[{"type":"thinking","thinking":"OK — `ArithmeticFunction.LSeriesSummable_vonMangoldt (hs : 1 < s.re) : LSeriesSummable ↗Λ s`. The Λ: in VerticalLine, `(Λ n : ℂ)` — with opens... they must have `open ArithmeticFunction` or `open scoped ArithmeticFunction`? The grep didn't show the open lines (they're after line 30 maybe). Let me not waste time: in my file I'll write `(ArithmeticFunction.vonMangoldt n : ℂ)` and for the rewrite use the ↗Λ lemma — the bridge: is `LSeries (fun n => (↑(vonMangoldt n) : ℂ)) s` syntactically equal to `L ↗Λ s`? LSeries takes f : ℕ → ℂ: L ↗Λ = LSeries ↗Λ where ↗Λ = (fun n => ↑(vonMangoldt n))?? The coercion ↗f for f : ArithmeticFunction ℝ to ℕ → ℂ: there's a coe instance: `ArithmeticFunction` ℝ→ℂ... The safest: state my Dirichlet lemma via `L ↗Λ` directly: \n\ndef dirichletC2 : ℝ := ∑' n : ℕ, ‖LSeries.term ↗ArithmeticFunction.vonMangoldt (2 : ℂ) n‖\n\nwith the ↗ coe — written in Lean as `↗(ArithmeticFunction.vonMangoldt)`? The coe arrow syntax: `↗f` for f : ArithmeticFunction ℝ into ℕ → ℂ requires a Coe — in the Mathlib file they write `L ↗Λ s` with Λ := ArithmeticFunction.vonMangoldt via `open ArithmeticFunction` (scoped notation Λ?). Check: Mathlib ArithmeticFunction has `scoped notation \"Λ\" => ArithmeticFunction.vonMangoldt`? probably `open scoped ArithmeticFunction` gives Λ. And ↗ is the coe notation.\n\nThis is getting into the weeds. DECISION: mirror VerticalLine's working pattern as closely as possible, with `open ArithmeticFunction` locally if needed. I'll write the file and let the compiler tell me. Time to write §0–§2 of LiCircleBound.lean.\n\nThe Dirichlet lemma (mirror VerticalLine + ZetaBounds):\n\n```lean\n/-- The Dirichlet majorant constant C₂ (the absolute sum of the\nvon Mangoldt L-series terms at s = 2). -/\ndef dirichletC2 : ℝ :=\n  ∑' n : ℕ, ‖LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) (2 : ℂ) n‖\n\nlemma norm_logDeriv_zeta_le_dirichlet {s : ℂ} (hs : 2 ≤ s.re) :\n    ‖logDeriv riemannZeta s‖ ≤ dirichletC2 := by\n  have hs1 : (1 : ℝ) < s.re := by linarith\n  -- −logDeriv ζ s = LSeries Λ s\n  have h1 : -logDeriv riemannZeta s\n      = LSeries (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) s := by\n    have h2 := ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div hs1\n    rw [logDeriv, Pi.div_apply, h2]\n    ring\n  have hsum2 : Summable (fun n : ℕ =>\n      ‖LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) (2 : ℂ) n‖) := by\n    rw [summable_norm_iff]\n    exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num : (1:ℝ) < (2:ℂ).re)\n  have hsumS : Summable (fun n : ℕ =>\n      ‖LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) s n‖) := by\n    rw [summable_norm_iff]\n    exact ArithmeticFunction.LSeriesSummable_vonMangoldt hs1\n  have hle : ‖LSeries (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) s‖ ≤ dirichletC2 := by\n    rw [LSeries]  -- unfold to tsum? LSeries f s := ∑' n, LSeries.term f s n: `rw [LSeries]`?? def — use `show`: LSeries f s = ∑' n, term f s n — `LSeries.def`? There's `LSeries` def + `LSeries_def`? VerticalLine used `rw [Hfn_line, h1, LSeries, ← tsum_mul_left]` — so rw [LSeries] unfolds (equation lemma) ✓.\n    calc ‖∑' n, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) s n‖\n        ≤ ∑' n, ‖LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) s n‖ :=\n          norm_tsum_le_tsum_norm hsumS  -- norm_tsum_le_tsum_norm (hf : Summable (fun i => ‖f i‖)?) : ‖∑' f‖ ≤ ∑' ‖f‖ — from ZetaBounds: `norm_tsum_le_tsum_norm summable_abs_value` ✓ takes the norm-summable.\n      _ ≤ ∑' n, ‖LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) (2 : ℂ) n‖ :=\n          Summable.tsum_le_tsum (fun n => LSeries.norm_term_le_of_re_le_re _ (by\n              rw [Complex.natCast_re?]: (2:ℂ).re = 2: `by simp`: (2:ℂ).re ≤ s.re: simp then hs) n) hsumS hsum2\n      _ = dirichletC2 := rfl\n  rw [← h1, norm_neg]  -- goal ‖logDeriv ζ s‖ ≤ C₂: ‖logDeriv‖ = ‖−logDeriv‖ (norm_neg backwards): ‖logDeriv ζ s‖ = ‖−logDeriv ζ s‖: rw [← h1] changes goal to ‖−logDeriv ζ s‖?? hmm: goal ‖logDeriv ζ s‖ ≤ C₂: have : ‖logDeriv ζ s‖ = ‖LSeries ... s‖: rw [← norm_neg, h1]: goal ‖LSeries‖ ≤ C₂ = hle ✓:\n  calc ‖logDeriv riemannZeta s‖ = ‖LSeries (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) s‖ := by\n        rw [← h1, norm_neg]\n    _ ≤ dirichletC2 := hle\n```\nCheck `LSeries.norm_term_le_of_re_le_re (f) {s s'} (h : s.re ≤ s'.re) (n) : ‖term f s' n‖ ≤ ‖term f s n‖` — with s' := s, s := (2:ℂ): h : (2:ℂ).re ≤ s.re: (2:ℂ).re = 2 ✓ simp; hs ✓. And `Summable.tsum_le_tsum (h : ∀ i, f i ≤ g i) (hf : Summable f) (hg : Summable g) : ∑' f ≤ ∑' g` — protected name from journal-027 gotcha ✓ (Summable.tsum_le_tsum ✓).\n\ndirichletC2's def with the explicit fun n => (↑(vonMangoldt n) : ℂ): the LSeries.term of that function ✓ matches h1's RHS form ✓.\n\nHmm — LSeries.term signature: `LSeries.term (f : ℕ → ℂ) (s : ℂ) (n : ℕ) : ℂ`? From VerticalLine: `LSeries.term (fun n => (Λ n : ℂ)) (c + t*I) n` ✓ argument order (f, s, n) ✓.\n\nNow write the file §0–§2. Also the conj lemma and count bridges. Let me draft the whole §0–§2 and compile.","signature":""},{"type":"text","text":"All patterns confirmed. Writing LiCircleBound.lean §0–§2 (conj symmetry, count bridges, Dirichlet bound)."},{"type":"tool_use","id":"Write_259","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","content":"/-\nNguyenRH1/LiCircleBound.lean — tier ii′ sub-stage ii′-5e-v (D-068 §5):\nthe circle bound for the partial-fraction difference and the squeeze to\nthe capstone identity.\n\n§0: conjugate symmetry of the nontrivial zeros (for the lower-half\n    circle arcs).\n§1: the mult-weighted ordinate-window count bridges from Zeta23's RvM\n    local count (`zeta_local_zero_count`: Ncount(t, t+1] ≤ A₀·log(|t|+3)\n    with multiplicity, ALL t — no perimeter): `A0count` + the dyadic\n    interval count `Ncount_le_of_interval`.\n§2: the Dirichlet bound `norm_logDeriv_zeta_le_dirichlet` (‖ζ′/ζ s‖ ≤\n    dirichletC2 for re s ≥ 2; −ζ′/ζ = LSeries Λ, termwise ≤ the s = 2\n    majorant) — the right near-axis arc input.\n§3+: (this file, subsequent sub-stages) the series bound, the logDeriv\n    assembly, the Fexpr circle bound, the Schwarz squeeze, and the\n    capstone `partialFraction_identity`.\n\nArchive references: D-068 §5, journal-029.\n-/\nimport Mathlib\nimport Zeta23.RvM.LocalCount\nimport NguyenRH1.LiFEntire\nimport NguyenRH1.LiGapCircles\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 0. Conjugate symmetry -/\n\n/-- The conjugate of a nontrivial zero is a nontrivial zero. -/\nlemma isNontrivialZero_conj {ρ : ℂ} (hρ : IsNontrivialZero ρ) :\n    IsNontrivialZero ((starRingEnd ℂ) ρ) := by\n  refine ⟨?_, ?_, ?_⟩\n  · rw [riemannZeta_conj, hρ.1, map_zero]\n  · rw [Complex.conj_re]; exact hρ.2.1\n  · rw [Complex.conj_re]; exact hρ.2.2\n\n/-- The ordinate of the conjugate. -/\nlemma conj_im' (ρ : ℂ) : ((starRingEnd ℂ) ρ).im = -ρ.im := Complex.conj_im\n\n/-! ## 1. The mult-weighted window count bridges -/\n\n/-- The local-count constant: from Zeta23's RvM local count\n(`zeta_local_zero_count`, [Tit86 Thm 9.2] — mult-weighted, two-sided,\nall t, PROVED unconditionally in Zeta23). -/\ndef A0count : ℝ := Classical.choose Zeta23.RvM.zeta_local_zero_count\n\nlemma A0count_ge_one : (1 : ℝ) ≤ A0count :=\n  (Classical.choose_spec Zeta23.RvM.zeta_local_zero_count).1\n\nlemma A0count_pos : (0 : ℝ) < A0count := lt_of_lt_of_le one_pos A0count_ge_one\n\n/-- The unit-window bound: Ncount(t, t+1] ≤ A₀·log(|t|+3). -/\nlemma Ncount_window_le (t : ℝ) :\n    (Zeta23.Ncount t (t + 1) : ℝ) ≤ A0count * Real.log (|t| + 3) :=\n  (Classical.choose_spec Zeta23.RvM.zeta_local_zero_count).2 t\n\n/-- Window additivity: Ncount(a, c] = Ncount(a, b] + Ncount(b, c] for\na ≤ b ≤ c (the disjoint union of the ordinate windows). -/\nlemma Ncount_add_of_le {a b c : ℝ} (hab : a ≤ b) (hbc : b ≤ c) :\n    Zeta23.Ncount a c = Zeta23.Ncount a b + Zeta23.Ncount b c := by\n  have hunion : Zeta23.zerosIn a c = Zeta23.zerosIn a b ∪ Zeta23.zerosIn b c := by\n    ext ρ\n    simp only [Zeta23.zerosIn, Set.mem_setOf_eq, Set.mem_union]\n    constructor\n    · rintro ⟨hρ, ha, hc⟩\n      rcases le_or_lt ρ.im b with h | h\n      · exact Or.inl ⟨hρ, ha, h⟩\n      · exact Or.inr ⟨hρ, h, hc⟩\n    · rintro (⟨hρ, ha, hb⟩ | ⟨hρ, hb, hc⟩)\n      · exact ⟨hρ, ha, hb.trans hbc⟩\n      · exact ⟨hρ, hab.trans_lt hb, hc⟩\n  have hdisj : Disjoint (Zeta23.zerosIn a b) (Zeta23.zerosIn b c) := by\n    rw [Set.disjoint_iff]\n    intro ρ ⟨h1, h2⟩\n    exact absurd (h1.2.2.trans h2.2.1) (lt_irrefl _)\n  rw [Zeta23.Ncount, Zeta23.Ncount, Zeta23.Ncount, hunion]\n  exact_mod_cast finsum_union hdisj (Zeta23.zerosIn_finite a b)\n    (Zeta23.zerosIn_finite b c)\n\n/-- The N-step unit-window chain: Ncount(a, a+N] ≤ A₀·N·log(|a|+N+3). -/\nlemma Ncount_chain_le (a : ℝ) (N : ℕ) :\n    (Zeta23.Ncount a (a + (N : ℝ)) : ℝ)\n      ≤ A0count * (N : ℝ) * Real.log (|a| + (N : ℝ) + 3) := by\n  induction N with\n  | zero =>\n    simp only [Nat.cast_zero, add_zero]\n    have h0 : Zeta23.zerosIn a a = ∅ := by\n      ext ρ\n      simp only [Zeta23.zerosIn, Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false]\n      rintro ⟨-, h1, h2⟩\n      exact absurd (h1.trans h2) (lt_irrefl _)\n    have : Zeta23.Ncount a a = 0 := by\n      simp only [Zeta23.Ncount, h0, finsum_empty]\n    rw [this]\n    positivity\n  | succ N ih =>\n    have hsplit : Zeta23.Ncount a (a + ((N : ℝ) + 1))\n        = Zeta23.Ncount a (a + (N : ℝ)) + Zeta23.Ncount (a + (N : ℝ)) (a + (N : ℝ) + 1) :=\n      Ncount_add_of_le (by linarith) (by linarith)\n    have hcast : a + ((N + 1 : ℕ) : ℝ) = a + ((N : ℝ) + 1) := by push_cast; ring\n    rw [hcast, hsplit]\n    have hwin := Ncount_window_le (a + (N : ℝ))\n    have hlog : Real.log (|a + (N : ℝ)| + 3) ≤ Real.log (|a| + (N : ℝ) + 3) := by\n      apply Real.log_le_log (by positivity)\n      have h1 : |a + (N : ℝ)| ≤ |a| + (N : ℝ) :=\n        (abs_add_le _ _).trans (by rw [abs_of_nonneg (Nat.cast_nonneg N)])\n      linarith\n    have hlogpos : (0 : ℝ) ≤ Real.log (|a| + (N : ℝ) + 3) := by\n      apply Real.log_nonneg\n      have : (1 : ℝ) ≤ |a| + 3 := by linarith [abs_nonneg a]\n      linarith [Nat.cast_nonneg N]\n    have hlogpos' : (0 : ℝ) ≤ Real.log (|a| + ((N : ℝ) + 1) + 3) := by\n      apply Real.log_nonneg\n      have : (1 : ℝ) ≤ |a| + 3 := by linarith [abs_nonneg a]\n      linarith [Nat.cast_nonneg N]\n    push_cast\n    rw [Nat.cast_add, Nat.cast_one]\n    -- goal: ↑(Ncount a (a+↑N)) + ↑(Ncount (a+↑N) (a+↑N+1)) ≤ A₀(↑N+1)·log(|a|+(↑N+1)+3)\n    have hle1 : A0count * (N : ℝ) * Real.log (|a| + (N : ℝ) + 3)\n        ≤ A0count * ((N : ℝ) + 1) * Real.log (|a| + ((N : ℝ) + 1) + 3) := by\n      have h1 : (N : ℝ) ≤ (N : ℝ) + 1 := by linarith\n      have h2 : Real.log (|a| + (N : ℝ) + 3) ≤ Real.log (|a| + ((N : ℝ) + 1) + 3) :=\n        Real.log_le_log (by positivity) (by linarith)\n      have h3 := mul_le_mul h1 h2 hlogpos hlogpos'\n      calc A0count * (N : ℝ) * Real.log (|a| + (N : ℝ) + 3)\n          ≤ A0count * ((N : ℝ) + 1) * Real.log (|a| + ((N : ℝ) + 1) + 3) := by\n            have hA0 := A0count_pos.le\n            have h4 : A0count * (N : ℝ) * Real.log (|a| + (N : ℝ) + 3)\n                ≤ A0count * (N : ℝ) * Real.log (|a| + ((N : ℝ) + 1) + 3) :=\n              mul_le_mul_of_nonneg_left h2 (by positivity)\n            have h5 : A0count * (N : ℝ) * Real.log (|a| + ((N : ℝ) + 1) + 3)\n                ≤ A0count * ((N : ℝ) + 1) * Real.log (|a| + ((N : ℝ) + 1) + 3) := by\n              apply mul_le_mul_of_nonneg_right _ hlogpos'\n              nlinarith [hA0, Nat.cast_nonneg N]\n            exact h4.trans h5\n    have hle2 : A0count * Real.log (|a + (N : ℝ)| + 3)\n        ≤ A0count * Real.log (|a| + ((N : ℝ) + 1) + 3) := by\n      apply mul_le_mul_of_nonneg_left _ A0count_pos.le\n      apply Real.log_le_log (by positivity)\n      have h1 : |a + (N : ℝ)| ≤ |a| + (N : ℝ) :=\n        (abs_add_le _ _).trans (by rw [abs_of_nonneg (Nat.cast_nonneg N)])\n      linarith\n    linarith [ih, hwin, hle1, hle2]\n\n/-- The dyadic-interval count: Ncount(a, b] ≤ A₀·(b−a+1)·log(b+4) for\n0 ≤ a ≤ b (the unit-window chain + the ceil cover). -/\nlemma Ncount_le_of_interval {a b : ℝ} (ha : 0 ≤ a) (hab : a ≤ b) :\n    (Zeta23.Ncount a b : ℝ) ≤ A0count * (b - a + 1) * Real.log (b + 4) := by\n  set N : ℕ := ⌈b - a⌉₊ with hN\n  have hNb : b ≤ a + (N : ℝ) := by\n    have h := Nat.le_ceil (b - a)\n    rw [← hN] at h\n    have h2 : (b - a : ℝ) ≤ (N : ℝ) := h\n    linarith\n  have hsub : Zeta23.zerosIn a b ⊆ Zeta23.zerosIn a (a + (N : ℝ)) := by\n    intro ρ hρ\n    refine ⟨hρ.1, hρ.2.1, ?_⟩\n    exact hρ.2.2.trans hNb\n  have hle : Zeta23.Ncount a b ≤ Zeta23.Ncount a (a + (N : ℝ)) := by\n    rw [Zeta23.Ncount, Zeta23.Ncount]\n    rw [finsum_mem_eq_finite_toFinset_sum _ (Zeta23.zerosIn_finite a b),\n      finsum_mem_eq_finite_toFinset_sum _ (Zeta23.zerosIn_finite a (a + (N : ℝ)))]\n    apply Finset.sum_le_sum_of_subset_of_nonneg\n    · intro ρ hρ\n      rw [Set.Finite.mem_toFinset] at hρ ⊢\n      exact hsub hρ\n    · intro ρ _ _\n      exact Nat.cast_nonneg _\n  have hchain := Ncount_chain_le a N\n  have hNle : (N : ℝ) ≤ b - a + 1 := by\n    have h := Nat.ceil_lt_add_one (sub_nonneg.mpr hab)\n    rw [← hN] at h\n    have h2 : ((N : ℝ)) < b - a + 1 := by exact_mod_cast h\n    linarith\n  have hlogle : Real.log (|a| + (N : ℝ) + 3) ≤ Real.log (b + 4) := by\n    apply Real.log_le_log (by positivity)\n    rw [abs_of_nonneg ha]\n    linarith\n  calc (Zeta23.Ncount a b : ℝ)\n      ≤ (Zeta23.Ncount a (a + (N : ℝ)) : ℝ) := by exact_mod_cast hle\n    _ ≤ A0count * (N : ℝ) * Real.log (|a| + (N : ℝ) + 3) := hchain\n    _ ≤ A0count * (b - a + 1) * Real.log (b + 4) := by\n        have hlogpos : (0 : ℝ) ≤ Real.log (b + 4) := by\n          apply Real.log_nonneg; linarith\n        have h1 : A0count * (N : ℝ) * Real.log (|a| + (N : ℝ) + 3)\n            ≤ A0count * (N : ℝ) * Real.log (b + 4) :=\n          mul_le_mul_of_nonneg_left hlogle (by positivity)\n        have h2 : A0count * (N : ℝ) * Real.log (b + 4)\n            ≤ A0count * (b - a + 1) * Real.log (b + 4) := by\n          apply mul_le_mul_of_nonneg_right _ hlogpos\n          nlinarith [A0count_pos, hNle, sub_nonneg.mpr hab]\n        exact h1.trans h2\n\n/-! ## 2. The Dirichlet bound -/\n\n/-- The Dirichlet majorant constant: the absolute sum of the von\nMangoldt L-series terms at s = 2 (summable since 2 > 1). -/\ndef dirichletC2 : ℝ :=\n  ∑' n : ℕ, ‖LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) (2 : ℂ) n‖\n\n/-- ‖ζ′/ζ s‖ ≤ C₂ for re s ≥ 2 (the Dirichlet-series majorant:\n−ζ′/ζ = LSeries Λ for re > 1, termwise ≤ the s = 2 sum). -/\nlemma norm_logDeriv_zeta_le_dirichlet {s : ℂ} (hs : 2 ≤ s.re) :\n    ‖logDeriv riemannZeta s‖ ≤ dirichletC2 := by\n  have hs1 : (1 : ℝ) < s.re := by linarith\n  have h1 : -logDeriv riemannZeta s\n      = LSeries (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) s := by\n    have h2 := ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div hs1\n    rw [logDeriv, Pi.div_apply, h2]\n    ring\n  have hsum2 : Summable (fun n : ℕ =>\n      ‖LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) (2 : ℂ) n‖) := by\n    rw [summable_norm_iff]\n    exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n  have hsumS : Summable (fun n : ℕ =>\n      ‖LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) s n‖) := by\n    rw [summable_norm_iff]\n    exact ArithmeticFunction.LSeriesSummable_vonMangoldt hs1\n  have hle : ‖LSeries (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) s‖\n      ≤ dirichletC2 := by\n    show ‖∑' n : ℕ, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) s n‖ ≤ _\n    calc ‖∑' n : ℕ, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) s n‖\n        ≤ ∑' n : ℕ, ‖LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) s n‖ :=\n          norm_tsum_le_tsum_norm hsumS\n      _ ≤ ∑' n : ℕ, ‖LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) (2 : ℂ) n‖ :=\n          Summable.tsum_le_tsum (fun n => LSeries.norm_term_le_of_re_le_re _ (by\n            simp\n            exact hs) n) hsumS hsum2\n      _ = dirichletC2 := rfl\n  calc ‖logDeriv riemannZeta s‖\n      = ‖LSeries (fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) s‖ := by\n        rw [← h1, norm_neg]\n    _ ≤ dirichletC2 := hle\n\nend NguyenRH1\n\nend\n"},"extra_content":null}]