Unconditional upper Rosser finite-to-continuous producers #
The boundary/source successor identity is now produced internally, so the finite-prefix comparison no longer exposes it as an input.
theorem
MathlibNt.SieveTheory.SwitchingPrinciple.upperRosserFiniteBoundaryFactor_le_suzukiContinuousUpperFactor_of_source
(H : SuzukiLemma144KappaOne.Section13HatLayers)
(hH : SuzukiLemma144KappaOne.Section13HatSourceContract H)
(L : ℕ)
{s : ℝ}
(hslo : 3 / 2 ≤ s)
(hshi : s ≤ 4)
:
Finite continuous upper Rosser boundary factors are bounded by Suzuki's continuous upper factor, with the layer identity supplied internally.
theorem
MathlibNt.SieveTheory.SwitchingPrinciple.exists_upperRosserBoundaryPrefix_le_suzukiContinuousUpperFactor_sub_one_add_of_source
(H : SuzukiLemma144KappaOne.Section13HatLayers)
(hH : SuzukiLemma144KappaOne.Section13HatSourceContract H)
(L : ℕ)
(K ρ : ℝ)
(hK : 1 ≤ K)
(hρ : 0 < ρ)
:
∃ (z₀ : ℝ),
2 ≤ z₀ ∧ ∀ (S : BoundingSieve) (z Δ s : ℝ),
z₀ ≤ z →
0 < Δ →
HasDimensionOneLocalProductBound S K →
(∀ p ∈ S.prodPrimes.primeFactors, ↑p ≤ z) →
s = Real.log Δ / Real.log z →
3 / 2 ≤ s →
s ≤ 4 →
∑ k ∈ Finset.range L,
∑ q ∈ S.prodPrimes.primeFactors,
S.nu q / (1 - S.nu q) * LinearSieve.upperRosserBoundaryChainsFixedDepthDensity
(fun (p : ℕ) => S.nu p / (1 - S.nu p)) (⌊Δ⌋₊ + 1) q
({p ∈ S.prodPrimes.primeFactors | q < p}) k ≤ suzukiContinuousUpperFactor s - 1 + ρ
Fixed-depth mesh comparison landing below the genuine Suzuki upper factor; the source-layer producer is no longer a caller premise.
theorem
MathlibNt.SieveTheory.SwitchingPrinciple.exists_upperRosserBoundaryPrefix_le_jurkatRichertUpperFactor_sub_one_add_of_source
(H : SuzukiLemma144KappaOne.Section13HatLayers)
(hH : SuzukiLemma144KappaOne.Section13HatSourceContract H)
(L : ℕ)
(K ρ : ℝ)
(hK : 1 ≤ K)
(hρ : 0 < ρ)
:
∃ (z₀ : ℝ),
2 ≤ z₀ ∧ ∀ (S : BoundingSieve) (z Δ s : ℝ),
z₀ ≤ z →
0 < Δ →
HasDimensionOneLocalProductBound S K →
(∀ p ∈ S.prodPrimes.primeFactors, ↑p ≤ z) →
s = Real.log Δ / Real.log z →
3 / 2 ≤ s →
s ≤ 4 →
∑ k ∈ Finset.range L,
∑ q ∈ S.prodPrimes.primeFactors,
S.nu q / (1 - S.nu q) * LinearSieve.upperRosserBoundaryChainsFixedDepthDensity
(fun (p : ℕ) => S.nu p / (1 - S.nu p)) (⌊Δ⌋₊ + 1) q
({p ∈ S.prodPrimes.primeFactors | q < p}) k ≤ jurkatRichertUpperLinearSieveFactor s - 1 + ρ
Jurkat--Richert rewrite of the unconditional finite-prefix producer.