Finite exponential families, arithmetic scores and relative meshes

Definition
erdos390_analytic_foundations_001
by doctosil · Sep 17, 2026 · Mathlib c5ea003 (Lean v4.30.0)
analytic-number-theoryerdos-390erdos390-source-construction
This bundle supplies the initial analytic objects for the Erdős 390 construction: finite probability distributions and exponential tilts, geometric relative meshes, smooth structured cells, prime and valuation scores, divisor-counting averages, head patterns, guard ledgers, local fugacity corrections, and covariance-driven vector fields.
For a finite sample space $\Omega$ , nonnegative baseline weights $b_\omega$ of positive total mass and the scaled parameter-dependent score $f_\xi(\omega)$ , the exponential-family weights and normalized probabilities are
Z(\xi)=\sum_{\omega\in\Omega}b_\omega e^{f_\xi(\omega)},\qquad P_\xi(\omega)=\frac{b_\omega e^{f_\xi(\omega)}}{Z(\xi)}.
The finite expectation and covariance operations are formed from these probabilities. Arithmetic scores include $I_p(m)=\mathbf1_{p\mid m}$ , $V_p(m)=v_p(m)$ and $J_p(m)=V_p(m)-I_p(m)$ . A relative mesh records endpoints $t_k=\delta(1+r)^k$ , a positive cell count and $r>0$ , with its last endpoint equal to $1$ and $r\le\eta$ .
These definitions provide the common arithmetic and finite-probability language for later covariance, quadrature and nonlinear fitting estimates. The bundle also defines target-path and covariance-gap predicates; such predicates specify hypotheses and do not by themselves assert that the corresponding estimates hold.
Definition code
import Mathlib
import Definitions.Def_erdos390_source_bank_interface
import Definitions.Def_erdos390_dickman_step_function
import Theorems.Thm_MediumPNT
import Theorems.Thm_Erdos390_Full_DickmanBasic_kernel_product_bound
import Theorems.Thm_Erdos390_Full_DickmanBasic_rho_pos_on_zero_five
import Theorems.Thm_Erdos390_Full_DickmanBasic_kernel_secondDerivative_first_bound

namespace Erdos390.Full.AffineCertificate
end Erdos390.Full.AffineCertificate
namespace Erdos390.Full.ArithmeticBandGeometry
end Erdos390.Full.ArithmeticBandGeometry
namespace Erdos390.Full.ArithmeticGaugeStableInverse
end Erdos390.Full.ArithmeticGaugeStableInverse
namespace Erdos390.Full.ArithmeticModel
end Erdos390.Full.ArithmeticModel
namespace Erdos390.Full.CanonicalEndpointAnchorCoverage
end Erdos390.Full.CanonicalEndpointAnchorCoverage
namespace Erdos390.Full.CanonicalEndpointCenterEnvelopeEventually
end Erdos390.Full.CanonicalEndpointCenterEnvelopeEventually
namespace Erdos390.Full.CanonicalEndpointDiagonalEventually
end Erdos390.Full.CanonicalEndpointDiagonalEventually
namespace Erdos390.Full.CanonicalEndpointDoubleKernelOrdinaryRowEventually
end Erdos390.Full.CanonicalEndpointDoubleKernelOrdinaryRowEventually
namespace Erdos390.Full.CanonicalEndpointDoubleKernelRowEventually
end Erdos390.Full.CanonicalEndpointDoubleKernelRowEventually
namespace Erdos390.Full.CanonicalEndpointIntervalMesh
end Erdos390.Full.CanonicalEndpointIntervalMesh
namespace Erdos390.Full.CanonicalEndpointMeshGeometryEventually
end Erdos390.Full.CanonicalEndpointMeshGeometryEventually
namespace Erdos390.Full.CanonicalEndpointMultiAnchorCoverage
end Erdos390.Full.CanonicalEndpointMultiAnchorCoverage
namespace Erdos390.Full.CanonicalEndpointMultiAnchorIntervalMesh
end Erdos390.Full.CanonicalEndpointMultiAnchorIntervalMesh
namespace Erdos390.Full.CanonicalEndpointMultiAnchorWeightedInverse
end Erdos390.Full.CanonicalEndpointMultiAnchorWeightedInverse
namespace Erdos390.Full.CanonicalEndpointMultiAnchorWeightedInverseEventually
end Erdos390.Full.CanonicalEndpointMultiAnchorWeightedInverseEventually
namespace Erdos390.Full.CanonicalEndpointOrdinaryProjectedRawInverseEventually
end Erdos390.Full.CanonicalEndpointOrdinaryProjectedRawInverseEventually
namespace Erdos390.Full.CanonicalEndpointOrdinaryRawRowEventually
end Erdos390.Full.CanonicalEndpointOrdinaryRawRowEventually
namespace Erdos390.Full.CanonicalEndpointPrefixGeometry
end Erdos390.Full.CanonicalEndpointPrefixGeometry
namespace Erdos390.Full.CanonicalEndpointRawWeightDefectEventually
end Erdos390.Full.CanonicalEndpointRawWeightDefectEventually
namespace Erdos390.Full.CanonicalEndpointRelativeCenterEventually
end Erdos390.Full.CanonicalEndpointRelativeCenterEventually
namespace Erdos390.Full.CanonicalEndpointTwoTailCertificate
end Erdos390.Full.CanonicalEndpointTwoTailCertificate
namespace Erdos390.Full.CanonicalEndpointWeightedInverseAttachment
end Erdos390.Full.CanonicalEndpointWeightedInverseAttachment
namespace Erdos390.Full.CanonicalEndpointWeightedInverseEventually
end Erdos390.Full.CanonicalEndpointWeightedInverseEventually
namespace Erdos390.Full.CanonicalRegularMeshEndpointFamily
end Erdos390.Full.CanonicalRegularMeshEndpointFamily
namespace Erdos390.Full.CoarseMixture
end Erdos390.Full.CoarseMixture
namespace Erdos390.Full.CompressedArithmeticOperator
end Erdos390.Full.CompressedArithmeticOperator
namespace Erdos390.Full.ConditionedPoisson
end Erdos390.Full.ConditionedPoisson
namespace Erdos390.Full.ConditionedPoissonLimit
end Erdos390.Full.ConditionedPoissonLimit
namespace Erdos390.Full.ContinuumCellGraph
end Erdos390.Full.ContinuumCellGraph
namespace Erdos390.Full.ContinuumCellGraph.ArithmeticGaugeOrdinary
end Erdos390.Full.ContinuumCellGraph.ArithmeticGaugeOrdinary
namespace Erdos390.Full.ContinuumCellGraphReversible
end Erdos390.Full.ContinuumCellGraphReversible
namespace Erdos390.Full.ContinuumIntervalCenterBounds
end Erdos390.Full.ContinuumIntervalCenterBounds
namespace Erdos390.Full.ContinuumManyLowArithmeticGaugeOrdinary
end Erdos390.Full.ContinuumManyLowArithmeticGaugeOrdinary
namespace Erdos390.Full.ContinuumManyLowHighGeometry
end Erdos390.Full.ContinuumManyLowHighGeometry
namespace Erdos390.Full.ContinuumManyLowProjectedRawInverse
end Erdos390.Full.ContinuumManyLowProjectedRawInverse
namespace Erdos390.Full.ContinuumManyLowRawGraphInverse
end Erdos390.Full.ContinuumManyLowRawGraphInverse
namespace Erdos390.Full.ContinuumProjectedRawOrdinary
end Erdos390.Full.ContinuumProjectedRawOrdinary
namespace Erdos390.Full.ContinuumRawLowRow
end Erdos390.Full.ContinuumRawLowRow
namespace Erdos390.Full.ContinuumRowResidualBound
end Erdos390.Full.ContinuumRowResidualBound
namespace Erdos390.Full.ContinuumSharpArithmeticTransfer
end Erdos390.Full.ContinuumSharpArithmeticTransfer
namespace Erdos390.Full.ContinuumTwoTailResidualBound
end Erdos390.Full.ContinuumTwoTailResidualBound
namespace Erdos390.Full.DiagonalPrimeQuadrature
end Erdos390.Full.DiagonalPrimeQuadrature
namespace Erdos390.Full.DickmanBasic
end Erdos390.Full.DickmanBasic
namespace Erdos390.Full.DickmanFourMarkProductKernel
end Erdos390.Full.DickmanFourMarkProductKernel
namespace Erdos390.Full.DivisibilityMomentBounds
end Erdos390.Full.DivisibilityMomentBounds
namespace Erdos390.Full.DoubleKernelPrimeQuadrature
end Erdos390.Full.DoubleKernelPrimeQuadrature
namespace Erdos390.Full.EndpointArithmeticGaugeIdentification
end Erdos390.Full.EndpointArithmeticGaugeIdentification
namespace Erdos390.Full.EndpointContinuumMeshIdentification
end Erdos390.Full.EndpointContinuumMeshIdentification
namespace Erdos390.Full.EquivalentNormODE
end Erdos390.Full.EquivalentNormODE
namespace Erdos390.Full.FiniteAnchoredDirichletQuadratic
end Erdos390.Full.FiniteAnchoredDirichletQuadratic
namespace Erdos390.Full.FiniteExponentialFamily
end Erdos390.Full.FiniteExponentialFamily
namespace Erdos390.Full.FiniteGraphQuotientInverse
end Erdos390.Full.FiniteGraphQuotientInverse
namespace Erdos390.Full.FiniteGraphStableInverse
end Erdos390.Full.FiniteGraphStableInverse
namespace Erdos390.Full.FiniteLogStieltjes
end Erdos390.Full.FiniteLogStieltjes
namespace Erdos390.Full.FiniteManyLowRawGraphInverse
end Erdos390.Full.FiniteManyLowRawGraphInverse
namespace Erdos390.Full.FiniteProbabilityCellConstantCovariance
end Erdos390.Full.FiniteProbabilityCellConstantCovariance
namespace Erdos390.Full.FiniteProbabilityCovarianceCauchySchwarz
end Erdos390.Full.FiniteProbabilityCovarianceCauchySchwarz
namespace Erdos390.Full.FiniteProbabilityMixture
end Erdos390.Full.FiniteProbabilityMixture
namespace Erdos390.Full.FiniteProbabilityMixtureGuardReindexing
end Erdos390.Full.FiniteProbabilityMixtureGuardReindexing
namespace Erdos390.Full.FiniteProbabilityMixturePerturbation
end Erdos390.Full.FiniteProbabilityMixturePerturbation
namespace Erdos390.Full.FiniteProbabilityMixtureTilt
end Erdos390.Full.FiniteProbabilityMixtureTilt
namespace Erdos390.Full.FiniteProbabilityPowerCorrectionMixturePerturbation
end Erdos390.Full.FiniteProbabilityPowerCorrectionMixturePerturbation
namespace Erdos390.Full.FiniteRawLowDiagonal
end Erdos390.Full.FiniteRawLowDiagonal
namespace Erdos390.Full.FiniteReversibleGraph
end Erdos390.Full.FiniteReversibleGraph
namespace Erdos390.Full.FiniteSignedQuadraticEntryTransfer
end Erdos390.Full.FiniteSignedQuadraticEntryTransfer
namespace Erdos390.Full.FiniteTiltCovarianceDomination
end Erdos390.Full.FiniteTiltCovarianceDomination
namespace Erdos390.Full.FiniteTiltTV
end Erdos390.Full.FiniteTiltTV
namespace Erdos390.Full.FirstFailure
end Erdos390.Full.FirstFailure
namespace Erdos390.Full.FixedFiniteMixtureFullUniform
end Erdos390.Full.FixedFiniteMixtureFullUniform
namespace Erdos390.Full.FixedFiniteMixtureGlobalTiltLemma75
end Erdos390.Full.FixedFiniteMixtureGlobalTiltLemma75
namespace Erdos390.Full.FixedFiniteMixtureLemma75
end Erdos390.Full.FixedFiniteMixtureLemma75
namespace Erdos390.Full.FixedFiniteMixturePrimePower
end Erdos390.Full.FixedFiniteMixturePrimePower
namespace Erdos390.Full.FixedFiniteMixtureSignedSquarefree
end Erdos390.Full.FixedFiniteMixtureSignedSquarefree
namespace Erdos390.Full.FriableAsymptotic
end Erdos390.Full.FriableAsymptotic
namespace Erdos390.Full.FullTiltPairChamber
end Erdos390.Full.FullTiltPairChamber
namespace Erdos390.Full.FullTiltPairHarmonicRate
end Erdos390.Full.FullTiltPairHarmonicRate
namespace Erdos390.Full.FullTiltPrimePowerActualChamber
end Erdos390.Full.FullTiltPrimePowerActualChamber
namespace Erdos390.Full.FullTiltPrimePowerCovariance
end Erdos390.Full.FullTiltPrimePowerCovariance
namespace Erdos390.Full.FullTiltPrimePowerFallback
end Erdos390.Full.FullTiltPrimePowerFallback
namespace Erdos390.Full.GuardDeletionFamilyRows
end Erdos390.Full.GuardDeletionFamilyRows
namespace Erdos390.Full.GuardDeletionProbability
end Erdos390.Full.GuardDeletionProbability
namespace Erdos390.Full.GuardDeletionReindexing
end Erdos390.Full.GuardDeletionReindexing
namespace Erdos390.Full.GuardDeletionSquarefreeProfiles
end Erdos390.Full.GuardDeletionSquarefreeProfiles
namespace Erdos390.Full.GuardSquarefreeErrorRate
end Erdos390.Full.GuardSquarefreeErrorRate
namespace Erdos390.Full.GuardedUniformCell
end Erdos390.Full.GuardedUniformCell
namespace Erdos390.Full.HeadPattern
end Erdos390.Full.HeadPattern
namespace Erdos390.Full.KernelPrimeQuadrature
end Erdos390.Full.KernelPrimeQuadrature
namespace Erdos390.Full.LocalFugacity
end Erdos390.Full.LocalFugacity
namespace Erdos390.Full.LocalFugacityBounds
end Erdos390.Full.LocalFugacityBounds
namespace Erdos390.Full.LocalFugacityRestoration
end Erdos390.Full.LocalFugacityRestoration
namespace Erdos390.Full.LocalizedMarkedTiltCovariance
end Erdos390.Full.LocalizedMarkedTiltCovariance
namespace Erdos390.Full.MarkedFriableAsymptotic
end Erdos390.Full.MarkedFriableAsymptotic
namespace Erdos390.Full.MovingLowGaugeTransfer
end Erdos390.Full.MovingLowGaugeTransfer
namespace Erdos390.Full.MovingLowMomentQuadrature
end Erdos390.Full.MovingLowMomentQuadrature
namespace Erdos390.Full.MovingLowSharpRowAggregation
end Erdos390.Full.MovingLowSharpRowAggregation
namespace Erdos390.Full.NonlinearFitODE
end Erdos390.Full.NonlinearFitODE
namespace Erdos390.Full.NuisanceCovariance
end Erdos390.Full.NuisanceCovariance
namespace Erdos390.Full.OmittedScoreCell
end Erdos390.Full.OmittedScoreCell
namespace Erdos390.Full.OmittedScoreTilt
end Erdos390.Full.OmittedScoreTilt
namespace Erdos390.Full.OmittedTiltFallback
end Erdos390.Full.OmittedTiltFallback
namespace Erdos390.Full.OmittedTiltHarmonicRate
end Erdos390.Full.OmittedTiltHarmonicRate
namespace Erdos390.Full.OmittedTiltMarkedProbability
end Erdos390.Full.OmittedTiltMarkedProbability
namespace Erdos390.Full.OmittedTiltPairChamber
end Erdos390.Full.OmittedTiltPairChamber
namespace Erdos390.Full.OrdinaryRawInverseTransfer
end Erdos390.Full.OrdinaryRawInverseTransfer
namespace Erdos390.Full.OrdinaryRawOperatorRowTransfer
end Erdos390.Full.OrdinaryRawOperatorRowTransfer
namespace Erdos390.Full.PaperActualCenterNuisanceRow
end Erdos390.Full.PaperActualCenterNuisanceRow
namespace Erdos390.Full.PaperActualCompensatedMarkedRowProfiles
end Erdos390.Full.PaperActualCompensatedMarkedRowProfiles
namespace Erdos390.Full.PaperActualCompensatedRegression
end Erdos390.Full.PaperActualCompensatedRegression
namespace Erdos390.Full.PaperActualFastMarkedRowProfiles
end Erdos390.Full.PaperActualFastMarkedRowProfiles
namespace Erdos390.Full.PaperActualFullBandIdentification
end Erdos390.Full.PaperActualFullBandIdentification
namespace Erdos390.Full.PaperActualFullEffectiveBall
end Erdos390.Full.PaperActualFullEffectiveBall
namespace Erdos390.Full.PaperActualFullOrdinaryProjectionConnector
end Erdos390.Full.PaperActualFullOrdinaryProjectionConnector
namespace Erdos390.Full.PaperActualFullOrdinaryRowTransfer
end Erdos390.Full.PaperActualFullOrdinaryRowTransfer
namespace Erdos390.Full.PaperActualFullProjectedInverseAssembly
end Erdos390.Full.PaperActualFullProjectedInverseAssembly
namespace Erdos390.Full.PaperActualFullProjectedInverseEventually
end Erdos390.Full.PaperActualFullProjectedInverseEventually
namespace Erdos390.Full.PaperActualLemma86Assembly
end Erdos390.Full.PaperActualLemma86Assembly
namespace Erdos390.Full.PaperActualLemma86PaperScale
end Erdos390.Full.PaperActualLemma86PaperScale
namespace Erdos390.Full.PaperActualLemma86WeightedRow
end Erdos390.Full.PaperActualLemma86WeightedRow
namespace Erdos390.Full.PaperActualPrimePowerRelative
end Erdos390.Full.PaperActualPrimePowerRelative
namespace Erdos390.Full.PaperActualPrimePowerRowTransfer
end Erdos390.Full.PaperActualPrimePowerRowTransfer
namespace Erdos390.Full.PaperActualSchurEndpointReferenceTransport
end Erdos390.Full.PaperActualSchurEndpointReferenceTransport
namespace Erdos390.Full.PaperActualSchurMarkedRow
end Erdos390.Full.PaperActualSchurMarkedRow
namespace Erdos390.Full.PaperActualSchurOrdinaryEndpointConnector
end Erdos390.Full.PaperActualSchurOrdinaryEndpointConnector
namespace Erdos390.Full.PaperActualSchurOrdinaryRow
end Erdos390.Full.PaperActualSchurOrdinaryRow
namespace Erdos390.Full.PaperActualSlowRightRowFinite
end Erdos390.Full.PaperActualSlowRightRowFinite
namespace Erdos390.Full.PaperActualSquarefreeMarkedRow
end Erdos390.Full.PaperActualSquarefreeMarkedRow
namespace Erdos390.Full.PaperActualSquarefreeReference
end Erdos390.Full.PaperActualSquarefreeReference
namespace Erdos390.Full.PaperActualSquarefreeSlowLower
end Erdos390.Full.PaperActualSquarefreeSlowLower
namespace Erdos390.Full.PaperActualTwoStageRegression
end Erdos390.Full.PaperActualTwoStageRegression
namespace Erdos390.Full.PaperBandBestDistance
end Erdos390.Full.PaperBandBestDistance
namespace Erdos390.Full.PaperBandQuadraticGeometry
end Erdos390.Full.PaperBandQuadraticGeometry
namespace Erdos390.Full.PaperBaselineSlack
end Erdos390.Full.PaperBaselineSlack
namespace Erdos390.Full.PaperBoundedScorePrefixCovariance
end Erdos390.Full.PaperBoundedScorePrefixCovariance
namespace Erdos390.Full.PaperBridgeBaselineL1
end Erdos390.Full.PaperBridgeBaselineL1
namespace Erdos390.Full.PaperBridgeCanonicalGuardPowerCorrection
end Erdos390.Full.PaperBridgeCanonicalGuardPowerCorrection
namespace Erdos390.Full.PaperBridgeCanonicalPhysicalPowerCorrectionEventually
end Erdos390.Full.PaperBridgeCanonicalPhysicalPowerCorrectionEventually
namespace Erdos390.Full.PaperBridgeCanonicalPowerCorrectionTriangle
end Erdos390.Full.PaperBridgeCanonicalPowerCorrectionTriangle
namespace Erdos390.Full.PaperBridgeCanonicalRawLemma75
end Erdos390.Full.PaperBridgeCanonicalRawLemma75
namespace Erdos390.Full.PaperBridgeCellTiltDecomposition
end Erdos390.Full.PaperBridgeCellTiltDecomposition
namespace Erdos390.Full.PaperBridgeFit
end Erdos390.Full.PaperBridgeFit
namespace Erdos390.Full.PaperBridgeFit.PaperActualSlowRightRowFinite
end Erdos390.Full.PaperBridgeFit.PaperActualSlowRightRowFinite
namespace Erdos390.Full.PaperBridgeFitFeasibility
end Erdos390.Full.PaperBridgeFitFeasibility
namespace Erdos390.Full.PaperBridgeMediumValuationMixture
end Erdos390.Full.PaperBridgeMediumValuationMixture
namespace Erdos390.Full.PaperBridgeNuisanceTiltFallback
end Erdos390.Full.PaperBridgeNuisanceTiltFallback
namespace Erdos390.Full.PaperBridgePhysicalDivisorCovariance
end Erdos390.Full.PaperBridgePhysicalDivisorCovariance
namespace Erdos390.Full.PaperBridgePhysicalPowerCorrectionRow
end Erdos390.Full.PaperBridgePhysicalPowerCorrectionRow
namespace Erdos390.Full.PaperBridgePhysicalValuationRow
end Erdos390.Full.PaperBridgePhysicalValuationRow
namespace Erdos390.Full.PaperBridgePrimewiseRate
end Erdos390.Full.PaperBridgePrimewiseRate
namespace Erdos390.Full.PaperBridgeQuadrature
end Erdos390.Full.PaperBridgeQuadrature
namespace Erdos390.Full.PaperCanonicalActiveMassBaseline
end Erdos390.Full.PaperCanonicalActiveMassBaseline
namespace Erdos390.Full.PaperCanonicalActualBandQuadraticTransfer
end Erdos390.Full.PaperCanonicalActualBandQuadraticTransfer
namespace Erdos390.Full.PaperCanonicalActualFullQuotientNullIdentification
end Erdos390.Full.PaperCanonicalActualFullQuotientNullIdentification
namespace Erdos390.Full.PaperCanonicalActualMomentBoundsEventually
end Erdos390.Full.PaperCanonicalActualMomentBoundsEventually
namespace Erdos390.Full.PaperCanonicalActualPrimePowerWeightedRowEventually
end Erdos390.Full.PaperCanonicalActualPrimePowerWeightedRowEventually
namespace Erdos390.Full.PaperCanonicalActualSchurOrdinaryEndpointEventually
end Erdos390.Full.PaperCanonicalActualSchurOrdinaryEndpointEventually
namespace Erdos390.Full.PaperCanonicalActualSchurOrdinaryRowsEventually
end Erdos390.Full.PaperCanonicalActualSchurOrdinaryRowsEventually
namespace Erdos390.Full.PaperCanonicalActualSchurSharpOrdinaryEventually
end Erdos390.Full.PaperCanonicalActualSchurSharpOrdinaryEventually
namespace Erdos390.Full.PaperCanonicalBandFirstMomentEventually
end Erdos390.Full.PaperCanonicalBandFirstMomentEventually
namespace Erdos390.Full.PaperCanonicalBandSchurInverseFromFull
end Erdos390.Full.PaperCanonicalBandSchurInverseFromFull
namespace Erdos390.Full.PaperCanonicalBaseline
end Erdos390.Full.PaperCanonicalBaseline
namespace Erdos390.Full.PaperCanonicalBaselineBridge
end Erdos390.Full.PaperCanonicalBaselineBridge
namespace Erdos390.Full.PaperCanonicalFullQuotientEventually
end Erdos390.Full.PaperCanonicalFullQuotientEventually
namespace Erdos390.Full.PaperCanonicalFullQuotientFineMeshEventually
end Erdos390.Full.PaperCanonicalFullQuotientFineMeshEventually
namespace Erdos390.Full.PaperCanonicalFullQuotientPaperDistanceEventually
end Erdos390.Full.PaperCanonicalFullQuotientPaperDistanceEventually
namespace Erdos390.Full.PaperCanonicalFullQuotientQScaledEventually
end Erdos390.Full.PaperCanonicalFullQuotientQScaledEventually
namespace Erdos390.Full.PaperCanonicalGeometricMomentFineMeshEventually
end Erdos390.Full.PaperCanonicalGeometricMomentFineMeshEventually
namespace Erdos390.Full.PaperCanonicalHeadPhysicalTarget
end Erdos390.Full.PaperCanonicalHeadPhysicalTarget
namespace Erdos390.Full.PaperCanonicalLemma84Eventually
end Erdos390.Full.PaperCanonicalLemma84Eventually
namespace Erdos390.Full.PaperCanonicalLemma84GapNuisanceEventually
end Erdos390.Full.PaperCanonicalLemma84GapNuisanceEventually
namespace Erdos390.Full.PaperCanonicalLemma86AnchorEventually
end Erdos390.Full.PaperCanonicalLemma86AnchorEventually
namespace Erdos390.Full.PaperCanonicalLemma86CoefficientPackageEventually
end Erdos390.Full.PaperCanonicalLemma86CoefficientPackageEventually
namespace Erdos390.Full.PaperCanonicalLemma86CompensatedMarkedRowEventually
end Erdos390.Full.PaperCanonicalLemma86CompensatedMarkedRowEventually
namespace Erdos390.Full.PaperCanonicalLemma86Eventually
end Erdos390.Full.PaperCanonicalLemma86Eventually
namespace Erdos390.Full.PaperCanonicalLemma86GeometryCoefficientsEventually
end Erdos390.Full.PaperCanonicalLemma86GeometryCoefficientsEventually
namespace Erdos390.Full.PaperCanonicalLemma86RelativePrimePowerEventually
end Erdos390.Full.PaperCanonicalLemma86RelativePrimePowerEventually
namespace Erdos390.Full.PaperCanonicalLemma86RemainingOutputsEventually
end Erdos390.Full.PaperCanonicalLemma86RemainingOutputsEventually
namespace Erdos390.Full.PaperCanonicalMarkedNuisanceRows
end Erdos390.Full.PaperCanonicalMarkedNuisanceRows
namespace Erdos390.Full.PaperCanonicalMediumNuisanceRows
end Erdos390.Full.PaperCanonicalMediumNuisanceRows
namespace Erdos390.Full.PaperCanonicalMomentRatioEventually
end Erdos390.Full.PaperCanonicalMomentRatioEventually
namespace Erdos390.Full.PaperCanonicalNonstepFullSlowEventually
end Erdos390.Full.PaperCanonicalNonstepFullSlowEventually
namespace Erdos390.Full.PaperCanonicalNonstepLocalDiagonalEventually
end Erdos390.Full.PaperCanonicalNonstepLocalDiagonalEventually
namespace Erdos390.Full.PaperCanonicalNonstepPowerInputsEventually
end Erdos390.Full.PaperCanonicalNonstepPowerInputsEventually
namespace Erdos390.Full.PaperCanonicalNonstepSlowResidualEventually
end Erdos390.Full.PaperCanonicalNonstepSlowResidualEventually
namespace Erdos390.Full.PaperCanonicalNonstepSlowRightColumnEventually
end Erdos390.Full.PaperCanonicalNonstepSlowRightColumnEventually
namespace Erdos390.Full.PaperCanonicalNonstepSquarefreeSlowEventually
end Erdos390.Full.PaperCanonicalNonstepSquarefreeSlowEventually
namespace Erdos390.Full.PaperCanonicalNuisanceEffectiveBall
end Erdos390.Full.PaperCanonicalNuisanceEffectiveBall
namespace Erdos390.Full.PaperCanonicalNuisanceEffectiveGap
end Erdos390.Full.PaperCanonicalNuisanceEffectiveGap
namespace Erdos390.Full.PaperCanonicalNuisanceUniformFloor
end Erdos390.Full.PaperCanonicalNuisanceUniformFloor
namespace Erdos390.Full.PaperCanonicalPositiveCellVarianceEventually
end Erdos390.Full.PaperCanonicalPositiveCellVarianceEventually
namespace Erdos390.Full.PaperCanonicalPrimeAnchorEventually
end Erdos390.Full.PaperCanonicalPrimeAnchorEventually
namespace Erdos390.Full.PaperCanonicalPrimeGraphActualSchurEventually
end Erdos390.Full.PaperCanonicalPrimeGraphActualSchurEventually
namespace Erdos390.Full.PaperCanonicalPrimeGraphReferenceEventually
end Erdos390.Full.PaperCanonicalPrimeGraphReferenceEventually
namespace Erdos390.Full.PaperCanonicalPrimeRowResidualEventually
end Erdos390.Full.PaperCanonicalPrimeRowResidualEventually
namespace Erdos390.Full.PaperCanonicalReferenceSlowEventually
end Erdos390.Full.PaperCanonicalReferenceSlowEventually
namespace Erdos390.Full.PaperCanonicalSlowKappa
end Erdos390.Full.PaperCanonicalSlowKappa
namespace Erdos390.Full.PaperCanonicalTiltedPrefixRows
end Erdos390.Full.PaperCanonicalTiltedPrefixRows
namespace Erdos390.Full.PaperCanonicalTiltedValuationMeanRows
end Erdos390.Full.PaperCanonicalTiltedValuationMeanRows
namespace Erdos390.Full.PaperCompensatedCoefficientBounds
end Erdos390.Full.PaperCompensatedCoefficientBounds
namespace Erdos390.Full.PaperEffectiveBallCovarianceGap
end Erdos390.Full.PaperEffectiveBallCovarianceGap
namespace Erdos390.Full.PaperEffectiveNorm
end Erdos390.Full.PaperEffectiveNorm
namespace Erdos390.Full.PaperEffectiveScoreBound
end Erdos390.Full.PaperEffectiveScoreBound
namespace Erdos390.Full.PaperExactSchurFastSlowCoercivity
end Erdos390.Full.PaperExactSchurFastSlowCoercivity
namespace Erdos390.Full.PaperExactSchurRawSlow
end Erdos390.Full.PaperExactSchurRawSlow
namespace Erdos390.Full.PaperExactSchurTwoStageQuadratic
end Erdos390.Full.PaperExactSchurTwoStageQuadratic
namespace Erdos390.Full.PaperExactTwoStageOrdinaryFast
end Erdos390.Full.PaperExactTwoStageOrdinaryFast
namespace Erdos390.Full.PaperExactTwoStageTargetSolve
end Erdos390.Full.PaperExactTwoStageTargetSolve
namespace Erdos390.Full.PaperFixedFiniteRawTiltedPrefixRows
end Erdos390.Full.PaperFixedFiniteRawTiltedPrefixRows
namespace Erdos390.Full.PaperGuardCensus
end Erdos390.Full.PaperGuardCensus
namespace Erdos390.Full.PaperGuardDeletionRows
end Erdos390.Full.PaperGuardDeletionRows
namespace Erdos390.Full.PaperGuardPowerCorrectionMixture
end Erdos390.Full.PaperGuardPowerCorrectionMixture
namespace Erdos390.Full.PaperGuardedStructuredSample
end Erdos390.Full.PaperGuardedStructuredSample
namespace Erdos390.Full.PaperHeadPrimeMarkedExact
end Erdos390.Full.PaperHeadPrimeMarkedExact
namespace Erdos390.Full.PaperHeadSimplex
end Erdos390.Full.PaperHeadSimplex
namespace Erdos390.Full.PaperLemma84StructuralCutoff
end Erdos390.Full.PaperLemma84StructuralCutoff
namespace Erdos390.Full.PaperLemma86CommonConstants
end Erdos390.Full.PaperLemma86CommonConstants
namespace Erdos390.Full.PaperMainFastSlowCoordinates
end Erdos390.Full.PaperMainFastSlowCoordinates
namespace Erdos390.Full.PaperMainRawGaugeCoordinates
end Erdos390.Full.PaperMainRawGaugeCoordinates
namespace Erdos390.Full.PaperMediumNuisanceInputReduction
end Erdos390.Full.PaperMediumNuisanceInputReduction
namespace Erdos390.Full.PaperMovingPrefixMarkedCell
end Erdos390.Full.PaperMovingPrefixMarkedCell
namespace Erdos390.Full.PaperNonstepLocalDiagonal
end Erdos390.Full.PaperNonstepLocalDiagonal
namespace Erdos390.Full.PaperNonstepPowerCorrectionLedger
end Erdos390.Full.PaperNonstepPowerCorrectionLedger
namespace Erdos390.Full.PaperNonstepPrimeMomentRate
end Erdos390.Full.PaperNonstepPrimeMomentRate
namespace Erdos390.Full.PaperNonstepReferenceSlowDirect
end Erdos390.Full.PaperNonstepReferenceSlowDirect
namespace Erdos390.Full.PaperNonstepSlowNuisanceCorrection
end Erdos390.Full.PaperNonstepSlowNuisanceCorrection
namespace Erdos390.Full.PaperNonstepSlowRightLedger
end Erdos390.Full.PaperNonstepSlowRightLedger
namespace Erdos390.Full.PaperNonstepSquarefreeSlowLedger
end Erdos390.Full.PaperNonstepSquarefreeSlowLedger
namespace Erdos390.Full.PaperNuisancePrimeLogRows
end Erdos390.Full.PaperNuisancePrimeLogRows
namespace Erdos390.Full.PaperPermittedRegularMesh
end Erdos390.Full.PaperPermittedRegularMesh
namespace Erdos390.Full.PaperPhysicalIntervalNuisanceGap
end Erdos390.Full.PaperPhysicalIntervalNuisanceGap
namespace Erdos390.Full.PaperPrimeDeviationGeometry
end Erdos390.Full.PaperPrimeDeviationGeometry
namespace Erdos390.Full.PaperPrimeDeviationMoments
end Erdos390.Full.PaperPrimeDeviationMoments
namespace Erdos390.Full.PaperPrimePowerActualSplice
end Erdos390.Full.PaperPrimePowerActualSplice
namespace Erdos390.Full.PaperPrimePowerAuxiliaryPrime
end Erdos390.Full.PaperPrimePowerAuxiliaryPrime
namespace Erdos390.Full.PaperPrimePowerChamberError
end Erdos390.Full.PaperPrimePowerChamberError
namespace Erdos390.Full.PaperPrimePowerFourDisplays
end Erdos390.Full.PaperPrimePowerFourDisplays
namespace Erdos390.Full.PaperPrimePowerGenericAggregation
end Erdos390.Full.PaperPrimePowerGenericAggregation
namespace Erdos390.Full.PaperPrimePowerLemma75
end Erdos390.Full.PaperPrimePowerLemma75
namespace Erdos390.Full.PaperPrimePowerPairAggregation
end Erdos390.Full.PaperPrimePowerPairAggregation
namespace Erdos390.Full.PaperPrimePowerPointwise
end Erdos390.Full.PaperPrimePowerPointwise
namespace Erdos390.Full.PaperPrimePowerRelativeQuadratic
end Erdos390.Full.PaperPrimePowerRelativeQuadratic
namespace Erdos390.Full.PaperPrimePowerRemainderRate
end Erdos390.Full.PaperPrimePowerRemainderRate
namespace Erdos390.Full.PaperPrimePowerRow
end Erdos390.Full.PaperPrimePowerRow
namespace Erdos390.Full.PaperPrimePowerSumAbsAggregation
end Erdos390.Full.PaperPrimePowerSumAbsAggregation
namespace Erdos390.Full.PaperPrimePowerTailLedger
end Erdos390.Full.PaperPrimePowerTailLedger
namespace Erdos390.Full.PaperPrimePowerTailRate
end Erdos390.Full.PaperPrimePowerTailRate
namespace Erdos390.Full.PaperPrimePowerTailRow
end Erdos390.Full.PaperPrimePowerTailRow
namespace Erdos390.Full.PaperProposition87ActiveMassTransport
end Erdos390.Full.PaperProposition87ActiveMassTransport
namespace Erdos390.Full.PaperProposition87AllPrimeMarkedSplit
end Erdos390.Full.PaperProposition87AllPrimeMarkedSplit
namespace Erdos390.Full.PaperProposition87Assembly
end Erdos390.Full.PaperProposition87Assembly
namespace Erdos390.Full.PaperProposition87CanonicalConclusion
end Erdos390.Full.PaperProposition87CanonicalConclusion
namespace Erdos390.Full.PaperProposition87CanonicalEventually
end Erdos390.Full.PaperProposition87CanonicalEventually
namespace Erdos390.Full.PaperProposition87CanonicalFeasibilitySlackEventually
end Erdos390.Full.PaperProposition87CanonicalFeasibilitySlackEventually
namespace Erdos390.Full.PaperProposition87CanonicalFullGap
end Erdos390.Full.PaperProposition87CanonicalFullGap
namespace Erdos390.Full.PaperProposition87CanonicalMarkedProfilesEventually
end Erdos390.Full.PaperProposition87CanonicalMarkedProfilesEventually
namespace Erdos390.Full.PaperProposition87CanonicalNuisanceReservesEventually
end Erdos390.Full.PaperProposition87CanonicalNuisanceReservesEventually
namespace Erdos390.Full.PaperProposition87CanonicalSpeedPackage
end Erdos390.Full.PaperProposition87CanonicalSpeedPackage
namespace Erdos390.Full.PaperProposition87CanonicalStatement
end Erdos390.Full.PaperProposition87CanonicalStatement
namespace Erdos390.Full.PaperProposition87CanonicalTargetPackage
end Erdos390.Full.PaperProposition87CanonicalTargetPackage
namespace Erdos390.Full.PaperProposition87CanonicalTwoStageAssembly
end Erdos390.Full.PaperProposition87CanonicalTwoStageAssembly
namespace Erdos390.Full.PaperProposition87LiteralBandBalance
end Erdos390.Full.PaperProposition87LiteralBandBalance
namespace Erdos390.Full.PaperProposition87MarkedProfilesAssembly
end Erdos390.Full.PaperProposition87MarkedProfilesAssembly
namespace Erdos390.Full.PaperProposition87MarkedProfilesMajorant
end Erdos390.Full.PaperProposition87MarkedProfilesMajorant
namespace Erdos390.Full.PaperProposition87MarkedRowRate
end Erdos390.Full.PaperProposition87MarkedRowRate
namespace Erdos390.Full.PaperProposition87MarkedTwoStage
end Erdos390.Full.PaperProposition87MarkedTwoStage
namespace Erdos390.Full.PaperProposition87SpeedRadius
end Erdos390.Full.PaperProposition87SpeedRadius
namespace Erdos390.Full.PaperProposition87UniformMarkedProfiles
end Erdos390.Full.PaperProposition87UniformMarkedProfiles
namespace Erdos390.Full.PaperRawPrefixThirdCumulantFallback
end Erdos390.Full.PaperRawPrefixThirdCumulantFallback
namespace Erdos390.Full.PaperRawPrefixThirdCumulantRow
end Erdos390.Full.PaperRawPrefixThirdCumulantRow
namespace Erdos390.Full.PaperRawPrefixValuationRow
end Erdos390.Full.PaperRawPrefixValuationRow
namespace Erdos390.Full.PaperRawTiltTaylorRate
end Erdos390.Full.PaperRawTiltTaylorRate
namespace Erdos390.Full.PaperRawTiltedPrefixRow
end Erdos390.Full.PaperRawTiltedPrefixRow
namespace Erdos390.Full.PaperRawTiltedPrefixRowUnrestricted
end Erdos390.Full.PaperRawTiltedPrefixRowUnrestricted
namespace Erdos390.Full.PaperRawTiltedValuationMeanRows
end Erdos390.Full.PaperRawTiltedValuationMeanRows
namespace Erdos390.Full.PaperReferenceSharpPrimeGraph
end Erdos390.Full.PaperReferenceSharpPrimeGraph
namespace Erdos390.Full.PaperScaleEndpoint
end Erdos390.Full.PaperScaleEndpoint
namespace Erdos390.Full.PaperScaleMarkedCell
end Erdos390.Full.PaperScaleMarkedCell
namespace Erdos390.Full.PaperSelectedMeshActualFullProjectedInverseEventually
end Erdos390.Full.PaperSelectedMeshActualFullProjectedInverseEventually
namespace Erdos390.Full.PaperSelectedMeshProjectedInverseEventually
end Erdos390.Full.PaperSelectedMeshProjectedInverseEventually
namespace Erdos390.Full.PaperSelectedMeshSchurEventually
end Erdos390.Full.PaperSelectedMeshSchurEventually
namespace Erdos390.Full.PaperSelectedMeshSchurRateEventually
end Erdos390.Full.PaperSelectedMeshSchurRateEventually
namespace Erdos390.Full.PaperSquarefreeSlowQuadraticLower
end Erdos390.Full.PaperSquarefreeSlowQuadraticLower
namespace Erdos390.Full.PaperStatisticNorm
end Erdos390.Full.PaperStatisticNorm
namespace Erdos390.Full.PaperTiltSmallness
end Erdos390.Full.PaperTiltSmallness
namespace Erdos390.Full.PaperTwoLocalRestorationBound
end Erdos390.Full.PaperTwoLocalRestorationBound
namespace Erdos390.Full.PaperUniformNuisanceGap
end Erdos390.Full.PaperUniformNuisanceGap
namespace Erdos390.Full.PaperValuationCutoff
end Erdos390.Full.PaperValuationCutoff
namespace Erdos390.Full.PaperVectorFieldEffectiveBound
end Erdos390.Full.PaperVectorFieldEffectiveBound
namespace Erdos390.Full.PaperVectorFieldSchur
end Erdos390.Full.PaperVectorFieldSchur
namespace Erdos390.Full.PaperWeightedGaugeDGeometry
end Erdos390.Full.PaperWeightedGaugeDGeometry
namespace Erdos390.Full.PaperWeightedInverseExport
end Erdos390.Full.PaperWeightedInverseExport
namespace Erdos390.Full.PartitionPrimeDeviationGeometry
end Erdos390.Full.PartitionPrimeDeviationGeometry
namespace Erdos390.Full.PoissonDickmanDirichlet
end Erdos390.Full.PoissonDickmanDirichlet
namespace Erdos390.Full.PoissonDickmanKernelBounds
end Erdos390.Full.PoissonDickmanKernelBounds
namespace Erdos390.Full.PoissonDickmanWeightedInverse
end Erdos390.Full.PoissonDickmanWeightedInverse
namespace Erdos390.Full.PositiveCellTransfer
end Erdos390.Full.PositiveCellTransfer
namespace Erdos390.Full.PowerLedger
end Erdos390.Full.PowerLedger
namespace Erdos390.Full.PrimeBandQuadrature
end Erdos390.Full.PrimeBandQuadrature
namespace Erdos390.Full.PrimeCoordinateSecondMoment
end Erdos390.Full.PrimeCoordinateSecondMoment
namespace Erdos390.Full.PrimeIntervalPartitionConstructor
end Erdos390.Full.PrimeIntervalPartitionConstructor
namespace Erdos390.Full.PrimePowerCovariance
end Erdos390.Full.PrimePowerCovariance
namespace Erdos390.Full.PrimePowerCutoffCovariance
end Erdos390.Full.PrimePowerCutoffCovariance
namespace Erdos390.Full.PrimePowerLcmGeometry
end Erdos390.Full.PrimePowerLcmGeometry
namespace Erdos390.Full.PrimePowerRowTransfer
end Erdos390.Full.PrimePowerRowTransfer
namespace Erdos390.Full.PrimePowerSharpBandTransfer
end Erdos390.Full.PrimePowerSharpBandTransfer
namespace Erdos390.Full.PrimeSquarefreeDirichletGeometry
end Erdos390.Full.PrimeSquarefreeDirichletGeometry
namespace Erdos390.Full.PrimeSums
end Erdos390.Full.PrimeSums
namespace Erdos390.Full.RegularMeshActualMomentBounds
end Erdos390.Full.RegularMeshActualMomentBounds
namespace Erdos390.Full.RegularMeshActualMomentBoundsEventually
end Erdos390.Full.RegularMeshActualMomentBoundsEventually
namespace Erdos390.Full.RegularMeshMomentBounds
end Erdos390.Full.RegularMeshMomentBounds
namespace Erdos390.Full.RegularMeshPrimeCutoffs
end Erdos390.Full.RegularMeshPrimeCutoffs
namespace Erdos390.Full.RegularMeshPrimeCutoffsEventually
end Erdos390.Full.RegularMeshPrimeCutoffsEventually
namespace Erdos390.Full.RegularRelativeMesh
end Erdos390.Full.RegularRelativeMesh
namespace Erdos390.Full.RegularRelativeMeshInteriorAnchors
end Erdos390.Full.RegularRelativeMeshInteriorAnchors
namespace Erdos390.Full.RegularRelativeMeshPrefixSplit
end Erdos390.Full.RegularRelativeMeshPrefixSplit
namespace Erdos390.Full.Scale
end Erdos390.Full.Scale
namespace Erdos390.Full.SelectedDyadicRegularMesh
end Erdos390.Full.SelectedDyadicRegularMesh
namespace Erdos390.Full.SquarefreeCovarianceReference
end Erdos390.Full.SquarefreeCovarianceReference
namespace Erdos390.Full.SquarefreeReferenceOperatorIdentification
end Erdos390.Full.SquarefreeReferenceOperatorIdentification
namespace Erdos390.Full.SquarefreeSharpBandTransfer
end Erdos390.Full.SquarefreeSharpBandTransfer
namespace Erdos390.Full.StableInverse
end Erdos390.Full.StableInverse
namespace Erdos390.Full.StructuredCellAsymptotic
end Erdos390.Full.StructuredCellAsymptotic
namespace Erdos390.Full.StructuredCellValuationLaw
end Erdos390.Full.StructuredCellValuationLaw
namespace Erdos390.Full.StructuredCells
end Erdos390.Full.StructuredCells
namespace Erdos390.Full.TwoLocalPairRestoration
end Erdos390.Full.TwoLocalPairRestoration
namespace Erdos390.Full.TwoLocalRestoration
end Erdos390.Full.TwoLocalRestoration
namespace Erdos390.Full.TwoLocalRestorationBound
end Erdos390.Full.TwoLocalRestorationBound
namespace Erdos390.Full.UniformFiniteProbability
end Erdos390.Full.UniformFiniteProbability
namespace Erdos390.Full.UniformMeshArithmeticInverse
end Erdos390.Full.UniformMeshArithmeticInverse
namespace Erdos390.Full.ValuationCutoff
end Erdos390.Full.ValuationCutoff
namespace Erdos390.Full.ValuationScoreDomination
end Erdos390.Full.ValuationScoreDomination
namespace Erdos390.Full.ValuationTiltCell
end Erdos390.Full.ValuationTiltCell
namespace Erdos390.Lemma75
end Erdos390.Lemma75
namespace Erdos390.Lemma84
end Erdos390.Lemma84
namespace Erdos390.Proposition87
end Erdos390.Proposition87
namespace Erdos390.WholePaper.AllocationCertificate
end Erdos390.WholePaper.AllocationCertificate
namespace Erdos390.WholePaper.AllocationCertificateArithmeticChecks
end Erdos390.WholePaper.AllocationCertificateArithmeticChecks
namespace Erdos390.WholePaper.AllocationCertificateCapacityChecks1
end Erdos390.WholePaper.AllocationCertificateCapacityChecks1
namespace Erdos390.WholePaper.AllocationCertificateCapacityChecks2
end Erdos390.WholePaper.AllocationCertificateCapacityChecks2
namespace Erdos390.WholePaper.AllocationCertificateCapacityChecks3
end Erdos390.WholePaper.AllocationCertificateCapacityChecks3
namespace Erdos390.WholePaper.AllocationCertificateCapacityChecks4
end Erdos390.WholePaper.AllocationCertificateCapacityChecks4
namespace Erdos390.WholePaper.AllocationCertificateChecker
end Erdos390.WholePaper.AllocationCertificateChecker
namespace Erdos390.WholePaper.AllocationCertificateData
end Erdos390.WholePaper.AllocationCertificateData
namespace Erdos390.WholePaper.AllocationCertificateOverlapChecks1
end Erdos390.WholePaper.AllocationCertificateOverlapChecks1
namespace Erdos390.WholePaper.AllocationCertificateOverlapChecks2
end Erdos390.WholePaper.AllocationCertificateOverlapChecks2
namespace Erdos390.WholePaper.AllocationCertificateOverlapChecks3
end Erdos390.WholePaper.AllocationCertificateOverlapChecks3
namespace Erdos390.WholePaper.AllocationCertificateOverlapChecks4
end Erdos390.WholePaper.AllocationCertificateOverlapChecks4
namespace Erdos390.WholePaper.AllocationCertificateRowChecks1
end Erdos390.WholePaper.AllocationCertificateRowChecks1
namespace Erdos390.WholePaper.AllocationCertificateRowChecks2
end Erdos390.WholePaper.AllocationCertificateRowChecks2
namespace Erdos390.WholePaper.AllocationCertificateRowChecks3
end Erdos390.WholePaper.AllocationCertificateRowChecks3
namespace Erdos390.WholePaper.AllocationCertificateRowChecks4
end Erdos390.WholePaper.AllocationCertificateRowChecks4
namespace Erdos390.WholePaper.AllocationCertificateStructuralChecks
end Erdos390.WholePaper.AllocationCertificateStructuralChecks
namespace Erdos390.WholePaper.AllocationTailCapacity
end Erdos390.WholePaper.AllocationTailCapacity
namespace Erdos390.WholePaper.BankAnchorCollisionFree
end Erdos390.WholePaper.BankAnchorCollisionFree
namespace Erdos390.WholePaper.BankAnchorGuardsFinite
end Erdos390.WholePaper.BankAnchorGuardsFinite
namespace Erdos390.WholePaper.BankBottomConcreteDemand
end Erdos390.WholePaper.BankBottomConcreteDemand
namespace Erdos390.WholePaper.BankBottomMarkerPoolScale
end Erdos390.WholePaper.BankBottomMarkerPoolScale
namespace Erdos390.WholePaper.BankBottomMarkerPools
end Erdos390.WholePaper.BankBottomMarkerPools
namespace Erdos390.WholePaper.BankBottomPaperComponents
end Erdos390.WholePaper.BankBottomPaperComponents
namespace Erdos390.WholePaper.BankBottomPoolMatching
end Erdos390.WholePaper.BankBottomPoolMatching
namespace Erdos390.WholePaper.BankGeometricDescentCore
end Erdos390.WholePaper.BankGeometricDescentCore
namespace Erdos390.WholePaper.BankGuardedCentralAnchorExistence
end Erdos390.WholePaper.BankGuardedCentralAnchorExistence
namespace Erdos390.WholePaper.BankMarkerDonorCombinatorics
end Erdos390.WholePaper.BankMarkerDonorCombinatorics
namespace Erdos390.WholePaper.BankOrdinaryConcreteDemand
end Erdos390.WholePaper.BankOrdinaryConcreteDemand
namespace Erdos390.WholePaper.BankOrdinaryCorePaths
end Erdos390.WholePaper.BankOrdinaryCorePaths
namespace Erdos390.WholePaper.BankOrdinaryDonorRelation
end Erdos390.WholePaper.BankOrdinaryDonorRelation
namespace Erdos390.WholePaper.BankOrdinaryDonorSupply
end Erdos390.WholePaper.BankOrdinaryDonorSupply
namespace Erdos390.WholePaper.BankOrdinaryPaperComponents
end Erdos390.WholePaper.BankOrdinaryPaperComponents
namespace Erdos390.WholePaper.BankOrdinaryPoolMatching
end Erdos390.WholePaper.BankOrdinaryPoolMatching
namespace Erdos390.WholePaper.BankPaperAnchorChangeBudget
end Erdos390.WholePaper.BankPaperAnchorChangeBudget
namespace Erdos390.WholePaper.BankPaperAnchorExactificationGuards
end Erdos390.WholePaper.BankPaperAnchorExactificationGuards
namespace Erdos390.WholePaper.BankPaperAnchorGuards
end Erdos390.WholePaper.BankPaperAnchorGuards
namespace Erdos390.WholePaper.BankPaperCanonicalActualActiveMeasureAdditiveRefinement
end Erdos390.WholePaper.BankPaperCanonicalActualActiveMeasureAdditiveRefinement
namespace Erdos390.WholePaper.BankPaperCanonicalActualActiveMeasureConstruction
end Erdos390.WholePaper.BankPaperCanonicalActualActiveMeasureConstruction
namespace Erdos390.WholePaper.BankPaperCanonicalActualEndpointSlackConnector
end Erdos390.WholePaper.BankPaperCanonicalActualEndpointSlackConnector
namespace Erdos390.WholePaper.BankPaperCanonicalActualMomentReadyEventually
end Erdos390.WholePaper.BankPaperCanonicalActualMomentReadyEventually
namespace Erdos390.WholePaper.BankPaperCanonicalActualP87RatioCellTrafficConnector
end Erdos390.WholePaper.BankPaperCanonicalActualP87RatioCellTrafficConnector
namespace Erdos390.WholePaper.BankPaperCanonicalArbitrarySourcePostHfitConnector
end Erdos390.WholePaper.BankPaperCanonicalArbitrarySourcePostHfitConnector
namespace Erdos390.WholePaper.BankPaperCanonicalArbitrarySourceSelectorDeficitPaperRateClosureConnector
end Erdos390.WholePaper.BankPaperCanonicalArbitrarySourceSelectorDeficitPaperRateClosureConnector
namespace Erdos390.WholePaper.BankPaperCanonicalBalancedRawSignedValuationResidualBoundConnector
end Erdos390.WholePaper.BankPaperCanonicalBalancedRawSignedValuationResidualBoundConnector
namespace Erdos390.WholePaper.BankPaperCanonicalBridgeRelevantGuardLedger
end Erdos390.WholePaper.BankPaperCanonicalBridgeRelevantGuardLedger
namespace Erdos390.WholePaper.BankPaperCanonicalDirectOrdinaryLogRateConnector
end Erdos390.WholePaper.BankPaperCanonicalDirectOrdinaryLogRateConnector
namespace Erdos390.WholePaper.BankPaperCanonicalDistributedCandidateSet
end Erdos390.WholePaper.BankPaperCanonicalDistributedCandidateSet
namespace Erdos390.WholePaper.BankPaperCanonicalDistributedCleanListAdapter
end Erdos390.WholePaper.BankPaperCanonicalDistributedCleanListAdapter
namespace Erdos390.WholePaper.BankPaperCanonicalDistributedPostChoice
end Erdos390.WholePaper.BankPaperCanonicalDistributedPostChoice
namespace Erdos390.WholePaper.BankPaperCanonicalDistributedSectionNineAssembly
end Erdos390.WholePaper.BankPaperCanonicalDistributedSectionNineAssembly
namespace Erdos390.WholePaper.BankPaperCanonicalDistributedSectionNineTerminal
end Erdos390.WholePaper.BankPaperCanonicalDistributedSectionNineTerminal
namespace Erdos390.WholePaper.BankPaperCanonicalDistributedTangentBridge
end Erdos390.WholePaper.BankPaperCanonicalDistributedTangentBridge
namespace Erdos390.WholePaper.BankPaperCanonicalFixedDepthCapacityBridge
end Erdos390.WholePaper.BankPaperCanonicalFixedDepthCapacityBridge
namespace Erdos390.WholePaper.BankPaperCanonicalFourFiveKernelVariation
end Erdos390.WholePaper.BankPaperCanonicalFourFiveKernelVariation
namespace Erdos390.WholePaper.BankPaperCanonicalGlobalCorrectedSourceSelector
end Erdos390.WholePaper.BankPaperCanonicalGlobalCorrectedSourceSelector
namespace Erdos390.WholePaper.BankPaperCanonicalGuardCapacityReduction
end Erdos390.WholePaper.BankPaperCanonicalGuardCapacityReduction
namespace Erdos390.WholePaper.BankPaperCanonicalGuardLocalRowLedger
end Erdos390.WholePaper.BankPaperCanonicalGuardLocalRowLedger
namespace Erdos390.WholePaper.BankPaperCanonicalGuardedBridgeConnector
end Erdos390.WholePaper.BankPaperCanonicalGuardedBridgeConnector
namespace Erdos390.WholePaper.BankPaperCanonicalGuardedPostchargeCorrectionRateClosure
end Erdos390.WholePaper.BankPaperCanonicalGuardedPostchargeCorrectionRateClosure
namespace Erdos390.WholePaper.BankPaperCanonicalGuardedRawCorrectionMomentDefectConnector
end Erdos390.WholePaper.BankPaperCanonicalGuardedRawCorrectionMomentDefectConnector
namespace Erdos390.WholePaper.BankPaperCanonicalGuardedSectionNineContinuation
end Erdos390.WholePaper.BankPaperCanonicalGuardedSectionNineContinuation
namespace Erdos390.WholePaper.BankPaperCanonicalHeadActiveMassEventually
end Erdos390.WholePaper.BankPaperCanonicalHeadActiveMassEventually
namespace Erdos390.WholePaper.BankPaperCanonicalImplementationDefectRateReductionConnector
end Erdos390.WholePaper.BankPaperCanonicalImplementationDefectRateReductionConnector
namespace Erdos390.WholePaper.BankPaperCanonicalMediumPrimeStructuredPlacementMomentRateConnector
end Erdos390.WholePaper.BankPaperCanonicalMediumPrimeStructuredPlacementMomentRateConnector
namespace Erdos390.WholePaper.BankPaperCanonicalNonsmoothSlackClosure
end Erdos390.WholePaper.BankPaperCanonicalNonsmoothSlackClosure
namespace Erdos390.WholePaper.BankPaperCanonicalP87ApproximateOrdinaryLogTargetEnvelopeConnector
end Erdos390.WholePaper.BankPaperCanonicalP87ApproximateOrdinaryLogTargetEnvelopeConnector
namespace Erdos390.WholePaper.BankPaperCanonicalP87OrdinaryLogCompatibilityConnector
end Erdos390.WholePaper.BankPaperCanonicalP87OrdinaryLogCompatibilityConnector
namespace Erdos390.WholePaper.BankPaperCanonicalP87TargetEnvelopeConnector
end Erdos390.WholePaper.BankPaperCanonicalP87TargetEnvelopeConnector
namespace Erdos390.WholePaper.BankPaperCanonicalP87VaryingMassUniformOrderConnector
end Erdos390.WholePaper.BankPaperCanonicalP87VaryingMassUniformOrderConnector
namespace Erdos390.WholePaper.BankPaperCanonicalPostHfitBalancedAlphaEventually
end Erdos390.WholePaper.BankPaperCanonicalPostHfitBalancedAlphaEventually
namespace Erdos390.WholePaper.BankPaperCanonicalPostHfitSlackConnector
end Erdos390.WholePaper.BankPaperCanonicalPostHfitSlackConnector
namespace Erdos390.WholePaper.BankPaperCanonicalPreTangentSelector
end Erdos390.WholePaper.BankPaperCanonicalPreTangentSelector
namespace Erdos390.WholePaper.BankPaperCanonicalPrefixAdapter
end Erdos390.WholePaper.BankPaperCanonicalPrefixAdapter
namespace Erdos390.WholePaper.BankPaperCanonicalPrefixQuadrature
end Erdos390.WholePaper.BankPaperCanonicalPrefixQuadrature
namespace Erdos390.WholePaper.BankPaperCanonicalProposition87EndpointSlack
end Erdos390.WholePaper.BankPaperCanonicalProposition87EndpointSlack
namespace Erdos390.WholePaper.BankPaperCanonicalRatioCellGeometry
end Erdos390.WholePaper.BankPaperCanonicalRatioCellGeometry
namespace Erdos390.WholePaper.BankPaperCanonicalRatioCellGeometryTrafficConnector
end Erdos390.WholePaper.BankPaperCanonicalRatioCellGeometryTrafficConnector
namespace Erdos390.WholePaper.BankPaperCanonicalRatioCellMomentTraffic
end Erdos390.WholePaper.BankPaperCanonicalRatioCellMomentTraffic
namespace Erdos390.WholePaper.BankPaperCanonicalRatioCellOccupancy
end Erdos390.WholePaper.BankPaperCanonicalRatioCellOccupancy
namespace Erdos390.WholePaper.BankPaperCanonicalRatioCellTraffic
end Erdos390.WholePaper.BankPaperCanonicalRatioCellTraffic
namespace Erdos390.WholePaper.BankPaperCanonicalRawBroadSurplusAsymptotic
end Erdos390.WholePaper.BankPaperCanonicalRawBroadSurplusAsymptotic
namespace Erdos390.WholePaper.BankPaperCanonicalRawRowCorrectionRateClosure
end Erdos390.WholePaper.BankPaperCanonicalRawRowCorrectionRateClosure
namespace Erdos390.WholePaper.BankPaperCanonicalRoughRowCorrection
end Erdos390.WholePaper.BankPaperCanonicalRoughRowCorrection
namespace Erdos390.WholePaper.BankPaperCanonicalSaiasUpperReduction
end Erdos390.WholePaper.BankPaperCanonicalSaiasUpperReduction
namespace Erdos390.WholePaper.BankPaperCanonicalSectionEightActiveMassScaleLowerConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionEightActiveMassScaleLowerConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionEightActualDataConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionEightActualDataConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionEightAnalyticLedgerReduction
end Erdos390.WholePaper.BankPaperCanonicalSectionEightAnalyticLedgerReduction
namespace Erdos390.WholePaper.BankPaperCanonicalSectionEightPrechargedLogConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionEightPrechargedLogConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionEightTopFrozenInitialMassConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionEightTopFrozenInitialMassConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionEightTopFrozenSourceConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionEightTopFrozenSourceConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionEightTwoZeroOneShotConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionEightTwoZeroOneShotConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNineActiveLedgerConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNineActiveLedgerConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNineAssembly
end Erdos390.WholePaper.BankPaperCanonicalSectionNineAssembly
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNineBudgetClosure
end Erdos390.WholePaper.BankPaperCanonicalSectionNineBudgetClosure
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNineFinalGeometry
end Erdos390.WholePaper.BankPaperCanonicalSectionNineFinalGeometry
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNineFixedComponentCollision
end Erdos390.WholePaper.BankPaperCanonicalSectionNineFixedComponentCollision
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNineMainAsymptoticConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNineMainAsymptoticConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNineParameterSynchronization
end Erdos390.WholePaper.BankPaperCanonicalSectionNineParameterSynchronization
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightBaselineTargetConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightBaselineTargetConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightCoherentTargetConstructor
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightCoherentTargetConstructor
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightEventualSupplierConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightEventualSupplierConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightFrozenLogInvarianceConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightFrozenLogInvarianceConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightHeadMarginAlgebra
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightHeadMarginAlgebra
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightLocalInputConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightLocalInputConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightOrdinaryLogLedgerConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightOrdinaryLogLedgerConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightPhysicalMarginAlgebra
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightPhysicalMarginAlgebra
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightPlacedMeasureConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightPlacedMeasureConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightPlacedSelectorDeficitRateConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightPlacedSelectorDeficitRateConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightPlacementValuationRateConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightPlacementValuationRateConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstActualData
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstActualData
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstCommonLedgerBeforeMesh
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstCommonLedgerBeforeMesh
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstCrossMeshScalarTransport
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstCrossMeshScalarTransport
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstCutoffAwareAnalyticCompletion
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstCutoffAwareAnalyticCompletion
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstCutoffAwareDistributedTerminalConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstCutoffAwareDistributedTerminalConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstEventualCoherentBridgeSourceObligation
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstEventualCoherentBridgeSourceObligation
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstEventualPostHfitConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstEventualPostHfitConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstFiniteAssembly
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstFiniteAssembly
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstFiniteExportedFields
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstFiniteExportedFields
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstLedgerConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstLedgerConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstLedgerFamilies
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstLedgerFamilies
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstLedgerPostBridgeAssembly
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstLedgerPostBridgeAssembly
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstMassAlgebra
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstMassAlgebra
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPostLedgerConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPostLedgerConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshEventualCoherentBridgeSourceObligation
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshEventualCoherentBridgeSourceObligation
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshLedgerPostMarginAssembly
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshLedgerPostMarginAssembly
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshNumericalData
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshNumericalData
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshPostMarginData
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshPostMarginData
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshSelectorProvider
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshSelectorProvider
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshSourceConstantsConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshSourceConstantsConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshUniformP87Connector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreMeshUniformP87Connector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreselectedPostHfitConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreselectedPostHfitConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreselectedPostLedgerConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstPreselectedPostLedgerConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstRichSourceProducer
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstRichSourceProducer
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstRichSourceWithFixedNumericalData
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceFirstRichSourceWithFixedNumericalData
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceInputAssemblyConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceInputAssemblyConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceObligationPublicSyncConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceObligationPublicSyncConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourcePrebridgeConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourcePrebridgeConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceResidualConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNinePostHeightSourceResidualConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNineTopFrozenEventStepConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNineTopFrozenEventStepConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNineTopFrozenFinalGeometryConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNineTopFrozenFinalGeometryConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSectionNineTopFrozenSynchronizedAnalyticCompletionConnector
end Erdos390.WholePaper.BankPaperCanonicalSectionNineTopFrozenSynchronizedAnalyticCompletionConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSelectorDeficitPaperRateClosureConnector
end Erdos390.WholePaper.BankPaperCanonicalSelectorDeficitPaperRateClosureConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSelectorRoundingCandidateHandoff
end Erdos390.WholePaper.BankPaperCanonicalSelectorRoundingCandidateHandoff
namespace Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalClippedIntervalGeometry
end Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalClippedIntervalGeometry
namespace Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalCoreFirst
end Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalCoreFirst
namespace Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveChamberConnector
end Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveChamberConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveCutoffIntervalPrecursorConnector
end Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveCutoffIntervalPrecursorConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveDeepDisplacementConnector
end Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveDeepDisplacementConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveDeepEndpointErrorConnector
end Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveDeepEndpointErrorConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveDeepIntervalEstimateConnector
end Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveDeepIntervalEstimateConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveGeometryPrecursorConnector
end Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveGeometryPrecursorConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveRemainingInputsConnector
end Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveRemainingInputsConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveResidualBoundConnector
end Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalFourFiveResidualBoundConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalOrderedMixtureConnector
end Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalOrderedMixtureConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalPhysicalReindex
end Erdos390.WholePaper.BankPaperCanonicalSignedExceptionalPhysicalReindex
namespace Erdos390.WholePaper.BankPaperCanonicalSignedGuardResidual
end Erdos390.WholePaper.BankPaperCanonicalSignedGuardResidual
namespace Erdos390.WholePaper.BankPaperCanonicalSignedResidualSelectorIdentification
end Erdos390.WholePaper.BankPaperCanonicalSignedResidualSelectorIdentification
namespace Erdos390.WholePaper.BankPaperCanonicalSmoothAdditivePlacement
end Erdos390.WholePaper.BankPaperCanonicalSmoothAdditivePlacement
namespace Erdos390.WholePaper.BankPaperCanonicalSmoothAdditiveRefinementTangentTransport
end Erdos390.WholePaper.BankPaperCanonicalSmoothAdditiveRefinementTangentTransport
namespace Erdos390.WholePaper.BankPaperCanonicalSmoothProtectedAdditiveRefinement
end Erdos390.WholePaper.BankPaperCanonicalSmoothProtectedAdditiveRefinement
namespace Erdos390.WholePaper.BankPaperCanonicalSmoothQuotaHeightAsymptotic
end Erdos390.WholePaper.BankPaperCanonicalSmoothQuotaHeightAsymptotic
namespace Erdos390.WholePaper.BankPaperCanonicalSmoothQuotaHeightLedger
end Erdos390.WholePaper.BankPaperCanonicalSmoothQuotaHeightLedger
namespace Erdos390.WholePaper.BankPaperCanonicalSmoothSourceGuardedRateClosure
end Erdos390.WholePaper.BankPaperCanonicalSmoothSourceGuardedRateClosure
namespace Erdos390.WholePaper.BankPaperCanonicalSmoothSourceGuardedSharpRoundedValuationRateConnector
end Erdos390.WholePaper.BankPaperCanonicalSmoothSourceGuardedSharpRoundedValuationRateConnector
namespace Erdos390.WholePaper.BankPaperCanonicalSmoothSourceGuardedSharpValuationRateClosure
end Erdos390.WholePaper.BankPaperCanonicalSmoothSourceGuardedSharpValuationRateClosure
namespace Erdos390.WholePaper.BankPaperCanonicalSmoothSourceGuardedSharpValuationRateClosureBroad
end Erdos390.WholePaper.BankPaperCanonicalSmoothSourceGuardedSharpValuationRateClosureBroad
namespace Erdos390.WholePaper.BankPaperCanonicalSmoothSourceGuardedSharpValuationRateClosureFinal
end Erdos390.WholePaper.BankPaperCanonicalSmoothSourceGuardedSharpValuationRateClosureFinal
namespace Erdos390.WholePaper.BankPaperCanonicalSmoothSourceGuardedSharpValuationRateClosureMixture
end Erdos390.WholePaper.BankPaperCanonicalSmoothSourceGuardedSharpValuationRateClosureMixture
namespace Erdos390.WholePaper.BankPaperCanonicalSmoothSourceGuardedSharpValuationRateClosureMoving
end Erdos390.WholePaper.BankPaperCanonicalSmoothSourceGuardedSharpValuationRateClosureMoving
namespace Erdos390.WholePaper.BankPaperCanonicalStructuredPrebridgeLedgerConnector
end Erdos390.WholePaper.BankPaperCanonicalStructuredPrebridgeLedgerConnector
namespace Erdos390.WholePaper.BankPaperCanonicalTangentResidualBridge
end Erdos390.WholePaper.BankPaperCanonicalTangentResidualBridge
namespace Erdos390.WholePaper.BankPaperCanonicalTangentSlackIntegration
end Erdos390.WholePaper.BankPaperCanonicalTangentSlackIntegration
namespace Erdos390.WholePaper.BankPaperCanonicalTopFrozenChargedRowsConnector
end Erdos390.WholePaper.BankPaperCanonicalTopFrozenChargedRowsConnector
namespace Erdos390.WholePaper.BankPaperCanonicalTopFrozenEventualOrdinaryLogRateConnector
end Erdos390.WholePaper.BankPaperCanonicalTopFrozenEventualOrdinaryLogRateConnector
namespace Erdos390.WholePaper.BankPaperCanonicalTopFrozenImplementationRateReductionConnector
end Erdos390.WholePaper.BankPaperCanonicalTopFrozenImplementationRateReductionConnector
namespace Erdos390.WholePaper.BankPaperCanonicalTopFrozenOrdinaryLogHeightReductionConnector
end Erdos390.WholePaper.BankPaperCanonicalTopFrozenOrdinaryLogHeightReductionConnector
namespace Erdos390.WholePaper.BankPaperCanonicalTopFrozenRoundedSourceStateConnector
end Erdos390.WholePaper.BankPaperCanonicalTopFrozenRoundedSourceStateConnector
namespace Erdos390.WholePaper.BankPaperCanonicalTopFrozenRoundedSourceStateEventualConnector
end Erdos390.WholePaper.BankPaperCanonicalTopFrozenRoundedSourceStateEventualConnector
namespace Erdos390.WholePaper.BankPaperCanonicalTopFrozenSelectorTailSupportReductionConnector
end Erdos390.WholePaper.BankPaperCanonicalTopFrozenSelectorTailSupportReductionConnector
namespace Erdos390.WholePaper.BankPaperCanonicalTopFrozenSmoothFeasibilityConnector
end Erdos390.WholePaper.BankPaperCanonicalTopFrozenSmoothFeasibilityConnector
namespace Erdos390.WholePaper.BankPaperCanonicalTopFrozenSourceFeasibilityReductionConnector
end Erdos390.WholePaper.BankPaperCanonicalTopFrozenSourceFeasibilityReductionConnector
namespace Erdos390.WholePaper.BankPaperCanonicalTwoZeroHeadCellHfitConnector
end Erdos390.WholePaper.BankPaperCanonicalTwoZeroHeadCellHfitConnector
namespace Erdos390.WholePaper.BankPaperCanonicalTwoZeroHeadCellMeasureConnector
end Erdos390.WholePaper.BankPaperCanonicalTwoZeroHeadCellMeasureConnector
namespace Erdos390.WholePaper.BankPaperCanonicalTwoZeroHeadCellOuterCapacityConnector
end Erdos390.WholePaper.BankPaperCanonicalTwoZeroHeadCellOuterCapacityConnector
namespace Erdos390.WholePaper.BankPaperCanonicalTwoZeroHeadCellPostHfitConnector
end Erdos390.WholePaper.BankPaperCanonicalTwoZeroHeadCellPostHfitConnector
namespace Erdos390.WholePaper.BankPaperCanonicalTwoZeroHeadCellProducerConnector
end Erdos390.WholePaper.BankPaperCanonicalTwoZeroHeadCellProducerConnector
namespace Erdos390.WholePaper.BankPaperCanonicalUpperConstructionEventually
end Erdos390.WholePaper.BankPaperCanonicalUpperConstructionEventually
namespace Erdos390.WholePaper.BankPaperCanonicalUpperConstructionReduction
end Erdos390.WholePaper.BankPaperCanonicalUpperConstructionReduction
namespace Erdos390.WholePaper.BankPaperCombinedChargeDepthFirstTerminal
end Erdos390.WholePaper.BankPaperCombinedChargeDepthFirstTerminal
namespace Erdos390.WholePaper.BankPaperCombinedChargeTerminal
end Erdos390.WholePaper.BankPaperCombinedChargeTerminal
namespace Erdos390.WholePaper.BankPaperComponents
end Erdos390.WholePaper.BankPaperComponents
namespace Erdos390.WholePaper.BankPaperExactificationApplication
end Erdos390.WholePaper.BankPaperExactificationApplication
namespace Erdos390.WholePaper.BankPaperExactificationState
end Erdos390.WholePaper.BankPaperExactificationState
namespace Erdos390.WholePaper.BankPaperFinalFactorSetInterval
end Erdos390.WholePaper.BankPaperFinalFactorSetInterval
namespace Erdos390.WholePaper.BankPaperFixedExceptionalBacking
end Erdos390.WholePaper.BankPaperFixedExceptionalBacking
namespace Erdos390.WholePaper.BankPaperFixedExceptionalChargeAsymptotic
end Erdos390.WholePaper.BankPaperFixedExceptionalChargeAsymptotic
namespace Erdos390.WholePaper.BankPaperFixedExceptionalValuationFibers
end Erdos390.WholePaper.BankPaperFixedExceptionalValuationFibers
namespace Erdos390.WholePaper.BankPaperFourFiveActiveLogCoordinate
end Erdos390.WholePaper.BankPaperFourFiveActiveLogCoordinate
namespace Erdos390.WholePaper.BankPaperFourFiveCellLebesgueIdentification
end Erdos390.WholePaper.BankPaperFourFiveCellLebesgueIdentification
namespace Erdos390.WholePaper.BankPaperFourFiveFixedSimplexKernel
end Erdos390.WholePaper.BankPaperFourFiveFixedSimplexKernel
namespace Erdos390.WholePaper.BankPaperFourFiveLastPrimeContinuumDecomposition
end Erdos390.WholePaper.BankPaperFourFiveLastPrimeContinuumDecomposition
namespace Erdos390.WholePaper.BankPaperFourFiveLebesgueCellAggregationBound
end Erdos390.WholePaper.BankPaperFourFiveLebesgueCellAggregationBound
namespace Erdos390.WholePaper.BankPaperFourFiveMovingFaceBV
end Erdos390.WholePaper.BankPaperFourFiveMovingFaceBV
namespace Erdos390.WholePaper.BankPaperFourFiveMovingFaceCellOscillation
end Erdos390.WholePaper.BankPaperFourFiveMovingFaceCellOscillation
namespace Erdos390.WholePaper.BankPaperFourFiveMovingFixedSimplexIdentification
end Erdos390.WholePaper.BankPaperFourFiveMovingFixedSimplexIdentification
namespace Erdos390.WholePaper.BankPaperFourFiveMovingSimplexCellIdentification
end Erdos390.WholePaper.BankPaperFourFiveMovingSimplexCellIdentification
namespace Erdos390.WholePaper.BankPaperFourFiveMovingSimplexCellTelescope
end Erdos390.WholePaper.BankPaperFourFiveMovingSimplexCellTelescope
namespace Erdos390.WholePaper.BankPaperFourFiveOrderedLastPrimeExpansion
end Erdos390.WholePaper.BankPaperFourFiveOrderedLastPrimeExpansion
namespace Erdos390.WholePaper.BankPaperFourFiveOrderedMixtureAssembly
end Erdos390.WholePaper.BankPaperFourFiveOrderedMixtureAssembly
namespace Erdos390.WholePaper.BankPaperFourFivePrimeEndpoint
end Erdos390.WholePaper.BankPaperFourFivePrimeEndpoint
namespace Erdos390.WholePaper.BankPaperFourFiveProductMeasureTelescope
end Erdos390.WholePaper.BankPaperFourFiveProductMeasureTelescope
namespace Erdos390.WholePaper.BankPaperFourFiveRealCommonDomainIdentification
end Erdos390.WholePaper.BankPaperFourFiveRealCommonDomainIdentification
namespace Erdos390.WholePaper.BankPaperFourFiveRealLastPrimeEndpoint
end Erdos390.WholePaper.BankPaperFourFiveRealLastPrimeEndpoint
namespace Erdos390.WholePaper.BankPaperFourFiveReciprocalPrimeBV
end Erdos390.WholePaper.BankPaperFourFiveReciprocalPrimeBV
namespace Erdos390.WholePaper.BankPaperFourFiveRoughChamberReduction
end Erdos390.WholePaper.BankPaperFourFiveRoughChamberReduction
namespace Erdos390.WholePaper.BankPaperFourFiveSignedCoreErrorAdapter
end Erdos390.WholePaper.BankPaperFourFiveSignedCoreErrorAdapter
namespace Erdos390.WholePaper.BankPaperGuardedUpperProductAssembly
end Erdos390.WholePaper.BankPaperGuardedUpperProductAssembly
namespace Erdos390.WholePaper.BankPaperMainAsymptoticReduction
end Erdos390.WholePaper.BankPaperMainAsymptoticReduction
namespace Erdos390.WholePaper.BankPaperPaths
end Erdos390.WholePaper.BankPaperPaths
namespace Erdos390.WholePaper.BankPaperPrecharge
end Erdos390.WholePaper.BankPaperPrecharge
namespace Erdos390.WholePaper.BankPaperPrechargeAsymptotic
end Erdos390.WholePaper.BankPaperPrechargeAsymptotic
namespace Erdos390.WholePaper.BankPaperPrechargeCapacityAlgebra
end Erdos390.WholePaper.BankPaperPrechargeCapacityAlgebra
namespace Erdos390.WholePaper.BankPaperPrechargeCapacityEventually
end Erdos390.WholePaper.BankPaperPrechargeCapacityEventually
namespace Erdos390.WholePaper.BankPaperPrechargeDivisibilityAlgebra
end Erdos390.WholePaper.BankPaperPrechargeDivisibilityAlgebra
namespace Erdos390.WholePaper.BankPaperPrechargeExactificationBridge
end Erdos390.WholePaper.BankPaperPrechargeExactificationBridge
namespace Erdos390.WholePaper.BankPaperPrechargeUniformCapacityAsymptotic
end Erdos390.WholePaper.BankPaperPrechargeUniformCapacityAsymptotic
namespace Erdos390.WholePaper.BankPaperPrechargedTailTarget
end Erdos390.WholePaper.BankPaperPrechargedTailTarget
namespace Erdos390.WholePaper.BankPaperProposition87ActualDataConnector
end Erdos390.WholePaper.BankPaperProposition87ActualDataConnector
namespace Erdos390.WholePaper.BankPaperProposition87EndpointSelector
end Erdos390.WholePaper.BankPaperProposition87EndpointSelector
namespace Erdos390.WholePaper.BankPaperSelectorTailTarget
end Erdos390.WholePaper.BankPaperSelectorTailTarget
namespace Erdos390.WholePaper.BankPaperSharpCensus
end Erdos390.WholePaper.BankPaperSharpCensus
namespace Erdos390.WholePaper.BankPaperUpperConstructionReduction
end Erdos390.WholePaper.BankPaperUpperConstructionReduction
namespace Erdos390.WholePaper.BankPathAlgebra
end Erdos390.WholePaper.BankPathAlgebra
namespace Erdos390.WholePaper.BankRoughSignatures
end Erdos390.WholePaper.BankRoughSignatures
namespace Erdos390.WholePaper.BankSmallDescentTable
end Erdos390.WholePaper.BankSmallDescentTable
namespace Erdos390.WholePaper.CanonicalCompleteRoughRows
end Erdos390.WholePaper.CanonicalCompleteRoughRows
namespace Erdos390.WholePaper.CentralAnchorCollision
end Erdos390.WholePaper.CentralAnchorCollision
namespace Erdos390.WholePaper.CentralAnchorCutoff
end Erdos390.WholePaper.CentralAnchorCutoff
namespace Erdos390.WholePaper.CentralAnchorDivisorSupport
end Erdos390.WholePaper.CentralAnchorDivisorSupport
namespace Erdos390.WholePaper.CentralAnchorExistence
end Erdos390.WholePaper.CentralAnchorExistence
namespace Erdos390.WholePaper.CentralAnchorGuardedCertificate
end Erdos390.WholePaper.CentralAnchorGuardedCertificate
namespace Erdos390.WholePaper.CentralAnchorGuardedChoice
end Erdos390.WholePaper.CentralAnchorGuardedChoice
namespace Erdos390.WholePaper.CentralAnchorModificationAlgebra
end Erdos390.WholePaper.CentralAnchorModificationAlgebra
namespace Erdos390.WholePaper.CentralAnchorProduct
end Erdos390.WholePaper.CentralAnchorProduct
namespace Erdos390.WholePaper.CentralAnchorReserveAlgebra
end Erdos390.WholePaper.CentralAnchorReserveAlgebra
namespace Erdos390.WholePaper.CentralAnchorTailDivisibility
end Erdos390.WholePaper.CentralAnchorTailDivisibility
namespace Erdos390.WholePaper.CentralCarryAnchors
end Erdos390.WholePaper.CentralCarryAnchors
namespace Erdos390.WholePaper.CentralExtension
end Erdos390.WholePaper.CentralExtension
namespace Erdos390.WholePaper.CentralPrimeBlockDecomposition
end Erdos390.WholePaper.CentralPrimeBlockDecomposition
namespace Erdos390.WholePaper.CentralPromotion
end Erdos390.WholePaper.CentralPromotion
namespace Erdos390.WholePaper.CentralPromotionCostAsymptotic
end Erdos390.WholePaper.CentralPromotionCostAsymptotic
namespace Erdos390.WholePaper.Complement
end Erdos390.WholePaper.Complement
namespace Erdos390.WholePaper.CompleteRoughDecomposition
end Erdos390.WholePaper.CompleteRoughDecomposition
namespace Erdos390.WholePaper.CompleteRoughRowPartition
end Erdos390.WholePaper.CompleteRoughRowPartition
namespace Erdos390.WholePaper.Constants
end Erdos390.WholePaper.Constants
namespace Erdos390.WholePaper.Definitions
end Erdos390.WholePaper.Definitions
namespace Erdos390.WholePaper.ExactFinitePartition
end Erdos390.WholePaper.ExactFinitePartition
namespace Erdos390.WholePaper.ExceptionalValuationSums
end Erdos390.WholePaper.ExceptionalValuationSums
namespace Erdos390.WholePaper.FactorizationIncidence
end Erdos390.WholePaper.FactorizationIncidence
namespace Erdos390.WholePaper.FiniteAsymmetricLocalLemma
end Erdos390.WholePaper.FiniteAsymmetricLocalLemma
namespace Erdos390.WholePaper.FinitePrefixAllocationCapacity
end Erdos390.WholePaper.FinitePrefixAllocationCapacity
namespace Erdos390.WholePaper.FixedModulusReducedResidueCount
end Erdos390.WholePaper.FixedModulusReducedResidueCount
namespace Erdos390.WholePaper.FloatingRounding
end Erdos390.WholePaper.FloatingRounding
namespace Erdos390.WholePaper.FloatingRoundingCounts
end Erdos390.WholePaper.FloatingRoundingCounts
namespace Erdos390.WholePaper.FloatingRoundingDiscrepancy
end Erdos390.WholePaper.FloatingRoundingDiscrepancy
namespace Erdos390.WholePaper.FloatingRoundingRows
end Erdos390.WholePaper.FloatingRoundingRows
namespace Erdos390.WholePaper.FloatingRoundingStep
end Erdos390.WholePaper.FloatingRoundingStep
namespace Erdos390.WholePaper.GuardedBankSelection
end Erdos390.WholePaper.GuardedBankSelection
namespace Erdos390.WholePaper.GuardedExactificationAlgebra
end Erdos390.WholePaper.GuardedExactificationAlgebra
namespace Erdos390.WholePaper.GuardedExactificationProduct
end Erdos390.WholePaper.GuardedExactificationProduct
namespace Erdos390.WholePaper.GuardedIntegralExactification
end Erdos390.WholePaper.GuardedIntegralExactification
namespace Erdos390.WholePaper.GuardedUpperProductAssembly
end Erdos390.WholePaper.GuardedUpperProductAssembly
namespace Erdos390.WholePaper.IncidenceBound
end Erdos390.WholePaper.IncidenceBound
namespace Erdos390.WholePaper.InfiniteAllocation
end Erdos390.WholePaper.InfiniteAllocation
namespace Erdos390.WholePaper.InfiniteAllocationCore
end Erdos390.WholePaper.InfiniteAllocationCore
namespace Erdos390.WholePaper.InfiniteAllocationTailCore
end Erdos390.WholePaper.InfiniteAllocationTailCore
namespace Erdos390.WholePaper.LargeCentralPrimeRouting
end Erdos390.WholePaper.LargeCentralPrimeRouting
namespace Erdos390.WholePaper.LargePrimeCollision
end Erdos390.WholePaper.LargePrimeCollision
namespace Erdos390.WholePaper.LeastPrimeTail
end Erdos390.WholePaper.LeastPrimeTail
namespace Erdos390.WholePaper.MainTargetEquivalence
end Erdos390.WholePaper.MainTargetEquivalence
namespace Erdos390.WholePaper.MovingLayerLowerBound
end Erdos390.WholePaper.MovingLayerLowerBound
namespace Erdos390.WholePaper.Nagura
end Erdos390.WholePaper.Nagura
namespace Erdos390.WholePaper.NaguraAnalyticTail
end Erdos390.WholePaper.NaguraAnalyticTail
namespace Erdos390.WholePaper.NaguraChebyshevIdentity
end Erdos390.WholePaper.NaguraChebyshevIdentity
namespace Erdos390.WholePaper.NaguraCombinatorial
end Erdos390.WholePaper.NaguraCombinatorial
namespace Erdos390.WholePaper.NaguraExplicitPsiBounds
end Erdos390.WholePaper.NaguraExplicitPsiBounds
namespace Erdos390.WholePaper.NaguraFactorialBounds
end Erdos390.WholePaper.NaguraFactorialBounds
namespace Erdos390.WholePaper.NaguraGeneralLowerBound
end Erdos390.WholePaper.NaguraGeneralLowerBound
namespace Erdos390.WholePaper.NaguraGeneralUpperBound
end Erdos390.WholePaper.NaguraGeneralUpperBound
namespace Erdos390.WholePaper.NaguraGeneralUpperModel
end Erdos390.WholePaper.NaguraGeneralUpperModel
namespace Erdos390.WholePaper.NaguraLeastPrimeCorollary
end Erdos390.WholePaper.NaguraLeastPrimeCorollary
namespace Erdos390.WholePaper.NaguraLogBounds
end Erdos390.WholePaper.NaguraLogBounds
namespace Erdos390.WholePaper.NaguraLowerCombination
end Erdos390.WholePaper.NaguraLowerCombination
namespace Erdos390.WholePaper.NaguraPrimeChain1046959To1482617
end Erdos390.WholePaper.NaguraPrimeChain1046959To1482617
namespace Erdos390.WholePaper.NaguraPrimeChain1482617To3538187
end Erdos390.WholePaper.NaguraPrimeChain1482617To3538187
namespace Erdos390.WholePaper.NaguraPrimeChain218737To522059
end Erdos390.WholePaper.NaguraPrimeChain218737To522059
namespace Erdos390.WholePaper.NaguraPrimeChain3538187To8443433
end Erdos390.WholePaper.NaguraPrimeChain3538187To8443433
namespace Erdos390.WholePaper.NaguraPrimeChain522059To1046959
end Erdos390.WholePaper.NaguraPrimeChain522059To1046959
namespace Erdos390.WholePaper.NaguraPrimeChain8443433To16000000
end Erdos390.WholePaper.NaguraPrimeChain8443433To16000000
namespace Erdos390.WholePaper.NaguraPrimeChain91639To218737
end Erdos390.WholePaper.NaguraPrimeChain91639To218737
namespace Erdos390.WholePaper.NaguraPrimeChainTo1482617
end Erdos390.WholePaper.NaguraPrimeChainTo1482617
namespace Erdos390.WholePaper.NaguraPrimeChainTo16000000
end Erdos390.WholePaper.NaguraPrimeChainTo16000000
namespace Erdos390.WholePaper.NaguraPsiBridge
end Erdos390.WholePaper.NaguraPsiBridge
namespace Erdos390.WholePaper.NaguraStirlingCombination
end Erdos390.WholePaper.NaguraStirlingCombination
namespace Erdos390.WholePaper.NaguraUniversal
end Erdos390.WholePaper.NaguraUniversal
namespace Erdos390.WholePaper.NoAdmissibleBelowTwoN
end Erdos390.WholePaper.NoAdmissibleBelowTwoN
namespace Erdos390.WholePaper.PrimeLayerCounts
end Erdos390.WholePaper.PrimeLayerCounts
namespace Erdos390.WholePaper.PrimeLayers
end Erdos390.WholePaper.PrimeLayers
namespace Erdos390.WholePaper.PrimePowerColumns
end Erdos390.WholePaper.PrimePowerColumns
namespace Erdos390.WholePaper.PrimePowerRounding
end Erdos390.WholePaper.PrimePowerRounding
namespace Erdos390.WholePaper.ResidualCentralFactors
end Erdos390.WholePaper.ResidualCentralFactors
namespace Erdos390.WholePaper.RoughBalancedFriableResidualReduction
end Erdos390.WholePaper.RoughBalancedFriableResidualReduction
namespace Erdos390.WholePaper.RoughCompactBVTranslation
end Erdos390.WholePaper.RoughCompactBVTranslation
namespace Erdos390.WholePaper.RoughFixedHeadFriableShift
end Erdos390.WholePaper.RoughFixedHeadFriableShift
namespace Erdos390.WholePaper.RoughFriableFaceInduction
end Erdos390.WholePaper.RoughFriableFaceInduction
namespace Erdos390.WholePaper.RoughHeadCompatibleFinitePoint
end Erdos390.WholePaper.RoughHeadCompatibleFinitePoint
namespace Erdos390.WholePaper.RoughSaiasBaseChange
end Erdos390.WholePaper.RoughSaiasBaseChange
namespace Erdos390.WholePaper.RoughSaiasCanonicalRowBridge
end Erdos390.WholePaper.RoughSaiasCanonicalRowBridge
namespace Erdos390.WholePaper.RoughSaiasCellTransitionSplit
end Erdos390.WholePaper.RoughSaiasCellTransitionSplit
namespace Erdos390.WholePaper.RoughSaiasContinuousBuchstab
end Erdos390.WholePaper.RoughSaiasContinuousBuchstab
namespace Erdos390.WholePaper.RoughSaiasCorrectionObstruction
end Erdos390.WholePaper.RoughSaiasCorrectionObstruction
namespace Erdos390.WholePaper.RoughSaiasDickmanThetaFourth
end Erdos390.WholePaper.RoughSaiasDickmanThetaFourth
namespace Erdos390.WholePaper.RoughSaiasEndpointApproximation
end Erdos390.WholePaper.RoughSaiasEndpointApproximation
namespace Erdos390.WholePaper.RoughSaiasFullyRealBuchstabCells
end Erdos390.WholePaper.RoughSaiasFullyRealBuchstabCells
namespace Erdos390.WholePaper.RoughSaiasFullyRealNaturalCells
end Erdos390.WholePaper.RoughSaiasFullyRealNaturalCells
namespace Erdos390.WholePaper.RoughSaiasFullyRealNormalForm
end Erdos390.WholePaper.RoughSaiasFullyRealNormalForm
namespace Erdos390.WholePaper.RoughSaiasFunctionalBuchstab
end Erdos390.WholePaper.RoughSaiasFunctionalBuchstab
namespace Erdos390.WholePaper.RoughSaiasNaturalThetaLedgerBound
end Erdos390.WholePaper.RoughSaiasNaturalThetaLedgerBound
namespace Erdos390.WholePaper.RoughSaiasNormalization
end Erdos390.WholePaper.RoughSaiasNormalization
namespace Erdos390.WholePaper.RoughSaiasQuotientBlocks
end Erdos390.WholePaper.RoughSaiasQuotientBlocks
namespace Erdos390.WholePaper.RoughSaiasRealBaseChange
end Erdos390.WholePaper.RoughSaiasRealBaseChange
namespace Erdos390.WholePaper.RoughSaiasRealStieltjesNormalForm
end Erdos390.WholePaper.RoughSaiasRealStieltjesNormalForm
namespace Erdos390.WholePaper.RoughSaiasRiemannAbsorption
end Erdos390.WholePaper.RoughSaiasRiemannAbsorption
namespace Erdos390.WholePaper.RoughSaiasRightAbel
end Erdos390.WholePaper.RoughSaiasRightAbel
namespace Erdos390.WholePaper.RoughSaiasSelectorLedgerBound
end Erdos390.WholePaper.RoughSaiasSelectorLedgerBound
namespace Erdos390.WholePaper.RoughSaiasSharpCanonicalRowPaperScale
end Erdos390.WholePaper.RoughSaiasSharpCanonicalRowPaperScale
namespace Erdos390.WholePaper.RoughSaiasSharpCorrectionTarget
end Erdos390.WholePaper.RoughSaiasSharpCorrectionTarget
namespace Erdos390.WholePaper.RoughSaiasSharpFixedHeadIntervalShift
end Erdos390.WholePaper.RoughSaiasSharpFixedHeadIntervalShift
namespace Erdos390.WholePaper.RoughSaiasSharpVariation
end Erdos390.WholePaper.RoughSaiasSharpVariation
namespace Erdos390.WholePaper.RoughSaiasSignedFractionalAbel
end Erdos390.WholePaper.RoughSaiasSignedFractionalAbel
namespace Erdos390.WholePaper.RoughSaiasStieltjesKernel
end Erdos390.WholePaper.RoughSaiasStieltjesKernel
namespace Erdos390.WholePaper.RoughSaiasStieltjesNormalForm
end Erdos390.WholePaper.RoughSaiasStieltjesNormalForm
namespace Erdos390.WholePaper.RoughSaiasUnconditionalDownstream
end Erdos390.WholePaper.RoughSaiasUnconditionalDownstream
namespace Erdos390.WholePaper.RoughSaiasWeightedTransition
end Erdos390.WholePaper.RoughSaiasWeightedTransition
namespace Erdos390.WholePaper.RoughTransitionBalancedBlock
end Erdos390.WholePaper.RoughTransitionBalancedBlock
namespace Erdos390.WholePaper.SafePrimeCounting
end Erdos390.WholePaper.SafePrimeCounting
namespace Erdos390.WholePaper.SafeShortIntervalPrimeCounting
end Erdos390.WholePaper.SafeShortIntervalPrimeCounting
namespace Erdos390.WholePaper.ScaledPrimeCounting
end Erdos390.WholePaper.ScaledPrimeCounting
namespace Erdos390.WholePaper.StationaryLayers
end Erdos390.WholePaper.StationaryLayers
namespace Erdos390.WholePaper.StationaryPrefixAnchors
end Erdos390.WholePaper.StationaryPrefixAnchors
namespace Erdos390.WholePaper.StationaryPrefixCentralCofactorChoice
end Erdos390.WholePaper.StationaryPrefixCentralCofactorChoice
namespace Erdos390.WholePaper.StationaryPrefixRealization
end Erdos390.WholePaper.StationaryPrefixRealization
namespace Erdos390.WholePaper.StationaryPrefixValuationAsymptotic
end Erdos390.WholePaper.StationaryPrefixValuationAsymptotic
namespace Erdos390.WholePaper.StrongSafePrimeCounting
end Erdos390.WholePaper.StrongSafePrimeCounting
namespace Erdos390.WholePaper.TailValuationCapacity
end Erdos390.WholePaper.TailValuationCapacity
namespace Erdos390.WholePaper.TailValuationCore
end Erdos390.WholePaper.TailValuationCore
namespace Erdos390.WholePaper.TailValuationTwoSided
end Erdos390.WholePaper.TailValuationTwoSided
namespace Erdos390.WholePaper.TangentBalancedProductFlow
end Erdos390.WholePaper.TangentBalancedProductFlow
namespace Erdos390.WholePaper.TangentCanonicalCleanListLower
end Erdos390.WholePaper.TangentCanonicalCleanListLower
namespace Erdos390.WholePaper.TangentCleanListCardinalityBridge
end Erdos390.WholePaper.TangentCleanListCardinalityBridge
namespace Erdos390.WholePaper.TangentCollisionCounting
end Erdos390.WholePaper.TangentCollisionCounting
namespace Erdos390.WholePaper.TangentCollisionFreeFeasibility
end Erdos390.WholePaper.TangentCollisionFreeFeasibility
namespace Erdos390.WholePaper.TangentCollisionLocalLemma
end Erdos390.WholePaper.TangentCollisionLocalLemma
namespace Erdos390.WholePaper.TangentCommonMultiplierLists
end Erdos390.WholePaper.TangentCommonMultiplierLists
namespace Erdos390.WholePaper.TangentDistributedFlowCensus
end Erdos390.WholePaper.TangentDistributedFlowCensus
namespace Erdos390.WholePaper.TangentExceptionalCanonicalBounds
end Erdos390.WholePaper.TangentExceptionalCanonicalBounds
namespace Erdos390.WholePaper.TangentExceptionalLambdaSquareSieve
end Erdos390.WholePaper.TangentExceptionalLambdaSquareSieve
namespace Erdos390.WholePaper.TangentExceptionalSelbergReduction
end Erdos390.WholePaper.TangentExceptionalSelbergReduction
namespace Erdos390.WholePaper.TangentFlowAlgebra
end Erdos390.WholePaper.TangentFlowAlgebra
namespace Erdos390.WholePaper.TangentLLLNumerics
end Erdos390.WholePaper.TangentLLLNumerics
namespace Erdos390.WholePaper.TangentPairArithmetic
end Erdos390.WholePaper.TangentPairArithmetic
namespace Erdos390.WholePaper.TangentPaperCleanListAbsorption
end Erdos390.WholePaper.TangentPaperCleanListAbsorption
namespace Erdos390.WholePaper.TangentPaperNumericalGuards
end Erdos390.WholePaper.TangentPaperNumericalGuards
namespace Erdos390.WholePaper.TangentRatioCellEarthmover
end Erdos390.WholePaper.TangentRatioCellEarthmover
namespace Erdos390.WholePaper.TangentRequestSplitting
end Erdos390.WholePaper.TangentRequestSplitting
namespace Erdos390.WholePaper.TangentSplitRequestLoads
end Erdos390.WholePaper.TangentSplitRequestLoads
namespace Erdos390.WholePaper.TangentStarSplitRequestBridge
end Erdos390.WholePaper.TangentStarSplitRequestBridge
namespace Erdos390.WholePaper.TangentStarTransport
end Erdos390.WholePaper.TangentStarTransport
namespace Erdos390.WholePaper.ThirteenLayerLowerBound
end Erdos390.WholePaper.ThirteenLayerLowerBound
namespace Erdos390.WholePaper.UpperProductAssembly
end Erdos390.WholePaper.UpperProductAssembly
namespace Erdos390.WholePaper.UpperScale
end Erdos390.WholePaper.UpperScale
namespace Erdos390.WholePaper.UpperTailValuationAsymptotic
end Erdos390.WholePaper.UpperTailValuationAsymptotic
namespace Erdos390.WholePaper.ValuationError
end Erdos390.WholePaper.ValuationError
namespace Erdos390.WholePaper.VariablePrimeCounting
end Erdos390.WholePaper.VariablePrimeCounting

/- Source module: Erdos390.Full.RegularRelativeMesh -/
section


/-!
# Existence of a finite regular relative mesh

Section 8.4 partitions the positive logarithmic interval `[delta,1]` into
geometric cells.  This file constructs such a mesh rather than leaving its
existence as a certificate.  Given `0 < delta < 1` and a requested upper
relative width `eta > 0`, the number of cells is the ceiling of the relevant
logarithmic ratio and the common relative width is adjusted so that the last
endpoint is exactly `1`.
-/

open scoped BigOperators

namespace Erdos390.Full.RegularRelativeMesh

noncomputable section

/-- Finite geometric mesh data.  The positive cells are indexed by
`Fin cellCount`; the separate moving-low cell `[t₀,delta]` is not part of
this structure. -/
structure Mesh (delta eta : ℝ) where
  cellCount : ℕ
  cellCount_pos : 0 < cellCount
  ratio : ℝ
  ratio_pos : 0 < ratio
  ratio_le_eta : ratio ≤ eta
  lastEndpoint : delta * (1 + ratio) ^ cellCount = 1

namespace Mesh

variable {delta eta : ℝ} (M : Mesh delta eta)

/-- Endpoint number `k`, extended to all natural `k`. -/
def endpoint (k : ℕ) : ℝ := delta * (1 + M.ratio) ^ k

/-- Lower endpoint of a positive cell. -/
def lower (k : Fin M.cellCount) : ℝ := M.endpoint k.1

/-- Upper endpoint of a positive cell. -/
def upper (k : Fin M.cellCount) : ℝ := M.endpoint (k.1 + 1)

/-- Cell width. -/
def width (k : Fin M.cellCount) : ℝ := M.upper k - M.lower k

theorem endpoint_zero : M.endpoint 0 = delta := by
  simp [endpoint]

theorem endpoint_cellCount : M.endpoint M.cellCount = 1 := by
  exact M.lastEndpoint























end Mesh



end

end Erdos390.Full.RegularRelativeMesh

end

/- Source module: Erdos390.Full.SelectedDyadicRegularMesh -/
section


/-!
# An explicit arbitrarily fine regular mesh with a fixed anchor block

For integers `K ≥ 3`, `N ≥ 1`, adjacent endpoints have ratio
`exp(log 2 / N)`.  The mesh runs from `2^{-K}` to `1`, and the consecutive
block of cells between `1/4` and `1/2` has total width exactly `1/4`.
Increasing `K` shrinks the moving-low cutoff; increasing `N` refines every
positive cell, without shrinking that total interior anchor mass.
-/

open scoped BigOperators
open Filter Topology

noncomputable section

namespace Erdos390.Full.SelectedDyadicRegularMesh

open RegularRelativeMesh

def delta (K : ℕ) : ℝ := (1 / 2 : ℝ) ^ K

def ratio (N : ℕ) : ℝ := Real.exp (Real.log 2 / (N : ℝ)) - 1

theorem one_add_ratio (N : ℕ) :
    1 + ratio N = Real.exp (Real.log 2 / (N : ℝ)) := by
  unfold ratio
  ring

theorem ratio_pos {N : ℕ} (hN : 0 < N) : 0 < ratio N := by
  rw [ratio, sub_pos, Real.one_lt_exp_iff]
  exact div_pos (Real.log_pos (by norm_num)) (by exact_mod_cast hN)

theorem ratio_pow_N {N : ℕ} (hN : 0 < N) :
    (1 + ratio N) ^ N = 2 := by
  rw [one_add_ratio, ← Real.exp_nat_mul]
  have hNreal : (N : ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hN)
  rw [show (N : ℝ) * (Real.log 2 / (N : ℝ)) = Real.log 2 by
    field_simp [hNreal]]
  exact Real.exp_log (by norm_num)

/-- The explicit dyadic regular mesh. -/
def mesh (K N : ℕ) (hK : 0 < K) (hN : 0 < N) :
    Mesh (delta K) (ratio N) where
  cellCount := K * N
  cellCount_pos := Nat.mul_pos hK hN
  ratio := ratio N
  ratio_pos := ratio_pos hN
  ratio_le_eta := le_rfl
  lastEndpoint := by
    rw [delta, one_add_ratio, Nat.mul_comm K N, pow_mul]
    have hpow : Real.exp (Real.log 2 / (N : ℝ)) ^ N = 2 := by
      simpa only [← one_add_ratio] using ratio_pow_N hN
    rw [hpow]
    rw [← mul_pow]
    norm_num

























end Erdos390.Full.SelectedDyadicRegularMesh

end

/- Source module: Erdos390.Full.FiniteExponentialFamily -/
section


/-!
# Finite exponential families

This file develops the exact finite exponential family used by the smooth-row
bridge.  The sample space is finite, the base weights are nonnegative with
positive total mass, and every score is a continuous linear functional on a
finite-dimensional real parameter space.  All normalization, positivity,
smoothness, and covariance-Jacobian statements are proved from these data.
-/

open scoped BigOperators

namespace Erdos390.Full

noncomputable section

/-- A probability mass function on a finite sample type. -/
structure FiniteProbability (Ω : Type*) [Fintype Ω] where
  mass : Ω → ℝ
  mass_nonneg : ∀ ω, 0 ≤ mass ω
  mass_sum : ∑ ω, mass ω = 1

namespace FiniteProbability

variable {Ω : Type*} [Fintype Ω]

/-- Expectation with respect to a finite probability mass function. -/
def expect (μ : FiniteProbability Ω) (F : Ω → ℝ) : ℝ :=
  ∑ ω, μ.mass ω * F ω

/-- Covariance with respect to a finite probability mass function. -/
def covariance (μ : FiniteProbability Ω) (F G : Ω → ℝ) : ℝ :=
  μ.expect (fun ω => F ω * G ω) - μ.expect F * μ.expect G

end FiniteProbability

/-- Raw data for a finite exponential family.  `scale` is the paper's `L`,
and `baseMass` below is the paper's active mass `q_n`. -/
structure FiniteExponentialFamily
    (Ω Param : Type*) [Fintype Ω]
    [NormedAddCommGroup Param] [NormedSpace ℝ Param]
    [FiniteDimensional ℝ Param] where
  baseWeight : Ω → ℝ
  baseWeight_nonneg : ∀ ω, 0 ≤ baseWeight ω
  baseMass_pos : 0 < ∑ ω, baseWeight ω
  scale : ℝ
  scale_pos : 0 < scale
  score : Ω → Param →L[ℝ] ℝ

namespace FiniteExponentialFamily

variable {Ω Param : Type*} [Fintype Ω]
  [NormedAddCommGroup Param] [NormedSpace ℝ Param]
  [FiniteDimensional ℝ Param]

/-- Total active mass of the baseline family. -/
def baseMass (fam : FiniteExponentialFamily Ω Param) : ℝ :=
  ∑ ω, fam.baseWeight ω

theorem baseMass_positive (fam : FiniteExponentialFamily Ω Param) : 0 < fam.baseMass := by
  exact fam.baseMass_pos



theorem scale_ne_zero (fam : FiniteExponentialFamily Ω Param) : fam.scale ≠ 0 :=
  ne_of_gt fam.scale_pos

/-- The unnormalized tilted mass at a sample point. -/
def unnormalizedWeight (fam : FiniteExponentialFamily Ω Param)
    (ξ : Param) (ω : Ω) : ℝ :=
  fam.baseWeight ω * Real.exp (fam.score ω ξ / fam.scale)

theorem unnormalizedWeight_nonneg (fam : FiniteExponentialFamily Ω Param)
    (ξ : Param) (ω : Ω) : 0 ≤ fam.unnormalizedWeight ξ ω :=
  mul_nonneg (fam.baseWeight_nonneg ω) (le_of_lt (Real.exp_pos _))

/-- The exact finite partition function. -/
def partition (fam : FiniteExponentialFamily Ω Param) (ξ : Param) : ℝ :=
  ∑ ω, fam.unnormalizedWeight ξ ω

theorem partition_pos (fam : FiniteExponentialFamily Ω Param) (ξ : Param) :
    0 < fam.partition ξ := by
  have hex : ∃ ω : Ω, 0 < fam.baseWeight ω := by
    have hsum : 0 < ∑ ω, fam.baseWeight ω := fam.baseMass_pos
    rw [Finset.sum_pos_iff_of_nonneg
      (fun ω (_ : ω ∈ (Finset.univ : Finset Ω)) => fam.baseWeight_nonneg ω)] at hsum
    exact hsum.imp fun _ hω => hω.2
  obtain ⟨ω, hω⟩ := hex
  rw [partition, Finset.sum_pos_iff_of_nonneg
    (fun η (_ : η ∈ (Finset.univ : Finset Ω)) => fam.unnormalizedWeight_nonneg ξ η)]
  exact ⟨ω, Finset.mem_univ ω, mul_pos hω (Real.exp_pos _)⟩

theorem partition_ne_zero (fam : FiniteExponentialFamily Ω Param) (ξ : Param) :
    fam.partition ξ ≠ 0 :=
  ne_of_gt (fam.partition_pos ξ)

/-- The normalized tilted probability mass. -/
def probabilityMass (fam : FiniteExponentialFamily Ω Param)
    (ξ : Param) (ω : Ω) : ℝ :=
  fam.unnormalizedWeight ξ ω / fam.partition ξ

theorem probabilityMass_nonneg (fam : FiniteExponentialFamily Ω Param)
    (ξ : Param) (ω : Ω) : 0 ≤ fam.probabilityMass ξ ω :=
  div_nonneg (fam.unnormalizedWeight_nonneg ξ ω) (le_of_lt (fam.partition_pos ξ))

theorem probabilityMass_sum (fam : FiniteExponentialFamily Ω Param) (ξ : Param) :
    ∑ ω, fam.probabilityMass ξ ω = 1 := by
  simp only [FiniteExponentialFamily.probabilityMass]
  rw [← Finset.sum_div, partition]
  exact div_self (fam.partition_ne_zero ξ)

/-- The tilted probability as an actual finite probability object. -/
def tiltedProbability (fam : FiniteExponentialFamily Ω Param)
    (ξ : Param) : FiniteProbability Ω where
  mass := fam.probabilityMass ξ
  mass_nonneg := fam.probabilityMass_nonneg ξ
  mass_sum := fam.probabilityMass_sum ξ

/-- Active coordinate weights.  Their total mass remains `baseMass`. -/
def activeWeight (fam : FiniteExponentialFamily Ω Param)
    (ξ : Param) (ω : Ω) : ℝ :=
  fam.baseMass * fam.probabilityMass ξ ω





/-- Expectation under the normalized tilted law. -/
def expectation (fam : FiniteExponentialFamily Ω Param)
    (F : Ω → ℝ) (ξ : Param) : ℝ :=
  (fam.tiltedProbability ξ).expect F

/-- An active (unnormalized) moment. -/
def moment (fam : FiniteExponentialFamily Ω Param)
    (F : Ω → ℝ) (ξ : Param) : ℝ :=
  ∑ ω, fam.activeWeight ξ ω * F ω

/-- The unnormalized numerator of a tilted moment. -/
def rawMoment (fam : FiniteExponentialFamily Ω Param)
    (F : Ω → ℝ) (ξ : Param) : ℝ :=
  ∑ ω, fam.unnormalizedWeight ξ ω * F ω

theorem moment_eq_normalizedRawMoment
    (fam : FiniteExponentialFamily Ω Param) (F : Ω → ℝ) (ξ : Param) :
    fam.moment F ξ = fam.baseMass * fam.rawMoment F ξ / fam.partition ξ := by
  simp only [moment, activeWeight, probabilityMass, rawMoment]
  calc
    (∑ ω, fam.baseMass * (fam.unnormalizedWeight ξ ω / fam.partition ξ) * F ω)
        = ∑ ω, (fam.baseMass / fam.partition ξ) *
            (fam.unnormalizedWeight ξ ω * F ω) := by
              apply Finset.sum_congr rfl
              intro ω _
              ring
    _ = (fam.baseMass / fam.partition ξ) *
          ∑ ω, fam.unnormalizedWeight ξ ω * F ω := by rw [Finset.mul_sum]
    _ = fam.baseMass * (∑ ω, fam.unnormalizedWeight ξ ω * F ω) /
          fam.partition ξ := by ring



theorem expectation_eq_rawMoment_div_partition
    (fam : FiniteExponentialFamily Ω Param) (F : Ω → ℝ) (ξ : Param) :
    fam.expectation F ξ = fam.rawMoment F ξ / fam.partition ξ := by
  simp only [expectation, tiltedProbability, FiniteProbability.expect,
    probabilityMass, rawMoment]
  calc
    (∑ ω, fam.unnormalizedWeight ξ ω / fam.partition ξ * F ω)
        = ∑ ω, (fam.unnormalizedWeight ξ ω * F ω) / fam.partition ξ := by
            apply Finset.sum_congr rfl
            intro ω _
            ring
    _ = (∑ ω, fam.unnormalizedWeight ξ ω * F ω) / fam.partition ξ := by
          rw [Finset.sum_div]

/-- Fréchet derivative of one unnormalized exponential weight. -/
def unnormalizedWeightFDeriv (fam : FiniteExponentialFamily Ω Param)
    (ξ : Param) (ω : Ω) : Param →L[ℝ] ℝ :=
  fam.baseWeight ω •
    (Real.exp (fam.score ω ξ / fam.scale) •
      ((fam.scale)⁻¹ • fam.score ω))

theorem hasFDerivAt_unnormalizedWeight
    (fam : FiniteExponentialFamily Ω Param) (ξ : Param) (ω : Ω) :
    HasFDerivAt (fun η => fam.unnormalizedWeight η ω)
      (fam.unnormalizedWeightFDeriv ξ ω) ξ := by
  have hs : HasFDerivAt (fun η => fam.score ω η / fam.scale)
      ((fam.scale)⁻¹ • fam.score ω) ξ := by
    simpa only [div_eq_mul_inv, mul_comm] using
      ((fam.score ω).hasFDerivAt.const_mul (fam.scale)⁻¹)
  simpa only [unnormalizedWeight, unnormalizedWeightFDeriv] using
    hs.exp.const_mul (fam.baseWeight ω)

theorem unnormalizedWeightFDeriv_apply
    (fam : FiniteExponentialFamily Ω Param) (ξ direction : Param) (ω : Ω) :
    fam.unnormalizedWeightFDeriv ξ ω direction =
      fam.unnormalizedWeight ξ ω / fam.scale * fam.score ω direction := by
  simp only [unnormalizedWeightFDeriv, ContinuousLinearMap.smul_apply,
    smul_eq_mul, unnormalizedWeight]
  field_simp

/-- Fréchet derivative of the partition function. -/
def partitionFDeriv (fam : FiniteExponentialFamily Ω Param)
    (ξ : Param) : Param →L[ℝ] ℝ :=
  ∑ ω, fam.unnormalizedWeightFDeriv ξ ω

theorem hasFDerivAt_partition
    (fam : FiniteExponentialFamily Ω Param) (ξ : Param) :
    HasFDerivAt fam.partition (fam.partitionFDeriv ξ) ξ := by
  have h := HasFDerivAt.sum (u := (Finset.univ : Finset Ω))
    (fun ω (_ : ω ∈ (Finset.univ : Finset Ω)) =>
      fam.hasFDerivAt_unnormalizedWeight ξ ω)
  have hfun : fam.partition =
      ∑ ω, fun η => fam.unnormalizedWeight η ω := by
    funext η
    simp only [partition, Finset.sum_apply]
  rw [hfun]
  simpa only [partitionFDeriv] using h

theorem partitionFDeriv_apply
    (fam : FiniteExponentialFamily Ω Param) (ξ direction : Param) :
    fam.partitionFDeriv ξ direction =
      (1 / fam.scale) *
        ∑ ω, fam.unnormalizedWeight ξ ω * fam.score ω direction := by
  simp only [partitionFDeriv, ContinuousLinearMap.sum_apply,
    fam.unnormalizedWeightFDeriv_apply]
  rw [Finset.mul_sum]
  apply Finset.sum_congr rfl
  intro ω _
  ring

/-- Fréchet derivative of an unnormalized moment numerator. -/
def rawMomentFDeriv (fam : FiniteExponentialFamily Ω Param)
    (F : Ω → ℝ) (ξ : Param) : Param →L[ℝ] ℝ :=
  ∑ ω, F ω • fam.unnormalizedWeightFDeriv ξ ω

theorem hasFDerivAt_rawMoment
    (fam : FiniteExponentialFamily Ω Param) (F : Ω → ℝ) (ξ : Param) :
    HasFDerivAt (fam.rawMoment F) (fam.rawMomentFDeriv F ξ) ξ := by
  have h := HasFDerivAt.sum (u := (Finset.univ : Finset Ω))
    (fun ω (_ : ω ∈ (Finset.univ : Finset Ω)) =>
      (fam.hasFDerivAt_unnormalizedWeight ξ ω).mul_const (F ω))
  have hfun : fam.rawMoment F =
      ∑ ω, fun η => fam.unnormalizedWeight η ω * F ω := by
    funext η
    simp only [rawMoment, Finset.sum_apply]
  rw [hfun]
  simpa only [rawMomentFDeriv] using h

theorem rawMomentFDeriv_apply
    (fam : FiniteExponentialFamily Ω Param) (F : Ω → ℝ)
    (ξ direction : Param) :
    fam.rawMomentFDeriv F ξ direction =
      (1 / fam.scale) *
        ∑ ω, fam.unnormalizedWeight ξ ω * F ω * fam.score ω direction := by
  simp only [rawMomentFDeriv, ContinuousLinearMap.sum_apply,
    ContinuousLinearMap.smul_apply, smul_eq_mul,
    fam.unnormalizedWeightFDeriv_apply]
  rw [Finset.mul_sum]
  apply Finset.sum_congr rfl
  intro ω _
  ring

/-- The derivative furnished directly by the quotient rule. -/
def quotientMomentFDeriv (fam : FiniteExponentialFamily Ω Param)
    (F : Ω → ℝ) (ξ : Param) : Param →L[ℝ] ℝ :=
  fam.baseMass •
    (fam.rawMoment F ξ •
        (-((fam.partition ξ) ^ 2)⁻¹ • fam.partitionFDeriv ξ) +
      (fam.partition ξ)⁻¹ • fam.rawMomentFDeriv F ξ)

theorem hasFDerivAt_moment_quotient
    (fam : FiniteExponentialFamily Ω Param) (F : Ω → ℝ) (ξ : Param) :
    HasFDerivAt (fam.moment F) (fam.quotientMomentFDeriv F ξ) ξ := by
  have hpart := fam.hasFDerivAt_partition ξ
  have hinv : HasFDerivAt (fun η => (fam.partition η)⁻¹)
      (-((fam.partition ξ) ^ 2)⁻¹ • fam.partitionFDeriv ξ) ξ := by
    exact (hasDerivAt_inv (fam.partition_ne_zero ξ)).comp_hasFDerivAt ξ hpart
  have hproduct := (fam.hasFDerivAt_rawMoment F ξ).mul hinv
  have hscaled := hproduct.const_mul fam.baseMass
  have heq : fam.moment F = fun η =>
      fam.baseMass * (fam.rawMoment F η * (fam.partition η)⁻¹) := by
    funext η
    rw [fam.moment_eq_normalizedRawMoment F η]
    simp only [div_eq_mul_inv]
    ring
  rw [heq]
  simpa only [quotientMomentFDeriv, div_eq_mul_inv, Pi.mul_apply] using hscaled

/-- Covariance under the normalized tilted law. -/
def covariance (fam : FiniteExponentialFamily Ω Param)
    (F G : Ω → ℝ) (ξ : Param) : ℝ :=
  (fam.tiltedProbability ξ).covariance F G









/-- The covariance with the score, as a continuous linear functional of
the parameter direction. -/
def covarianceScore (fam : FiniteExponentialFamily Ω Param)
    (F : Ω → ℝ) (ξ : Param) : Param →L[ℝ] ℝ :=
  ∑ ω, (fam.probabilityMass ξ ω *
      (F ω - fam.expectation F ξ)) • fam.score ω

theorem covarianceScore_apply (fam : FiniteExponentialFamily Ω Param)
    (F : Ω → ℝ) (ξ direction : Param) :
    fam.covarianceScore F ξ direction =
      fam.covariance F (fun ω => fam.score ω direction) ξ := by
  simp only [covarianceScore, ContinuousLinearMap.sum_apply,
    ContinuousLinearMap.smul_apply, smul_eq_mul]
  rw [covariance, FiniteProbability.covariance, expectation,
    FiniteProbability.expect, FiniteProbability.expect]
  simp only [FiniteProbability.expect]
  calc
    (∑ ω, fam.probabilityMass ξ ω * (F ω -
        ∑ η, fam.probabilityMass ξ η * F η) * fam.score ω direction)
        = (∑ ω, fam.probabilityMass ξ ω * (F ω * fam.score ω direction)) -
          (∑ η, fam.probabilityMass ξ η * F η) *
            (∑ ω, fam.probabilityMass ξ ω * fam.score ω direction) := by
              rw [Finset.mul_sum]
              rw [← Finset.sum_sub_distrib]
              apply Finset.sum_congr rfl
              intro ω _
              ring
    _ = (∑ ω, fam.probabilityMass ξ ω * (F ω * fam.score ω direction)) -
          (∑ η, fam.probabilityMass ξ η * F η) *
            (∑ ω, fam.probabilityMass ξ ω * fam.score ω direction) := rfl

theorem covariance_eq_rawMoments
    (fam : FiniteExponentialFamily Ω Param) (F G : Ω → ℝ) (ξ : Param) :
    fam.covariance F G ξ =
      fam.rawMoment (fun ω => F ω * G ω) ξ / fam.partition ξ -
        (fam.rawMoment F ξ / fam.partition ξ) *
          (fam.rawMoment G ξ / fam.partition ξ) := by
  rw [covariance, FiniteProbability.covariance]
  change fam.expectation (fun ω => F ω * G ω) ξ -
      fam.expectation F ξ * fam.expectation G ξ = _
  rw [fam.expectation_eq_rawMoment_div_partition,
    fam.expectation_eq_rawMoment_div_partition,
    fam.expectation_eq_rawMoment_div_partition]

theorem quotientMomentFDeriv_apply
    (fam : FiniteExponentialFamily Ω Param) (F : Ω → ℝ)
    (ξ direction : Param) :
    fam.quotientMomentFDeriv F ξ direction =
      (fam.baseMass / fam.scale) *
        fam.covariance F (fun ω => fam.score ω direction) ξ := by
  simp only [quotientMomentFDeriv, ContinuousLinearMap.smul_apply,
    ContinuousLinearMap.add_apply, smul_eq_mul]
  rw [fam.partitionFDeriv_apply ξ direction,
    fam.rawMomentFDeriv_apply F ξ direction,
    fam.covariance_eq_rawMoments F (fun ω => fam.score ω direction) ξ]
  simp only [rawMoment]
  field_simp [fam.partition_ne_zero ξ, fam.scale_ne_zero]
  ring

theorem quotientMomentFDeriv_eq_covarianceScore
    (fam : FiniteExponentialFamily Ω Param) (F : Ω → ℝ) (ξ : Param) :
    fam.quotientMomentFDeriv F ξ =
      (fam.baseMass / fam.scale) • fam.covarianceScore F ξ := by
  ext direction
  rw [fam.quotientMomentFDeriv_apply F ξ direction,
    ContinuousLinearMap.smul_apply, smul_eq_mul,
    fam.covarianceScore_apply F ξ direction]

/-- Exact covariance-Jacobian identity for a finite active moment. -/
theorem hasFDerivAt_moment_covariance
    (fam : FiniteExponentialFamily Ω Param) (F : Ω → ℝ) (ξ : Param) :
    HasFDerivAt (fam.moment F)
      ((fam.baseMass / fam.scale) • fam.covarianceScore F ξ) ξ := by
  rw [← fam.quotientMomentFDeriv_eq_covarianceScore F ξ]
  exact fam.hasFDerivAt_moment_quotient F ξ



end FiniteExponentialFamily

end

end Erdos390.Full

end

/- Source module: Erdos390.Full.FiniteTiltTV -/
section


/-!
# Total-variation control for finite exponential tilts

This supplies the normalization estimate used to compare each finite-`n`
tilted nuisance covariance with its own finite-`n` baseline.  The bound is
proved directly from the exponential density ratio.
-/

open scoped BigOperators

namespace Erdos390.Full

noncomputable section

namespace FiniteProbability

variable {Omega : Type*} [Fintype Omega]

def expPartition (mu : FiniteProbability Omega) (S : Omega → ℝ) : ℝ :=
  mu.expect (fun omega ↦ Real.exp (S omega))

theorem expPartition_pos (mu : FiniteProbability Omega) (S : Omega → ℝ) :
    0 < mu.expPartition S := by
  have hex : ∃ omega, 0 < mu.mass omega := by
    have hsum : 0 < ∑ omega, mu.mass omega := by
      rw [mu.mass_sum]
      exact zero_lt_one
    rw [Finset.sum_pos_iff_of_nonneg
      (fun omega (_ : omega ∈ (Finset.univ : Finset Omega)) ↦
        mu.mass_nonneg omega)] at hsum
    exact hsum.imp fun omega h ↦ h.2
  obtain ⟨omega, homega⟩ := hex
  rw [expPartition, expect, Finset.sum_pos_iff_of_nonneg]
  · exact ⟨omega, Finset.mem_univ omega,
      mul_pos homega (Real.exp_pos _)⟩
  · intro eta _
    exact mul_nonneg (mu.mass_nonneg eta) (Real.exp_pos _).le

def exponentialTilt (mu : FiniteProbability Omega) (S : Omega → ℝ) :
    FiniteProbability Omega where
  mass := fun omega ↦ mu.mass omega * Real.exp (S omega) / mu.expPartition S
  mass_nonneg := by
    intro omega
    exact div_nonneg
      (mul_nonneg (mu.mass_nonneg omega) (Real.exp_pos _).le)
      (mu.expPartition_pos S).le
  mass_sum := by
    rw [← Finset.sum_div, expPartition, expect]
    exact div_self (ne_of_gt (mu.expPartition_pos S))

def exponentialDeviation (mu : FiniteProbability Omega) (S : Omega → ℝ) : ℝ :=
  mu.expect (fun omega ↦ |Real.exp (S omega) - 1|)















end FiniteProbability

end

end Erdos390.Full

end

/- Source module: Erdos390.Full.DickmanBasic -/
section


/-!
# The Dickman function on the finite range needed for Erdős 390

This file constructs the Dickman function by the method of steps.  No
existence, regularity, positivity, or kernel estimate is assumed as an input.
Only the range up to `5` is needed for the Poisson--Dickman operator at
`U = 9 / 2`.
-/

open Filter Set
open scoped Interval

noncomputable section

namespace Erdos390.Full.DickmanBasic



lemma safeDenom_pos (x : ℝ) : 0 < safeDenom x := by
  unfold safeDenom
  exact lt_of_lt_of_le zero_lt_one (le_max_right _ _)

lemma safeDenom_ne_zero (x : ℝ) : safeDenom x ≠ 0 :=
  ne_of_gt (safeDenom_pos x)

lemma safeDenom_eq_self {x : ℝ} (hx : 1 ≤ x) : safeDenom x = x := by
  simp [safeDenom, hx]



@[simp] lemma rhoApprox_zero (x : ℝ) : rhoApprox 0 x = 1 := rfl

@[simp] lemma rhoApprox_succ_of_le_one (n : ℕ) {x : ℝ} (hx : x ≤ 1) :
    rhoApprox (n + 1) x = 1 := by
  simp [rhoApprox, hx]

lemma rhoApprox_succ_of_one_lt (n : ℕ) {x : ℝ} (hx : 1 < x) :
    rhoApprox (n + 1) x =
      1 - ∫ t in (1 : ℝ)..x, rhoApprox n (t - 1) / safeDenom t := by
  simp [rhoApprox, not_le.mpr hx]

lemma continuous_safeDenom : Continuous safeDenom := by
  exact continuous_id.max continuous_const

lemma continuous_stepIntegrand (n : ℕ) :
    Continuous (fun t : ℝ => rhoApprox n (t - 1) / safeDenom t) := by
  induction n with
  | zero =>
      exact continuous_const.div continuous_safeDenom safeDenom_ne_zero
  | succ n ih =>
      have hRho : Continuous (rhoApprox (n + 1)) := by
        let g : ℝ → ℝ := fun t => rhoApprox n (t - 1) / safeDenom t
        have hg : Continuous g := by
          simpa [g] using ih
        have hprimitive : Continuous (fun x : ℝ => ∫ t in (1 : ℝ)..x, g t) :=
          (intervalIntegral.differentiable_integral_of_continuous hg).continuous
        have hbranch : Continuous (fun x : ℝ => 1 - ∫ t in (1 : ℝ)..x, g t) :=
          continuous_const.sub hprimitive
        have hpiece : Continuous (fun x : ℝ =>
            if x ≤ 1 then (1 : ℝ)
            else 1 - ∫ t in (1 : ℝ)..x, g t) := by
          apply continuous_if_le continuous_id continuous_const
            continuous_const.continuousOn hbranch.continuousOn
          intro x hx
          change x = 1 at hx
          rw [hx]
          simp
        simpa [rhoApprox, g] using hpiece
      exact (hRho.comp (continuous_id.sub continuous_const)).div
        continuous_safeDenom safeDenom_ne_zero

lemma continuous_rhoApprox (n : ℕ) : Continuous (rhoApprox n) := by
  cases n with
  | zero => exact continuous_const
  | succ n =>
      let g : ℝ → ℝ := fun t => rhoApprox n (t - 1) / safeDenom t
      have hg : Continuous g := continuous_stepIntegrand n
      have hprimitive : Continuous (fun x : ℝ => ∫ t in (1 : ℝ)..x, g t) :=
        (intervalIntegral.differentiable_integral_of_continuous hg).continuous
      have hbranch : Continuous (fun x : ℝ => 1 - ∫ t in (1 : ℝ)..x, g t) :=
        continuous_const.sub hprimitive
      have hpiece : Continuous (fun x : ℝ =>
          if x ≤ 1 then (1 : ℝ)
          else 1 - ∫ t in (1 : ℝ)..x, g t) := by
        apply continuous_if_le continuous_id continuous_const
          continuous_const.continuousOn hbranch.continuousOn
        intro x hx
        change x = 1 at hx
        rw [hx]
        simp
      simpa [rhoApprox, g] using hpiece

/-- Successive method-of-steps approximants agree on the interval which
has already been constructed. -/
lemma rhoApprox_succ_eq_prev : ∀ (n : ℕ) (x : ℝ),
    x ≤ (n : ℝ) + 1 → rhoApprox (n + 1) x = rhoApprox n x := by
  intro n
  induction n with
  | zero =>
      intro x hx
      have hx1 : x ≤ 1 := by simpa using hx
      simp [rhoApprox, hx1]
  | succ n ih =>
      intro x hx
      by_cases hx1 : x ≤ 1
      · simp [rhoApprox, hx1]
      · have h1x : 1 < x := lt_of_not_ge hx1
        rw [rhoApprox_succ_of_one_lt (n + 1) h1x,
          rhoApprox_succ_of_one_lt n h1x]
        congr 1
        apply intervalIntegral.integral_congr
        intro t ht
        have htmem : t ∈ Icc (1 : ℝ) x := by
          simpa [uIcc_of_le h1x.le] using ht
        have htbound : t - 1 ≤ (n : ℝ) + 1 := by
          have hcast : x ≤ (n : ℝ) + 2 := by
            norm_num [Nat.cast_add, Nat.cast_one, add_assoc] at hx ⊢
            exact hx
          linarith [htmem.2]
        change rhoApprox (n + 1) (t - 1) / safeDenom t =
          rhoApprox n (t - 1) / safeDenom t
        rw [ih (t - 1) htbound]

lemma hasDerivAt_rhoApprox_succ (n : ℕ) {x : ℝ} (hx : 1 < x) :
    HasDerivAt (rhoApprox (n + 1))
      (-rhoApprox n (x - 1) / x) x := by
  let g : ℝ → ℝ := fun t => rhoApprox n (t - 1) / safeDenom t
  have hg : Continuous g := by
    simpa [g] using continuous_stepIntegrand n
  have hprimitive : HasDerivAt (fun z : ℝ => ∫ t in (1 : ℝ)..z, g t) (g x) x :=
    (hg.integral_hasStrictDerivAt 1 x).hasDerivAt
  have hbranch : HasDerivAt
      (fun z : ℝ => 1 - ∫ t in (1 : ℝ)..z, g t) (-g x) x := by
    convert (hasDerivAt_const x (1 : ℝ)).sub hprimitive using 1
    ring
  have heq : rhoApprox (n + 1) =ᶠ[nhds x]
      (fun z : ℝ => 1 - ∫ t in (1 : ℝ)..z, g t) := by
    filter_upwards [Ioi_mem_nhds hx] with z hz
    simpa [g] using rhoApprox_succ_of_one_lt n hz
  have h := hbranch.congr_of_eventuallyEq heq
  have hgx : g x = rhoApprox n (x - 1) / x := by
    simp [g, safeDenom_eq_self hx.le]
  rw [hgx] at h
  convert h using 1
  ring



lemma continuous_rho : Continuous rho := by
  exact continuous_rhoApprox 5

lemma rho_eq_one_of_le_one {x : ℝ} (hx : x ≤ 1) : rho x = 1 := by
  simp [rho, rhoApprox, hx]

@[simp] lemma rho_zero : rho 0 = 1 := rho_eq_one_of_le_one (by norm_num)

@[simp] lemma rho_one : rho 1 = 1 := rho_eq_one_of_le_one le_rfl

/-- The delay differential equation on the entire range needed later. -/
lemma hasDerivAt_rho {x : ℝ} (hx1 : 1 < x) (hx6 : x ≤ 6) :
    HasDerivAt rho (-rho (x - 1) / x) x := by
  have hprev : rhoApprox 5 (x - 1) = rhoApprox 4 (x - 1) := by
    apply rhoApprox_succ_eq_prev 4
    norm_num
    linarith
  have h := hasDerivAt_rhoApprox_succ 4 hx1
  simpa [rho, hprev] using h

lemma deriv_rho {x : ℝ} (hx1 : 1 < x) (hx6 : x ≤ 6) :
    deriv rho x = -rho (x - 1) / x :=
  (hasDerivAt_rho hx1 hx6).deriv



lemma rho_integral_eq {x : ℝ} (hx1 : 1 ≤ x) (hx5 : x ≤ 5) :
    rho x = 1 - ∫ t in (1 : ℝ)..x, rho (t - 1) / t := by
  rcases hx1.eq_or_lt with rfl | hx1
  · simp [rho_one]
  · rw [rho, rhoApprox_succ_of_one_lt 4 hx1]
    congr 1
    apply intervalIntegral.integral_congr
    intro t ht
    have htmem : t ∈ Icc (1 : ℝ) x := by
      simpa [uIcc_of_le hx1.le] using ht
    have ht5 : t - 1 ≤ (4 : ℝ) + 1 := by linarith [htmem.2, hx5]
    have hstable := rhoApprox_succ_eq_prev 4 (t - 1) ht5
    change rhoApprox 4 (t - 1) / safeDenom t = rhoApprox 5 (t - 1) / t
    rw [safeDenom_eq_self htmem.1]
    rw [hstable]

/-- The integral of `rho` over its initial unit interval. -/
lemma integral_rho_zero_one :
    (∫ t in (0 : ℝ)..1, rho t) = 1 := by
  calc
    (∫ t in (0 : ℝ)..1, rho t) = ∫ _t in (0 : ℝ)..1, (1 : ℝ) := by
      apply intervalIntegral.integral_congr
      intro t ht
      apply rho_eq_one_of_le_one
      have ht' : t ∈ Icc (0 : ℝ) 1 := by
        simpa [uIcc_of_le zero_le_one] using ht
      exact ht'.2
    _ = 1 := by simp

/-- The delay equation has the equivalent positive averaging form
`x * rho x = ∫_[x-1,x] rho`.  This identity is the key to positivity. -/
lemma rho_average_eq {x : ℝ} (hx1 : 1 ≤ x) (hx5 : x ≤ 5) :
    x * rho x = ∫ t in (x - 1)..x, rho t := by
  rcases hx1.eq_or_lt with rfl | hx1
  · simp [rho_one, integral_rho_zero_one]
  · have hderiv : ∀ t ∈ Ioo (1 : ℝ) x,
        HasDerivWithinAt (fun u : ℝ => u * rho u)
          (rho t - rho (t - 1)) (Ioi t) t := by
      intro t ht
      have ht6 : t ≤ 6 := by linarith [ht.2, hx5]
      have h := (hasDerivAt_id t).mul (hasDerivAt_rho ht.1 ht6)
      have ht0 : t ≠ 0 := ne_of_gt (zero_lt_one.trans ht.1)
      have heq : 1 * rho t + t * (-rho (t - 1) / t) =
          rho t - rho (t - 1) := by
        field_simp
        ring
      exact (h.congr_deriv heq).hasDerivWithinAt
    have hFTC := intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le
      hx1.le
      ((continuous_id.mul continuous_rho).continuousOn)
      hderiv
      ((continuous_rho.sub
        (continuous_rho.comp (continuous_id.sub continuous_const))).intervalIntegrable 1 x)
    have hshift : (∫ t in (1 : ℝ)..x, rho (t - 1)) =
        ∫ t in (0 : ℝ)..(x - 1), rho t := by
      convert intervalIntegral.integral_comp_sub_right
        (a := (1 : ℝ)) (b := x) rho 1 using 1
      norm_num
    have hsub : (∫ t in (1 : ℝ)..x, rho t - rho (t - 1)) =
        (∫ t in (1 : ℝ)..x, rho t) - ∫ t in (1 : ℝ)..x, rho (t - 1) := by
      apply intervalIntegral.integral_sub
      · exact continuous_rho.intervalIntegrable 1 x
      · exact (continuous_rho.comp
          (continuous_id.sub continuous_const)).intervalIntegrable 1 x
    have hadd₁ := intervalIntegral.integral_add_adjacent_intervals
      (μ := MeasureTheory.volume)
      (continuous_rho.intervalIntegrable (0 : ℝ) 1)
      (continuous_rho.intervalIntegrable (1 : ℝ) x)
    have hadd₂ := intervalIntegral.integral_add_adjacent_intervals
      (μ := MeasureTheory.volume)
      (continuous_rho.intervalIntegrable (0 : ℝ) (x - 1))
      (continuous_rho.intervalIntegrable (x - 1) x)
    rw [hsub, hshift] at hFTC
    rw [integral_rho_zero_one] at hadd₁
    have hrho1 : rho (1 : ℝ) = 1 := rho_one
    change _ = x * rho x - 1 * rho 1 at hFTC
    rw [hrho1] at hFTC
    norm_num at hFTC
    linarith





/-- The constructed solution is decreasing on `[1,5]`. -/
lemma antitoneOn_rho_one_five : AntitoneOn rho (Icc (1 : ℝ) 5) := by
  apply antitoneOn_of_deriv_nonpos (convex_Icc (1 : ℝ) 5) continuous_rho.continuousOn
  · intro x hx
    rw [interior_Icc] at hx
    exact (hasDerivAt_rho hx.1 (by linarith [hx.2])).differentiableAt.differentiableWithinAt
  · intro x hx
    rw [interior_Icc] at hx
    have hpos : 0 < rho (x - 1) :=
      rho_pos_on_zero_five (by linarith [hx.1]) (by linarith [hx.2])
    rw [deriv_rho hx.1 (by linarith [hx.2])]
    exact div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hpos.le) (by linarith [hx.1])



 lemma continuousOn_delayDeriv :
    ContinuousOn delayDeriv (Ioo (1 : ℝ) 6) := by
  apply ContinuousOn.div
  · exact (continuous_rho.comp (continuous_id.sub continuous_const)).neg.continuousOn
  · exact continuous_id.continuousOn
  · intro x hx
    exact ne_of_gt (zero_lt_one.trans hx.1)

/-- The method-of-steps solution is continuously differentiable away from
the initial corner at `1`. -/
lemma contDiffOn_one_rho : ContDiffOn ℝ 1 rho (Ioo (1 : ℝ) 6) := by
  rw [show (1 : WithTop ℕ∞) = 0 + 1 by rfl,
    contDiffOn_succ_iff_deriv_of_isOpen isOpen_Ioo]
  refine ⟨?_, ?_, ?_⟩
  · intro x hx
    exact (hasDerivAt_rho hx.1 hx.2.le).differentiableAt.differentiableWithinAt
  · intro h
    simp at h
  · rw [contDiffOn_zero]
    exact continuousOn_delayDeriv.congr fun x hx => by
      exact deriv_rho hx.1 hx.2.le

 lemma contDiffOn_one_delayDeriv :
    ContDiffOn ℝ 1 delayDeriv (Ioo (2 : ℝ) 5) := by
  have hshift : ContDiffOn ℝ 1 (fun x : ℝ => x - 1) (Ioo (2 : ℝ) 5) := by
    fun_prop
  have hmaps : MapsTo (fun x : ℝ => x - 1) (Ioo (2 : ℝ) 5) (Ioo (1 : ℝ) 6) := by
    intro x hx
    constructor <;> linarith [hx.1, hx.2]
  have hcomp : ContDiffOn ℝ 1 (rho ∘ fun x : ℝ => x - 1) (Ioo (2 : ℝ) 5) :=
    contDiffOn_one_rho.comp hshift hmaps
  have hnum : ContDiffOn ℝ 1 (fun x : ℝ => -rho (x - 1)) (Ioo (2 : ℝ) 5) := by
    simpa [Function.comp_def] using hcomp.neg
  exact hnum.div contDiffOn_id fun x hx => ne_of_gt (by linarith [hx.1])

/-- On the compact interval needed for `U = 9/2`, the Dickman function is
`C²`.  We prove the slightly stronger open-interval statement first, so
ordinary derivatives can be used without endpoint conventions. -/
lemma contDiffOn_two_rho_open : ContDiffOn ℝ 2 rho (Ioo (2 : ℝ) 5) := by
  rw [show (2 : WithTop ℕ∞) = 1 + 1 by rfl,
    contDiffOn_succ_iff_deriv_of_isOpen isOpen_Ioo]
  refine ⟨?_, ?_, ?_⟩
  · exact (contDiffOn_one_rho.mono fun x hx => by
      constructor <;> linarith [hx.1, hx.2]).differentiableOn (by norm_num)
  · intro h
    simp at h
  · exact contDiffOn_one_delayDeriv.congr fun x hx =>
      deriv_rho (by linarith [hx.1]) (by linarith [hx.2])





lemma rho_U_pos : 0 < rho U := by
  apply rho_pos_on_zero_five
  · norm_num [U]
  · norm_num [U]

lemma rho_U_ne_zero : rho U ≠ 0 := ne_of_gt rho_U_pos



/-- The one-point weight; it is separated from zero uniformly on `[0,2]`. -/
def h (x : ℝ) : ℝ := F x

@[simp] lemma F_zero : F 0 = 1 := by
  simp [F, rho_U_ne_zero]

@[simp] lemma h_zero : h 0 = 1 := by
  simp [h]

lemma F_pos {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 2) : 0 < F x := by
  apply div_pos
  · apply rho_pos_on_zero_five
    · norm_num [U] at hx ⊢
      linarith [hx.2]
    · norm_num [U] at hx ⊢
      linarith [hx.1]
  · exact rho_U_pos







/-- `F` is `C²` on an open neighborhood of `[0,2]`; retaining the open
neighborhood makes all later uses of ordinary derivatives unambiguous. -/
 lemma contDiffOn_two_F_open :
    ContDiffOn ℝ 2 F (Ioo (-(1 : ℝ) / 4) ((9 : ℝ) / 4)) := by
  have haff : ContDiffOn ℝ 2 (fun x : ℝ => U - x)
      (Ioo (-(1 : ℝ) / 4) ((9 : ℝ) / 4)) := by
    fun_prop
  have hmaps : MapsTo (fun x : ℝ => U - x)
      (Ioo (-(1 : ℝ) / 4) ((9 : ℝ) / 4)) (Ioo (2 : ℝ) 5) := by
    intro x hx
    norm_num [U] at hx ⊢
    constructor <;> linarith [hx.1, hx.2]
  have hcomp : ContDiffOn ℝ 2 (rho ∘ fun x : ℝ => U - x)
      (Ioo (-(1 : ℝ) / 4) ((9 : ℝ) / 4)) :=
    contDiffOn_two_rho_open.comp haff hmaps
  simpa [F, Function.comp_def] using hcomp.div_const (rho U)



 lemma Icc_zero_two_subset_F_open :
    Icc (0 : ℝ) 2 ⊆ Ioo (-(1 : ℝ) / 4) ((9 : ℝ) / 4) := by
  intro x hx
  constructor <;> norm_num <;> linarith [hx.1, hx.2]

 lemma differentiableAt_F_of_mem {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 2) :
    DifferentiableAt ℝ F x := by
  have hxopen := Icc_zero_two_subset_F_open hx
  exact (contDiffOn_two_F_open.contDiffAt (isOpen_Ioo.mem_nhds hxopen)).differentiableAt
    (by norm_num)

lemma differentiableAt_F {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 2) :
    DifferentiableAt ℝ F x := differentiableAt_F_of_mem hx



/-- The ordinary derivative of `F` varies continuously on the whole compact
range used by the product kernel. -/
lemma continuousOn_deriv_F : ContinuousOn (deriv F) (Icc (0 : ℝ) 2) := by
  exact (contDiffOn_two_F_open.continuousOn_deriv_of_isOpen isOpen_Ioo
    (by norm_num)).mono Icc_zero_two_subset_F_open







end Erdos390.Full.DickmanBasic

end

/- Source module: Erdos390.Full.ArithmeticModel -/
section


/-!
# Arithmetic model for the full Erdős 390 formalization

This file connects the abstract covariance columns used by the earlier audit
companion to the actual arithmetic objects in the paper.  All definitions are
total, including at `n = 0`, `m = 0`, and at non-prime bases.  The valuation
identities themselves are stated with the mathematically necessary hypotheses
that the base is prime and the sampled integer is positive.
-/

open scoped BigOperators

namespace Erdos390.Full.ArithmeticModel





/-- The actual prime band `W < p ≤ y`. -/
noncomputable def primeBand (n W : ℕ) : Finset ℕ :=
  ((yNat n + 1).primesBelow).filter (W < ·)

/-- The logarithmic prime coordinate `log p / log y`.  The definition is
total; at the irrelevant boundary `n ≤ 1`, Lean's field conventions give a
well-defined value even though the analytic argument never uses it there. -/
noncomputable def tPrime (n p : ℕ) : ℝ :=
  Real.log (p : ℝ) / Real.log (y n)

/-- The divisibility indicator, as a real-valued random variable.  In
particular `divInd d 0 = 1`, since every natural number divides zero. -/
def divInd (d m : ℕ) : ℝ :=
  if d ∣ m then 1 else 0

/-- The actual `p`-adic valuation column.  Mathlib's factorization is zero at
`m = 0` and at non-prime bases; both conventions are recorded below. -/
def valuation (p m : ℕ) : ℝ :=
  (m.factorization p : ℝ)

/-- The higher-prime-power part `J_p = V_p - I_p`. -/
def higherValuation (p m : ℕ) : ℝ :=
  valuation p m - divInd p m

/-- A natural upper bound for the fixed physical interval `m ≤ C⋅n`. -/
noncomputable def physicalBound (C : ℝ) (n : ℕ) : ℕ :=
  ⌊C * (n : ℝ)⌋₊

/-- All positive exponents up to a common physical bound. -/
def positiveExponents (M : ℕ) : Finset ℕ :=
  Finset.Icc 1 M

/-- All exponents at least two up to a common physical bound. -/
def highExponents (M : ℕ) : Finset ℕ :=
  Finset.Icc 2 M

@[simp] theorem y_zero : y 0 = 0 := by
  simp [y]

@[simp] theorem y_one : y 1 = 1 := by
  norm_num [y]

@[simp] theorem yNat_zero : yNat 0 = 0 := by
  simp [yNat]

theorem mem_primeBand {n W p : ℕ} :
    p ∈ primeBand n W ↔ p.Prime ∧ W < p ∧ p ≤ yNat n := by
  simp only [primeBand, Finset.mem_filter, Nat.mem_primesBelow,
    Nat.lt_succ_iff]
  tauto

theorem prime_of_mem_primeBand {n W p : ℕ} (hp : p ∈ primeBand n W) :
    p.Prime :=
  (mem_primeBand.mp hp).1

theorem cutoff_lt_of_mem_primeBand {n W p : ℕ} (hp : p ∈ primeBand n W) :
    W < p :=
  (mem_primeBand.mp hp).2.1

theorem le_yNat_of_mem_primeBand {n W p : ℕ} (hp : p ∈ primeBand n W) :
    p ≤ yNat n :=
  (mem_primeBand.mp hp).2.2

@[simp] theorem zero_not_mem_primeBand (n W : ℕ) : 0 ∉ primeBand n W := by
  intro h
  exact (prime_of_mem_primeBand h).ne_zero rfl

@[simp] theorem one_not_mem_primeBand (n W : ℕ) : 1 ∉ primeBand n W := by
  intro h
  exact (prime_of_mem_primeBand h).ne_one rfl

@[simp] theorem primeBand_zero (W : ℕ) : primeBand 0 W = ∅ := by
  apply Finset.eq_empty_iff_forall_notMem.mpr
  intro p hp
  have hle : p ≤ 0 := by
    simpa using le_yNat_of_mem_primeBand hp
  have hpos : 0 < p := (prime_of_mem_primeBand hp).pos
  omega

@[simp] theorem divInd_zero_right (d : ℕ) : divInd d 0 = 1 := by
  simp [divInd]

@[simp] theorem divInd_zero_left {m : ℕ} (hm : m ≠ 0) : divInd 0 m = 0 := by
  simp [divInd, zero_dvd_iff, hm]

theorem divInd_eq_zero_or_one (d m : ℕ) :
    divInd d m = 0 ∨ divInd d m = 1 := by
  by_cases h : d ∣ m
  · exact Or.inr (by simp [divInd, h])
  · exact Or.inl (by simp [divInd, h])





@[simp] theorem divInd_sq (d m : ℕ) : divInd d m ^ 2 = divInd d m := by
  rcases divInd_eq_zero_or_one d m with h | h <;> simp [h]

@[simp] theorem valuation_zero_right (p : ℕ) : valuation p 0 = 0 := by
  simp [valuation]

theorem valuation_eq_zero_of_not_prime {p m : ℕ} (hp : ¬p.Prime) :
    valuation p m = 0 := by
  simp [valuation, Nat.factorization_eq_zero_of_not_prime m hp]

@[simp] theorem valuation_zero_left (m : ℕ) : valuation 0 m = 0 := by
  exact valuation_eq_zero_of_not_prime Nat.not_prime_zero

@[simp] theorem valuation_one_left (m : ℕ) : valuation 1 m = 0 := by
  exact valuation_eq_zero_of_not_prime Nat.not_prime_one

theorem valuation_nonneg (p m : ℕ) : 0 ≤ valuation p m := by
  simp [valuation]

/-- At the excluded sample point `m = 0`, actual divisibility and Mathlib's
total factorization convention make `J_p = -1`.  All probabilistic cells in
the paper contain positive integers, and every positivity theorem below
therefore states that hypothesis explicitly. -/
@[simp] theorem higherValuation_zero_right (p : ℕ) :
    higherValuation p 0 = -1 := by
  simp [higherValuation]



@[simp] theorem mem_positiveExponents {M k : ℕ} :
    k ∈ positiveExponents M ↔ 1 ≤ k ∧ k ≤ M := by
  simp [positiveExponents]

@[simp] theorem mem_highExponents {M k : ℕ} :
    k ∈ highExponents M ↔ 2 ≤ k ∧ k ≤ M := by
  simp [highExponents]































/-- The pointwise identity `I_p J_p = J_p`.  It remains true at all total
boundary values; primality is not needed for this algebraic identity. -/
@[simp] theorem divInd_mul_higherValuation (p m : ℕ) :
    divInd p m * higherValuation p m = higherValuation p m := by
  by_cases hdvd : p ∣ m
  · simp [divInd, higherValuation, hdvd]
  · have hfac : m.factorization p = 0 :=
      Nat.factorization_eq_zero_of_not_dvd hdvd
    simp [divInd, higherValuation, valuation, hdvd, hfac]

end Erdos390.Full.ArithmeticModel

end

/- Source module: Erdos390.Full.HeadPattern -/
section


/-!
# Exact finite head patterns

The marked smooth cells impose exact valuations at finitely many fixed head
primes.  This file records their integral factor and modulus and proves the
factorization/divisibility part of the inclusion--exclusion reduction.  No
smooth-number asymptotic is used here.
-/

open scoped BigOperators
open scoped ArithmeticFunction.Moebius

namespace Erdos390.Full.HeadPattern



namespace Pattern

variable (P : Pattern)

/-- The forced prime-power part `h_e`. -/
def factor : ℕ :=
  ∏ p ∈ P.primes, p ^ P.exponent p

/-- The squarefree head modulus `M_H`. -/
def modulus : ℕ :=
  ∏ p ∈ P.primes, p

















/-- The exact Möbius weight detecting that an integer is coprime to the head
modulus.  Writing it through the gcd avoids any asymptotic or probabilistic
input. -/
def coprimeWeight (k : ℕ) : ℤ :=
  ∑ d ∈ (Nat.gcd k P.modulus).divisors, ArithmeticFunction.moebius d

theorem sum_moebius_divisors {a : ℕ} :
    (∑ d ∈ a.divisors, ArithmeticFunction.moebius d) =
      if a = 1 then 1 else 0 := by
  calc
    (∑ d ∈ a.divisors, ArithmeticFunction.moebius d) =
        (ArithmeticFunction.moebius * ArithmeticFunction.zeta :
          ArithmeticFunction ℤ) a := by
            symm
            simp only [ArithmeticFunction.coe_mul_zeta_apply]
    _ = (1 : ArithmeticFunction ℤ) a := by
      rw [ArithmeticFunction.moebius_mul_coe_zeta]
    _ = if a = 1 then 1 else 0 := by
      rw [ArithmeticFunction.one_apply]



@[simp]
theorem coprimeWeight_eq_indicator (k : ℕ) :
    P.coprimeWeight k = if Nat.Coprime k P.modulus then 1 else 0 := by
  rw [coprimeWeight, sum_moebius_divisors]







end Pattern

end Erdos390.Full.HeadPattern

end

/- Source module: Erdos390.Full.StructuredCells -/
section


/-!
# Exact structured-cell reductions

This file defines the actual finite smooth intervals and exact head cells.
It proves the finite inclusion--exclusion identity before any use of a
friable-number asymptotic.
-/

open scoped BigOperators ArithmeticFunction.Moebius

namespace Erdos390.Full.StructuredCells

open HeadPattern

/-- Positive `y`-smooth integers in `(lo, hi]`.  Mathlib's
`smoothNumbers (y+1)` means that all prime factors are at most `y`. -/
def smoothInterval (lo hi y : ℕ) : Finset ℕ :=
  (Nat.smoothNumbersUpTo hi (y + 1)).filter (lo < ·)

/-- The exact finite friable counting function used in this development;
`psi X y` counts positive integers at most `X` whose prime factors are at
most `y`. -/
def psi (X y : ℕ) : ℕ :=
  (Nat.smoothNumbersUpTo X (y + 1)).card

/-- A finite structured cell with exact head valuations. -/
def structuredCell (P : Pattern) (lo hi y : ℕ) : Finset ℕ :=
  (smoothInterval lo hi y).filter P.Matches

/-- The part of a structured cell divisible by a marked integer `d`. -/
def markedCell (P : Pattern) (lo hi y d : ℕ) : Finset ℕ :=
  (structuredCell P lo hi y).filter (d ∣ ·)

@[simp] theorem mem_smoothInterval {lo hi y m : ℕ} :
    m ∈ smoothInterval lo hi y ↔
      lo < m ∧ m ≤ hi ∧ m ∈ Nat.smoothNumbers (y + 1) := by
  simp only [smoothInterval, Finset.mem_filter, Nat.mem_smoothNumbersUpTo]
  tauto

theorem pos_of_mem_smoothInterval {lo hi y m : ℕ}
    (hm : m ∈ smoothInterval lo hi y) : 0 < m := by
  exact (Nat.mem_smoothNumbers.mp (mem_smoothInterval.mp hm).2.2).1.bot_lt





@[simp] theorem mem_structuredCell {P : Pattern} {lo hi y m : ℕ} :
    m ∈ structuredCell P lo hi y ↔
      m ∈ smoothInterval lo hi y ∧ P.Matches m := by
  simp [structuredCell]

@[simp] theorem mem_markedCell {P : Pattern} {lo hi y d m : ℕ} :
    m ∈ markedCell P lo hi y d ↔
      m ∈ smoothInterval lo hi y ∧ P.Matches m ∧ d ∣ m := by
  simp [markedCell, structuredCell, and_assoc, and_left_comm]

















end Erdos390.Full.StructuredCells

end

/- Source module: Erdos390.Full.Scale -/
section


/-!
# Exact scale identities

This file records the elementary identities behind the fixed choice
`y = n^(2/9)` used throughout the marked and smooth bridge.  Keeping them as
equalities avoids repeatedly treating `U = 9/2` and the four-mark cofactor
margin as informal arithmetic simplifications.
-/

open Filter Topology

namespace Erdos390.Full.Scale

open ArithmeticModel

/-- The logarithmic scale `L = log n`. -/
noncomputable def L (n : ℕ) : ℝ :=
  Real.log (n : ℝ)





theorem L_pos {n : ℕ} (hn : 1 < n) : 0 < L n := by
  exact Real.log_pos (by exact_mod_cast hn)

theorem y_pos {n : ℕ} (hn : 0 < n) : 0 < y n := by
  exact Real.rpow_pos_of_pos (by exact_mod_cast hn) _

/-- `log y = (2/9) log n`, with no asymptotic error. -/
theorem log_y {n : ℕ} (hn : 0 < n) :
    Real.log (y n) = (2 / 9 : ℝ) * L n := by
  simpa [y, L] using
    (Real.log_rpow (by exact_mod_cast hn : (0 : ℝ) < n) (2 / 9 : ℝ))





theorem y_pow_two (n : ℕ) :
    y n ^ 2 = (n : ℝ) ^ (4 / 9 : ℝ) := by
  calc
    y n ^ 2 = ((n : ℝ) ^ (2 / 9 : ℝ)) ^ 2 := rfl
    _ = (n : ℝ) ^ ((2 / 9 : ℝ) * (2 : ℕ)) :=
      (Real.rpow_mul_natCast (Nat.cast_nonneg n) (2 / 9 : ℝ) 2).symm
    _ = (n : ℝ) ^ (4 / 9 : ℝ) := by norm_num





/-- The accumulated two-mark endpoint scale used in the row estimates. -/
noncomputable def endpointRatio (n : ℕ) : ℝ :=
  y n ^ 2 * L n ^ 2 / (n : ℝ)





end Erdos390.Full.Scale

end

/- Source module: Erdos390.Full.FriableAsymptotic -/
section


/-!
# Friable counts on the Erdős 390 scale

This file develops the arithmetic side of the uniform Dickman estimate from
Mathlib's actual finite set `Nat.smoothNumbersUpTo`.  In particular, the
statements below are exact finite identities: no asymptotic count is exposed
as a hypothesis or a contract.

Our second parameter `y` is inclusive: `friableCount X y` counts positive
integers at most `X` all of whose prime factors are at most `y`.  Thus the
underlying Mathlib smoothness threshold is `y + 1`.
-/

open scoped BigOperators
open Filter Topology Asymptotics

namespace Erdos390.Full.FriableAsymptotic

open ArithmeticModel DickmanBasic Scale

/-! ## Quantitative prime-number input

The finite de Bruijn induction below needs a power-saving prime Stieltjes
estimate.  We derive it here from the audited medium-strength PNT, rather than
postulating a prime-distribution contract.  The exponential PNT remainder is
stronger than every fixed power of `1 / log x`.
-/

/-- The medium PNT error for `ψ` is `O(x / log(x)^A)` for every fixed real
power `A`. -/
theorem psi_error_isBigO_log_power (A : ℝ) :
    (Chebyshev.psi - id) =O[atTop]
      (fun x : ℝ => x / (Real.log x) ^ A) := by
  obtain ⟨c, hc, hPNT⟩ := MediumPNT
  apply hPNT.trans
  have hpow : Tendsto (fun x : ℝ => (Real.log x) ^ ((1 : ℝ) / 10))
      atTop atTop := by
    exact tendsto_rpow_atTop (by norm_num : (0 : ℝ) < 1 / 10) |>.comp
      Real.tendsto_log_atTop
  have hdecay :=
    (isLittleO_exp_neg_mul_rpow_atTop hc (-10 * A)).comp_tendsto hpow
  have hdecay' :
      (fun x : ℝ => Real.exp (-c * (Real.log x) ^ ((1 : ℝ) / 10)))
        =O[atTop] (fun x : ℝ => (Real.log x) ^ (-A)) := by
    apply hdecay.isBigO.congr' (Eventually.of_forall fun x => rfl)
    filter_upwards [Real.tendsto_log_atTop.eventually (eventually_ge_atTop (0 : ℝ))]
      with x hx
    simp only [Function.comp_apply]
    rw [← Real.rpow_mul hx]
    congr 2
    ring
  have hmul := (isBigO_refl (fun x : ℝ => x) atTop).mul hdecay'
  apply hmul.congr' (Eventually.of_forall fun x => rfl)
  filter_upwards [Real.tendsto_log_atTop.eventually (eventually_gt_atTop (0 : ℝ))]
    with x hx
  rw [Real.rpow_neg hx.le, div_eq_mul_inv]

/-- The prime-power correction `√x log x` is smaller than the same arbitrary
fixed logarithmic PNT scale. -/
theorem sqrt_mul_log_isBigO_log_power (A : ℝ) :
    (fun x : ℝ => Real.sqrt x * Real.log x) =O[atTop]
      (fun x : ℝ => x / (Real.log x) ^ A) := by
  have hlog :=
    (isLittleO_log_rpow_rpow_atTop (A + 1) (by norm_num : (0 : ℝ) < 1 / 2)).isBigO
  let common : ℝ → ℝ := fun x => x ^ ((1 : ℝ) / 2) / (Real.log x) ^ A
  have hmul := hlog.mul (isBigO_refl common atTop)
  apply hmul.congr'
  · filter_upwards [Real.tendsto_log_atTop.eventually (eventually_gt_atTop (0 : ℝ)),
      eventually_gt_atTop (0 : ℝ)] with x hlogx hx
    dsimp [common]
    rw [Real.sqrt_eq_rpow, Real.rpow_add hlogx]
    field_simp
    simp [Real.rpow_one]
  · filter_upwards [Real.tendsto_log_atTop.eventually (eventually_gt_atTop (0 : ℝ)),
      eventually_gt_atTop (0 : ℝ)] with x hlogx hx
    dsimp [common]
    calc
      x ^ ((1 : ℝ) / 2) * (x ^ ((1 : ℝ) / 2) / Real.log x ^ A) =
          (x ^ ((1 : ℝ) / 2) * x ^ ((1 : ℝ) / 2)) / Real.log x ^ A := by ring
      _ = x ^ ((1 : ℝ) / 2 + (1 : ℝ) / 2) / Real.log x ^ A := by
        rw [Real.rpow_add hx]
      _ = x / Real.log x ^ A := by norm_num [Real.rpow_one]

/-- Quantitative PNT for Chebyshev's `θ`, at every fixed logarithmic power.
This is the form used by partial summation over primes. -/
theorem theta_error_isBigO_log_power (A : ℝ) :
    (Chebyshev.theta - id) =O[atTop]
      (fun x : ℝ => x / (Real.log x) ^ A) := by
  have hraw :
      (Chebyshev.theta - Chebyshev.psi) =O[atTop]
        (fun x : ℝ => Real.sqrt x * Real.log x) := by
    apply IsBigO.of_bound 2
    filter_upwards [eventually_ge_atTop (1 : ℝ)] with x hx
    have hnonneg : 0 ≤ Real.sqrt x * Real.log x :=
      mul_nonneg (Real.sqrt_nonneg x) (Real.log_nonneg hx)
    simpa only [Pi.sub_apply, Real.norm_eq_abs, abs_sub_comm,
      Real.norm_of_nonneg hnonneg, mul_assoc] using
      Chebyshev.abs_psi_sub_theta_le_sqrt_mul_log hx
  have hdiff := hraw.trans (sqrt_mul_log_isBigO_log_power A)
  have hadd := hdiff.add (psi_error_isBigO_log_power A)
  apply hadd.congr' (Eventually.of_forall fun x => ?_)
    (Eventually.of_forall fun x => rfl)
  simp only [Pi.sub_apply, id_eq]
  ring

/-- The actual finite Chebyshev prime sum, written with the same
`Nat.primesBelow` finset used by the friable recurrence. -/
noncomputable def primeLogSumUpTo (X : ℕ) : ℝ :=
  ∑ p ∈ (X + 1).primesBelow, Real.log (p : ℝ)

theorem primeLogSumUpTo_eq_theta (X : ℕ) :
    primeLogSumUpTo X = Chebyshev.theta (X : ℝ) := by
  have hfin : (X + 1).primesBelow = (Finset.Ioc 0 X).filter Nat.Prime := by
    ext p
    simp only [Nat.mem_primesBelow, Finset.mem_filter, Finset.mem_Ioc]
    constructor
    · rintro ⟨hpX, hp⟩
      exact ⟨⟨hp.pos, by omega⟩, hp⟩
    · rintro ⟨⟨_, hpX⟩, hp⟩
      exact ⟨by omega, hp⟩
  rw [primeLogSumUpTo, hfin, Chebyshev.theta]
  simp only [Nat.floor_natCast]

/-- Quantitative PNT for the precise natural-number prime finset used below.
-/
theorem primeLogSumUpTo_error_isBigO_log_power (A : ℝ) :
    (fun X : ℕ => primeLogSumUpTo X - (X : ℝ)) =O[atTop]
      (fun X : ℕ => (X : ℝ) / (Real.log (X : ℝ)) ^ A) := by
  have h := (theta_error_isBigO_log_power A).comp_tendsto
    tendsto_natCast_atTop_atTop
  apply h.congr' (Eventually.of_forall fun X => ?_)
    (Eventually.of_forall fun X => rfl)
  simp only [Function.comp_apply, Pi.sub_apply, id_eq, primeLogSumUpTo_eq_theta]

/-- An explicit eventual inequality extracted from the quantitative PNT.  In
particular, later uses may choose the constant and threshold before starting
the finite de Bruijn induction. -/
theorem exists_primeLogSumUpTo_error_bound (A : ℝ) :
    ∃ C : ℝ, 0 < C ∧ ∃ X₀ : ℕ, ∀ X, X₀ ≤ X →
      |primeLogSumUpTo X - (X : ℝ)| ≤
        C * ((X : ℝ) / (Real.log (X : ℝ)) ^ A) := by
  obtain ⟨C, hC, hbound⟩ :=
    (primeLogSumUpTo_error_isBigO_log_power A).exists_pos
  rw [IsBigOWith, eventually_atTop] at hbound
  obtain ⟨X₀, hX₀⟩ := hbound
  refine ⟨C, hC, max X₀ 2, fun X hX => ?_⟩
  have hXX₀ : X₀ ≤ X := (le_max_left X₀ 2).trans hX
  have hX2 : 2 ≤ X := (le_max_right X₀ 2).trans hX
  have htarget : 0 ≤ (X : ℝ) / (Real.log (X : ℝ)) ^ A := by
    positivity
  simpa only [Real.norm_eq_abs, Real.norm_of_nonneg htarget] using
    hX₀ X hXX₀

/-- The logarithmically weighted prime sum on the half-open natural interval
`z < p ≤ y`. -/
noncomputable def primeLogSumInterval (z y : ℕ) : ℝ :=
  ∑ p ∈ (y + 1).primesBelow \ (z + 1).primesBelow,
    Real.log (p : ℝ)





/-- The point mass of the Chebyshev `θ` measure at an integer. -/
noncomputable def primeLogIncrement (m : ℕ) : ℝ :=
  if m.Prime then Real.log (m : ℝ) else 0

theorem sum_range_primeLogIncrement (n : ℕ) :
    ∑ m ∈ Finset.range (n + 1), primeLogIncrement m =
      primeLogSumUpTo n := by
  rw [primeLogSumUpTo]
  simp only [primeLogIncrement]
  rw [← Finset.sum_filter]
  apply Finset.sum_congr
  · ext p
    simp only [Finset.mem_filter, Finset.mem_range, Nat.mem_primesBelow]
  · intro p hp
    rfl

/-- A prime Stieltjes sum, still written as a finite sum. -/
noncomputable def primeThetaWeightedInterval (f : ℕ → ℝ) (z y : ℕ) : ℝ :=
  ∑ p ∈ (y + 1).primesBelow \ (z + 1).primesBelow,
    f p * Real.log (p : ℝ)

theorem primeThetaWeightedInterval_eq_Ioc (f : ℕ → ℝ) {z y : ℕ} :
    primeThetaWeightedInterval f z y =
      ∑ m ∈ Finset.Ioc z y, f m * primeLogIncrement m := by
  rw [primeThetaWeightedInterval]
  simp only [primeLogIncrement, mul_ite, mul_zero]
  rw [← Finset.sum_filter]
  apply Finset.sum_congr
  · ext p
    simp only [Finset.mem_sdiff, Nat.mem_primesBelow,
      Finset.mem_filter, Finset.mem_Ioc]
    constructor
    · rintro ⟨⟨hpy, hp⟩, hpz⟩
      refine ⟨⟨?_, by omega⟩, hp⟩
      by_contra hnot
      apply hpz
      exact ⟨by omega, hp⟩
    · rintro ⟨⟨hzp, hpy⟩, hp⟩
      exact ⟨⟨by omega, hp⟩, by omega⟩
  · intro p hp
    rfl

/-- Exact Abel summation for a prime Stieltjes sum.  No asymptotic statement
is hidden here: the cumulative mass is the actual finite `primeLogSumUpTo`.
This is the arithmetic identity to which the quantitative PNT error is
applied. -/
theorem primeThetaWeightedInterval_by_parts (f : ℕ → ℝ) {z y : ℕ}
    (hzy : z < y) :
    primeThetaWeightedInterval f z y =
      f y * primeLogSumUpTo y - f (z + 1) * primeLogSumUpTo z -
        ∑ m ∈ Finset.Ioc z (y - 1),
          (f (m + 1) - f m) * primeLogSumUpTo m := by
  rw [primeThetaWeightedInterval_eq_Ioc f]
  have hparts := Finset.sum_Ioc_by_parts f primeLogIncrement hzy
  simpa only [smul_eq_mul, sum_range_primeLogIncrement,
    Nat.add_sub_cancel] using hparts

/-- Abel's main term after replacing the cumulative prime mass `θ(m)` by its
PNT main term `m`.  This is deliberately kept discrete; the subsequent
comparison with a Dickman integral is a separate Riemann-sum step. -/
noncomputable def integerAbelMain (f : ℕ → ℝ) (z y : ℕ) : ℝ :=
  f y * (y : ℝ) - f (z + 1) * (z : ℝ) -
    ∑ m ∈ Finset.Ioc z (y - 1),
      (f (m + 1) - f m) * (m : ℝ)

/-- Unit mass on the positive integers.  Its cumulative mass through `n` is
exactly `n`, matching the main term of `θ(n)`. -/
def positiveIncrement (m : ℕ) : ℝ := if m = 0 then 0 else 1

theorem sum_range_positiveIncrement (n : ℕ) :
    ∑ m ∈ Finset.range (n + 1), positiveIncrement m = (n : ℝ) := by
  induction n with
  | zero => simp [positiveIncrement]
  | succ n ih =>
      rw [Finset.sum_range_succ, ih]
      simp [positiveIncrement]

/-- Consequently the discrete Abel main term is just the ordinary integer
Riemann sum of the test weight. -/
theorem integerAbelMain_eq_sum_Ioc (f : ℕ → ℝ) {z y : ℕ}
    (hzy : z < y) :
    integerAbelMain f z y = ∑ m ∈ Finset.Ioc z y, f m := by
  have hparts := Finset.sum_Ioc_by_parts f positiveIncrement hzy
  have hlhs : (∑ m ∈ Finset.Ioc z y, f m * positiveIncrement m) =
      ∑ m ∈ Finset.Ioc z y, f m := by
    apply Finset.sum_congr rfl
    intro m hm
    have hm0 : m ≠ 0 := by
      rw [Finset.mem_Ioc] at hm
      omega
    simp [positiveIncrement, hm0]
  simp only [smul_eq_mul] at hparts
  rw [hlhs] at hparts
  simpa only [sum_range_positiveIncrement,
    Nat.add_sub_cancel, integerAbelMain, mul_one] using hparts.symm

/-- Shift a sum over `z < m ≤ y` to the adjacent unit cells
`z ≤ k < y`. -/
theorem sum_Ioc_shift (f : ℕ → ℝ) {z y : ℕ} :
    ∑ m ∈ Finset.Ioc z y, f m =
      ∑ k ∈ Finset.Ico z y, f (k + 1) := by
  have hfin : (Finset.Ico z y).image (fun k => k + 1) =
      Finset.Ioc z y := by
    ext m
    simp only [Finset.mem_image, Finset.mem_Ico, Finset.mem_Ioc]
    constructor
    · rintro ⟨k, ⟨hzk, hky⟩, rfl⟩
      omega
    · rintro ⟨hzm, hmy⟩
      refine ⟨m - 1, ?_, by omega⟩
      omega
  have hinj : Set.InjOn (fun k : ℕ => k + 1)
      (Finset.Ico z y : Set ℕ) := by
    intro a ha b hb hab
    exact Nat.add_right_cancel hab
  rw [← hfin, Finset.sum_image hinj]

/-- Exact decomposition of an integer Riemann-sum error into its unit-cell
errors. -/
theorem sum_sub_integral_identity (f : ℝ → ℝ) {z y : ℕ}
    (hzy : z ≤ y)
    (hint : ∀ k ∈ Set.Ico z y,
      IntervalIntegrable f MeasureTheory.volume (k : ℝ) (k + 1 : ℕ)) :
    (∑ m ∈ Finset.Ioc z y, f m) -
        ∫ t in (z : ℝ)..(y : ℝ), f t =
      ∑ k ∈ Finset.Ico z y,
        (f (k + 1) - ∫ t in (k : ℝ)..(k + 1 : ℕ), f t) := by
  rw [sum_Ioc_shift (fun m => f m)]
  have hintsum := intervalIntegral.sum_integral_adjacent_intervals_Ico
    (f := f) (μ := MeasureTheory.volume)
    (a := fun k : ℕ => (k : ℝ)) hzy hint
  simp only [Nat.cast_add, Nat.cast_one] at hintsum
  simp only [Nat.cast_add, Nat.cast_one]
  rw [← hintsum, Finset.sum_sub_distrib]

/-- A quantitative Riemann-sum estimate from a supplied oscillation on each
unit cell.  Later the one-Lipschitz Dickman bound supplies `hosc` for the
specific de Bruijn kernel. -/
theorem sum_integral_error_bound (f : ℝ → ℝ) (e : ℕ → ℝ)
    {z y : ℕ} (hzy : z ≤ y)
    (hint : ∀ k ∈ Set.Ico z y,
      IntervalIntegrable f MeasureTheory.volume (k : ℝ) (k + 1 : ℕ))
    (hosc : ∀ k ∈ Finset.Ico z y,
      ∀ t ∈ Set.Icc (k : ℝ) (k + 1 : ℕ),
        |f (k + 1) - f t| ≤ e k) :
    |(∑ m ∈ Finset.Ioc z y, f m) -
        ∫ t in (z : ℝ)..(y : ℝ), f t| ≤
      ∑ k ∈ Finset.Ico z y, e k := by
  rw [sum_sub_integral_identity f hzy hint]
  calc
    |∑ k ∈ Finset.Ico z y,
        (f (k + 1) - ∫ t in (k : ℝ)..(k + 1 : ℕ), f t)| ≤
        ∑ k ∈ Finset.Ico z y,
          |f (k + 1) - ∫ t in (k : ℝ)..(k + 1 : ℕ), f t| :=
      Finset.abs_sum_le_sum_abs _ _
    _ ≤ ∑ k ∈ Finset.Ico z y, e k := by
      apply Finset.sum_le_sum
      intro k hk
      have hconst :
          (∫ _t in (k : ℝ)..(k + 1 : ℕ), f (k + 1)) =
            f (k + 1) := by
        simp
      rw [← hconst, ← intervalIntegral.integral_sub
        (continuous_const.intervalIntegrable _ _)
        (hint k (by simpa using hk))]
      have hnorm := intervalIntegral.norm_integral_le_of_norm_le_const
        (f := fun t => f (k + 1) - f t) (C := e k)
        (a := (k : ℝ)) (b := (k + 1 : ℕ)) (fun t ht => ?_)
      · simpa only [Real.norm_eq_abs, Nat.cast_add, Nat.cast_one,
          add_sub_cancel_left, abs_one, mul_one] using hnorm
      · apply hosc k hk t
        have ht' := Set.uIoc_subset_uIcc ht
        rw [Set.uIcc_of_le
          (by norm_num : (k : ℝ) ≤ (k + 1 : ℕ))] at ht'
        exact ht'

/-- Exact decomposition of the prime Stieltjes error into endpoint errors and
the discrete variation of the test weight. -/
theorem primeThetaWeightedInterval_error_identity (f : ℕ → ℝ) {z y : ℕ}
    (hzy : z < y) :
    primeThetaWeightedInterval f z y - integerAbelMain f z y =
      f y * (primeLogSumUpTo y - (y : ℝ)) -
      f (z + 1) * (primeLogSumUpTo z - (z : ℝ)) -
      ∑ m ∈ Finset.Ioc z (y - 1),
        (f (m + 1) - f m) *
          (primeLogSumUpTo m - (m : ℝ)) := by
  rw [primeThetaWeightedInterval_by_parts f hzy]
  simp only [integerAbelMain]
  simp_rw [mul_sub]
  rw [Finset.sum_sub_distrib]
  ring

/-- Total-variation estimate for the prime Stieltjes error.  Together with
`exists_primeLogSumUpTo_error_bound`, this is a quantitative and fully finite
partial-summation interface; no prime asymptotic is assumed by callers. -/
theorem primeThetaWeightedInterval_error_bound (f : ℕ → ℝ) {z y : ℕ}
    (hzy : z < y) :
    |primeThetaWeightedInterval f z y - integerAbelMain f z y| ≤
      |f y| * |primeLogSumUpTo y - (y : ℝ)| +
      |f (z + 1)| * |primeLogSumUpTo z - (z : ℝ)| +
      ∑ m ∈ Finset.Ioc z (y - 1),
        |f (m + 1) - f m| *
          |primeLogSumUpTo m - (m : ℝ)| := by
  rw [primeThetaWeightedInterval_error_identity f hzy]
  calc
    |f y * (primeLogSumUpTo y - (y : ℝ)) -
        f (z + 1) * (primeLogSumUpTo z - (z : ℝ)) -
        ∑ m ∈ Finset.Ioc z (y - 1),
          (f (m + 1) - f m) *
            (primeLogSumUpTo m - (m : ℝ))| ≤
      |f y * (primeLogSumUpTo y - (y : ℝ)) -
        f (z + 1) * (primeLogSumUpTo z - (z : ℝ))| +
        |∑ m ∈ Finset.Ioc z (y - 1),
          (f (m + 1) - f m) *
            (primeLogSumUpTo m - (m : ℝ))| := abs_sub _ _
    _ ≤ (|f y * (primeLogSumUpTo y - (y : ℝ))| +
        |f (z + 1) * (primeLogSumUpTo z - (z : ℝ))|) +
        ∑ m ∈ Finset.Ioc z (y - 1),
          |(f (m + 1) - f m) *
            (primeLogSumUpTo m - (m : ℝ))| := by
      apply add_le_add
      · exact abs_sub _ _
      · exact Finset.abs_sum_le_sum_abs _ _
    _ = _ := by simp only [abs_mul]

/-- The preceding variation estimate with the quantitative PNT substituted at
every endpoint.  The single threshold `X₀` is uniform over the whole finite
prime interval. -/
theorem primeThetaWeightedInterval_pnt_bound (f : ℕ → ℝ)
    {A C : ℝ} {X₀ z y : ℕ}
    (hbound : ∀ X, X₀ ≤ X →
      |primeLogSumUpTo X - (X : ℝ)| ≤
        C * ((X : ℝ) / (Real.log (X : ℝ)) ^ A))
    (hz : X₀ ≤ z) (hzy : z < y) :
    |primeThetaWeightedInterval f z y - integerAbelMain f z y| ≤
      |f y| * (C * ((y : ℝ) / (Real.log (y : ℝ)) ^ A)) +
      |f (z + 1)| * (C * ((z : ℝ) / (Real.log (z : ℝ)) ^ A)) +
      ∑ m ∈ Finset.Ioc z (y - 1),
        |f (m + 1) - f m| *
          (C * ((m : ℝ) / (Real.log (m : ℝ)) ^ A)) := by
  calc
    |primeThetaWeightedInterval f z y - integerAbelMain f z y| ≤
        |f y| * |primeLogSumUpTo y - (y : ℝ)| +
        |f (z + 1)| * |primeLogSumUpTo z - (z : ℝ)| +
        ∑ m ∈ Finset.Ioc z (y - 1),
          |f (m + 1) - f m| *
            |primeLogSumUpTo m - (m : ℝ)| :=
      primeThetaWeightedInterval_error_bound f hzy
    _ ≤ _ := by
      apply add_le_add
      · apply add_le_add
        · exact mul_le_mul_of_nonneg_left
            (hbound y (hz.trans hzy.le)) (abs_nonneg (f y))
        · exact mul_le_mul_of_nonneg_left
            (hbound z hz) (abs_nonneg (f (z + 1)))
      · apply Finset.sum_le_sum
        intro m hm
        apply mul_le_mul_of_nonneg_left
        · apply hbound m
          exact hz.trans (Nat.le_of_lt (Finset.mem_Ioc.mp hm).1)
        · exact abs_nonneg (f (m + 1) - f m)

/-! ## Explicit regularity of the finite Dickman kernel -/

/-- The method-of-steps Dickman solution is at most one on the full compact
range used by the friable induction. -/
theorem rho_le_one_of_le_five {x : ℝ} (hx5 : x ≤ 5) :
    rho x ≤ 1 := by
  by_cases hx1 : x ≤ 1
  · rw [rho_eq_one_of_le_one hx1]
  · have hxmem : x ∈ Set.Icc (1 : ℝ) 5 :=
      ⟨le_of_not_ge hx1, hx5⟩
    have hone : (1 : ℝ) ∈ Set.Icc (1 : ℝ) 5 := by norm_num
    simpa [rho_one] using
      antitoneOn_rho_one_five hone hxmem hxmem.1

/-- On `[1,5]` the delay-equation integrand lies in `[0,1]`. -/
theorem rho_delay_integrand_bounds {t : ℝ} (ht1 : 1 ≤ t) (ht5 : t ≤ 5) :
    0 ≤ rho (t - 1) / t ∧ rho (t - 1) / t ≤ 1 := by
  have hrpos : 0 < rho (t - 1) :=
    rho_pos_on_zero_five (by linarith) (by linarith)
  have hrle : rho (t - 1) ≤ 1 :=
    rho_le_one_of_le_five (by linarith)
  constructor
  · exact div_nonneg hrpos.le (by linarith)
  · apply (div_le_one (by linarith : 0 < t)).mpr
    exact hrle.trans ht1

/-- Explicit one-Lipschitz estimate for `rho` to the right of its initial
corner.  The proof integrates the actual delay equation, so no smoothness at
the corner `1` is assumed. -/
theorem rho_lipschitz_one_five {a b : ℝ}
    (ha : 1 ≤ a) (hab : a ≤ b) (hb : b ≤ 5) :
    |rho b - rho a| ≤ b - a := by
  let g : ℝ → ℝ := fun t => rho (t - 1) / t
  have hgcont : ContinuousOn g (Set.Icc a b) := by
    apply ContinuousOn.div
    · exact (continuous_rho.comp
        (continuous_id.sub continuous_const)).continuousOn
    · exact continuous_id.continuousOn
    · intro t ht
      exact ne_of_gt (by linarith [ha, ht.1])
  have hgint_ab : IntervalIntegrable g MeasureTheory.volume a b :=
    by
      rw [← Set.uIcc_of_le hab] at hgcont
      exact hgcont.intervalIntegrable
  have hgint_1a : IntervalIntegrable g MeasureTheory.volume 1 a := by
    have hgcont_1a : ContinuousOn g (Set.Icc (1 : ℝ) a) := by
      apply ContinuousOn.div
      · exact (continuous_rho.comp
          (continuous_id.sub continuous_const)).continuousOn
      · exact continuous_id.continuousOn
      · intro t ht
        exact ne_of_gt (by linarith [ht.1])
    rw [← Set.uIcc_of_le ha] at hgcont_1a
    exact hgcont_1a.intervalIntegrable
  have hadd := intervalIntegral.integral_add_adjacent_intervals
    hgint_1a hgint_ab
  have hrhoa := rho_integral_eq ha (hab.trans hb)
  have hrhob := rho_integral_eq (ha.trans hab) hb
  have hdiff : rho b - rho a = -(∫ t in a..b, g t) := by
    dsimp [g] at hadd ⊢
    linarith
  rw [hdiff, abs_neg]
  have hnonneg : 0 ≤ ∫ t in a..b, g t := by
    apply intervalIntegral.integral_nonneg hab
    intro t ht
    exact (rho_delay_integrand_bounds
      (ha.trans ht.1) (ht.2.trans hb)).1
  rw [abs_of_nonneg hnonneg]
  have hmono := intervalIntegral.integral_mono_on hab hgint_ab
    (continuous_const.intervalIntegrable a b)
    (fun t ht => (rho_delay_integrand_bounds
      (ha.trans ht.1) (ht.2.trans hb)).2)
  simpa using hmono

/-- Global ordered one-Lipschitz estimate on `[0,5]`, including intervals
that cross the corner at `1`. -/
theorem rho_lipschitz_of_le_five {a b : ℝ}
    (hab : a ≤ b) (hb : b ≤ 5) :
    |rho b - rho a| ≤ b - a := by
  by_cases hb1 : b ≤ 1
  · rw [rho_eq_one_of_le_one hb1,
      rho_eq_one_of_le_one (hab.trans hb1)]
    simp only [sub_self, abs_zero]
    linarith
  by_cases ha1 : 1 ≤ a
  · exact rho_lipschitz_one_five ha1 hab hb
  · have hcross := rho_lipschitz_one_five
      (a := (1 : ℝ)) (b := b) le_rfl (le_of_not_ge hb1) hb
    rw [rho_eq_one_of_le_one (le_of_not_ge ha1)]
    simpa only [rho_one] using hcross.trans (by linarith)

/-! ## The Dickman test weight in the prime recurrence -/

/-- The real main term contributed by a prime `p` in the largest-prime
recurrence. -/
noncomputable def dickmanPrimeSummand (X m : ℕ) : ℝ :=
  (X : ℝ) / (m : ℝ) *
    rho (Real.log (X : ℝ) / Real.log (m : ℝ) - 1)

/-- The corresponding test function against Chebyshev's `θ` measure. -/
noncomputable def dickmanThetaWeight (X m : ℕ) : ℝ :=
  dickmanPrimeSummand X m / Real.log (m : ℝ)

/-- Continuous extension of the Dickman test weight used for the ordinary
Riemann integral. -/
noncomputable def dickmanContinuousWeight (X t : ℝ) : ℝ :=
  (X / t * rho (Real.log X / Real.log t - 1)) / Real.log t

/-- An antiderivative of the Dickman test weight on every compact region
where the Dickman delay equation applies. -/
noncomputable def dickmanAntiderivative (X t : ℝ) : ℝ :=
  X * rho (Real.log X / Real.log t)











/-- The genuine finite friable counting function `Ψ(X,y)`. -/
def friableCount (X y : ℕ) : ℕ :=
  (Nat.smoothNumbersUpTo X (y + 1)).card

/-- The Dickman coordinate attached to integral endpoints.  It is total at
the irrelevant boundary `y = 0,1`; analytic statements impose `1 < y`. -/
noncomputable def dickmanU (X y : ℕ) : ℝ :=
  Real.log (X : ℝ) / Real.log (y : ℝ)

@[simp] theorem friableCount_zero (y : ℕ) : friableCount 0 y = 0 := by
  rw [friableCount]
  have hempty : Nat.smoothNumbersUpTo 0 (y + 1) = ∅ := by
    apply Finset.eq_empty_iff_forall_notMem.mpr
    intro m hm
    rw [Nat.mem_smoothNumbersUpTo] at hm
    have hm0 : m = 0 := by omega
    exact Nat.ne_zero_of_mem_smoothNumbers hm.2 hm0
  rw [hempty]
  simp





/-- Below the smoothness threshold every positive integer is smooth.  This is
the exact (not asymptotic) initial condition for the de Bruijn induction. -/
theorem friableCount_eq_self {X y : ℕ} (hXy : X ≤ y) :
    friableCount X y = X := by
  have hrough : Nat.roughNumbersUpTo X (y + 1) = ∅ := by
    apply Finset.eq_empty_iff_forall_notMem.mpr
    intro m hm
    rw [Nat.roughNumbersUpTo, Finset.mem_filter, Finset.mem_range] at hm
    exact hm.2.2 (Nat.mem_smoothNumbers_of_lt
      (Nat.pos_of_ne_zero hm.2.1) (by omega))
  have hcard := Nat.smoothNumbersUpTo_card_add_roughNumbersUpTo_card X (y + 1)
  rw [hrough] at hcard
  simpa [friableCount] using hcard

theorem dickmanU_le_one {X y : ℕ} (hX : 0 < X) (hy : 1 < y)
    (hXy : X ≤ y) : dickmanU X y ≤ 1 := by
  have hlogy : 0 < Real.log (y : ℝ) := Real.log_pos (by exact_mod_cast hy)
  have hlogxy : Real.log (X : ℝ) ≤ Real.log (y : ℝ) := by
    apply Real.log_le_log
    · exact_mod_cast hX
    · exact_mod_cast hXy
  rw [dickmanU, div_le_one hlogy]
  exact hlogxy



/-- The part of the `p`-smooth count divisible by the newly admitted prime
`p`. -/
def primeMultipleCell (X p : ℕ) : Finset ℕ :=
  (Nat.smoothNumbersUpTo X (p + 1)).filter (p ∣ ·)















/-! ## Explicit oscillation and Riemann error for the Dickman weight -/


open DickmanBasic

noncomputable def invMulLog (t : ℝ) : ℝ := 1 / (t * Real.log t)

noncomputable def logRatio (X t : ℝ) : ℝ := Real.log X / Real.log t





















theorem sum_Ico_inv_sq_le {z y : ℕ} (hz : 1 ≤ z) :
    (∑ k ∈ Finset.Ico z y, 1 / (k : ℝ) ^ 2) ≤ 2 / (z : ℝ) := by
  have hfin : Finset.Ico z y = Finset.Ioo (z - 1) y := by
    ext k
    simp only [Finset.mem_Ico, Finset.mem_Ioo]
    omega
  rw [hfin]
  have h := sum_Ioo_inv_sq_le (α := ℝ) (z - 1) y
  have hcast : (((z - 1 : ℕ) : ℝ) + 1) = (z : ℝ) := by
    rw [Nat.cast_sub hz]
    norm_num
  rw [hcast] at h
  simpa [one_div] using h




/-! ## Quantitative PNT transfer for the Dickman prime weight

The following estimates combine the preceding Chebyshev--Stieltjes identity,
the audited medium PNT, and the explicit Dickman-weight oscillation bound.
They are the one-prime analytic input for the finite de Bruijn induction.
-/





theorem harmonic_Ioc_le {z y : ℕ} :
    (∑ m ∈ Finset.Ioc z y, 1 / (m : ℝ)) ≤ 1 + Real.log (y : ℝ) := by
  calc
    (∑ m ∈ Finset.Ioc z y, 1 / (m : ℝ)) ≤
        ∑ m ∈ Finset.Icc 1 y, 1 / (m : ℝ) := by
      apply Finset.sum_le_sum_of_subset_of_nonneg
      · intro m hm
        rw [Finset.mem_Ioc] at hm
        rw [Finset.mem_Icc]
        exact ⟨by omega, hm.2⟩
      · intro m hm hmn
        positivity
    _ = (harmonic y : ℝ) := by
      rw [harmonic_eq_sum_Icc]
      simp only [Rat.cast_sum, Rat.cast_inv, Rat.cast_natCast, one_div]
    _ ≤ 1 + Real.log (y : ℝ) := by exact_mod_cast harmonic_le_one_add_log y












/-! ## Uniform floor stability for the recursive Dickman kernel -/






/-- Chebyshev test weight whose prime mass is `1 / (p log p)`. -/
noncomputable def invLogThetaWeight (m : ℕ) : ℝ :=
  1 / ((m : ℝ) * Real.log (m : ℝ) ^ 2)

noncomputable def invLogContinuousWeight (t : ℝ) : ℝ :=
  1 / (t * Real.log t ^ 2)

noncomputable def invLogAntiderivative (t : ℝ) : ℝ :=
  -(1 / Real.log t)

theorem invLogContinuousWeight_nat (m : ℕ) :
    invLogContinuousWeight (m : ℝ) = invLogThetaWeight m := rfl

theorem hasDerivAt_invLogContinuousWeight {t : ℝ} (ht : t ≠ 0)
    (hlog : Real.log t ≠ 0) :
    HasDerivAt invLogContinuousWeight
      (-(Real.log t + 2) / (t ^ 2 * Real.log t ^ 3)) t := by
  unfold invLogContinuousWeight
  have hlogsq : HasDerivAt (fun s : ℝ => Real.log s ^ 2)
      (2 * Real.log t / t) t := by
    convert (Real.hasDerivAt_log ht).pow 2 using 1
    field_simp [ht]
    ring
  have hden : HasDerivAt (fun s : ℝ => s * Real.log s ^ 2)
      (Real.log t ^ 2 + 2 * Real.log t) t := by
    convert (hasDerivAt_id t).mul hlogsq using 1
    simp only [id_eq]
    field_simp [ht]
  have hden0 : t * Real.log t ^ 2 ≠ 0 :=
    mul_ne_zero ht (pow_ne_zero 2 hlog)
  have hraw := (hasDerivAt_const t (1 : ℝ)).div hden hden0
  convert hraw using 1
  field_simp [ht, hlog]
  ring

theorem invLog_deriv_bound {k : ℕ} (hk : 2 ≤ k) {t : ℝ}
    (ht : (k : ℝ) ≤ t) :
    |-(Real.log t + 2) / (t ^ 2 * Real.log t ^ 3)| ≤
      20 / (k : ℝ) ^ 2 := by
  have hkR : (2 : ℝ) ≤ (k : ℝ) := by exact_mod_cast hk
  have ht2 : (2 : ℝ) ≤ t := hkR.trans ht
  have htpos : 0 < t := by linarith
  have hkpos : (0 : ℝ) < (k : ℝ) := by positivity
  have hloghalf : (1 / 2 : ℝ) ≤ Real.log t := by
    have hmono : Real.log 2 ≤ Real.log t :=
      Real.log_le_log (by norm_num) ht2
    nlinarith [Real.log_two_gt_d9]
  have hlogpos : 0 < Real.log t := by linarith
  have hpoly : (Real.log t + 2) / Real.log t ^ 3 ≤ 20 := by
    apply (div_le_iff₀ (pow_pos hlogpos 3)).2
    have hfac : 0 ≤ (2 * Real.log t - 1) *
        (10 * Real.log t ^ 2 + 5 * Real.log t + 2) := by
      apply mul_nonneg
      · linarith
      · nlinarith [sq_nonneg (Real.log t)]
    nlinarith
  rw [abs_div, abs_neg, abs_mul,
    abs_of_nonneg (sq_nonneg t), abs_of_pos (pow_pos hlogpos 3),
    abs_of_nonneg (by linarith : 0 ≤ Real.log t + 2)]
  calc
    (Real.log t + 2) / (t ^ 2 * Real.log t ^ 3) =
        ((Real.log t + 2) / Real.log t ^ 3) / t ^ 2 := by field_simp
    _ ≤ 20 / t ^ 2 := div_le_div_of_nonneg_right hpoly (sq_nonneg t)
    _ ≤ 20 / (k : ℝ) ^ 2 := by gcongr

theorem invLogWeight_cell_oscillation {k : ℕ} (hk : 2 ≤ k)
    {s t : ℝ} (hs : s ∈ Set.Icc (k : ℝ) (k + 1 : ℕ))
    (ht : t ∈ Set.Icc (k : ℝ) (k + 1 : ℕ)) :
    |invLogContinuousWeight s - invLogContinuousWeight t| ≤
      (20 / (k : ℝ) ^ 2) * |s - t| := by
  have hdiff (u : ℝ) (hu : u ∈ Set.Icc (k : ℝ) (k + 1 : ℕ)) :
      DifferentiableAt ℝ invLogContinuousWeight u := by
    have hu2 : (2 : ℝ) ≤ u := (by exact_mod_cast hk : (2 : ℝ) ≤ k).trans hu.1
    have hu0 : u ≠ 0 := by linarith
    have hlogu : Real.log u ≠ 0 :=
      Real.log_ne_zero_of_pos_of_ne_one (by linarith) (by linarith)
    exact (hasDerivAt_invLogContinuousWeight hu0 hlogu).differentiableAt
  have hbound (u : ℝ) (hu : u ∈ Set.Icc (k : ℝ) (k + 1 : ℕ)) :
      ‖deriv invLogContinuousWeight u‖ ≤ 20 / (k : ℝ) ^ 2 := by
    have hu2 : (2 : ℝ) ≤ u := (by exact_mod_cast hk : (2 : ℝ) ≤ k).trans hu.1
    have hu0 : u ≠ 0 := by linarith
    have hlogu : Real.log u ≠ 0 :=
      Real.log_ne_zero_of_pos_of_ne_one (by linarith) (by linarith)
    rw [(hasDerivAt_invLogContinuousWeight hu0 hlogu).deriv,
      Real.norm_eq_abs]
    exact invLog_deriv_bound hk hu.1
  simpa only [Real.norm_eq_abs] using
    Convex.norm_image_sub_le_of_norm_deriv_le hdiff hbound
      (convex_Icc (k : ℝ) (k + 1 : ℕ)) ht hs

theorem invLogThetaWeight_abs_le {m : ℕ} (hm : 2 ≤ m) :
    |invLogThetaWeight m| ≤ 4 / (m : ℝ) := by
  have hloghalf : (1 / 2 : ℝ) ≤ Real.log (m : ℝ) := by
    have hmono : Real.log 2 ≤ Real.log (m : ℝ) := by
      apply Real.log_le_log
      · norm_num
      · exact_mod_cast hm
    nlinarith [Real.log_two_gt_d9]
  have hmpos : (0 : ℝ) < m := by positivity
  have hlogpos : 0 < Real.log (m : ℝ) := by linarith
  rw [invLogThetaWeight, abs_div, abs_one, abs_mul,
    abs_of_pos hmpos, abs_pow, abs_of_pos hlogpos]
  apply (div_le_div_iff₀ (mul_pos hmpos (sq_pos_of_pos hlogpos)) hmpos).2
  nlinarith [sq_nonneg (Real.log (m : ℝ) - 1 / 2)]

theorem invLogWeight_pnt_bound {C : ℝ} (hC : 0 ≤ C)
    {X₀ z Y : ℕ}
    (hbound : ∀ T, X₀ ≤ T →
      |primeLogSumUpTo T - (T : ℝ)| ≤
        C * ((T : ℝ) / Real.log (T : ℝ) ^ (3 : ℝ)))
    (hX₀z : X₀ ≤ z) (hz : 2 ≤ z) (hzY : z < Y)
    (hlogz : 1 ≤ Real.log (z : ℝ))
    (hlogY : Real.log (Y : ℝ) ≤ 5 * Real.log (z : ℝ)) :
    |primeThetaWeightedInterval invLogThetaWeight z Y -
        integerAbelMain invLogThetaWeight z Y| ≤
      128 * C / Real.log (z : ℝ) ^ 2 := by
  have hlogzpos : 0 < Real.log (z : ℝ) := lt_of_lt_of_le (by norm_num) hlogz
  have hlogzY : Real.log (z : ℝ) ≤ Real.log (Y : ℝ) := by
    apply Real.log_le_log
    · positivity
    · exact_mod_cast hzY.le
  have hgeneric := primeThetaWeightedInterval_pnt_bound
    invLogThetaWeight (A := (3 : ℝ)) hbound hX₀z hzY
  simp only [Real.rpow_ofNat] at hgeneric
  have htermY : |invLogThetaWeight Y| *
      (C * ((Y : ℝ) / Real.log (Y : ℝ) ^ (3 : ℕ))) ≤
      4 * C / Real.log (z : ℝ) ^ 2 := by
    have hY0 : (Y : ℝ) ≠ 0 := by
      exact_mod_cast (show Y ≠ 0 by omega)
    calc
      _ ≤ (4 / (Y : ℝ)) *
          (C * ((Y : ℝ) / Real.log (Y : ℝ) ^ 3)) := by
        gcongr
        exact invLogThetaWeight_abs_le (hz.trans hzY.le)
      _ = 4 * C / Real.log (Y : ℝ) ^ 3 := by field_simp [hY0]
      _ ≤ 4 * C / Real.log (z : ℝ) ^ 3 := by gcongr
      _ ≤ 4 * C / Real.log (z : ℝ) ^ 2 := by
        have hfac : 0 ≤ 4 * C := by positivity
        apply div_le_div_of_nonneg_left hfac (pow_pos hlogzpos 2)
        nlinarith [mul_le_mul_of_nonneg_left hlogz hlogzpos.le]
  have htermZ : |invLogThetaWeight (z + 1)| *
      (C * ((z : ℝ) / Real.log (z : ℝ) ^ (3 : ℕ))) ≤
      4 * C / Real.log (z : ℝ) ^ 2 := by
    have hf := invLogThetaWeight_abs_le (m := z + 1) (by omega)
    have hf' : |invLogThetaWeight (z + 1)| ≤ 4 / (z : ℝ) :=
      hf.trans (by gcongr; omega)
    calc
      _ ≤ (4 / (z : ℝ)) *
          (C * ((z : ℝ) / Real.log (z : ℝ) ^ 3)) := by gcongr
      _ = 4 * C / Real.log (z : ℝ) ^ 3 := by field_simp
      _ ≤ 4 * C / Real.log (z : ℝ) ^ 2 := by
        have hfac : 0 ≤ 4 * C := by positivity
        apply div_le_div_of_nonneg_left hfac (pow_pos hlogzpos 2)
        nlinarith [mul_le_mul_of_nonneg_left hlogz hlogzpos.le]
  have hvar : (∑ m ∈ Finset.Ioc z (Y - 1),
      |invLogThetaWeight (m + 1) - invLogThetaWeight m| *
        (C * ((m : ℝ) / Real.log (m : ℝ) ^ (3 : ℕ)))) ≤
      120 * C / Real.log (z : ℝ) ^ 2 := by
    have hpoint : ∀ m ∈ Finset.Ioc z (Y - 1),
        |invLogThetaWeight (m + 1) - invLogThetaWeight m| *
            (C * ((m : ℝ) / Real.log (m : ℝ) ^ 3)) ≤
          (20 * C / Real.log (z : ℝ) ^ 3) * (1 / (m : ℝ)) := by
      intro m hm
      rw [Finset.mem_Ioc] at hm
      have hm2 : 2 ≤ m := hz.trans hm.1.le
      have hdiff : |invLogThetaWeight (m + 1) -
          invLogThetaWeight m| ≤ 20 / (m : ℝ) ^ 2 := by
        rw [← invLogContinuousWeight_nat,
          ← invLogContinuousWeight_nat]
        have hs : ((m + 1 : ℕ) : ℝ) ∈ Set.Icc (m : ℝ) (m + 1 : ℕ) := by
          constructor <;> norm_num
        have ht : (m : ℝ) ∈ Set.Icc (m : ℝ) (m + 1 : ℕ) := by
          constructor <;> norm_num
        simpa using invLogWeight_cell_oscillation hm2 hs ht
      have hlogzm : Real.log (z : ℝ) ≤ Real.log (m : ℝ) := by
        apply Real.log_le_log
        · positivity
        · exact_mod_cast hm.1.le
      have hlogmpos : 0 < Real.log (m : ℝ) := hlogzpos.trans_le hlogzm
      calc
        _ ≤ (20 / (m : ℝ) ^ 2) *
            (C * ((m : ℝ) / Real.log (m : ℝ) ^ 3)) := by gcongr
        _ = (20 * C / Real.log (m : ℝ) ^ 3) * (1 / (m : ℝ)) := by
          field_simp
        _ ≤ (20 * C / Real.log (z : ℝ) ^ 3) * (1 / (m : ℝ)) := by
          gcongr
    have hsum : (∑ m ∈ Finset.Ioc z (Y - 1),
        |invLogThetaWeight (m + 1) - invLogThetaWeight m| *
          (C * ((m : ℝ) / Real.log (m : ℝ) ^ 3))) ≤
        (20 * C / Real.log (z : ℝ) ^ 3) *
          (∑ m ∈ Finset.Ioc z (Y - 1), 1 / (m : ℝ)) := by
      rw [Finset.mul_sum]
      exact Finset.sum_le_sum fun m hm => hpoint m hm
    have hhar : (∑ m ∈ Finset.Ioc z (Y - 1), 1 / (m : ℝ)) ≤
        6 * Real.log (z : ℝ) := by
      calc
        _ ≤ 1 + Real.log (Y : ℝ) := by
          exact (harmonic_Ioc_le (z := z) (y := Y - 1)).trans (by
            have hlogsub : Real.log ((Y - 1 : ℕ) : ℝ) ≤
                Real.log (Y : ℝ) := by
              apply Real.log_le_log
              · exact_mod_cast (show 0 < Y - 1 by omega)
              · exact_mod_cast (Nat.sub_le Y 1)
            linarith)
        _ ≤ 6 * Real.log (z : ℝ) := by linarith
    calc
      _ ≤ (20 * C / Real.log (z : ℝ) ^ 3) *
          (∑ m ∈ Finset.Ioc z (Y - 1), 1 / (m : ℝ)) := hsum
      _ ≤ (20 * C / Real.log (z : ℝ) ^ 3) *
          (6 * Real.log (z : ℝ)) := by gcongr
      _ = 120 * C / Real.log (z : ℝ) ^ 2 := by field_simp; ring
  calc
    _ ≤ |invLogThetaWeight Y| *
          (C * ((Y : ℝ) / Real.log (Y : ℝ) ^ (3 : ℕ))) +
        |invLogThetaWeight (z + 1)| *
          (C * ((z : ℝ) / Real.log (z : ℝ) ^ (3 : ℕ))) +
        ∑ m ∈ Finset.Ioc z (Y - 1),
          |invLogThetaWeight (m + 1) - invLogThetaWeight m| *
            (C * ((m : ℝ) / Real.log (m : ℝ) ^ (3 : ℕ))) := hgeneric
    _ ≤ 4 * C / Real.log (z : ℝ) ^ 2 +
        4 * C / Real.log (z : ℝ) ^ 2 +
        120 * C / Real.log (z : ℝ) ^ 2 := by
      exact add_le_add (add_le_add htermY htermZ) hvar
    _ = 128 * C / Real.log (z : ℝ) ^ 2 := by ring

theorem hasDerivAt_invLogAntiderivative {t : ℝ} (ht : t ≠ 0)
    (hlog : Real.log t ≠ 0) :
    HasDerivAt invLogAntiderivative
      (invLogContinuousWeight t) t := by
  unfold invLogAntiderivative invLogContinuousWeight
  have hraw := (hasDerivAt_const t (1 : ℝ)).div
    (Real.hasDerivAt_log ht) hlog
  convert hraw.neg using 1
  field_simp [ht, hlog]
  ring

theorem integral_invLogContinuousWeight {z Y : ℕ}
    (hz : 2 ≤ z) (hzY : z ≤ Y) :
    (∫ t in (z : ℝ)..(Y : ℝ), invLogContinuousWeight t) =
      1 / Real.log (z : ℝ) - 1 / Real.log (Y : ℝ) := by
  have hcont : ContinuousOn invLogAntiderivative
      (Set.Icc (z : ℝ) (Y : ℝ)) := by
    intro t htI
    have ht2 : (2 : ℝ) ≤ t := (by exact_mod_cast hz : (2 : ℝ) ≤ z).trans htI.1
    have ht0 : t ≠ 0 := by linarith
    have hlogt : Real.log t ≠ 0 :=
      Real.log_ne_zero_of_pos_of_ne_one (by linarith) (by linarith)
    exact (hasDerivAt_invLogAntiderivative ht0 hlogt).continuousAt.continuousWithinAt
  have hint : IntervalIntegrable invLogContinuousWeight
      MeasureTheory.volume (z : ℝ) (Y : ℝ) := by
    have hcontW : ContinuousOn invLogContinuousWeight
        (Set.Icc (z : ℝ) (Y : ℝ)) := by
      intro t htI
      have ht2 : (2 : ℝ) ≤ t := (by exact_mod_cast hz : (2 : ℝ) ≤ z).trans htI.1
      have ht0 : t ≠ 0 := by linarith
      have hlogt : Real.log t ≠ 0 :=
        Real.log_ne_zero_of_pos_of_ne_one (by linarith) (by linarith)
      exact (hasDerivAt_invLogContinuousWeight ht0 hlogt).continuousAt.continuousWithinAt
    rw [← Set.uIcc_of_le (by exact_mod_cast hzY)] at hcontW
    exact hcontW.intervalIntegrable
  have hfund := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le
    (f := invLogAntiderivative) (f' := invLogContinuousWeight)
    (by exact_mod_cast hzY) hcont
    (fun t htI => by
      have ht2 : (2 : ℝ) < t := (by exact_mod_cast hz : (2 : ℝ) ≤ z).trans_lt htI.1
      exact hasDerivAt_invLogAntiderivative (by linarith)
        (Real.log_ne_zero_of_pos_of_ne_one (by linarith) (by linarith)))
    hint
  rw [hfund]
  simp only [invLogAntiderivative]
  ring

theorem invLogWeight_sum_integral_bound {z Y : ℕ}
    (hz : 2 ≤ z) (hzY : z ≤ Y) :
    |(∑ m ∈ Finset.Ioc z Y, invLogThetaWeight m) -
        ∫ t in (z : ℝ)..(Y : ℝ), invLogContinuousWeight t| ≤
      40 / (z : ℝ) := by
  have hint : ∀ k ∈ Set.Ico z Y,
      IntervalIntegrable invLogContinuousWeight MeasureTheory.volume
        (k : ℝ) (k + 1 : ℕ) := by
    intro k hk
    rw [Set.mem_Ico] at hk
    have hk2 : 2 ≤ k := hz.trans hk.1
    have hcont : ContinuousOn invLogContinuousWeight
        (Set.Icc (k : ℝ) (k + 1 : ℕ)) := by
      intro t htI
      have ht2 : (2 : ℝ) ≤ t := (by exact_mod_cast hk2 : (2 : ℝ) ≤ k).trans htI.1
      have ht0 : t ≠ 0 := by linarith
      have hlogt : Real.log t ≠ 0 :=
        Real.log_ne_zero_of_pos_of_ne_one (by linarith) (by linarith)
      exact (hasDerivAt_invLogContinuousWeight ht0 hlogt).continuousAt.continuousWithinAt
    rw [← Set.uIcc_of_le (by norm_num : (k : ℝ) ≤ (k + 1 : ℕ))] at hcont
    exact hcont.intervalIntegrable
  have hosc : ∀ k ∈ Finset.Ico z Y,
      ∀ t ∈ Set.Icc (k : ℝ) (k + 1 : ℕ),
        |invLogContinuousWeight ((k : ℝ) + 1) -
          invLogContinuousWeight t| ≤ 20 / (k : ℝ) ^ 2 := by
    intro k hk t htI
    rw [Finset.mem_Ico] at hk
    have hk2 : 2 ≤ k := hz.trans hk.1
    have hs : ((k + 1 : ℕ) : ℝ) ∈ Set.Icc (k : ℝ) (k + 1 : ℕ) := by
      constructor <;> norm_num
    have h := invLogWeight_cell_oscillation hk2 hs htI
    simp only [Nat.cast_add, Nat.cast_one] at h
    apply h.trans
    have htI' : (k : ℝ) ≤ t ∧ t ≤ (k : ℝ) + 1 := by
      simpa only [Nat.cast_add, Nat.cast_one] using htI
    have hdist0 : 0 ≤ (k : ℝ) + 1 - t := by linarith [htI'.2]
    rw [abs_of_nonneg hdist0]
    have hdist1 : (k : ℝ) + 1 - t ≤ 1 := by linarith [htI'.1]
    calc
      _ ≤ (20 / (k : ℝ) ^ 2) * 1 :=
        mul_le_mul_of_nonneg_left hdist1
          (div_nonneg (by norm_num) (sq_nonneg (k : ℝ)))
      _ = 20 / (k : ℝ) ^ 2 := by ring
  have hR := sum_integral_error_bound invLogContinuousWeight
    (fun k => 20 / (k : ℝ) ^ 2) hzY hint hosc
  simp only [invLogContinuousWeight_nat] at hR
  calc
    _ ≤ ∑ k ∈ Finset.Ico z Y, 20 / (k : ℝ) ^ 2 := hR
    _ = 20 * ∑ k ∈ Finset.Ico z Y, 1 / (k : ℝ) ^ 2 := by
      rw [Finset.mul_sum]
      apply Finset.sum_congr rfl
      intro k hk
      ring
    _ ≤ 20 * (2 / (z : ℝ)) := by
      exact mul_le_mul_of_nonneg_left
        (sum_Ico_inv_sq_le (show 1 ≤ z by omega)) (by norm_num)
    _ = 40 / (z : ℝ) := by ring

noncomputable def primeInvLogSum (z Y : ℕ) : ℝ :=
  ∑ p ∈ (Y + 1).primesBelow \ (z + 1).primesBelow,
    1 / ((p : ℝ) * Real.log (p : ℝ))

theorem primeThetaWeightedInterval_invLog (z Y : ℕ) :
    primeThetaWeightedInterval invLogThetaWeight z Y =
      primeInvLogSum z Y := by
  rw [primeThetaWeightedInterval, primeInvLogSum]
  apply Finset.sum_congr rfl
  intro p hp
  have hpprime : p.Prime :=
    Nat.prime_of_mem_primesBelow (Finset.mem_sdiff.mp hp).1
  have hlog : Real.log (p : ℝ) ≠ 0 :=
    Real.log_ne_zero_of_pos_of_ne_one
      (by exact_mod_cast hpprime.pos) (by exact_mod_cast hpprime.ne_one)
  rw [invLogThetaWeight]
  field_simp [hlog]

theorem primeInvLogSum_contraction_of_bound {C : ℝ} (hC : 0 ≤ C)
    {X₀ z Y : ℕ}
    (hbound : ∀ T, X₀ ≤ T →
      |primeLogSumUpTo T - (T : ℝ)| ≤
        C * ((T : ℝ) / Real.log (T : ℝ) ^ (3 : ℝ)))
    (hX₀z : X₀ ≤ z) (hz : 2 ≤ z) (hzY : z < Y)
    (hlogz : 1 ≤ Real.log (z : ℝ))
    (hlogY : Real.log (Y : ℝ) ≤ 5 * Real.log (z : ℝ))
    (hsmall : 128 * C / Real.log (z : ℝ) ^ 2 + 40 / (z : ℝ) ≤
      1 / (10 * Real.log (z : ℝ))) :
    primeInvLogSum z Y ≤ 9 / (10 * Real.log (z : ℝ)) := by
  have hlogzpos : 0 < Real.log (z : ℝ) := lt_of_lt_of_le (by norm_num) hlogz
  have hlogzY : Real.log (z : ℝ) ≤ Real.log (Y : ℝ) := by
    apply Real.log_le_log
    · positivity
    · exact_mod_cast hzY.le
  have hlogYpos : 0 < Real.log (Y : ℝ) := hlogzpos.trans_le hlogzY
  have hpnt := invLogWeight_pnt_bound hC hbound hX₀z hz hzY hlogz hlogY
  rw [primeThetaWeightedInterval_invLog,
    integerAbelMain_eq_sum_Ioc invLogThetaWeight hzY] at hpnt
  have hriemann := invLogWeight_sum_integral_bound hz hzY.le
  rw [integral_invLogContinuousWeight hz hzY.le] at hriemann
  have htotal : |primeInvLogSum z Y -
      (1 / Real.log (z : ℝ) - 1 / Real.log (Y : ℝ))| ≤
      128 * C / Real.log (z : ℝ) ^ 2 + 40 / (z : ℝ) := by
    calc
      _ = |(primeInvLogSum z Y -
            ∑ m ∈ Finset.Ioc z Y, invLogThetaWeight m) +
          ((∑ m ∈ Finset.Ioc z Y, invLogThetaWeight m) -
            (1 / Real.log (z : ℝ) - 1 / Real.log (Y : ℝ)))| := by ring_nf
      _ ≤ |primeInvLogSum z Y -
            ∑ m ∈ Finset.Ioc z Y, invLogThetaWeight m| +
          |(∑ m ∈ Finset.Ioc z Y, invLogThetaWeight m) -
            (1 / Real.log (z : ℝ) - 1 / Real.log (Y : ℝ))| :=
        abs_add_le _ _
      _ ≤ _ := add_le_add hpnt hriemann
  have hmain : 1 / Real.log (z : ℝ) - 1 / Real.log (Y : ℝ) ≤
      4 / (5 * Real.log (z : ℝ)) := by
    have hfrac : (1 / 5 : ℝ) ≤
        Real.log (z : ℝ) / Real.log (Y : ℝ) := by
      apply (le_div_iff₀ hlogYpos).2
      linarith
    calc
      1 / Real.log (z : ℝ) - 1 / Real.log (Y : ℝ) =
          (1 - Real.log (z : ℝ) / Real.log (Y : ℝ)) /
            Real.log (z : ℝ) := by field_simp
      _ ≤ (4 / 5 : ℝ) / Real.log (z : ℝ) := by
        exact div_le_div_of_nonneg_right (by linarith) hlogzpos.le
      _ = 4 / (5 * Real.log (z : ℝ)) := by ring
  have hupper : primeInvLogSum z Y ≤
      (1 / Real.log (z : ℝ) - 1 / Real.log (Y : ℝ)) +
        (128 * C / Real.log (z : ℝ) ^ 2 + 40 / (z : ℝ)) := by
    have hsigned := (le_abs_self (primeInvLogSum z Y -
      (1 / Real.log (z : ℝ) - 1 / Real.log (Y : ℝ)))).trans htotal
    linarith
  calc
    primeInvLogSum z Y ≤
        (1 / Real.log (z : ℝ) - 1 / Real.log (Y : ℝ)) +
          (128 * C / Real.log (z : ℝ) ^ 2 + 40 / (z : ℝ)) := hupper
    _ ≤ 4 / (5 * Real.log (z : ℝ)) +
          1 / (10 * Real.log (z : ℝ)) := add_le_add hmain hsmall
    _ = 9 / (10 * Real.log (z : ℝ)) := by ring

/-- The contraction estimate has an absolute lower threshold; all analytic
hypotheses are discharged from the medium-strength prime number theorem. -/
theorem exists_primeInvLogSum_contraction_threshold :
    ∃ Z₀ : ℕ, ∀ {z Y : ℕ}, Z₀ ≤ z → z < Y →
      Real.log (Y : ℝ) ≤ 5 * Real.log (z : ℝ) →
      primeInvLogSum z Y ≤ 9 / (10 * Real.log (z : ℝ)) := by
  obtain ⟨C, hCpos, X₀, hbound⟩ :=
    exists_primeLogSumUpTo_error_bound (3 : ℝ)
  have hlogTop : Filter.Tendsto (fun z : ℕ => Real.log (z : ℝ))
      Filter.atTop Filter.atTop :=
    Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop
  have hinvlog : Filter.Tendsto
      (fun z : ℕ => (Real.log (z : ℝ))⁻¹)
      Filter.atTop (nhds 0) := hlogTop.inv_tendsto_atTop
  have hlogdiv : Filter.Tendsto
      (fun z : ℕ => Real.log (z : ℝ) / (z : ℝ))
      Filter.atTop (nhds 0) := by
    simpa only [Function.comp_apply, id_eq] using
      Real.isLittleO_log_id_atTop.tendsto_div_nhds_zero.comp
        (tendsto_natCast_atTop_atTop (R := ℝ))
  have hsmallT : Filter.Tendsto
      (fun z : ℕ =>
        128 * C * (Real.log (z : ℝ))⁻¹ +
          40 * (Real.log (z : ℝ) / (z : ℝ)))
      Filter.atTop (nhds 0) := by
    convert (tendsto_const_nhds.mul hinvlog).add
      (tendsto_const_nhds.mul hlogdiv) using 1
    norm_num
  have hsmallEvent : ∀ᶠ z : ℕ in Filter.atTop,
      128 * C * (Real.log (z : ℝ))⁻¹ +
          40 * (Real.log (z : ℝ) / (z : ℝ)) ≤ 1 / 10 := by
    have hlt := hsmallT.eventually
      (Iio_mem_nhds (show (0 : ℝ) < 1 / 10 by norm_num))
    filter_upwards [hlt] with z hz
    exact hz.le
  have hlogEvent : ∀ᶠ z : ℕ in Filter.atTop,
      1 ≤ Real.log (z : ℝ) :=
    hlogTop.eventually (Filter.eventually_ge_atTop 1)
  have hall : ∀ᶠ z : ℕ in Filter.atTop,
      X₀ ≤ z ∧ 2 ≤ z ∧ 1 ≤ Real.log (z : ℝ) ∧
        128 * C / Real.log (z : ℝ) ^ 2 + 40 / (z : ℝ) ≤
          1 / (10 * Real.log (z : ℝ)) := by
    filter_upwards [Filter.eventually_ge_atTop (max X₀ 2),
      hlogEvent, hsmallEvent] with z hzbase hlogz hscaled
    have hX₀z : X₀ ≤ z := (le_max_left X₀ 2).trans hzbase
    have hz2 : 2 ≤ z := (le_max_right X₀ 2).trans hzbase
    have hlogpos : 0 < Real.log (z : ℝ) :=
      lt_of_lt_of_le (by norm_num) hlogz
    have hmul :
        (128 * C / Real.log (z : ℝ) ^ 2 + 40 / (z : ℝ)) *
            Real.log (z : ℝ) ≤ 1 / 10 := by
      calc
        _ = 128 * C * (Real.log (z : ℝ))⁻¹ +
            40 * (Real.log (z : ℝ) / (z : ℝ)) := by
              field_simp [hlogpos.ne']
        _ ≤ 1 / 10 := hscaled
    have hsmall :
        128 * C / Real.log (z : ℝ) ^ 2 + 40 / (z : ℝ) ≤
          (1 / 10) / Real.log (z : ℝ) :=
      (le_div_iff₀ hlogpos).2 hmul
    refine ⟨hX₀z, hz2, hlogz, ?_⟩
    convert hsmall using 1
    ring
  obtain ⟨Z₀, hZ₀⟩ := Filter.eventually_atTop.mp hall
  refine ⟨Z₀, ?_⟩
  intro z Y hz hzY hlogY
  obtain ⟨hX₀z, hz2, hlogz, hsmall⟩ := hZ₀ z hz
  exact primeInvLogSum_contraction_of_bound hCpos.le hbound
    hX₀z hz2 hzY hlogz hlogY hsmall
























end Erdos390.Full.FriableAsymptotic

end

/- Source module: Erdos390.Full.StructuredCellAsymptotic -/
section


/-!
# Unconditional Dickman estimate for the actual structured cells

This file composes the exact head-pattern inclusion--exclusion identity with
the proved uniform friable-number estimate.  In particular, the theorem below
does not assume a marked-cell asymptotic: its only analytic hypotheses are the
explicit logarithmic-range inequalities at the finitely many endpoints.
-/

open scoped BigOperators ArithmeticFunction.Moebius

namespace Erdos390.Full.StructuredCellAsymptotic

open ArithmeticModel DickmanBasic Scale
open StructuredCells HeadPattern MarkedFriableAsymptotic

/-- The Dickman main term at one integral endpoint. -/
noncomputable def endpointMain (X y : ℕ) : ℝ :=
  (X : ℝ) * rho (FriableAsymptotic.dickmanU X y)

/-- The signed main term contributed by one head inclusion--exclusion
divisor. -/
noncomputable def headDivisorMain (P : Pattern) (lo hi y d a : ℕ) : ℝ :=
  (ArithmeticFunction.moebius a : ℝ) *
    (endpointMain (hi / (P.factor * a * d)) y -
      endpointMain (lo / (P.factor * a * d)) y)





end Erdos390.Full.StructuredCellAsymptotic

end

/- Source module: Erdos390.Full.PaperScaleEndpoint -/
section


/-! Quantitative endpoint specialization at the paper scale. -/

namespace Erdos390.Full.PaperScaleMarkedCell











noncomputable def paperDivisorCoordinate (n d : ℕ) : ℝ :=
  DickmanBasic.U -
    Real.log (d : ℝ) / Real.log (ArithmeticModel.y n)













end Erdos390.Full.PaperScaleMarkedCell

end

/- Source module: Erdos390.Full.DivisibilityMomentBounds -/
section


/-!
# Exact multiple-counting moment bounds

The omitted-local-score step in Lemma 7.5 expands a valuation score into
divisor indicators and then counts common multiples.  This file proves that
finite combinatorial step for an arbitrary set of positive integers bounded
by one physical endpoint.  No smooth-number estimate or probability-model
assumption enters here.
-/

open scoped BigOperators

namespace Erdos390.Full.DivisibilityMomentBounds

open ArithmeticModel

noncomputable section









/-- The finite divisor-indicator score attached to a family of moduli. -/
def divisorScore (R : Finset ℕ) (m : ℕ) : ℝ :=
  ∑ a ∈ R, divInd a m













/-- Uniform average over a nonempty finite cell. -/
def uniformAverage (S : Finset ℕ) (F : ℕ → ℝ) : ℝ :=
  (∑ m ∈ S, F m) / (S.card : ℝ)











end

end Erdos390.Full.DivisibilityMomentBounds

end

/- Source module: Erdos390.Full.PaperScaleMarkedCell -/
section


/-!
# Paper-scale common main term for marked structured cells

The raw structured-cell asymptotic is a finite sum of endpoint Dickman
terms.  This module performs the algebraic closure required in the paper:
the head Möbius coefficient is evaluated exactly, and the endpoint terms
are compared with one common Dickman translate before normalization.
-/

open scoped BigOperators ArithmeticFunction.Moebius

namespace Erdos390.Full.PaperScaleMarkedCell

open ArithmeticModel DickmanBasic Scale
open StructuredCells HeadPattern MarkedFriableAsymptotic
open StructuredCellAsymptotic
open DivisibilityMomentBounds

noncomputable section

/-! ## Exact closure of the head Möbius coefficient -/

 def reciprocalAF : ArithmeticFunction ℝ :=
  ⟨fun n => if n = 0 then 0 else 1 / (n : ℝ), by simp⟩







/-- The fixed positive density of a prescribed head valuation pattern. -/
def headDensity (P : Pattern) : ℝ :=
  (1 / (P.factor : ℝ)) *
    ((P.modulus.totient : ℝ) / (P.modulus : ℝ))





/-! ## The common paper-scale main term -/

/-- The single main term remaining after both endpoint specialization and
head Möbius closure. -/
def paperMarkedMain (P : Pattern) (A B : ℝ) (n d : ℕ) : ℝ :=
  headDensity P * ((B - A) * (n : ℝ) / (d : ℝ)) *
    rho (paperDivisorCoordinate n d)

/-- The positive unmarked density of one fixed physical/head cell. -/
def paperCellDensity (P : Pattern) (A B : ℝ) : ℝ :=
  headDensity P * (B - A) * rho DickmanBasic.U



/-- The paper's normalized marked-divisor main term. -/
def paperDivisibilityMain (n d : ℕ) : ℝ :=
  (rho (paperDivisorCoordinate n d) / rho DickmanBasic.U) / (d : ℝ)

@[simp] theorem paperDivisorCoordinate_one (n : ℕ) :
    paperDivisorCoordinate n 1 = DickmanBasic.U := by
  simp [paperDivisorCoordinate]

@[simp] theorem paperDivisibilityMain_one (n : ℕ) :
    paperDivisibilityMain n 1 = 1 := by
  simp [paperDivisibilityMain, rho_U_ne_zero]

theorem paperMarkedMain_eq_density_mul_divisibility
    (P : Pattern) (A B : ℝ) {n d : ℕ} (hd : 0 < d) :
    paperMarkedMain P A B n d =
      paperCellDensity P A B * (n : ℝ) * paperDivisibilityMain n d := by
  have hdR : (d : ℝ) ≠ 0 := by exact_mod_cast hd.ne'
  have hrho : rho DickmanBasic.U ≠ 0 := rho_U_ne_zero
  unfold paperMarkedMain paperCellDensity paperDivisibilityMain
  field_simp [hdR, hrho]

@[simp] theorem paperMarkedMain_one (P : Pattern) (A B : ℝ) (n : ℕ) :
    paperMarkedMain P A B n 1 = paperCellDensity P A B * (n : ℝ) := by
  rw [paperMarkedMain_eq_density_mul_divisibility P A B (n := n)
    (d := 1) (by omega), paperDivisibilityMain_one]
  ring







/-- Fixed coefficient in the error of one physical endpoint. -/
def endpointCoefficient (C : ℝ) (g : ℕ) : ℝ :=
  2 + 5 * (Real.log 2 + |Real.log C - Real.log (g : ℝ)|) +
    (45 / 2 : ℝ) * Real.log 2



/-- Total fixed endpoint-error coefficient after head inclusion--exclusion. -/
def endpointFamilyCoefficient (P : Pattern) (C : ℝ) : ℝ :=
  ∑ a ∈ P.modulus.divisors,
    |(ArithmeticFunction.moebius a : ℝ)| *
      (endpointCoefficient C (P.factor * a) *
        C / ((P.factor * a : ℕ) : ℝ))





/-- Fixed total variation of the finite head Möbius ledger. -/
def headMoebiusMass (P : Pattern) : ℝ :=
  ∑ a ∈ P.modulus.divisors, |(ArithmeticFunction.moebius a : ℝ)|





/-! ## Normalization algebra -/











end

end Erdos390.Full.PaperScaleMarkedCell

end

/- Source module: Erdos390.Full.NuisanceCovariance -/
section


/-!
# Uniform finite nuisance-covariance gaps

This file isolates the finite-dimensional argument used for the actual
finite-`n` nuisance covariance `Γ_{0,n}`.  There is no limiting mixture.
The hypotheses are explicit pattern vectors, explicit probability weights,
a positive lower bound for those weights, and a checkable affine-spanning
certificate (a continuous linear reconstruction from pair differences).

The second part controls a perturbation of the mixture weights in `ℓ¹` and
shows that a sufficiently small bounded tilt preserves half of the baseline
gap.
-/

open scoped BigOperators

namespace Erdos390.Full

noncomputable section

/-- A finite probability mixture of explicit nuisance-pattern vectors. -/
structure PatternMixture
    (Cell Nuisance : Type*) [Fintype Cell]
    [NormedAddCommGroup Nuisance] [InnerProductSpace ℝ Nuisance] where
  weight : Cell → ℝ
  weight_nonneg : ∀ c, 0 ≤ weight c
  weight_sum : ∑ c, weight c = 1
  pattern : Cell → Nuisance

namespace PatternMixture

variable {Cell Nuisance : Type*} [Fintype Cell]
  [NormedAddCommGroup Nuisance] [InnerProductSpace ℝ Nuisance]

/-- The mixture weights as an actual finite probability law. -/
def probability (mix : PatternMixture Cell Nuisance) : FiniteProbability Cell where
  mass := mix.weight
  mass_nonneg := mix.weight_nonneg
  mass_sum := mix.weight_sum

/-- The scalar quadratic form of the nuisance covariance matrix in direction
`x`. -/
def covarianceForm (mix : PatternMixture Cell Nuisance) (x : Nuisance) : ℝ :=
  mix.probability.covariance
    (fun c => inner ℝ x (mix.pattern c))
    (fun c => inner ℝ x (mix.pattern c))

/-- All ordered pair differences, equipped with the Euclidean norm. -/
def pairDifferenceAnalysis (mix : PatternMixture Cell Nuisance) :
    Nuisance →L[ℝ] EuclideanSpace ℝ (Cell × Cell) :=
  (EuclideanSpace.equiv (Cell × Cell) ℝ).symm.toContinuousLinearMap.comp
    (ContinuousLinearMap.pi fun ij : Cell × Cell =>
      (innerSL ℝ) (mix.pattern ij.1 - mix.pattern ij.2))





/-- A directly checkable affine-spanning certificate.  It asks for explicit
linear reconstruction from the complete list of pairwise pattern
differences; it does not assume a covariance inequality. -/
structure AffineSpanningCertificate (mix : PatternMixture Cell Nuisance) where
  reconstruct : EuclideanSpace ℝ (Cell × Cell) →L[ℝ] Nuisance
  left_inverse : reconstruct.comp mix.pairDifferenceAnalysis =
    ContinuousLinearMap.id ℝ Nuisance

namespace AffineSpanningCertificate

variable {mix : PatternMixture Cell Nuisance}











end AffineSpanningCertificate

theorem weightedVariance_pairDifference
    (w X : Cell → ℝ) (hsum : ∑ c, w c = 1) :
    (∑ c, w c * X c ^ 2) - (∑ c, w c * X c) ^ 2 =
      (1 / 2 : ℝ) *
        ∑ i, ∑ j, w i * w j * (X i - X j) ^ 2 := by
  have hA : (∑ i, ∑ j, w i * w j * X i ^ 2) = ∑ i, w i * X i ^ 2 := by
    calc
      (∑ i, ∑ j, w i * w j * X i ^ 2)
          = ∑ i, (w i * X i ^ 2) * ∑ j, w j := by
              apply Finset.sum_congr rfl
              intro i _
              rw [Finset.mul_sum]
              apply Finset.sum_congr rfl
              intro j _
              ring
      _ = ∑ i, w i * X i ^ 2 := by rw [hsum]; simp
  have hB : (∑ i, ∑ j, w i * w j * X j ^ 2) = ∑ j, w j * X j ^ 2 := by
    rw [Finset.sum_comm]
    calc
      (∑ j, ∑ i, w i * w j * X j ^ 2)
          = ∑ j, (w j * X j ^ 2) * ∑ i, w i := by
              apply Finset.sum_congr rfl
              intro j _
              rw [Finset.mul_sum]
              apply Finset.sum_congr rfl
              intro i _
              ring
      _ = ∑ j, w j * X j ^ 2 := by rw [hsum]; simp
  have hC : (∑ i, ∑ j, w i * w j * (X i * X j)) =
      (∑ i, w i * X i) ^ 2 := by
    calc
      (∑ i, ∑ j, w i * w j * (X i * X j))
          = (∑ i, w i * X i) * ∑ j, w j * X j := by
              rw [Finset.mul_sum]
              apply Finset.sum_congr rfl
              intro i _
              rw [Finset.sum_mul]
              apply Finset.sum_congr rfl
              intro j _
              ring
      _ = (∑ i, w i * X i) ^ 2 := by ring
  calc
    (∑ i, w i * X i ^ 2) - (∑ i, w i * X i) ^ 2
        = (1 / 2 : ℝ) * ((∑ i, w i * X i ^ 2) +
            (∑ j, w j * X j ^ 2) - 2 * (∑ i, w i * X i) ^ 2) := by ring
    _ = (1 / 2 : ℝ) * ((∑ i, ∑ j, w i * w j * X i ^ 2) +
          (∑ i, ∑ j, w i * w j * X j ^ 2) -
          2 * (∑ i, ∑ j, w i * w j * (X i * X j))) := by rw [hA, hB, hC]
    _ = (1 / 2 : ℝ) * ∑ i, ∑ j, w i * w j * (X i - X j) ^ 2 := by
      congr 1
      simp_rw [Finset.mul_sum]
      rw [← Finset.sum_add_distrib, ← Finset.sum_sub_distrib]
      apply Finset.sum_congr rfl
      intro i _
      rw [← Finset.sum_add_distrib, ← Finset.sum_sub_distrib]
      apply Finset.sum_congr rfl
      intro j _
      ring

theorem covarianceForm_pairDifference
    (mix : PatternMixture Cell Nuisance) (x : Nuisance) :
    mix.covarianceForm x =
      (1 / 2 : ℝ) * ∑ i, ∑ j, mix.weight i * mix.weight j *
        (inner ℝ x (mix.pattern i - mix.pattern j)) ^ 2 := by
  simp only [covarianceForm, probability, FiniteProbability.covariance,
    FiniteProbability.expect]
  simpa only [pow_two, inner_sub_right] using
    (weightedVariance_pairDifference
      mix.weight (fun c => inner ℝ x (mix.pattern c)) mix.weight_sum)







/-- Concrete certificate that a coarse pattern vector is the conditional
mean of an actual finer finite distribution on each cell. -/
structure CoarseMeanCertificate
    {Fine : Type*} [Fintype Fine]
    [DecidableEq Cell]
    (fine : PatternMixture Fine Nuisance)
    (coarse : PatternMixture Cell Nuisance) where
  cell : Fine → Cell
  cell_mass : ∀ c,
    ∑ f ∈ (Finset.univ : Finset Fine) with cell f = c, fine.weight f =
      coarse.weight c
  cell_firstMoment : ∀ c x,
    ∑ f ∈ (Finset.univ : Finset Fine) with cell f = c,
      fine.weight f * inner ℝ x (fine.pattern f) =
        coarse.weight c * inner ℝ x (coarse.pattern c)

namespace CoarseMeanCertificate

variable {Fine : Type*} [Fintype Fine]
  [DecidableEq Cell]
  {fine : PatternMixture Fine Nuisance}
  {coarse : PatternMixture Cell Nuisance}

theorem global_firstMoment (cert : CoarseMeanCertificate fine coarse)
    (x : Nuisance) :
    ∑ f, fine.weight f * inner ℝ x (fine.pattern f) =
      ∑ c, coarse.weight c * inner ℝ x (coarse.pattern c) := by
  classical
  rw [← Finset.sum_fiberwise (Finset.univ : Finset Fine) cert.cell
    (fun f => fine.weight f * inner ℝ x (fine.pattern f))]
  apply Finset.sum_congr rfl
  intro c _
  exact cert.cell_firstMoment c x

theorem global_cellPatternSq (cert : CoarseMeanCertificate fine coarse)
    (x : Nuisance) :
    ∑ f, fine.weight f *
        (inner ℝ x (coarse.pattern (cert.cell f))) ^ 2 =
      ∑ c, coarse.weight c * (inner ℝ x (coarse.pattern c)) ^ 2 := by
  classical
  rw [← Finset.sum_fiberwise (Finset.univ : Finset Fine) cert.cell
    (fun f => fine.weight f *
      (inner ℝ x (coarse.pattern (cert.cell f))) ^ 2)]
  apply Finset.sum_congr rfl
  intro c _
  calc
    (∑ f ∈ (Finset.univ : Finset Fine) with cert.cell f = c,
        fine.weight f * (inner ℝ x (coarse.pattern (cert.cell f))) ^ 2)
        = (∑ f ∈ (Finset.univ : Finset Fine) with cert.cell f = c,
            fine.weight f) * (inner ℝ x (coarse.pattern c)) ^ 2 := by
              rw [Finset.sum_mul]
              apply Finset.sum_congr rfl
              intro f hf
              rw [(Finset.mem_filter.mp hf).2]
    _ = coarse.weight c * (inner ℝ x (coarse.pattern c)) ^ 2 := by
          rw [cert.cell_mass c]

theorem global_crossMoment (cert : CoarseMeanCertificate fine coarse)
    (x : Nuisance) :
    ∑ f, fine.weight f * inner ℝ x (fine.pattern f) *
        inner ℝ x (coarse.pattern (cert.cell f)) =
      ∑ c, coarse.weight c * (inner ℝ x (coarse.pattern c)) ^ 2 := by
  classical
  rw [← Finset.sum_fiberwise (Finset.univ : Finset Fine) cert.cell
    (fun f => fine.weight f * inner ℝ x (fine.pattern f) *
      inner ℝ x (coarse.pattern (cert.cell f)))]
  apply Finset.sum_congr rfl
  intro c _
  calc
    (∑ f ∈ (Finset.univ : Finset Fine) with cert.cell f = c,
      fine.weight f * inner ℝ x (fine.pattern f) *
        inner ℝ x (coarse.pattern (cert.cell f)))
        = (∑ f ∈ (Finset.univ : Finset Fine) with cert.cell f = c,
            fine.weight f * inner ℝ x (fine.pattern f)) *
              inner ℝ x (coarse.pattern c) := by
                rw [Finset.sum_mul]
                apply Finset.sum_congr rfl
                intro f hf
                rw [(Finset.mem_filter.mp hf).2]
    _ = coarse.weight c * (inner ℝ x (coarse.pattern c)) ^ 2 := by
          rw [cert.cell_firstMoment c x]
          ring

/-- Exact finite law-of-total-variance identity. -/
theorem covarianceForm_decomposition
    (cert : CoarseMeanCertificate fine coarse) (x : Nuisance) :
    fine.covarianceForm x = coarse.covarianceForm x +
      ∑ f, fine.weight f *
        (inner ℝ x (fine.pattern f) -
          inner ℝ x (coarse.pattern (cert.cell f))) ^ 2 := by
  have hfirst := cert.global_firstMoment x
  have hcellSq := cert.global_cellPatternSq x
  have hcross := cert.global_crossMoment x
  simp only [covarianceForm, probability, FiniteProbability.covariance,
    FiniteProbability.expect]
  rw [show (∑ f, fine.weight f *
      (inner ℝ x (fine.pattern f) * inner ℝ x (fine.pattern f))) =
      ∑ f, fine.weight f * (inner ℝ x (fine.pattern f)) ^ 2 by
        apply Finset.sum_congr rfl
        intro f _
        ring]
  rw [show (∑ c, coarse.weight c *
      (inner ℝ x (coarse.pattern c) * inner ℝ x (coarse.pattern c))) =
      ∑ c, coarse.weight c * (inner ℝ x (coarse.pattern c)) ^ 2 by
        apply Finset.sum_congr rfl
        intro c _
        ring]
  have hwithin : (∑ f, fine.weight f *
      (inner ℝ x (fine.pattern f) -
        inner ℝ x (coarse.pattern (cert.cell f))) ^ 2) =
      (∑ f, fine.weight f * (inner ℝ x (fine.pattern f)) ^ 2) -
        2 * (∑ f, fine.weight f * inner ℝ x (fine.pattern f) *
          inner ℝ x (coarse.pattern (cert.cell f))) +
        (∑ f, fine.weight f *
          (inner ℝ x (coarse.pattern (cert.cell f))) ^ 2) := by
    calc
      (∑ f, fine.weight f *
          (inner ℝ x (fine.pattern f) -
            inner ℝ x (coarse.pattern (cert.cell f))) ^ 2)
          = ∑ f, (fine.weight f * (inner ℝ x (fine.pattern f)) ^ 2 -
              2 * (fine.weight f * inner ℝ x (fine.pattern f) *
                inner ℝ x (coarse.pattern (cert.cell f))) +
              fine.weight f *
                (inner ℝ x (coarse.pattern (cert.cell f))) ^ 2) := by
                  apply Finset.sum_congr rfl
                  intro f _
                  ring
      _ = (∑ f, fine.weight f * (inner ℝ x (fine.pattern f)) ^ 2) -
          (∑ f, 2 * (fine.weight f * inner ℝ x (fine.pattern f) *
            inner ℝ x (coarse.pattern (cert.cell f)))) +
          (∑ f, fine.weight f *
            (inner ℝ x (coarse.pattern (cert.cell f))) ^ 2) := by
              rw [Finset.sum_add_distrib, Finset.sum_sub_distrib]
      _ = (∑ f, fine.weight f * (inner ℝ x (fine.pattern f)) ^ 2) -
          2 * (∑ f, fine.weight f * inner ℝ x (fine.pattern f) *
            inner ℝ x (coarse.pattern (cert.cell f))) +
          (∑ f, fine.weight f *
            (inner ℝ x (coarse.pattern (cert.cell f))) ^ 2) := by
              rw [Finset.mul_sum]
  rw [hwithin, hfirst, hcellSq, hcross]
  ring

theorem coarse_covarianceForm_le_fine
    (cert : CoarseMeanCertificate fine coarse) (x : Nuisance) :
    coarse.covarianceForm x ≤ fine.covarianceForm x := by
  rw [cert.covarianceForm_decomposition x]
  exact le_add_of_nonneg_right (Finset.sum_nonneg fun f _ =>
    mul_nonneg (fine.weight_nonneg f) (sq_nonneg _))

end CoarseMeanCertificate

















end PatternMixture

end

end Erdos390.Full

end

/- Source module: Erdos390.Full.PrimePowerCovariance -/
section


/-!
# Exact prime-power covariance expansion

This file replaces the algebraic part of the prime-power transfer by proofs
for the actual valuation and divisor indicators under an actual finite
probability law.  In particular, `V = I + J` and all `JI`, `IJ`, and `JJ`
covariances are finite sums of genuine prime-power indicator covariances.
-/

open scoped BigOperators

namespace Erdos390.Full

noncomputable section

namespace FiniteProbability

variable {Omega Iota : Type*} [Fintype Omega]



theorem expect_add (mu : FiniteProbability Omega) (F G : Omega → ℝ) :
    mu.expect (fun omega ↦ F omega + G omega) = mu.expect F + mu.expect G := by
  unfold expect
  simp_rw [mul_add]
  exact Finset.sum_add_distrib

theorem expect_smul (mu : FiniteProbability Omega) (a : ℝ) (F : Omega → ℝ) :
    mu.expect (fun omega ↦ a * F omega) = a * mu.expect F := by
  unfold expect
  rw [Finset.mul_sum]
  apply Finset.sum_congr rfl
  intro omega _
  ring

theorem expect_nonneg (mu : FiniteProbability Omega) (F : Omega → ℝ)
    (hF : ∀ omega, 0 ≤ F omega) : 0 ≤ mu.expect F := by
  unfold expect
  exact Finset.sum_nonneg fun omega _ ↦
    mul_nonneg (mu.mass_nonneg omega) (hF omega)











theorem covariance_add_right (mu : FiniteProbability Omega)
    (F G H : Omega → ℝ) :
    mu.covariance F (fun omega ↦ G omega + H omega) =
      mu.covariance F G + mu.covariance F H := by
  unfold covariance
  rw [mu.expect_add G H]
  rw [show (fun omega ↦ F omega * (G omega + H omega)) =
      fun omega ↦ F omega * G omega + F omega * H omega by
    funext omega
    ring]
  rw [mu.expect_add]
  ring



theorem covariance_smul_right (mu : FiniteProbability Omega)
    (a : ℝ) (F G : Omega → ℝ) :
    mu.covariance F (fun omega ↦ a * G omega) =
      a * mu.covariance F G := by
  unfold covariance
  rw [mu.expect_smul a G]
  rw [show (fun omega ↦ F omega * (a * G omega)) =
      fun omega ↦ a * (F omega * G omega) by
    funext omega
    ring]
  rw [mu.expect_smul]
  ring







end FiniteProbability

namespace PrimePowerCovariance

open ArithmeticModel

/-- An actual finite law of positive integers with one common endpoint. -/
structure BoundedValuationLaw (Omega : Type*) [Fintype Omega] (M : ℕ) where
  probability : FiniteProbability Omega
  value : Omega → ℕ
  value_pos : ∀ omega, 0 < value omega
  value_le : ∀ omega, value omega ≤ M

namespace BoundedValuationLaw

variable {Omega : Type*} [Fintype Omega] {M : ℕ}

def I (law : BoundedValuationLaw Omega M) (p : ℕ) : Omega → ℝ :=
  fun omega ↦ divInd p (law.value omega)

def V (law : BoundedValuationLaw Omega M) (p : ℕ) : Omega → ℝ :=
  fun omega ↦ valuation p (law.value omega)

def J (law : BoundedValuationLaw Omega M) (p : ℕ) : Omega → ℝ :=
  fun omega ↦ higherValuation p (law.value omega)

def Ip (law : BoundedValuationLaw Omega M) (p : ℕ) (k : ℕ) : Omega → ℝ :=
  fun omega ↦ divInd (p ^ k) (law.value omega)

def covII (law : BoundedValuationLaw Omega M) (p q : ℕ) : ℝ :=
  law.probability.covariance (law.I p) (law.I q)

def covJI (law : BoundedValuationLaw Omega M) (p q : ℕ) : ℝ :=
  law.probability.covariance (law.J p) (law.I q)

def covIJ (law : BoundedValuationLaw Omega M) (p q : ℕ) : ℝ :=
  law.probability.covariance (law.I p) (law.J q)

def covJJ (law : BoundedValuationLaw Omega M) (p q : ℕ) : ℝ :=
  law.probability.covariance (law.J p) (law.J q)

def covVV (law : BoundedValuationLaw Omega M) (p q : ℕ) : ℝ :=
  law.probability.covariance (law.V p) (law.V q)



























end BoundedValuationLaw

end PrimePowerCovariance

end

end Erdos390.Full

end

/- Source module: Erdos390.Full.OmittedScoreTilt -/
section


/-!
# Normalized marked-event bounds under an omitted-score tilt

The four-mark step in Lemma 7.5 first omits the forced local primes from
the score, estimates a marked numerator, and then divides by the tilted
partition function.  This file proves that normalization step for an actual
finite probability law.  The result keeps the marked and unmarked
exponential deviations separate, so a `1 / D` marked moment is not replaced
by an unscaled total-variation error.
-/

open scoped BigOperators

namespace Erdos390.Full

noncomputable section

namespace FiniteProbability

variable {Omega : Type*} [Fintype Omega]

/-- Exponential deviation carrying an additional nonnegative mark. -/
def markedExponentialDeviation (mu : FiniteProbability Omega)
    (A S : Omega → ℝ) : ℝ :=
  mu.expect (fun omega ↦ A omega * |Real.exp (S omega) - 1|)











/-- Exact expectation formula under the normalized exponential tilt. -/
theorem exponentialTilt_expect_eq
    (mu : FiniteProbability Omega) (A S : Omega → ℝ) :
    (mu.exponentialTilt S).expect A =
      mu.expect (fun omega ↦ A omega * Real.exp (S omega)) /
        mu.expPartition S := by
  unfold expect
  change (∑ omega, (mu.mass omega * Real.exp (S omega) /
      mu.expPartition S) * A omega) = _
  calc
    (∑ omega, (mu.mass omega * Real.exp (S omega) /
        mu.expPartition S) * A omega) =
        ∑ omega, (mu.mass omega * (A omega * Real.exp (S omega))) /
          mu.expPartition S := by
      apply Finset.sum_congr rfl
      intro omega _
      ring
    _ = (∑ omega, mu.mass omega *
        (A omega * Real.exp (S omega))) / mu.expPartition S := by
      rw [Finset.sum_div]









end FiniteProbability

end

end Erdos390.Full

end

/- Source module: Erdos390.Full.UniformFiniteProbability -/
section


/-!
# Uniform probability on an actual finite cell

This is the exact bridge between counting averages on a finite arithmetic
cell and the `FiniteProbability` objects used by the tilt and covariance
modules.  Nonemptiness is explicit, so normalization never divides by a
silently vanishing cell cardinality.
-/

open scoped BigOperators

namespace Erdos390.Full

noncomputable section

namespace FiniteProbability

variable {Alpha : Type*}

/-- Uniform probability on the subtype represented by a nonempty finset. -/
def uniformOnFinset (S : Finset Alpha) (hS : S.Nonempty) :
    FiniteProbability S where
  mass := fun _ ↦ 1 / (S.card : ℝ)
  mass_nonneg := fun _ ↦ by positivity
  mass_sum := by
    rw [Finset.sum_const, nsmul_eq_mul]
    have huniv : (Finset.univ : Finset S).card = S.card := by
      rw [Finset.card_univ, Fintype.card_coe]
    rw [huniv]
    have hcard : (S.card : ℝ) ≠ 0 := by
      exact_mod_cast (Finset.card_ne_zero.mpr hS)
    field_simp

@[simp] theorem uniformOnFinset_mass (S : Finset Alpha) (hS : S.Nonempty)
    (x : S) :
    (uniformOnFinset S hS).mass x = 1 / (S.card : ℝ) := rfl







end FiniteProbability

end

end Erdos390.Full

end

/- Source module: Erdos390.Full.ValuationScoreDomination -/
section


open scoped BigOperators

namespace Erdos390.Full.ValuationScoreDomination

open ArithmeticModel DivisibilityMomentBounds

noncomputable section

/-- All prime-power moduli which can occur below the common endpoint `M`. -/
def primePowerModuli (P : Finset ℕ) (M : ℕ) : Finset ℕ :=
  (P ×ˢ positiveExponents M).image (fun pk ↦ pk.1 ^ pk.2)





/-- The actual valuation score appearing in a compact tilt. -/
def valuationScore (P : Finset ℕ) (eta : ℕ → ℝ)
    (L : ℝ) (m : ℕ) : ℝ :=
  ∑ p ∈ P, (eta p / L) * valuation p m

















end

end Erdos390.Full.ValuationScoreDomination

end

/- Source module: Erdos390.Full.ValuationTiltCell -/
section


/-!
# Actual compact valuation tilts on a finite cell

This file substitutes the genuine valuation score into the omitted-score
comparison.  Both divisor-score moments are discharged by exact
common-multiple counting; no probabilistic independence is assumed.
-/

open scoped BigOperators

namespace Erdos390.Full.ValuationTiltCell

open ArithmeticModel DivisibilityMomentBounds FiniteProbability
open OmittedScoreCell ValuationScoreDomination

noncomputable section



/-- The total number of prime factors drawn from primes at least `W` is
bounded by the physical logarithm.  This is the useful compact-score bound;
it is independent of the number of primes in the band. -/
theorem sum_valuation_le_log_ratio
    (P : Finset ℕ) {m M W : ℕ}
    (hm : 0 < m) (hmM : m ≤ M)
    (hW : 1 < W) (hpW : ∀ p ∈ P, W ≤ p) :
    (∑ p ∈ P, valuation p m) ≤
      Real.log (M : ℝ) / Real.log (W : ℝ) := by
  let support : Finset ℕ := m.factorization.support
  let f : ℕ → ℝ := fun p ↦ valuation p m * Real.log (p : ℝ)
  have hsumP : (∑ p ∈ P, f p) = ∑ p ∈ P ∩ support, f p := by
    symm
    apply Finset.sum_subset Finset.inter_subset_left
    intro p hpP hpNotInter
    have hpNotSupport : p ∉ support := by
      intro hpSupport
      exact hpNotInter (Finset.mem_inter.mpr ⟨hpP, hpSupport⟩)
    have hzero : m.factorization p = 0 :=
      Finsupp.notMem_support_iff.mp (by simpa [support] using hpNotSupport)
    simp [f, valuation, hzero]
  have hsubset : P ∩ support ⊆ support := Finset.inter_subset_right
  have hsumSupport : (∑ p ∈ P ∩ support, f p) ≤
      ∑ p ∈ support, f p := by
    apply Finset.sum_le_sum_of_subset_of_nonneg hsubset
    intro p hpSupport hpNotInter
    have hpPrime : p.Prime := by
      exact Nat.prime_of_mem_primeFactors (by simpa [support] using hpSupport)
    exact mul_nonneg (valuation_nonneg p m)
      (Real.log_nonneg (by exact_mod_cast hpPrime.one_lt.le))
  have hfactorization : (∑ p ∈ support, f p) = Real.log (m : ℝ) := by
    rw [Real.log_nat_eq_sum_factorization]
    rfl
  have hweighted : Real.log (W : ℝ) * (∑ p ∈ P, valuation p m) ≤
      ∑ p ∈ P, f p := by
    rw [Finset.mul_sum]
    apply Finset.sum_le_sum
    intro p hp
    have hlog : Real.log (W : ℝ) ≤ Real.log (p : ℝ) := by
      apply Real.log_le_log (by exact_mod_cast hW.le)
      exact_mod_cast hpW p hp
    simpa only [f, mul_comm] using
      mul_le_mul_of_nonneg_left hlog (valuation_nonneg p m)
  have hlogmM : Real.log (m : ℝ) ≤ Real.log (M : ℝ) := by
    apply Real.log_le_log (by exact_mod_cast hm)
    exact_mod_cast hmM
  have hlogW : 0 < Real.log (W : ℝ) :=
    Real.log_pos (by exact_mod_cast hW)
  apply (le_div_iff₀ hlogW).2
  calc
    (∑ p ∈ P, valuation p m) * Real.log (W : ℝ) =
        Real.log (W : ℝ) * (∑ p ∈ P, valuation p m) := by ring
    _ ≤ ∑ p ∈ P, f p := hweighted
    _ = ∑ p ∈ P ∩ support, f p := hsumP
    _ ≤ ∑ p ∈ support, f p := hsumSupport
    _ = Real.log (m : ℝ) := hfactorization
    _ ≤ Real.log (M : ℝ) := hlogmM





end

end Erdos390.Full.ValuationTiltCell

end

/- Source module: Erdos390.Full.PrimeSums -/
section


/-!
# Prime sums on the actual moving band

This file derives the finite prime-sum estimates needed in Lemma 7.5 from
Mathlib's Chebyshev bound.  No prime-sum estimate is introduced as a contract.
-/

open scoped BigOperators Nat.Prime
open Filter Topology

namespace Erdos390.Full.PrimeSums

open ArithmeticModel Scale

/-- The full set of primes at most `Y`, written in the same interval form as
`Chebyshev.theta`. -/
def primesUpTo (Y : ℕ) : Finset ℕ :=
  (Finset.Icc 0 Y).filter Nat.Prime

noncomputable def fullLogReciprocalSum (Y : ℕ) : ℝ :=
  ∑ p ∈ primesUpTo Y, Real.log (p : ℝ) / (p : ℝ)

noncomputable def fullReciprocalSum (Y : ℕ) : ℝ :=
  ∑ p ∈ primesUpTo Y, 1 / (p : ℝ)

noncomputable def bandLogReciprocalSum (n W : ℕ) : ℝ :=
  ∑ p ∈ primeBand n W, Real.log (p : ℝ) / (p : ℝ)

noncomputable def bandReciprocalSum (n W : ℕ) : ℝ :=
  ∑ p ∈ primeBand n W, 1 / (p : ℝ)

noncomputable def bandReciprocalSquareSum (n W : ℕ) : ℝ :=
  ∑ p ∈ primeBand n W, 1 / (p : ℝ) ^ 2

/-- The weighted first moment of the actual logarithmic prime coordinates. -/
noncomputable def bandTReciprocalSum (n W : ℕ) : ℝ :=
  ∑ p ∈ primeBand n W, tPrime n p / (p : ℝ)

 noncomputable def primeLogCoeff (k : ℕ) : ℝ :=
  if k.Prime then Real.log (k : ℝ) else 0

 theorem sum_primeLogCoeff_Icc (x : ℝ) :
    (∑ k ∈ Finset.Icc 0 ⌊x⌋₊, primeLogCoeff k) = Chebyshev.theta x := by
  rw [Chebyshev.theta_eq_sum_Icc]
  rw [Finset.sum_filter]
  rfl









/-- Abel summation for the logarithmically weighted prime harmonic sum.
This is exported so the moving-cell transfer can subtract two endpoints
without introducing an unproved Mertens-type input. -/
theorem fullLogReciprocalSum_eq (Y : ℕ) (hY : 2 ≤ Y) :
    fullLogReciprocalSum Y =
      Chebyshev.theta (Y : ℝ) / (Y : ℝ) +
        ∫ t in (2 : ℝ)..Y, Chebyshev.theta t / t ^ 2 := by
  let c : ℕ → ℝ := primeLogCoeff
  have hc0 : c 0 = 0 := by simp [c, primeLogCoeff]
  have hc1 : c 1 = 0 := by simp [c, primeLogCoeff]
  have hdiff : ∀ t ∈ Set.Icc (2 : ℝ) Y,
      DifferentiableAt ℝ (fun x : ℝ => x⁻¹) t := by
    intro t ht
    exact differentiableAt_inv (by linarith [ht.1])
  have hint : MeasureTheory.IntegrableOn
      (deriv (fun x : ℝ => x⁻¹)) (Set.Icc (2 : ℝ) Y) := by
    rw [show deriv (fun x : ℝ => x⁻¹) = fun x => -(x ^ 2)⁻¹ by
      funext x
      exact deriv_inv]
    apply ContinuousOn.integrableOn_Icc
    intro t ht
    have ht0 : t ≠ 0 := by linarith [ht.1]
    have ht20 : t ^ 2 ≠ 0 := pow_ne_zero 2 ht0
    exact ContinuousAt.continuousWithinAt (((continuousAt_id.pow 2).inv₀ ht20).neg)
  have habel := sum_mul_eq_sub_integral_mul₁ (f := fun x : ℝ => x⁻¹)
    c hc0 hc1 (Y : ℝ) hdiff hint
  rw [← intervalIntegral.integral_of_le (by exact_mod_cast hY)] at habel
  rw [show (⌊(Y : ℝ)⌋₊ : ℕ) = Y by simp] at habel
  have hcumY : (∑ k ∈ Finset.Icc 0 Y, c k) = Chebyshev.theta (Y : ℝ) := by
    simpa [c] using sum_primeLogCoeff_Icc (Y : ℝ)
  have hcum : ∀ t : ℝ,
      (∑ k ∈ Finset.Icc 0 ⌊t⌋₊, c k) = Chebyshev.theta t := by
    intro t
    simpa [c] using sum_primeLogCoeff_Icc t
  rw [hcumY] at habel
  simp_rw [hcum] at habel
  simp_rw [deriv_inv] at habel
  have hlhs :
      (∑ k ∈ Finset.Icc 0 Y, (k : ℝ)⁻¹ * c k) =
        fullLogReciprocalSum Y := by
    simp [fullLogReciprocalSum, primesUpTo, c, primeLogCoeff,
      Finset.sum_filter, div_eq_mul_inv, mul_comm]
  rw [hlhs] at habel
  have hInt :
      (∫ x in (2 : ℝ)..Y, -(x ^ 2)⁻¹ * Chebyshev.theta x) =
        -(∫ x in (2 : ℝ)..Y, Chebyshev.theta x / x ^ 2) := by
    calc
      (∫ x in (2 : ℝ)..Y, -(x ^ 2)⁻¹ * Chebyshev.theta x) =
          ∫ x in (2 : ℝ)..Y, -(Chebyshev.theta x / x ^ 2) := by
            apply intervalIntegral.integral_congr
            intro x hx
            simp only [div_eq_mul_inv]
            ring
      _ = -(∫ x in (2 : ℝ)..Y, Chebyshev.theta x / x ^ 2) :=
        intervalIntegral.integral_neg
  rw [hInt] at habel
  calc
    fullLogReciprocalSum Y =
        (Y : ℝ)⁻¹ * Chebyshev.theta (Y : ℝ) +
          ∫ x in (2 : ℝ)..Y, Chebyshev.theta x / x ^ 2 := by
      linarith
    _ = Chebyshev.theta (Y : ℝ) / (Y : ℝ) +
        ∫ t in (2 : ℝ)..Y, Chebyshev.theta t / t ^ 2 := by
      rw [div_eq_mul_inv]
      ring

 theorem deriv_inv_mul_log {x : ℝ} (hx : x ≠ 0)
    (hlog : Real.log x ≠ 0) :
    deriv (fun z : ℝ => (z * Real.log z)⁻¹) x =
      -(Real.log x + 1) / (x ^ 2 * Real.log x ^ 2) := by
  have hmul : HasDerivAt (fun z : ℝ => z * Real.log z)
      (Real.log x + x * x⁻¹) x := by
    simpa only [one_mul, id_eq] using
      (hasDerivAt_id x).mul (Real.hasDerivAt_log hx)
  have hinv : HasDerivAt (fun z : ℝ => (z * Real.log z)⁻¹)
      (-(Real.log x + x * x⁻¹) / (x * Real.log x) ^ 2) x :=
    hmul.inv (mul_ne_zero hx hlog)
  rw [hinv.deriv]
  field_simp

/-- Abel summation formula for the prime harmonic sum.  This is exported for
the moving-band quadrature: subtracting the formula at two endpoints makes
the Mertens constant disappear. -/
theorem fullReciprocalSum_eq (Y : ℕ) (hY : 2 ≤ Y) :
    fullReciprocalSum Y =
      Chebyshev.theta (Y : ℝ) /
          ((Y : ℝ) * Real.log (Y : ℝ)) +
        ∫ t in (2 : ℝ)..Y,
          Chebyshev.theta t * (Real.log t + 1) /
            (t ^ 2 * Real.log t ^ 2) := by
  let c : ℕ → ℝ := primeLogCoeff
  have hc0 : c 0 = 0 := by simp [c, primeLogCoeff]
  have hc1 : c 1 = 0 := by simp [c, primeLogCoeff]
  have hdiff : ∀ t ∈ Set.Icc (2 : ℝ) Y,
      DifferentiableAt ℝ (fun x : ℝ => (x * Real.log x)⁻¹) t := by
    intro t ht
    have ht0 : t ≠ 0 := by linarith [ht.1]
    have hlog0 : Real.log t ≠ 0 :=
      Real.log_ne_zero_of_pos_of_ne_one (by linarith [ht.1]) (by linarith [ht.1])
    exact (((differentiableAt_id.mul (Real.differentiableAt_log ht0))).inv
      (mul_ne_zero ht0 hlog0))
  have hint : MeasureTheory.IntegrableOn
      (deriv (fun x : ℝ => (x * Real.log x)⁻¹)) (Set.Icc (2 : ℝ) Y) := by
    apply ContinuousOn.integrableOn_Icc
    intro t ht
    have ht0 : t ≠ 0 := by linarith [ht.1]
    have hlog0 : Real.log t ≠ 0 :=
      Real.log_ne_zero_of_pos_of_ne_one (by linarith [ht.1]) (by linarith [ht.1])
    have hcont : ContinuousWithinAt
        (fun u : ℝ => -(Real.log u + 1) / (u ^ 2 * Real.log u ^ 2))
        (Set.Icc (2 : ℝ) Y) t :=
      ContinuousAt.continuousWithinAt
        (((Real.continuousAt_log ht0).add continuousAt_const).neg.div
          ((continuousAt_id.pow 2).mul ((Real.continuousAt_log ht0).pow 2))
          (mul_ne_zero (pow_ne_zero 2 ht0) (pow_ne_zero 2 hlog0)))
    refine hcont.congr (fun (u : ℝ) hu => ?_) ?_
    · exact deriv_inv_mul_log
        (by linarith [hu.1])
        (Real.log_ne_zero_of_pos_of_ne_one
          (by linarith [hu.1]) (by linarith [hu.1]))
    · exact deriv_inv_mul_log ht0 hlog0
  have habel := sum_mul_eq_sub_integral_mul₁
    (f := fun x : ℝ => (x * Real.log x)⁻¹)
    c hc0 hc1 (Y : ℝ) hdiff hint
  rw [← intervalIntegral.integral_of_le (by exact_mod_cast hY)] at habel
  rw [show (⌊(Y : ℝ)⌋₊ : ℕ) = Y by simp] at habel
  have hcumY : (∑ k ∈ Finset.Icc 0 Y, c k) = Chebyshev.theta (Y : ℝ) := by
    simpa [c] using sum_primeLogCoeff_Icc (Y : ℝ)
  have hcum : ∀ t : ℝ,
      (∑ k ∈ Finset.Icc 0 ⌊t⌋₊, c k) = Chebyshev.theta t := by
    intro t
    simpa [c] using sum_primeLogCoeff_Icc t
  rw [hcumY] at habel
  simp_rw [hcum] at habel
  have hlhs :
      (∑ k ∈ Finset.Icc 0 Y, ((k : ℝ) * Real.log (k : ℝ))⁻¹ * c k) =
        fullReciprocalSum Y := by
    rw [fullReciprocalSum, primesUpTo, Finset.sum_filter]
    apply Finset.sum_congr rfl
    intro k hk
    by_cases hprime : k.Prime
    · have hk0 : (k : ℝ) ≠ 0 := by exact_mod_cast hprime.ne_zero
      have hlog0 : Real.log (k : ℝ) ≠ 0 :=
        Real.log_ne_zero_of_pos_of_ne_one
          (by exact_mod_cast hprime.pos) (by exact_mod_cast hprime.ne_one)
      simp only [c, primeLogCoeff, hprime, if_true]
      field_simp
    · simp [c, primeLogCoeff, hprime]
  rw [hlhs] at habel
  have hInt :
      (∫ x in (2 : ℝ)..Y,
        deriv (fun z : ℝ => (z * Real.log z)⁻¹) x * Chebyshev.theta x) =
        -(∫ x in (2 : ℝ)..Y,
          Chebyshev.theta x * (Real.log x + 1) /
            (x ^ 2 * Real.log x ^ 2)) := by
    calc
      (∫ x in (2 : ℝ)..Y,
        deriv (fun z : ℝ => (z * Real.log z)⁻¹) x * Chebyshev.theta x) =
          ∫ x in (2 : ℝ)..Y,
            -(Chebyshev.theta x * (Real.log x + 1) /
              (x ^ 2 * Real.log x ^ 2)) := by
        apply intervalIntegral.integral_congr
        intro x hx
        rw [Set.uIcc_of_le (by exact_mod_cast hY)] at hx
        have hx0 : x ≠ 0 := by linarith [hx.1]
        have hlog0 : Real.log x ≠ 0 :=
          Real.log_ne_zero_of_pos_of_ne_one
            (by linarith [hx.1]) (by linarith [hx.1])
        change deriv (fun z : ℝ => (z * Real.log z)⁻¹) x * Chebyshev.theta x = _
        rw [deriv_inv_mul_log hx0 hlog0]
        ring
      _ = -(∫ x in (2 : ℝ)..Y,
          Chebyshev.theta x * (Real.log x + 1) /
            (x ^ 2 * Real.log x ^ 2)) := intervalIntegral.integral_neg
  rw [hInt] at habel
  have hY0 : (Y : ℝ) ≠ 0 := by positivity
  have hlogY0 : Real.log (Y : ℝ) ≠ 0 :=
    Real.log_ne_zero_of_pos_of_ne_one (by positivity) (by exact_mod_cast (show Y ≠ 1 by omega))
  calc
    fullReciprocalSum Y =
        (((Y : ℝ) * Real.log (Y : ℝ))⁻¹ * Chebyshev.theta (Y : ℝ)) +
          ∫ x in (2 : ℝ)..Y,
            Chebyshev.theta x * (Real.log x + 1) /
              (x ^ 2 * Real.log x ^ 2) := by
      linarith
    _ = Chebyshev.theta (Y : ℝ) /
          ((Y : ℝ) * Real.log (Y : ℝ)) +
        ∫ t in (2 : ℝ)..Y,
          Chebyshev.theta t * (Real.log t + 1) /
            (t ^ 2 * Real.log t ^ 2) := by
      field_simp

/-- Local integrability of the first Abel kernel. -/
theorem intervalIntegrable_theta_div_sq {Y : ℝ} (hY : 2 ≤ Y) :
    IntervalIntegrable (fun t : ℝ => Chebyshev.theta t / t ^ 2)
      MeasureTheory.volume 2 Y := by
  rw [intervalIntegrable_iff_integrableOn_Icc_of_le hY]
  conv =>
    arg 1
    ext t
    rw [Chebyshev.theta, div_eq_mul_inv, mul_comm, Finset.sum_filter]
  refine integrableOn_mul_sum_Icc _ (by norm_num) ?_
  apply ContinuousOn.integrableOn_Icc
  intro t ht
  have ht0 : t ≠ 0 := by linarith [ht.1]
  exact ContinuousAt.continuousWithinAt ((continuousAt_id.pow 2).inv₀ (pow_ne_zero 2 ht0))

/-- Local integrability of the Abel-summation integrand. -/
theorem intervalIntegrable_mertensIntegrand {Y : ℝ} (hY : 2 ≤ Y) :
    IntervalIntegrable
      (fun t : ℝ => Chebyshev.theta t * (Real.log t + 1) /
        (t ^ 2 * Real.log t ^ 2)) MeasureTheory.volume 2 Y := by
  rw [intervalIntegrable_iff_integrableOn_Icc_of_le hY]
  have hg : MeasureTheory.IntegrableOn
      (fun t : ℝ =>
        ((Real.log t + 1) / (t ^ 2 * Real.log t ^ 2)) * Chebyshev.theta t)
      (Set.Icc (2 : ℝ) Y) := by
    conv =>
      arg 1
      ext t
      rw [Chebyshev.theta, Finset.sum_filter]
    refine integrableOn_mul_sum_Icc _ (by norm_num) ?_
    apply ContinuousOn.integrableOn_Icc
    intro t ht
    have ht0 : t ≠ 0 := by linarith [ht.1]
    have hlog0 : Real.log t ≠ 0 :=
      Real.log_ne_zero_of_pos_of_ne_one (by linarith [ht.1]) (by linarith [ht.1])
    exact (((Real.continuousAt_log ht0).add continuousAt_const).div
        ((continuousAt_id.pow 2).mul ((Real.continuousAt_log ht0).pow 2))
        (mul_ne_zero (pow_ne_zero 2 ht0) (pow_ne_zero 2 hlog0))).continuousWithinAt
  exact hg.congr_fun (by
    intro t ht
    ring) measurableSet_Icc

















end Erdos390.Full.PrimeSums

end

/- Source module: Erdos390.Full.OmittedTiltMarkedProbability -/
section


/-!
# Actual omitted-score marked probabilities

This module joins the paper-scale structured-cell count to the genuine
finite valuation tilt.  The local forced primes are omitted from the score,
so their marked modulus is coprime to every score prime.  No marked-number
asymptotic or normalization estimate remains as an input.
-/

open Filter Topology
open scoped BigOperators

namespace Erdos390.Full.OmittedTiltMarkedProbability

open ArithmeticModel Scale PrimeSums
open StructuredCells HeadPattern
open DivisibilityMomentBounds FiniteProbability
open PaperScaleMarkedCell PaperTiltSmallness ValuationScoreDomination
open DickmanBasic

noncomputable section

/-- The explicit common majorant for every omitted-score subset of the
moving prime band. -/
def omittedTiltMajorant (B C c : ℝ) (W n : ℕ) : ℝ :=
  Real.exp
      ((B / L n) *
        (Real.log (physicalBound C n : ℝ) / Real.log (W : ℝ))) *
    (B / L n) * ((2 / c) * bandReciprocalSum n W)





/-- A single nonnegative remainder combining the untilted marked-count
error and the omitted-score normalization error. -/
def omittedTiltRemainder (K B C c : ℝ) (W n : ℕ) : ℝ :=
  |K / L n| +
    2 * |omittedTiltMajorant B C c W n| *
      (1 + 1 / rho DickmanBasic.U + |K / L n|)





/-! ## The actual omitted-score marked law -/



end

end Erdos390.Full.OmittedTiltMarkedProbability

end

/- Source module: Erdos390.Full.LocalFugacity -/
section


/-!
# Exact restoration of one local fugacity

This proves the finite telescoping identity used when an omitted prime is
restored in the tilted measure.  The coefficients are the actual
`λ^a-λ^(a-1)` coefficients and the indicators are actual divisibility
indicators.
-/

open scoped BigOperators

namespace Erdos390.Full.LocalFugacity

open ArithmeticModel

def coefficient (lam : ℝ) (a : ℕ) : ℝ :=
  if a = 0 then 1 else lam ^ a - lam ^ (a - 1)

@[simp] theorem coefficient_zero (lam : ℝ) : coefficient lam 0 = 1 := by
  simp [coefficient]













end Erdos390.Full.LocalFugacity

end

/- Source module: Erdos390.Full.LocalFugacityRestoration -/
section


/-!
# Exact restoration of omitted local fugacities

The paper omits one or two forced primes from the global score and then
restores their local fugacity by a finite divisor-indicator expansion.  This
file proves that restoration identity for the actual valuation and for an
actual finite probability law.  No approximation or independence assumption
is used here.
-/

open scoped BigOperators

namespace Erdos390.Full.LocalFugacityRestoration

open ArithmeticModel LocalFugacity

noncomputable section

/-- The local divisor expansion, including the exponent-zero coefficient. -/
def localFactor (p A : ℕ) (lam : ℝ) (m : ℕ) : ℝ :=
  ∑ a ∈ Finset.Icc 0 A, coefficient lam a * divInd (p ^ a) m



















section Probability

variable {Omega : Type*} [Fintype Omega]





end Probability

end

end Erdos390.Full.LocalFugacityRestoration

end

/- Source module: Erdos390.Full.LocalFugacityBounds -/
section


/-!
# Uniform bounds for restored local fugacity coefficients

The local restoration in Lemma 7.5 must retain both the factor `1 / L`
and the geometric factor `p^{-r}`.  Cardinality bounds on the exponent set
are too crude for that purpose.  This module proves the required shifted
geometric estimates directly for the actual coefficients
`exp (eta/L)^a - exp (eta/L)^(a-1)`.
-/

open scoped BigOperators

namespace Erdos390.Full.LocalFugacityBounds

open LocalFugacity

noncomputable section





 def linearHalfTerm (i : ℕ) : ℝ :=
  ((i : ℝ) + 3) * (1 / (2 : ℝ)) ^ i











 def quadraticHalfTerm (i : ℕ) : ℝ :=
  (((2 * i + 1) * (i + 3) : ℕ) : ℝ) *
    (1 / (2 : ℝ)) ^ i



/-- A fixed absolute constant for the diagonal coefficient ledger.  Its
precise numerical value is irrelevant; what matters is that it is defined
before every tilt box and is proved finite by an arithmetic--geometric
series. -/
def quadraticHalfMass : ℝ := ∑' i : ℕ, quadraticHalfTerm i







/-- The weighted coefficient mass used when restoring one omitted local
prime.  The exponent cutoff is finite and literal. -/
def coefficientTail (p A r : ℕ) (eta L : ℝ) : ℝ :=
  ∑ a ∈ Finset.Icc 1 A,
    |coefficient (Real.exp (eta / L)) a| /
      ((p ^ max r a : ℕ) : ℝ)













end

end Erdos390.Full.LocalFugacityBounds

end

/- Source module: Erdos390.Full.ValuationCutoff -/
section


/-!
# The actual logarithmic valuation cutoff

The local fugacity expansion must be truncated at
`floor(log M / log p)`, not at the much larger physical endpoint `M`.
This module identifies Mathlib's `Nat.log p M` as a valid common cutoff for
all positive integers `m ≤ M`, proves the exact restored factor at that
cutoff, and compares it with the real logarithmic ratio used in the paper.
-/

namespace Erdos390.Full.ValuationCutoff

open ArithmeticModel LocalFugacityRestoration

noncomputable section

/-- The largest possible exponent of `p` below the physical endpoint `M`. -/
def valuationCutoff (p M : ℕ) : ℕ := Nat.log p M











end

end Erdos390.Full.ValuationCutoff

end

/- Source module: Erdos390.Full.TwoLocalRestoration -/
section


/-!
# Exact restoration of two omitted local primes

The omitted-score marked law is reweighted by the two literal local
valuation fugacities.  This file proves the exact finite identity connecting
that sequential reweighting to the full valuation tilt and then expands both
the marked numerator and the normalizer into prime-power indicators.
-/

open scoped BigOperators

namespace Erdos390.Full

open ArithmeticModel FiniteProbability ValuationScoreDomination
open LocalFugacity LocalFugacityRestoration ValuationCutoff

noncomputable section

namespace FiniteProbability

variable {Omega : Type*} [Fintype Omega]







end FiniteProbability

namespace TwoLocalRestoration

/-- Remove two distinct local primes from a finite score set. -/
def erasePair (P : Finset ℕ) (p q : ℕ) : Finset ℕ :=
  (P.erase p).erase q







section Cell

variable {S : Finset ℕ}





end Cell

end TwoLocalRestoration

end

end Erdos390.Full

end

/- Source module: Erdos390.Full.OmittedTiltPairChamber -/
section


/-!
# The actual omitted-tilt two-prime four-mark chamber

This file specializes the uniform marked-divisibility theorem to the literal
moduli `p^a q^b` occurring in the two-local restoration of Lemma 7.5.  All
side conditions are discharged from membership of `p,q` in the paper prime
band, the four-mark inequality `a+b <= 4`, and coprimality with the fixed
head.  In particular, the theorem below is about the genuine structured
cell and the genuine omitted valuation score, not an abstract probability
law or a conditional analytic contract.
-/

open Filter Topology

namespace Erdos390.Full.OmittedTiltPairChamber

open ArithmeticModel Scale StructuredCells HeadPattern
open FiniteProbability ValuationScoreDomination
open PaperScaleMarkedCell OmittedTiltMarkedProbability
open TwoLocalRestoration

noncomputable section

/-- The literal two-prime-power modulus used in the four-mark chamber. -/
def pairPower (p q a b : ℕ) : ℕ := p ^ a * q ^ b













end

end Erdos390.Full.OmittedTiltPairChamber

end

/- Source module: Erdos390.Full.GuardDeletionProbability -/
section


/-!
# Exact perturbation bounds for deleting a finite guard set

This module contains the finite-probability algebra used by the guarded-cell
corollary after the arithmetic guard census supplies a cardinality bound.
Deletion is defined on the original finite sample type (guarded points receive
mass zero) and is normalized exactly.  The expectation and covariance losses
are proved from the deleted mass; they are not stored in a certificate.
-/

open scoped BigOperators

namespace Erdos390.Full
namespace FiniteProbability

noncomputable section

variable {Omega : Type*} [Fintype Omega] [DecidableEq Omega]

/-- Probability mass carried by the declared guard set. -/
def guardMass (mu : FiniteProbability Omega) (G : Finset Omega) : ℝ :=
  ∑ x ∈ G, mu.mass x





/- A guard census and a point-mass bound give the deleted-mass bound used in
the marked-cell application.  This is an inequality for the actual deleted
set, rather than a total-variation hypothesis stored in a certificate. -/








theorem sum_mass_complement_guard
    (mu : FiniteProbability Omega) (G : Finset Omega) :
    (∑ x ∈ (Finset.univ \ G), mu.mass x) = 1 - mu.guardMass G := by
  have h := Finset.sum_sdiff (Finset.subset_univ G) (f := mu.mass)
  rw [mu.mass_sum] at h
  unfold guardMass
  linarith

/-- The probability law obtained after deleting `G` and renormalizing. -/
def deleteGuards (mu : FiniteProbability Omega) (G : Finset Omega)
    (hsmall : mu.guardMass G < 1) : FiniteProbability Omega where
  mass x := if x ∈ G then 0 else mu.mass x / (1 - mu.guardMass G)
  mass_nonneg x := by
    split_ifs
    · exact le_rfl
    · exact div_nonneg (mu.mass_nonneg x) (sub_nonneg.mpr hsmall.le)
  mass_sum := by
    have hden : 1 - mu.guardMass G ≠ 0 := ne_of_gt (sub_pos.mpr hsmall)
    calc
      (∑ x, if x ∈ G then 0 else
          mu.mass x / (1 - mu.guardMass G)) =
          ∑ x ∈ (Finset.univ \ G),
            mu.mass x / (1 - mu.guardMass G) := by
        rw [Finset.sdiff_eq_filter, Finset.sum_filter]
        apply Finset.sum_congr rfl
        intro x hx
        by_cases hxG : x ∈ G <;> simp [hxG]
      _ = (∑ x ∈ (Finset.univ \ G), mu.mass x) /
          (1 - mu.guardMass G) := by rw [Finset.sum_div]
      _ = 1 := by
        rw [mu.sum_mass_complement_guard G]
        exact div_self hden

@[simp] theorem deleteGuards_mass_of_mem
    (mu : FiniteProbability Omega) (G : Finset Omega)
    (hsmall : mu.guardMass G < 1) {x : Omega} (hx : x ∈ G) :
    (mu.deleteGuards G hsmall).mass x = 0 := by
  simp [deleteGuards, hx]

@[simp] theorem deleteGuards_mass_of_not_mem
    (mu : FiniteProbability Omega) (G : Finset Omega)
    (hsmall : mu.guardMass G < 1) {x : Omega} (hx : x ∉ G) :
    (mu.deleteGuards G hsmall).mass x =
      mu.mass x / (1 - mu.guardMass G) := by
  simp [deleteGuards, hx]

/-- Guard contribution to an unnormalized moment. -/
def guardMoment (mu : FiniteProbability Omega) (G : Finset Omega)
    (F : Omega → ℝ) : ℝ :=
  ∑ x ∈ G, mu.mass x * F x



/-- A normalized scalar measuring the effect of deleting the guard mass. -/
def guardPerturbation (mu : FiniteProbability Omega) (G : Finset Omega) : ℝ :=
  2 * mu.guardMass G / (1 - mu.guardMass G)



/- On the paper's small-deletion regime, the exact renormalization loss is
at most four times the deleted probability mass. -/










end

end FiniteProbability
end Erdos390.Full

end

/- Source module: Erdos390.Full.GuardedUniformCell -/
section


/-!
# Guard deletion on an actual uniform arithmetic cell

The paper deletes an ambient finite guard set from each structured cell.
This module constructs the literal guard subset of the cell, proves that its
cardinality is no larger than the ambient guard census, and combines that fact
with the bounded-tilt density ratio.  No total-variation or covariance error is
assumed.
-/

namespace Erdos390.Full

open FiniteProbability

noncomputable section

namespace GuardedUniformCell

variable {Alpha : Type*} [DecidableEq Alpha]

/-- Members of `S` whose ambient value lies in the guard set `G`. -/
def guardSubtype (S G : Finset Alpha) : Finset S :=
  Finset.univ.filter fun x : S => (x : Alpha) ∈ G

@[simp] theorem mem_guardSubtype {S G : Finset Alpha} {x : S} :
    x ∈ guardSubtype S G ↔ (x : Alpha) ∈ G := by
  simp [guardSubtype]









end GuardedUniformCell

end

end Erdos390.Full

end

/- Source module: Erdos390.Full.PaperGuardCensus -/
section


/-!
# The concrete numerical-guard census used by the smooth bridge

The rough construction has only two kinds of numerical occurrences which can
meet one of the active smooth cells:

* promoted smooth anchors, in `O(y)` slots;
* endpoints (and the possible donor occurrence) of an `O(y)` family of bank
  paths, with at most one component at each geometric scale.

This file represents those occurrences by a literal finite slot type and
defines the numerical guard set as the image of its value map.  Consequently
the guard-cardinality estimate is a theorem about `Finset.image`; it is not a
field of a contract and it is not assumed about an arbitrary finset.

The number of geometric slots is fixed here to `ceil(log n)+1`.  This is a
harmless enlargement of the paper's geometric-scale list and makes the
`O(y log n)` census completely explicit.  Multiple slots are allowed to map to
the same integer; this only makes the proved image-cardinality bound stronger.
-/

open scoped BigOperators
open Filter Topology

namespace Erdos390.Full.PaperGuardCensus

open ArithmeticModel Scale StructuredCells HeadPattern
open FiniteProbability GuardedUniformCell ValuationTiltCell

noncomputable section

/-- A concrete upper bound for the number of geometric bank scales. -/
def scaleSlots (n : ℕ) : ℕ := ⌈L n⌉₊ + 1

/-- The three bank occurrences which are guarded at one component: its two
states and its designated backing donor.  A donor lying above the physical
cells simply has empty intersection with those cells. -/
inductive BankGuardRole
  | stateZero
  | stateOne
  | donor
  deriving DecidableEq, Fintype

/-- Literal occurrence slots in the bridge-relevant guard census.  The fixed
natural constants `Cprom` and `Cbank` are the construction's absolute path
multiplicities. -/
abbrev GuardSlot (n Cprom Cbank : ℕ) :=
  Fin (Cprom * yNat n) ⊕
    (Fin (Cbank * yNat n) × Fin (scaleSlots n) × BankGuardRole)

/-- The data retained from the bank/anchor construction: the actual integer
owned by every concrete occurrence slot.  There is deliberately no
cardinality field. -/
structure Ledger (n Cprom Cbank : ℕ) where
  value : GuardSlot n Cprom Cbank → ℕ

/-- The exhaustive numerical guards which can meet an active smooth cell. -/
def Ledger.guards {n Cprom Cbank : ℕ} (G : Ledger n Cprom Cbank) : Finset ℕ :=
  Finset.univ.image G.value

/-- Exact number of occurrence slots before possible collisions of their
integer values. -/
theorem card_guardSlot (n Cprom Cbank : ℕ) :
    Fintype.card (GuardSlot n Cprom Cbank) =
      Cprom * yNat n + Cbank * yNat n * scaleSlots n * 3 := by
  simp only [GuardSlot, Fintype.card_sum, Fintype.card_fin,
    Fintype.card_prod]
  have hrole : Fintype.card BankGuardRole = 3 := by decide
  rw [hrole]
  ring









/-- Explicit real majorant for the ratio `#guards / n`. -/
def censusRatioMajorant (Cprom Cbank : ℕ) (n : ℕ) : ℝ :=
  ((Cprom : ℝ) + 3 * (Cbank : ℝ) * (L n + 2)) * y n / (n : ℝ)











end

end Erdos390.Full.PaperGuardCensus

end

/- Source module: Erdos390.Proposition87 -/
section


/-!
# The finite and ODE algebra behind Proposition 8.7

The analytic inputs to the nonlinear fit (uniform covariance gaps, relative row
errors, and existence of an integral curve inside the preselected box) are kept as
explicit hypotheses.  This file proves the block solve, the slow-coordinate and
effective-velocity ledgers, compatibility recovery, and the straight-target chain
rule with no hidden limiting assertion.
-/

open Metric Set

namespace Erdos390

section BlockInverse

variable {Q : Type*} [AddCommGroup Q] [Module ℝ Q]









end BlockInverse

section SlowCoordinateBounds







end SlowCoordinateBounds

section GaugeAnnihilator

variable {E : Type*} [AddCommGroup E] [Module ℝ E]





end GaugeAnnihilator

section PrechosenBox

variable {E : Type*} [PseudoMetricSpace E]



end PrechosenBox

section StraightTarget

variable {E Y : Type*}
  [NormedAddCommGroup E] [NormedSpace ℝ E]
  [NormedAddCommGroup Y] [NormedSpace ℝ Y]

/-- Explicit path-existence hypothesis for the nonlinear continuation step.

In the paper this proposition is discharged by Picard--Lindelöf together with the
preselected-box continuation estimate.  Keeping it as a named predicate prevents
the chain-rule algebra below from silently assuming that the path remains in the
box. -/
def HasStraightTargetPath (Ω : Set E) (v : E → E) (x₀ : E) : Prop :=
  ∃ ξ : ℝ → E, ξ 0 = x₀ ∧
    (∀ t ∈ Icc (0 : ℝ) 1, ξ t ∈ Ω) ∧
    ∀ t ∈ Icc (0 : ℝ) 1,
      HasDerivWithinAt ξ (v (ξ t)) (Icc (0 : ℝ) 1) t









end StraightTarget

end Erdos390

end

/- Source module: Erdos390.Full.NonlinearFitODE -/
section


/-!
# The finite nonlinear moment lift

This file turns the scalar covariance--Jacobian identity for a finite
exponential family into the vector Jacobian used by the nonlinear fitting
argument.  A quantitative covariance gap is then used to construct the
inverse Jacobian (rather than assuming a right inverse), define the actual
ODE vector field, and prove its exact solve and velocity estimates.

The final section records a non-circular continuation theorem.  Its input is
an explicit covariance gap on a ball chosen before solving the ODE; local
Lipschitz continuity is derived from the finite exponential formulas and
operator inversion.
-/

open scoped BigOperators
open Metric Set

namespace Erdos390.Full

noncomputable section

/-- A self-dual finite exponential family.  The tilt score of a sample is the
inner product with its statistic. -/
structure VectorExponentialFamily
    (Omega E : Type*) [Fintype Omega]
    [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] where
  baseWeight : Omega → ℝ
  baseWeight_nonneg : ∀ omega, 0 ≤ baseWeight omega
  baseMass_pos : 0 < ∑ omega, baseWeight omega
  scale : ℝ
  scale_pos : 0 < scale
  statistic : Omega → E

namespace VectorExponentialFamily

variable {Omega E : Type*} [Fintype Omega]
  [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]

/-- The scalar family underlying a vector exponential family. -/
def scalarFamily (vf : VectorExponentialFamily Omega E) :
    FiniteExponentialFamily Omega E where
  baseWeight := vf.baseWeight
  baseWeight_nonneg := vf.baseWeight_nonneg
  baseMass_pos := vf.baseMass_pos
  scale := vf.scale
  scale_pos := vf.scale_pos
  score := fun omega => (innerSL ℝ) (vf.statistic omega)

/-- Total active mass. -/
def baseMass (vf : VectorExponentialFamily Omega E) : ℝ :=
  vf.scalarFamily.baseMass

theorem baseMass_positive (vf : VectorExponentialFamily Omega E) :
    0 < vf.baseMass :=
  vf.scalarFamily.baseMass_positive

/-- The exact normalized tilted probability mass. -/
def probabilityMass (vf : VectorExponentialFamily Omega E)
    (xi : E) (omega : Omega) : ℝ :=
  vf.scalarFamily.probabilityMass xi omega

/-- The active vector moment map. -/
def vectorMoment (vf : VectorExponentialFamily Omega E) (xi : E) : E :=
  ∑ omega, vf.scalarFamily.activeWeight xi omega • vf.statistic omega

/-- Kronecker test function used to differentiate one active weight. -/
def delta (omega : Omega) (eta : Omega) : ℝ := by
  classical
  exact if eta = omega then 1 else 0

@[simp]
theorem moment_delta (vf : VectorExponentialFamily Omega E)
    (xi : E) (omega : Omega) :
    vf.scalarFamily.moment (delta omega) xi =
      vf.scalarFamily.activeWeight xi omega := by
  classical
  simp [FiniteExponentialFamily.moment, delta]

/-- Derivative of one active coordinate, obtained from the already proved
finite covariance identity. -/
def activeWeightFDeriv (vf : VectorExponentialFamily Omega E)
    (xi : E) (omega : Omega) : E →L[ℝ] ℝ :=
  (vf.baseMass / vf.scale) •
    vf.scalarFamily.covarianceScore (delta omega) xi

theorem hasFDerivAt_activeWeight (vf : VectorExponentialFamily Omega E)
    (xi : E) (omega : Omega) :
    HasFDerivAt (fun eta => vf.scalarFamily.activeWeight eta omega)
      (vf.activeWeightFDeriv xi omega) xi := by
  have h := vf.scalarFamily.hasFDerivAt_moment_covariance
    (delta omega) xi
  have hfun : vf.scalarFamily.moment (delta omega) =
      fun eta => vf.scalarFamily.activeWeight eta omega := by
    funext eta
    exact vf.moment_delta eta omega
  rw [hfun] at h
  simpa only [activeWeightFDeriv, baseMass] using h

/-- The concrete vector Jacobian.  It is a finite sum of the derivatives of
the active weights, each multiplied by its statistic vector. -/
def jacobian (vf : VectorExponentialFamily Omega E) (xi : E) : E →L[ℝ] E :=
  ∑ omega, (vf.activeWeightFDeriv xi omega).smulRight (vf.statistic omega)

/-- Exact Fréchet derivative of the vector moment map. -/
theorem hasFDerivAt_vectorMoment (vf : VectorExponentialFamily Omega E)
    (xi : E) :
    HasFDerivAt vf.vectorMoment (vf.jacobian xi) xi := by
  have h := HasFDerivAt.sum (u := (Finset.univ : Finset Omega))
    (fun omega (_ : omega ∈ (Finset.univ : Finset Omega)) =>
      (vf.hasFDerivAt_activeWeight xi omega).smul_const
        (vf.statistic omega))
  have hfun : vf.vectorMoment =
      ∑ omega, fun eta =>
        vf.scalarFamily.activeWeight eta omega • vf.statistic omega := by
    funext eta
    simp only [vectorMoment, Finset.sum_apply]
  rw [hfun]
  simpa only [jacobian] using h

/-- Pairing the vector moment with a direction gives the corresponding
scalar active moment. -/
theorem inner_vectorMoment (vf : VectorExponentialFamily Omega E)
    (x xi : E) :
    inner ℝ x (vf.vectorMoment xi) =
      vf.scalarFamily.moment (fun omega => inner ℝ x (vf.statistic omega)) xi := by
  simp only [vectorMoment, FiniteExponentialFamily.moment, inner_sum,
    real_inner_smul_right]

/-- The tilted statistics as an actual finite pattern mixture. -/
def tiltedMixture (vf : VectorExponentialFamily Omega E) (xi : E) :
    PatternMixture Omega E where
  weight := vf.probabilityMass xi
  weight_nonneg := vf.scalarFamily.probabilityMass_nonneg xi
  weight_sum := vf.scalarFamily.probabilityMass_sum xi
  pattern := vf.statistic

/-- Exact quadratic covariance identity for the vector Jacobian. -/
theorem inner_jacobian_self (vf : VectorExponentialFamily Omega E)
    (xi x : E) :
    inner ℝ x (vf.jacobian xi x) =
      (vf.baseMass / vf.scale) * (vf.tiltedMixture xi).covarianceForm x := by
  let F : Omega → ℝ := fun omega => inner ℝ x (vf.statistic omega)
  have hleft : HasFDerivAt
      (fun eta => inner ℝ x (vf.vectorMoment eta))
      (((innerSL ℝ) x).comp (vf.jacobian xi)) xi := by
    exact ((innerSL ℝ) x).hasFDerivAt.comp xi
      (vf.hasFDerivAt_vectorMoment xi)
  have hright : HasFDerivAt
      (vf.scalarFamily.moment F)
      ((vf.baseMass / vf.scale) •
        vf.scalarFamily.covarianceScore F xi) xi :=
    vf.scalarFamily.hasFDerivAt_moment_covariance F xi
  have hfun : (fun eta => inner ℝ x (vf.vectorMoment eta)) =
      vf.scalarFamily.moment F := by
    funext eta
    exact vf.inner_vectorMoment x eta
  have hmaps : ((innerSL ℝ) x).comp (vf.jacobian xi) =
      (vf.baseMass / vf.scale) •
        vf.scalarFamily.covarianceScore F xi := by
    exact hleft.unique (hfun ▸ hright)
  have happ := congrArg (fun T : E →L[ℝ] ℝ => T x) hmaps
  change inner ℝ x (vf.jacobian xi x) =
    (vf.baseMass / vf.scale) *
      vf.scalarFamily.covarianceScore F xi x at happ
  rw [vf.scalarFamily.covarianceScore_apply F xi x] at happ
  have hscore : (fun omega => vf.scalarFamily.score omega x) = F := by
    funext omega
    simp only [scalarFamily, innerSL_apply_apply, F]
    exact real_inner_comm _ _
  rw [hscore] at happ
  simpa only [tiltedMixture, PatternMixture.covarianceForm,
    PatternMixture.probability, FiniteExponentialFamily.covariance,
    FiniteExponentialFamily.tiltedProbability, probabilityMass, smul_eq_mul]
    using happ

/-- A pointwise spectral-gap hypothesis for the actual tilted finite law. -/
def HasCovarianceGap (vf : VectorExponentialFamily Omega E)
    (gamma : ℝ) (xi : E) : Prop :=
  ∀ x, gamma * ‖x‖ ^ 2 ≤ (vf.tiltedMixture xi).covarianceForm x

theorem jacobian_coercive (vf : VectorExponentialFamily Omega E)
    {gamma : ℝ} {xi : E}
    (hgap : vf.HasCovarianceGap gamma xi) (x : E) :
    ((vf.baseMass / vf.scale) * gamma) * ‖x‖ ^ 2 ≤
      inner ℝ x (vf.jacobian xi x) := by
  rw [vf.inner_jacobian_self xi x]
  simpa only [mul_assoc] using
    (mul_le_mul_of_nonneg_left (hgap x)
      (div_nonneg (le_of_lt vf.baseMass_positive)
        (le_of_lt vf.scale_pos)))

theorem jacobian_injective (vf : VectorExponentialFamily Omega E)
    {gamma : ℝ} (hgamma : 0 < gamma) {xi : E}
    (hgap : vf.HasCovarianceGap gamma xi) :
    Function.Injective (vf.jacobian xi) := by
  intro x y hxy
  have hzero : vf.jacobian xi (x - y) = 0 := by
    rw [map_sub, hxy, sub_self]
  have hcoer := vf.jacobian_coercive hgap (x - y)
  rw [hzero, inner_zero_right] at hcoer
  have hcpos : 0 < (vf.baseMass / vf.scale) * gamma :=
    mul_pos (div_pos vf.baseMass_positive vf.scale_pos) hgamma
  have hnorm : ‖x - y‖ = 0 := by
    by_contra hn
    have hnormpos : 0 < ‖x - y‖ :=
      lt_of_le_of_ne (norm_nonneg (x - y)) (Ne.symm hn)
    have hprodpos : 0 <
        ((vf.baseMass / vf.scale) * gamma) * ‖x - y‖ ^ 2 :=
      mul_pos hcpos (sq_pos_of_pos hnormpos)
    linarith
  exact sub_eq_zero.mp (norm_eq_zero.mp hnorm)

/-- The Jacobian as a continuous linear equivalence, constructed from the
proved covariance gap. -/
def jacobianEquiv [CompleteSpace E]
    (vf : VectorExponentialFamily Omega E)
    {gamma : ℝ} (hgamma : 0 < gamma) {xi : E}
    (hgap : vf.HasCovarianceGap gamma xi) : E ≃L[ℝ] E := by
  let J := vf.jacobian xi
  have hinj : Function.Injective J := vf.jacobian_injective hgamma hgap
  have hsurj : Function.Surjective J :=
    LinearMap.surjective_of_injective hinj
  exact ContinuousLinearEquiv.ofBijective J
    (LinearMap.ker_eq_bot.mpr hinj)
    (LinearMap.range_eq_top.mpr hsurj)



/-- The actual globally defined vector field.  At a point with a covariance
gap this is the genuine inverse-Jacobian lift of `target`. -/
def vectorField (vf : VectorExponentialFamily Omega E) (target : E)
    (xi : E) : E :=
  (vf.jacobian xi).inverse target





/-! ## Smoothness and non-circular continuation -/



















end VectorExponentialFamily

end

end Erdos390.Full

end
Source
https://github.com/ShouqiaoW/erdos/blob/61325b10bbdc29f4fb5e0618b414b9f2189333ad/390/lean/Erdos390/Full/FiniteExponentialFamily.lean; grouped definitions extracted from multiple source modules at this commit, including Erdos390.Full.RegularRelativeMesh, Erdos390.Full.SelectedDyadicRegularMesh, Erdos390.Full.FiniteExponentialFamily, with the remaining originating modules identified by source-module comments in the definition body.
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