import Erdos1135.ND.Fourier.FixedTotalAmbientPolynomial import Erdos1135.Tao.Fourier.OscillationAlgebra import Erdos1135.Tao.Section6.GatePartition import Erdos1135.Tao.Section6.HighRegimeMixing /-! # Raw Fixed-Total Ambient Aggregation This leaf transports the additive raw-slice estimate to one fixed ambient group, intersects Tao's enlarged local union with a fixed full valuation total, and sums the resulting disjoint source slices. The exact local box costs two powers of the ambient level. The output is still an unnormalized defective submass. It neither replaces the full fixed-total vector nor charges an exceptional event or divides by an endpoint mass. -/ open scoped BigOperators namespace Erdos1135 namespace ND open Tao noncomputable section /-- The additive raw fixed-total slice rate transported to one fixed ambient group. Terminal tail length zero is included. -/ theorem exists_ndSection6FixedTotalAmbientOscillation_le_fixed_n (CA : ℝ) (hCA : 17 ≤ CA) (Q : ℕ) : ∃ D : ℝ, 0 ≤ D ∧ ∃ N0 : ℕ, ∀ n k l L m : ℕ, N0 ≤ n → k + 1 ≤ n → m ≤ n → 9 * n ≤ 10 * m → Tao.taoZModPowOscillation m n (ndSection6FixedTotalAmbientSubmass CA n k l L) ≤ D / (n : ℝ) ^ (Q + 3) := by obtain ⟨D, hD, N0, hslice⟩ := exists_ndSection6FixedTotalAmbientOscillation_le CA hCA Q refine ⟨D, hD, N0, ?_⟩ intro n k l L m hn hk hmn hhigh let T := n - (k + 1) have hambient : T + (k + 1) = n := by dsimp [T] exact Nat.sub_add_cancel hk have hbound := hslice T k l L m (by simpa only [hambient] using hn) (by simpa only [hambient] using hmn) (by simpa only [hambient] using hhigh) rw [hambient] at hbound exact hbound /-- Tao's enlarged local source gate intersected with one full valuation total. The total selector is inside the source gate and is not normalized. -/ noncomputable def ndSection6FixedTotalLocalUnionGate (CA : ℝ) (n L : ℕ) (full : List ℕ+) : Prop := Tao.taoSection6LocalUnionGate CA n full ∧ Tao.taoTupleWeight full = L /-- Raw accepted residue mass of the fixed-total enlarged local union. -/ noncomputable def ndSection6FixedTotalLocalUnionSubmass (CA : ℝ) (n L : ℕ) (y : ZMod (3 ^ n)) : ℝ := Tao.taoGatedSubmass (Tao.geom2PNatListPMF n) (ndSection6FixedTotalLocalUnionGate CA n L) (Tao.taoSection7OffsetZMod n) y /-- The common total selector distributes through the exact disjoint local gate partition. This identity is valid also for the empty box at `n = 0`. -/ theorem ndSection6FixedTotalLocalUnionSubmass_eq_sum_sum (CA : ℝ) (n L : ℕ) (y : ZMod (3 ^ n)) : ndSection6FixedTotalLocalUnionSubmass CA n L y = ∑ k : Fin n, ∑ l : Fin (2 * n), ndSection6FixedTotalAmbientSubmass CA n k l L y := by classical have hunion : ndSection6FixedTotalLocalUnionGate CA n L = fun full => ∃ i ∈ (Finset.univ : Finset (Tao.taoSection6LocalGateIndex n)), Tao.taoSection6LocalGate CA n i full ∧ Tao.taoTupleWeight full = L := by funext full apply propext change ((∃ i : Tao.taoSection6LocalGateIndex n, Tao.taoSection6LocalGate CA n i full) ∧ Tao.taoTupleWeight full = L) ↔ ∃ i ∈ (Finset.univ : Finset (Tao.taoSection6LocalGateIndex n)), Tao.taoSection6LocalGate CA n i full ∧ Tao.taoTupleWeight full = L constructor · rintro ⟨⟨i, hi⟩, hweight⟩ exact ⟨i, Finset.mem_univ i, hi, hweight⟩ · rintro ⟨i, _hi, hgate, hweight⟩ exact ⟨⟨i, hgate⟩, hweight⟩ unfold ndSection6FixedTotalLocalUnionSubmass rw [hunion] calc Tao.taoGatedSubmass (Tao.geom2PNatListPMF n) (fun full => ∃ i ∈ (Finset.univ : Finset (Tao.taoSection6LocalGateIndex n)), Tao.taoSection6LocalGate CA n i full ∧ Tao.taoTupleWeight full = L) (Tao.taoSection7OffsetZMod n) y = ∑ i ∈ (Finset.univ : Finset (Tao.taoSection6LocalGateIndex n)), Tao.taoGatedSubmass (Tao.geom2PNatListPMF n) (fun full => Tao.taoSection6LocalGate CA n i full ∧ Tao.taoTupleWeight full = L) (Tao.taoSection7OffsetZMod n) y := by apply Tao.taoGatedSubmass_finset_union_eq_sum intro i _hi j _hj hij full hifull hjfull exact (Tao.taoSection6LocalGate_disjoint hij full) ⟨hifull.1, hjfull.1⟩ _ = ∑ i : Tao.taoSection6LocalGateIndex n, ndSection6FixedTotalAmbientSubmass CA n i.1 i.2 L y := by apply Finset.sum_congr rfl intro i _hi rfl _ = ∑ k : Fin n, ∑ l : Fin (2 * n), ndSection6FixedTotalAmbientSubmass CA n k l L y := Fintype.sum_prod_type _ /-- The raw fixed-total enlarged local union has the exact two-parameter-box polynomial rate. The full box spends precisely two powers of `n`. -/ theorem exists_ndSection6FixedTotalLocalUnionOscillation_le (CA : ℝ) (hCA : 17 ≤ CA) (Q : ℕ) : ∃ D : ℝ, 0 ≤ D ∧ ∃ N0 : ℕ, 1 ≤ N0 ∧ ∀ n L m : ℕ, N0 ≤ n → m ≤ n → 9 * n ≤ 10 * m → Tao.taoZModPowOscillation m n (ndSection6FixedTotalLocalUnionSubmass CA n L) ≤ (2 * D) / (n : ℝ) ^ (Q + 1) := by obtain ⟨D, hD, Nslice, hslice⟩ := exists_ndSection6FixedTotalAmbientOscillation_le_fixed_n CA hCA Q refine ⟨D, hD, max Nslice 1, le_max_right _ _, ?_⟩ intro n L m hn hmn hhigh have hnSlice : Nslice ≤ n := (le_max_left _ _).trans hn have hnOne : 1 ≤ n := (le_max_right _ _).trans hn let B : ℝ := D / (n : ℝ) ^ (Q + 3) calc Tao.taoZModPowOscillation m n (ndSection6FixedTotalLocalUnionSubmass CA n L) = Tao.taoZModPowOscillation m n (fun y => ∑ k : Fin n, ∑ l : Fin (2 * n), ndSection6FixedTotalAmbientSubmass CA n k l L y) := by congr 1 funext y exact ndSection6FixedTotalLocalUnionSubmass_eq_sum_sum CA n L y _ ≤ ∑ k : Fin n, Tao.taoZModPowOscillation m n (fun y => ∑ l : Fin (2 * n), ndSection6FixedTotalAmbientSubmass CA n k l L y) := Tao.taoZModPowOscillation_sum_le_sum m n (fun k : Fin n => fun y => ∑ l : Fin (2 * n), ndSection6FixedTotalAmbientSubmass CA n k l L y) _ ≤ ∑ k : Fin n, ∑ l : Fin (2 * n), Tao.taoZModPowOscillation m n (ndSection6FixedTotalAmbientSubmass CA n k l L) := by apply Finset.sum_le_sum intro k _hk exact Tao.taoZModPowOscillation_sum_le_sum m n (fun l : Fin (2 * n) => ndSection6FixedTotalAmbientSubmass CA n k l L) _ ≤ ∑ _k : Fin n, ∑ _l : Fin (2 * n), B := by apply Finset.sum_le_sum intro k _hk apply Finset.sum_le_sum intro l _hl exact hslice n k l L m hnSlice (Nat.succ_le_of_lt k.isLt) hmn hhigh _ = (((n * (2 * n) : ℕ) : ℝ) * B) := by simp [Nat.cast_mul] ring _ = (2 * D) / (n : ℝ) ^ (Q + 1) := by exact Tao.taoSection6_box_mul_div_pow_eq Q n D hnOne end end ND end Erdos1135