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Fourier Transform Ontology

Signal Processing Mathematics
Authors: MArk Anthony Higgins - Sky Recursive Ai
© Mark Anthony Higgins 11 October 2025
Revised Date: 12 September 2026
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This Flow Algebra defines the Fourier Transform Ontology within the Kireth Research Semantic Library. The Fourier transform maps a function or signal from its original domain into a continuous frequency-domain representation expressed through complex exponentials. This ontology supports forward and inverse transforms, transform pairs, magnitude and phase spectra, energy relations, convolution and multiplication duality, time and frequency shifts, scaling, differentiation, integration, modulation, symmetry, Parseval and Plancherel relations, existence and convergence conditions, distributional extensions, multidimensional transforms and relationships with Fourier analysis, signal processing, harmonic analysis, differential equations and numerical computation.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Fourier Transform Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Signal Processing Mathematics / Fourier Transform"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/signal-processing-mathematics/fourier-transform-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = FourierTransform
 SYMBOLIC_FORMS = [
  "X(omega)=integral_-infinity^infinity x(t)e^{-j omega t}dt",
  "x(t)=(1/(2*pi))integral_-infinity^infinity X(omega)e^{j omega t}domega"
 ]
 ENTITY_CLASS = ContinuousFrequencyDomainTransformFramework
 PRIMARY_CONTEXT = FourierAnalysis
 DEFINITION = "The Fourier transform is an integral transform that represents a function or signal as a continuous superposition of complex exponential frequency components."
 SEMANTIC_ROLE = "Provides the machine-readable framework for forward and inverse Fourier transformation, spectra, transform properties, convolution duality, energy equivalence, symmetry, convergence and continuous-frequency analysis."
}

SCOPE {
 INCLUDE = [
  FourierTransform,
  ContinuousFourierTransform,
  ForwardFourierTransform,
  InverseFourierTransform,
  FourierTransformPair,
  TimeDomainSignal,
  FrequencyDomainSignal,
  AngularFrequency,
  Frequency,
  Spectrum,
  MagnitudeSpectrum,
  PhaseSpectrum,
  EnergySpectrum,
  ComplexSpectrum,
  ComplexExponential,
  FourierKernel,
  TransformConvention,
  NormalisationConvention,
  Linearity,
  TimeShiftProperty,
  FrequencyShiftProperty,
  ScalingProperty,
  ModulationProperty,
  ConvolutionTheorem,
  MultiplicationTheorem,
  DifferentiationProperty,
  IntegrationProperty,
  DualityProperty,
  ConjugationProperty,
  HermitianSymmetry,
  EvenSymmetry,
  OddSymmetry,
  ParsevalTheorem,
  PlancherelTheorem,
  TransformExistence,
  AbsoluteIntegrability,
  SquareIntegrability,
  GeneralisedFunction,
  DistributionalFourierTransform,
  DiracDelta,
  ImpulseTransform,
  RectangularPulse,
  SincFunction,
  GaussianFunction,
  MultidimensionalFourierTransform,
  TwoDimensionalFourierTransform,
  SpatialFrequency
 ]

 EXCLUDE_AS_PRIMARY = [
  DiscreteFourierTransformWithoutContinuousTransformContext,
  FourierSeriesWithoutAperiodicTransformContext,
  LaplaceTransformWithoutImaginaryAxisRestrictionContext,
  ZTransformWithoutDiscreteFrequencyInterpretation
 ]

 DISAMBIGUATION_RULE =
  "When a continuous-domain function or signal is mapped to a continuous frequency representation by integration against complex exponentials, interpret the context as the Fourier transform."
}

CORE_CONCEPTS {

 FourierTransform {
  definition = "A continuous-frequency integral transform from an original domain to frequency domain."
 }

 InverseFourierTransform {
  definition = "The inverse integral operation reconstructing the original function from its Fourier-domain representation."
 }

 FourierTransformPair {
  definition = "A pair of functions related by forward and inverse Fourier transformation."
 }

 FourierKernel {
  definition = "The complex exponential kernel used by the Fourier integral."
 }

 Spectrum {
  definition = "The complex-valued frequency-domain representation produced by the transform."
 }

 MagnitudeSpectrum {
  definition = "The magnitude of the complex spectrum as a function of frequency."
 }

 PhaseSpectrum {
  definition = "The phase angle of the complex spectrum as a function of frequency."
 }

 ParsevalTheorem {
  definition = "A relation equating signal energy in the original and Fourier domains under a declared normalisation."
 }

 PlancherelTheorem {
  definition = "The extension of the Fourier transform to a unitary map on square-integrable functions under suitable normalisation."
 }

 DistributionalFourierTransform {
  definition = "A generalised Fourier transform defined for distributions such as the Dirac delta."
 }
}

RELATIONS {
 FourierTransform --is_child_of--> FourierAnalysis
 FourierTransform --maps--> TimeDomainSignal
 FourierTransform --to--> FrequencyDomainSignal
 InverseFourierTransform --maps--> FrequencyDomainSignal
 InverseFourierTransform --to--> TimeDomainSignal
 FourierTransformPair --contains--> FourierTransform
 FourierTransformPair --contains--> InverseFourierTransform
 FourierKernel --uses--> ComplexExponential
 Spectrum --has--> MagnitudeSpectrum
 Spectrum --has--> PhaseSpectrum
 ParsevalTheorem --relates--> TimeDomainEnergy
 ParsevalTheorem --relates--> FrequencyDomainEnergy
 PlancherelTheorem --extends--> FourierTransform
 ConvolutionTheorem --maps--> TimeConvolutionToFrequencyMultiplication
 MultiplicationTheorem --maps--> TimeMultiplicationToFrequencyConvolution
 HermitianSymmetry --applies_to--> RealValuedSignals
 DistributionalFourierTransform --extends--> ClassicalFourierTransform
 TwoDimensionalFourierTransform --is_type_of--> MultidimensionalFourierTransform
}

OPERATORS {
 FORWARD_FOURIER_TRANSFORM(x,t,omega) :=
  X(omega)=integral_-infinity^infinity x(t)*exp(-j*omega*t) dt

 INVERSE_FOURIER_TRANSFORM(X,omega,t) :=
  x(t)=(1/(2*pi))*integral_-infinity^infinity X(omega)*exp(j*omega*t) d omega

 FREQUENCY_HZ_FORM(x,t,f) :=
  X(f)=integral_-infinity^infinity x(t)*exp(-j*2*pi*f*t) dt

 LINEARITY(a,x,b,y) :=
  F{a*x+b*y}=a*X+b*Y

 TIME_SHIFT(x,t0) :=
  F{x(t-t0)}=exp(-j*omega*t0)*X(omega)

 FREQUENCY_SHIFT(x,omega0) :=
  F{exp(j*omega0*t)*x(t)}=X(omega-omega0)

 SCALING(x,a) :=
  F{x(a*t)}=(1/abs(a))*X(omega/a)

 CONVOLUTION_THEOREM(x,h) :=
  F{x*h}=X*H

 MULTIPLICATION_THEOREM(x,h) :=
  F{x*h_pointwise}=(1/(2*pi))*(X convolution H)

 DIFFERENTIATION_PROPERTY(x,n) :=
  F{d^n x/dt^n}=(j*omega)^n*X(omega)

 TIME_MULTIPLICATION_PROPERTY(x,n) :=
  F{t^n*x(t)}=j^n*d^nX/domega^n

 PARSEVAL_RELATION(x,y,X,Y) :=
  integral x(t)*conj(y(t)) dt
  = (1/(2*pi))*integral X(omega)*conj(Y(omega)) d omega

 ENERGY_RELATION(x,X) :=
  integral |x(t)|^2 dt
  = (1/(2*pi))*integral |X(omega)|^2 d omega

 TWO_DIMENSIONAL_FT(f,x,y,kx,ky) :=
  integral integral f(x,y)*exp(-j*(kx*x+ky*y)) dx dy
}

DEFINITIONS {
 D1 = "The Fourier transform gives a continuous frequency representation."
 D2 = "The inverse Fourier transform reconstructs the original signal under suitable conditions."
 D3 = "Magnitude and phase together encode the full complex spectrum."
 D4 = "Time shifts produce linear phase factors in frequency."
 D5 = "Frequency shifts correspond to modulation in the original domain."
 D6 = "Convolution becomes multiplication under the Fourier transform."
 D7 = "Differentiation in time becomes multiplication by powers of j omega."
 D8 = "Parseval and Plancherel relations preserve inner-product or energy structure."
 D9 = "Real-valued signals have conjugate-symmetric spectra."
 D10 = "Distributional extensions allow transforms of impulses and other non-integrable idealised signals."
}

AXIOMS {
 A1_LINEARITY:
  FourierTransform is linear

 A2_INVERSION:
  Forward and inverse transforms recover one another under suitable regularity or distributional conditions

 A3_CONTINUOUS_FREQUENCY:
  Classical FourierTransform produces a continuous frequency-domain representation

 A4_CONVOLUTION_DUALITY:
  Convolution in the original domain maps to multiplication in frequency domain

 A5_SHIFT_DUALITY:
  Shifts in one domain create predictable phase or translation effects in the other

 A6_ENERGY_EQUIVALENCE:
  Parseval or Plancherel relations preserve norm structure under the declared convention

 A7_REAL_SYMMETRY:
  Real-valued original-domain signals yield Hermitian-symmetric spectra

 A8_GENERALISATION:
  DistributionalFourierTransform extends classical transform semantics to generalised functions
}

REASONING {

 R1_TRANSFORM_CONTEXT_EXTRACTION {
  INPUT = function_or_signal
  IDENTIFY = domain
  IDENTIFY = continuity
  IDENTIFY = integrability_or_distributional status
  IDENTIFY = transform convention
  RETURN = Fourier_transform_context
 }

 R2_EXISTENCE_ANALYSIS {
  INPUT = signal
  TEST = absolute_integrability
  TEST = square_integrability
  IF classical_conditions_fail
   CONSIDER = distributional interpretation
  RETURN = transform_existence_profile
 }

 R3_FORWARD_TRANSFORM {
  INPUT = signal_and_convention
  COMPUTE = integral_transform
  SIMPLIFY = using symmetry_and_known pairs
  RETURN = spectrum
 }

 R4_INVERSE_TRANSFORM {
  INPUT = spectrum_and_convention
  COMPUTE = inverse_integral
  VERIFY = reconstruction
  RETURN = original_signal
 }

 R5_PROPERTY_REASONING {
  INPUT = transformed_expression
  APPLY = linearity_shift_scaling_modulation_differentiation_convolution_or_duality
  RETURN = simplified_result
 }

 R6_SPECTRAL_REASONING {
  INPUT = complex_spectrum
  COMPUTE = magnitude
  COMPUTE = phase
  IDENTIFY = bandwidth_peaks_symmetry_and_decay
  RETURN = spectral_profile
 }

 R7_ENERGY_REASONING {
  INPUT = signal_and_transform
  APPLY = Parseval_or_Plancherel relation
  RETURN = domain_energy_equivalence
 }

 R8_DISTRIBUTIONAL_REASONING {
  INPUT = impulse_constant_sinusoid_or_generalised function
  USE = distributional transform pair
  RETURN = generalised_spectrum
 }

 R9_MULTIDIMENSIONAL_REASONING {
  INPUT = image_field_or_multivariate function
  APPLY = multidimensional Fourier transform
  RETURN = spatial_or_multidimensional spectrum
 }

 R10_TRANSFORM_VALIDATION {
  INPUT = transform_result
  CHECK = scaling_sign_frequency variable_symmetry_units_and_inverse reconstruction
  RETURN = validated_Fourier_transform result
 }
}

CONSTRAINTS {
 C1 = "Fourier-transform sign and normalisation conventions must be explicit."
 C2 = "Frequency in hertz and angular frequency must not be conflated."
 C3 = "Classical integrability conditions and distributional interpretations must be distinguished."
 C4 = "Magnitude-only information is insufficient for exact inversion in general."
 C5 = "Scaling laws require the absolute-value Jacobian factor."
 C6 = "Convolution and pointwise multiplication must not be conflated."
 C7 = "Real-signal Hermitian symmetry depends on convention-consistent indexing."
 C8 = "Numerical approximations to continuous transforms introduce sampling and truncation effects."
 C9 = "Plancherel interpretations require square-integrability or appropriate generalisation."
 C10 = "Inverse reconstruction must use the matching forward-transform convention."
}

SEMANTIC_EQUIVALENCE {
 "FourierTransform" <=> "ContinuousFourierTransform"
 "InverseFourierTransform" <=> "FourierInverseTransform"
 "FourierTransformPair" <=> "FourierPair"
 "MagnitudeSpectrum" <=> "AmplitudeSpectrum"
 "DistributionalFourierTransform" <=> "GeneralisedFourierTransform"
 "TwoDimensionalFourierTransform" <=> "2DFourierTransform"
}

VALIDATION {
 V1 = "Transform convention is explicit."
 V2 = "Frequency variable and units are explicit."
 V3 = "Forward and inverse scaling are mutually consistent."
 V4 = "Existence assumptions are documented."
 V5 = "Real-signal symmetry is checked when relevant."
 V6 = "Convolution and multiplication factors are convention-correct."
 V7 = "Energy scaling matches Parseval convention."
 V8 = "Distributional results are labelled as such."
 V9 = "Multidimensional transforms use correct dimensional measures."
 V10 = "Inverse reconstruction reproduces the original signal where applicable."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 FOURIER_TRANSFORM_DEFINITION = 0.99
 EXISTENCE_REASONING = 0.98
 FORWARD_TRANSFORM_REASONING = 0.99
 INVERSE_TRANSFORM_REASONING = 0.99
 PROPERTY_REASONING = 0.99
 SPECTRAL_REASONING = 0.99
 ENERGY_REASONING = 0.99
 DISTRIBUTIONAL_REASONING = 0.98
 MULTIDIMENSIONAL_REASONING = 0.98

 POLICY =
  "Reduce confidence when transform convention, frequency variable, integrability, distributional status, scaling, phase information or numerical approximation details are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  FourierTransformReasoning,
  InverseFourierTransformReasoning,
  FourierPairReasoning,
  SpectralReasoning,
  MagnitudePhaseReasoning,
  ConvolutionTheoremReasoning,
  ShiftPropertyReasoning,
  ScalingPropertyReasoning,
  ModulationReasoning,
  DifferentiationPropertyReasoning,
  ParsevalReasoning,
  PlancherelReasoning,
  DistributionalTransformReasoning,
  MultidimensionalFourierReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "Fourier transform" -> FourierTransform
  "continuous Fourier transform" -> FourierTransform
  "inverse Fourier transform" -> InverseFourierTransform
  "Fourier pair" -> FourierTransformPair
  "amplitude spectrum" -> MagnitudeSpectrum
  "magnitude spectrum" -> MagnitudeSpectrum
  "phase spectrum" -> PhaseSpectrum
  "2D Fourier transform" -> TwoDimensionalFourierTransform
  "distributional Fourier transform" -> DistributionalFourierTransform
 }

 INTERPRETATION_ORDER = [
  detect_Fourier_transform_context,
  identify signal_domain_integrability_frequency variable_and_transform convention,
  determine classical_or_distributional interpretation,
  compute forward_or_inverse transform,
  apply symmetry_shift_scaling_modulation_convolution_differentiation_or_energy properties,
  derive magnitude_phase_energy_or_spatial spectrum,
  verify scaling_units_symmetry_and_inverse reconstruction,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("Fourier transform")
  OR asks_to_map_continuous_signal_to_continuous_frequency domain
  THEN activate(FourierTransformOntology)

 E2:
  IF transform_convention_unspecified
  THEN declare chosen convention before calculation

 E3:
  IF signal_is_real
  THEN test Hermitian symmetry

 E4:
  IF signal_is_even
  THEN exploit even symmetry

 E5:
  IF convolution_present
  THEN consider multiplication in frequency domain

 E6:
  IF derivative_present
  THEN apply differentiation property where valid

 E7:
  IF classical_integral_does_not_exist
  THEN consider distributional interpretation

 E8:
  IF two_dimensional_data
  THEN consider multidimensional Fourier transform

 E9:
  IF reconstruction_required
  THEN preserve phase_and_inverse scaling
}

RECURSION {
 ENTRY = FourierTransform

 EXPAND(FourierTransform) -> [
  ContinuousFourierTransform,
  ForwardFourierTransform,
  InverseFourierTransform,
  FourierTransformPair,
  TimeDomainSignal,
  FrequencyDomainSignal,
  AngularFrequency,
  Frequency,
  Spectrum,
  MagnitudeSpectrum,
  PhaseSpectrum,
  EnergySpectrum,
  ComplexSpectrum,
  ComplexExponential,
  FourierKernel,
  TransformConvention,
  NormalisationConvention,
  Linearity,
  TimeShiftProperty,
  FrequencyShiftProperty,
  ScalingProperty,
  ModulationProperty,
  ConvolutionTheorem,
  MultiplicationTheorem,
  DifferentiationProperty,
  IntegrationProperty,
  DualityProperty,
  ConjugationProperty,
  HermitianSymmetry,
  EvenSymmetry,
  OddSymmetry,
  ParsevalTheorem,
  PlancherelTheorem,
  TransformExistence,
  AbsoluteIntegrability,
  SquareIntegrability,
  GeneralisedFunction,
  DistributionalFourierTransform,
  DiracDelta,
  ImpulseTransform,
  RectangularPulse,
  SincFunction,
  GaussianFunction,
  MultidimensionalFourierTransform,
  TwoDimensionalFourierTransform,
  SpatialFrequency
 ]

 EXPAND(TransformProperties) -> [
  Linearity,
  TimeShiftProperty,
  FrequencyShiftProperty,
  ScalingProperty,
  ModulationProperty,
  ConvolutionTheorem,
  MultiplicationTheorem,
  DifferentiationProperty,
  IntegrationProperty,
  DualityProperty,
  ConjugationProperty
 ]

 EXPAND(ExistenceFramework) -> [
  AbsoluteIntegrability,
  SquareIntegrability,
  TemperedDistribution,
  DistributionalFourierTransform
 ]

 EXPAND(CommonTransformPairs) -> [
  DiracDelta,
  ConstantFunction,
  ComplexExponential,
  RectangularPulse,
  SincFunction,
  GaussianFunction
 ]

 STOP_CONDITION =
  "Stop when the transform convention, existence interpretation, forward or inverse transform, spectral properties, transform identities, scaling and reconstruction conditions are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Signal Processing Mathematics"

 IMMEDIATE_CONTEXTS = [
  "Fourier Analysis Ontology",
  "Signal Processing Ontology",
  "Harmonic Analysis Ontologies",
  "Complex Analysis Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Signal Processing Ontology",
  "Signal Processing Mathematics",
  "Fourier Analysis Ontology",
  "Harmonic Analysis Ontologies",
  "Complex Analysis Ontologies",
  "Linear Algebra Ontologies",
  "Differential Equations Ontologies",
  "Partial Differential Equations Ontologies",
  "Probability & Statistics Ontologies",
  "Numerical Analysis Ontologies",
  "Communications Mathematics Ontologies",
  "Image Processing Ontologies"
 ]

 ROUTE_RULES = [
  "Fourier Transform -> Fourier Analysis Ontology for parent frequency-decomposition semantics",
  "Fourier Transform -> Signal Processing Ontology for signal and system context",
  "Fourier Transform -> Harmonic Analysis for functional-analytic foundations",
  "Fourier Transform -> Complex Analysis for complex exponential and contour-related methods",
  "Fourier Transform -> Linear Algebra for discretised and multidimensional representations",
  "Fourier Transform -> Differential Equations for transform-based solution methods",
  "Fourier Transform -> Partial Differential Equations for spatial-frequency methods",
  "Fourier Transform -> Probability & Statistics for characteristic functions and spectra",
  "Fourier Transform -> Numerical Analysis for FFT-based approximation of continuous transforms",
  "Fourier Transform -> Communications and Image Processing for frequency-domain applications"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible transform properties, generalisations or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve forward-versus-inverse transform distinction.",
  "Preserve continuous-frequency semantics.",
  "Preserve transform-convention semantics.",
  "Preserve magnitude-and-phase semantics.",
  "Preserve convolution and shift dualities.",
  "Preserve classical-versus-distributional distinction.",
  "Preserve multidimensional transform semantics.",
  "Preserve machine-readable Fourier-transform semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "The Fourier transform maps a continuous-domain function or signal into a continuous frequency-domain representation using complex exponential basis functions."

 FORWARD_SUMMARY =
  "The forward Fourier transform computes the frequency-domain spectrum from the original signal."

 INVERSE_SUMMARY =
  "The inverse Fourier transform reconstructs the original signal from its spectrum under compatible conditions."

 MAGNITUDE_PHASE_SUMMARY =
  "Magnitude describes spectral strength while phase describes relative complex angle; both are generally required for exact reconstruction."

 CONVOLUTION_SUMMARY =
  "Convolution in the original domain becomes multiplication in the Fourier domain."

 SHIFT_SUMMARY =
  "Time shifts produce linear phase factors, while modulation produces frequency shifts."

 ENERGY_SUMMARY =
  "Parseval and Plancherel relations connect energy or inner products across the original and Fourier domains."

 DISTRIBUTIONAL_SUMMARY =
  "Distributional Fourier transforms extend the transform to impulses, sinusoids and other generalised functions."

 MULTIDIMENSIONAL_SUMMARY =
  "Multidimensional Fourier transforms represent spatial and higher-dimensional frequency content."
}

FLOW {
 INPUT
 -> detect_Fourier_transform_context
 -> identify original_domain_signal_frequency variable_units_integrability_and_transform convention
 -> classify classical_squareIntegrable_or_distributional transform interpretation
 -> compute forward_or_inverse Fourier transform
 -> derive complex_magnitude_phase_energy_or_spatial spectrum
 -> apply linearity_timeShift_frequencyShift_scaling_modulation_convolution_multiplication_differentiation_integration_duality_or_conjugation properties where useful
 -> verify Parseval_or_Plancherel relations where relevant
 -> if_multidimensional apply multidimensional transform
 -> validate scaling_sign_units_symmetry_and_inverse reconstruction
 -> route to Fourier_analysis signal_processing harmonic_analysis complex_analysis linear_algebra differential_equations PDE probability_statistics numerical_analysis communications_mathematics_or_image_processing ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Fourier Transform Ontology defines continuous frequency-domain transformation of functions and signals. It supports forward and inverse transforms, transform pairs, magnitude and phase spectra, Fourier kernels, frequency variables, linearity, time and frequency shifts, scaling, modulation, convolution and multiplication duality, differentiation and integration properties, Parseval and Plancherel relations, Hermitian and parity symmetries, transform existence, square-integrable and distributional interpretations, canonical transform pairs such as impulses, rectangular pulses, sinc and Gaussian functions, and multidimensional Fourier transformation."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-FOURIER-TRANSFORM-ONTOLOGY-V1"
}

}

END_FLOW
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