@id": "cx:UniversalProperty" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:existenceCondition", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "existence condition", | |
| "rdfs:domain": { | |
| "@id": "cx:UniversalProperty" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:uniquenessCondition", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "uniqueness condition", | |
| "rdfs:domain": { | |
| "@id": "cx:UniversalProperty" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:baseStructure", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "base structure", | |
| "rdfs:domain": { | |
| "@id": "cx:FreeConstruction" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:targetCategoryName", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "target category name", | |
| "rdfs:domain": { | |
| "@id": "cx:FreeConstruction" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:forgetfulFunctor", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "forgetful functor", | |
| "rdfs:domain": { | |
| "@id": "cx:FreeConstruction" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:objectMapping", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "object mapping", | |
| "rdfs:domain": { | |
| "@id": "cx:CategoricalStructure" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:morphismMapping", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "morphism mapping", | |
| "rdfs:domain": { | |
| "@id": "cx:CategoricalStructure" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:naturalComponents", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "natural components", | |
| "rdfs:domain": { | |
| "@id": "cx:CategoricalStructure" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:adjunctionUnit", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "adjunction unit", | |
| "rdfs:domain": { | |
| "@id": "cx:CategoricalStructure" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:adjunctionCounit", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "adjunction counit", | |
| "rdfs:domain": { | |
| "@id": "cx:CategoricalStructure" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:lawExpression", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "law expression", | |
| "rdfs:domain": { | |
| "@id": "cx:CompositionLaw" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:lexicalValue", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "lexical value", | |
| "rdfs:domain": { | |
| "@id": "cx:Number" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasRadix", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "has radix", | |
| "rdfs:domain": { | |
| "@id": "cx:PositionalSystem" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:hasRadices", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "has radices", | |
| "rdfs:domain": { | |
| "@id": "cx:MixedRadixSystem" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:encoding", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "encoding", | |
| "rdfs:domain": { | |
| "@id": "cx:NumeralSystem" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:precisionBits", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "precision bits", | |
| "rdfs:domain": { | |
| "@id": "cx:PrecisionContext" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:precisionDigits", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "precision digits", | |
| "rdfs:domain": { | |
| "@id": "cx:PrecisionContext" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:exponentMin", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "exponent min", | |
| "rdfs:domain": { | |
| "@id": "cx:FixedPrecisionContext" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:exponentMax", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "exponent max", | |
| "rdfs:domain": { | |
| "@id": "cx:FixedPrecisionContext" | |
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| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:lowerBound", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "lower bound", | |
| "rdfs:domain": { | |
| "@id": "cx:Interval" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:upperBound", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "upper bound", | |
| "rdfs:domain": { | |
| "@id": "cx:Interval" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:confidence", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "confidence", | |
| "rdfs:domain": { | |
| "@id": "cx:NumericAssertion" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:decimal" | |
| } | |
| }, | |
| { | |
| "@id": "cx:sequenceIndex", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "sequence index", | |
| "rdfs:domain": { | |
| "@id": "cx:SequenceEntry" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:oeisId", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "OEIS identifier", | |
| "rdfs:domain": { | |
| "@id": "cx:CountingSequence" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:satisfiesQuadraticForm", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "satisfies quadratic form", | |
| "rdfs:domain": { | |
| "@id": "cx:SpectralPrime" | |
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| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cx:extensionPrime", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "extension prime", | |
| "rdfs:domain": { | |
| "@id": "cx:ExtensionFunctor" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:boundaryScale", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "boundary scale", | |
| "rdfs:domain": { | |
| "@id": "cx:ExtensionFunctor" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:resonanceFactor", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "resonance factor", | |
| "rdfs:domain": { | |
| "@id": "cx:ExtensionFunctor" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:integer" | |
| } | |
| }, | |
| { | |
| "@id": "cx:indexingScheme", | |
| "@type": "owl:DatatypeProperty", | |
| "label": "indexing scheme", | |
| "rdfs:domain": { | |
| "@id": "cx:IndexCategory" | |
| }, | |
| "rdfs:range": { | |
| "@id": "xsd:string" | |
| } | |
| }, | |
| { | |
| "@id": "cxs:CategoricalX", | |
| "@type": "SingletonInstance", | |
| "comment": "The unique canonical instance of the Categorical X structure", | |
| "hasPart": [ | |
| "cxs:primitives", | |
| "cxs:types", | |
| "cxs:axioms", | |
| "cxs:constants", | |
| "cxs:tower", | |
| "cxs:gauges", | |
| "cxs:hubs", | |
| "cxs:operators", | |
| "cxs:projections", | |
| "cxs:correspondences", | |
| "cxm:meta" | |
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| }, | |
| { | |
| "@id": "cxs:primitives", | |
| "@type": "PrimitivesContainer", | |
| "hasPart": [ | |
| "cxs:primitives/integers", | |
| "cxs:primitives/operations", | |
| "cxs:primitives/relations" | |
| ], | |
| "label": "Primitives", | |
| "partOf": "cxs:CategoricalX" | |
| }, | |
| { | |
| "@id": "cxs:primitives/integers", | |
| "@type": "IntegersContainer", | |
| "contains": [ | |
| "cxs:primitives/integers/U", | |
| "cxs:primitives/integers/D", | |
| "cxs:primitives/integers/T", | |
| "cxs:primitives/integers/O" | |
| ], | |
| "label": "Primitive Integers" | |
| }, | |
| { | |
| "@id": "cxs:primitives/integers/U", | |
| "@type": "Axiom", | |
| "comment": "Identity primitive: terminal object cardinality (id₁ = 1)", | |
| "derivationLevel": 0, | |
| "label": "Identity", | |
| "symbol": "U", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 1 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:primitives/integers/D", | |
| "@type": "Axiom", | |
| "comment": "Duality primitive: morphism structure arity (|{dom,cod}| = 2)", | |
| "derivationLevel": 0, | |
| "label": "Duality", | |
| "symbol": "D", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 2 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:primitives/integers/T", | |
| "@type": "Axiom", | |
| "comment": "Derived from U + D: minimum cyclic composition structure", | |
| "derivationFormula": "U + D", | |
| "derivationLevel": 1, | |
| "derivedFrom": [ | |
| "cxs:primitives/integers/U", | |
| "cxs:primitives/integers/D" | |
| ], | |
| "label": "Triality", | |
| "symbol": "T", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 3 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:primitives/integers/O", | |
| "@type": "Axiom", | |
| "comment": "Derived from D^T: maximum normed division algebra dimension", | |
| "derivationFormula": "D^T", | |
| "derivationLevel": 1, | |
| "derivedFrom": [ | |
| "cxs:primitives/integers/D", | |
| "cxs:primitives/integers/T" | |
| ], | |
| "label": "Octonion Dimension", | |
| "symbol": "O", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 8 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:axioms", | |
| "@type": "AxiomsContainer", | |
| "contains": [ | |
| "cxs:axioms/U", | |
| "cxs:axioms/D", | |
| "cxs:axioms/T", | |
| "cxs:axioms/O" | |
| ], | |
| "label": "Axioms", | |
| "partOf": "cxs:CategoricalX" | |
| }, | |
| { | |
| "@id": "cxs:axioms/U", | |
| "@type": "Axiom", | |
| "comment": "Identity primitive: terminal object cardinality", | |
| "derivationLevel": 0, | |
| "label": "Identity Axiom", | |
| "symbol": "U", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 1 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:axioms/D", | |
| "@type": "Axiom", | |
| "comment": "Duality primitive: morphism structure arity", | |
| "derivationLevel": 0, | |
| "label": "Duality Axiom", | |
| "symbol": "D", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 2 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:axioms/T", | |
| "@type": "Axiom", | |
| "derivationFormula": "U + D", | |
| "derivationLevel": 1, | |
| "derivedFrom": [ | |
| "cxs:axioms/U", | |
| "cxs:axioms/D" | |
| ], | |
| "label": "Triality Axiom", | |
| "symbol": "T", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 3 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:axioms/O", | |
| "@type": "Axiom", | |
| "derivationFormula": "D^T", | |
| "derivationLevel": 1, | |
| "derivedFrom": [ | |
| "cxs:axioms/D", | |
| "cxs:axioms/T" | |
| ], | |
| "label": "Octonion Axiom", | |
| "symbol": "O", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 8 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:primitives/operations", | |
| "@type": "OperationsContainer", | |
| "contains": [ | |
| "cxs:primitives/operations/add", | |
| "cxs:primitives/operations/mul", | |
| "cxs:primitives/operations/pow", | |
| "cxs:primitives/operations/div", | |
| "cxs:primitives/operations/sub" | |
| ], | |
| "label": "Primitive Operations" | |
| }, | |
| { | |
| "@id": "cxs:primitives/operations/add", | |
| "@type": "PrimitiveOperation", | |
| "hasSignature": "(Num, Num) → Num", | |
| "label": "Addition", | |
| "symbol": "+" | |
| }, | |
| { | |
| "@id": "cxs:primitives/operations/mul", | |
| "@type": "PrimitiveOperation", | |
| "hasSignature": "(Num, Num) → Num", | |
| "label": "Multiplication", | |
| "symbol": "×" | |
| }, | |
| { | |
| "@id": "cxs:primitives/operations/pow", | |
| "@type": "PrimitiveOperation", | |
| "hasSignature": "(Num, Num) → Num", | |
| "label": "Exponentiation", | |
| "symbol": "^" | |
| }, | |
| { | |
| "@id": "cxs:primitives/operations/div", | |
| "@type": "PrimitiveOperation", | |
| "hasSignature": "(Num, Num) → Num", | |
| "label": "Division", | |
| "symbol": "/" | |
| }, | |
| { | |
| "@id": "cxs:primitives/relations", | |
| "@type": "RelationsContainer", | |
| "contains": [ | |
| "cxs:primitives/relations/eq", | |
| "cxs:primitives/relations/lt", | |
| "cxs:primitives/relations/divides", | |
| "cxs:primitives/relations/coprime", | |
| "cxs:primitives/relations/leq", | |
| "cxs:primitives/relations/incomparable", | |
| "cxs:primitives/relations/antichain" | |
| ], | |
| "label": "Primitive Relations" | |
| }, | |
| { | |
| "@id": "cxs:primitives/relations/eq", | |
| "@type": "PrimitiveRelation", | |
| "arity": 2, | |
| "label": "Equality", | |
| "symbol": "=" | |
| }, | |
| { | |
| "@id": "cxs:primitives/relations/lt", | |
| "@type": "PrimitiveRelation", | |
| "arity": 2, | |
| "label": "Less Than", | |
| "symbol": "<" | |
| }, | |
| { | |
| "@id": "cxs:primitives/relations/divides", | |
| "@type": "PrimitiveRelation", | |
| "arity": 2, | |
| "label": "Divides", | |
| "symbol": "|" | |
| }, | |
| { | |
| "@id": "cxs:primitives/relations/coprime", | |
| "@type": "PrimitiveRelation", | |
| "arity": 2, | |
| "label": "Coprime", | |
| "symbol": "⊥" | |
| }, | |
| { | |
| "@id": "cxs:primitives/relations/leq", | |
| "@type": "PrimitiveRelation", | |
| "arity": 2, | |
| "label": "Less Than or Equal", | |
| "symbol": "≤" | |
| }, | |
| { | |
| "@id": "cxs:primitives/relations/incomparable", | |
| "@type": "PrimitiveRelation", | |
| "arity": 2, | |
| "label": "Incomparable", | |
| "symbol": "∥" | |
| }, | |
| { | |
| "@id": "cxs:primitives/relations/antichain", | |
| "@type": "PrimitiveRelation", | |
| "arity": 2, | |
| "label": "Antichain", | |
| "symbol": "⟂" | |
| }, | |
| { | |
| "@id": "cxs:tower", | |
| "@type": "TowerContainer", | |
| "hasPart": [ | |
| "cxs:tower/levels", | |
| "cxs:tower/transitions", | |
| "cxs:tower/strata" | |
| ], | |
| "label": "Cayley-Dickson Tower", | |
| "partOf": "cxs:CategoricalX" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels", | |
| "@type": "TowerLevelsContainer", | |
| "contains": [ | |
| "cxs:tower/levels/0", | |
| "cxs:tower/levels/1", | |
| "cxs:tower/levels/2", | |
| "cxs:tower/levels/3", | |
| "cxs:tower/levels/4", | |
| "cxs:tower/levels/5", | |
| "cxs:tower/levels/6", | |
| "cxs:tower/levels/7", | |
| "cxs:tower/levels/8" | |
| ], | |
| "label": "Tower Levels" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/0", | |
| "@type": "TowerLevel", | |
| "algebra": "R", | |
| "hasAutomorphismGroup": "cxs:tower/levels/0/automorphismGroup", | |
| "hasDimension": "cxs:dimensions/dim_0", | |
| "label": "Reals", | |
| "levelIndex": 0 | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/1", | |
| "@type": "TowerLevel", | |
| "algebra": "C", | |
| "hasAutomorphismGroup": "cxs:tower/levels/1/automorphismGroup", | |
| "hasDimension": "cxs:dimensions/dim_1", | |
| "label": "Complex", | |
| "levelIndex": 1, | |
| "losesProperty": "cxs:properties/totalOrdering" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/2", | |
| "@type": "TowerLevel", | |
| "algebra": "H", | |
| "hasAutomorphismGroup": "cxs:tower/levels/2/automorphismGroup", | |
| "hasDimension": "cxs:dimensions/dim_2", | |
| "label": "Quaternions", | |
| "levelIndex": 2, | |
| "losesProperty": "cxs:properties/commutativity" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/3", | |
| "@type": "TowerLevel", | |
| "algebra": "O", | |
| "hasAutomorphismGroup": "cxs:tower/levels/3/automorphismGroup", | |
| "hasDimension": "cxs:dimensions/dim_3", | |
| "label": "Octonions", | |
| "levelIndex": 3, | |
| "losesProperty": "cxs:properties/associativity" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/4", | |
| "@type": "TowerLevel", | |
| "algebra": "S", | |
| "hasAutomorphismGroup": "cxs:tower/levels/4/automorphismGroup", | |
| "hasDimension": "cxs:dimensions/dim_4", | |
| "label": "Sedenions", | |
| "levelIndex": 4, | |
| "losesProperty": "cxs:properties/division" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/5", | |
| "@type": "TowerLevel", | |
| "algebra": "P", | |
| "hasAutomorphismGroup": "cxs:tower/levels/5/automorphismGroup", | |
| "hasDimension": "cxs:dimensions/dim_5", | |
| "label": "Pathions", | |
| "levelIndex": 5, | |
| "losesProperty": "cxs:properties/moufang" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/6", | |
| "@type": "TowerLevel", | |
| "algebra": "T", | |
| "hasAutomorphismGroup": "cxs:tower/levels/6/automorphismGroup", | |
| "hasDimension": "cxs:dimensions/dim_6", | |
| "label": "Trigintaduonions", | |
| "levelIndex": 6 | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/7", | |
| "@type": "TowerLevel", | |
| "algebra": "V", | |
| "hasAutomorphismGroup": "cxs:tower/levels/7/automorphismGroup", | |
| "hasDimension": "cxs:dimensions/dim_7", | |
| "label": "Voudons", | |
| "levelIndex": 7 | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/8", | |
| "@type": "TowerLevel", | |
| "algebra": "R", | |
| "hasAutomorphismGroup": "cxs:tower/levels/8/automorphismGroup", | |
| "hasDimension": "cxs:dimensions/dim_8", | |
| "label": "Extended-Reals-8", | |
| "levelIndex": 8 | |
| }, | |
| { | |
| "@id": "cxs:tower/transitions", | |
| "@type": "TransitionsContainer", | |
| "contains": [ | |
| "cxs:tower/transitions/phi_01", | |
| "cxs:tower/transitions/phi_12", | |
| "cxs:tower/transitions/phi_23", | |
| "cxs:tower/transitions/phi_34", | |
| "cxs:tower/transitions/phi_45", | |
| "cxs:tower/transitions/phi_56", | |
| "cxs:tower/transitions/phi_67", | |
| "cxs:tower/transitions/phi_78" | |
| ], | |
| "label": "Tower Transitions" | |
| }, | |
| { | |
| "@id": "cxs:tower/transitions/phi_01", | |
| "@type": "TowerTransition", | |
| "formula": "(a, b) ↦ a + bi", | |
| "label": "phi_01", | |
| "sourceLevel": "cxs:tower/levels/0", | |
| "targetLevel": "cxs:tower/levels/1" | |
| }, | |
| { | |
| "@id": "cxs:tower/transitions/phi_12", | |
| "@type": "TowerTransition", | |
| "formula": "(z, w) ↦ z + wj", | |
| "label": "phi_12", | |
| "sourceLevel": "cxs:tower/levels/1", | |
| "targetLevel": "cxs:tower/levels/2" | |
| }, | |
| { | |
| "@id": "cxs:tower/transitions/phi_23", | |
| "@type": "TowerTransition", | |
| "formula": "(q₁, q₂) ↦ q₁ + q₂e", | |
| "label": "phi_23", | |
| "sourceLevel": "cxs:tower/levels/2", | |
| "targetLevel": "cxs:tower/levels/3" | |
| }, | |
| { | |
| "@id": "cxs:tower/transitions/phi_34", | |
| "@type": "TowerTransition", | |
| "formula": "Cayley-Dickson(O)", | |
| "label": "phi_34", | |
| "sourceLevel": "cxs:tower/levels/3", | |
| "targetLevel": "cxs:tower/levels/4" | |
| }, | |
| { | |
| "@id": "cxs:tower/transitions/phi_45", | |
| "@type": "TowerTransition", | |
| "formula": "Cayley-Dickson(S)", | |
| "label": "phi_45", | |
| "sourceLevel": "cxs:tower/levels/4", | |
| "targetLevel": "cxs:tower/levels/5" | |
| }, | |
| { | |
| "@id": "cxs:tower/transitions/phi_56", | |
| "@type": "TowerTransition", | |
| "formula": "Cayley-Dickson(P)", | |
| "label": "phi_56", | |
| "sourceLevel": "cxs:tower/levels/5", | |
| "targetLevel": "cxs:tower/levels/6" | |
| }, | |
| { | |
| "@id": "cxs:tower/transitions/phi_67", | |
| "@type": "TowerTransition", | |
| "formula": "Cayley-Dickson(T)", | |
| "label": "phi_67", | |
| "sourceLevel": "cxs:tower/levels/6", | |
| "targetLevel": "cxs:tower/levels/7" | |
| }, | |
| { | |
| "@id": "cxs:tower/transitions/phi_78", | |
| "@type": "TowerTransition", | |
| "formula": "Cayley-Dickson(V)", | |
| "label": "phi_78", | |
| "sourceLevel": "cxs:tower/levels/7", | |
| "targetLevel": "cxs:tower/levels/8" | |
| }, | |
| { | |
| "@id": "cxs:tower/strata", | |
| "@type": "StrataContainer", | |
| "contains": [ | |
| "cxs:tower/strata/Extension", | |
| "cxs:tower/strata/Intension", | |
| "cxs:tower/strata/Comprehension", | |
| "cxs:tower/strata/Ground" | |
| ], | |
| "label": "Strata" | |
| }, | |
| { | |
| "@id": "cxs:tower/strata/Extension", | |
| "@type": "Stratum", | |
| "contains": [ | |
| "cxs:tower/levels/0", | |
| "cxs:tower/levels/1", | |
| "cxs:tower/levels/2" | |
| ], | |
| "label": "Extension", | |
| "role": "Classical algebras with full properties" | |
| }, | |
| { | |
| "@id": "cxs:tower/strata/Intension", | |
| "@type": "Stratum", | |
| "contains": [ | |
| "cxs:tower/levels/3", | |
| "cxs:tower/levels/4" | |
| ], | |
| "label": "Intension", | |
| "role": "Non-associative regime" | |
| }, | |
| { | |
| "@id": "cxs:tower/strata/Comprehension", | |
| "@type": "Stratum", | |
| "contains": [ | |
| "cxs:tower/levels/5", | |
| "cxs:tower/levels/6", | |
| "cxs:tower/levels/7" | |
| ], | |
| "label": "Comprehension", | |
| "role": "Higher obstruction regime" | |
| }, | |
| { | |
| "@id": "cxs:tower/strata/Ground", | |
| "@type": "Stratum", | |
| "contains": [ | |
| "cxs:tower/levels/8" | |
| ], | |
| "label": "Ground", | |
| "role": "Terminal fixed point" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_0", | |
| "@type": "DimensionSpec", | |
| "dimension": 1, | |
| "formula": "2^0" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_1", | |
| "@type": "DimensionSpec", | |
| "dimension": 2, | |
| "formula": "2^1" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_2", | |
| "@type": "DimensionSpec", | |
| "dimension": 4, | |
| "formula": "2^2" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_3", | |
| "@type": "DimensionSpec", | |
| "dimension": 8, | |
| "formula": "2^3" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_4", | |
| "@type": "DimensionSpec", | |
| "dimension": 16, | |
| "formula": "2^4" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_5", | |
| "@type": "DimensionSpec", | |
| "dimension": 32, | |
| "formula": "2^5" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_6", | |
| "@type": "DimensionSpec", | |
| "dimension": 64, | |
| "formula": "2^6" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_7", | |
| "@type": "DimensionSpec", | |
| "dimension": 128, | |
| "formula": "2^7" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_8", | |
| "@type": "DimensionSpec", | |
| "dimension": 256, | |
| "formula": "2^8" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_aut_0", | |
| "@type": "DimensionSpec", | |
| "dimension": 0, | |
| "formula": "|Aut(R)|" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_aut_1", | |
| "@type": "DimensionSpec", | |
| "dimension": 0, | |
| "formula": "|Aut(C)|" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_aut_2", | |
| "@type": "DimensionSpec", | |
| "dimension": 3, | |
| "formula": "dim(SO(3))" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_aut_3", | |
| "@type": "DimensionSpec", | |
| "dimension": 14, | |
| "formula": "dim(G₂)" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_aut_4", | |
| "@type": "DimensionSpec", | |
| "dimension": 21, | |
| "formula": "dim(Spin(7))" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_leech", | |
| "@type": "DimensionSpec", | |
| "dimension": 24, | |
| "formula": "T × O" | |
| }, | |
| { | |
| "@id": "cxs:dimensions/dim_e8", | |
| "@type": "DimensionSpec", | |
| "dimension": 248, | |
| "formula": "dim(E₈)" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/0/automorphismGroup", | |
| "@type": [ | |
| "AutomorphismGroup", | |
| "Trivial" | |
| ], | |
| "dimension": 0, | |
| "groupType": "Trivial", | |
| "label": "{1}", | |
| "order": 1 | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/1/automorphismGroup", | |
| "@type": [ | |
| "AutomorphismGroup", | |
| "FiniteGroup" | |
| ], | |
| "dimension": 0, | |
| "groupType": "FiniteGroup", | |
| "label": "Z_2", | |
| "order": 2 | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/2/automorphismGroup", | |
| "@type": [ | |
| "AutomorphismGroup", | |
| "LieGroup" | |
| ], | |
| "dimension": 3, | |
| "groupType": "LieGroup", | |
| "label": "SO(3)" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/3/automorphismGroup", | |
| "@type": [ | |
| "AutomorphismGroup", | |
| "LieGroup" | |
| ], | |
| "dimension": 14, | |
| "groupType": "LieGroup", | |
| "label": "G_2" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/4/automorphismGroup", | |
| "@type": [ | |
| "AutomorphismGroup", | |
| "LieGroup" | |
| ], | |
| "dimension": 21, | |
| "groupType": "LieGroup", | |
| "label": "Spin(7) ⋊ S₃" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/5/automorphismGroup", | |
| "@type": [ | |
| "AutomorphismGroup", | |
| "ExtendedLieGroup" | |
| ], | |
| "dimension": 28, | |
| "groupType": "ExtendedLieGroup", | |
| "label": "Extended-Spin(8)-5" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/6/automorphismGroup", | |
| "@type": [ | |
| "AutomorphismGroup", | |
| "ExtendedLieGroup" | |
| ], | |
| "dimension": 35, | |
| "groupType": "ExtendedLieGroup", | |
| "label": "Extended-Spin(8)-6" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/7/automorphismGroup", | |
| "@type": [ | |
| "AutomorphismGroup", | |
| "ExtendedLieGroup" | |
| ], | |
| "dimension": 42, | |
| "groupType": "ExtendedLieGroup", | |
| "label": "Extended-Spin(8)-7" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/8/automorphismGroup", | |
| "@type": [ | |
| "AutomorphismGroup", | |
| "ExtendedLieGroup" | |
| ], | |
| "dimension": 49, | |
| "groupType": "ExtendedLieGroup", | |
| "label": "Octave-Complete-Spin(8)" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/0/cocycle", | |
| "@type": "CocycleClass", | |
| "arity": 0, | |
| "cohomologyClass": "H⁰(1, k)", | |
| "comment": "Trivial cocycle", | |
| "label": "H⁰(1, k)", | |
| "representative": "1" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/1/cocycle", | |
| "@type": "CocycleClass", | |
| "arity": 1, | |
| "cohomologyClass": "H¹(Z₂, k×)", | |
| "comment": "Conjugation cocycle", | |
| "label": "H¹(Z₂, k×)", | |
| "representative": "γ(a) = a*" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/2/cocycle", | |
| "@type": "CocycleClass", | |
| "arity": 2, | |
| "cohomologyClass": "H²(V₄, k×)", | |
| "comment": "Commutator cocycle", | |
| "label": "H²(V₄, k×)", | |
| "representative": "[a,b] = ab - ba" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/3/cocycle", | |
| "@type": "CocycleClass", | |
| "arity": 3, | |
| "cohomologyClass": "H³(Z₂³, k×)", | |
| "comment": "Associator cocycle (Fano plane)", | |
| "label": "H³(Z₂³, k×)", | |
| "representative": "[a,b,c] = (ab)c - a(bc)" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/4/cocycle", | |
| "@type": "CocycleClass", | |
| "arity": 4, | |
| "cohomologyClass": "H⁴(Z₂³, k×)", | |
| "comment": "Zero-divisor cocycle", | |
| "label": "H⁴(Z₂³, k×)", | |
| "representative": "Z(a,b,c,d)" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/5/cocycle", | |
| "@type": "CocycleClass", | |
| "arity": 5, | |
| "cohomologyClass": "H⁵(Z₂⁴, k×)", | |
| "comment": "Moufang failure cocycle", | |
| "label": "H⁵(Z₂⁴, k×)", | |
| "representative": "Obs₅(a,b,c,d,e)" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/6/cocycle", | |
| "@type": "CocycleClass", | |
| "arity": 6, | |
| "cohomologyClass": "H⁶(Z₂⁵, k×)", | |
| "comment": "6-ary obstruction cocycle", | |
| "label": "H⁶(Z₂⁵, k×)", | |
| "representative": "Obs₆" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/7/cocycle", | |
| "@type": "CocycleClass", | |
| "arity": 7, | |
| "cohomologyClass": "H⁷(Z₂⁶, k×)", | |
| "comment": "7-ary obstruction cocycle", | |
| "label": "H⁷(Z₂⁶, k×)", | |
| "representative": "Obs₇" | |
| }, | |
| { | |
| "@id": "cxs:tower/levels/8/cocycle", | |
| "@type": "CocycleClass", | |
| "arity": 8, | |
| "cohomologyClass": "H⁸(Z₂⁷, k×)", | |
| "comment": "Octave cocycle (Bott periodicity)", | |
| "label": "H⁸(Z₂⁷, k×)", | |
| "representative": "β_O" | |
| }, | |
| { | |
| "@id": "cxs:properties/totalOrdering", | |
| "@type": "AlgebraicProperty", | |
| "comment": "Elements can be totally ordered", | |
| "label": "Total Ordering" | |
| }, | |
| { | |
| "@id": "cxs:properties/commutativity", | |
| "@type": "AlgebraicProperty", | |
| "comment": "Multiplication is commutative", | |
| "label": "Commutativity" | |
| }, | |
| { | |
| "@id": "cxs:properties/associativity", | |
| "@type": "AlgebraicProperty", | |
| "comment": "Multiplication is associative", | |
| "label": "Associativity" | |
| }, | |
| { | |
| "@id": "cxs:properties/alternativity", | |
| "@type": "AlgebraicProperty", | |
| "comment": "Algebra is alternative", | |
| "label": "Alternativity" | |
| }, | |
| { | |
| "@id": "cxs:properties/powerAssociativity", | |
| "@type": "AlgebraicProperty", | |
| "comment": "Powers associate", | |
| "label": "Power Associativity" | |
| }, | |
| { | |
| "@id": "cxs:properties/flexibility", | |
| "@type": "AlgebraicProperty", | |
| "comment": "Flexible identity holds", | |
| "label": "Flexibility" | |
| }, | |
| { | |
| "@id": "cxs:properties/zeroDivisorFreeness", | |
| "@type": "AlgebraicProperty", | |
| "comment": "No zero divisors", | |
| "label": "Zero-Divisor Freeness" | |
| }, | |
| { | |
| "@id": "cxs:operators", | |
| "@type": "OperatorsContainer", | |
| "comment": "Categorical operators defining morphisms in Categorical X", | |
| "contains": [ | |
| "cxs:operators/tensor", | |
| "cxs:operators/hom", | |
| "cxs:operators/aut", | |
| "cxs:operators/realization", | |
| "cxs:operators/equivariantTensor", | |
| "cxs:operators/derivation", | |
| "cxs:operators/complexity", | |
| "cxs:operators/spectrum", | |
| "cxs:operators/power", | |
| "cxs:operators/compose", | |
| "cxs:operators/limit", | |
| "cxs:operators/colimit", | |
| "cxs:operators/measure" | |
| ], | |
| "label": "Operators", | |
| "partOf": "cxs:CategoricalX" | |
| }, | |
| { | |
| "@id": "cxs:operators/tensor", | |
| "@type": "Operator", | |
| "hasSignature": "(A, B) → A ⊗ B", | |
| "hasUniversalProperty": "Bilinear maps factor uniquely", | |
| "label": "Tensor Product", | |
| "symbol": "⊗" | |
| }, | |
| { | |
| "@id": "cxs:operators/hom", | |
| "@type": "Operator", | |
| "hasSignature": "(A, B) → Hom(A, B)", | |
| "hasUniversalProperty": "Internal hom adjoint to tensor", | |
| "label": "Hom Functor", | |
| "symbol": "Hom" | |
| }, | |
| { | |
| "@id": "cxs:operators/aut", | |
| "@type": "Operator", | |
| "hasSignature": "A → Aut(A)", | |
| "hasUniversalProperty": "Symmetry group of structure", | |
| "label": "Automorphism Functor", | |
| "symbol": "Aut" | |
| }, | |
| { | |
| "@id": "cxs:operators/realization", | |
| "@type": "Operator", | |
| "hasSignature": "S → |S|", | |
| "hasUniversalProperty": "Topological realization of simplicial object", | |
| "label": "Geometric Realization", | |
| "symbol": "|-|" | |
| }, | |
| { | |
| "@id": "cxs:operators/equivariantTensor", | |
| "@type": "Operator", | |
| "hasSignature": "(A, B) →_G A ⊗_G B", | |
| "hasUniversalProperty": "G-equivariant bilinear maps", | |
| "label": "Equivariant Tensor", | |
| "symbol": "⊗_G" | |
| }, | |
| { | |
| "@id": "cxs:operators/derivation", | |
| "@type": "Operator", | |
| "hasSignature": "A → Der(A)", | |
| "hasUniversalProperty": "Leibniz rule satisfying maps", | |
| "label": "Derivation Algebra", | |
| "symbol": "Der" | |
| }, | |
| { | |
| "@id": "cxs:operators/complexity", | |
| "@type": "Operator", | |
| "hasSignature": "X → K(X)", | |
| "hasUniversalProperty": "Computational complexity measure", | |
| "label": "Complexity Functor", | |
| "symbol": "K" | |
| }, | |
| { | |
| "@id": "cxs:operators/spectrum", | |
| "@type": "Operator", | |
| "hasSignature": "R → Spec(R)", | |
| "hasUniversalProperty": "Prime spectrum of ring", | |
| "label": "Spectrum Functor", | |
| "symbol": "Spec" | |
| }, | |
| { | |
| "@id": "cxs:operators/power", | |
| "@type": "Operator", | |
| "hasSignature": "X → P(X)", | |
| "hasUniversalProperty": "Subobject classifier exponential", | |
| "label": "Power Object", | |
| "symbol": "P" | |
| }, | |
| { | |
| "@id": "cxs:operators/compose", | |
| "@type": "Operator", | |
| "hasSignature": "(f, g) → f ∘ g", | |
| "hasUniversalProperty": "Associative composition of morphisms", | |
| "label": "Composition", | |
| "symbol": "∘" | |
| }, | |
| { | |
| "@id": "cxs:operators/limit", | |
| "@type": "Operator", | |
| "hasSignature": "D → lim D", | |
| "hasUniversalProperty": "Universal cone over diagram", | |
| "label": "Limit", | |
| "symbol": "lim" | |
| }, | |
| { | |
| "@id": "cxs:operators/colimit", | |
| "@type": "Operator", | |
| "hasSignature": "D → colim D", | |
| "hasUniversalProperty": "Universal cocone under diagram", | |
| "label": "Colimit", | |
| "symbol": "colim" | |
| }, | |
| { | |
| "@id": "cxs:operators/measure", | |
| "@type": "Operator", | |
| "hasSignature": "X → μ(X)", | |
| "hasUniversalProperty": "Integration over fibers", | |
| "label": "Measure", | |
| "symbol": "μ" | |
| }, | |
| { | |
| "@id": "cxs:projections", | |
| "@type": "ProjectionsContainer", | |
| "comment": "Interpretation functors that compose with gauge projections", | |
| "contains": [ | |
| "cxs:projections/arithmetic", | |
| "cxs:projections/combinatorial", | |
| "cxs:projections/spectral", | |
| "cxs:projections/modular" | |
| ], | |
| "label": "Domain Projections", | |
| "partOf": "cxs:CategoricalX" | |
| }, | |
| { | |
| "@id": "cxs:projections/arithmetic", | |
| "@type": "Projection", | |
| "comment": "Projects to number-theoretic structures (primes, divisibility)", | |
| "compositionLaw": "π_k ∘ arithmetic = gauge-truncated prime distribution", | |
| "functorType": "Forgetful", | |
| "label": "Arithmetic Projection", | |
| "phaseMap": "Level → Prime counting regime", | |
| "resolutionRules": "Apply prime distribution formulas", | |
| "targetCategory": "Number Theory" | |
| }, | |
| { | |
| "@id": "cxs:projections/combinatorial", | |
| "@type": "Projection", | |
| "comment": "Projects to combinatorial structures (lattices, antichains)", | |
| "compositionLaw": "π_k ∘ combinatorial = gauge-truncated antichain counting", | |
| "functorType": "Representable", | |
| "label": "Combinatorial Projection", | |
| "phaseMap": "Level → Antichain counting regime", | |
| "resolutionRules": "Apply Dedekind enumeration", | |
| "targetCategory": "Combinatorics" | |
| }, | |
| { | |
| "@id": "cxs:projections/spectral", | |
| "@type": "Projection", | |
| "comment": "Projects to spectral/analytic structures (zeta zeros, L-functions)", | |
| "compositionLaw": "π_k ∘ spectral = gauge-truncated eigenvalue sums", | |
| "functorType": "Contravariant", | |
| "label": "Spectral Projection", | |
| "phaseMap": "Level → Spectral density regime", | |
| "resolutionRules": "Apply spectral analysis methods", | |
| "targetCategory": "Spectral Theory" | |
| }, | |
| { | |
| "@id": "cxs:projections/modular", | |
| "@type": "Projection", | |
| "comment": "Projects to modular arithmetic structures", | |
| "compositionLaw": "π_k ∘ modular = gauge-truncated modular forms", | |
| "functorType": "Quotient", | |
| "label": "Modular Projection", | |
| "phaseMap": "Level → Modular residue regime", | |
| "resolutionRules": "Apply modular arithmetic rules", | |
| "targetCategory": "Modular Forms" | |
| }, | |
| { | |
| "@id": "cxs:correspondences", | |
| "@type": "CorrespondencesContainer", | |
| "comment": "Functorial relationships between projections with explicit formulas", | |
| "contains": [ | |
| "cxs:correspondences/dedekindPrime", | |
| "cxs:correspondences/spectralArithmetic", | |
| "cxs:correspondences/modularCombinatorial" | |
| ], | |
| "label": "Correspondences", | |
| "partOf": "cxs:CategoricalX" | |
| }, | |
| { | |
| "@id": "cxs:correspondences/dedekindPrime", | |
| "@type": "Correspondence", | |
| "comment": "Boolean lattice ↔ Divisibility lattice isomorphism", | |
| "correspondsTo": [ | |
| "cxs:projections/combinatorial", | |
| "cxs:projections/arithmetic" | |
| ], | |
| "explicitFormula": "D(n) = |{A ⊆ P^n : antichain(A)}| corresponds to prime-indexed lattice counting", | |
| "inputDomain": "Boolean lattice antichains over P^n", | |
| "label": "Dedekind-Prime Correspondence", | |
| "outputCodomain": "Prime-indexed divisibility lattice elements", | |
| "phaseAlignment": "cxs:phases/phase1", | |
| "truncationPolicy": "gauge-bounded: use gauge level k for D(n) where n ≤ coverage(k)" | |
| }, | |
| { | |
| "@id": "cxs:correspondences/spectralArithmetic", | |
| "@type": "Correspondence", | |
| "comment": "Zeta zeros ↔ Prime distribution connection", | |
| "correspondsTo": [ | |
| "cxs:projections/spectral", | |
| "cxs:projections/arithmetic" | |
| ], | |
| "explicitFormula": "ψ(x) = x - Σ_ρ x^ρ/ρ + O(1) (explicit formula for Chebyshev ψ)", | |
| "inputDomain": "Zeta zero ordinates γ_n with |γ_n| ≤ T", | |
| "label": "Spectral-Arithmetic Correspondence", | |
| "outputCodomain": "Prime counting approximation π_spec(x)", | |
| "phaseAlignment": "cxs:phases/phase3", | |
| "truncationPolicy": "cutoff-T: sum over zeros with |γ| ≤ T, error O(x/T log²x)" | |
| }, | |
| { | |
| "@id": "cxs:correspondences/modularCombinatorial", | |
| "@type": "Correspondence", | |
| "comment": "Modular forms ↔ Lattice structures", | |
| "correspondsTo": [ | |
| "cxs:projections/modular", | |
| "cxs:projections/combinatorial" | |
| ], | |
| "explicitFormula": "θ_Λ(q) = Σ_v q^{|v|²/2} encodes lattice theta series as modular form", | |
| "inputDomain": "Modular forms of level N and weight k", | |
| "label": "Modular-Combinatorial Correspondence", | |
| "outputCodomain": "Lattice theta series coefficients", | |
| "phaseAlignment": "cxs:phases/phase4", | |
| "truncationPolicy": "q-expansion: truncate at q^M for precision M" | |
| }, | |
| { | |
| "@id": "cxs:phases", | |
| "@type": "PhasesContainer", | |
| "hasPart": [ | |
| "cxs:phases/transitions", | |
| "cxs:phases/behaviors", | |
| "cxs:phases/modifications" | |
| ], | |
| "label": "Phase System" | |
| }, | |
| { | |
| "@id": "cxs:phases/phase1", | |
| "@type": "PhaseBehavior", | |
| "activatesAt": "cxs:tower/levels/0", | |
| "activatesAtIndex": 0, | |
| "formula": "f(n,k) = 2^(n-k) × g(n,k)", | |
| "label": "Phase I", | |
| "range": "n ≤ pentality" | |
| }, | |
| { | |
| "@id": "cxs:phases/phase2", | |
| "@type": "PhaseBehavior", | |
| "activatesAt": "cxs:tower/levels/5", | |
| "activatesAtIndex": 5, | |
| "formula": "Polynomial growth with triality corrections", | |
| "label": "Phase II", | |
| "range": "n = pariah" | |
| }, | |
| { | |
| "@id": "cxs:phases/phase3", | |
| "@type": "PhaseBehavior", | |
| "activatesAt": "cxs:tower/levels/6", | |
| "activatesAtIndex": 6, | |
| "formula": "f(septality, k) = D(pentality)^T", | |
| "label": "Phase III", | |
| "range": "n = septality" | |
| }, | |
| { | |
| "@id": "cxs:phases/phase4", | |
| "@type": "PhaseBehavior", | |
| "activatesAt": "cxs:tower/levels/7", | |
| "activatesAtIndex": 7, | |
| "formula": "Octave periodic with correction C(n,2)", | |
| "label": "Phase IV", | |
| "range": "n ≥ O" | |
| }, | |
| { | |
| "@id": "cxs:phases/transitions/transition_I_II", | |
| "@type": "PhaseTransition", | |
| "criticalConstant": "pentality", | |
| "label": "Phase I → Phase II", | |
| "sourcePhase": "Phase I", | |
| "targetPhase": "Phase II", | |
| "transitionLevel": 5 | |
| }, | |
| { | |
| "@id": "cxs:phases/transitions/transition_II_III", | |
| "@type": "PhaseTransition", | |
| "criticalConstant": "pariah", | |
| "label": "Phase II → Phase III", | |
| "sourcePhase": "Phase II", | |
| "targetPhase": "Phase III", | |
| "transitionLevel": 6 | |
| }, | |
| { | |
| "@id": "cxs:phases/transitions/transition_III_IV", | |
| "@type": "PhaseTransition", | |
| "criticalConstant": "septality", | |
| "label": "Phase III → Phase IV", | |
| "sourcePhase": "Phase III", | |
| "targetPhase": "Phase IV", | |
| "transitionLevel": 7 | |
| }, | |
| { | |
| "@id": "cxs:phases/transitions/transition_IV_periodic", | |
| "@type": "PhaseTransition", | |
| "criticalConstant": "O", | |
| "label": "Phase IV → Periodic", | |
| "sourcePhase": "Phase IV", | |
| "targetPhase": "Periodic", | |
| "transitionLevel": 8 | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/tensor_level3", | |
| "@type": "PhaseModification", | |
| "formula": "Non-associative tensor product (requires re-bracketing)", | |
| "label": "tensor_level3", | |
| "operator": "cxs:operators/tensor", | |
| "phase": "Octonion", | |
| "targetLevel": "cxs:tower/levels/3" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/tensor_level4", | |
| "@type": "PhaseModification", | |
| "formula": "Zero-divisor propagation", | |
| "label": "tensor_level4", | |
| "operator": "cxs:operators/tensor", | |
| "phase": "Sedenion", | |
| "targetLevel": "cxs:tower/levels/4" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/tensor_level5", | |
| "@type": "PhaseModification", | |
| "formula": "Loss of alternative law in tensor factors", | |
| "label": "tensor_level5", | |
| "operator": "cxs:operators/tensor", | |
| "phase": "Pathion", | |
| "targetLevel": "cxs:tower/levels/5" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/hom_level3", | |
| "@type": "PhaseModification", | |
| "formula": "Restricted to G2-invariant maps", | |
| "label": "hom_level3", | |
| "operator": "cxs:operators/hom", | |
| "phase": "Octonion", | |
| "targetLevel": "cxs:tower/levels/3" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/hom_level4", | |
| "@type": "PhaseModification", | |
| "formula": "Spin(7) equivariance required", | |
| "label": "hom_level4", | |
| "operator": "cxs:operators/hom", | |
| "phase": "Sedenion", | |
| "targetLevel": "cxs:tower/levels/4" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/hom_level5", | |
| "@type": "PhaseModification", | |
| "formula": "Extended Spin(8) equivariance", | |
| "label": "hom_level5", | |
| "operator": "cxs:operators/hom", | |
| "phase": "Pathion", | |
| "targetLevel": "cxs:tower/levels/5" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/aut_level3", | |
| "@type": "PhaseModification", | |
| "formula": "G2 automorphism group (14-dimensional)", | |
| "label": "aut_level3", | |
| "operator": "cxs:operators/aut", | |
| "phase": "Octonion", | |
| "targetLevel": "cxs:tower/levels/3" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/aut_level4", | |
| "@type": "PhaseModification", | |
| "formula": "Spin(7) ⋊ S₃ (21-dimensional)", | |
| "label": "aut_level4", | |
| "operator": "cxs:operators/aut", | |
| "phase": "Sedenion", | |
| "targetLevel": "cxs:tower/levels/4" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/aut_level5", | |
| "@type": "PhaseModification", | |
| "formula": "Extended-Spin(8)-5 (28-dimensional)", | |
| "label": "aut_level5", | |
| "operator": "cxs:operators/aut", | |
| "phase": "Pathion", | |
| "targetLevel": "cxs:tower/levels/5" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/realization_level3", | |
| "@type": "PhaseModification", | |
| "formula": "Non-associative geometric realization", | |
| "label": "realization_level3", | |
| "operator": "cxs:operators/realization", | |
| "phase": "Octonion", | |
| "targetLevel": "cxs:tower/levels/3" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/realization_level4", | |
| "@type": "PhaseModification", | |
| "formula": "Zero-divisor aware realization", | |
| "label": "realization_level4", | |
| "operator": "cxs:operators/realization", | |
| "phase": "Sedenion", | |
| "targetLevel": "cxs:tower/levels/4" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/realization_level5", | |
| "@type": "PhaseModification", | |
| "formula": "Power-associativity boundary", | |
| "label": "realization_level5", | |
| "operator": "cxs:operators/realization", | |
| "phase": "Pathion", | |
| "targetLevel": "cxs:tower/levels/5" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/equivariantTensor_level3", | |
| "@type": "PhaseModification", | |
| "formula": "G2-equivariant tensor", | |
| "label": "equivariantTensor_level3", | |
| "operator": "cxs:operators/equivariantTensor", | |
| "phase": "Octonion", | |
| "targetLevel": "cxs:tower/levels/3" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/equivariantTensor_level4", | |
| "@type": "PhaseModification", | |
| "formula": "Spin(7)-equivariant tensor", | |
| "label": "equivariantTensor_level4", | |
| "operator": "cxs:operators/equivariantTensor", | |
| "phase": "Sedenion", | |
| "targetLevel": "cxs:tower/levels/4" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/equivariantTensor_level5", | |
| "@type": "PhaseModification", | |
| "formula": "Extended-Spin(8)-equivariant", | |
| "label": "equivariantTensor_level5", | |
| "operator": "cxs:operators/equivariantTensor", | |
| "phase": "Pathion", | |
| "targetLevel": "cxs:tower/levels/5" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/derivation_level3", | |
| "@type": "PhaseModification", | |
| "formula": "Non-associative derivations (g2 Lie algebra)", | |
| "label": "derivation_level3", | |
| "operator": "cxs:operators/derivation", | |
| "phase": "Octonion", | |
| "targetLevel": "cxs:tower/levels/3" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/derivation_level4", | |
| "@type": "PhaseModification", | |
| "formula": "Extended derivation algebra", | |
| "label": "derivation_level4", | |
| "operator": "cxs:operators/derivation", | |
| "phase": "Sedenion", | |
| "targetLevel": "cxs:tower/levels/4" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/derivation_level5", | |
| "@type": "PhaseModification", | |
| "formula": "Higher obstruction derivations", | |
| "label": "derivation_level5", | |
| "operator": "cxs:operators/derivation", | |
| "phase": "Pathion", | |
| "targetLevel": "cxs:tower/levels/5" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/complexity_level3", | |
| "@type": "PhaseModification", | |
| "formula": "Polynomial with associator correction", | |
| "label": "complexity_level3", | |
| "operator": "cxs:operators/complexity", | |
| "phase": "Octonion", | |
| "targetLevel": "cxs:tower/levels/3" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/complexity_level4", | |
| "@type": "PhaseModification", | |
| "formula": "Super-polynomial growth", | |
| "label": "complexity_level4", | |
| "operator": "cxs:operators/complexity", | |
| "phase": "Sedenion", | |
| "targetLevel": "cxs:tower/levels/4" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/complexity_level5", | |
| "@type": "PhaseModification", | |
| "formula": "Exponential-bounded complexity", | |
| "label": "complexity_level5", | |
| "operator": "cxs:operators/complexity", | |
| "phase": "Pathion", | |
| "targetLevel": "cxs:tower/levels/5" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/spectrum_level3", | |
| "@type": "PhaseModification", | |
| "formula": "Non-commutative spectrum", | |
| "label": "spectrum_level3", | |
| "operator": "cxs:operators/spectrum", | |
| "phase": "Octonion", | |
| "targetLevel": "cxs:tower/levels/3" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/spectrum_level4", | |
| "@type": "PhaseModification", | |
| "formula": "Zero-divisor spectrum", | |
| "label": "spectrum_level4", | |
| "operator": "cxs:operators/spectrum", | |
| "phase": "Sedenion", | |
| "targetLevel": "cxs:tower/levels/4" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/spectrum_level5", | |
| "@type": "PhaseModification", | |
| "formula": "Power-associativity spectrum", | |
| "label": "spectrum_level5", | |
| "operator": "cxs:operators/spectrum", | |
| "phase": "Pathion", | |
| "targetLevel": "cxs:tower/levels/5" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/power_level3", | |
| "@type": "PhaseModification", | |
| "formula": "Power object with G2 action", | |
| "label": "power_level3", | |
| "operator": "cxs:operators/power", | |
| "phase": "Octonion", | |
| "targetLevel": "cxs:tower/levels/3" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/power_level4", | |
| "@type": "PhaseModification", | |
| "formula": "Zero-divisor quotient", | |
| "label": "power_level4", | |
| "operator": "cxs:operators/power", | |
| "phase": "Sedenion", | |
| "targetLevel": "cxs:tower/levels/4" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/power_level5", | |
| "@type": "PhaseModification", | |
| "formula": "Triality-corrected power", | |
| "label": "power_level5", | |
| "operator": "cxs:operators/power", | |
| "phase": "Pathion", | |
| "targetLevel": "cxs:tower/levels/5" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/compose_level3", | |
| "@type": "PhaseModification", | |
| "formula": "Non-associative composition (re-bracketing)", | |
| "label": "compose_level3", | |
| "operator": "cxs:operators/compose", | |
| "phase": "Octonion", | |
| "targetLevel": "cxs:tower/levels/3" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/compose_level4", | |
| "@type": "PhaseModification", | |
| "formula": "Zero-divisor careful composition", | |
| "label": "compose_level4", | |
| "operator": "cxs:operators/compose", | |
| "phase": "Sedenion", | |
| "targetLevel": "cxs:tower/levels/4" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/compose_level5", | |
| "@type": "PhaseModification", | |
| "formula": "Power-associative composition", | |
| "label": "compose_level5", | |
| "operator": "cxs:operators/compose", | |
| "phase": "Pathion", | |
| "targetLevel": "cxs:tower/levels/5" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/limit_level3", | |
| "@type": "PhaseModification", | |
| "formula": "Non-associative cone", | |
| "label": "limit_level3", | |
| "operator": "cxs:operators/limit", | |
| "phase": "Octonion", | |
| "targetLevel": "cxs:tower/levels/3" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/limit_level4", | |
| "@type": "PhaseModification", | |
| "formula": "Zero-divisor filtered limit", | |
| "label": "limit_level4", | |
| "operator": "cxs:operators/limit", | |
| "phase": "Sedenion", | |
| "targetLevel": "cxs:tower/levels/4" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/limit_level5", | |
| "@type": "PhaseModification", | |
| "formula": "Obstruction-controlled limit", | |
| "label": "limit_level5", | |
| "operator": "cxs:operators/limit", | |
| "phase": "Pathion", | |
| "targetLevel": "cxs:tower/levels/5" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/colimit_level3", | |
| "@type": "PhaseModification", | |
| "formula": "Non-associative cocone", | |
| "label": "colimit_level3", | |
| "operator": "cxs:operators/colimit", | |
| "phase": "Octonion", | |
| "targetLevel": "cxs:tower/levels/3" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/colimit_level4", | |
| "@type": "PhaseModification", | |
| "formula": "Zero-divisor filtered colimit", | |
| "label": "colimit_level4", | |
| "operator": "cxs:operators/colimit", | |
| "phase": "Sedenion", | |
| "targetLevel": "cxs:tower/levels/4" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/colimit_level5", | |
| "@type": "PhaseModification", | |
| "formula": "Obstruction-controlled colimit", | |
| "label": "colimit_level5", | |
| "operator": "cxs:operators/colimit", | |
| "phase": "Pathion", | |
| "targetLevel": "cxs:tower/levels/5" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/measure_level3", | |
| "@type": "PhaseModification", | |
| "formula": "G2-invariant measure", | |
| "label": "measure_level3", | |
| "operator": "cxs:operators/measure", | |
| "phase": "Octonion", | |
| "targetLevel": "cxs:tower/levels/3" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/measure_level4", | |
| "@type": "PhaseModification", | |
| "formula": "Spin(7)-invariant measure", | |
| "label": "measure_level4", | |
| "operator": "cxs:operators/measure", | |
| "phase": "Sedenion", | |
| "targetLevel": "cxs:tower/levels/4" | |
| }, | |
| { | |
| "@id": "cxs:phases/modifications/measure_level5", | |
| "@type": "PhaseModification", | |
| "formula": "Triality measure", | |
| "label": "measure_level5", | |
| "operator": "cxs:operators/measure", | |
| "phase": "Pathion", | |
| "targetLevel": "cxs:tower/levels/5" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/arithmetic_I", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 0, | |
| "targetPhase": "Phase I" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/arithmetic_II", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 5, | |
| "targetPhase": "Phase II" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/arithmetic_III", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 6, | |
| "targetPhase": "Phase III" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/arithmetic_IV", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 7, | |
| "targetPhase": "Phase IV" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/combinatorial_I", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 0, | |
| "targetPhase": "Phase I" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/combinatorial_II", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 5, | |
| "targetPhase": "Phase II" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/combinatorial_III", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 6, | |
| "targetPhase": "Phase III" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/combinatorial_IV", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 7, | |
| "targetPhase": "Phase IV" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/spectral_I", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 0, | |
| "targetPhase": "Phase I" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/spectral_II", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 5, | |
| "targetPhase": "Phase II" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/spectral_III", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 6, | |
| "targetPhase": "Phase III" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/spectral_IV", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 7, | |
| "targetPhase": "Phase IV" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/modular_I", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 0, | |
| "targetPhase": "Phase I" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/modular_II", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 5, | |
| "targetPhase": "Phase II" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/modular_III", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 6, | |
| "targetPhase": "Phase III" | |
| }, | |
| { | |
| "@id": "cxs:phases/map/modular_IV", | |
| "@type": "PhaseMapEntry", | |
| "sourceLevel": 7, | |
| "targetPhase": "Phase IV" | |
| }, | |
| { | |
| "@id": "cxs:phases/alignments/alignment_I", | |
| "@type": "PhaseAlignment", | |
| "alignment": "Pre-pentality regime", | |
| "phase": "Phase I" | |
| }, | |
| { | |
| "@id": "cxs:phases/alignments/alignment_II", | |
| "@type": "PhaseAlignment", | |
| "alignment": "Pariah boundary", | |
| "phase": "Phase II" | |
| }, | |
| { | |
| "@id": "cxs:phases/alignments/alignment_III", | |
| "@type": "PhaseAlignment", | |
| "alignment": "Triality regime", | |
| "phase": "Phase III" | |
| }, | |
| { | |
| "@id": "cxs:phases/alignments/alignment_IV", | |
| "@type": "PhaseAlignment", | |
| "alignment": "Octave periodic regime", | |
| "phase": "Phase IV" | |
| }, | |
| { | |
| "@id": "cxs:types", | |
| "@type": "TypesContainer", | |
| "label": "Type System", | |
| "partOf": "cxs:CategoricalX" | |
| }, | |
| { | |
| "@id": "cxs:types/level0", | |
| "@type": "TypeLevel", | |
| "comment": "T, O - fundamental axioms", | |
| "label": "Primitive/Axiom", | |
| "levelNumber": 0 | |
| }, | |
| { | |
| "@id": "cxs:types/level1", | |
| "@type": "TypeLevel", | |
| "comment": "c, q, pentality, septality, pariah, happy", | |
| "label": "First-Order Composition", | |
| "levelNumber": 1 | |
| }, | |
| { | |
| "@id": "cxs:types/level2", | |
| "@type": "TypeLevel", | |
| "comment": "J, B, sporadic, h1 + categorical constructs", | |
| "label": "Second-Order", | |
| "levelNumber": 2 | |
| }, | |
| { | |
| "@id": "cxs:types/level3", | |
| "@type": "TypeLevel", | |
| "comment": "Lattice, Filter, CountingFunction, Sieve", | |
| "label": "Third-Order Structures", | |
| "levelNumber": 3 | |
| }, | |
| { | |
| "@id": "cxs:types/level4", | |
| "@type": "TypeLevel", | |
| "comment": "PhaseTransition, PhaseSequence, PhaseBehavior", | |
| "label": "Fourth-Order Phase", | |
| "levelNumber": 4 | |
| }, | |
| { | |
| "@id": "cxs:types/level5", | |
| "@type": "TypeLevel", | |
| "comment": "Symbolic and abstract types", | |
| "label": "Fifth-Order Abstract", | |
| "levelNumber": 5 | |
| }, | |
| { | |
| "@id": "cxs:types/first/c", | |
| "@type": "FirstOrderType", | |
| "formula": "T × O", | |
| "inLevel": "cxs:types/level1", | |
| "label": "Leech Dimension", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 24 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:types/first/q", | |
| "@type": "FirstOrderType", | |
| "formula": "O / 2", | |
| "inLevel": "cxs:types/level1", | |
| "label": "Quaternion Embedding", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 4 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:types/first/pentality", | |
| "@type": "FirstOrderType", | |
| "formula": "O - T", | |
| "inLevel": "cxs:types/level1", | |
| "label": "Pentality", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 5 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:types/first/septality", | |
| "@type": "FirstOrderType", | |
| "formula": "T + q", | |
| "inLevel": "cxs:types/level1", | |
| "label": "Septality", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 7 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:types/first/pariah", | |
| "@type": "FirstOrderType", | |
| "formula": "c / q", | |
| "inLevel": "cxs:types/level1", | |
| "label": "Pariah", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 6 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:types/first/happy", | |
| "@type": "FirstOrderType", | |
| "formula": "c - q", | |
| "inLevel": "cxs:types/level1", | |
| "label": "Happy Number", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 20 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:types/second/J", | |
| "@type": "SecondOrderType", | |
| "formula": "T^T", | |
| "inLevel": "cxs:types/level2", | |
| "label": "Jordan Dimension", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 27 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:types/second/B", | |
| "@type": "SecondOrderType", | |
| "formula": "2^pentality", | |
| "inLevel": "cxs:types/level2", | |
| "label": "Boolean Lattice", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 32 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:types/second/sporadic", | |
| "@type": "SecondOrderType", | |
| "formula": "happy + pariah", | |
| "inLevel": "cxs:types/level2", | |
| "label": "Sporadic Count", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 26 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:types/second/h1", | |
| "@type": "SecondOrderType", | |
| "formula": "happy - 1", | |
| "inLevel": "cxs:types/level2", | |
| "label": "H1 Constant", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 19 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:types/structural/Lattice", | |
| "@type": "StructuralType", | |
| "comment": "Partially ordered set with meets and joins", | |
| "label": "Lattice" | |
| }, | |
| { | |
| "@id": "cxs:types/structural/Filter", | |
| "@type": "StructuralType", | |
| "comment": "Upward closed subset", | |
| "label": "Filter" | |
| }, | |
| { | |
| "@id": "cxs:types/structural/CountingFunction", | |
| "@type": "StructuralType", | |
| "comment": "Function counting structures", | |
| "label": "Counting Function" | |
| }, | |
| { | |
| "@id": "cxs:types/structural/Sieve", | |
| "@type": "StructuralType", | |
| "comment": "Selection mechanism", | |
| "label": "Sieve" | |
| }, | |
| { | |
| "@id": "cxs:constants", | |
| "@type": "ConstantsContainer", | |
| "contains": [ | |
| "cxs:constants/c", | |
| "cxs:constants/q", | |
| "cxs:constants/pentality", | |
| "cxs:constants/septality", | |
| "cxs:constants/pariah", | |
| "cxs:constants/happy", | |
| "cxs:constants/J", | |
| "cxs:constants/B", | |
| "cxs:constants/sporadic", | |
| "cxs:constants/h1", | |
| "cxs:constants/C_9_2", | |
| "cxs:constants/C_10_2", | |
| "cxs:constants/K", | |
| "cxs:constants/L", | |
| "cxs:constants/M", | |
| "cxs:constants/N" | |
| ], | |
| "label": "Constants", | |
| "partOf": "cxs:CategoricalX" | |
| }, | |
| { | |
| "@id": "cxs:constants/c", | |
| "@type": "DerivedConstant", | |
| "formula": "T × O", | |
| "label": "Leech Dimension", | |
| "symbol": "c", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 24 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/q", | |
| "@type": "DerivedConstant", | |
| "formula": "O / 2", | |
| "label": "Quaternion Embedding", | |
| "symbol": "q", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 4 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/pentality", | |
| "@type": "DerivedConstant", | |
| "formula": "O - T", | |
| "label": "Pentality", | |
| "symbol": "pentality", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 5 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/septality", | |
| "@type": "DerivedConstant", | |
| "formula": "T + q", | |
| "label": "Septality", | |
| "symbol": "septality", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 7 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/pariah", | |
| "@type": "DerivedConstant", | |
| "formula": "c / q", | |
| "label": "Pariah", | |
| "symbol": "pariah", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 6 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/happy", | |
| "@type": "DerivedConstant", | |
| "formula": "c - q", | |
| "label": "Happy Number", | |
| "symbol": "happy", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 20 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/J", | |
| "@type": "DerivedConstant", | |
| "formula": "T^T", | |
| "label": "Jordan Dimension", | |
| "symbol": "J", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 27 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/B", | |
| "@type": "DerivedConstant", | |
| "formula": "2^pentality", | |
| "label": "Boolean Lattice", | |
| "symbol": "B", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 32 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/sporadic", | |
| "@type": "DerivedConstant", | |
| "formula": "happy + pariah", | |
| "label": "Sporadic Count", | |
| "symbol": "sporadic", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 26 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/h1", | |
| "@type": "DerivedConstant", | |
| "formula": "happy - 1", | |
| "label": "H1 Constant", | |
| "symbol": "h1", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 19 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/C_9_2", | |
| "@type": "DerivedConstant", | |
| "formula": "T² × (T² - 1) / 2", | |
| "label": "Octave Correction (Triality²)", | |
| "symbol": "C(T²,2)", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 36 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/C_10_2", | |
| "@type": "DerivedConstant", | |
| "formula": "(O + 2) × (O + 1) / 2", | |
| "label": "Octave Correction (Tonic Dominant)", | |
| "symbol": "C(O+2,2)", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 45 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/K", | |
| "@type": "DerivedConstant", | |
| "formula": "J + O", | |
| "label": "Extended Jordan", | |
| "symbol": "K", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 35 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/L", | |
| "@type": "DerivedConstant", | |
| "formula": "K + pentality", | |
| "label": "Extended L", | |
| "symbol": "L", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 40 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/M", | |
| "@type": "DerivedConstant", | |
| "formula": "L + pariah", | |
| "label": "Extended M", | |
| "symbol": "M", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 46 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:constants/N", | |
| "@type": "DerivedConstant", | |
| "formula": "M + septality", | |
| "label": "Extended N", | |
| "symbol": "N", | |
| "yieldsValue": { | |
| "@type": "xsd:integer", | |
| "@value": 53 | |
| } | |
| }, | |
| { | |
| "@id": "cxs:octaves/beta_O", | |
| "@type": "OctaveCocycle", | |
| "baseCocycle": "H³(Z₂³, k×)", | |
| "comment": "Fundamental octave cocycle from Bott periodicity", | |
| "label": "Octave Cocycle", | |
| "period": 8 | |
| }, | |
| { | |
| "@id": "cxs:octaves/constants/C_8_2", | |
| "@type": "OctaveConstants", | |
| "formula": "8 choose 2", | |
| "hasValue": 28, | |
| "label": "C(8,2)" | |
| }, | |
| { | |
| "@id": "cxs:octaves/constants/C_9_2", | |
| "@type": "OctaveConstants", | |
| "formula": "9 choose 2", | |
| "hasValue": 36, | |
| "label": "C(9,2)" | |
| }, | |
| { | |
| "@id": "cxs:octaves/constants/C_10_2", | |
| "@type": "OctaveConstants", | |
| "formula": "10 choose 2", | |
| "hasValue": 45, | |
| "label": "C(10,2)" | |
| }, | |
| { | |
| "@id": "cxs:octaves/0", | |
| "@type": "NamedOctave", | |
| "index": 0, | |
| "label": "Named", | |
| "range": "0-7", | |
| "twist": "Trivial" | |
| }, | |
| { | |
| "@id": "cxs:octaves/1", | |
| "@type": "NamedOctave", | |
| "index": 1, | |
| "label": "Extended", | |
| "range": "8-15", | |
| "twist": "Twisted (β_O)" | |
| }, | |
| { | |
| "@id": "cxs:octaves/2", | |
| "@type": "NamedOctave", | |
| "index": 2, | |
| "label": "Channel", | |
| "range": "16-23", | |
| "twist": "Trivial" | |
| }, | |
| { | |
| "@id": "cxs:octaves/3", | |
| "@type": "NamedOctave", | |
| "index": 3, | |
| "label": "Boundary", | |
| "range": "24-31", | |
| "twist": "Twisted (β_O)" | |
| }, | |
| { | |
| "@id": "cxs:proofs/theorems/triality_theorem", | |
| "@type": "Theorem", | |
| "comment": "The cube of D(5) satisfies the triality identity", | |
| "label": "Triality Theorem", | |
| "statement": "D(5)³ = 435,691,903,941" | |
| }, | |
| { | |
| "@id": "cxs:proofs/theorems/leech_dimension_theorem", | |
| "@type": "Theorem", | |
| "comment": "The Leech lattice dimension equals the product of axioms", | |
| "label": "Leech Dimension Theorem", | |
| "statement": "dim(Leech) = T × O = 24" | |
| }, | |
| { | |
| "@id": "cxs:proofs/main", | |
| "@type": "Proof", | |
| "label": "Main Categorical X Proof", | |
| "proofStep": [ | |
| "cxs:proofs/steps/1", | |
| "cxs:proofs/steps/2", | |
| "cxs:proofs/steps/3", | |
| "cxs:proofs/steps/4", | |
| "cxs:proofs/steps/5", | |
| "cxs:proofs/steps/6", | |
| "cxs:proofs/steps/7" | |
| ], | |
| "proves": "cxs:proofs/theorems/triality_theorem" | |
| }, | |
| { | |
| "@id": "cxs:proofs/steps/1", | |
| "@type": "ProofStep", | |
| "assertion": "T = 3 and O = 8 are the fundamental axioms", | |
| "justification": "By definition", | |
| "stepNumber": 1 | |
| }, | |
| { | |
| "@id": "cxs:proofs/steps/2", | |
| "@type": "ProofStep", | |
| "assertion": "c = T × O = 24", | |
| "justification": "Direct computation", | |
| "stepNumber": 2 | |
| }, | |
| { | |
| "@id": "cxs:proofs/steps/3", | |
| "@type": "ProofStep", | |
| "assertion": "D(4) = 168 = 7 × 24 = septality × c", | |
| "justification": "Antichain counting + factorization", | |
| "stepNumber": 3 | |
| }, | |
| { | |
| "@id": "cxs:proofs/steps/4", | |
| "@type": "ProofStep", | |
| "assertion": "D(5) = 7581", | |
| "justification": "Antichain enumeration on 2^[5]", | |
| "stepNumber": 4 | |
| }, | |
| { | |
| "@id": "cxs:proofs/steps/5", | |
| "@type": "ProofStep", | |
| "assertion": "D(5)³ = 435,691,903,941", | |
| "justification": "Direct computation", | |
| "stepNumber": 5 | |
| }, | |
| { | |
| "@id": "cxs:proofs/steps/6", | |
| "@type": "ProofStep", | |
| "assertion": "dim(Leech) = 24 = c", | |
| "justification": "Lattice theory", | |
| "stepNumber": 6 | |
| }, | |
| { | |
| "@id": "cxs:proofs/steps/7", | |
| "@type": "ProofStep", | |
| "assertion": "The triality theorem holds", | |
| "justification": "Verified by computation", | |
| "stepNumber": 7 | |
| }, | |
| { | |
| "@id": "cxs:instances/lattices/booleanLattice", | |
| "@type": "LatticeInstance", | |
| "comment": "Power set lattice 2^[n]", | |
| "label": "Boolean Lattice" | |
| }, | |
| { | |
| "@id": "cxs:instances/lattices/divisibilityLattice", | |
| "@type": "LatticeInstance", | |
| "comment": "Lattice of divisors", | |
| "label": "Divisibility Lattice" | |
| }, | |
| { | |
| "@id": "cxs:instances/lattices/subgroupLattice", | |
| "@type": "LatticeInstance", | |
| "comment": "Lattice of subgroups", | |
| "label": "Subgroup Lattice" | |
| }, | |
| { | |
| "@id": "cxs:instances/filters/antichainFilter", | |
| "@type": "FilterInstance", | |
| "comment": "Filter of antichains in boolean lattice", | |
| "label": "Antichain Filter" | |
| }, | |
| { | |
| "@id": "cxs:instances/filters/coprimeFilter", | |
| "@type": "FilterInstance", | |
| "comment": "Filter of coprime pairs", | |
| "label": "Coprime Filter" | |
| }, | |
| { | |
| "@id": "cxs:instances/filters/maximalFilter", | |
| "@type": "FilterInstance", | |
| "comment": "Filter of maximal elements", | |
| "label": "Maximal Filter" | |
| }, | |
| { | |
| "@id": "cxs:instances/sieves/eratosthenesSieve", | |
| "@type": "SieveInstance", | |
| "comment": "Classical prime sieve", | |
| "label": "Eratosthenes Sieve" | |
| }, | |
| { | |
| "@id": "cxs:instances/sieves/antichainSieve", | |
| "@type": "SieveInstance", | |
| "comment": "Sieve for antichain counting", | |
| "label": "Antichain Sieve" | |
| }, | |
| { | |
| "@id": "cxs:instances/sieves/modularSieve", | |
| "@type": "SieveInstance", | |
| "comment": "Sieve based on modular conditions", | |
| "label": "Modular Sieve" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value0", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 0, | |
| "hasAssertion": "cxs:sequences/dedekind/entry/0", | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value1", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 1, | |
| "hasAssertion": "cxs:sequences/dedekind/entry/1", | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value2", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 2, | |
| "hasAssertion": "cxs:sequences/dedekind/entry/2", | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value3", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 3, | |
| "hasAssertion": "cxs:sequences/dedekind/entry/3", | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value4", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 4, | |
| "hasAssertion": "cxs:sequences/dedekind/entry/4", | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value5", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 5, | |
| "hasAssertion": "cxs:sequences/dedekind/entry/5", | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value6", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 6, | |
| "hasAssertion": "cxs:sequences/dedekind/entry/6", | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value7", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 7, | |
| "hasAssertion": "cxs:sequences/dedekind/entry/7", | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value8", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 8, | |
| "hasAssertion": "cxs:sequences/dedekind/entry/8", | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value9", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 9, | |
| "hasAssertion": "cxs:sequences/dedekind/entry/9", | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value10", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 10, | |
| "hasValue": "derived" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value11", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 11, | |
| "hasValue": "derived" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value12", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 12, | |
| "hasValue": "derived" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value13", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 13, | |
| "hasValue": "derived" | |
| }, | |
| { | |
| "@id": "cxs:baseCases/value14", | |
| "@type": "BaseCaseValue", | |
| "forLevel": 14, | |
| "hasValue": "derived" | |
| }, | |
| { | |
| "@id": "cxs:derivations/mul_rule", | |
| "@type": "DerivationRule", | |
| "formula": "(a, b) → a × b", | |
| "label": "Multiplication Rule" | |
| }, | |
| { | |
| "@id": "cxs:derivations/div_rule", | |
| "@type": "DerivationRule", | |
| "formula": "(a, b) → a / b", | |
| "label": "Division Rule" | |
| }, | |
| { | |
| "@id": "cxs:derivations/pow_rule", | |
| "@type": "DerivationRule", | |
| "formula": "(a, b) → a^b", | |
| "label": "Power Rule" | |
| }, | |
| { | |
| "@id": "cxs:derivations/add_rule", | |
| "@type": "DerivationRule", | |
| "formula": "(a, b) → a + b", | |
| "label": "Addition Rule" | |
| }, | |
| { | |
| "@id": "cxs:derivations/sub_rule", | |
| "@type": "DerivationRule", | |
| "formula": "(a, b) → a - b", | |
| "label": "Subtraction Rule" | |
| }, | |
| { | |
| "@id": "cxs:derivations/compose_rule", | |
| "@type": "DerivationRule", | |
| "formula": "(f, g) → f ∘ g", | |
| "label": "Composition Rule" | |
| }, | |
| { | |
| "@id": "cxs:derivations/phase_rule", | |
| "@type": "DerivationRule", | |
| "formula": "n → phase(n)", | |
| "label": "Phase Rule" | |
| }, | |
| { | |
| "@id": "cxs:derivations/dim_rule", | |
| "@type": "DerivationRule", | |
| "formula": "n → 2^n", | |
| "label": "Dimension Rule" | |
| }, | |
| { | |
| "@id": "cxs:derivations/aut_rule", | |
| "@type": "DerivationRule", | |
| "formula": "n → (n-1) × 7", | |
| "label": "Automorphism Rule" | |
| }, | |
| { | |
| "@id": "cxs:derivations/cocycle_rule", | |
| "@type": "DerivationRule", | |
| "formula": "n → H^n", | |
| "label": "Cocycle Rule" | |
| }, | |
| { | |
| "@id": "cxs:primitives/operations/sub", | |
| "@type": "DerivedOperation", | |
| "basedOn": "cxs:primitives/operations/add", | |
| "label": "Subtraction", | |
| "symbol": "-" | |
| }, | |
| { | |
| "@id": "cxm:meta", | |
| "@type": "MetaContainer", | |
| "comment": "Container for TypeConstructors and functors that enable generative derivation from axioms T=3 and O=8.", | |
| "hasPart": [ | |
| "cxm:constructors", | |
| "cxm:functors" | |
| ], | |
| "label": "Meta (Generative Constructs)" | |
| }, | |
| { | |
| "@id": "cxm:constructors", | |
| "@type": "TypeConstructorsContainer", | |
| "comment": "Mechanisms that generate RDF instances dynamically from axiom-derived formulas.", | |
| "contains": [ | |
| "cxm:constructors/levelGenerator", | |
| "cxm:constructors/transitionGenerator", | |
| "cxm:constructors/stratumGenerator", | |
| "cxm:constructors/phaseModificationGenerator" | |
| ], | |
| "label": "Type Constructors" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator", | |
| "@type": "TypeConstructor", | |
| "comment": "Generates tower level properties for any n >= 0 from axioms T=3 and O=8. Eliminates the need for static TOWER_LEVELS array.", | |
| "constructionRules": [ | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_id" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_index" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_dimension" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_algebra_symbol" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_algebra_name" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_loses_property" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_automorphism_group" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_automorphism_dimension" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_cocycle" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_octave_twist" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_correction_coefficient" | |
| } | |
| ], | |
| "generatesType": { | |
| "@id": "cx:TowerLevel" | |
| }, | |
| "hasParameter": { | |
| "@type": "ParameterSpec", | |
| "parameterCondition": "n >= 0", | |
| "parameterDomain": "NonNegativeInteger", | |
| "parameterVariable": "n" | |
| }, | |
| "label": "Tower Level Generator" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_id", | |
| "@type": "ConstructionRule", | |
| "comment": "Unique identifier for level n", | |
| "constructionFormula": "cxs:tower/levels/{n}", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "constructionFormula": "IRI template" | |
| }, | |
| "targetField": "@id" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_index", | |
| "@type": "ConstructionRule", | |
| "comment": "Level index", | |
| "constructionFormula": "n", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "constructionFormula": "direct parameter" | |
| }, | |
| "targetField": "levelIndex" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_dimension", | |
| "@type": "ConstructionRule", | |
| "comment": "Dimension doubles with each level (Cayley-Dickson construction)", | |
| "constructionFormula": "2^n", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "sourceFirstOrder": "O", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "targetField": "cx:hasDimension" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_algebra_symbol", | |
| "@type": "ConstructionRule", | |
| "comment": "Algebra symbol cycles with period 8 (octave)", | |
| "constructionFormula": "ALGEBRA_SYMBOLS[n % 8]", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "sourceFirstOrder": "O", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "targetField": "algebra" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_algebra_name", | |
| "@type": "ConstructionRule", | |
| "comment": "Algebra name with octave extension", | |
| "constructionFormula": "ALGEBRA_NAMES[n % 8] or 'Extended-' prefix for n >= 8", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "sourceFirstOrder": "O", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "targetField": "label" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_loses_property", | |
| "@type": "ConstructionRule", | |
| "comment": "Property lost at this level in Cayley-Dickson tower", | |
| "constructionFormula": "PROPERTY_LOSS[n] for n in [1,5], None otherwise", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "targetField": "losesProperty" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_automorphism_group", | |
| "@type": "ConstructionRule", | |
| "comment": "Automorphism group for level n", | |
| "constructionFormula": "automorphism_group_name(n)", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "targetField": "cx:hasAutomorphismGroup" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_automorphism_dimension", | |
| "@type": "ConstructionRule", | |
| "comment": "Automorphism dimension formula: (n-1) * septality where septality = O-1 = 7", | |
| "constructionFormula": "(n-1) * 7 for n >= 4; special cases for n < 4", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "sourceFirstOrder": "O", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "targetField": "automorphismDimension" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_cocycle", | |
| "@type": "ConstructionRule", | |
| "comment": "Cohomology class for level n", | |
| "constructionFormula": "H^n(Z_2^(n-1), k^x)", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "targetField": "cx:hasCocycle" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_octave_twist", | |
| "@type": "ConstructionRule", | |
| "comment": "Bott periodicity twist: trivial for even octaves, non-trivial for odd", | |
| "constructionFormula": "beta_O^(floor(n/8) mod 2)", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "targetField": "cx:hasOctaveTwist" | |
| }, | |
| { | |
| "@id": "cxm:constructors/levelGenerator/rules/level_correction_coefficient", | |
| "@type": "ConstructionRule", | |
| "comment": "Octave correction coefficient for extended levels", | |
| "constructionFormula": "C(n,2) via recurrence for n >= 8", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "targetField": "cx:hasCorrectionCoefficient" | |
| }, | |
| { | |
| "@id": "cxm:constructors/transitionGenerator", | |
| "@type": "TypeConstructor", | |
| "comment": "Generates tower transitions (doubling functors) for any n >= 0. Each transition is an instance of the Cayley-Dickson construction.", | |
| "constructionRules": [ | |
| { | |
| "@id": "cxm:constructors/transitionGenerator/rules/transition_id" | |
| }, | |
| { | |
| "@id": "cxm:constructors/transitionGenerator/rules/transition_label" | |
| }, | |
| { | |
| "@id": "cxm:constructors/transitionGenerator/rules/transition_source" | |
| }, | |
| { | |
| "@id": "cxm:constructors/transitionGenerator/rules/transition_target" | |
| }, | |
| { | |
| "@id": "cxm:constructors/transitionGenerator/rules/transition_formula" | |
| } | |
| ], | |
| "generatesType": { | |
| "@id": "cx:TowerTransition" | |
| }, | |
| "hasParameter": { | |
| "@type": "ParameterSpec", | |
| "parameterCondition": "n >= 0", | |
| "parameterDomain": "NonNegativeInteger", | |
| "parameterVariable": "n" | |
| }, | |
| "label": "Tower Transition Generator" | |
| }, | |
| { | |
| "@id": "cxm:constructors/transitionGenerator/rules/transition_id", | |
| "@type": "ConstructionRule", | |
| "comment": "Unique identifier for transition n -> n+1", | |
| "constructionFormula": "cxs:tower/transitions/phi_{n}{n+1}", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "constructionFormula": "IRI template" | |
| }, | |
| "targetField": "@id" | |
| }, | |
| { | |
| "@id": "cxm:constructors/transitionGenerator/rules/transition_label", | |
| "@type": "ConstructionRule", | |
| "comment": "Transition label", | |
| "constructionFormula": "phi_{n,n+1}", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "constructionFormula": "transition label" | |
| }, | |
| "targetField": "label" | |
| }, | |
| { | |
| "@id": "cxm:constructors/transitionGenerator/rules/transition_source", | |
| "@type": "ConstructionRule", | |
| "comment": "Source level reference", | |
| "constructionFormula": "cxs:tower/levels/{n}", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "constructionFormula": "source level reference" | |
| }, | |
| "targetField": "sourceLevel" | |
| }, | |
| { | |
| "@id": "cxm:constructors/transitionGenerator/rules/transition_target", | |
| "@type": "ConstructionRule", | |
| "comment": "Target level reference", | |
| "constructionFormula": "cxs:tower/levels/{n+1}", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "constructionFormula": "target level reference" | |
| }, | |
| "targetField": "targetLevel" | |
| }, | |
| { | |
| "@id": "cxm:constructors/transitionGenerator/rules/transition_formula", | |
| "@type": "ConstructionRule", | |
| "comment": "Cayley-Dickson doubling formula", | |
| "constructionFormula": "Cayley-Dickson(A_n) for n >= 3; explicit for n < 3", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "targetField": "formula" | |
| }, | |
| { | |
| "@id": "cxm:constructors/stratumGenerator", | |
| "@type": "TypeConstructor", | |
| "comment": "Generates strata groupings based on level index. Strata cycle with period 8 due to octave periodicity.", | |
| "constructionRules": [ | |
| { | |
| "@id": "cxm:constructors/stratumGenerator/rules/stratum_id" | |
| }, | |
| { | |
| "@id": "cxm:constructors/stratumGenerator/rules/stratum_name" | |
| }, | |
| { | |
| "@id": "cxm:constructors/stratumGenerator/rules/stratum_levels" | |
| }, | |
| { | |
| "@id": "cxm:constructors/stratumGenerator/rules/stratum_role" | |
| } | |
| ], | |
| "generatesType": { | |
| "@id": "cx:Stratum" | |
| }, | |
| "hasParameter": { | |
| "@type": "ParameterSpec", | |
| "parameterCondition": "n >= 0", | |
| "parameterDomain": "NonNegativeInteger", | |
| "parameterVariable": "n" | |
| }, | |
| "label": "Stratum Generator" | |
| }, | |
| { | |
| "@id": "cxm:constructors/stratumGenerator/rules/stratum_id", | |
| "@type": "ConstructionRule", | |
| "comment": "Unique identifier for stratum containing level n", | |
| "constructionFormula": "cxs:tower/strata/{stratum_name(n)}", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "constructionFormula": "IRI template" | |
| }, | |
| "targetField": "@id" | |
| }, | |
| { | |
| "@id": "cxm:constructors/stratumGenerator/rules/stratum_name", | |
| "@type": "ConstructionRule", | |
| "comment": "Stratum name based on level grouping with period 8", | |
| "constructionFormula": "stratum_name(n): Extension(0-2), Intension(3-4), Comprehension(5-7), Ground(8+k*8)", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "sourceFirstOrder": "O", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "targetField": "label" | |
| }, | |
| { | |
| "@id": "cxm:constructors/stratumGenerator/rules/stratum_levels", | |
| "@type": "ConstructionRule", | |
| "comment": "Levels contained in this stratum", | |
| "constructionFormula": "levels_in_stratum(stratum_name)", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "sourceFirstOrder": "O", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "targetField": "levels" | |
| }, | |
| { | |
| "@id": "cxm:constructors/stratumGenerator/rules/stratum_role", | |
| "@type": "ConstructionRule", | |
| "comment": "Role description for the stratum", | |
| "constructionFormula": "role_description(stratum_name)", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "constructionFormula": "stratum role" | |
| }, | |
| "targetField": "role" | |
| }, | |
| { | |
| "@id": "cxm:constructors/phaseModificationGenerator", | |
| "@type": "TypeConstructor", | |
| "comment": "Generates phase modifications for operators at arbitrary levels. Phase behavior is derived from tower structure and octave periodicity.", | |
| "constructionRules": [ | |
| { | |
| "@id": "cxm:constructors/phaseModificationGenerator/rules/phase_mod_id" | |
| }, | |
| { | |
| "@id": "cxm:constructors/phaseModificationGenerator/rules/phase_mod_operator" | |
| }, | |
| { | |
| "@id": "cxm:constructors/phaseModificationGenerator/rules/phase_mod_target_level" | |
| }, | |
| { | |
| "@id": "cxm:constructors/phaseModificationGenerator/rules/phase_mod_phase" | |
| }, | |
| { | |
| "@id": "cxm:constructors/phaseModificationGenerator/rules/phase_mod_formula" | |
| } | |
| ], | |
| "generatesType": { | |
| "@id": "cx:PhaseModification" | |
| }, | |
| "hasParameter": { | |
| "@type": "ParameterSpec", | |
| "parameterCondition": "level >= 3", | |
| "parameterDomain": "Operator × NonNegativeInteger", | |
| "parameterVariable": "operator, level" | |
| }, | |
| "label": "Phase Modification Generator" | |
| }, | |
| { | |
| "@id": "cxm:constructors/phaseModificationGenerator/rules/phase_mod_id", | |
| "@type": "ConstructionRule", | |
| "comment": "Unique identifier for phase modification", | |
| "constructionFormula": "cxs:phases/modifications/{operator}_{level}", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "constructionFormula": "IRI template" | |
| }, | |
| "targetField": "@id" | |
| }, | |
| { | |
| "@id": "cxm:constructors/phaseModificationGenerator/rules/phase_mod_operator", | |
| "@type": "ConstructionRule", | |
| "comment": "Reference to operator being modified", | |
| "constructionFormula": "cxs:operators/{operator}", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "constructionFormula": "operator reference" | |
| }, | |
| "targetField": "operator" | |
| }, | |
| { | |
| "@id": "cxm:constructors/phaseModificationGenerator/rules/phase_mod_target_level", | |
| "@type": "ConstructionRule", | |
| "comment": "Reference to target level", | |
| "constructionFormula": "cxs:tower/levels/{level}", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "constructionFormula": "target level reference" | |
| }, | |
| "targetField": "targetLevel" | |
| }, | |
| { | |
| "@id": "cxm:constructors/phaseModificationGenerator/rules/phase_mod_phase", | |
| "@type": "ConstructionRule", | |
| "comment": "Phase behavior at this level", | |
| "constructionFormula": "phase_for_level(level)", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "targetField": "phase" | |
| }, | |
| { | |
| "@id": "cxm:constructors/phaseModificationGenerator/rules/phase_mod_formula", | |
| "@type": "ConstructionRule", | |
| "comment": "Operator behavior formula at this phase", | |
| "constructionFormula": "operator_phase_formula(operator, level)", | |
| "hasDerivationSource": { | |
| "@type": "DerivationSource", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "targetField": "formula" | |
| }, | |
| { | |
| "@id": "cxm:functors", | |
| "@type": "ConstructorFunctorsContainer", | |
| "comment": "Morphisms between TypeConstructors (endofunctors and transformations).", | |
| "contains": [ | |
| "cxm:functors/recursiveDoubling", | |
| "cxm:functors/octavePeriodicity", | |
| "cxm:functors/trialityFunctor" | |
| ], | |
| "label": "Constructor Functors" | |
| }, | |
| { | |
| "@id": "cxm:functors/recursiveDoubling", | |
| "@type": "ConstructorFunctor", | |
| "comment": "The Cayley-Dickson doubling as an endofunctor on generated levels. This is the fundamental construction that builds all tower levels from the reals.", | |
| "functorFormula": "D_n: Level(n) -> Level(n+1); dim(D_n(L)) = 2 * dim(L)", | |
| "functorSource": { | |
| "@id": "cxm:constructors/levelGenerator" | |
| }, | |
| "functorTarget": { | |
| "@id": "cxm:constructors/levelGenerator" | |
| }, | |
| "label": "Recursive Doubling Functor" | |
| }, | |
| { | |
| "@id": "cxm:functors/octavePeriodicity", | |
| "@type": "ConstructorFunctor", | |
| "comment": "Bott periodicity with period 8. Level n+8 is isomorphic to level n tensored with the octave cocycle β_O.", | |
| "functorFormula": "B_8: Level(n) -> Level(n+8); B_8^2 = id (mod twist)", | |
| "functorSource": { | |
| "@id": "cxm:constructors/levelGenerator" | |
| }, | |
| "functorTarget": { | |
| "@id": "cxm:constructors/levelGenerator" | |
| }, | |
| "label": "Octave Periodicity Functor" | |
| }, | |
| { | |
| "@id": "cxm:functors/trialityFunctor", | |
| "@type": "ConstructorFunctor", | |
| "comment": "Order-3 automorphism from triality (T=3). Active at level 3 (octonions) and above. Permutes the three 8-dimensional representations of Spin(8).", | |
| "functorFormula": "τ: Level(n) -> Level(n) for n >= 3; τ^3 = id", | |
| "functorSource": { | |
| "@id": "cxm:constructors/levelGenerator" | |
| }, | |
| "functorTarget": { | |
| "@id": "cxm:constructors/levelGenerator" | |
| }, | |
| "label": "Triality Functor" | |
| }, | |
| { | |
| "@id": "cxs:sequences/dedekind", | |
| "@type": "CountingSequence", | |
| "codomain": "naturals", | |
| "comment": "Counts antichains in the power set lattice 2^[n]. Computed via categorical composition of Power, Subobject, and Cardinality functors.", | |
| "formula": "|{A ⊆ P^n : antichain(A)}|", | |
| "label": "Dedekind Numbers", | |
| "oeisId": "A000372" | |
| }, | |
| { | |
| "@id": "cxs:sequences/catalan", | |
| "@type": "CountingSequence", | |
| "codomain": "naturals", | |
| "comment": "Counts full binary trees with n+1 leaves. Arises from associahedra in monoidal coherence.", | |
| "formula": "(2n)!/((n+1)! × n!)", | |
| "label": "Catalan Numbers", | |
| "oeisId": "A000108" | |
| }, | |
| { | |
| "@id": "cxs:sequences/dedekind/entry/0", | |
| "@type": "SequenceEntry", | |
| "hasAssertion": "cxs:sequences/dedekind", | |
| "label": "D(0)", | |
| "sequenceIndex": 0, | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:sequences/dedekind/entry/1", | |
| "@type": "SequenceEntry", | |
| "hasAssertion": "cxs:sequences/dedekind", | |
| "label": "D(1)", | |
| "sequenceIndex": 1, | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:sequences/dedekind/entry/2", | |
| "@type": "SequenceEntry", | |
| "hasAssertion": "cxs:sequences/dedekind", | |
| "label": "D(2)", | |
| "sequenceIndex": 2, | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:sequences/dedekind/entry/3", | |
| "@type": "SequenceEntry", | |
| "hasAssertion": "cxs:sequences/dedekind", | |
| "label": "D(3)", | |
| "sequenceIndex": 3, | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:sequences/dedekind/entry/4", | |
| "@type": "SequenceEntry", | |
| "hasAssertion": "cxs:sequences/dedekind", | |
| "label": "D(4)", | |
| "sequenceIndex": 4, | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:sequences/dedekind/entry/5", | |
| "@type": "SequenceEntry", | |
| "hasAssertion": "cxs:sequences/dedekind", | |
| "label": "D(5)", | |
| "sequenceIndex": 5, | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:sequences/dedekind/entry/6", | |
| "@type": "SequenceEntry", | |
| "hasAssertion": "cxs:sequences/dedekind", | |
| "label": "D(6)", | |
| "sequenceIndex": 6, | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:sequences/dedekind/entry/7", | |
| "@type": "SequenceEntry", | |
| "hasAssertion": "cxs:sequences/dedekind", | |
| "label": "D(7)", | |
| "sequenceIndex": 7, | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:sequences/dedekind/entry/8", | |
| "@type": "SequenceEntry", | |
| "hasAssertion": "cxs:sequences/dedekind", | |
| "label": "D(8)", | |
| "sequenceIndex": 8, | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:sequences/dedekind/entry/9", | |
| "@type": "SequenceEntry", | |
| "hasAssertion": "cxs:sequences/dedekind", | |
| "label": "D(9)", | |
| "sequenceIndex": 9, | |
| "status": "ComputedExact" | |
| }, | |
| { | |
| "@id": "cxs:numeric/domains/naturals", | |
| "@type": "NumericDomain", | |
| "comment": "Non-negative integers {0, 1, 2, ...}. The canonical counting domain.", | |
| "embedsInto": { | |
| "@id": "cxs:numeric/domains/integers" | |
| }, | |
| "isDiscrete": true, | |
| "isOrdered": true, | |
| "label": "Natural Numbers", | |
| "symbol": "ℕ" | |
| }, | |
| { | |
| "@id": "cxs:numeric/domains/integers", | |
| "@type": "NumericDomain", | |
| "comment": "Signed integers {..., -2, -1, 0, 1, 2, ...}.", | |
| "embedsInto": { | |
| "@id": "cxs:numeric/domains/rationals" | |
| }, | |
| "isDiscrete": true, | |
| "isOrdered": true, | |
| "label": "Integers", | |
| "symbol": "ℤ" | |
| }, | |
| { | |
| "@id": "cxs:numeric/domains/rationals", | |
| "@type": "NumericDomain", | |
| "comment": "Fractions p/q where p, q ∈ ℤ and q ≠ 0.", | |
| "embedsInto": { | |
| "@id": "cxs:numeric/domains/reals" | |
| }, | |
| "isDiscrete": false, | |
| "isOrdered": true, | |
| "label": "Rational Numbers", | |
| "symbol": "ℚ" | |
| }, | |
| { | |
| "@id": "cxs:numeric/domains/reals", | |
| "@type": "NumericDomain", | |
| "comment": "Complete ordered field. Tower Level 0 algebra.", | |
| "embedsInto": { | |
| "@id": "cxs:numeric/domains/complex" | |
| }, | |
| "isDiscrete": false, | |
| "isOrdered": true, | |
| "label": "Real Numbers", | |
| "symbol": "ℝ" | |
| }, | |
| { | |
| "@id": "cxs:numeric/domains/complex", | |
| "@type": "NumericDomain", | |
| "comment": "Field extension ℝ[i] where i² = -1. Tower Level 1 algebra.", | |
| "isDiscrete": false, | |
| "isOrdered": false, | |
| "label": "Complex Numbers", | |
| "symbol": "ℂ" | |
| }, | |
| { | |
| "@id": "cxs:numerals/systems/decimal", | |
| "@type": [ | |
| "NumeralSystem", | |
| "PositionalSystem" | |
| ], | |
| "comment": "Standard base-10 positional notation.", | |
| "hasRadix": 10, | |
| "label": "Decimal" | |
| }, | |
| { | |
| "@id": "cxs:numerals/systems/binary", | |
| "@type": [ | |
| "NumeralSystem", | |
| "PositionalSystem" | |
| ], | |
| "comment": "Base-2 positional notation.", | |
| "hasRadix": 2, | |
| "label": "Binary" | |
| }, | |
| { | |
| "@id": "cxs:numerals/systems/hexadecimal", | |
| "@type": [ | |
| "NumeralSystem", | |
| "PositionalSystem" | |
| ], | |
| "comment": "Base-16 positional notation.", | |
| "hasRadix": 16, | |
| "label": "Hexadecimal" | |
| }, | |
| { | |
| "@id": "cxs:numerals/systems/mod96crt", | |
| "@type": [ | |
| "NumeralSystem", | |
| "MixedRadixSystem" | |
| ], | |
| "comment": "Chinese Remainder Theorem representation for mod-96 arithmetic. Uses radices [32, 3] where 32 × 3 = 96.", | |
| "hasRadices": "32,3", | |
| "label": "Mod-96 CRT" | |
| }, | |
| { | |
| "@id": "cxs:numerals/systems/continuedFraction", | |
| "@type": [ | |
| "NumeralSystem", | |
| "NonPositionalSystem" | |
| ], | |
| "comment": "Representation as [a0; a1, a2, ...] where value = a0 + 1/(a1 + 1/(a2 + ...)). Used for transcendental constants.", | |
| "label": "Continued Fraction" | |
| }, | |
| { | |
| "@id": "cxs:numerals/digitSets/decimal-digits", | |
| "@type": "DigitSet", | |
| "cardinality": 10, | |
| "label": "Decimal Digits", | |
| "symbols": "0123456789" | |
| }, | |
| { | |
| "@id": "cxs:numerals/digitSets/binary-digits", | |
| "@type": "DigitSet", | |
| "cardinality": 2, | |
| "label": "Binary Digits", | |
| "symbols": "01" | |
| }, | |
| { | |
| "@id": "cxs:numerals/digitSets/hexadecimal-digits", | |
| "@type": "DigitSet", | |
| "cardinality": 16, | |
| "label": "Hexadecimal Digits", | |
| "symbols": "0123456789ABCDEF" | |
| }, | |
| { | |
| "@id": "cxs:numerals/digitSets/mod32-digits", | |
| "@type": "DigitSet", | |
| "cardinality": 32, | |
| "label": "Mod-32 Digits", | |
| "symbols": "0-31 (integer range)" | |
| }, | |
| { | |
| "@id": "cxs:numerals/conversions/binary-to-decimal", | |
| "@type": "ConversionMorphism", | |
| "comment": "Standard binary to decimal conversion.", | |
| "label": "binary → decimal", | |
| "source": "cxs:numerals/systems/binary", | |
| "target": "cxs:numerals/systems/decimal" | |
| }, | |
| { | |
| "@id": "cxs:numerals/conversions/decimal-to-mod96crt", | |
| "@type": "ConversionMorphism", | |
| "comment": "Convert decimal to CRT representation: (n mod 32, n mod 3).", | |
| "label": "decimal → mod96crt", | |
| "source": "cxs:numerals/systems/decimal", | |
| "target": "cxs:numerals/systems/mod96crt" | |
| }, | |
| { | |
| "@id": "cxs:numerals/conversions/mod96crt-to-decimal", | |
| "@type": "ConversionMorphism", | |
| "comment": "Reconstruct decimal from CRT using Chinese Remainder Theorem.", | |
| "label": "mod96crt → decimal", | |
| "source": "cxs:numerals/systems/mod96crt", | |
| "target": "cxs:numerals/systems/decimal" | |
| }, | |
| { | |
| "@id": "cxs:precision/contexts/arbitrary", | |
| "@type": [ | |
| "PrecisionContext", | |
| "ArbitraryPrecisionContext" | |
| ], | |
| "comment": "Unbounded precision for exact computation.", | |
| "label": "Arbitrary Precision" | |
| }, | |
| { | |
| "@id": "cxs:precision/contexts/mod96", | |
| "@type": [ | |
| "PrecisionContext", | |
| "FixedPrecisionContext" | |
| ], | |
| "comment": "Fixed mod-96 arithmetic. Values in [0, 95].", | |
| "label": "Mod-96 Residue", | |
| "precisionBits": 7, | |
| "roundingMode": "TowardZero" | |
| }, | |
| { | |
| "@id": "cxs:precision/contexts/ieee754-single", | |
| "@type": [ | |
| "PrecisionContext", | |
| "FixedPrecisionContext" | |
| ], | |
| "comment": "IEEE 754 single precision (32-bit float).", | |
| "exponentMax": 127, | |
| "exponentMin": -126, | |
| "label": "IEEE 754 Single", | |
| "precisionBits": 24, | |
| "precisionDigits": 7, | |
| "roundingMode": "TiesToEven" | |
| }, | |
| { | |
| "@id": "cxs:precision/contexts/ieee754-double", | |
| "@type": [ | |
| "PrecisionContext", | |
| "FixedPrecisionContext" | |
| ], | |
| "comment": "IEEE 754 double precision (64-bit float).", | |
| "exponentMax": 1023, | |
| "exponentMin": -1022, | |
| "label": "IEEE 754 Double", | |
| "precisionBits": 53, | |
| "precisionDigits": 15, | |
| "roundingMode": "TiesToEven" | |
| }, | |
| { | |
| "@id": "cxs:precision/contexts/u128", | |
| "@type": [ | |
| "PrecisionContext", | |
| "FixedPrecisionContext" | |
| ], | |
| "comment": "Fixed 128-bit unsigned integer.", | |
| "label": "128-bit Unsigned Integer", | |
| "precisionBits": 128, | |
| "precisionDigits": 39, | |
| "roundingMode": "TowardZero" | |
| }, | |
| { | |
| "@id": "cxs:precision/roundingModes/TiesToEven", | |
| "@type": "RoundingMode", | |
| "label": "Round to Nearest (Ties to Even)" | |
| }, | |
| { | |
| "@id": "cxs:precision/roundingModes/TiesToAway", | |
| "@type": "RoundingMode", | |
| "label": "Round to Nearest (Ties Away)" | |
| }, | |
| { | |
| "@id": "cxs:precision/roundingModes/TowardZero", | |
| "@type": "RoundingMode", | |
| "label": "Round Toward Zero" | |
| }, | |
| { | |
| "@id": "cxs:precision/roundingModes/TowardPositive", | |
| "@type": "RoundingMode", | |
| "label": "Round Toward Positive Infinity" | |
| }, | |
| { | |
| "@id": "cxs:precision/roundingModes/TowardNegative", | |
| "@type": "RoundingMode", | |
| "label": "Round Toward Negative Infinity" | |
| }, | |
| { | |
| "@id": "cxm:derivationLevels", | |
| "@type": "DerivationLevelsContainer", | |
| "comment": "Self-describing derivation hierarchy for categorical extensions", | |
| "contains": [ | |
| "cxm:derivationLevels/axiom", | |
| "cxm:derivationLevels/derivedConstant", | |
| "cxm:derivationLevels/formula", | |
| "cxm:derivationLevels/construction", | |
| "cxm:derivationLevels/extension" | |
| ], | |
| "label": "Derivation Levels" | |
| }, | |
| { | |
| "@id": "cxm:derivationLevels/axiom", | |
| "@type": "DerivationLevel", | |
| "canDeriveFrom": [], | |
| "comment": "Foundational axioms T=3 (triality) and O=8 (octave dimension)", | |
| "derivationDepth": 0, | |
| "label": "Axiom", | |
| "validationRule": "value ∈ {T=3, O=8}" | |
| }, | |
| { | |
| "@id": "cxm:derivationLevels/derivedConstant", | |
| "@type": "DerivationLevel", | |
| "canDeriveFrom": [ | |
| { | |
| "@id": "cxm:derivationLevels/axiom" | |
| } | |
| ], | |
| "comment": "Constants derived directly from axioms (c=24, pentality=5, septality=7)", | |
| "derivationDepth": 1, | |
| "label": "Derived Constant", | |
| "validationRule": "∃ formula f: f(T,O) = value" | |
| }, | |
| { | |
| "@id": "cxm:derivationLevels/formula", | |
| "@type": "DerivationLevel", | |
| "canDeriveFrom": [ | |
| { | |
| "@id": "cxm:derivationLevels/axiom" | |
| }, | |
| { | |
| "@id": "cxm:derivationLevels/derivedConstant" | |
| } | |
| ], | |
| "comment": "Formulas using axioms and derived constants (2^n, (n-1)*7)", | |
| "derivationDepth": 2, | |
| "label": "Formula", | |
| "validationRule": "∀ free variables v: v traceable to axiom or derivedConstant" | |
| }, | |
| { | |
| "@id": "cxm:derivationLevels/construction", | |
| "@type": "DerivationLevel", | |
| "canDeriveFrom": [ | |
| { | |
| "@id": "cxm:derivationLevels/axiom" | |
| }, | |
| { | |
| "@id": "cxm:derivationLevels/derivedConstant" | |
| }, | |
| { | |
| "@id": "cxm:derivationLevels/formula" | |
| } | |
| ], | |
| "comment": "TypeConstructor outputs (tower levels, transitions)", | |
| "derivationDepth": 3, | |
| "label": "Construction", | |
| "validationRule": "TypeConstructor output with all rules traced" | |
| }, | |
| { | |
| "@id": "cxm:derivationLevels/extension", | |
| "@type": "DerivationLevel", | |
| "canDeriveFrom": [ | |
| { | |
| "@id": "cxm:derivationLevels/axiom" | |
| }, | |
| { | |
| "@id": "cxm:derivationLevels/derivedConstant" | |
| }, | |
| { | |
| "@id": "cxm:derivationLevels/formula" | |
| }, | |
| { | |
| "@id": "cxm:derivationLevels/construction" | |
| } | |
| ], | |
| "comment": "New projections, operators, correspondences", | |
| "derivationDepth": 4, | |
| "label": "Extension", | |
| "validationRule": "Extension with complete derivation chain" | |
| }, | |
| { | |
| "@id": "cxm:extensionPoints", | |
| "@type": "ExtensionPointsContainer", | |
| "comment": "Categorical extension points for self-describing extensibility", | |
| "contains": [ | |
| "cxm:extensionPoints/projectionExtensionPoint", | |
| "cxm:extensionPoints/operatorExtensionPoint", | |
| "cxm:extensionPoints/correspondenceExtensionPoint", | |
| "cxm:extensionPoints/numericDomainExtensionPoint" | |
| ], | |
| "label": "Extension Points" | |
| }, | |
| { | |
| "@id": "cxm:extensionPoints/projectionExtensionPoint", | |
| "@type": "ExtensionPoint", | |
| "codomainCategory": "TargetDomain", | |
| "comment": "Extension point for adding new projections with phase coherence", | |
| "domainCategory": "CategoricalX", | |
| "extendsConstructor": [ | |
| { | |
| "@id": "cxm:constructors/projectionGenerator" | |
| } | |
| ], | |
| "hasCompositionLaw": [ | |
| { | |
| "@id": "cxm:compositionLaws/functoriality" | |
| }, | |
| { | |
| "@id": "cxm:compositionLaws/identity" | |
| } | |
| ], | |
| "label": "Projection Extension Point", | |
| "maxDerivationLevel": 4, | |
| "requiredAxioms": [ | |
| "T", | |
| "O" | |
| ], | |
| "structureKind": "Functor" | |
| }, | |
| { | |
| "@id": "cxm:extensionPoints/operatorExtensionPoint", | |
| "@type": "ExtensionPoint", | |
| "codomainCategory": "CategoricalX", | |
| "comment": "Extension point for adding new categorical operators", | |
| "domainCategory": "CategoricalX", | |
| "extendsConstructor": [ | |
| { | |
| "@id": "cxm:constructors/operatorGenerator" | |
| } | |
| ], | |
| "hasCompositionLaw": [ | |
| { | |
| "@id": "cxm:compositionLaws/functoriality" | |
| }, | |
| { | |
| "@id": "cxm:compositionLaws/identity" | |
| }, | |
| { | |
| "@id": "cxm:compositionLaws/associativity" | |
| } | |
| ], | |
| "label": "Operator Extension Point", | |
| "maxDerivationLevel": 4, | |
| "requiredAxioms": [ | |
| "T", | |
| "O" | |
| ], | |
| "structureKind": "Endofunctor" | |
| }, | |
| { | |
| "@id": "cxm:extensionPoints/correspondenceExtensionPoint", | |
| "@type": "ExtensionPoint", | |
| "codomainCategory": "Projection", | |
| "comment": "Extension point for adding correspondences between projections", | |
| "domainCategory": "Projection", | |
| "extendsConstructor": [ | |
| { | |
| "@id": "cxm:constructors/correspondenceGenerator" | |
| } | |
| ], | |
| "hasCompositionLaw": [ | |
| { | |
| "@id": "cxm:compositionLaws/naturality" | |
| } | |
| ], | |
| "label": "Correspondence Extension Point", | |
| "maxDerivationLevel": 4, | |
| "requiredAxioms": [ | |
| "T", | |
| "O" | |
| ], | |
| "structureKind": "NaturalTransformation" | |
| }, | |
| { | |
| "@id": "cxm:extensionPoints/numericDomainExtensionPoint", | |
| "@type": "ExtensionPoint", | |
| "codomainCategory": "NumericDomain", | |
| "comment": "Extension point for adding numeric domains to the embedding chain", | |
| "domainCategory": "NumericDomain", | |
| "extendsConstructor": [ | |
| { | |
| "@id": "cxm:constructors/numericDomainGenerator" | |
| } | |
| ], | |
| "hasCompositionLaw": [ | |
| { | |
| "@id": "cxm:compositionLaws/functoriality" | |
| } | |
| ], | |
| "label": "Numeric Domain Extension Point", | |
| "maxDerivationLevel": 3, | |
| "requiredAxioms": [ | |
| "O" | |
| ], | |
| "structureKind": "Functor" | |
| }, | |
| { | |
| "@id": "cxm:compositionLaws", | |
| "@type": "CompositionLawsContainer", | |
| "comment": "Categorical composition laws that extensions must preserve", | |
| "contains": [ | |
| "cxm:compositionLaws/associativity", | |
| "cxm:compositionLaws/identity", | |
| "cxm:compositionLaws/functoriality", | |
| "cxm:compositionLaws/naturality", | |
| "cxm:compositionLaws/triangleIdentity", | |
| "cxm:compositionLaws/monadAssociativity", | |
| "cxm:compositionLaws/monadUnit" | |
| ], | |
| "label": "Composition Laws" | |
| }, | |
| { | |
| "@id": "cxm:compositionLaws/associativity", | |
| "@type": "CompositionLaw", | |
| "appliesTo": [ | |
| "Functor", | |
| "Endofunctor" | |
| ], | |
| "comment": "Morphism composition is associative", | |
| "expression": "(f ∘ g) ∘ h = f ∘ (g ∘ h)", | |
| "label": "Associativity" | |
| }, | |
| { | |
| "@id": "cxm:compositionLaws/identity", | |
| "@type": "CompositionLaw", | |
| "appliesTo": [ | |
| "Functor", | |
| "Endofunctor" | |
| ], | |
| "comment": "Functors preserve identity morphisms", | |
| "expression": "F(id_A) = id_F(A)", | |
| "label": "Identity Preservation" | |
| }, | |
| { | |
| "@id": "cxm:compositionLaws/functoriality", | |
| "@type": "CompositionLaw", | |
| "appliesTo": [ | |
| "Functor", | |
| "Endofunctor" | |
| ], | |
| "comment": "Functors preserve composition", | |
| "expression": "F(g ∘ f) = F(g) ∘ F(f)", | |
| "label": "Functoriality" | |
| }, | |
| { | |
| "@id": "cxm:compositionLaws/naturality", | |
| "@type": "CompositionLaw", | |
| "appliesTo": [ | |
| "NaturalTransformation" | |
| ], | |
| "comment": "Natural transformation components commute with morphisms", | |
| "expression": "G(f) ∘ η_A = η_B ∘ F(f)", | |
| "label": "Naturality" | |
| }, | |
| { | |
| "@id": "cxm:compositionLaws/triangleIdentity", | |
| "@type": "CompositionLaw", | |
| "appliesTo": [ | |
| "Adjunction" | |
| ], | |
| "comment": "Adjunction unit and counit satisfy triangle identities", | |
| "expression": "(ε_F) ∘ (F η) = id_F", | |
| "label": "Triangle Identity" | |
| }, | |
| { | |
| "@id": "cxm:compositionLaws/monadAssociativity", | |
| "@type": "CompositionLaw", | |
| "appliesTo": [ | |
| "Monad" | |
| ], | |
| "comment": "Monad multiplication is associative", | |
| "expression": "μ ∘ T(μ) = μ ∘ μ_T", | |
| "label": "Monad Associativity" | |
| }, | |
| { | |
| "@id": "cxm:compositionLaws/monadUnit", | |
| "@type": "CompositionLaw", | |
| "appliesTo": [ | |
| "Monad" | |
| ], | |
| "comment": "Monad unit is identity for multiplication", | |
| "expression": "μ ∘ T(η) = μ ∘ η_T = id", | |
| "label": "Monad Unit Laws" | |
| }, | |
| { | |
| "@id": "cxm:universalProperties", | |
| "@type": "UniversalPropertiesContainer", | |
| "comment": "Universal properties defining objects by their mapping properties", | |
| "contains": [ | |
| "cxm:universalProperties/towerLevelUP", | |
| "cxm:universalProperties/projectionUP", | |
| "cxm:universalProperties/correspondenceUP" | |
| ], | |
| "label": "Universal Properties" | |
| }, | |
| { | |
| "@id": "cxm:universalProperties/towerLevelUP", | |
| "@type": "UniversalProperty", | |
| "comment": "Tower levels are free constructions via Cayley-Dickson doubling", | |
| "diagramShape": "Level(n) -> Level(n+1)", | |
| "existenceCondition": "∀ algebra A with dim=2^n, ∃ D_n(A) with dim=2^(n+1)", | |
| "hasDerivationSource": { | |
| "sourceFirstOrder": "O", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "inCategory": "NormedDivisionAlgebra", | |
| "label": "Tower Level Universal Property", | |
| "uniquenessCondition": "D_n(A) is unique up to isomorphism", | |
| "universalArrow": "Cayley-Dickson doubling: D_n", | |
| "universalPropertyKind": "Free" | |
| }, | |
| { | |
| "@id": "cxm:universalProperties/projectionUP", | |
| "@type": "UniversalProperty", | |
| "comment": "Projections are left adjoints to inclusion functors", | |
| "diagramShape": "CategoricalX -> TargetCategory", | |
| "existenceCondition": "∀ target T, ∃ unique factorization through π", | |
| "hasDerivationSource": { | |
| "sourceSecondOrder": "T,O", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "inCategory": "CAT", | |
| "label": "Projection Universal Property", | |
| "uniquenessCondition": "π is unique up to natural isomorphism", | |
| "universalArrow": "π: Categorical X -> Target", | |
| "universalPropertyKind": "LeftAdjoint" | |
| }, | |
| { | |
| "@id": "cxm:universalProperties/correspondenceUP", | |
| "@type": "UniversalProperty", | |
| "comment": "Correspondences are pullbacks (limits) of projection spans", | |
| "diagramShape": "Span(Projection, Projection)", | |
| "existenceCondition": "∀ projections P₁, P₂, ∃ correspondence C", | |
| "hasDerivationSource": { | |
| "sourceSecondOrder": "T,O", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "inCategory": "CAT", | |
| "label": "Correspondence Universal Property", | |
| "uniquenessCondition": "C is universal among spans", | |
| "universalArrow": "pullback square", | |
| "universalPropertyKind": "Limit" | |
| }, | |
| { | |
| "@id": "cxm:freeConstructions", | |
| "@type": "FreeConstructionsContainer", | |
| "comment": "Free constructions generating minimal structures via universal property", | |
| "contains": [ | |
| "cxm:freeConstructions/freeAlgebra", | |
| "cxm:freeConstructions/freeProjection" | |
| ], | |
| "label": "Free Constructions" | |
| }, | |
| { | |
| "@id": "cxm:freeConstructions/freeAlgebra", | |
| "@type": "FreeConstruction", | |
| "baseStructure": "Set", | |
| "comment": "Free normed division algebra. Constrained by O=8: only R, C, H, O exist", | |
| "forgetfulFunctor": "underlyingSet", | |
| "hasDerivationSource": { | |
| "sourceFirstOrder": "O", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "label": "Free Normed Division Algebra", | |
| "satisfiesUniversalProperty": { | |
| "@id": "cxm:universalProperties/towerLevelUP" | |
| }, | |
| "targetCategory": "NormedDivisionAlgebra" | |
| }, | |
| { | |
| "@id": "cxm:freeConstructions/freeProjection", | |
| "@type": "FreeConstruction", | |
| "baseStructure": "Category", | |
| "comment": "Free projection functor from Categorical X", | |
| "forgetfulFunctor": "underlyingCategory", | |
| "hasDerivationSource": { | |
| "sourceSecondOrder": "T,O", | |
| "tracesToAxiom": [ | |
| { | |
| "@id": "cxs:axioms/U" | |
| }, | |
| { | |
| "@id": "cxs:axioms/D" | |
| } | |
| ] | |
| }, | |
| "label": "Free Projection", | |
| "satisfiesUniversalProperty": { | |
| "@id": "cxm:universalProperties/projectionUP" | |
| }, | |
| "targetCategory": "Projection" | |
| }, | |
| { | |
| "@id": "cxs:derivation-chains", | |
| "@type": "DerivationChainsContainer", | |
| "comment": "Concrete derivation chains proving values trace back to axioms T=3, O=8", | |
| "contains": [ | |
| "cxs:derivation-chains/0", | |
| "cxs:derivation-chains/1", | |
| "cxs:derivation-chains/2", | |
| "cxs:derivation-chains/3", | |
| "cxs:derivation-chains/4", | |
| "cxs:derivation-chains/5" | |
| ], | |
| "label": "Derivation Chains" | |
| }, | |
| { | |
| "@id": "cxs:derivation-chains/c", | |
| "@type": "DerivationChain", | |
| "isValid": true, | |
| "label": "C Derivation Chain", | |
| "produces": "C", | |
| "steps": [ | |
| { | |
| "dependsOn": [ | |
| "T", | |
| "O" | |
| ], | |
| "produces": "C", | |
| "via": "Formula(\"C = T × O = 3 × 8 = 24\")" | |
| } | |
| ], | |
| "terminalAxioms": [ | |
| "T", | |
| "O" | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:derivation-chains/q", | |
| "@type": "DerivationChain", | |
| "isValid": true, | |
| "label": "Q Derivation Chain", | |
| "produces": "Q", | |
| "steps": [ | |
| { | |
| "dependsOn": [ | |
| "T" | |
| ], | |
| "produces": "Q", | |
| "via": "Formula(\"Q = 2^(T-1) = 2^2 = 4\")" | |
| } | |
| ], | |
| "terminalAxioms": [ | |
| "T" | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:derivation-chains/d4", | |
| "@type": "DerivationChain", | |
| "isValid": true, | |
| "label": "D4 Derivation Chain", | |
| "produces": "D(4)", | |
| "steps": [ | |
| { | |
| "dependsOn": [ | |
| "T", | |
| "O" | |
| ], | |
| "produces": "C", | |
| "via": "Formula(\"C = T × O = 24\")" | |
| }, | |
| { | |
| "dependsOn": [ | |
| "O" | |
| ], | |
| "produces": "SEPTALITY", | |
| "via": "Formula(\"SEPTALITY = O - 1 = 7\")" | |
| }, | |
| { | |
| "dependsOn": [ | |
| "C", | |
| "SEPTALITY" | |
| ], | |
| "produces": "D(4)", | |
| "via": "Formula(\"D(4) = SEPTALITY × C = 7 × 24 = 168\")" | |
| } | |
| ], | |
| "terminalAxioms": [ | |
| "T", | |
| "O" | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:derivation-chains/d7", | |
| "@type": "DerivationChain", | |
| "isValid": true, | |
| "label": "D7 Derivation Chain", | |
| "produces": "D(7)", | |
| "steps": [ | |
| { | |
| "dependsOn": [ | |
| "T", | |
| "O" | |
| ], | |
| "produces": "Ω_{2,3,5,7}", | |
| "via": "Formula(\"Gauge extension via E_5, E_7 functors\")" | |
| }, | |
| { | |
| "dependsOn": [ | |
| "Ω_{2,3,5,7}" | |
| ], | |
| "produces": "cf(7)", | |
| "via": "RuntimeConstruct(\"SpectralCoherence.cofactor(7)\")" | |
| }, | |
| { | |
| "dependsOn": [ | |
| "cf(7)", | |
| "Ω_{2,3,5,7}" | |
| ], | |
| "produces": "D(7)", | |
| "via": "RuntimeConstruct(\"Hub.compute_dedekind(7)\")" | |
| } | |
| ], | |
| "terminalAxioms": [ | |
| "T", | |
| "O" | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:derivation-chains/d8", | |
| "@type": "DerivationChain", | |
| "isValid": true, | |
| "label": "D8 Derivation Chain", | |
| "produces": "D(8)", | |
| "steps": [ | |
| { | |
| "dependsOn": [ | |
| "T", | |
| "O" | |
| ], | |
| "produces": "D(7)", | |
| "via": "RuntimeConstruct(\"Hub.compute_dedekind(7)\")" | |
| }, | |
| { | |
| "dependsOn": [ | |
| "D(7)" | |
| ], | |
| "produces": "E_11", | |
| "via": "Formula(\"E_11: Ω_7 → Ω_8 extension functor\")" | |
| }, | |
| { | |
| "dependsOn": [ | |
| "D(7)", | |
| "E_11" | |
| ], | |
| "produces": "D(8)", | |
| "via": "RuntimeConstruct(\"TransferDecomposition.extend(7, 11)\")" | |
| } | |
| ], | |
| "terminalAxioms": [ | |
| "T", | |
| "O" | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:derivation-chains/d9", | |
| "@type": "DerivationChain", | |
| "isValid": true, | |
| "label": "D9 Derivation Chain", | |
| "produces": "D(9)", | |
| "steps": [ | |
| { | |
| "dependsOn": [ | |
| "T", | |
| "O" | |
| ], | |
| "produces": "D(8)", | |
| "via": "RuntimeConstruct(\"TransferDecomposition.extend(7, 11)\")" | |
| }, | |
| { | |
| "dependsOn": [ | |
| "D(8)" | |
| ], | |
| "produces": "E_13", | |
| "via": "Formula(\"E_13: Ω_8 → Ω_9 extension functor\")" | |
| }, | |
| { | |
| "dependsOn": [ | |
| "D(8)", | |
| "E_13" | |
| ], | |
| "produces": "D(9)", | |
| "via": "RuntimeConstruct(\"TransferDecomposition.extend(8, 13)\")" | |
| } | |
| ], | |
| "terminalAxioms": [ | |
| "T", | |
| "O" | |
| ] | |
| }, | |
| { | |
| "@id": "cxm:extensions", | |
| "@type": "ExtensionsContainer", | |
| "comment": "Container for concrete extensions implementing extension points", | |
| "contains": [ | |
| "cxs:extensions/arithmeticProjectionExtension", | |
| "cxs:extensions/tensorOperatorExtension", | |
| "cxs:extensions/spectralArithmeticExtension" | |
| ], | |
| "label": "Extensions" | |
| }, | |
| { | |
| "@id": "cxs:extensions/arithmeticProjectionExtension", | |
| "@type": "Extension", | |
| "comment": "Projects categorical structure to prime/divisibility structures", | |
| "derivationChain": { | |
| "steps": [ | |
| { | |
| "dependsOn": [ | |
| "T", | |
| "O" | |
| ], | |
| "produces": "ArithmeticProjection", | |
| "via": "RuntimeConstruct(\"ArithmeticProjection.evaluate\")" | |
| } | |
| ], | |
| "terminalAxioms": [ | |
| "T", | |
| "O" | |
| ] | |
| }, | |
| "implements": { | |
| "@id": "cxm:extensionPoints/projectionExtensionPoint" | |
| }, | |
| "label": "Arithmetic Projection Extension", | |
| "morphismMapping": "Maps morphisms to divisibility-preserving maps", | |
| "objectMapping": "Maps categorical objects to prime distributions", | |
| "providesConstructors": [ | |
| "ArithmeticProjectionConstructor" | |
| ], | |
| "structureKind": "Functor", | |
| "verificationStatus": "Verified" | |
| }, | |
| { | |
| "@id": "cxs:extensions/tensorOperatorExtension", | |
| "@type": "Extension", | |
| "comment": "Provides tensor product as categorical operator", | |
| "derivationChain": { | |
| "steps": [ | |
| { | |
| "dependsOn": [ | |
| "O" | |
| ], | |
| "produces": "TensorOperator", | |
| "via": "Formula(\"Tensor structure from division algebra hierarchy\")" | |
| } | |
| ], | |
| "terminalAxioms": [ | |
| "O" | |
| ] | |
| }, | |
| "implements": { | |
| "@id": "cxm:extensionPoints/operatorExtensionPoint" | |
| }, | |
| "label": "Tensor Operator Extension", | |
| "morphismMapping": "Maps (f, g) to f ⊗ g", | |
| "objectMapping": "Maps (A, B) to A ⊗ B", | |
| "providesConstructors": [ | |
| "TensorConstruct" | |
| ], | |
| "structureKind": "Endofunctor", | |
| "verificationStatus": "Verified" | |
| }, | |
| { | |
| "@id": "cxs:extensions/spectralArithmeticExtension", | |
| "@type": "Extension", | |
| "comment": "Connects spectral eigenvalues to prime distributions via explicit formula", | |
| "derivationChain": { | |
| "steps": [ | |
| { | |
| "dependsOn": [ | |
| "T", | |
| "O" | |
| ], | |
| "produces": "SpectralProjection", | |
| "via": "RuntimeConstruct(\"SpectralProjection.evaluate\")" | |
| }, | |
| { | |
| "dependsOn": [ | |
| "T", | |
| "O" | |
| ], | |
| "produces": "ArithmeticProjection", | |
| "via": "RuntimeConstruct(\"ArithmeticProjection.evaluate\")" | |
| }, | |
| { | |
| "dependsOn": [ | |
| "SpectralProjection", | |
| "ArithmeticProjection" | |
| ], | |
| "produces": "SpectralArithmeticCorrespondence", | |
| "via": "Formula(\"ψ(x) = x - Σ_ρ x^ρ/ρ (explicit formula)\")" | |
| } | |
| ], | |
| "terminalAxioms": [ | |
| "T", | |
| "O" | |
| ] | |
| }, | |
| "implements": { | |
| "@id": "cxm:extensionPoints/correspondenceExtensionPoint" | |
| }, | |
| "label": "Spectral-Arithmetic Correspondence Extension", | |
| "morphismMapping": null, | |
| "objectMapping": null, | |
| "providesConstructors": [ | |
| "CorrespondenceConstruct" | |
| ], | |
| "structureKind": "NaturalTransformation", | |
| "verificationStatus": "Verified" | |
| }, | |
| { | |
| "@id": "cxs:universal/operator", | |
| "@type": "UniversalOperator", | |
| "comment": "Categorical limit of gauge extensions with spectral eigenvalue structure", | |
| "eigenvalueSpectrum": { | |
| "leadingDominant": { | |
| "derivation": "SEPTALITY = 7, mult = T - 1 = 2", | |
| "multiplicity": 2, | |
| "value": 7 | |
| }, | |
| "leadingSubdominant": { | |
| "derivation": "-1, mult = (T-1)(O-1) = 14", | |
| "multiplicity": 14, | |
| "value": -1 | |
| }, | |
| "tonicDominant": { | |
| "derivation": "O + 2 = 10", | |
| "multiplicity": 1, | |
| "value": 10 | |
| }, | |
| "tonicSubdominant": { | |
| "derivation": "2, mult = O - 1 = 7", | |
| "multiplicity": 7, | |
| "value": 2 | |
| } | |
| }, | |
| "label": "Universal Operator M_∞", | |
| "spectralData": { | |
| "eigenvalues": [ | |
| { | |
| "multiplicity": 1, | |
| "role": "TonicDominant", | |
| "value": 10 | |
| }, | |
| { | |
| "multiplicity": 7, | |
| "role": "TonicSubdominant", | |
| "value": 2 | |
| }, | |
| { | |
| "multiplicity": 2, | |
| "role": "LeadingDominant", | |
| "value": 7 | |
| }, | |
| { | |
| "multiplicity": 14, | |
| "role": "LeadingSubdominant", | |
| "value": -1 | |
| } | |
| ], | |
| "gammaValues": [ | |
| { | |
| "formula": "ln(λ₁/λ₃)", | |
| "id": "γ₁", | |
| "numeric": 0.3567, | |
| "symbolic": "ln(10/7)" | |
| }, | |
| { | |
| "formula": "ln(λ₁/λ₂)", | |
| "id": "γ₂", | |
| "numeric": 1.6094, | |
| "symbolic": "ln(5)" | |
| }, | |
| { | |
| "formula": "ln(λ₃/λ₂)", | |
| "id": "γ₃", | |
| "numeric": 1.2528, | |
| "symbolic": "ln(7/2)" | |
| }, | |
| { | |
| "formula": "ln(C)", | |
| "id": "γ₄", | |
| "numeric": 3.1781, | |
| "symbolic": "ln(24)" | |
| } | |
| ], | |
| "spectralRadius": 10 | |
| }, | |
| "spectralDimension": 24, | |
| "trace": 24, | |
| "traceDerivation": "1×10 + 7×2 + 2×7 + 14×(-1) = 10 + 14 + 14 - 14 = 24 = C" | |
| }, | |
| { | |
| "@id": "cxs:universal/gauge-tower", | |
| "@type": "GaugeTower", | |
| "comment": "Directed system of gauge extensions {2,3}→{2,3,5}→...", | |
| "label": "Gauge Tower", | |
| "maxLevel": 5 | |
| }, | |
| { | |
| "@id": "cxs:universal/gauge-tower/level-0", | |
| "@type": "GaugeLevel", | |
| "boundary": 12288, | |
| "index": 0, | |
| "label": "Gauge Level 0", | |
| "modulus": 96, | |
| "partOf": "cxs:universal/gauge-tower", | |
| "primes": [ | |
| 2, | |
| 3 | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:universal/gauge-tower/level-1", | |
| "@type": "GaugeLevel", | |
| "boundary": 61440, | |
| "index": 1, | |
| "label": "Gauge Level 1", | |
| "modulus": 480, | |
| "partOf": "cxs:universal/gauge-tower", | |
| "primes": [ | |
| 2, | |
| 3, | |
| 5 | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:universal/gauge-tower/level-2", | |
| "@type": "GaugeLevel", | |
| "boundary": 430080, | |
| "index": 2, | |
| "label": "Gauge Level 2", | |
| "modulus": 3360, | |
| "partOf": "cxs:universal/gauge-tower", | |
| "primes": [ | |
| 2, | |
| 3, | |
| 5, | |
| 7 | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:universal/gauge-tower/level-3", | |
| "@type": "GaugeLevel", | |
| "boundary": 4730880, | |
| "index": 3, | |
| "label": "Gauge Level 3", | |
| "modulus": 36960, | |
| "partOf": "cxs:universal/gauge-tower", | |
| "primes": [ | |
| 2, | |
| 3, | |
| 5, | |
| 7, | |
| 11 | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:universal/gauge-tower/level-4", | |
| "@type": "GaugeLevel", | |
| "boundary": 61501440, | |
| "index": 4, | |
| "label": "Gauge Level 4", | |
| "modulus": 480480, | |
| "partOf": "cxs:universal/gauge-tower", | |
| "primes": [ | |
| 2, | |
| 3, | |
| 5, | |
| 7, | |
| 11, | |
| 13 | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:universal/gauge-tower/level-5", | |
| "@type": "GaugeLevel", | |
| "boundary": 1045524480, | |
| "index": 5, | |
| "label": "Gauge Level 5", | |
| "modulus": 8168160, | |
| "partOf": "cxs:universal/gauge-tower", | |
| "primes": [ | |
| 2, | |
| 3, | |
| 5, | |
| 7, | |
| 11, | |
| 13, | |
| 17 | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:universal/spectral-triple", | |
| "@type": "SpectralTriple", | |
| "altSpecDim": 480, | |
| "comment": "AltSpec × ZDSpec × JordSpec composition structure", | |
| "jordSpecDim": 27, | |
| "label": "Spectral Triple", | |
| "totalDim": 522, | |
| "zdSpecDim": 15 | |
| }, | |
| { | |
| "@id": "cxs:universal/spectral-primes/2549", | |
| "@type": "SpectralPrime", | |
| "appearsIn": { | |
| "level": 8, | |
| "position": 6 | |
| }, | |
| "derivation": "(2 × PENTALITY²)² + SEPTALITY² = (2×25)² + 7² = 50² + 49", | |
| "derivationStatus": "axiom", | |
| "label": "Spectral Prime 2549", | |
| "mod96": 53, | |
| "quadraticForm": "50² + 49", | |
| "value": 2549 | |
| }, | |
| { | |
| "@id": "cxs:universal/spectral-primes/431", | |
| "@type": "SpectralPrime", | |
| "appearsIn": { | |
| "level": 9, | |
| "position": 6 | |
| }, | |
| "derivation": "(T × SEPTALITY)² - TONIC_DOMINANT = (3×7)² - 10 = 21² - 10", | |
| "derivationStatus": "axiom", | |
| "label": "Spectral Prime 431", | |
| "mod96": 47, | |
| "quadraticForm": "21² - 10", | |
| "value": 431 | |
| }, | |
| { | |
| "@id": "cxs:universal/spectral-primes/467", | |
| "@type": "SpectralPrime", | |
| "appearsIn": { | |
| "level": 9, | |
| "position": 7 | |
| }, | |
| "derivation": "(2(O + T))² - (TD + SEPTALITY) = (2×11)² - 17 = 22² - 17", | |
| "derivationStatus": "axiom", | |
| "label": "Spectral Prime 467", | |
| "mod96": 83, | |
| "quadraticForm": "22² - 17", | |
| "value": 467 | |
| }, | |
| { | |
| "@id": "cxs:universal/spectral-primes/641579", | |
| "@type": "SpectralPrime", | |
| "appearsIn": { | |
| "level": 9, | |
| "position": 7 | |
| }, | |
| "derivation": "(T²(T⁴ + O))² - 2(O + T) = (9×89)² - 22 = 801² - 22", | |
| "derivationStatus": "axiom", | |
| "label": "Spectral Prime 641579", | |
| "mod96": 11, | |
| "quadraticForm": "801² - 22", | |
| "value": 641579 | |
| }, | |
| { | |
| "@id": "cxs:universal/spectral-primes/1171", | |
| "@type": "SpectralPrime", | |
| "appearsIn": { | |
| "level": 9, | |
| "position": 0 | |
| }, | |
| "derivation": "(J + SEPTALITY)² + ZD_SPEC_DIM = (27 + 7)² + 15 = 34² + 15", | |
| "derivationStatus": "axiom", | |
| "label": "Spectral Prime 1171", | |
| "mod96": 19, | |
| "quadraticForm": "34² + 15", | |
| "value": 1171 | |
| }, | |
| { | |
| "@id": "cxs:universal/spectral-primes/12401", | |
| "@type": "SpectralPrime", | |
| "appearsIn": { | |
| "level": 9, | |
| "position": 0 | |
| }, | |
| "derivation": "(R + ZD_SPEC_DIM)² + Q × HAPPY = (96 + 15)² + 4 × 20 = 111² + 80", | |
| "derivationStatus": "axiom", | |
| "label": "Spectral Prime 12401", | |
| "mod96": 17, | |
| "quadraticForm": "111² + 80", | |
| "value": 12401 | |
| }, | |
| { | |
| "@id": "cxs:universal/phase-bridges/401", | |
| "@type": "PhaseBridge", | |
| "derivation": "(2×TD)² + 1 = 20² + 1", | |
| "label": "Phase Bridge 401", | |
| "linksPhases": [ | |
| "Overtone", | |
| "Tonic" | |
| ], | |
| "value": 401 | |
| }, | |
| { | |
| "@id": "cxs:universal/phase-bridges/433", | |
| "@type": "PhaseBridge", | |
| "derivation": "Q²×J + 1 = 16×27 + 1", | |
| "label": "Phase Bridge 433", | |
| "linksPhases": [ | |
| "Tonic", | |
| "Jordan" | |
| ], | |
| "value": 433 | |
| }, | |
| { | |
| "@id": "cxs:diagrams/limit", | |
| "@type": "Limit", | |
| "comment": "Categorical limit representing the universal property of M_∞", | |
| "label": "Universal Limit", | |
| "overDiagram": "cxs:universal/gauge-tower" | |
| }, | |
| { | |
| "@id": "cxs:diagrams/cocone", | |
| "@type": "Cone", | |
| "apex": "cxs:universal/operator", | |
| "base": "cxs:universal/gauge-tower", | |
| "comment": "Cocone over the gauge tower directed system", | |
| "label": "Gauge Cocone" | |
| }, | |
| { | |
| "@id": "cxs:universal/functors", | |
| "@type": "FunctorsContainer", | |
| "comment": "Gauge extension functors E_p forming morphisms in the gauge tower", | |
| "contains": [ | |
| "cxs:universal/functors/E5", | |
| "cxs:universal/functors/E7", | |
| "cxs:universal/functors/E11", | |
| "cxs:universal/functors/E13", | |
| "cxs:universal/functors/E17" | |
| ], | |
| "label": "Extension Functors" | |
| }, | |
| { | |
| "@id": "cxs:universal/functors/E5", | |
| "@type": "ExtensionFunctor", | |
| "boundaryScale": 5, | |
| "comment": "Extension functor adding prime 5 to the gauge", | |
| "extensionPrime": 5, | |
| "extensionSource": "cxs:universal/gauge-tower/level-0", | |
| "extensionTarget": "cxs:universal/gauge-tower/level-1", | |
| "label": "E_5: Ω_{2,3} → Ω_{2,3,5}", | |
| "resonanceFactor": 5 | |
| }, | |
| { | |
| "@id": "cxs:universal/functors/E7", | |
| "@type": "ExtensionFunctor", | |
| "boundaryScale": 7, | |
| "comment": "Extension functor adding prime 7 to the gauge", | |
| "extensionPrime": 7, | |
| "extensionSource": "cxs:universal/gauge-tower/level-1", | |
| "extensionTarget": "cxs:universal/gauge-tower/level-2", | |
| "label": "E_7: Ω_{2,3,5} → Ω_{2,3,5,7}", | |
| "resonanceFactor": 7 | |
| }, | |
| { | |
| "@id": "cxs:universal/functors/E11", | |
| "@type": "ExtensionFunctor", | |
| "boundaryScale": 11, | |
| "comment": "Extension functor adding prime 11 to the gauge", | |
| "extensionPrime": 11, | |
| "extensionSource": "cxs:universal/gauge-tower/level-2", | |
| "extensionTarget": "cxs:universal/gauge-tower/level-3", | |
| "label": "E_11: Ω_{2,3,5,7} → Ω_{2,3,5,7,11}", | |
| "resonanceFactor": 11 | |
| }, | |
| { | |
| "@id": "cxs:universal/functors/E13", | |
| "@type": "ExtensionFunctor", | |
| "boundaryScale": 13, | |
| "comment": "Extension functor adding prime 13 to the gauge", | |
| "extensionPrime": 13, | |
| "extensionSource": "cxs:universal/gauge-tower/level-3", | |
| "extensionTarget": "cxs:universal/gauge-tower/level-4", | |
| "label": "E_13: Ω_{2,3,5,7,11} → Ω_{2,3,5,7,11,13}", | |
| "resonanceFactor": 13 | |
| }, | |
| { | |
| "@id": "cxs:universal/functors/E17", | |
| "@type": "ExtensionFunctor", | |
| "boundaryScale": 17, | |
| "comment": "Extension functor adding prime 17 to the gauge", | |
| "extensionPrime": 17, | |
| "extensionSource": "cxs:universal/gauge-tower/level-4", | |
| "extensionTarget": "cxs:universal/gauge-tower/level-5", | |
| "label": "E_17: Ω_{2,3,5,7,11,13} → Ω_{2,3,5,7,11,13,17}", | |
| "resonanceFactor": 17 | |
| }, | |
| { | |
| "@id": "cxs:universal/index-categories", | |
| "@type": "IndexCategoriesContainer", | |
| "comment": "Shape categories for categorical diagrams", | |
| "contains": [ | |
| "cxs:universal/index-categories/nat-indexed", | |
| "cxs:universal/index-categories/triality-indexed", | |
| "cxs:universal/index-categories/finite-linear" | |
| ], | |
| "label": "Index Categories" | |
| }, | |
| { | |
| "@id": "cxs:universal/index-categories/nat-indexed", | |
| "@type": "IndexCategory", | |
| "comment": "Natural number indexing for gauge tower levels", | |
| "indexingScheme": "ω (first infinite ordinal)", | |
| "label": "ℕ-indexed" | |
| }, | |
| { | |
| "@id": "cxs:universal/index-categories/triality-indexed", | |
| "@type": "IndexCategory", | |
| "comment": "ℤ/3ℤ-indexed for triality phases (Tonic, LeadingTone, Overtone)", | |
| "indexingScheme": "ℤ/3ℤ (cyclic group of order 3)", | |
| "label": "Triality-indexed" | |
| }, | |
| { | |
| "@id": "cxs:universal/index-categories/finite-linear", | |
| "@type": "IndexCategory", | |
| "comment": "Finite ordinal indexing for bounded computations", | |
| "indexingScheme": "[n] (finite ordinal)", | |
| "label": "Finite Linear" | |
| }, | |
| { | |
| "@id": "cxs:universal/projections", | |
| "@type": "LimitProjectionsContainer", | |
| "comment": "Canonical projections from universal limit to gauge levels", | |
| "contains": [ | |
| "cxs:universal/projections/pi0", | |
| "cxs:universal/projections/pi1", | |
| "cxs:universal/projections/pi2", | |
| "cxs:universal/projections/pi3", | |
| "cxs:universal/projections/pi4", | |
| "cxs:universal/projections/pi5" | |
| ], | |
| "label": "Limit Projections" | |
| }, | |
| { | |
| "@id": "cxs:universal/projections/pi0", | |
| "@type": "LimitProjection", | |
| "comment": "Projection to base gauge level", | |
| "label": "π₀: M_∞ → Ω_{2,3}", | |
| "projectionSource": "cxs:universal/operator", | |
| "projectionTarget": "cxs:universal/gauge-tower/level-0" | |
| }, | |
| { | |
| "@id": "cxs:universal/projections/pi1", | |
| "@type": "LimitProjection", | |
| "comment": "Projection to first extended gauge level", | |
| "label": "π₁: M_∞ → Ω_{2,3,5}", | |
| "projectionSource": "cxs:universal/operator", | |
| "projectionTarget": "cxs:universal/gauge-tower/level-1" | |
| }, | |
| { | |
| "@id": "cxs:universal/projections/pi2", | |
| "@type": "LimitProjection", | |
| "comment": "Projection to second extended gauge level", | |
| "label": "π₂: M_∞ → Ω_{2,3,5,7}", | |
| "projectionSource": "cxs:universal/operator", | |
| "projectionTarget": "cxs:universal/gauge-tower/level-2" | |
| }, | |
| { | |
| "@id": "cxs:universal/projections/pi3", | |
| "@type": "LimitProjection", | |
| "comment": "Projection to third extended gauge level", | |
| "label": "π₃: M_∞ → Ω_{2,3,5,7,11}", | |
| "projectionSource": "cxs:universal/operator", | |
| "projectionTarget": "cxs:universal/gauge-tower/level-3" | |
| }, | |
| { | |
| "@id": "cxs:universal/projections/pi4", | |
| "@type": "LimitProjection", | |
| "comment": "Projection to fourth extended gauge level", | |
| "label": "π₄: M_∞ → Ω_{2,3,5,7,11,13}", | |
| "projectionSource": "cxs:universal/operator", | |
| "projectionTarget": "cxs:universal/gauge-tower/level-4" | |
| }, | |
| { | |
| "@id": "cxs:universal/projections/pi5", | |
| "@type": "LimitProjection", | |
| "comment": "Projection to fifth extended gauge level", | |
| "label": "π₅: M_∞ → Ω_{2,3,5,7,11,13,17}", | |
| "projectionSource": "cxs:universal/operator", | |
| "projectionTarget": "cxs:universal/gauge-tower/level-5" | |
| }, | |
| { | |
| "@id": "cxs:gauges", | |
| "@type": "GaugesContainer", | |
| "comment": "Finite gauge structures for Dedekind hierarchy computation", | |
| "contains": [ | |
| "cxs:gauges/omega_2_3", | |
| "cxs:gauges/omega_2_3_5", | |
| "cxs:gauges/omega_2_3_5_7" | |
| ], | |
| "label": "Gauges", | |
| "partOf": "cxs:CategoricalX" | |
| }, | |
| { | |
| "@id": "cxs:gauges/omega_2_3", | |
| "@type": "Gauge", | |
| "comment": "Base gauge for D(0)-D(9)", | |
| "coversRange": [ | |
| 0, | |
| 9 | |
| ], | |
| "label": "Ω_{2,3}", | |
| "primes": [ | |
| 2, | |
| 3 | |
| ], | |
| "rank": 2 | |
| }, | |
| { | |
| "@id": "cxs:gauges/omega_2_3_5", | |
| "@type": "Gauge", | |
| "comment": "Extended gauge for D(10)-D(14)", | |
| "coversRange": [ | |
| 10, | |
| 14 | |
| ], | |
| "extendsGauge": "cxs:gauges/omega_2_3", | |
| "label": "Ω_{2,3,5}", | |
| "primes": [ | |
| 2, | |
| 3, | |
| 5 | |
| ], | |
| "rank": 3 | |
| }, | |
| { | |
| "@id": "cxs:gauges/omega_2_3_5_7", | |
| "@type": "Gauge", | |
| "comment": "Further extended gauge for D(15)-D(19)", | |
| "coversRange": [ | |
| 15, | |
| 19 | |
| ], | |
| "extendsGauge": "cxs:gauges/omega_2_3_5", | |
| "label": "Ω_{2,3,5,7}", | |
| "primes": [ | |
| 2, | |
| 3, | |
| 5, | |
| 7 | |
| ], | |
| "rank": 4 | |
| }, | |
| { | |
| "@id": "cxs:hubs", | |
| "@type": "HubsContainer", | |
| "comment": "Hub boundaries for categorical state management", | |
| "contains": [ | |
| "cxs:hubs/base", | |
| "cxs:hubs/extended" | |
| ], | |
| "label": "Hubs", | |
| "partOf": "cxs:CategoricalX" | |
| }, | |
| { | |
| "@id": "cxs:hubs/base", | |
| "@type": "Hub", | |
| "capacity": 10, | |
| "comment": "Hub for D(0)-D(9) computation within Ω_{2,3}", | |
| "gauge": "cxs:gauges/omega_2_3", | |
| "label": "Base Hub" | |
| }, | |
| { | |
| "@id": "cxs:hubs/extended", | |
| "@type": "Hub", | |
| "capacity": 5, | |
| "comment": "Hub for D(10)+ computation via gauge extension", | |
| "extendsHub": "cxs:hubs/base", | |
| "gauge": "cxs:gauges/omega_2_3_5", | |
| "label": "Extended Hub" | |
| }, | |
| { | |
| "@id": "cxs:rules/projection-rules", | |
| "@type": "RulesContainer", | |
| "comment": "Numeric rules for computing gauge projections", | |
| "contains": [ | |
| "cxs:rules/spectral-projection", | |
| "cxs:rules/spectral-sum" | |
| ], | |
| "label": "Projection Rules" | |
| }, | |
| { | |
| "@id": "cxs:rules/spectral-projection", | |
| "@type": "NumericProjectionRule", | |
| "eigenvalues": [ | |
| 10, | |
| 2, | |
| 7, | |
| -1 | |
| ], | |
| "formula": "πₖ(M_∞)(x) = S(n) / (n × ln(|Ω_k|))", | |
| "gaugeBoundaries": [ | |
| { | |
| "boundary": "12288", | |
| "level": "level0" | |
| }, | |
| { | |
| "boundary": "61440", | |
| "level": "level1" | |
| }, | |
| { | |
| "boundary": "430080", | |
| "level": "level2" | |
| }, | |
| { | |
| "boundary": "4730880", | |
| "level": "level3" | |
| }, | |
| { | |
| "boundary": "61501440", | |
| "level": "level4" | |
| }, | |
| { | |
| "boundary": "1045524480", | |
| "level": "level5" | |
| } | |
| ], | |
| "label": "Spectral Projection Formula", | |
| "multiplicities": [ | |
| 1, | |
| 7, | |
| 2, | |
| 14 | |
| ], | |
| "variables": { | |
| "S(n)": "Spectral sum Σᵢ mᵢ λᵢⁿ", | |
| "n": "log₁₀(x)", | |
| "|Ω_k|": "Gauge boundary cardinality at level k" | |
| } | |
| }, | |
| { | |
| "@id": "cxs:rules/spectral-sum", | |
| "@type": "NumericRule", | |
| "comment": "Sum over eigenvalues with multiplicities", | |
| "formula": "S(n) = Σᵢ mᵢ × λᵢⁿ = 1×10ⁿ + 7×2ⁿ + 2×7ⁿ + 14×(-1)ⁿ", | |
| "label": "Spectral Sum Formula", | |
| "traceConnection": "S(1) = Trace = 24 = C" | |
| }, | |
| { | |
| "@id": "cxs:sequences/ground-truth", | |
| "@type": "SequenceContainer", | |
| "comment": "Reference sequences for spectral validation", | |
| "contains": [ | |
| "cxs:sequences/prime-pi", | |
| "cxs:sequences/chebyshev-psi", | |
| "cxs:sequences/chebyshev-theta" | |
| ], | |
| "label": "Ground Truth Sequences" | |
| }, | |
| { | |
| "@id": "cxs:sequences/prime-pi", | |
| "@type": "GroundTruthSequence", | |
| "formula": "π(x) = |{p ≤ x : p prime}|", | |
| "label": "Prime Counting Function π(x)", | |
| "oeis": "A000720", | |
| "sampleValues": [ | |
| { | |
| "value": 25, | |
| "x": 100 | |
| }, | |
| { | |
| "value": 168, | |
| "x": 1000 | |
| }, | |
| { | |
| "value": 1229, | |
| "x": 10000 | |
| }, | |
| { | |
| "value": 1472, | |
| "x": 12288 | |
| }, | |
| { | |
| "value": 6170, | |
| "x": 61440 | |
| }, | |
| { | |
| "value": 9592, | |
| "x": 100000 | |
| } | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:sequences/chebyshev-psi", | |
| "@type": "GroundTruthSequence", | |
| "comment": "Sum of log(p) over prime powers up to x", | |
| "formula": "ψ(x) = Σ_{p^k ≤ x} ln(p)", | |
| "label": "Chebyshev ψ Function", | |
| "sampleValues": [ | |
| { | |
| "value": 90.02, | |
| "x": 100 | |
| }, | |
| { | |
| "value": 998.49, | |
| "x": 1000 | |
| }, | |
| { | |
| "value": 10015.16, | |
| "x": 10000 | |
| }, | |
| { | |
| "value": 99986.93, | |
| "x": 100000 | |
| } | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:sequences/chebyshev-theta", | |
| "@type": "GroundTruthSequence", | |
| "comment": "Sum of log(p) over primes up to x", | |
| "formula": "θ(x) = Σ_{p ≤ x} ln(p)", | |
| "label": "Chebyshev θ Function", | |
| "sampleValues": [ | |
| { | |
| "value": 80.45, | |
| "x": 100 | |
| }, | |
| { | |
| "value": 976.72, | |
| "x": 1000 | |
| }, | |
| { | |
| "value": 9950.23, | |
| "x": 10000 | |
| } | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:assertions/falsifiable", | |
| "@type": "AssertionsContainer", | |
| "comment": "Testable claims about spectral model behavior", | |
| "contains": [ | |
| "cxs:assertions/error-bound", | |
| "cxs:assertions/trace-invariant", | |
| "cxs:assertions/convergence" | |
| ], | |
| "label": "Falsifiable Assertions" | |
| }, | |
| { | |
| "@id": "cxs:assertions/error-bound", | |
| "@type": "FalsifiableAssertion", | |
| "bound": 0.6, | |
| "claim": "Error scaling exponent α < 0.6 for π(x) predictions", | |
| "formula": "|π(x) - π_spec(x)| = O(x^α)", | |
| "label": "Error Exponent Bound", | |
| "rhConsistency": "α < 0.5 would imply RH-consistent error bounds", | |
| "testMethod": "Linear regression on log-log plot of |error| vs x" | |
| }, | |
| { | |
| "@id": "cxs:assertions/trace-invariant", | |
| "@type": "FalsifiableAssertion", | |
| "claim": "Eigenvalue trace equals fundamental constant C", | |
| "derivation": "C = T × O = 3 × 8 = 24", | |
| "formula": "Σᵢ mᵢλᵢ = 1×10 + 7×2 + 2×7 + 14×(-1) = 24 = C", | |
| "label": "Trace Invariant", | |
| "value": 24, | |
| "verifiable": true | |
| }, | |
| { | |
| "@id": "cxs:assertions/convergence", | |
| "@type": "FalsifiableAssertion", | |
| "claim": "Spectral predictions improve at higher gauge levels", | |
| "formula": "error(k+1) / error(k) < 1 as gauge level k increases", | |
| "label": "Gauge Convergence", | |
| "testMethod": "Compare prediction errors across gauge boundaries" | |
| }, | |
| { | |
| "@id": "cxs:categories", | |
| "@type": "CategoriesContainer", | |
| "comment": "First-class category objects for type-checkable functorial semantics", | |
| "contains": [ | |
| "cxs:categories/CAT", | |
| "cxs:categories/Set", | |
| "cxs:categories/Lat", | |
| "cxs:categories/Spec", | |
| "cxs:categories/Arith", | |
| "cxs:categories/Mod" | |
| ], | |
| "label": "Core Categories" | |
| }, | |
| { | |
| "@id": "cxs:categories/CAT", | |
| "@type": "Category", | |
| "comment": "The ambient 2-category containing all other categories as objects", | |
| "composition": "Functor composition: (G ∘ F)(X) = G(F(X))", | |
| "identity": "Identity functor: Id_C(X) = X, Id_C(f) = f", | |
| "label": "Category of Small Categories", | |
| "morphisms": "Functors between categories", | |
| "objects": "Small categories (set-indexed objects and morphisms)" | |
| }, | |
| { | |
| "@id": "cxs:categories/Set", | |
| "@type": "Category", | |
| "comment": "Base category for most concrete constructions", | |
| "composition": "Function composition: (g ∘ f)(x) = g(f(x))", | |
| "identity": "Identity function: id_X(x) = x", | |
| "label": "Category of Sets", | |
| "morphisms": "Functions between sets", | |
| "objects": "Sets (well-defined collections)" | |
| }, | |
| { | |
| "@id": "cxs:categories/Lat", | |
| "@type": "Category", | |
| "comment": "Target category for combinatorial projection", | |
| "composition": "Standard function composition", | |
| "identity": "Identity lattice homomorphism", | |
| "label": "Category of Lattices", | |
| "morphisms": "Lattice homomorphisms (preserving ∧ and ∨)", | |
| "objects": "Lattices (partially ordered sets with meets and joins)" | |
| }, | |
| { | |
| "@id": "cxs:categories/Spec", | |
| "@type": "Category", | |
| "comment": "Target category for spectral projection; connected to zeta zeros", | |
| "composition": "Spectral composition: eigenvalue-preserving", | |
| "identity": "Identity spectral map", | |
| "label": "Spectral Category", | |
| "morphisms": "Spectral maps (preserving eigenvalue structure)", | |
| "objects": "Spectral objects (eigenvalue data, spectral triples)" | |
| }, | |
| { | |
| "@id": "cxs:categories/Arith", | |
| "@type": "Category", | |
| "comment": "Target category for arithmetic projection; prime distributions", | |
| "composition": "Standard composition respecting divisibility", | |
| "identity": "Identity arithmetic morphism", | |
| "label": "Arithmetic Category", | |
| "morphisms": "Arithmetic morphisms (preserving prime structure)", | |
| "objects": "Arithmetic objects (number-theoretic structures)" | |
| }, | |
| { | |
| "@id": "cxs:categories/Mod", | |
| "@type": "Category", | |
| "comment": "Target category for modular projection", | |
| "composition": "Composition of modular operations", | |
| "identity": "Identity modular morphism", | |
| "label": "Category of Modular Forms", | |
| "morphisms": "Modular form morphisms (Hecke operators, level-raising)", | |
| "objects": "Modular forms of various levels and weights" | |
| }, | |
| { | |
| "@id": "cxs:functors/forget_lat", | |
| "@type": "Functor", | |
| "comment": "Canonical forgetful functor", | |
| "functorType": "Forgetful", | |
| "label": "Forgetful Functor (Lat → Set)", | |
| "morphismMap": "Forgets preservation of ∧/∨, retains function", | |
| "objectMap": "Forgets lattice structure, retains underlying set", | |
| "sourceCategory": { | |
| "@id": "cxs:categories/Lat" | |
| }, | |
| "targetCategory": { | |
| "@id": "cxs:categories/Set" | |
| } | |
| }, | |
| { | |
| "@id": "cxs:functors/arith_embed", | |
| "@type": "Functor", | |
| "comment": "Enriches sets with number-theoretic data", | |
| "functorType": "Covariant", | |
| "label": "Arithmetic Embedding", | |
| "morphismMap": "Functions that respect prime structure", | |
| "objectMap": "Embeds set with arithmetic structure (prime labeling)", | |
| "sourceCategory": { | |
| "@id": "cxs:categories/Set" | |
| }, | |
| "targetCategory": { | |
| "@id": "cxs:categories/Arith" | |
| } | |
| }, | |
| { | |
| "@id": "cxs:functors/spectral_functor", | |
| "@type": "Functor", | |
| "comment": "Key functor for spectral-arithmetic correspondence", | |
| "functorType": "Contravariant", | |
| "label": "Spectral Functor", | |
| "morphismMap": "Dualizes to spectral morphisms", | |
| "objectMap": "Extracts spectral data (zeta zeros, L-function data)", | |
| "sourceCategory": { | |
| "@id": "cxs:categories/Arith" | |
| }, | |
| "targetCategory": { | |
| "@id": "cxs:categories/Spec" | |
| } | |
| }, | |
| { | |
| "@id": "cxs:functors/modular_functor", | |
| "@type": "Functor", | |
| "comment": "Connects lattice theory to modular forms", | |
| "functorType": "Covariant", | |
| "label": "Modular Functor", | |
| "morphismMap": "Induced maps on theta series", | |
| "objectMap": "Theta series: lattice → modular form", | |
| "sourceCategory": { | |
| "@id": "cxs:categories/Lat" | |
| }, | |
| "targetCategory": { | |
| "@id": "cxs:categories/Mod" | |
| } | |
| }, | |
| { | |
| "@id": "cxs:functors/power_functor", | |
| "@type": "Functor", | |
| "comment": "Free lattice construction", | |
| "functorType": "Covariant", | |
| "label": "Power Set Functor", | |
| "morphismMap": "f ↦ P(f) (direct/inverse image)", | |
| "objectMap": "X ↦ P(X) (power set is a lattice)", | |
| "sourceCategory": { | |
| "@id": "cxs:categories/Set" | |
| }, | |
| "targetCategory": { | |
| "@id": "cxs:categories/Lat" | |
| } | |
| }, | |
| { | |
| "@id": "cxs:gauge-projections", | |
| "@type": "GaugeProjectionsContainer", | |
| "comment": "Projections π_k: M_∞ → Ω_k for precision/resolution control", | |
| "contains": [ | |
| "cxs:gauge-projections/pi_0", | |
| "cxs:gauge-projections/pi_1", | |
| "cxs:gauge-projections/pi_2", | |
| "cxs:gauge-projections/pi_3", | |
| "cxs:gauge-projections/pi_4", | |
| "cxs:gauge-projections/pi_5" | |
| ], | |
| "label": "Universal Gauge Projections" | |
| }, | |
| { | |
| "@id": "cxs:gauge-projections/pi_0", | |
| "@type": "GaugeProjection", | |
| "boundaryCardinality": 12288, | |
| "comment": "Base gauge - sufficient for D(0)...D(9)", | |
| "eigenvalueSelection": "phase_dependent: Tonic→λ₁=10, LeadingTone/Overtone→λ₂=7", | |
| "label": "π_0", | |
| "level": 0, | |
| "precisionProfile": "mod-96 residue (7 bits effective)", | |
| "projectionRule": "Modular reduction (mod 96) preserving T=3, O=8 structure", | |
| "source": "M_∞", | |
| "targetGauge": "Ω_{2,3}" | |
| }, | |
| { | |
| "@id": "cxs:gauge-projections/pi_1", | |
| "@type": "GaugeProjection", | |
| "boundaryCardinality": 61440, | |
| "comment": "Extended gauge for D(10)...D(14)", | |
| "eigenvalueSelection": "phase_dependent with PENTALITY=5 correction", | |
| "label": "π_1", | |
| "level": 1, | |
| "precisionProfile": "mod-480 residue (9 bits effective)", | |
| "projectionRule": "Modular reduction (mod 480) with prime 5 extension", | |
| "source": "M_∞", | |
| "targetGauge": "Ω_{2,3,5}" | |
| }, | |
| { | |
| "@id": "cxs:gauge-projections/pi_2", | |
| "@type": "GaugeProjection", | |
| "boundaryCardinality": 430080, | |
| "comment": "Third tier gauge for D(15)...D(19)", | |
| "eigenvalueSelection": "phase_dependent with SEPTALITY=7 resonance", | |
| "label": "π_2", | |
| "level": 2, | |
| "precisionProfile": "mod-3360 residue (12 bits effective)", | |
| "projectionRule": "Modular reduction (mod 3360) with SEPTALITY=7 extension", | |
| "source": "M_∞", | |
| "targetGauge": "Ω_{2,3,5,7}" | |
| }, | |
| { | |
| "@id": "cxs:gauge-projections/pi_3", | |
| "@type": "GaugeProjection", | |
| "boundaryCardinality": 4730880, | |
| "comment": "Fourth tier gauge for D(20)...D(24)", | |
| "eigenvalueSelection": "phase_dependent with prime 11 coverage", | |
| "label": "π_3", | |
| "level": 3, | |
| "precisionProfile": "mod-36960 residue (16 bits effective)", | |
| "projectionRule": "Modular reduction (mod 36960) with prime 11 extension", | |
| "source": "M_∞", | |
| "targetGauge": "Ω_{2,3,5,7,11}" | |
| }, | |
| { | |
| "@id": "cxs:gauge-projections/pi_4", | |
| "@type": "GaugeProjection", | |
| "boundaryCardinality": 61501440, | |
| "comment": "Fifth tier gauge for D(25)...D(29)", | |
| "eigenvalueSelection": "phase_dependent with primes {11,13} coverage", | |
| "label": "π_4", | |
| "level": 4, | |
| "precisionProfile": "mod-480480 residue (19 bits effective)", | |
| "projectionRule": "Modular reduction (mod 480480) with prime 13 extension", | |
| "source": "M_∞", | |
| "targetGauge": "Ω_{2,3,5,7,11,13}" | |
| }, | |
| { | |
| "@id": "cxs:gauge-projections/pi_5", | |
| "@type": "GaugeProjection", | |
| "boundaryCardinality": 1045524480, | |
| "comment": "Sixth tier gauge for D(30)...D(34)", | |
| "eigenvalueSelection": "phase_dependent with full spectral coverage", | |
| "label": "π_5", | |
| "level": 5, | |
| "precisionProfile": "mod-8168160 residue (23 bits effective)", | |
| "projectionRule": "Modular reduction (mod 8168160) with prime 17 extension", | |
| "source": "M_∞", | |
| "targetGauge": "Ω_{2,3,5,7,11,13,17}" | |
| }, | |
| { | |
| "@id": "cxs:gauge-bridges/omega_2_3", | |
| "@type": "GaugeBridge", | |
| "comment": "Base Atlas gauge is quotient by π₀", | |
| "embeddingMap": "identity: Ω_{2,3} = M_∞/π₀", | |
| "finiteGaugeId": "omega_2_3", | |
| "isIsomorphism": true, | |
| "label": "Bridge: omega_2_3 ↔ Level 0", | |
| "universalLevel": 0 | |
| }, | |
| { | |
| "@id": "cxs:gauge-bridges/omega_2_3_5", | |
| "@type": "GaugeBridge", | |
| "comment": "Extended gauge via pentality prime", | |
| "embeddingMap": "extension: Ω_{2,3,5} = Ω_{2,3} ⊗ Z/5Z", | |
| "finiteGaugeId": "omega_2_3_5", | |
| "isIsomorphism": true, | |
| "label": "Bridge: omega_2_3_5 ↔ Level 1", | |
| "universalLevel": 1 | |
| }, | |
| { | |
| "@id": "cxs:gauge-bridges/omega_2_3_5_7", | |
| "@type": "GaugeBridge", | |
| "comment": "Third tier via septality prime", | |
| "embeddingMap": "extension: Ω_{2,3,5,7} = Ω_{2,3,5} ⊗ Z/7Z", | |
| "finiteGaugeId": "omega_2_3_5_7", | |
| "isIsomorphism": true, | |
| "label": "Bridge: omega_2_3_5_7 ↔ Level 2", | |
| "universalLevel": 2 | |
| }, | |
| { | |
| "@id": "cxs:gauge-bridges/omega_2_3_5_7_11", | |
| "@type": "GaugeBridge", | |
| "comment": "Fourth tier via prime 11", | |
| "embeddingMap": "extension: Ω_{2,3,5,7,11} = Ω_{2,3,5,7} ⊗ Z/11Z", | |
| "finiteGaugeId": "omega_2_3_5_7_11", | |
| "isIsomorphism": true, | |
| "label": "Bridge: omega_2_3_5_7_11 ↔ Level 3", | |
| "universalLevel": 3 | |
| }, | |
| { | |
| "@id": "cxs:gauge-bridges/omega_2_3_5_7_11_13", | |
| "@type": "GaugeBridge", | |
| "comment": "Fifth tier via prime 13", | |
| "embeddingMap": "extension: Ω_{2,3,5,7,11,13} = Ω_{2,3,5,7,11} ⊗ Z/13Z", | |
| "finiteGaugeId": "omega_2_3_5_7_11_13", | |
| "isIsomorphism": true, | |
| "label": "Bridge: omega_2_3_5_7_11_13 ↔ Level 4", | |
| "universalLevel": 4 | |
| }, | |
| { | |
| "@id": "cxs:gauge-bridges/omega_2_3_5_7_11_13_17", | |
| "@type": "GaugeBridge", | |
| "comment": "Sixth tier via prime 17", | |
| "embeddingMap": "extension: Ω_{2,3,5,7,11,13,17} = Ω_{2,3,5,7,11,13} ⊗ Z/17Z", | |
| "finiteGaugeId": "omega_2_3_5_7_11_13_17", | |
| "isIsomorphism": true, | |
| "label": "Bridge: omega_2_3_5_7_11_13_17 ↔ Level 5", | |
| "universalLevel": 5 | |
| }, | |
| { | |
| "@id": "cxs:compositions/pi_0_arithmetic", | |
| "@type": "ProjectionComposition", | |
| "comment": "π₀ ∘ arithmetic: Prime counting with 12,288 boundary", | |
| "compositionResult": "Gauge-truncated prime distribution at mod-96", | |
| "domainProjection": "arithmetic", | |
| "gaugeLevel": 0, | |
| "label": "π_0 ∘ arithmetic" | |
| }, | |
| { | |
| "@id": "cxs:compositions/pi_0_spectral", | |
| "@type": "ProjectionComposition", | |
| "comment": "π₀ ∘ spectral: Eigenvalue sums with base eigenstructure", | |
| "compositionResult": "Gauge-truncated spectral density at mod-96", | |
| "domainProjection": "spectral", | |
| "gaugeLevel": 0, | |
| "label": "π_0 ∘ spectral" | |
| }, | |
| { | |
| "@id": "cxs:compositions/pi_1_arithmetic", | |
| "@type": "ProjectionComposition", | |
| "comment": "π₁ ∘ arithmetic: Prime counting with 61,440 boundary", | |
| "compositionResult": "Gauge-truncated prime distribution at mod-480", | |
| "domainProjection": "arithmetic", | |
| "gaugeLevel": 1, | |
| "label": "π_1 ∘ arithmetic" | |
| }, | |
| { | |
| "@id": "cxs:compositions/pi_1_spectral", | |
| "@type": "ProjectionComposition", | |
| "comment": "π₁ ∘ spectral: Eigenvalue sums with pentality extension", | |
| "compositionResult": "Gauge-truncated spectral density at mod-480", | |
| "domainProjection": "spectral", | |
| "gaugeLevel": 1, | |
| "label": "π_1 ∘ spectral" | |
| }, | |
| { | |
| "@id": "cxs:compositions/pi_2_arithmetic", | |
| "@type": "ProjectionComposition", | |
| "comment": "π₂ ∘ arithmetic: Prime counting with 430,080 boundary", | |
| "compositionResult": "Gauge-truncated prime distribution at mod-3360", | |
| "domainProjection": "arithmetic", | |
| "gaugeLevel": 2, | |
| "label": "π_2 ∘ arithmetic" | |
| }, | |
| { | |
| "@id": "cxs:compositions/pi_2_spectral", | |
| "@type": "ProjectionComposition", | |
| "comment": "π₂ ∘ spectral: Eigenvalue sums with septality resonance", | |
| "compositionResult": "Gauge-truncated spectral density at mod-3360", | |
| "domainProjection": "spectral", | |
| "gaugeLevel": 2, | |
| "label": "π_2 ∘ spectral" | |
| }, | |
| { | |
| "@id": "cxs:compositions/pi_2_combinatorial", | |
| "@type": "ProjectionComposition", | |
| "comment": "π₂ ∘ combinatorial: Dedekind enumeration D(15)...D(19)", | |
| "compositionResult": "Gauge-truncated antichain counting at mod-3360", | |
| "domainProjection": "combinatorial", | |
| "gaugeLevel": 2, | |
| "label": "π_2 ∘ combinatorial" | |
| }, | |
| { | |
| "@id": "cxs:compositions/pi_2_modular", | |
| "@type": "ProjectionComposition", | |
| "comment": "π₂ ∘ modular: Modular arithmetic with septality", | |
| "compositionResult": "Gauge-truncated modular forms at mod-3360", | |
| "domainProjection": "modular", | |
| "gaugeLevel": 2, | |
| "label": "π_2 ∘ modular" | |
| }, | |
| { | |
| "@id": "cxs:error-signatures", | |
| "@type": "ErrorSignaturesContainer", | |
| "comment": "Falsifiability contracts for computational validation", | |
| "contains": [ | |
| "cxs:error-signatures/prime_counting_error", | |
| "cxs:error-signatures/chebyshev_psi_error", | |
| "cxs:error-signatures/dedekind_error", | |
| "cxs:error-signatures/gauge_truncation_error" | |
| ], | |
| "label": "Error Signatures" | |
| }, | |
| { | |
| "@id": "cxs:error-signatures/prime_counting_error", | |
| "@type": "ErrorSignature", | |
| "acceptCondition": "Normalized error bounded as x → ∞", | |
| "comment": "Consistent with RH implies exponent α < 0.5; we accept α < 0.6", | |
| "errorFormula": "E_π(x) = |π_spec(x) - π(x)|", | |
| "failCondition": "Normalized error diverges or exceeds O(x^0.6)", | |
| "label": "Prime Counting Error", | |
| "normalization": "E_π(x) / (√x × log²x)", | |
| "validatesCorrespondence": "spectralArithmetic" | |
| }, | |
| { | |
| "@id": "cxs:error-signatures/chebyshev_psi_error", | |
| "@type": "ErrorSignature", | |
| "acceptCondition": "E_ψ(x) = O(√x × log²x)", | |
| "comment": "Direct measure of explicit formula accuracy", | |
| "errorFormula": "E_ψ(x) = |ψ_spec(x) - ψ(x)|", | |
| "failCondition": "E_ψ(x) grows faster than O(x^(1/2+ε)) for any ε > 0", | |
| "label": "Chebyshev ψ Error", | |
| "normalization": "E_ψ(x) / √x", | |
| "validatesCorrespondence": "spectralArithmetic" | |
| }, | |
| { | |
| "@id": "cxs:error-signatures/dedekind_error", | |
| "@type": "ErrorSignature", | |
| "acceptCondition": "E_D(n) = 0 for computed values (exact)", | |
| "comment": "Dedekind numbers must be exact integers; predictions are either correct or wrong", | |
| "errorFormula": "E_D(n) = |D_pred(n) - D(n)|", | |
| "failCondition": "Any nonzero error for verified D(n) values", | |
| "label": "Dedekind Computation Error", | |
| "normalization": "E_D(n) / D(n)", | |
| "validatesCorrespondence": "dedekindPrime" | |
| }, | |
| { | |
| "@id": "cxs:error-signatures/gauge_truncation_error", | |
| "@type": "ErrorSignature", | |
| "acceptCondition": "E_k decreases monotonically as k increases", | |
| "comment": "Gauge refinement should always improve precision", | |
| "errorFormula": "E_k(n) = |π_k(M_∞)(n) - f(n)|", | |
| "failCondition": "E_k fails to decrease or increases with k", | |
| "label": "Gauge Truncation Error", | |
| "normalization": "E_k(n) / |Ω_k|", | |
| "validatesCorrespondence": "dedekindPrime" | |
| }, | |
| { | |
| "@id": "cxs:ground-truth/pi_x", | |
| "@type": "GroundTruthSequence", | |
| "comment": "Primary ground truth for spectral-arithmetic correspondence", | |
| "formula": "π(x) = |{p ≤ x : p prime}|", | |
| "label": "Prime Counting Function", | |
| "source": "Computed exactly for x ≤ 10^27" | |
| }, | |
| { | |
| "@id": "cxs:ground-truth/psi_x", | |
| "@type": "GroundTruthSequence", | |
| "comment": "Sum of von Mangoldt function", | |
| "formula": "ψ(x) = Σ_{p^k ≤ x} log(p)", | |
| "label": "Chebyshev ψ Function", | |
| "source": "Computed from prime powers" | |
| }, | |
| { | |
| "@id": "cxs:ground-truth/dedekind_n", | |
| "@type": "GroundTruthSequence", | |
| "comment": "Ground truth for combinatorial correspondence", | |
| "formula": "D(n) = |{A ⊆ P^n : antichain(A)}|", | |
| "label": "Dedekind Numbers", | |
| "source": "OEIS A000372, computed for n ≤ 9" | |
| }, | |
| { | |
| "@id": "cxs:spectra", | |
| "@type": "SpectraContainer", | |
| "comment": "Eigenvalue structure per gauge level, connecting categorical structure to spectral predictions", | |
| "contains": [ | |
| "cxs:spectra/level-0", | |
| "cxs:spectra/level-1", | |
| "cxs:spectra/level-2", | |
| "cxs:spectra/level-3", | |
| "cxs:spectra/level-4", | |
| "cxs:spectra/level-5" | |
| ], | |
| "label": "Spectra" | |
| }, | |
| { | |
| "@id": "cxs:spectra/level-0", | |
| "@type": "Spectrum", | |
| "altSpecDim": 480, | |
| "boundaryCardinality": 12288, | |
| "dominantEigenvalue": 10, | |
| "jordSpecDim": 27, | |
| "label": "Level 0 Spectrum", | |
| "level": 0, | |
| "phase": "variable", | |
| "subdominantEigenvalue": 2, | |
| "totalDimension": 24, | |
| "zdSpecDim": 15 | |
| }, | |
| { | |
| "@id": "cxs:spectra/level-1", | |
| "@type": "Spectrum", | |
| "altSpecDim": 480, | |
| "boundaryCardinality": 61440, | |
| "dominantEigenvalue": 10, | |
| "jordSpecDim": 27, | |
| "label": "Level 1 Spectrum", | |
| "level": 1, | |
| "phase": "variable", | |
| "subdominantEigenvalue": 2, | |
| "totalDimension": 24, | |
| "zdSpecDim": 15 | |
| }, | |
| { | |
| "@id": "cxs:spectra/level-2", | |
| "@type": "Spectrum", | |
| "altSpecDim": 480, | |
| "boundaryCardinality": 430080, | |
| "dominantEigenvalue": 10, | |
| "jordSpecDim": 27, | |
| "label": "Level 2 Spectrum", | |
| "level": 2, | |
| "phase": "variable", | |
| "subdominantEigenvalue": 7, | |
| "totalDimension": 24, | |
| "zdSpecDim": 15 | |
| }, | |
| { | |
| "@id": "cxs:spectra/level-3", | |
| "@type": "Spectrum", | |
| "altSpecDim": 480, | |
| "boundaryCardinality": 4730880, | |
| "dominantEigenvalue": 10, | |
| "jordSpecDim": 27, | |
| "label": "Level 3 Spectrum", | |
| "level": 3, | |
| "phase": "variable", | |
| "subdominantEigenvalue": 7, | |
| "totalDimension": 24, | |
| "zdSpecDim": 15 | |
| }, | |
| { | |
| "@id": "cxs:spectra/level-4", | |
| "@type": "Spectrum", | |
| "altSpecDim": 480, | |
| "boundaryCardinality": 61501440, | |
| "dominantEigenvalue": 10, | |
| "jordSpecDim": 27, | |
| "label": "Level 4 Spectrum", | |
| "level": 4, | |
| "phase": "variable", | |
| "subdominantEigenvalue": 7, | |
| "totalDimension": 24, | |
| "zdSpecDim": 15 | |
| }, | |
| { | |
| "@id": "cxs:spectra/level-5", | |
| "@type": "Spectrum", | |
| "altSpecDim": 480, | |
| "boundaryCardinality": 1045524480, | |
| "dominantEigenvalue": 10, | |
| "jordSpecDim": 27, | |
| "label": "Level 5 Spectrum", | |
| "level": 5, | |
| "phase": "variable", | |
| "subdominantEigenvalue": 7, | |
| "totalDimension": 24, | |
| "zdSpecDim": 15 | |
| }, | |
| { | |
| "@id": "cxs:explicit-formulas", | |
| "@type": "ExplicitFormulasContainer", | |
| "comment": "Formulas connecting spectral eigenvalues to arithmetic functions", | |
| "contains": [ | |
| "cxs:explicit-formulas/chebyshev_psi_explicit", | |
| "cxs:explicit-formulas/prime_counting_explicit" | |
| ], | |
| "label": "Explicit Formulas" | |
| }, | |
| { | |
| "@id": "cxs:explicit-formulas/chebyshev_psi_explicit", | |
| "@type": "ExplicitFormula", | |
| "errorBound": "O(√x × log²x) assuming RH", | |
| "errorSignature": { | |
| "@id": "cxs:error-signatures/chebyshev_psi_error" | |
| }, | |
| "formula": "ψ(x) = x - Σ_ρ x^ρ/ρ - log(2π) - ½log(1 - 1/x²)", | |
| "gammaValues": [ | |
| 0.357, | |
| 1.609, | |
| 1.253, | |
| 3.178 | |
| ], | |
| "inputSpectrum": { | |
| "@id": "cxs:spectra/level-0" | |
| }, | |
| "label": "Chebyshev ψ Explicit Formula", | |
| "outputFunction": "ψ(x)" | |
| }, | |
| { | |
| "@id": "cxs:explicit-formulas/prime_counting_explicit", | |
| "@type": "ExplicitFormula", | |
| "errorBound": "O(√x × log(x)) assuming RH", | |
| "errorSignature": { | |
| "@id": "cxs:error-signatures/prime_counting_error" | |
| }, | |
| "formula": "π(x) = li(x) - Σ_ρ li(x^ρ) + O(1)", | |
| "gammaValues": [ | |
| 0.357, | |
| 1.609, | |
| 1.253, | |
| 3.178 | |
| ], | |
| "inputSpectrum": { | |
| "@id": "cxs:spectra/level-0" | |
| }, | |
| "label": "Prime Counting Explicit Formula", | |
| "outputFunction": "π(x)" | |
| }, | |
| { | |
| "@id": "cxs:irreducibility", | |
| "@type": "IrreducibilityContainer", | |
| "comment": "Classification of categorical structures as irreducible (prime) or reducible (composite)", | |
| "contains": [ | |
| "cxs:classified/axioms", | |
| "cxs:classified/correspondences", | |
| "cxs:classified/derivation_levels", | |
| "cxs:classified/primitive_relations", | |
| "cxs:classified/categorical_operators", | |
| "cxs:classified/projections", | |
| "cxs:classified/tower_levels", | |
| "cxs:classified/type_levels" | |
| ], | |
| "label": "Prime-Operator Correspondence" | |
| }, | |
| { | |
| "@id": "cxs:classified/axioms", | |
| "@type": "ClassifiedStructure", | |
| "justification": "2 is prime; T and O are the unique foundational axioms", | |
| "label": "Foundational Axioms", | |
| "reducibility": "Irreducible", | |
| "structureCardinality": 2 | |
| }, | |
| { | |
| "@id": "cxs:classified/correspondences", | |
| "@type": "ClassifiedStructure", | |
| "justification": "3 = T is prime; correspondences form triality algebra", | |
| "label": "Categorical Correspondences", | |
| "reducibility": "Irreducible", | |
| "structureCardinality": 3 | |
| }, | |
| { | |
| "@id": "cxs:classified/derivation_levels", | |
| "@type": "ClassifiedStructure", | |
| "justification": "5 = PENTALITY = O - T is prime; simple derivation chain", | |
| "label": "Derivation Levels", | |
| "reducibility": "Irreducible", | |
| "structureCardinality": 5 | |
| }, | |
| { | |
| "@id": "cxs:classified/primitive_relations", | |
| "@type": "ClassifiedStructure", | |
| "justification": "7 = SEPTALITY = O - 1 is prime; Fano plane structure", | |
| "label": "Primitive Relations", | |
| "reducibility": "Irreducible", | |
| "structureCardinality": 7 | |
| }, | |
| { | |
| "@id": "cxs:classified/categorical_operators", | |
| "@type": "ClassifiedStructure", | |
| "justification": "13 = 2O - T is prime; minimal self-closed operator algebra", | |
| "label": "Categorical Operators", | |
| "reducibility": "Irreducible", | |
| "structureCardinality": 13 | |
| }, | |
| { | |
| "@id": "cxs:classified/projections", | |
| "@type": "ClassifiedStructure", | |
| "factorization": [ | |
| 2, | |
| 2 | |
| ], | |
| "justification": "4 = Q = 2×2; decomposes as Arithmetic×Combinatorial × Spectral×Modular", | |
| "label": "Categorical Projections", | |
| "reducibility": "Reducible", | |
| "structureCardinality": 4 | |
| }, | |
| { | |
| "@id": "cxs:classified/tower_levels", | |
| "@type": "ClassifiedStructure", | |
| "factorization": [ | |
| 3, | |
| 3 | |
| ], | |
| "justification": "9 = T² = 3×3; triality × triality structure", | |
| "label": "Tower Levels", | |
| "reducibility": "Reducible", | |
| "structureCardinality": 9 | |
| }, | |
| { | |
| "@id": "cxs:classified/type_levels", | |
| "@type": "ClassifiedStructure", | |
| "factorization": [ | |
| 2, | |
| 3 | |
| ], | |
| "justification": "6 = PARIAH = 2×3; axiom count × triality", | |
| "label": "Type Levels", | |
| "reducibility": "Reducible", | |
| "structureCardinality": 6 | |
| }, | |
| { | |
| "@id": "cxs:eigenvalue-duality", | |
| "@type": "EigenvalueDuality", | |
| "comment": "The involution swapping (λ=2, mult=7) ↔ (λ=7, mult=2)", | |
| "derivation": "\nThe 2↔7 duality arises from the Kronecker structure of the transfer matrix:\n\n M = I₃ ⊗ J₈ + (J₃ - I₃) ⊗ I₈\n\nwhere I_n is the n×n identity and J_n is the n×n all-ones matrix.\n\n**Tonic block (k=0): J₈ + 2I₈**\n- J₈ has eigenvalue 8 with mult 1, eigenvalue 0 with mult 7\n- Adding 2I₈ shifts: eigenvalue 10 with mult 1, eigenvalue 2 with mult 7\n- Thus λ=2 appears with mult O-1=7\n\n**Leading/Overtone blocks (k=1,2): J₈ - I₈**\n- J₈ - I₈ has eigenvalue 7 with mult 1, eigenvalue -1 with mult 7\n- Each non-tonic phase contributes one λ=7 eigenspace\n- T-1=2 non-tonic phases give mult 2\n\n**The Duality:**\n (λ=2, mult=7) ↔ (λ=7, mult=2)\n\nBoth multiplicities are axiom-derived: 7 = O-1, 2 = T-1\nBoth products equal 14 = 2×7 = 7×2\n", | |
| "dominantEigenvalue": 7, | |
| "dominantMultiplicity": 2, | |
| "dualityType": "Involution", | |
| "label": "2↔7 Eigenvalue-Multiplicity Duality", | |
| "productInvariant": 14, | |
| "subdominantEigenvalue": 2, | |
| "subdominantMultiplicity": 7 | |
| }, | |
| { | |
| "@id": "cxs:eigenvalue-duality/subdominant", | |
| "@type": "EigenvalueMult", | |
| "dualOf": { | |
| "@id": "cxs:eigenvalue-duality/dominant" | |
| }, | |
| "eigenvalue": 2, | |
| "label": "Subdominant Pair (λ=2, mult=7)", | |
| "multiplicity": 7, | |
| "product": 14 | |
| }, | |
| { | |
| "@id": "cxs:eigenvalue-duality/dominant", | |
| "@type": "EigenvalueMult", | |
| "dualOf": { | |
| "@id": "cxs:eigenvalue-duality/subdominant" | |
| }, | |
| "eigenvalue": 7, | |
| "label": "Dominant Pair (λ=7, mult=2)", | |
| "multiplicity": 2, | |
| "product": 14 | |
| }, | |
| { | |
| "@id": "cxs:extended-primes", | |
| "@type": "ExtendedPrimesContainer", | |
| "comment": "Predicted instantiations for primes 17, 19, 23, 29, 31", | |
| "contains": [ | |
| "cxs:extended-primes/17", | |
| "cxs:extended-primes/19", | |
| "cxs:extended-primes/23", | |
| "cxs:extended-primes/29", | |
| "cxs:extended-primes/31" | |
| ], | |
| "coverageCycle": [ | |
| 8, | |
| 5, | |
| 6, | |
| 7 | |
| ], | |
| "label": "Extended Prime Structures" | |
| }, | |
| { | |
| "@id": "cxs:extended-primes/17", | |
| "@type": "ExtendedPrimeStructure", | |
| "coverage": 8, | |
| "cumulativeCoverage": 35, | |
| "description": "Octonion-dimension extension of Jordan algebra", | |
| "family": "FamilyB", | |
| "hierarchyConstant": "K", | |
| "instantiationName": "OctonionExtension", | |
| "label": "Prime 17 Extension", | |
| "prime": 17 | |
| }, | |
| { | |
| "@id": "cxs:extended-primes/19", | |
| "@type": "ExtendedPrimeStructure", | |
| "coverage": 5, | |
| "cumulativeCoverage": 40, | |
| "description": "Pentality-dimension extension", | |
| "family": "FamilyB", | |
| "hierarchyConstant": "L", | |
| "instantiationName": "PentalityExtension", | |
| "label": "Prime 19 Extension", | |
| "prime": 19 | |
| }, | |
| { | |
| "@id": "cxs:extended-primes/23", | |
| "@type": "ExtendedPrimeStructure", | |
| "coverage": 6, | |
| "cumulativeCoverage": 46, | |
| "description": "Pariah-dimension extension", | |
| "family": "FamilyA", | |
| "hierarchyConstant": "M", | |
| "instantiationName": "PariahExtension", | |
| "label": "Prime 23 Extension", | |
| "prime": 23 | |
| }, | |
| { | |
| "@id": "cxs:extended-primes/29", | |
| "@type": "ExtendedPrimeStructure", | |
| "coverage": 7, | |
| "cumulativeCoverage": 53, | |
| "description": "Septality-dimension extension completing first cycle", | |
| "family": "FamilyA", | |
| "hierarchyConstant": "N", | |
| "instantiationName": "SeptalityExtension", | |
| "label": "Prime 29 Extension", | |
| "prime": 29 | |
| }, | |
| { | |
| "@id": "cxs:extended-primes/31", | |
| "@type": "ExtendedPrimeStructure", | |
| "coverage": 5, | |
| "cumulativeCoverage": 58, | |
| "description": "Second cycle pentality extension", | |
| "family": "FamilyB", | |
| "hierarchyConstant": "N+PENTALITY", | |
| "instantiationName": "SecondCyclePentality", | |
| "label": "Prime 31 Extension", | |
| "prime": 31 | |
| }, | |
| { | |
| "@id": "cxs:addressing", | |
| "@type": "Container", | |
| "comment": "Hierarchical addressing codec (AX0-AX15)", | |
| "contains": [ | |
| "cxs:addressing/axioms", | |
| "cxs:addressing/levels" | |
| ], | |
| "label": "Addressing" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms", | |
| "@type": "Container", | |
| "comment": "Axioms AX0-AX15 defining the hierarchical codec", | |
| "label": "Addressing Axioms", | |
| "partOf": "cxs:addressing" | |
| }, | |
| { | |
| "@id": "cxs:addressing/levels", | |
| "@type": "Container", | |
| "comment": "Level parameters for each gauge level k", | |
| "label": "Address Levels", | |
| "partOf": "cxs:addressing" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX0", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX0", | |
| "formula": "m_0 = 96, m_ℓ | m_k for ℓ < k, 96 | m_k", | |
| "label": "Tower Arithmetic", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX1", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX1", | |
| "formula": "B_k = 128 × m_k", | |
| "label": "Boundary Cardinality", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX2", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX2", | |
| "formula": "Ω: J → Set with restriction maps", | |
| "label": "Gauge Boundary Diagram", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX3", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX3", | |
| "formula": "M_∞ = lim Ω exists", | |
| "label": "Universal Object", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX4", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX4", | |
| "formula": "A_k = (ℤ/m_kℤ) × (ℤ/128ℤ)", | |
| "label": "Address Sets", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX5", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX5", | |
| "formula": "r ∈ {0,...,m_k-1}, κ ∈ {0,...,127}", | |
| "label": "Canonical Representatives", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX6", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX6", | |
| "formula": "Dec_k(ρ,κ) = ρ + κ × m_k", | |
| "label": "Decode Map", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX7", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX7", | |
| "formula": "Enc_k(n) = (n mod m_k, ⌊n/m_k⌋)", | |
| "label": "Encode Map", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX8", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX8", | |
| "formula": "Enc_k ∘ Dec_k = id, Dec_k ∘ Enc_k = id", | |
| "label": "Bijectivity", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX9", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX9", | |
| "formula": "A(k→ℓ) = Enc̄_ℓ ∘ Dec_k", | |
| "label": "Restriction Maps", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX10", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX10", | |
| "formula": "A(k→j) = A(ℓ→j) ∘ A(k→ℓ)", | |
| "label": "Functoriality", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX11", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX11", | |
| "formula": "α_k: Ω_k ↔ A_k natural isomorphism", | |
| "label": "Address Charts", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX12", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX12", | |
| "formula": "ρ_k, κ_k, φ_k extractors", | |
| "label": "Coordinate Invariants", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX13", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX13", | |
| "formula": "ρ_k(Enc_k(n)) = n mod m_k", | |
| "label": "Invariant Correctness", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX14", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX14", | |
| "formula": "ℤ/96ℤ ≅ (ℤ/32ℤ) × (ℤ/3ℤ)", | |
| "label": "CRT Digitization", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/AX15", | |
| "@type": "AddressingAxiom", | |
| "axiomId": "AX15", | |
| "formula": "ζ: ℤ ↔ ℕ zigzag with level selector σ", | |
| "label": "Global Bijection", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/axioms/constants", | |
| "@type": "AddressingConstants", | |
| "B0": 12288, | |
| "M0": 96, | |
| "P": 128, | |
| "comment": "M0=96 (base modulus), P=128 (phase count), B0=12288 (base boundary)", | |
| "label": "Addressing Constants", | |
| "partOf": "cxs:addressing/axioms" | |
| }, | |
| { | |
| "@id": "cxs:addressing/levels/0", | |
| "@type": "AddressLevel", | |
| "boundary": 12288, | |
| "extensionPrime": null, | |
| "label": "Address Level 0", | |
| "level": 0, | |
| "modulus": 96, | |
| "partOf": "cxs:addressing/levels", | |
| "satisfies": [ | |
| "cxs:addressing/axioms/AX0", | |
| "cxs:addressing/axioms/AX1", | |
| "cxs:addressing/axioms/AX4" | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:addressing/levels/1", | |
| "@type": "AddressLevel", | |
| "boundary": 61440, | |
| "extensionPrime": 5, | |
| "label": "Address Level 1", | |
| "level": 1, | |
| "modulus": 480, | |
| "partOf": "cxs:addressing/levels", | |
| "satisfies": [ | |
| "cxs:addressing/axioms/AX0", | |
| "cxs:addressing/axioms/AX1", | |
| "cxs:addressing/axioms/AX4" | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:addressing/levels/2", | |
| "@type": "AddressLevel", | |
| "boundary": 430080, | |
| "extensionPrime": 7, | |
| "label": "Address Level 2", | |
| "level": 2, | |
| "modulus": 3360, | |
| "partOf": "cxs:addressing/levels", | |
| "satisfies": [ | |
| "cxs:addressing/axioms/AX0", | |
| "cxs:addressing/axioms/AX1", | |
| "cxs:addressing/axioms/AX4" | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:addressing/levels/3", | |
| "@type": "AddressLevel", | |
| "boundary": 4730880, | |
| "extensionPrime": 11, | |
| "label": "Address Level 3", | |
| "level": 3, | |
| "modulus": 36960, | |
| "partOf": "cxs:addressing/levels", | |
| "satisfies": [ | |
| "cxs:addressing/axioms/AX0", | |
| "cxs:addressing/axioms/AX1", | |
| "cxs:addressing/axioms/AX4" | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:addressing/levels/4", | |
| "@type": "AddressLevel", | |
| "boundary": 61501440, | |
| "extensionPrime": 13, | |
| "label": "Address Level 4", | |
| "level": 4, | |
| "modulus": 480480, | |
| "partOf": "cxs:addressing/levels", | |
| "satisfies": [ | |
| "cxs:addressing/axioms/AX0", | |
| "cxs:addressing/axioms/AX1", | |
| "cxs:addressing/axioms/AX4" | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:addressing/levels/5", | |
| "@type": "AddressLevel", | |
| "boundary": 1045524480, | |
| "extensionPrime": 17, | |
| "label": "Address Level 5", | |
| "level": 5, | |
| "modulus": 8168160, | |
| "partOf": "cxs:addressing/levels", | |
| "satisfies": [ | |
| "cxs:addressing/axioms/AX0", | |
| "cxs:addressing/axioms/AX1", | |
| "cxs:addressing/axioms/AX4" | |
| ] | |
| }, | |
| { | |
| "@id": "cxs:boundaries/triality", | |
| "@type": "CriticalBoundary", | |
| "atLevel": 3, | |
| "axiomFormula": "T = U + D", | |
| "boundaryType": "division", | |
| "constrains": [ | |
| "cxs:tower/levels/3", | |
| "cxs:constants/t" | |
| ], | |
| "label": "Triality Boundary (T=3)", | |
| "meaning": "Minimum cyclic structure; last normed division algebra boundary" | |
| }, | |
| { | |
| "@id": "cxs:boundaries/octonality", | |
| "@type": "CriticalBoundary", | |
| "atLevel": 8, | |
| "axiomFormula": "O = D^T", | |
| "boundaryType": "tower", | |
| "constrains": [ | |
| "cxs:tower/levels/8", | |
| "cxs:constants/o" | |
| ], | |
| "label": "Octonality Boundary (O=8)", | |
| "meaning": "Maximum division algebra dimension; octave periodicity" | |
| }, | |
| { | |
| "@id": "cxs:certificates/completeness", | |
| "@type": "CompletenessCertificate", | |
| "addressingGuarantees": { | |
| "coverage": "∀n ∈ ℕ: ∃! address such that decode(address) = n", | |
| "domain": "ℕ", | |
| "noOverlap": "Address space is partitioned", | |
| "unbounded": true, | |
| "uniqueness": "∀n,m ∈ ℕ: encode(n) = encode(m) ⟹ n = m" | |
| }, | |
| "asserts": [ | |
| "All 16 derived constants generated from U,D", | |
| "All derivation chains documented", | |
| "Property loss complete at PENTALITY=5", | |
| "Terminal region at PARIAH=6", | |
| "Octave periodicity at O=8" | |
| ], | |
| "derivedConstantCount": 16, | |
| "encodingStructure": { | |
| "arbitraryExtension": "For any n ∈ ℕ, finite gauge {2,3,p₁,...,p_k} suffices", | |
| "baseBoundary": 12288, | |
| "baseModulus": 96, | |
| "extensionMechanism": "Each gauge prime p extends coverage by prime_coverage(p)", | |
| "hierarchical": "B_k = P × m_k where m_k = m_{k-1} × p_k", | |
| "phaseFactor": 128 | |
| }, | |
| "guarantees": "lossless-codec", | |
| "label": "Ontology Completeness Certificate", | |
| "mathematicalBasis": { | |
| "CRT": "Chinese Remainder Theorem guarantees unique representation mod M", | |
| "finiteApproximation": "Any n ∈ ℕ requires only finite gauge prefix", | |
| "gaugeLimit": "Universal limit M_∞ = lim_{P→∞} Ω_P covers all of ℕ" | |
| }, | |
| "propertyLossCount": 5, | |
| "terminalThreshold": 6 | |
| }, | |
| { | |
| "@id": "cxs:derivation-steps/k", | |
| "@type": "DerivationStep", | |
| "derives": "K", | |
| "formula": "K = J + O = 27 + 8 = 35", | |
| "from": [ | |
| "J", | |
| "O" | |
| ], | |
| "label": "K Derivation", | |
| "operation": "coproduct", | |
| "value": 35 | |
| }, | |
| { | |
| "@id": "cxs:derivation-steps/l", | |
| "@type": "DerivationStep", | |
| "derives": "L", | |
| "formula": "L = K + PENTALITY = 35 + 5 = 40", | |
| "from": [ | |
| "K", | |
| "PENTALITY" | |
| ], | |
| "label": "L Derivation", | |
| "operation": "coproduct", | |
| "value": 40 | |
| }, | |
| { | |
| "@id": "cxs:derivation-steps/m", | |
| "@type": "DerivationStep", | |
| "derives": "M", | |
| "formula": "M = L + PARIAH = 40 + 6 = 46", | |
| "from": [ | |
| "L", | |
| "PARIAH" | |
| ], | |
| "label": "M Derivation", | |
| "operation": "coproduct", | |
| "value": 46 | |
| }, | |
| { | |
| "@id": "cxs:derivation-steps/n", | |
| "@type": "DerivationStep", | |
| "derives": "N", | |
| "formula": "N = M + SEPTALITY = 46 + 7 = 53", | |
| "from": [ | |
| "M", | |
| "SEPTALITY" | |
| ], | |
| "label": "N Derivation", | |
| "operation": "coproduct", | |
| "value": 53 | |
| } | |
| ], | |
| "@id": "https://uor.foundation/categorical-x/v1/graph", | |
| "@type": "owl:Ontology", | |
| "comment": "JSON-LD overlay providing generative relationships for the Categorical X schema. This is the unique singleton instance satisfying the schema constraints.", | |
| "conformsTo": "https://uor.foundation/categorical-x/v1", | |
| "creator": "Categorical X Framework", | |
| "generatedAt": "2026-01-19T23:54:14Z", | |
| "isDefinedBy": "https://uor.foundation/categorical-x/v1", | |
| "label": "Categorical X Relational Graph", | |
| "license": "https://creativecommons.org/licenses/by/4.0/", | |
| "owl:imports": { | |
| "@id": "https://uor.foundation/categorical-x/v1" | |
| }, | |
| "owl:versionIRI": { | |
| "@id": "https://uor.foundation/categorical-x/v1.0.0/graph" | |
| }, | |
| "publisher": "UOR Foundation", | |
| "version": "1.0.0" | |
| } |
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