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Actual versus medium-only power correction under the same weights

Proved
Erdos390.Full.PaperBridgeFit.BridgeData.sum_abs_actual_powerCorrection_sub_physicalMedium_le_compact

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

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

For a valuation law λ\lambdaλ, put Δλ(p,q)=Cov⁡λ(vp,vq)−Cov⁡λ(1p∣m,1q∣m)\Delta_\lambda(p,q)=\operatorname{Cov}_\lambda(v_p,v_q)-\operatorname{Cov}_\lambda(\mathbf1_{p\mid m},\mathbf1_{q\mid m})Δλ​(p,q)=Covλ​(vp​,vq​)−Covλ​(1p∣m​,1q∣m​). Let BBB have nonempty head set and W>1W>1W>1, and fix a tilt parameter ξ\xiξ. Suppose positive cell densities ρc\rho_cρc​ obey #Sc≥ρchic\#S_c\ge\rho_c\mathrm{hi}_c#Sc​≥ρc​hic​. Let A,Aphys,Kphys,G≥0A,A_{\rm phys},K_{\rm phys},G\ge0A,Aphys​,Kphys​,G≥0 bound, respectively, every effective prime coefficient, the physical tilt coordinate, the absolute physical score, and each reciprocal-event constant

exp⁡ ⁣(2AB.Llog⁡hiclog⁡W)/ρc≤G.\exp\!\left(2\frac A{B.L}\frac{\log\mathrm{hi}_c}{\log W}\right)/\rho_c\le G.exp(2B.LA​logWloghic​​)/ρc​≤G.

Assume 8AphysKphys/B.L≤18A_{\rm phys}K_{\rm phys}/B.L\le18Aphys​Kphys​/B.L≤1. Then

∑q∣Δactual(p,q)−Δmedium(p,q)∣≤physicalPowerCorrectionMixtureBudget(AphysKphys/B.L,G,n,W,p).\sum_q|\Delta_{\rm actual}(p,q)-\Delta_{\rm medium}(p,q)|\le\mathrm{physicalPowerCorrectionMixtureBudget}(A_{\rm phys}K_{\rm phys}/B.L,G,n,W,p).q∑​∣Δactual​(p,q)−Δmedium​(p,q)∣≤physicalPowerCorrectionMixtureBudget(Aphys​Kphys​/B.L,G,n,W,p).

The reference removes only the residual physical score, retaining the actual reweighted component probabilities.

This compact declaration is definitionally identical to the original fully quantified statement; the named proposition preserves every hypothesis.

Preamble
import Definitions.Def_erdos390_analytic_compact_goals_003

universe u_1 u_2
Formal statement
theorem Erdos390.Full.PaperBridgeFit.BridgeData.sum_abs_actual_powerCorrection_sub_physicalMedium_le_compact : Erdos390.AnalyticCompactBatch003Goal001.{u_1, u_2} := by sorry
Source
https://github.com/ShouqiaoW/erdos/blob/61325b10bbdc29f4fb5e0618b414b9f2189333ad/390/lean/Erdos390/Full/PaperBridgePhysicalPowerCorrectionRow.lean#L1347-L1467

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