import Erdos1135.Tao.Fourier.DecayStatement import Erdos1135.Tao.Probability.Pascal /-! # Section 7 Character Surface This module records the statement-only bridge between Tao's Section 7 character sum `S_chi(n)` and the existing primitive-frequency DFT surface. It proves no Fourier decay and no source bridge from the recursive `syracPMF` definition. -/ open scoped BigOperators open scoped ZMod namespace Erdos1135 namespace Tao private noncomputable def prefixWeight (as : List ℕ+) (k : ℕ) : ℕ := taoTupleWeight (as.take k) /-- The argument of Tao's Section 7 character sum, equation (7.2), reduced in `ZMod (3^n)`. For a valuation list `as = [a_1, ..., a_n]`, this is `sum_{i=0}^{n-1} 3^i * 2^{-a_[1,i+1]}`. -/ noncomputable def taoSection7OffsetZMod (n : ℕ) (as : List ℕ+) : ZMod (3 ^ n) := ∑ i ∈ Finset.range n, (3 : ZMod (3 ^ n)) ^ i * (((2 : ZMod (3 ^ n)) ^ prefixWeight as (i + 1))⁻¹) /-- One fixed-modulus summand in Tao's Section 7 offset. -/ noncomputable def taoSection7OffsetSummand (N : ℕ) (as : List ℕ+) (i : ℕ) : ZMod (3 ^ N) := (3 : ZMod (3 ^ N)) ^ i * (((2 : ZMod (3 ^ N)) ^ taoTupleWeight (as.take (i + 1)))⁻¹) /-- The first `k` offset summands, all evaluated in the fixed ambient modulus `3^N`. -/ noncomputable def taoSection7OffsetPrefix (N k : ℕ) (as : List ℕ+) : ZMod (3 ^ N) := ∑ i ∈ Finset.range k, taoSection7OffsetSummand N as i theorem taoSection7OffsetZMod_eq_offsetPrefix (N : ℕ) (as : List ℕ+) : taoSection7OffsetZMod N as = taoSection7OffsetPrefix N N as := by rfl private noncomputable def section7CharacterTwoUnit (n : ℕ) : (ZMod (3 ^ n))ˣ := ZMod.unitOfCoprime 2 (Nat.Coprime.pow_right n (by decide : Nat.Coprime 2 3)) private theorem section7CharacterTwoUnit_coe (n : ℕ) : (section7CharacterTwoUnit n : ZMod (3 ^ n)) = 2 := by simp [section7CharacterTwoUnit] private theorem section7Character_cast_two_unit (n : ℕ) : (ZMod.castHom (Nat.pow_dvd_pow 3 (Nat.le_succ n)) (ZMod (3 ^ n))) (section7CharacterTwoUnit (n + 1) : ZMod (3 ^ (n + 1))) = (section7CharacterTwoUnit n : ZMod (3 ^ n)) := by rw [section7CharacterTwoUnit_coe, section7CharacterTwoUnit_coe] rw [ZMod.castHom_apply] exact ZMod.cast_natCast (Nat.pow_dvd_pow 3 (Nat.le_succ n)) 2 private theorem section7Character_cast_two_unit_pow_inv (n k : ℕ) : (ZMod.castHom (Nat.pow_dvd_pow 3 (Nat.le_succ n)) (ZMod (3 ^ n))) (((section7CharacterTwoUnit (n + 1)) ^ k)⁻¹ : (ZMod (3 ^ (n + 1)))ˣ) = (((section7CharacterTwoUnit n) ^ k)⁻¹ : (ZMod (3 ^ n))ˣ) := by change ((Units.map (ZMod.castHom (Nat.pow_dvd_pow 3 (Nat.le_succ n)) (ZMod (3 ^ n))) (((section7CharacterTwoUnit (n + 1)) ^ k)⁻¹) : (ZMod (3 ^ n))ˣ) : ZMod (3 ^ n)) = (((section7CharacterTwoUnit n) ^ k)⁻¹ : (ZMod (3 ^ n))ˣ) have hu : Units.map (ZMod.castHom (Nat.pow_dvd_pow 3 (Nat.le_succ n)) (ZMod (3 ^ n))) (section7CharacterTwoUnit (n + 1)) = section7CharacterTwoUnit n := by apply Units.ext exact section7Character_cast_two_unit n rw [map_inv, map_pow, hu] private theorem section7Character_castHom_inv_two_pow (n k : ℕ) : (ZMod.castHom (Nat.pow_dvd_pow 3 (Nat.le_succ n)) (ZMod (3 ^ n))) (((2 : ZMod (3 ^ (n + 1))) ^ k)⁻¹) = (((2 : ZMod (3 ^ n)) ^ k)⁻¹) := by calc (ZMod.castHom (Nat.pow_dvd_pow 3 (Nat.le_succ n)) (ZMod (3 ^ n))) (((2 : ZMod (3 ^ (n + 1))) ^ k)⁻¹) = (ZMod.castHom (Nat.pow_dvd_pow 3 (Nat.le_succ n)) (ZMod (3 ^ n))) ((((section7CharacterTwoUnit (n + 1)) ^ k : (ZMod (3 ^ (n + 1)))ˣ) : ZMod (3 ^ (n + 1)))⁻¹) := by simp [section7CharacterTwoUnit_coe] _ = (ZMod.castHom (Nat.pow_dvd_pow 3 (Nat.le_succ n)) (ZMod (3 ^ n))) (((((section7CharacterTwoUnit (n + 1)) ^ k)⁻¹ : (ZMod (3 ^ (n + 1)))ˣ) : ZMod (3 ^ (n + 1)))) := by rw [ZMod.inv_coe_unit] _ = (((((section7CharacterTwoUnit n) ^ k)⁻¹ : (ZMod (3 ^ n))ˣ) : ZMod (3 ^ n))) := by exact section7Character_cast_two_unit_pow_inv n k _ = (((2 : ZMod (3 ^ n)) ^ k)⁻¹) := by rw [← ZMod.inv_coe_unit] simp [section7CharacterTwoUnit_coe] private theorem section7Character_castHom_three_pow (n i : ℕ) : (ZMod.castHom (Nat.pow_dvd_pow 3 (Nat.le_succ n)) (ZMod (3 ^ n))) ((3 : ZMod (3 ^ (n + 1))) ^ i) = ((3 : ZMod (3 ^ n)) ^ i) := by rw [map_pow] congr 1 rw [ZMod.castHom_apply] exact ZMod.cast_natCast (Nat.pow_dvd_pow 3 (Nat.le_succ n)) 3 private theorem section7Character_cast_highOffset (n : ℕ) (as : List ℕ+) : (ZMod.castHom (Nat.pow_dvd_pow 3 (Nat.le_succ n)) (ZMod (3 ^ n))) (∑ i ∈ Finset.range n, (3 : ZMod (3 ^ (n + 1))) ^ i * (((2 : ZMod (3 ^ (n + 1))) ^ prefixWeight as (i + 1))⁻¹)) = taoSection7OffsetZMod n as := by unfold taoSection7OffsetZMod rw [map_sum] apply Finset.sum_congr rfl intro i _hi rw [map_mul, section7Character_castHom_three_pow, section7Character_castHom_inv_two_pow] private theorem section7Character_three_mul_val_eq_three_mul_of_cast_eq (n : ℕ) (e : ZMod (3 ^ (n + 1))) (x : ZMod (3 ^ n)) (h : (ZMod.castHom (Nat.pow_dvd_pow 3 (Nat.le_succ n)) (ZMod (3 ^ n))) e = x) : (3 : ZMod (3 ^ (n + 1))) * (x.val : ZMod (3 ^ (n + 1))) = 3 * e := by have hlow : (e.val : ZMod (3 ^ n)) = x := by rw [ZMod.castHom_apply] at h rw [ZMod.cast_eq_val] at h exact h have hmod : e.val ≡ x.val [MOD 3 ^ n] := by rw [← ZMod.natCast_eq_natCast_iff] rw [hlow, ZMod.natCast_zmod_val] have hmod3 : 3 * e.val ≡ 3 * x.val [MOD 3 ^ (n + 1)] := by simpa [pow_succ'] using Nat.ModEq.mul_left' 3 hmod rw [← ZMod.natCast_zmod_val e] simpa [Nat.cast_mul] using (ZMod.natCast_eq_natCast_iff (3 * x.val) (3 * e.val) (3 ^ (n + 1))).mpr hmod3.symm private theorem section7Character_inv_two_pow_add (n a b : ℕ) : (((2 : ZMod (3 ^ n)) ^ (a + b))⁻¹) = (((2 : ZMod (3 ^ n)) ^ a)⁻¹) * (((2 : ZMod (3 ^ n)) ^ b)⁻¹) := by calc (((2 : ZMod (3 ^ n)) ^ (a + b))⁻¹) = (((((section7CharacterTwoUnit n) ^ (a + b) : (ZMod (3 ^ n))ˣ) : ZMod (3 ^ n)))⁻¹) := by simp [section7CharacterTwoUnit_coe] _ = (((((section7CharacterTwoUnit n) ^ (a + b))⁻¹ : (ZMod (3 ^ n))ˣ) : ZMod (3 ^ n))) := by rw [ZMod.inv_coe_unit] _ = (((((section7CharacterTwoUnit n) ^ a)⁻¹ * ((section7CharacterTwoUnit n) ^ b)⁻¹ : (ZMod (3 ^ n))ˣ) : ZMod (3 ^ n))) := by congr 1 rw [pow_add] simp [mul_comm] _ = (((2 : ZMod (3 ^ n)) ^ a)⁻¹) * (((2 : ZMod (3 ^ n)) ^ b)⁻¹) := by rw [Units.val_mul, ← ZMod.inv_coe_unit, ← ZMod.inv_coe_unit] simp [section7CharacterTwoUnit_coe] private theorem prefixWeight_cons_succ (a : ℕ+) (as : List ℕ+) (i : ℕ) : prefixWeight (a :: as) (i + 1) = (a : ℕ) + prefixWeight as i := by simp [prefixWeight, taoTupleWeight, List.take_succ_cons] private theorem prefixWeight_zero (as : List ℕ+) : prefixWeight as 0 = 0 := by simp [prefixWeight, taoTupleWeight] /-- The Section 7 offset obeys the same cons recursion as `syracPMF`. -/ theorem taoSection7OffsetZMod_cons (n : ℕ) (a : ℕ+) (as : List ℕ+) : taoSection7OffsetZMod (n + 1) (a :: as) = syracStep n (taoSection7OffsetZMod n as) a := by let liftedTail : ZMod (3 ^ (n + 1)) := ∑ i ∈ Finset.range n, (3 : ZMod (3 ^ (n + 1))) ^ i * (((2 : ZMod (3 ^ (n + 1))) ^ prefixWeight as (i + 1))⁻¹) have htailCast : (ZMod.castHom (Nat.pow_dvd_pow 3 (Nat.le_succ n)) (ZMod (3 ^ n))) liftedTail = taoSection7OffsetZMod n as := by simpa [liftedTail] using section7Character_cast_highOffset n as have htailVal : (3 : ZMod (3 ^ (n + 1))) * ((taoSection7OffsetZMod n as).val : ZMod (3 ^ (n + 1))) = 3 * liftedTail := section7Character_three_mul_val_eq_three_mul_of_cast_eq n liftedTail (taoSection7OffsetZMod n as) htailCast have hsum : (∑ k ∈ Finset.range n, (3 : ZMod (3 ^ (n + 1))) ^ (k + 1) * ((((2 : ZMod (3 ^ (n + 1))) ^ (a : ℕ))⁻¹) * (((2 : ZMod (3 ^ (n + 1))) ^ prefixWeight as (k + 1))⁻¹))) = (3 : ZMod (3 ^ (n + 1))) * ((((2 : ZMod (3 ^ (n + 1))) ^ (a : ℕ))⁻¹) * (∑ i ∈ Finset.range n, (3 : ZMod (3 ^ (n + 1))) ^ i * (((2 : ZMod (3 ^ (n + 1))) ^ prefixWeight as (i + 1))⁻¹))) := by rw [Finset.mul_sum] rw [Finset.mul_sum] apply Finset.sum_congr rfl intro k _hk rw [pow_succ'] ring calc taoSection7OffsetZMod (n + 1) (a :: as) = (((2 : ZMod (3 ^ (n + 1))) ^ (a : ℕ))⁻¹) + (3 : ZMod (3 ^ (n + 1))) * ((((2 : ZMod (3 ^ (n + 1))) ^ (a : ℕ))⁻¹) * liftedTail) := by simp [taoSection7OffsetZMod, liftedTail, prefixWeight_cons_succ, section7Character_inv_two_pow_add, Finset.sum_range_succ', prefixWeight_zero] rw [hsum] ring _ = (((2 : ZMod (3 ^ (n + 1))) ^ (a : ℕ))⁻¹) * ((3 : ZMod (3 ^ (n + 1))) * ((taoSection7OffsetZMod n as).val : ZMod (3 ^ (n + 1))) + 1) := by rw [htailVal] ring _ = syracStep n (taoSection7OffsetZMod n as) a := by rfl /-- The source character value `chi_{n,xi}` applied to the Section 7 offset. The sign and product order cite the project-local DFT convention spine in `Fourier.Basic` through `taoForwardDFTKernel`. -/ noncomputable def taoSection7CharacterTerm (n : ℕ) (ξ : ZMod (3 ^ n)) (as : List ℕ+) : ℂ := taoForwardDFTKernel (taoSection7OffsetZMod n as) ξ /-- Tao's Section 7 character expectation `S_chi(n)`, expressed through the checked exact-length iid positive `Geom(2)` list PMF. -/ noncomputable def taoSection7SChi (n : ℕ) (ξ : ZMod (3 ^ n)) : ℂ := ∑' as : List ℕ+, ((geom2PNatListPMF n as).toReal : ℂ) * taoSection7CharacterTerm n ξ as /-- Statement target for the source bridge from the recursive `syracPMF` DFT coefficient to the Section 7 character expectation. -/ def TaoSection7CharacterBridgeStatement : Prop := ∀ n : ℕ, 1 ≤ n → ∀ ξ : ZMod (3 ^ n), zmodThreePrimitive n ξ → ZMod.dft (pmfComplexMass (syracPMF n)) ξ = taoSection7SChi n ξ /-- Polynomial decay statement for Tao Proposition 7.1 in the `S_chi` vocabulary. -/ def taoSection7CharacterPolynomialDecayAt (A : ℕ) (C : ℝ) : Prop := ∀ n : ℕ, 1 ≤ n → ∀ ξ : ZMod (3 ^ n), zmodThreePrimitive n ξ → ‖taoSection7SChi n ξ‖ ≤ C / (n : ℝ) ^ A /-- Statement-shaped target corresponding to Tao Proposition 7.1. -/ def TaoProp71CharacterDecayStatement : Prop := ∀ A : ℕ, 0 < A → ∃ C : ℝ, 0 ≤ C ∧ taoSection7CharacterPolynomialDecayAt A C theorem TaoProp71CharacterDecayStatement.bound (h : TaoProp71CharacterDecayStatement) {A : ℕ} (hA : 0 < A) : ∃ C : ℝ, 0 ≤ C ∧ taoSection7CharacterPolynomialDecayAt A C := h A hA /-- If the Section 7 character bridge is proved, then Tao's Proposition 7.1 statement implies the current primitive-frequency Proposition 1.17 surface. -/ theorem TaoProp117PrimitivePolynomialDecayStatement.of_section7_character (hbridge : TaoSection7CharacterBridgeStatement) (h71 : TaoProp71CharacterDecayStatement) : TaoProp117PrimitivePolynomialDecayStatement := by intro A hA rcases h71 A hA with ⟨C, hC_nonneg, hC⟩ refine ⟨C, hC_nonneg, ?_⟩ intro n hn ξ hξ rw [hbridge n hn ξ hξ] exact hC n hn ξ hξ /-- Section 7's finite pilot cites the shared R5 DFT convention spine. -/ theorem tao_dft_pmfComplexMass_apply_syracPMF_one (ξ : ZMod (3 ^ 1)) : ZMod.dft (pmfComplexMass (syracPMF 1)) ξ = ∑ x : ZMod (3 ^ 1), taoForwardDFTKernel x ξ * (((syracPMF 1 x).toReal : ℝ) : ℂ) := tao_dft_pmfComplexMass_apply (syracPMF 1) ξ /-- At level `n = 1`, an even positive valuation contributes residue `1`. -/ theorem taoSection7OffsetZMod_one_even (k : ℕ) : taoSection7OffsetZMod 1 [⟨2 * k + 2, by omega⟩] = (1 : ZMod (3 ^ 1)) := by simp [taoSection7OffsetZMod, prefixWeight, taoTupleWeight] rw [show 2 * k + 2 = 2 * (k + 1) by omega, pow_mul] change (4 : ZMod 3) ^ (k + 1) = 1 rw [show (4 : ZMod 3) = 1 by decide] simp /-- At level `n = 1`, an odd positive valuation contributes residue `2`. -/ theorem taoSection7OffsetZMod_one_odd (k : ℕ) : taoSection7OffsetZMod 1 [⟨2 * k + 1, by omega⟩] = (2 : ZMod (3 ^ 1)) := by simp [taoSection7OffsetZMod, prefixWeight, taoTupleWeight] have heven : (2 : ZMod (3 ^ 1)) ^ (2 * k) = 1 := by rw [pow_mul] change (4 : ZMod 3) ^ k = 1 rw [show (4 : ZMod 3) = 1 by decide] simp have hpow : (2 : ZMod (3 ^ 1)) ^ (2 * k + 1) = 2 := by calc (2 : ZMod (3 ^ 1)) ^ (2 * k + 1) = (2 : ZMod (3 ^ 1)) ^ (2 * k) * 2 := by exact pow_succ (2 : ZMod (3 ^ 1)) (2 * k) _ = 1 * 2 := by rw [heven] _ = 2 := by norm_num rw [hpow] change ((2 : ZMod 3)⁻¹ = 2) exact inv_eq_of_mul_eq_one_right (by decide : (2 : ZMod 3) * 2 = 1) theorem taoSection7CharacterTerm_one_even (k : ℕ) (ξ : ZMod (3 ^ 1)) : taoSection7CharacterTerm 1 ξ [⟨2 * k + 2, by omega⟩] = taoForwardDFTKernel 1 ξ := by simp [taoSection7CharacterTerm, taoSection7OffsetZMod_one_even k] theorem taoSection7CharacterTerm_one_odd (k : ℕ) (ξ : ZMod (3 ^ 1)) : taoSection7CharacterTerm 1 ξ [⟨2 * k + 1, by omega⟩] = taoForwardDFTKernel 2 ξ := by simp [taoSection7CharacterTerm, taoSection7OffsetZMod_one_odd k] /-- The landed `Syrac(Z/3Z)` DFT table is exactly the even/odd split predicted by the Section 7 character surface. -/ theorem dft_syracPMF_one_apply_table_section7_terms (ξ : ZMod (3 ^ 1)) : ZMod.dft (pmfComplexMass (syracPMF 1)) ξ = taoSection7CharacterTerm 1 ξ [⟨2, by norm_num⟩] • (((1 / 3 : ℝ) : ℂ)) + taoSection7CharacterTerm 1 ξ [⟨1, by norm_num⟩] • (((2 / 3 : ℝ) : ℂ)) := by rw [dft_syracPMF_one_apply_table] rw [taoSection7CharacterTerm_one_even 0, taoSection7CharacterTerm_one_odd 0] simp [taoForwardDFTKernel] private theorem taoSection7Char_add {n : ℕ} (ξ x y : ZMod (3 ^ n)) : taoForwardDFTKernel (x + y) ξ = taoForwardDFTKernel x ξ * taoForwardDFTKernel y ξ := by simp only [taoForwardDFTKernel] rw [← AddChar.map_add_eq_mul] congr 1 ring private theorem norm_taoForwardDFTKernel {n : ℕ} (x ξ : ZMod (3 ^ n)) : ‖taoForwardDFTKernel x ξ‖ = 1 := by simp [taoForwardDFTKernel, ZMod.stdAddChar] /-- The two-point conditional average from Tao equation (7.4) at `b = 3`. -/ noncomputable def taoSection7PairAverage (n : ℕ) (ξ x : ZMod (3 ^ n)) : ℂ := (1 / 2 : ℂ) * taoForwardDFTKernel ((5 : ZMod (3 ^ n)) * x) ξ + (1 / 2 : ℂ) * taoForwardDFTKernel ((7 : ZMod (3 ^ n)) * x) ξ /-- Algebraic part of Lemma 7.2: after conditioning on `b = 3`, the two-point average factors as `chi(5x) / 2 * (1 + chi(2x))`. -/ theorem taoSection7PairAverage_eq_factor (n : ℕ) (ξ x : ZMod (3 ^ n)) : taoSection7PairAverage n ξ x = taoForwardDFTKernel ((5 : ZMod (3 ^ n)) * x) ξ * ((1 / 2 : ℂ) * (1 + taoForwardDFTKernel ((2 : ZMod (3 ^ n)) * x) ξ)) := by unfold taoSection7PairAverage rw [show ((7 : ZMod (3 ^ n)) * x) = (5 : ZMod (3 ^ n)) * x + 2 * x by ring] rw [taoSection7Char_add] ring /-- The prefactor `chi(5x)` has norm one, so Lemma 7.2's local bound reduces to controlling `|1 + chi(2x)| / 2`. -/ theorem norm_taoSection7PairAverage_eq (n : ℕ) (ξ x : ZMod (3 ^ n)) : ‖taoSection7PairAverage n ξ x‖ = ‖(1 / 2 : ℂ) * (1 + taoForwardDFTKernel ((2 : ZMod (3 ^ n)) * x) ξ)‖ := by rw [taoSection7PairAverage_eq_factor, norm_mul, norm_taoForwardDFTKernel, one_mul] end Tao end Erdos1135