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One combined-charge capacity depth works for every admissible exceptional exponent

Proved
Erdos390.WholePaper.exists_depth_bankPaperCombinedChargeTerminal_uniform_deltaStar_compact

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

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

For every c>C0c>C_0c>C0​ there is a natural depth d≥201d\ge201d≥201 such that for every real δ∗\delta_*δ∗​ with 0<δ∗<1/180<\delta_*<1/180<δ∗​<1/18 and Cexc(c)δ∗/θexc≤(c−C0)/48C_{\rm exc}(c)\delta_*/\theta_{\rm exc}\le(c-C_0)/48Cexc​(c)δ∗​/θexc​≤(c−C0​)/48, the paper combined-charge terminal holds at depth d:

∃d≥201 ∀δ∗ admissible,eventually in n a compatible bank and guarded anchor certificate exist.\exists d\ge201\ \forall\delta_*\text{ admissible},\quad\text{eventually in }n\text{ a compatible bank and guarded anchor certificate exist}.∃d≥201 ∀δ∗​ admissible,eventually in n a compatible bank and guarded anchor certificate exist.

They satisfy anchor-times-base divisibility into the central tail product, base-bank and combined-selector-charge divisibility into the precharged target, retained reserve (c−C0)n/[24(p−1)log⁡n](c-C_0)n/[24(p-1)\log n](c−C0​)n/[24(p−1)logn] above the combined charge for primes p≤2d+1p\le2d+1p≤2d+1, and the exact selector-target-times-charge and precharged-target-times-anchor-divisor identities.

The depth is fixed before choosing the exceptional exponent; the eventual threshold may depend on that later exponent.

Preamble
import Definitions.Def_erdos390_remaining_analytic_propositions_008
Formal statement
theorem Erdos390.WholePaper.exists_depth_bankPaperCombinedChargeTerminal_uniform_deltaStar_compact : Erdos390.RemainingAnalyticGoal008_010 := by sorry
Source
https://github.com/ShouqiaoW/erdos/blob/61325b10bbdc29f4fb5e0618b414b9f2189333ad/390/lean/Erdos390/WholePaper/BankPaperCombinedChargeDepthFirstTerminal.lean#L74-L411

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