import Erdos1135.Tao.Fourier.Section7SourcePredicates /-! # Lemma 7.4 Source Theta Arithmetic This proof leaf develops the exact residue transport and centered no-wrap arithmetic behind Tao's local black-point adjacency claims. It does not use primitivity, construct triangle selectors, or import renewal infrastructure. -/ namespace Erdos1135 namespace Tao noncomputable section /-- Tao's signed real theta at a Section 7 lattice point. -/ noncomputable def taoSection7SourceTheta (n : ℕ) (xi : ZMod (3 ^ n)) (p : TaoSection7Point) : ℝ := taoSignedTheta (taoSection7ThetaResidue n xi p.j p.l) theorem taoSection7SourceBlackPoint_iff_geometryBlack (n : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) (p : TaoSection7Point) : taoSection7SourceBlackPoint n xi epsilon p ↔ TaoSection7Black epsilon (taoSection7SourceTheta n xi) p := by rfl theorem taoSection7SourceWhitePoint_iff_not_geometryBlack (n : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) (p : TaoSection7Point) : taoSection7SourceWhitePoint n xi epsilon p ↔ ¬ TaoSection7Black epsilon (taoSection7SourceTheta n xi) p := by exact taoSection7White_iff_not_black epsilon (taoSection7ThetaResidue n xi p.j p.l) theorem taoSection7SourceBlackInDomain_iff_geometryBlack (n J : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) (p : TaoSection7Point) : taoSection7SourceBlackInDomain n xi epsilon J p ↔ taoSection7SourcePointInDomain J p ∧ TaoSection7Black epsilon (taoSection7SourceTheta n xi) p := by rfl theorem taoSection7SourceBlackInDomain_geometryBlack {n J : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ} {p : TaoSection7Point} (hp : taoSection7SourceBlackInDomain n xi epsilon J p) : TaoSection7Black epsilon (taoSection7SourceTheta n xi) p := by exact (taoSection7SourceBlackPoint_iff_geometryBlack n xi epsilon p).1 hp.2 /-- Moving right multiplies the source theta residue by `9`. -/ theorem taoSection7ThetaResidue_right (n : ℕ) (xi : ZMod (3 ^ n)) (p : TaoSection7Point) : taoSection7ThetaResidue n xi p.right.j p.right.l = 9 * taoSection7ThetaResidue n xi p.j p.l := by have hj : 0 < (p.j : ℕ) := p.j.pos have hexp : 2 * (((p.j : ℕ) + 1) - 1) = 2 + 2 * ((p.j : ℕ) - 1) := by omega unfold taoSection7ThetaResidue simp only [TaoSection7Point.right_j, TaoSection7Point.right_l] rw [hexp, pow_add] norm_num ring /-- Moving down multiplies the source theta residue by `2`. -/ theorem taoSection7ThetaResidue_down (n : ℕ) (xi : ZMod (3 ^ n)) (p : TaoSection7Point) : taoSection7ThetaResidue n xi p.down.j p.down.l = 2 * taoSection7ThetaResidue n xi p.j p.l := by have htwo := two_mul_taoSection7TwoZPow_neg n (p.l - 1) have hpow : taoSection7TwoZPow n (1 - (p.l - 1)) = 2 * taoSection7TwoZPow n (1 - p.l) := by simpa [show -(p.l - 1) = 1 - p.l by ring] using htwo.symm unfold taoSection7ThetaResidue simp only [TaoSection7Point.down_j, TaoSection7Point.down_l] rw [hpow] ring /-- The source residue satisfies Tao's right/down linear relation. -/ theorem taoSection7ThetaResidue_eq_right_sub_four_down (n : ℕ) (xi : ZMod (3 ^ n)) (p : TaoSection7Point) : taoSection7ThetaResidue n xi p.j p.l = taoSection7ThetaResidue n xi p.right.j p.right.l - 4 * taoSection7ThetaResidue n xi p.down.j p.down.l := by rw [taoSection7ThetaResidue_right, taoSection7ThetaResidue_down] ring /-- Away from the left boundary, moving from the left neighbor to `p` multiplies the source theta residue by `9`. -/ theorem taoSection7ThetaResidue_eq_nine_mul_left (n : ℕ) (xi : ZMod (3 ^ n)) (p : TaoSection7Point) (h : 1 < (p.j : ℕ)) : taoSection7ThetaResidue n xi p.j p.l = 9 * taoSection7ThetaResidue n xi (p.left h).j (p.left h).l := by have hj : 0 < (p.j : ℕ) := p.j.pos have hexp : 2 * ((p.j : ℕ) - 1) = 2 + 2 * (((p.j : ℕ) - 1) - 1) := by omega unfold taoSection7ThetaResidue simp only [TaoSection7Point.left] rw [hexp, pow_add] norm_num ring /-- Any integral representative whose normalized ratio lies strictly inside the centered half-interval is the `valMinAbs` representative. The strict lower boundary is essential because `valMinAbs` uses `(-1/2, 1/2]`. -/ theorem taoSignedTheta_eq_intRatio_of_intCast_eq_of_abs_lt_half {N : ℕ} [NeZero N] (z : ZMod N) (m : ℤ) (hcast : (m : ZMod N) = z) (habs : |(m : ℝ) / (N : ℝ)| < (1 / 2 : ℝ)) : taoSignedTheta z = (m : ℝ) / (N : ℝ) := by have hN : (0 : ℝ) < (N : ℝ) := by exact_mod_cast NeZero.pos N have hbounds := (abs_lt.mp habs) have hlreal : (-(N : ℝ)) < (m : ℝ) * 2 := by have h := (lt_div_iff₀ hN).mp hbounds.1 nlinarith have hureal : (m : ℝ) * 2 < (N : ℝ) := by have h := (div_lt_iff₀ hN).mp hbounds.2 nlinarith have hl : (-(N : ℤ)) < m * 2 := by exact_mod_cast hlreal have hu : m * 2 ≤ (N : ℤ) := by have : m * 2 < (N : ℤ) := by exact_mod_cast hureal omega have hm : z.valMinAbs = m := (ZMod.valMinAbs_spec z m).2 ⟨hcast.symm, hl, hu⟩ rw [taoSignedTheta, hm] /-- Centered no-wrap rule for multiplying a residue by an integer. -/ theorem taoSignedTheta_int_mul_of_abs_lt_half {N : ℕ} [NeZero N] (k : ℤ) (z : ZMod N) (habs : |(k : ℝ) * taoSignedTheta z| < (1 / 2 : ℝ)) : taoSignedTheta ((k : ZMod N) * z) = (k : ℝ) * taoSignedTheta z := by let m : ℤ := k * z.valMinAbs have hcast : (m : ZMod N) = (k : ZMod N) * z := by dsimp [m] rw [Int.cast_mul, ZMod.coe_valMinAbs] have hratio : (m : ℝ) / (N : ℝ) = (k : ℝ) * taoSignedTheta z := by dsimp [m, taoSignedTheta] push_cast ring rw [taoSignedTheta_eq_intRatio_of_intCast_eq_of_abs_lt_half ((k : ZMod N) * z) m hcast (by simpa [hratio] using habs), hratio] /-- Centered no-wrap rule for Tao's modular relation `z = zRight - 4*zDown`. -/ theorem taoSignedTheta_eq_right_sub_four_down_of_abs_lt_half {N : ℕ} [NeZero N] (z zRight zDown : ZMod N) (hresidue : z = zRight - 4 * zDown) (habs : |taoSignedTheta zRight - 4 * taoSignedTheta zDown| < (1 / 2 : ℝ)) : taoSignedTheta z = taoSignedTheta zRight - 4 * taoSignedTheta zDown := by let m : ℤ := zRight.valMinAbs - 4 * zDown.valMinAbs have hcast : (m : ZMod N) = z := by dsimp [m] rw [Int.cast_sub, Int.cast_mul, ZMod.coe_valMinAbs, ZMod.coe_valMinAbs, hresidue] norm_num have hratio : (m : ℝ) / (N : ℝ) = taoSignedTheta zRight - 4 * taoSignedTheta zDown := by dsimp [m, taoSignedTheta] push_cast ring rw [taoSignedTheta_eq_intRatio_of_intCast_eq_of_abs_lt_half z m hcast (by simpa [hratio] using habs), hratio] /-- Exact real theta transport to the right whenever the multiplied value does not wrap around the centered representative interval. -/ theorem taoSection7SourceTheta_right_of_abs_lt_half (n : ℕ) (xi : ZMod (3 ^ n)) (p : TaoSection7Point) (habs : |9 * taoSection7SourceTheta n xi p| < (1 / 2 : ℝ)) : taoSection7SourceTheta n xi p.right = 9 * taoSection7SourceTheta n xi p := by unfold taoSection7SourceTheta rw [taoSection7ThetaResidue_right] simpa using taoSignedTheta_int_mul_of_abs_lt_half (N := 3 ^ n) 9 (taoSection7ThetaResidue n xi p.j p.l) habs /-- Exact real theta transport downward whenever doubling does not wrap. -/ theorem taoSection7SourceTheta_down_of_abs_lt_half (n : ℕ) (xi : ZMod (3 ^ n)) (p : TaoSection7Point) (habs : |2 * taoSection7SourceTheta n xi p| < (1 / 2 : ℝ)) : taoSection7SourceTheta n xi p.down = 2 * taoSection7SourceTheta n xi p := by unfold taoSection7SourceTheta rw [taoSection7ThetaResidue_down] simpa using taoSignedTheta_int_mul_of_abs_lt_half (N := 3 ^ n) 2 (taoSection7ThetaResidue n xi p.j p.l) habs /-- Exact real form of Tao's right/down relation under its centered smallness guard. -/ theorem taoSection7SourceTheta_eq_right_sub_four_down_of_abs_lt_half (n : ℕ) (xi : ZMod (3 ^ n)) (p : TaoSection7Point) (habs : |taoSection7SourceTheta n xi p.right - 4 * taoSection7SourceTheta n xi p.down| < (1 / 2 : ℝ)) : taoSection7SourceTheta n xi p = taoSection7SourceTheta n xi p.right - 4 * taoSection7SourceTheta n xi p.down := by unfold taoSection7SourceTheta at habs ⊢ exact taoSignedTheta_eq_right_sub_four_down_of_abs_lt_half (taoSection7ThetaResidue n xi p.j p.l) (taoSection7ThetaResidue n xi p.right.j p.right.l) (taoSection7ThetaResidue n xi p.down.j p.down.l) (taoSection7ThetaResidue_eq_right_sub_four_down n xi p) habs /-- Exact real transport from the left neighbor under its centered guard. -/ theorem taoSection7SourceTheta_eq_nine_mul_left_of_abs_lt_half (n : ℕ) (xi : ZMod (3 ^ n)) (p : TaoSection7Point) (h : 1 < (p.j : ℕ)) (habs : |9 * taoSection7SourceTheta n xi (p.left h)| < (1 / 2 : ℝ)) : taoSection7SourceTheta n xi p = 9 * taoSection7SourceTheta n xi (p.left h) := by unfold taoSection7SourceTheta rw [taoSection7ThetaResidue_eq_nine_mul_left n xi p h] simpa using taoSignedTheta_int_mul_of_abs_lt_half (N := 3 ^ n) 9 (taoSection7ThetaResidue n xi (p.left h).j (p.left h).l) habs /-- The source theta itself supplies Tao's three weak-black adjacency claims. No primitive-frequency or epsilon-smallness premise is needed by the record; each black-neighbor hypothesis supplies nonnegativity of `epsilon`. -/ theorem taoSection7SourceTheta_weakBlackAdjacency (n : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ) : TaoSection7WeakBlackAdjacency epsilon (taoSection7SourceTheta n xi) := by let theta := taoSection7SourceTheta n xi constructor · intro p hp hright have hp' : |theta p| ≤ (1 / 100 : ℝ) := hp have hguard : |9 * theta p| < (1 / 2 : ℝ) := by calc |9 * theta p| = 9 * |theta p| := by rw [abs_mul]; norm_num _ ≤ 9 * (1 / 100 : ℝ) := mul_le_mul_of_nonneg_left hp' (by norm_num) _ < 1 / 2 := by norm_num have htransport := taoSection7SourceTheta_right_of_abs_lt_half n xi p hguard change theta p.right = 9 * theta p at htransport have hright' : |theta p.right| ≤ epsilon := hright have hepsilon : 0 ≤ epsilon := (abs_nonneg _).trans hright' rw [htransport] at hright' have hscaled : 9 * |theta p| ≤ epsilon := by simpa [abs_mul] using hright' show |theta p| ≤ epsilon nlinarith [abs_nonneg (theta p)] · intro p hp hdown have hp' : |theta p| ≤ (1 / 100 : ℝ) := hp have hguard : |2 * theta p| < (1 / 2 : ℝ) := by calc |2 * theta p| = 2 * |theta p| := by rw [abs_mul]; norm_num _ ≤ 2 * (1 / 100 : ℝ) := mul_le_mul_of_nonneg_left hp' (by norm_num) _ < 1 / 2 := by norm_num have htransport := taoSection7SourceTheta_down_of_abs_lt_half n xi p hguard change theta p.down = 2 * theta p at htransport have hdown' : |theta p.down| ≤ epsilon := hdown have hepsilon : 0 ≤ epsilon := (abs_nonneg _).trans hdown' rw [htransport] at hdown' have hscaled : 2 * |theta p| ≤ epsilon := by simpa [abs_mul] using hdown' show |theta p| ≤ epsilon nlinarith [abs_nonneg (theta p)] · intro p hright hdown have hright' : |theta p.right| ≤ (1 / 100 : ℝ) := hright have hdown' : |theta p.down| ≤ (1 / 100 : ℝ) := hdown have hcandidate : |theta p.right - 4 * theta p.down| ≤ (5 / 100 : ℝ) := by calc |theta p.right - 4 * theta p.down| ≤ |theta p.right| + |4 * theta p.down| := abs_sub _ _ _ = |theta p.right| + 4 * |theta p.down| := by rw [abs_mul] norm_num _ ≤ (1 / 100 : ℝ) + 4 * (1 / 100 : ℝ) := add_le_add hright' (mul_le_mul_of_nonneg_left hdown' (by norm_num)) _ = 5 / 100 := by norm_num have hguard : |theta p.right - 4 * theta p.down| < (1 / 2 : ℝ) := hcandidate.trans_lt (by norm_num) have hrelation := taoSection7SourceTheta_eq_right_sub_four_down_of_abs_lt_half n xi p hguard change theta p = theta p.right - 4 * theta p.down at hrelation have hp05 : |theta p| ≤ (5 / 100 : ℝ) := by rw [hrelation] exact hcandidate have hrightGuard : |9 * theta p| < (1 / 2 : ℝ) := by calc |9 * theta p| = 9 * |theta p| := by rw [abs_mul]; norm_num _ ≤ 9 * (5 / 100 : ℝ) := mul_le_mul_of_nonneg_left hp05 (by norm_num) _ < 1 / 2 := by norm_num have hrightTransport := taoSection7SourceTheta_right_of_abs_lt_half n xi p hrightGuard change theta p.right = 9 * theta p at hrightTransport rw [hrightTransport] at hright' have hsharp : 9 * |theta p| ≤ (1 / 100 : ℝ) := by simpa [abs_mul] using hright' show |theta p| ≤ (1 / 100 : ℝ) nlinarith [abs_nonneg (theta p)] · intro p h hleft hdown have hleft' : |theta (p.left h)| ≤ (1 / 100 : ℝ) := hleft have hdown' : |theta p.down| ≤ (1 / 100 : ℝ) := hdown have hleftGuard : |9 * theta (p.left h)| < (1 / 2 : ℝ) := by calc |9 * theta (p.left h)| = 9 * |theta (p.left h)| := by rw [abs_mul] norm_num _ ≤ 9 * (1 / 100 : ℝ) := mul_le_mul_of_nonneg_left hleft' (by norm_num) _ < 1 / 2 := by norm_num have hleftTransport := taoSection7SourceTheta_eq_nine_mul_left_of_abs_lt_half n xi p h hleftGuard change theta p = 9 * theta (p.left h) at hleftTransport have hp09 : |theta p| ≤ (9 / 100 : ℝ) := by calc |theta p| = 9 * |theta (p.left h)| := by rw [hleftTransport, abs_mul] norm_num _ ≤ 9 * (1 / 100 : ℝ) := mul_le_mul_of_nonneg_left hleft' (by norm_num) _ = 9 / 100 := by ring have hdownGuard : |2 * theta p| < (1 / 2 : ℝ) := by calc |2 * theta p| = 2 * |theta p| := by rw [abs_mul]; norm_num _ ≤ 2 * (9 / 100 : ℝ) := mul_le_mul_of_nonneg_left hp09 (by norm_num) _ < 1 / 2 := by norm_num have hdownTransport := taoSection7SourceTheta_down_of_abs_lt_half n xi p hdownGuard change theta p.down = 2 * theta p at hdownTransport rw [hdownTransport] at hdown' have hsharp : 2 * |theta p| ≤ (1 / 100 : ℝ) := by simpa [abs_mul] using hdown' show |theta p| ≤ (1 / 100 : ℝ) nlinarith [abs_nonneg (theta p)] /-- Centered representatives minimize absolute numerator under natural scaling. -/ theorem natAbs_valMinAbs_nsmul_le {N : ℕ} (k : ℕ) (z : ZMod N) : (k • z).valMinAbs.natAbs ≤ k * z.valMinAbs.natAbs := by induction k with | zero => simp | succ k ih => rw [succ_nsmul] calc (k • z + z).valMinAbs.natAbs ≤ ((k • z).valMinAbs + z.valMinAbs).natAbs := ZMod.natAbs_valMinAbs_add_le (k • z) z _ ≤ (k • z).valMinAbs.natAbs + z.valMinAbs.natAbs := Int.natAbs_add_le _ _ _ ≤ k * z.valMinAbs.natAbs + z.valMinAbs.natAbs := Nat.add_le_add_right ih _ _ = (k + 1) * z.valMinAbs.natAbs := by ring /-- Natural residue multiplication scales the centered real representative by at most the same factor. Equality is not asserted because wrapping may occur. -/ theorem abs_taoSignedTheta_natCast_mul_le {N : ℕ} [NeZero N] (k : ℕ) (z : ZMod N) : |taoSignedTheta ((k : ZMod N) * z)| ≤ (k : ℝ) * |taoSignedTheta z| := by have hN : (0 : ℝ) < (N : ℝ) := by exact_mod_cast NeZero.pos N have hmul : (k : ZMod N) * z = k • z := by rw [mul_comm] exact (nsmul_eq_mul' z k).symm have hnumNat := natAbs_valMinAbs_nsmul_le (N := N) k z have hnumInt : |(k • z).valMinAbs| ≤ (k : ℤ) * |z.valMinAbs| := by rw [Int.abs_eq_natAbs, Int.abs_eq_natAbs] exact_mod_cast hnumNat have hnum : |((((k • z).valMinAbs : ℤ) : ℝ))| ≤ (k : ℝ) * |((z.valMinAbs : ℤ) : ℝ)| := by rw [← Int.cast_abs, ← Int.cast_abs] exact_mod_cast hnumInt rw [taoSignedTheta, taoSignedTheta, hmul, abs_div, abs_div, abs_of_pos hN] calc |(((k • z).valMinAbs : ℤ) : ℝ)| / (N : ℝ) ≤ ((k : ℝ) * |((z.valMinAbs : ℤ) : ℝ)|) / (N : ℝ) := div_le_div_of_nonneg_right hnum hN.le _ = (k : ℝ) * (|((z.valMinAbs : ℤ) : ℝ)| / (N : ℝ)) := by ring end end Tao end Erdos1135