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Canonical two-sided slow variance from fixed-witness base outputs

Proved
Erdos390.Full.PaperBridgeFit.BridgeData.exists_paperFineMesh_cutoff_eventually_canonical_lemma86_variance_of_base_compact

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

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

Fix cMesh>0 and Creg≥0. There are meshTol,γslow,Cvar>0 and W₀ preceding the mesh. Here a canonical bridge means the guarded canonical sample at the indicated n and W, with separated physical intervals, a nonempty remainder in each cell, and the canonical mesh prime partition. The finite nonempty head family has exactly the primes at most W in each head pattern. Physical interval endpoints lie between 1 and a fixed upper bound; guard ledgers have fixed promotion and bank parameters. The barycentric target has cell-mass margin at least a fixed positive marginFloor, and its baseline equals the bridge baseline. Conclusions are uniform on the closed effective-parameter ball of any fixed radius a ≥ 0. For W≥W₀ and a permitted mesh with δ>0 and w=δ+η≤meshTol, eventually the following holds for every point z and every raw-band equivalence e. Assume its actual regression satisfies ‖q‖sharp≤Creg w and the same partition has |dev|≤w, L₁≤7w, w²≤(456/cMesh²)V, V≤4w² and V≤centerEnergy. Then its two-stage compensated variance obeys the following two-sided bound. The result needs no additional analytic-error assumptions.

γsloww2≤V2(z,e)≤Cvarw2.\gamma_{\rm slow}w^2\le V_2(z,e)\le C_{\rm var}w^2.γslow​w2≤V2​(z,e)≤Cvar​w2.
Preamble
import Definitions.Def_erdos390_remaining_analytic_propositions_003

universe u_1
Formal statement
theorem Erdos390.Full.PaperBridgeFit.BridgeData.exists_paperFineMesh_cutoff_eventually_canonical_lemma86_variance_of_base_compact : Erdos390.RemainingAnalyticGoal003_002.{u_1} := by sorry
Source
https://github.com/ShouqiaoW/erdos/blob/61325b10bbdc29f4fb5e0618b414b9f2189333ad/390/lean/Erdos390/Full/PaperCanonicalLemma86RemainingOutputsEventually.lean#L157-L720

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