import Erdos1135.Tao.Renewal.QEndpointFreshOutsideEprimeWidth
import Erdos1135.Tao.Renewal.Prop78Case3Stopping
import Erdos1135.Tao.Renewal.Prop78Case3EStarSupportProducer

/-!
# Canonical Outside-Eprime Source Widths

This source-facing leaf packages the pure outside-`E'_p` width comparison
against the current-scale and exact offset-admissibility records.  The
natural gate `2 <= m` remains explicit.
-/

namespace Erdos1135
namespace Tao

noncomputable section

open TaoSection7Lemma710

namespace TaoSection7Case3SourceStoppingRun
namespace Lemma79TailExpectation

/-- Current-scale controls and exact offset admissibility supply both
quadratic kernel widths at the admissible event threshold. -/
theorem lemma79OutsideEprimeKernelWidths_of_currentScale_admissible
    {n m p fpGap R : ℕ}
    {old : TaoSection7Triangle} {entry : TaoSection7RenewalPoint}
    {M S sMin : ℝ}
    (hscale :
      TaoSection7Lemma710CurrentScaleControls
        n old entry.toPoint M S)
    (hm : 2 ≤ m)
    (hM : M = (m : ℝ))
    (hgapAlign : old.cornerL - entry.l = (fpGap : ℤ))
    (hadmissible :
      TaoSection7Case3EStarUsedOffsetAdmissible m p sMin)
    (hR : (R : ℝ) ≤ sMin / 10) :
    (R : ℝ) ^ 2 ≤ 1 + (fpGap : ℝ) ∧
      (sigmaSeparationScale sMin) ^ 2 ≤ 1 + (fpGap : ℝ) := by
  have hm_gt_one : 1 < m := by omega
  have hm_gt_one_real : (1 : ℝ) < (m : ℝ) := by
    exact_mod_cast hm_gt_one
  have hMlarge : 1 < M := by
    rw [hM]
    exact hm_gt_one_real
  have hdepth :
      old.verticalDepth entry.toPoint = (fpGap : ℤ) := by
    simpa [TaoSection7Triangle.verticalDepth,
      TaoSection7RenewalPoint.toPoint] using hgapAlign
  have hS_fpGap : S = (fpGap : ℝ) := by
    calc
      S = ((old.verticalDepth entry.toPoint : ℤ) : ℝ) :=
        hscale.S_def
      _ = ((fpGap : ℤ) : ℝ) :=
        congrArg (fun z : ℤ => (z : ℝ)) hdepth
      _ = (fpGap : ℝ) := by norm_num
  have hgap :
      M / (Real.log M) ^ 2 ≤ (fpGap : ℝ) := by
    calc
      M / (Real.log M) ^ 2 ≤ S := hscale.hMlower
      _ = (fpGap : ℝ) := hS_fpGap
  have hsMin_nonneg : 0 ≤ sMin :=
    le_trans (by norm_num) hadmissible.2.1
  have hsMin_le : sMin ≤ M ^ (2 / 5 : ℝ) := by
    calc
      sMin ≤ (m : ℝ) ^ (2 / 5 : ℝ) := hadmissible.2.2
      _ = M ^ (2 / 5 : ℝ) := by rw [hM]
  exact
    lemma79OutsideEprimeKernelWidths_of_sourceScale
      hMlarge hsMin_nonneg hsMin_le hgap hR

/-- The base-`Kcut` admissibility schedule specializes the source wrapper
at the literal large-triangle event threshold. -/
theorem lemma79OutsideEprimeKernelWidths_of_baseKcutSchedule
    {allowed : Finset ℕ}
    {n base m Kcut p fpGap R : ℕ}
    {old : TaoSection7Triangle} {entry : TaoSection7RenewalPoint}
    {M S : ℝ}
    (hscale :
      TaoSection7Lemma710CurrentScaleControls
        n old entry.toPoint M S)
    (hm : 2 ≤ m)
    (hM : M = (m : ℝ))
    (hgapAlign : old.cornerL - entry.l = (fpGap : ℤ))
    (hschedule :
      TaoSection7Case3BaseKcutAdmissibilitySchedule
        allowed base m Kcut)
    (hp : p ∈ allowed)
    (hR :
      (R : ℝ) ≤
        taoSection7Case3LargeTriangleBoundWithBase
          (base : ℝ) Kcut p / 10) :
    (R : ℝ) ^ 2 ≤ 1 + (fpGap : ℝ) ∧
      (sigmaSeparationScale
        (taoSection7Case3LargeTriangleBoundWithBase
          (base : ℝ) Kcut p)) ^ 2 ≤
        1 + (fpGap : ℝ) := by
  exact
    lemma79OutsideEprimeKernelWidths_of_currentScale_admissible
      hscale hm hM hgapAlign (hschedule.admissible hp) hR

end Lemma79TailExpectation
end TaoSection7Case3SourceStoppingRun

end

end Tao
end Erdos1135