Diagonal centering error in the sharp reference row

Proved

Erdos390.Full.PaperBridgeFit.BridgeData.abs_referenceCenteringDiagonal_le

by doctosil · Sep 17, 2026 · Mathlib c5ea003 (Lean v4.30.0)

analytic-number-theoryerdos-390erdos390-source-construction

Write $L_n=\log n$ , $y_n=n^{2/9}$ , $Y_n=\lfloor y_n\rfloor$ , and $\mathcal P_{n,W}=\{p\text{ prime}:W<p\le Y_n\}$ . Let $B$ be a bridge with finite head and band sets, and write $t_p=\log p/\log y_n$ , $h_i$ for harmonic band masses, $\alpha_i$ for band centers, and $j(p)$ for the band of $p$ . Let $R_{pr}$ be its squarefree Poisson–Dickman reference entry, $I_p=\mathbf1_{p\mid m}$ , and $\mu_\xi$ its tilted law. Write $d_p=\alpha_{j(p)}-t_p$ for the prime deviation. Let $w,C_F,R\ge0$ . Assume $|F(t_p)-F(\alpha_{j(p)})|\le C_F|d_p|$ and $|d_p|\le w$ for every band prime, and $|x_j|\le R$ for every band. Then

|E_i^{\rm diag}(x)|\le2C_FwR\qquad\text{for every }i,

where $E^{\rm diag}$ is the source diagonal centering error.

Exact fiber centering gives a relative estimate without dividing the final bound by the smallest band center.

Preamble

import Mathlib
import Definitions.Def_erdos390_source_bank_interface
import Definitions.Def_erdos390_dickman_step_function
import Theorems.Thm_MediumPNT
import Definitions.Def_erdos390_analytic_foundations_001
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
import Definitions.Def_erdos390_analytic_foundations_002
import Definitions.Def_erdos390_analytic_foundations_005

universe u_1 u_2

Formal statement

theorem Erdos390.Full.PaperBridgeFit.BridgeData.abs_referenceCenteringDiagonal_le :
∀ {Head : Type u_1} {Band : Type u_2} [inst : Fintype.{u_1} Head] [inst_1 : DecidableEq.{u_1 + 1} Head]
  [inst_2 : Fintype.{u_2} Band] [inst_3 : DecidableEq.{u_2 + 1} Band]
  (B : Erdos390.Full.PaperBridgeFit.BridgeData.{u_1, u_2} Head Band) {w CF R : Real},
  LE.le.{0} (0 : Real) w →
    LE.le.{0} (0 : Real) CF →
      LE.le.{0} (0 : Real) R →
        (∀
            (p :
              Erdos390.Full.ArithmeticBandGeometry.BandPrime
                (Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1}
                  (Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} B))
                (Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1}
                  (Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} B))),
            LE.le.{0}
              (abs.{0}
                (HSub.hSub.{0, 0, 0}
                  (Erdos390.Full.DickmanBasic.F
                    (Erdos390.Full.ArithmeticModel.tPrime
                      (Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1}
                        (Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} B))
                      ↑p))
                  (Erdos390.Full.DickmanBasic.F
                    (Erdos390.Full.PaperBridgeFit.BridgeData.bandCenter.{u_1, u_2} B
                      (Erdos390.Full.ArithmeticBandGeometry.Partition.band.{u_2}
                        (Erdos390.Full.PaperBridgeFit.BridgeData.partition.{u_1, u_2} B) p)))))
              (HMul.hMul.{0, 0, 0} CF
                (abs.{0} (Erdos390.Full.PaperBridgeFit.BridgeData.primeDeviation.{u_1, u_2} B p)))) →
          (∀
              (p :
                Erdos390.Full.ArithmeticBandGeometry.BandPrime
                  (Erdos390.Full.PaperBridgeFit.StructuredSampleData.n.{u_1}
                    (Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} B))
                  (Erdos390.Full.PaperBridgeFit.StructuredSampleData.W.{u_1}
                    (Erdos390.Full.PaperBridgeFit.BridgeData.sampleData.{u_1, u_2} B))),
              LE.le.{0} (abs.{0} (Erdos390.Full.PaperBridgeFit.BridgeData.primeDeviation.{u_1, u_2} B p)) w) →
            ∀ (x : Band → Real),
              (∀ (j : Band), LE.le.{0} (abs.{0} (x j)) R) →
                ∀ (i : Band),
                  LE.le.{0}
                    (abs.{0} (Erdos390.Full.PaperBridgeFit.BridgeData.referenceCenteringDiagonal.{u_1, u_2} B x i))
                    (HMul.hMul.{0, 0, 0} (HMul.hMul.{0, 0, 0} (HMul.hMul.{0, 0, 0} (2 : Real) CF) w) R) := by sorry

Source

https://github.com/ShouqiaoW/erdos/blob/61325b10bbdc29f4fb5e0618b414b9f2189333ad/390/lean/Erdos390/Full/PaperReferenceSharpPrimeGraph.lean#L628-L753

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