import Erdos1135.Tao.Renewal.QEndpointFreshOutsideEprimeNative
import Erdos1135.Tao.Renewal.QEndpointFreshOuterBadJBoundary

/-!
# Boundary Source Adapter for Canonical Outside-Eprime

This leaf packages the triangle selected by the Proposition 7.8 far-below
boundary branch as the exact current-scale input consumed by the canonical
fixed-offset estimate.
-/

namespace Erdos1135
namespace Tao

noncomputable section

namespace TaoSection7Case3SourceStoppingRun
namespace Lemma79TailExpectation

/-- The far-below boundary triangle supplies exact gap alignment and all
current-scale fields.  The gate `2 <= m` is required for `1 < M`. -/
theorem
    lemma79_exists_qmBoundaryFarBelow_entryTriangle_currentScaleControls
    {n m : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ}
    (hactive : TaoSection7Prop78ActiveCoverData n xi epsilon)
    {entry : TaoSection7RenewalPoint}
    (hboundary : taoSection7QmBoundary (n / 2) m entry)
    (hfar :
      TaoSection7QmBoundaryFarBelow
        (taoSection7Prop78BoundaryThreshold m)
        hactive.family entry)
    (hm : 2 ≤ m) :
    ∃ Delta ∈ hactive.family,
      Delta.Mem entry.toPoint ∧
      taoSection7Prop78BoundaryThreshold m <
        (lemma79EntryVerticalGap Delta entry.toPoint : ℝ) ∧
      Delta.cornerL - entry.l =
        (lemma79EntryVerticalGap Delta entry.toPoint : ℤ) ∧
      TaoSection7Lemma710CurrentScaleControls
        n Delta entry.toPoint (m : ℝ)
          (lemma79EntryVerticalGap Delta entry.toPoint : ℝ) := by
  rcases
      lemma79_exists_qmBoundaryFarBelow_entryTriangle_gap_bounds
        hactive hboundary hfar with
    ⟨Delta, hDelta, hmem, hfarGap, hgapUpper⟩
  have hdomain :=
    taoSection7QmBoundary_toPoint_sourceDomain hboundary
  have hsub : n / 2 - (entry.j : ℕ) = m := by
    unfold taoSection7QmBoundary at hboundary
    omega
  have hcurrent :
      taoSection7Lemma710CurrentM n entry.toPoint = (m : ℝ) := by
    unfold taoSection7Lemma710CurrentM
    simpa [TaoSection7RenewalPoint.toPoint] using
      congrArg (fun k : ℕ => (k : ℝ)) hsub
  have hgap :
      Delta.verticalDepth entry.toPoint =
        (lemma79EntryVerticalGap Delta entry.toPoint : ℤ) := by
    simpa [TaoSection7Triangle.verticalDepth] using
      lemma79EntryVerticalGap_coe_of_mem hmem
  have hgapAlign :
      Delta.cornerL - entry.l =
        (lemma79EntryVerticalGap Delta entry.toPoint : ℤ) := by
    simpa [TaoSection7RenewalPoint.toPoint] using
      lemma79EntryVerticalGap_coe_of_mem hmem
  have hSdef :
      (lemma79EntryVerticalGap Delta entry.toPoint : ℝ) =
        ((Delta.verticalDepth entry.toPoint : ℤ) : ℝ) := by
    rw [hgap]
    norm_num
  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 hMlower :
      (m : ℝ) / (Real.log (m : ℝ)) ^ 2 ≤
        (lemma79EntryVerticalGap Delta entry.toPoint : ℝ) := by
    unfold taoSection7Prop78BoundaryThreshold at hfarGap
    exact hfarGap.le
  refine ⟨Delta, hDelta, hmem, hfarGap, hgapAlign, ?_⟩
  exact
    { M_def := hcurrent.symm
      S_def := hSdef
      domain := hdomain
      M_large := hm_gt_one_real
      logM_pos := Real.log_pos hm_gt_one_real
      S_nonneg := Nat.cast_nonneg _
      hMlower := hMlower
      upper_752 := hgapUpper }

end Lemma79TailExpectation
end TaoSection7Case3SourceStoppingRun

end

end Tao
end Erdos1135