Sharp fixed-head shift budget at the canonical row scale

Proved

Erdos390.WholePaper.roughCanonicalBalancedSharpFixedHeadBudget_le

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

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

Write $L=\log n$ , $Y_n=\lfloor n^{2/9}\rfloor$ , $h=\lceil cn/\log n\rceil$ , and $K=K_0+1$ . Fix natural W and $K_0$ , real $\beta$ , and $c>0$ . Suppose $n,y\ge2$ , $L\ge1$ , $\log y\ge L/5$ , $h\le2cn/L$ , and $Kh\le n$ . Let row be any canonical complete rough row of the raw candidate set at cutoff y whose label $r\le n$ . The source all-row sharp fixed-head shift budget $\mathcal H$ , evaluated with the head-balanced alpha and beta, satisfies

\mathcal H\le C_{\rm head}(W,K_0,c,\beta)\left(\frac{\lfloor n/r\rfloor}{L^2}+1\right).

The row definition ensures $r>0$ , and $C_{\rm head}$ is the explicit sharp head-row scale constant.

This controls fixed-head shifts uniformly over all eligible canonical rows, including the divisor contributions.

Preamble

import Definitions.Def_erdos390_remaining_analytic_propositions_008
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_006
import Definitions.Def_erdos390_analytic_foundations_008
import Definitions.Def_erdos390_analytic_foundations_009

Formal statement

theorem Erdos390.WholePaper.roughCanonicalBalancedSharpFixedHeadBudget_le : Erdos390.RemainingAnalyticGoal008_016 := by sorry

Source

https://github.com/ShouqiaoW/erdos/blob/61325b10bbdc29f4fb5e0618b414b9f2189333ad/390/lean/Erdos390/WholePaper/RoughSaiasSharpCanonicalRowPaperScale.lean#L1524-L2084

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