Inspect dependencies
MathlibNt.SieveTheory.nat_lt_of_eq_ceil_iff · compiled type and proof/definition references.
theorem
MathlibNt.SieveTheory.suzukiVProduct_natCeil_eq
(S : BoundingSieve)
{x : ℝ}
{z : ℕ}
(hz : z = ⌈x⌉₊)
:
Suzuki's finite Euler product is unchanged when a strict real cutoff is replaced by its natural ceiling.
Inspect dependencies
MathlibNt.SieveTheory.suzukiVProduct_natCeil_eq · compiled type and proof/definition references.
theorem
MathlibNt.SieveTheory.suzukiLemmaEightSevenPrimeSum_natCeil_eq
(S : BoundingSieve)
(D w v x : ℝ)
(F : ℝ → ℝ)
{z : ℕ}
(hz : z = ⌈x⌉₊)
:
SwitchingPrinciple.suzukiLemmaEightSevenPrimeSum S D w v (↑z) F = SwitchingPrinciple.suzukiLemmaEightSevenPrimeSum S D w v x F
The Lemma-8.7 prime sum is likewise insensitive to replacing its Euler suffix cutoff by the natural ceiling.
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MathlibNt.SieveTheory.suzukiLemmaEightSevenPrimeSum_natCeil_eq · compiled type and proof/definition references.
theorem
MathlibNt.SieveTheory.sigmaEleven_add_sigmaTwelve_suzukiVProduct_le_finiteSourceLayer_add_qD_natCeil
{S : BoundingSieve}
{H : SwitchingPrinciple.SuzukiLemma144KappaOne.Section13HatLayers}
{β C K d Δ s τ σ : ℝ}
{N D z : ℕ}
(hH : SwitchingPrinciple.SuzukiLemma144KappaOne.Section13HatContract H β)
(hsdom : s ∈ SuzukiFiniteContinuousLayers.KappaOneModel.parityDomain β N)
(hτdom : τ - 1 ∈ SuzukiFiniteContinuousLayers.KappaOneModel.parityDomain β (N - 1))
(hsτ : s ≤ τ)
(hτσ : τ ≤ σ)
(hD : 1 < ↑D)
(_hz2 : 2 ≤ ↑z)
(hroot2 : 2 ≤ ↑D ^ (1 / s))
(hv2 : 2 ≤ ↑D ^ (1 / τ))
(hw2 : 2 ≤ ↑D ^ (1 / σ))
(hwv : ↑D ^ (1 / σ) ≤ ↑D ^ (1 / τ))
(hvroot : ↑D ^ (1 / τ) ≤ ↑D ^ (1 / s))
(hzceil : z = ⌈↑D ^ (1 / s)⌉₊)
(hτerr : H.betaHat + (SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth N).epsilon < τ)
(hK : 2 ≤ K)
(hlocal : SwitchingPrinciple.HasDimensionOneLocalProductBound S K)
(hii : SwitchingPrinciple.SuzukiLemma144KappaOne.Claim14_6_MonotoneQPremise H (↑D) d Δ σ)
(hC : 0 ≤ C)
(hΔ : 0 ≤ Δ)
(hCeil : ∀ p ∈ S.prodPrimes.primeFactors, ↑D ^ (1 / σ) ≤ ↑p → ↑p < ↑D ^ (1 / τ) → 2 ≤ p ∧ 2 * p ≤ D)
(hT :
∀ p ∈ S.prodPrimes.primeFactors,
↑D ^ (1 / σ) ≤ ↑p →
↑p < ↑D ^ (1 / τ) →
0 ≤ H.T (SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth (N - 1))
(SwitchingPrinciple.SuzukiLemma144Equation1410.inheritedCoordinate D p))
(h1413 :
∀ p ∈ S.prodPrimes.primeFactors,
↑D ^ (1 / σ) ≤ ↑p →
↑p < ↑D ^ (1 / τ) →
SwitchingPrinciple.SuzukiLemma144KappaOne.Claim14_13PointwisePremise H N (↑D) d Δ (Real.log ↑D / Real.log ↑p)
(↑D / ↑p))
:
SwitchingPrinciple.SuzukiLemma144Equation1410.sigmaEleven (SwitchingPrinciple.suzukiSupportedBelow S z) (⇑S.nu)
(fun (p : ℕ) => SwitchingPrinciple.suzukiVProduct S ↑p) (SwitchingPrinciple.suzukiVProduct S ↑z) β N D σ τ + SwitchingPrinciple.SuzukiLemma144Equation1410.sigmaTwelve S.prodPrimes.primeFactors (⇑S.nu)
(fun (p : ℕ) => SwitchingPrinciple.suzukiVProduct S ↑p)
(fun (n D' : ℕ) (x : ℝ) => SwitchingPrinciple.SuzukiLemma144KappaOne.errorEnvelope H n (↑D') d x)
(SwitchingPrinciple.suzukiVProduct S ↑z) C K Δ N D σ τ ≤ SwitchingPrinciple.suzukiVProduct S ↑z * (SuzukiFiniteContinuousLayers.finiteSourceLayer 1 β N s + 6 * K ^ 2 * SuzukiFiniteContinuousLayers.finiteSourceLayer 1 β (N - 1) (τ - 1) / Real.log (↑D ^ (1 / σ)) * (τ / s)) + C * Real.exp √K * SwitchingPrinciple.suzukiVProduct S ↑z * Real.log ↑D ^ (-Δ) * ((1 / s * ∫ (t : ℝ) in τ..σ, SwitchingPrinciple.SuzukiLemma144KappaOne.qD H
(SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth N).opposite (↑D) d Δ t) + 6 * K ^ 2 * SwitchingPrinciple.SuzukiLemma144KappaOne.qD H
(SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth N).opposite (↑D) d Δ τ / Real.log (↑D ^ (1 / σ)) * (τ / s))
Rounded version of the concrete middle provider. Its only additional datum
is the exact carrier identity z = ceil(D^(1/s)); no cutoff error is added.
Inspect dependencies
MathlibNt.SieveTheory.sigmaEleven_add_sigmaTwelve_suzukiVProduct_le_finiteSourceLayer_add_qD_natCeil · compiled type and proof/definition references.
theorem
MathlibNt.SieveTheory.caseII_endpoint_le_concrete_finiteSourceLayer_add_qD_natCeil
(S : BoundingSieve)
(H : SwitchingPrinciple.SuzukiLemma144KappaOne.Section13HatLayers)
{N D y : ℕ}
{β σ C C1 K ΘK Δ d B0 : ℝ}
(hH : SwitchingPrinciple.SuzukiLemma144KappaOne.Section13HatContract H β)
(hN : Odd N)
(hN2 : 2 ≤ N)
(hycube : ∀ p ∈ SwitchingPrinciple.suzukiSupportedBelow S y, p ^ 3 < D)
(hyceil : y = ⌈↑D ^ (1 / (β + 1))⌉₊)
(_hyDhalf : ↑y ≤ ↑D / 2)
(hβ1σ : β + 1 ≤ σ)
(hD : 1 < ↑D)
(hDlarge : β * Real.log 2 ≤ (β - 1) * Real.log ↑D)
(hwy : ↑D ^ (1 / σ) ≤ ↑D ^ (1 / (β + 1)))
(hy2 : 2 ≤ ↑y)
(hroot2 : 2 ≤ ↑D ^ (1 / (β + 1)))
(hw2 : 2 ≤ ↑D ^ (1 / σ))
(hEndpoint :
SwitchingPrinciple.SuzukiLemma144KappaOne.Claim14_5Regime β (↑D) σ C1 K ΘK σ →
∑ p ∈ SwitchingPrinciple.suzukiSupportedBelow S ⌈↑D ^ (1 / σ)⌉₊,
S.nu p * ∑ m ∈ sourceParityIndices (N - 1), suzukiSourceV S m (D ⌈/⌉ p) p ≤ B0)
(hnu :
∀
p ∈
SwitchingPrinciple.SuzukiLemma144Equation1410.sigmaOneCarrier (SwitchingPrinciple.suzukiSupportedBelow S y) D σ
(β + 1),
0 ≤ S.nu p)
(hC : 0 ≤ C)
(hlog :
∀
p ∈
SwitchingPrinciple.SuzukiLemma144Equation1410.sigmaOneCarrier (SwitchingPrinciple.suzukiSupportedBelow S y) D σ
(β + 1),
0 ≤ Real.log ↑(D ⌈/⌉ p))
(hSourceDomain :
∀
p ∈
SwitchingPrinciple.SuzukiLemma144Equation1410.sigmaOneCarrier (SwitchingPrinciple.suzukiSupportedBelow S y) D σ
(β + 1),
SwitchingPrinciple.SuzukiLemma144Equation1410.inheritedCoordinate D p ∈ SuzukiFiniteContinuousLayers.KappaOneModel.parityDomain β (N - 1) ∧ SwitchingPrinciple.SuzukiLemma144Equation1410.recursiveCoordinate D p ∈ SuzukiFiniteContinuousLayers.KappaOneModel.parityDomain β (N - 1))
(hClaim14_6_i :
∀
p ∈
SwitchingPrinciple.SuzukiLemma144Equation1410.sigmaOneCarrier (SwitchingPrinciple.suzukiSupportedBelow S y) D σ
(β + 1),
SwitchingPrinciple.SuzukiLemma144KappaOne.Claim14_6_MonotoneLambdaPremise H (↑(D ⌈/⌉ p)) d σ)
(hErrorDomain :
∀
p ∈
SwitchingPrinciple.SuzukiLemma144Equation1410.sigmaOneCarrier (SwitchingPrinciple.suzukiSupportedBelow S y) D σ
(β + 1),
SwitchingPrinciple.SuzukiLemma144Equation1410.inheritedCoordinate D p ∈ Set.Icc (H.betaHat + (SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth (N - 1)).epsilon) σ ∧ SwitchingPrinciple.SuzukiLemma144Equation1410.recursiveCoordinate D p ∈ Set.Icc (H.betaHat + (SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth (N - 1)).epsilon) σ)
(hIH :
SwitchingPrinciple.SuzukiLemma144Equation1410.NaturalCeilPointwiseInductionContract
(SwitchingPrinciple.suzukiSupportedBelow S y)
(fun (n D' p : ℕ) => ∑ m ∈ sourceParityIndices n, suzukiSourceV S m D' p)
(fun (p : ℕ) => SwitchingPrinciple.suzukiVProduct S ↑p)
(fun (n D' : ℕ) (x : ℝ) => SwitchingPrinciple.SuzukiLemma144KappaOne.errorEnvelope H n (↑D') d x) β C K Δ N D σ
(β + 1))
(hErrorThreshold : H.betaHat + (SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth N).epsilon < β + 1)
(hK : 2 ≤ K)
(hlocal : SwitchingPrinciple.HasDimensionOneLocalProductBound S K)
(hClaim14_6_ii : SwitchingPrinciple.SuzukiLemma144KappaOne.Claim14_6_MonotoneQPremise H (↑D) d Δ σ)
(hΔ : 0 ≤ Δ)
(hCeilFull : ∀ p ∈ S.prodPrimes.primeFactors, ↑D ^ (1 / σ) ≤ ↑p → ↑p < ↑D ^ (1 / (β + 1)) → 2 ≤ p ∧ 2 * p ≤ D)
(hT :
∀ p ∈ S.prodPrimes.primeFactors,
↑D ^ (1 / σ) ≤ ↑p →
↑p < ↑D ^ (1 / (β + 1)) →
0 ≤ H.T (SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth (N - 1))
(SwitchingPrinciple.SuzukiLemma144Equation1410.inheritedCoordinate D p))
(hClaim14_13 :
∀ p ∈ S.prodPrimes.primeFactors,
↑D ^ (1 / σ) ≤ ↑p →
↑p < ↑D ^ (1 / (β + 1)) →
SwitchingPrinciple.SuzukiLemma144KappaOne.Claim14_13PointwisePremise H N (↑D) d Δ (Real.log ↑D / Real.log ↑p)
(↑D / ↑p))
:
∑ n ∈ sourceParityIndices N, suzukiSourceV S n D y ≤ B0 + (SwitchingPrinciple.suzukiVProduct S ↑y * (SuzukiFiniteContinuousLayers.finiteSourceLayer 1 β N (β + 1) + 6 * K ^ 2 * SuzukiFiniteContinuousLayers.finiteSourceLayer 1 β (N - 1) β / Real.log (↑D ^ (1 / σ)) * ((β + 1) / (β + 1))) + C * Real.exp √K * SwitchingPrinciple.suzukiVProduct S ↑y * Real.log ↑D ^ (-Δ) * ((1 / (β + 1) * ∫ (t : ℝ) in β + 1..σ, SwitchingPrinciple.SuzukiLemma144KappaOne.qD H
(SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth N).opposite (↑D) d Δ t) + 6 * K ^ 2 * SwitchingPrinciple.SuzukiLemma144KappaOne.qD H
(SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth N).opposite (↑D) d Δ (β + 1) / Real.log (↑D ^ (1 / σ)) * ((β + 1) / (β + 1))))
Rounded Case-II endpoint. The source cutoff is the natural ceiling of the real Case-II power coordinate; no perfect-power identity and no cutoff error term is assumed.
Inspect dependencies
MathlibNt.SieveTheory.caseII_endpoint_le_concrete_finiteSourceLayer_add_qD_natCeil · compiled type and proof/definition references.