The actual row mean is bounded for positive cutoffs

Proved

OAI.PiExponent.DeterminantContradiction.actual_mean_le_theta_of_pos

by Eyal1990 · Oct 16, 2026 · Mathlib 0df444a (Lean v4.33.1)

number-theoryopenai-pi-exponent-017

For admissible fixed data, the actual row mean is at most theta whenever the cutoff is positive.

Preamble

import Definitions.Def_OAI_NumberTheory_PiExponent_Analysis_AnalyticAggregate_RemainderData

open scoped BigOperators Topology
open OAI.PiExponent OAI.PiExponent.DeterminantContradiction

Formal statement

theorem OAI.PiExponent.DeterminantContradiction.actual_mean_le_theta_of_pos
    {nu : ℝ} (d : FixedData nu) (H : ℝ) (hH : 0 < H) :
    actualMean d H ≤ d.base.theta := by sorry

Source

https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/PiExponent/Analysis/LiteralAnalyticSummand.lean#L177-L199

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