import Erdos1135.Tao.Section6.AmbientTailDFT
import Erdos1135.Tao.Section6.ConductorFrequency

/-!
# Section 6 Ambient Tail Decay

This leaf turns the checked reduced-conductor packet into the uniform ambient
tail Fourier bound used in Tao's fixed-slice collision estimate.  It retains
primitive-frequency decay and introduces no all-nonzero hypothesis.
-/

namespace Erdos1135
namespace Tao

noncomputable section

/-- Uniform tail-decay scale after weakening a reduced conductor `r` using
the checked comparison `n <= 20r`. -/
def taoSection6TailDecayDelta (C : ℝ) (B n : ℕ) : ℝ :=
  C * (20 : ℝ) ^ B / (n : ℝ) ^ B

theorem taoSection6TailDecayDelta_nonneg
    {C : ℝ} (hC : 0 ≤ C) (B n : ℕ) :
    0 ≤ taoSection6TailDecayDelta C B n := by
  unfold taoSection6TailDecayDelta
  positivity

/-- Primitive Syracuse decay at the checked reduced conductor gives a
uniform bound for every discarded ambient-tail frequency. -/
theorem syracPMFPrimitivePolynomialDecayAt.dft_taoSection6AmbientTailPMF_le
    {B : ℕ} {C : ℝ}
    (hdecay : syracPMFPrimitivePolynomialDecayAt B C)
    (hC : 0 ≤ C)
    {T k l m : ℕ}
    (hhead : k + 1 ≤ m)
    (hm : 9 * (T + (k + 1)) ≤ 10 * m)
    (hk : 20 * k ≤ 17 * (T + (k + 1)))
    (xi : ZMod (3 ^ (T + (k + 1))))
    (hxi : ¬zmodThreePowMultiple
      (T + (k + 1)) (T + (k + 1) - m) xi) :
    ‖ZMod.dft
        (pmfComplexMass
          (taoSection6AmbientTailPMF (k + 1) T l)) xi‖ ≤
      taoSection6TailDecayDelta C B (T + (k + 1)) := by
  rw [dft_taoSection6AmbientTailPMF]
  obtain ⟨r, j, hj, hrj, hfull, hr, hmkr, hnr, hbound⟩ :=
    hdecay.tail_scaled_frequency_le
      (n := T + (k + 1)) (m := m) (k := k) (T := T) (l := l)
      (by omega) hhead hm hk hxi
  have hn_pos : 0 < T + (k + 1) := by omega
  have hr_real_pos : 0 < (r : ℝ) := by exact_mod_cast hr
  have hn_real_pos : 0 < ((T + (k + 1) : ℕ) : ℝ) := by
    exact_mod_cast hn_pos
  have hnr_real :
      ((T + (k + 1) : ℕ) : ℝ) ≤ (20 : ℝ) * (r : ℝ) := by
    exact_mod_cast hnr
  have hpow :
      (((T + (k + 1) : ℕ) : ℝ) ^ B) ≤
        (20 : ℝ) ^ B * (r : ℝ) ^ B := by
    calc
      (((T + (k + 1) : ℕ) : ℝ) ^ B) ≤
          ((20 : ℝ) * (r : ℝ)) ^ B := by gcongr
      _ = (20 : ℝ) ^ B * (r : ℝ) ^ B := by rw [mul_pow]
  have hr_pow_pos : 0 < (r : ℝ) ^ B := pow_pos hr_real_pos B
  have hn_pow_pos :
      0 < (((T + (k + 1) : ℕ) : ℝ) ^ B) :=
    pow_pos hn_real_pos B
  calc
    ‖ZMod.dft (pmfComplexMass (syracPMF T))
        (taoSection6TailScaledFrequency
          (by omega : T ≤ T + (k + 1)) l xi)‖ ≤
      C / (r : ℝ) ^ B := hbound
    _ ≤ C * (20 : ℝ) ^ B /
        (((T + (k + 1) : ℕ) : ℝ) ^ B) := by
      apply (div_le_div_iff₀ hr_pow_pos hn_pow_pos).2
      calc
        C * (((T + (k + 1) : ℕ) : ℝ) ^ B) ≤
            C * ((20 : ℝ) ^ B * (r : ℝ) ^ B) :=
          mul_le_mul_of_nonneg_left hpow hC
        _ = (C * (20 : ℝ) ^ B) * (r : ℝ) ^ B := by ring
    _ = taoSection6TailDecayDelta C B (T + (k + 1)) := rfl

end

end Tao
end Erdos1135