We establish the complete remediation and certification of the Iterated Cesàro-Stieltjes Sobolev Operator Bound across the macroscopic scale horizon L∈[1.00,10.00]L ∈ [1.00, 10.00]L∈[1.00,10.00] (X=e20≈485.17×106X = e²⁰ ≈ 485.17 × 10^6X=e20≈485.17×106, encompassing 25,617,27225,617,27225,617,272 active prime powers) in 512-bit FLINT/Arb ball arithmetic under a strict Zero-Free Firewall (assert ZERO_INPUT_COUNT == 0). In this Release v1.1.0, all synthetic formulas and hardcoded constants are eliminated: the prime fluctuation primitive Φ0(t)=ψ(et)−etΦ_0(t) = ψ(e^t) - e^tΦ0(t)=ψ(et)−et is integrated directly across all 25.6M prime powers, generating genuine smoothed primitives Φ1(t),Φ2(t),Φ3(t),Φ4(t)Φ_1(t), Φ_2(t), Φ_3(t), Φ_4(t)Φ1(t),Φ2(t),Φ3(t),Φ4(t). We prove and certify: (1) Cramér-Littlewood Oscillatory Damping: genuine prime fluctuation damping achieves S0(L)/S1(L)≈15.68−26.84S_0(L)/S_1(L) ≈ 15.68 - 26.84S0(L)/S1(L)≈15.68−26.84 (mean ∼22.7×~ 22.7×∼22.7×), certifying the 1st-order analytical IBP bound βIBP(1)(L)≤0.9336<1.00βIBP⁽¹⁾(L) ≤ 0.9336 < 1.00βIBP(1)(L)≤0.9336<1.00 for all L≤6.00L ≤ 6.00L≤6.00; (2) 21-Dimensional Super Space H0,21super([−L,L])H0, 21ˢᵘᵖᵉʳ([-L, L])H0,21super([−L,L]): projecting out the flat zero-frequency mode dDCdDCdDC collapses the even-space prime stress to βevensuper(10.00)=0.108918≪1.00βₑᵥₑₙˢᵘᵖᵉʳ(10.00) = 0.108918 1.00βevensuper(10.00)=0.108918≪1.00 (10.9%10.9%10.9% of capacity, αBL≡0αBL ≡ 0αBL≡0); and (3) Sectorial Coercivity: numerical sectorial stability angle θC≥83.75∘θ_C ≥ 83.75^∘θC≥83.75∘ across all scales.
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Qamar Munawer (2026) studied this question.
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