← Session 35 — ii′-6-R: Lagarias 6.1 right-tail re-proof (E-032/D-073)
thinking (3132 chars)
I see the issues now:
1. **Line 1087 (`linarith` inside the by-block at 1085-1087)**: the `by simp; linarith` for `(by ... : ((2√↑n : ℂ)) ≤ (((2√↑n : ℝ) + t·I).re)` — hmm — `LSeries.norm_term_le_of_re_le_re _ (by simp; linarith) m` — the second arg is the re-comparison `s.re ≤ s'.re` — with s := (2√n : ℂ) and s' := σ+tI: needs `(2√n : ℂ).re ≤ ((2√n : ℝ) + t·I).re` — ((2√n:ℂ).re) = 2√n and ((σ+tI).re) = σ = 2√n — equal ✓ — the by simp gives the RHS .re = 2√n?? then linarith closes 2√n ≤ 2√n ✓ — hmm but the error is at 1087:12 "No goals to be solved" — meaning `simp` already closed the goal (the re-equality is rfl-simplifiable: ((2√↑n : ℂ)).re ≤ (((2√↑n:ℝ)+↑t·I).re) — simp reduces both sides' .re to 2√↑n: goal becomes 2√↑n ≤ 2√↑n — simp's le_refl closes it ✓ — then linarith has no goals ✗. Fix: `by simp` alone.
2. **Line 1119 (the e in X_le_three_pow)**: `rw [e]` failed? — no, 1119:68 was in hΛ's region... — actually looking at the previous error output: 1119:68 "unsolved goals" with context ending at hΛ — hmm, the context showed `hΛ : ‖↑(Λ m)‖ = Λ m` present — and 1087:12 "No goals" — hmm these are at DIFFERENT spots than I thought. The 1119:68 error — line 1119 col 68 is inside `e`'s statement (line 1117-1119) — the `field_simp [hm2.ne']` at 1122 — field_simp closed, no ring after ✓ — hmm. Actually 1119:68 might be the `rw [show (2·√↑n) = (2·√↑n−2)+2 by ring, Real.rpow_add hm2]` — the `by ring` proves the exponent identity ✓ — hmm. Wait, maybe 1119:68 is `have e : ... := by rw [...]; field_simp [hm2.ne']` — the field_simp at 1122 — no.
Let me just apply the known fixes and recompile to see what remains. The 1119:68 error with context ending at hΛ — the only tactic between hΛ (1108-1110) and the end of the lemma is at 1111 (`rw [hΛ]`) — "unsolved goals" after rw [hΛ]?? — the goal after `simp only [norm_term_eq, hXre, h2re]; split; next hm0` is `‖(fun n => (↑(Λ n)):ℂ) m‖ / ↑m ^ (2·√↑n) ≤ 3^(...)·(‖(fun n => (↑(Λ n)):ℂ) m‖ / ↑m ^ 2)` — rw [hΛ] rewrites `‖(ArithmeticFunction.vonMangoldt m : ℂ)‖` — but the goal has `‖(fun n => (↑(Λ m)):ℂ) m‖` — beta-redex — hΛ's LHS is `‖(ArithmeticFunction.vonMangoldt m : ℂ)‖` — the beta-redex `(fun n => (↑(Λ n)):ℂ) m` vs `(↑(Λ m):ℂ)` — NOT syntactically equal → rw [hΛ] fails OR rewrites nothing... — "unsolved goals" — hmm no, rw would say "pattern not found". The 1119:68 is at `e`'s have (1117-1122) — probably the field_simp's behavior. Let me just add `show` to beta-reduce before rw [hΛ], and fix the known issues, then recompile.
Actually — looking more carefully at the error list: 1119:68 was in the FIRST error batch with the context `n, hn, m, hm, hXre, h2re, hm0, hΛ` — that's exactly the state BEFORE `rw [hΛ]` at line 1111... — no wait, hΛ is PRESENT in the context, so it's AFTER hΛ's proof. The unsolved goals at 1119:68 — col 68 — line 1119 is `(ArithmeticFunction.vonMangoldt m / (m : ℝ) ^ (2 : ℝ)) := by` (e's RHS continuation) — the `field_simp [hm2.ne']` at 1122 — hmm col 68 of line 1119 doesn't align with 1122. The errors might be misaligned due to my earlier edits. Let me just recompile and look at the current first error precisely.
Applying the known fixes and recompiling to see the precise remaining errors.
tool: Edit ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"old_string": " Summable.tsum_le_tsum (fun m => LSeries.norm_term_le_of_re_le_re _ (by\n simp\n linarith) m) hsumS hsumX",
"new_string": " Summable.tsum_le_tsum (fun m => LSeries.norm_term_le_of_re_le_re _ (by\n simp) m) hsumS hsumX"
}
result 2026-08-26 09:29:27Z → 2026-08-26 09:29:27Z
{
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"oldString": " Summable.tsum_le_tsum (fun m => LSeries.norm_term_le_of_re_le_re _ (by\n simp\n linarith) m) hsumS hsumX",
"newString": " Summable.tsum_le_tsum (fun m => LSeries.norm_term_le_of_re_le_re _ (by\n simp) m) hsumS hsumX",
"originalFile": "/-\nNguyenRH1/LiDogLeg.lean \u2014 ii\u2032-6-DL-1 (D-074, session 040): the zero-free\ndog-leg deformation of the Lagarias horizontal right tail.\n\nTHE CONTENT (D-074 \u00a73.1\u20133.2): on the dog-leg rectangle [3, X_R]\u00d7[T, U]\n(T > 0), the Lagarias integrand Jlag n s = (s/(s\u22121))\u207f\u22121 \u00b7 (\u2212\u03b6\u2032/\u03b6(s)) is\nholomorphic UNCONDITIONALLY (s \u2260 1 since im > 0; \u03b6 s \u2260 0 for re s \u2265 3 > 1\nby Mathlib's Euler-product nonvanishing `riemannZeta_ne_zero_of_one_lt_re`\n\u2014 NO perimeter hypothesis, no zero-avoidance at U). Cauchy's theorem\n(`HolomorphicOn.vanishesOnRectangle`) then gives the EXACT deformation\nidentity\n H(3\u2192X_R, T) = H(3\u2192X_R, U) \u2212 V(X_R, T\u2192U) + V(3, T\u2192U)\n(the upper right tail = the high horizontal \u2212 the right drop + the left\nlift), plus the Jlag conjugation identity for the lower-side transfer.\n\nThis is the repair of Finding F-D (D-072): the broken low-path envelope\nis ABANDONED (not bounded) \u2014 the magnitude bound comes later from the\nthree dog-leg pieces, all in the unconditionally zero-free region.\n-/\nimport Mathlib\nimport NguyenRH1.LiSides\nimport NguyenRH1.LiTightR\nimport NguyenRH1.LiBasePoints\n\nopen Zeta23 Complex Filter Topology\nopen scoped Nat\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. Dog-leg analyticity (D-074 \u00a73.1; no perimeter) -/\n\n/-- The dog-leg corners (same real-part/imaginary-part form as rectZ/rectW). -/\ndef dogZ (T : \u211d) : \u2102 := ((3 : \u211d) : \u2102) + (T : \u211d) * Complex.I\ndef dogW (XR U : \u211d) : \u2102 := ((XR : \u211d) : \u2102) + (U : \u211d) * Complex.I\n\nlemma dogZ_re (T : \u211d) : (dogZ T).re = 3 := by simp [dogZ]\nlemma dogZ_im (T : \u211d) : (dogZ T).im = T := by simp [dogZ]\nlemma dogW_re (XR U : \u211d) : (dogW XR U).re = XR := by simp [dogW]\nlemma dogW_im (XR U : \u211d) : (dogW XR U).im = U := by simp [dogW]\n\n/-- Jlag is differentiable at every point with 3 \u2264 re, 0 < im:\n\u03b6 is zero-free for 1 < re (Mathlib `riemannZeta_ne_zero_of_one_lt_re`,\nUNCONDITIONAL \u2014 the Euler-product region) and s \u2260 1 (im \u2260 0).\nNO perimeter hypothesis. -/\ntheorem Jlag_differentiableAt_dogleg {n : \u2115} {s : \u2102}\n (hre : 3 \u2264 s.re) (him : 0 < s.im) :\n DifferentiableAt \u2102 (Jlag n) s := by\n apply Jlag_differentiableAt\n \u00b7 rintro rfl\n simp at him\n \u00b7 exact riemannZeta_ne_zero_of_one_lt_re (by linarith)\n\n/-- Jlag is holomorphic on the whole dog-leg rectangle\n[3, X_R]\u00d7[T, U] (T > 0). -/\ntheorem holomorphicOn_Jlag_dogleg {n : \u2115} {T U XR : \u211d}\n (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n HolomorphicOn (Jlag n) (Rectangle (dogZ T) (dogW XR U)) := by\n intro s hs\n rw [Rectangle, Complex.mem_reProdIm, dogZ_re, dogW_re, dogZ_im, dogW_im] at hs\n rw [Set.uIcc_of_le hXR, Set.uIcc_of_le hTU] at hs\n exact (Jlag_differentiableAt_dogleg hs.1.1 (by linarith [hs.2.1])).differentiableWithinAt\n\n/-- **The dog-leg rectangle integral vanishes** (Cauchy; the region is\nunconditionally zero-free \u2014 the residue sum is EMPTY, no perimeter). -/\ntheorem rectangleIntegral_Jlag_dogleg_eq_zero {n : \u2115} {T U XR : \u211d}\n (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n RectangleIntegral (Jlag n) (dogZ T) (dogW XR U) = 0 :=\n (holomorphicOn_Jlag_dogleg hT hTU hXR).vanishesOnRectangle subset_rfl\n\n/-- The (2\u03c0i)\u207b\u00b9-normalized form (API symmetry with rectInt'_Jlag). -/\ntheorem rectangleIntegral'_Jlag_dogleg_eq_zero {n : \u2115} {T U XR : \u211d}\n (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n RectangleIntegral' (Jlag n) (dogZ T) (dogW XR U) = 0 := by\n rw [RectangleIntegral', rectangleIntegral_Jlag_dogleg_eq_zero hT hTU hXR,\n smul_zero]\n\n/-- **THE EXACT DOG-LEG DEFORMATION IDENTITY** (D-074 \u00a73.2 (\u2605), the\ncapstone for ii\u2032-6-c): the low right-horizontal equals the high\nhorizontal minus the right drop plus the left lift. Orientations: all\nH-integrals left-to-right, all V-integrals upward (the main contour's\ntop side is their reversal, handled at the fold site). -/\ntheorem upper_right_horizontal_eq_dogleg {n : \u2115} {T U XR : \u211d}\n (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n HIntegral (Jlag n) 3 XR T =\n HIntegral (Jlag n) 3 XR U - VIntegral (Jlag n) XR T U +\n VIntegral (Jlag n) 3 T U := by\n have h0 := rectangleIntegral_Jlag_dogleg_eq_zero (n := n) hT hTU hXR\n unfold RectangleIntegral at h0\n rw [dogZ_re, dogZ_im, dogW_re, dogW_im] at h0\n linear_combination h0\n\n/-! ## 2. The conjugation identity (for the lower-side transfer, DL-5) -/\n\n/-- kFun is conjugation-symmetric (real-coefficient rational function). -/\ntheorem kFun_conj (n : \u2115) (s : \u2102) :\n kFun n ((starRingEnd \u2102) s) = (starRingEnd \u2102) (kFun n s) := by\n unfold kFun\n simp [map_sub, map_pow, map_one]\n\n/-- **Jlag is conjugation-symmetric** (from `riemannZeta_conj` +\n`iteratedDeriv_riemannZeta_conj`): Jlag n (s\u0304) = conj (Jlag n s).\nNeeded for the lower-horizontal transfer (DL-5). -/\ntheorem Jlag_conj (n : \u2115) {s : \u2102} (hs : s \u2260 1) :\n Jlag n ((starRingEnd \u2102) s) = (starRingEnd \u2102) (Jlag n s) := by\n have h1 : deriv riemannZeta ((starRingEnd \u2102) s) =\n (starRingEnd \u2102) (deriv riemannZeta s) := by\n have h := iteratedDeriv_riemannZeta_conj 1 hs\n rwa [iteratedDeriv_one] at h\n unfold Jlag\n rw [kFun_conj, _root_.riemannZeta_conj, h1]\n simp only [map_mul, map_neg, map_div\u2080]\n\n/-! ## 3. The left-lift bound (DL-2, D-074 \u00a73.3 V_left) -/\n\n/-- **Kernel on the lift**: \u2016kFun n (3+it)\u2016 \u2264 e^{5/2}+1 for n \u2264 t\u00b2\n(D-074 \u00a76: |(3+it)/(2+it)|\u00b2 = (9+t\u00b2)/(4+t\u00b2) = 1 + 5/(4+t\u00b2), whose n-th\npower is \u2264 e^{5n/(2(4+t\u00b2))} \u2264 e^{5/2}). -/\ntheorem kFun_three_add_it_norm_le_exp_five_halves_add_one {n : \u2115} {t : \u211d}\n (hT : (n : \u211d) \u2264 t ^ 2) :\n \u2016kFun n ((3 : \u211d) + t * Complex.I)\u2016 \u2264 Real.exp (5 / 2) + 1 := by\n rcases Nat.eq_zero_or_pos n with hn | hn\n \u00b7 subst hn\n simp only [kFun, pow_zero, sub_self, norm_zero]\n positivity\n \u00b7 have hnR : (1 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have ht2 : (0 : \u211d) < 4 + t ^ 2 := by positivity\n have hn4t2 : (n : \u211d) \u2264 4 + t ^ 2 := by linarith [hT]\n have hQpos : (0 : \u211d) \u2264 (9 + t ^ 2) / (4 + t ^ 2) := by positivity\n have hQle : (9 + t ^ 2) / (4 + t ^ 2) \u2264 Real.exp (5 / (n : \u211d)) := by\n have h1 : (9 + t ^ 2) / (4 + t ^ 2) = 1 + 5 / (4 + t ^ 2) := by\n field_simp [ht2.ne']\n ring\n rw [h1]\n have h2 : 5 / (4 + t ^ 2) \u2264 5 / (n : \u211d) :=\n (div_le_div_iff_of_pos_left (by norm_num) ht2 (by positivity)).mpr hn4t2\n have h3 := Real.add_one_le_exp (5 / (n : \u211d))\n linarith\n have hQn : ((9 + t ^ 2) / (4 + t ^ 2)) ^ n \u2264 Real.exp 5 := by\n calc ((9 + t ^ 2) / (4 + t ^ 2)) ^ n \u2264 (Real.exp (5 / (n : \u211d))) ^ n :=\n pow_le_pow_left\u2080 hQpos hQle n\n _ = Real.exp 5 := by\n rw [\u2190 Real.exp_nat_mul]\n congr 1\n have hn0 : (n : \u211d) \u2260 0 := by exact_mod_cast hn.ne'\n field_simp [hn0]\n have hns3 : Complex.normSq ((3 : \u211d) + t * Complex.I) = 9 + t ^ 2 := by\n simp [Complex.normSq_apply]\n ring\n have hns2 : Complex.normSq ((2 : \u211d) + t * Complex.I) = 4 + t ^ 2 := by\n simp [Complex.normSq_apply]\n ring\n have hnormA : \u2016((3 : \u211d) + t * Complex.I)\u2016 = Real.sqrt (9 + t ^ 2) := by\n have h1 : \u2016((3 : \u211d) + t * Complex.I)\u2016 =\n Real.sqrt (Complex.normSq ((3 : \u211d) + t * Complex.I)) := by\n rw [Complex.normSq_eq_norm_sq, Real.sqrt_sq (norm_nonneg _)]\n rw [h1, hns3]\n have hnormB : \u2016((2 : \u211d) + t * Complex.I)\u2016 = Real.sqrt (4 + t ^ 2) := by\n have h1 : \u2016((2 : \u211d) + t * Complex.I)\u2016 =\n Real.sqrt (Complex.normSq ((2 : \u211d) + t * Complex.I)) := by\n rw [Complex.normSq_eq_norm_sq, Real.sqrt_sq (norm_nonneg _)]\n rw [h1, hns2]\n have hnorm : \u2016((3 : \u211d) + t * Complex.I) / ((2 : \u211d) + t * Complex.I)\u2016 =\n Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2)) := by\n rw [norm_div, hnormA, hnormB,\n \u2190 Real.sqrt_div (by positivity : (0 : \u211d) \u2264 9 + t ^ 2)]\n have hs1 : ((3 : \u211d) + t * Complex.I) - 1 = (2 : \u211d) + t * Complex.I := by\n push_cast\n ring\n unfold kFun\n rw [hs1]\n calc \u2016(((3 : \u211d) + t * Complex.I) / ((2 : \u211d) + t * Complex.I)) ^ n - 1\u2016\n \u2264 \u2016(((3 : \u211d) + t * Complex.I) / ((2 : \u211d) + t * Complex.I)) ^ n\u2016 + \u2016(1 : \u2102)\u2016 :=\n norm_sub_le _ _\n _ = (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n + 1 := by\n rw [norm_pow, hnorm, norm_one]\n _ \u2264 Real.exp (5 / 2) + 1 := by\n have h4 : (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n \u2264 Real.exp (5 / 2) := by\n have hx0 : (0 : \u211d) \u2264 Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2)) :=\n Real.sqrt_nonneg _\n have hx : (0 : \u211d) \u2264 (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n :=\n pow_nonneg hx0 n\n have h5 : ((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2 \u2264 Real.exp 5 := by\n have e : ((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2 =\n ((9 + t ^ 2) / (4 + t ^ 2)) ^ n := by\n rw [\u2190 pow_mul, mul_comm n 2, pow_mul, Real.sq_sqrt hQpos]\n rw [e]\n exact hQn\n have h8 : (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n =\n Real.sqrt (((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2) := by\n rw [Real.sqrt_sq hx]\n rw [h8]\n calc Real.sqrt (((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2)\n \u2264 Real.sqrt (Real.exp 5) := Real.sqrt_le_sqrt h5\n _ = Real.exp (5 / 2) := (Real.exp_half 5).symm\n linarith [h4]\n\n/-- The Dirichlet majorant at s = 3 (summable since 3 > 1). -/\ndef dirichletC3 : \u211d :=\n \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016\n\n/-- Termwise 3-vs-2 comparison for the \u039b L-series: \u039b(n)/n\u00b3 \u2264 (1/2)\u00b7\u039b(n)/n\u00b2. -/\nlemma norm_term_vonMangoldt_three_le_half (n : \u2115) :\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016 \u2264\n (1 / 2) * \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016 := by\n have h3re : ((3 : \u2102).re) = 3 := rfl\n have h2re : ((2 : \u2102).re) = 2 := rfl\n simp only [LSeries.norm_term_eq, h3re, h2re]\n split\n \u00b7 norm_num\n \u00b7 next hn =>\n rcases eq_or_ne n 1 with rfl | hn1\n \u00b7 simp [ArithmeticFunction.vonMangoldt_apply_one]\n \u00b7 have hn2 : 2 \u2264 n := by omega\n have h\u039b : \u2016(ArithmeticFunction.vonMangoldt n : \u2102)\u2016 =\n ArithmeticFunction.vonMangoldt n :=\n Complex.norm_of_nonneg ArithmeticFunction.vonMangoldt_nonneg\n rw [h\u039b]\n have e : ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (3 : \u211d) =\n (1 / (n : \u211d)) * (ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d)) := by\n rw [show (3 : \u211d) = 2 + 1 by norm_num,\n Real.rpow_add (by positivity : (0 : \u211d) < (n : \u211d)), Real.rpow_one]\n field_simp [show (n : \u211d) \u2260 0 by positivity]\n have h12 : (1 : \u211d) / (n : \u211d) \u2264 1 / 2 :=\n one_div_le_one_div_of_le (by norm_num : (0 : \u211d) < 2)\n (by exact_mod_cast hn2 : (2 : \u211d) \u2264 (n : \u211d))\n have hpos : (0 : \u211d) \u2264 ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d) :=\n div_nonneg ArithmeticFunction.vonMangoldt_nonneg\n (Real.rpow_nonneg (by positivity : (0 : \u211d) \u2264 (n : \u211d)) _)\n calc ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (3 : \u211d)\n = (1 / (n : \u211d)) * (ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d)) := e\n _ \u2264 (1 / 2) * (ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d)) :=\n mul_le_mul_of_nonneg_right h12 hpos\n\n/-- dirichletC3 \u2264 1/2 (the termwise comparison + dirichletC2 \u2264 1) \u2014\nthe certified C_zeta3 = 0.5 of D-074 \u00a74. (A tighter 0.17 build is\npossible via a 6-prime partial sum + antitone-integral tail; the fold\nis indifferent to the constant, so the cheap certified 0.5 is charged.) -/\nlemma dirichletC3_le_half : dirichletC3 \u2264 1 / 2 := by\n have hsum3 : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n have hsum2 : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n calc dirichletC3\n = \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016 := rfl\n _ \u2264 \u2211' n : \u2115, (1 / 2) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016 :=\n Summable.tsum_le_tsum (fun n => norm_term_vonMangoldt_three_le_half n) hsum3\n (hsum2.mul_left (1 / 2))\n _ = (1 / 2) * dirichletC2 := by rw [dirichletC2, tsum_mul_left]\n _ \u2264 1 / 2 := by linarith [dirichletC2_le_one]\n\n/-- \u2016\u03b6\u2032/\u03b6 s\u2016 \u2264 dirichletC3 for re s \u2265 3 (the s = 3 clone of the\nDirichlet-series majorant). -/\nlemma norm_logDeriv_zeta_le_dirichletC3 {s : \u2102} (hs : 3 \u2264 s.re) :\n \u2016logDeriv riemannZeta s\u2016 \u2264 dirichletC3 := by\n have hs1 : (1 : \u211d) < s.re := by linarith\n have h1 : -logDeriv riemannZeta s\n = LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s := by\n have h2 := ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div hs1\n rw [logDeriv, Pi.div_apply, h2]\n ring\n have hsum3 : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n have hsumS : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt hs1\n have hle : \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s\u2016\n \u2264 dirichletC3 := by\n show \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016 \u2264 _\n calc \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016\n \u2264 \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016 :=\n norm_tsum_le_tsum_norm hsumS\n _ \u2264 \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016 :=\n Summable.tsum_le_tsum (fun n => LSeries.norm_term_le_of_re_le_re _ (by\n simp\n exact hs) n) hsumS hsum3\n _ = dirichletC3 := rfl\n calc \u2016logDeriv riemannZeta s\u2016\n = \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s\u2016 := by\n rw [\u2190 h1, norm_neg]\n _ \u2264 dirichletC3 := hle\n\n/-- \u2016\u03b6\u2032/\u03b6 s\u2016 \u2264 1/2 for re s \u2265 3 (the charged C_zeta3 = 0.5). -/\nlemma norm_logDeriv_zeta_le_half {s : \u2102} (hs : 3 \u2264 s.re) :\n \u2016logDeriv riemannZeta s\u2016 \u2264 1 / 2 :=\n (norm_logDeriv_zeta_le_dirichletC3 hs).trans dirichletC3_le_half\n\n/-- **The left-lift bound** (D-074 \u00a73.3 V_left): for 0 < T \u2264 U and\nn \u2264 T\u00b2, \u2016VIntegral (Jlag n) 3 T U\u2016 \u2264 (U \u2212 T)\u00b7(e^{5/2}+1)/2. The kernel\n\u2264 e^{5/2}+1 (the 1+5/(4+t\u00b2) computation), the zeta factor \u2264 1/2\n(certified, no perimeter). -/\ntheorem vIntegral_Jlag_three_norm_le {n : \u2115} {T U : \u211d}\n (hT0 : (0 : \u211d) < T) (hTU : T \u2264 U) (hTn : (n : \u211d) \u2264 T ^ 2) :\n \u2016VIntegral (Jlag n) 3 T U\u2016 \u2264 (U - T) * (Real.exp (5 / 2) + 1) / 2 := by\n have hnorm : \u2016VIntegral (Jlag n) 3 T U\u2016 =\n \u2016\u222b t in T..U, Jlag n ((3 : \u211d) + t * Complex.I)\u2016 := by\n unfold VIntegral\n rw [norm_smul, Complex.norm_I, one_mul]\n rw [hnorm]\n have hbound : \u2200 t \u2208 Set.Ioc T U,\n \u2016Jlag n ((3 : \u211d) + t * Complex.I)\u2016 \u2264 (Real.exp (5 / 2) + 1) * (1 / 2) := by\n intro t ht\n have htT : T < t := ht.1\n have ht2 : (n : \u211d) \u2264 t ^ 2 := by nlinarith [hTn, hT0]\n have hre : (3 : \u211d) \u2264 ((3 : \u211d) + t * Complex.I).re := by simp\n have h1 := kFun_three_add_it_norm_le_exp_five_halves_add_one ht2\n have h2 := norm_logDeriv_zeta_le_half (s := (3 : \u211d) + t * Complex.I) hre\n have hnorm3 : \u2016Jlag n ((3 : \u211d) + t * Complex.I)\u2016 =\n \u2016kFun n ((3 : \u211d) + t * Complex.I)\u2016 *\n \u2016logDeriv riemannZeta ((3 : \u211d) + t * Complex.I)\u2016 := by\n show \u2016kFun n ((3 : \u211d) + t * Complex.I) *\n (-logDeriv riemannZeta ((3 : \u211d) + t * Complex.I))\u2016 = _\n rw [norm_mul, norm_neg]\n rw [hnorm3]\n exact mul_le_mul h1 h2 (norm_nonneg _) (by positivity)\n have hint : \u2016\u222b t in T..U, Jlag n ((3 : \u211d) + t * Complex.I)\u2016 \u2264\n ((Real.exp (5 / 2) + 1) * (1 / 2)) * |U - T| := by\n apply intervalIntegral.norm_integral_le_of_norm_le_const\n intro t ht\n rw [Set.uIoc_of_le hTU] at ht\n exact hbound t ht\n have habs : |U - T| = U - T := abs_of_nonneg (by linarith)\n rw [habs] at hint\n calc \u2016\u222b t in T..U, Jlag n ((3 : \u211d) + t * Complex.I)\u2016\n \u2264 ((Real.exp (5 / 2) + 1) * (1 / 2)) * (U - T) := hint\n _ = (U - T) * (Real.exp (5 / 2) + 1) / 2 := by ring\n\n/-! ## 4. The high-horizontal stepping (DL-3, D-074 \u00a77 + D-073 \u00a74) -/\n\n/-- Termwise \u03c3-vs-2 comparison for the \u039b L-series (the (6.10) halving):\n\u039b(m)/m^\u03c3 \u2264 2^{\u2212(\u03c3\u22122)}\u00b7\u039b(m)/m\u00b2 for \u03c3 \u2265 2. -/\nlemma norm_term_vonMangoldt_le_two_pow {\u03c3 t : \u211d} (h\u03c3 : 2 \u2264 \u03c3) (m : \u2115) :\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) m\u2016 \u2264\n (2 : \u211d) ^ (-(\u03c3 - 2)) * \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016 := by\n have h\u03c3re : ((\u03c3 : \u211d) + t * Complex.I).re = \u03c3 := by simp\n have h2re : ((2 : \u2102).re) = 2 := rfl\n simp only [LSeries.norm_term_eq, h\u03c3re, h2re]\n split\n \u00b7 norm_num\n \u00b7 next hm =>\n have h\u039b : \u2016(ArithmeticFunction.vonMangoldt m : \u2102)\u2016 =\n ArithmeticFunction.vonMangoldt m :=\n Complex.norm_of_nonneg ArithmeticFunction.vonMangoldt_nonneg\n rw [h\u039b]\n rcases eq_or_ne m 1 with rfl | hm1\n \u00b7 simp [ArithmeticFunction.vonMangoldt_apply_one]\n \u00b7 have hm2 : 2 \u2264 m := by omega\n have e : ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ \u03c3 =\n (1 / (m : \u211d) ^ (\u03c3 - 2)) * (ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 : \u211d)) := by\n rw [show (\u03c3 : \u211d) = (\u03c3 - 2) + 2 by ring,\n Real.rpow_add (by positivity : (0 : \u211d) < (m : \u211d))]\n field_simp [show (m : \u211d) \u2260 0 by positivity]\n ring_nf\n rw [e]\n have hle : (1 : \u211d) / (m : \u211d) ^ (\u03c3 - 2) \u2264 (2 : \u211d) ^ (-(\u03c3 - 2)) := by\n rw [Real.rpow_neg (by norm_num : (0 : \u211d) \u2264 2), \u2190 one_div]\n apply one_div_le_one_div_of_le (Real.rpow_pos_of_pos (by norm_num : (0 : \u211d) < 2) _)\n exact Real.rpow_le_rpow (by norm_num : (0 : \u211d) \u2264 2)\n (by exact_mod_cast hm2 : (2 : \u211d) \u2264 (m : \u211d)) (by linarith : (0 : \u211d) \u2264 \u03c3 - 2)\n have hpos : (0 : \u211d) \u2264 ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 : \u211d) :=\n div_nonneg ArithmeticFunction.vonMangoldt_nonneg\n (Real.rpow_nonneg (by positivity : (0 : \u211d) \u2264 (m : \u211d)) _)\n exact mul_le_mul_of_nonneg_right hle hpos\n\n/-- \u2016\u03b6\u2032/\u03b6(\u03c3+it)\u2016 \u2264 2^{\u2212(\u03c3\u22122)}\u00b7dirichletC2 for \u03c3 \u2265 2 (the (6.10)\nhalving in LSeries form; dirichletC2 \u2264 1 makes the tail free). -/\nlemma norm_logDeriv_zeta_le_two_pow {\u03c3 t : \u211d} (h\u03c3 : 2 \u2264 \u03c3) :\n \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264\n (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := by\n have hs1 : (1 : \u211d) < ((\u03c3 : \u211d) + t * Complex.I).re := by simp; linarith\n have h1 : -logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\n = LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) ((\u03c3 : \u211d) + t * Complex.I) := 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 : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n have hsumS : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt hs1\n have hle : \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264 (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := by\n show \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) n\u2016 \u2264 _\n calc \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) n\u2016\n \u2264 \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) n\u2016 :=\n norm_tsum_le_tsum_norm hsumS\n _ \u2264 \u2211' n : \u2115, (2 : \u211d) ^ (-(\u03c3 - 2)) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016 :=\n Summable.tsum_le_tsum (fun n => norm_term_vonMangoldt_le_two_pow h\u03c3 n) hsumS\n (hsum2.mul_left _)\n _ = (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := by rw [dirichletC2, tsum_mul_left]\n calc \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\u2016\n = \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I)\u2016 := by\n rw [\u2190 h1, norm_neg]\n _ \u2264 (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := hle\n\n/-- **Kernel envelope** (D-074 \u00a77, the single-exponential form that\nreplaces the unit-step geometric series): \u2016(1+1/((\u03c3\u22121)+iU))\u207f\u2016 \u2264\ne^{(4/9)\u03c3} for \u03c3 \u2265 3 and n \u2264 (4/9)U\u00b2. From the (6.11) core\n\u20161+1/s\u2016 \u2264 1+(re s+1)/normSq s \u2264 1+\u03c3/U\u00b2. -/\ntheorem one_add_div_pow_norm_le_exp_four_ninths {n : \u2115} {\u03c3 U : \u211d}\n (h\u03c3 : 3 \u2264 \u03c3) (hU : 0 < U) (hnU : (n : \u211d) \u2264 (4 / 9) * U ^ 2) :\n \u2016(1 + 1/(((\u03c3 : \u211d) - 1) + U * Complex.I))\u2016 ^ n \u2264 Real.exp ((4 / 9) * \u03c3) := by\n have hs0 : (((\u03c3 : \u211d) - 1) + U * Complex.I) \u2260 0 := by\n intro h\n have him : (((\u03c3 : \u211d) - 1) + U * Complex.I).im = 0 := by rw [h]; simp\n simp at him\n linarith\n have hle := norm_one_add_one_div_le hs0\n have hre : (((\u03c3 : \u211d) - 1) + U * Complex.I).re + 1 = \u03c3 := by\n simp [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.I_im,\n Complex.sub_re, Complex.one_re]\n have hns : Complex.normSq (((\u03c3 : \u211d) - 1) + U * Complex.I) = (\u03c3 - 1) ^ 2 + U ^ 2 := by\n simp [Complex.normSq_apply]\n ring\n have hstep : ((((\u03c3 : \u211d) - 1) + U * Complex.I).re + 1) /\n Complex.normSq (((\u03c3 : \u211d) - 1) + U * Complex.I) \u2264 \u03c3 / U ^ 2 := by\n rw [hre, hns]\n have hU2 : (0 : \u211d) < U ^ 2 := by positivity\n have h1 : (0 : \u211d) \u2264 (\u03c3 - 1) ^ 2 := sq_nonneg _\n exact (div_le_div_iff_of_pos_left (by linarith : (0 : \u211d) < \u03c3)\n (by positivity) hU2).mpr (by linarith [h1])\n have hstep1 : 1 + ((((\u03c3 : \u211d) - 1) + U * Complex.I).re + 1) /\n Complex.normSq (((\u03c3 : \u211d) - 1) + U * Complex.I) \u2264 1 + \u03c3 / U ^ 2 := by\n linarith [hstep]\n have hpow1 : \u2016(1 + 1/(((\u03c3 : \u211d) - 1) + U * Complex.I))\u2016 ^ n \u2264 (1 + \u03c3 / U ^ 2) ^ n :=\n pow_le_pow_left\u2080 (norm_nonneg _) (hle.trans hstep1) n\n have hpow2 : (1 + \u03c3 / U ^ 2) ^ n \u2264 Real.exp ((n : \u211d) * (\u03c3 / U ^ 2)) := by\n calc (1 + \u03c3 / U ^ 2) ^ n \u2264 (Real.exp (\u03c3 / U ^ 2)) ^ n := by\n have h := Real.add_one_le_exp (\u03c3 / U ^ 2)\n exact pow_le_pow_left\u2080 (by positivity) (by linarith) n\n _ = Real.exp ((n : \u211d) * (\u03c3 / U ^ 2)) := (Real.exp_nat_mul (\u03c3 / U ^ 2) n).symm\n have hexp : (n : \u211d) * (\u03c3 / U ^ 2) \u2264 (4 / 9) * \u03c3 := by\n have hU2 : (0 : \u211d) < U ^ 2 := by positivity\n have h\u03c30 : (0 : \u211d) < \u03c3 := by linarith\n have h1 : (n : \u211d) / U ^ 2 \u2264 4 / 9 := by\n have h2 := div_le_div_of_nonneg_right hnU hU2.le\n rwa [mul_div_cancel_right\u2080 _ hU2.ne'] at h2\n rw [show (n : \u211d) * (\u03c3 / U ^ 2) = ((n : \u211d) / U ^ 2) * \u03c3 by ring]\n exact mul_le_mul_of_nonneg_right h1 h\u03c30.le\n calc \u2016(1 + 1/(((\u03c3 : \u211d) - 1) + U * Complex.I))\u2016 ^ n\n \u2264 Real.exp ((n : \u211d) * (\u03c3 / U ^ 2)) := hpow1.trans hpow2\n _ \u2264 Real.exp ((4 / 9) * \u03c3) := Real.exp_le_exp.mpr hexp\n\n/-! ## 5. The product envelope and the integral assembly (DL-3c/d) -/\n\n/-- **The product envelope** (D-074 \u00a77): for \u03c3 \u2265 3, 0 < t, n \u2264 (4/9)t\u00b2,\n\u2016Jlag n (\u03c3+it)\u2016 \u2264 (4\u00b7e^{\u2212c\u03c3} + 4\u00b72^{\u2212\u03c3})\u00b7dirichletC2, c = log 2 \u2212 4/9.\nThe kernel envelope (e^{(4/9)\u03c3}) times the zeta halving (2^{\u2212(\u03c3\u22122)})\ncollapses to the single exponential 4\u00b7e^{\u2212c\u03c3}: no geometric series. -/\ntheorem Jlag_dogleg_norm_le_envelope {n : \u2115} {\u03c3 t : \u211d}\n (h\u03c3 : 3 \u2264 \u03c3) (ht : 0 < t) (hnU : (n : \u211d) \u2264 (4 / 9) * t ^ 2) :\n \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) := by\n have hw1 : ((\u03c3 : \u211d) + t * Complex.I) - 1 = ((\u03c3 : \u211d) - 1) + t * Complex.I := by\n ring\n have hs1ne : (((\u03c3 : \u211d) - 1) + t * Complex.I) \u2260 0 := by\n intro h\n have him : (((\u03c3 : \u211d) - 1) + t * Complex.I).im = 0 := by rw [h]; simp\n simp at him\n linarith\n have hwid : ((\u03c3 : \u211d) + t * Complex.I) / (((\u03c3 : \u211d) + t * Complex.I) - 1) =\n 1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I) := by\n rw [hw1]\n field_simp [hs1ne]\n ring\n have hker := one_add_div_pow_norm_le_exp_four_ninths h\u03c3 ht hnU\n have hzeta := norm_logDeriv_zeta_le_two_pow (show (2 : \u211d) \u2264 \u03c3 by linarith) (t := t)\n have hk : \u2016kFun n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264 Real.exp ((4 / 9) * \u03c3) + 1 := by\n unfold kFun\n rw [hwid]\n calc \u2016(1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I)) ^ n - 1\u2016\n \u2264 \u2016(1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I)) ^ n\u2016 + \u2016(1 : \u2102)\u2016 := norm_sub_le _ _\n _ = \u2016(1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I))\u2016 ^ n + 1 := by rw [norm_pow, norm_one]\n _ \u2264 Real.exp ((4 / 9) * \u03c3) + 1 := by linarith [hker]\n have hprod : \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264\n (Real.exp ((4 / 9) * \u03c3) + 1) * ((2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2) := by\n have hn3 : \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016 =\n \u2016kFun n ((\u03c3 : \u211d) + t * Complex.I)\u2016 *\n \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\u2016 := by\n show \u2016kFun n ((\u03c3 : \u211d) + t * Complex.I) *\n (-logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I))\u2016 = _\n rw [norm_mul, norm_neg]\n rw [hn3]\n exact mul_le_mul hk hzeta (norm_nonneg _)\n (add_nonneg (Real.exp_nonneg _) (by norm_num))\n have h2b : (2 : \u211d) ^ (-\u03c3) = Real.exp (-(Real.log 2) * \u03c3) := by\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n congr 1\n ring\n have h22 : (2 : \u211d) ^ (2 : \u211d) = 4 := by\n rw [Real.rpow_two]\n norm_num\n have h2a : (2 : \u211d) ^ (-(\u03c3 - 2)) = 4 * (2 : \u211d) ^ (-\u03c3) := by\n rw [show (-(\u03c3 - 2)) = -\u03c3 + 2 by ring, Real.rpow_add (by norm_num : (0 : \u211d) < 2), h2b, h22]\n ring\n have hexp_alg : Real.exp ((4 / 9) * \u03c3) * Real.exp (-(Real.log 2) * \u03c3) =\n Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) := by\n rw [\u2190 Real.exp_add]\n congr 1\n ring\n have halg : (Real.exp ((4 / 9) * \u03c3) + 1) * ((2 : \u211d) ^ (-(\u03c3 - 2))) =\n 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3) := by\n rw [h2a, h2b]\n calc (Real.exp ((4 / 9) * \u03c3) + 1) * (4 * Real.exp (-(Real.log 2) * \u03c3))\n = 4 * (Real.exp ((4 / 9) * \u03c3) * Real.exp (-(Real.log 2) * \u03c3)) +\n 4 * Real.exp (-(Real.log 2) * \u03c3) := by ring\n _ = 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * Real.exp (-(Real.log 2) * \u03c3) := by\n rw [hexp_alg]\n calc \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016\n \u2264 (Real.exp ((4 / 9) * \u03c3) + 1) * ((2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2) := hprod\n _ = dirichletC2 * ((Real.exp ((4 / 9) * \u03c3) + 1) * (2 : \u211d) ^ (-(\u03c3 - 2))) := by ring\n _ = dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) := by\n rw [halg]\n\n/-- The FTC-bound \u222b\u2083^XR e^{\u2212c\u03c3}d\u03c3 \u2264 e^{\u22123c}/c for c > 0. -/\nlemma integral_exp_neg_le {c XR : \u211d} (hc : 0 < c) (_hXR : 3 \u2264 XR) :\n \u222b \u03c3 in (3 : \u211d)..XR, Real.exp (-c * \u03c3) \u2264 Real.exp (-3 * c) / c := by\n have hF : \u2200 x \u2208 Set.uIcc (3 : \u211d) XR,\n HasDerivAt (fun \u03c3 => -Real.exp (-c * \u03c3) / c) (Real.exp (-c * x)) x := by\n intro x hx\n have h2 : HasDerivAt (fun \u03c3 => Real.exp (-c * \u03c3)) (Real.exp (-c * x) * (-c)) x := by\n have h := ((hasDerivAt_id' x).const_mul (-c)).exp\n rwa [mul_one] at h\n have h3 := h2.const_mul (-1 / c)\n rw [show (-1 / c) * (Real.exp (-c * x) * (-c)) = Real.exp (-c * x) by\n field_simp [hc.ne']] at h3\n have h5 : (fun \u03c3 => (-1 / c) * Real.exp (-c * \u03c3)) =\n (fun \u03c3 => -Real.exp (-c * \u03c3) / c) := by\n funext \u03c3\n field_simp [hc.ne']\n rw [h5] at h3\n exact h3\n have hint : IntervalIntegrable (fun \u03c3 => Real.exp (-c * \u03c3)) MeasureTheory.volume 3 XR := by\n apply ContinuousOn.intervalIntegrable\n exact (Real.continuous_exp.comp (continuous_const.mul continuous_id)).continuousOn\n have e : Real.exp (-c * 3) = Real.exp (-3 * c) := by\n congr 1\n ring\n rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF hint, e]\n have h1 : (0 : \u211d) \u2264 Real.exp (-c * XR) := (Real.exp_pos _).le\n have h2 : (0 : \u211d) < c := hc\n rw [show -Real.exp (-c * XR) / c - -Real.exp (-3 * c) / c =\n (Real.exp (-3 * c) - Real.exp (-c * XR)) / c by\n field_simp [h2.ne']\n ring]\n rw [show Real.exp (-3 * c) / c = (Real.exp (-3 * c) - (0 : \u211d)) / c by ring]\n exact div_le_div_of_nonneg_right (by linarith [h1]) h2.le\n\n/-- The certified exponent-decay constant c = log 2 \u2212 4/9 \u2265 0.2487. -/\nlemma cdog_gt : (0.2487 : \u211d) \u2264 Real.log 2 - 4 / 9 := by\n have h := Real.log_two_gt_d9\n norm_num at h\n linarith\n\n/-- The certified e^{4/3} \u2264 3.8073 (the (1\u2212x/16)\u207b\u00b9\u2076 route + d9 e). -/\nlemma exp_four_thirds_le : Real.exp (4 / 3) \u2264 (3.8073 : \u211d) := by\n have he13 : Real.exp (1 / 3) \u2264 (1.4006 : \u211d) :=\n le_trans (exp_le_inv_sub_sixteen (by norm_num) (by norm_num)) (by norm_num)\n rw [show (4 / 3 : \u211d) = 1 + 1 / 3 by norm_num, Real.exp_add]\n calc Real.exp 1 * Real.exp (1 / 3) \u2264 2.7182818286 * 1.4006 :=\n mul_le_mul Real.exp_one_lt_d9.le he13 (Real.exp_pos _).le (by norm_num)\n _ \u2264 3.8073 := by norm_num\n\n/-- **THE HIGH-HORIZONTAL BOUND** (D-074 \u00a77 capstone; D-073 Cor 4.2's\nR(1.5) in single-exponential-envelope form): for 0 < U, n \u2264 (4/9)U\u00b2\n(i.e. U \u2265 (3/2)\u221an) and 3 \u2264 XR, \u2016HIntegral (Jlag n) 3 XR U\u2016 \u2264 8.38\n(charged; the true value \u2264 1.2 at n = 10\u2075, E-034 V5). Charged with\ndirichletC2 \u2264 1 \u2014 the tight R(1.5) = 5.94 would need a dirichletC2 \u2264\n0.57 build (documented as the tightening path; the fold is\nindifferent: \u0394 \u2248 +2.4 per side). -/\ntheorem hIntegral_Jlag_dogleg_norm_le {n : \u2115} {U XR : \u211d}\n (hU : 0 < U) (hnU : (n : \u211d) \u2264 (4 / 9) * U ^ 2) (hXR : 3 \u2264 XR) :\n \u2016HIntegral (Jlag n) 3 XR U\u2016 \u2264 (8.38 : \u211d) := by\n have hc : (0 : \u211d) < Real.log 2 - 4 / 9 := by\n have h := cdog_gt\n linarith\n have hpt : \u2200 \u03c3 \u2208 Set.Ioc 3 XR,\n \u2016Jlag n ((\u03c3 : \u211d) + U * Complex.I)\u2016 \u2264\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) :=\n fun \u03c3 h\u03c3 => Jlag_dogleg_norm_le_envelope (by linarith [h\u03c3.1]) hU hnU\n have hcontA : Continuous (fun \u03c3 : \u211d => Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) :=\n Real.continuous_exp.comp (continuous_const.mul continuous_id)\n have hcontB : Continuous (fun \u03c3 : \u211d => (2 : \u211d) ^ (-\u03c3)) := by\n have h1 : Continuous (fun \u03c3 : \u211d => Real.exp (-(Real.log 2) * \u03c3)) :=\n Real.continuous_exp.comp (continuous_const.mul continuous_id)\n have h2 : (fun \u03c3 : \u211d => (2 : \u211d) ^ (-\u03c3)) =\n (fun \u03c3 : \u211d => Real.exp (-(Real.log 2) * \u03c3)) := by\n funext \u03c3\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n congr 1\n ring\n rw [h2]\n exact h1\n have hIntA : IntervalIntegrable (fun \u03c3 : \u211d => 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3))\n MeasureTheory.volume 3 XR := by\n apply ContinuousOn.intervalIntegrable\n exact (hcontA.const_mul 4).continuousOn\n have hIntB : IntervalIntegrable (fun \u03c3 : \u211d => 4 * (2 : \u211d) ^ (-\u03c3)) MeasureTheory.volume 3 XR := by\n apply ContinuousOn.intervalIntegrable\n exact (hcontB.const_mul 4).continuousOn\n have hIntG : IntervalIntegrable (fun \u03c3 : \u211d =>\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)))\n MeasureTheory.volume 3 XR := by\n have h := (hIntA.add hIntB).const_mul dirichletC2\n convert h using 1\n -- step 1: \u2016H\u2016 \u2264 \u222b envelope\n have h1 : \u2016HIntegral (Jlag n) 3 XR U\u2016 \u2264\n \u222b \u03c3 in (3 : \u211d)..XR,\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) := by\n unfold HIntegral\n apply intervalIntegral.norm_integral_le_of_norm_le hXR _ hIntG\n exact Filter.Eventually.of_forall (fun \u03c3 h\u03c3 => hpt \u03c3 h\u03c3)\n -- step 2: \u222b envelope = dirichletC2\u00b7(4\u222be^{\u2212c\u03c3} + 4\u222b2^{\u2212\u03c3})\n have h2 : \u222b \u03c3 in (3 : \u211d)..XR,\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3))\n = dirichletC2 * ((\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3))) := by\n rw [intervalIntegral.integral_const_mul, intervalIntegral.integral_add hIntA hIntB]\n -- step 3: the two integral bounds\n have h3a : \u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) \u2264\n Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) := by\n rw [intervalIntegral.integral_const_mul]\n have h := integral_exp_neg_le hc hXR\n have he3c : Real.exp (-3 * (Real.log 2 - 4 / 9)) = Real.exp (4 / 3) / 8 := by\n have h3l2 : Real.exp (-(3 * Real.log 2)) = 1 / 8 := by\n have e1 : -(3 * Real.log 2) = Real.log (1 / 8) := by\n rw [show ((1 : \u211d) / 8) = (2 ^ 3 : \u211d)\u207b\u00b9 by norm_num, Real.log_inv, Real.log_pow,\n Nat.cast_ofNat]\n rw [e1, Real.exp_log (by norm_num : (0 : \u211d) < 1 / 8)]\n calc Real.exp (-3 * (Real.log 2 - 4 / 9))\n = Real.exp ((4 / 3) + -(3 * Real.log 2)) := by congr 1; ring\n _ = Real.exp (4 / 3) * Real.exp (-(3 * Real.log 2)) := by rw [Real.exp_add]\n _ = Real.exp (4 / 3) / 8 := by rw [h3l2]; ring\n rw [he3c] at h\n calc 4 * \u222b \u03c3 in (3 : \u211d)..XR, Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)\n \u2264 4 * ((Real.exp (4 / 3) / 8) / (Real.log 2 - 4 / 9)) :=\n mul_le_mul_of_nonneg_left h (by norm_num)\n _ = Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) := by\n field_simp [hc.ne']\n ring\n have h3b : \u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3) \u2264 1 / (2 * Real.log 2) := by\n have h2e : (fun \u03c3 : \u211d => 4 * (2 : \u211d) ^ (-\u03c3)) =\n (fun \u03c3 : \u211d => 4 * Real.exp (-(Real.log 2) * \u03c3)) := by\n funext \u03c3\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n congr 1\n ring\n rw [h2e, intervalIntegral.integral_const_mul]\n have h := integral_exp_neg_le (Real.log_pos (by norm_num : (1 : \u211d) < 2)) hXR\n have h3l2 : Real.exp (-3 * Real.log 2) = 1 / 8 := by\n have e1 : (-3 * Real.log 2) = Real.log (1 / 8) := by\n rw [show ((1 : \u211d) / 8) = (2 ^ 3 : \u211d)\u207b\u00b9 by norm_num, Real.log_inv, Real.log_pow,\n Nat.cast_ofNat]\n ring\n rw [e1, Real.exp_log (by norm_num : (0 : \u211d) < 1 / 8)]\n rw [h3l2] at h\n calc 4 * \u222b \u03c3 in (3 : \u211d)..XR, Real.exp (-(Real.log 2) * \u03c3)\n \u2264 4 * ((1 / 8) / Real.log 2) := by\n apply mul_le_mul_of_nonneg_left h (by norm_num : (0 : \u211d) \u2264 4)\n _ = 1 / (2 * Real.log 2) := by\n field_simp [(Real.log_pos (by norm_num : (1 : \u211d) < 2)).ne']\n norm_num\n -- step 4: the certified assembly\n have hlog2 : (0.6931471803 : \u211d) \u2264 Real.log 2 := Real.log_two_gt_d9.le\n have h48 : Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) \u2264 (7.6545 : \u211d) := by\n have hcge := cdog_gt\n have h1 : Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) \u2264 3.8073 / (2 * 0.2487) :=\n div_le_div\u2080 (by norm_num : (0 : \u211d) \u2264 3.8073) exp_four_thirds_le\n (by norm_num : (0 : \u211d) < 2 * 0.2487) (by linarith [hcge])\n calc Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9))\n \u2264 3.8073 / (2 * 0.2487) := h1\n _ \u2264 7.6545 := by norm_num\n have h72 : 1 / (2 * Real.log 2) \u2264 (0.7214 : \u211d) := by\n have h1 : 1 / (2 * Real.log 2) \u2264 1 / (2 * 0.6931471803) :=\n one_div_le_one_div_of_le (by positivity) (by linarith)\n calc 1 / (2 * Real.log 2) \u2264 1 / (2 * 0.6931471803) := h1\n _ \u2264 0.7214 := by norm_num\n have hC2 : dirichletC2 \u2264 1 := dirichletC2_le_one\n have hsum : dirichletC2 * ((\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3))) \u2264 (8.38 : \u211d) := by\n have hA0 : (0 : \u211d) \u2264 Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) := by positivity\n have hB0 : (0 : \u211d) \u2264 1 / (2 * Real.log 2) := by positivity\n have hIntAB : (0 : \u211d) \u2264 (\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3)) := by\n apply add_nonneg\n \u00b7 apply intervalIntegral.integral_nonneg hXR\n intro \u03c3 h\u03c3\n positivity\n \u00b7 apply intervalIntegral.integral_nonneg hXR\n intro \u03c3 h\u03c3\n positivity\n calc dirichletC2 * ((\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3)))\n \u2264 1 * (Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) + 1 / (2 * Real.log 2)) := by\n apply mul_le_mul hC2 (add_le_add h3a h3b) hIntAB (by norm_num)\n _ = Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) + 1 / (2 * Real.log 2) := by ring\n _ \u2264 7.6545 + 0.7214 := add_le_add h48 h72\n _ \u2264 8.38 := by norm_num\n calc \u2016HIntegral (Jlag n) 3 XR U\u2016\n \u2264 \u222b \u03c3 in (3 : \u211d)..XR,\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) := h1\n _ = dirichletC2 * ((\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3))) := h2\n _ \u2264 8.38 := hsum\n\n/-! ## 6. The far-right vertical drop (DL-4, D-074 \u00a78) -/\n\n/-- **Kernel at the far-right vertical** (D-074 \u00a78): for n \u2265 16 and\nt\u00b2 \u2265 n, \u2016kFun n (2\u221an+it)\u2016 \u2264 e^{\u221an/2}+1. (The ii\u2032-4 lemma\n`kFun_shift_norm_le_exp_sqrt_half_add_one` takes re s \u2265 2\u221an, i.e.\nre w \u2265 2\u221an+1 \u2014 this is the w = 2\u221an variant: |1+1/(w\u22121)| \u2264\n1+(2\u221an)/((2\u221an\u22121)\u00b2+t\u00b2) \u2264 1+1/(2\u221an) since 4\u221an\u22121 \u2264 t\u00b2.) -/\ntheorem kFun_right_vertical_norm_le_exp_sqrt_half_add_one {n : \u2115} {t : \u211d}\n (hn : 16 \u2264 n) (ht : (n : \u211d) \u2264 t ^ 2) :\n \u2016kFun n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n Real.exp (Real.sqrt (n : \u211d) / 2) + 1 := by\n have hnR : (16 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have hsq : (0 : \u211d) < Real.sqrt (n : \u211d) := Real.sqrt_pos.mpr (by linarith)\n have hw1 : (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1 =\n ((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I := by\n push_cast\n ring\n have hs0 : ((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I \u2260 0 := by\n intro h\n have him : (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I).im = 0 := by\n rw [h]; simp\n simp at him\n have ht0 : t = 0 := by\n have h1 : (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I).im = t := by simp\n linarith\n rw [ht0] at ht\n nlinarith [hnR]\n have hle := norm_one_add_one_div_le hs0\n have hre : (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I).re + 1 =\n 2 * Real.sqrt (n : \u211d) := by\n simp [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.I_im,\n Complex.sub_re, Complex.one_re]\n have hns : Complex.normSq (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) =\n (2 * Real.sqrt (n : \u211d) - 1) ^ 2 + t ^ 2 := by\n simp [Complex.normSq_apply]\n ring\n have hkey : (2 * Real.sqrt (n : \u211d)) /\n Complex.normSq (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) \u2264\n 1 / (2 * Real.sqrt (n : \u211d)) := by\n rw [hns]\n have hsqn : Real.sqrt (n : \u211d) ^ 2 = (n : \u211d) := Real.sq_sqrt (by linarith)\n have h1 : 4 * Real.sqrt (n : \u211d) - 1 \u2264 t ^ 2 := by\n have h2 : 4 * Real.sqrt (n : \u211d) \u2264 (n : \u211d) + 1 := by\n have h3 : (4 * Real.sqrt (n : \u211d)) ^ 2 \u2264 ((n : \u211d) + 1) ^ 2 := by\n rw [show (4 * Real.sqrt (n : \u211d)) ^ 2 = 16 * (Real.sqrt (n : \u211d) ^ 2) by ring, hsqn]\n nlinarith [sq_nonneg ((n : \u211d) - 1), hnR]\n nlinarith [hsq]\n nlinarith [ht, h2]\n rw [show (2 * Real.sqrt (n : \u211d) - 1) ^ 2 + t ^ 2 =\n 4 * (Real.sqrt (n : \u211d) ^ 2) - 4 * Real.sqrt (n : \u211d) + 1 + t ^ 2 by ring, hsqn]\n have h4 : (0 : \u211d) < 4 * (n : \u211d) - 4 * Real.sqrt (n : \u211d) + 1 + t ^ 2 := by\n nlinarith [ht, sq_nonneg t, hsq]\n rw [div_le_iff\u2080 h4,\n show (1 / (2 * Real.sqrt (n : \u211d))) * (4 * (n : \u211d) - 4 * Real.sqrt (n : \u211d) + 1 + t ^ 2) =\n (4 * (n : \u211d) - 4 * Real.sqrt (n : \u211d) + 1 + t ^ 2) / (2 * Real.sqrt (n : \u211d)) by\n field_simp [hsq.ne'],\n le_div_iff\u2080 (by linarith [hsq] : (0 : \u211d) < 2 * Real.sqrt (n : \u211d))]\n nlinarith [h1, ht, hsq, hsqn]\n have hstep : 1 + ((((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I).re + 1) /\n Complex.normSq (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) \u2264\n 1 + 1 / (2 * Real.sqrt (n : \u211d)) := by\n rw [hre]\n have h1 : (2 * Real.sqrt (n : \u211d)) /\n Complex.normSq (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) \u2264\n 1 / (2 * Real.sqrt (n : \u211d)) := hkey\n linarith [h1]\n have hpow1 : \u2016(1 + 1/(((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I))\u2016 ^ n \u2264\n (1 + 1 / (2 * Real.sqrt (n : \u211d))) ^ n :=\n pow_le_pow_left\u2080 (norm_nonneg _) (hle.trans hstep) n\n have hpow2 : (1 + 1 / (2 * Real.sqrt (n : \u211d))) ^ n \u2264\n Real.exp (Real.sqrt (n : \u211d) / 2) := by\n calc (1 + 1 / (2 * Real.sqrt (n : \u211d))) ^ n\n \u2264 (Real.exp (1 / (2 * Real.sqrt (n : \u211d)))) ^ n := by\n have h := Real.add_one_le_exp (1 / (2 * Real.sqrt (n : \u211d)))\n exact pow_le_pow_left\u2080 (by positivity) (by linarith) n\n _ = Real.exp ((n : \u211d) * (1 / (2 * Real.sqrt (n : \u211d)))) :=\n (Real.exp_nat_mul _ n).symm\n _ = Real.exp (Real.sqrt (n : \u211d) / 2) := by\n congr 1\n have hsqn : Real.sqrt (n : \u211d) ^ 2 = (n : \u211d) := Real.sq_sqrt (by linarith)\n field_simp [hsq.ne']\n nlinarith [hsqn]\n have hwid : (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) /\n ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1) =\n 1 + 1 / (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) := by\n rw [hw1]\n field_simp [hs0]\n push_cast\n ring\n unfold kFun\n rw [hwid]\n calc \u2016(1 + 1/(((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I)) ^ n - 1\u2016\n \u2264 \u2016(1 + 1/(((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I)) ^ n\u2016 + \u2016(1 : \u2102)\u2016 :=\n norm_sub_le _ _\n _ = \u2016(1 + 1/(((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I))\u2016 ^ n + 1 := by\n rw [norm_pow, norm_one]\n _ \u2264 Real.exp (Real.sqrt (n : \u211d) / 2) + 1 := by linarith [hpow1.trans hpow2]\n\n/-- **The far-right vertical drop** (D-074 \u00a78 V_right): for n \u2265 16,\n0 < T \u2264 U, T\u00b2 \u2265 n, \u2016VIntegral (Jlag n) (2\u221an) T U\u2016 \u2264\n(U\u2212T)\u00b7(e^{\u221an/2}+1)\u00b72^{\u2212(2\u221an\u22122)}\u00b7dirichletC2 \u2014 exponentially tiny. -/\ntheorem vIntegral_Jlag_right_norm_le {n : \u2115} {T U : \u211d}\n (hn : 16 \u2264 n) (hT0 : (0 : \u211d) < T) (hTU : T \u2264 U) (hTn : (n : \u211d) \u2264 T ^ 2) :\n \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T U\u2016 \u2264\n (U - T) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2) := by\n have hnorm : \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T U\u2016 =\n \u2016\u222b t in T..U, Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 := by\n unfold VIntegral\n rw [norm_smul, Complex.norm_I, one_mul]\n rw [hnorm]\n have hbound : \u2200 t \u2208 Set.Ioc T U,\n \u2016Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2) := by\n intro t ht\n have htT : T < t := ht.1\n have ht2 : (n : \u211d) \u2264 t ^ 2 := by nlinarith [hTn, hT0]\n have h1 := kFun_right_vertical_norm_le_exp_sqrt_half_add_one hn ht2\n have h2 := norm_logDeriv_zeta_le_two_pow\n (show (2 : \u211d) \u2264 2 * Real.sqrt (n : \u211d) by\n have hnR : (16 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have hsq : (4 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n have h1 : Real.sqrt 16 \u2264 Real.sqrt (n : \u211d) :=\n Real.sqrt_le_sqrt (by exact_mod_cast hn : (16 : \u211d) \u2264 (n : \u211d))\n have h2 : Real.sqrt 16 = 4 := by\n rw [Real.sqrt_eq_iff_eq_sq (by norm_num) (by norm_num)]; norm_num\n rwa [h2] at h1\n linarith) (t := t)\n have hnorm3 : \u2016Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 =\n \u2016kFun n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 *\n \u2016logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 := by\n show \u2016kFun n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) *\n (-logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I))\u2016 = _\n rw [norm_mul, norm_neg]\n rw [hnorm3]\n exact mul_le_mul h1 h2 (norm_nonneg _)\n (add_nonneg (Real.exp_nonneg _) (by norm_num))\n have hint : \u2016\u222b t in T..U, Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n ((Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2)) * |U - T| := by\n apply intervalIntegral.norm_integral_le_of_norm_le_const\n intro t ht\n rw [Set.uIoc_of_le hTU] at ht\n exact hbound t ht\n have habs : |U - T| = U - T := abs_of_nonneg (by linarith)\n rw [habs] at hint\n calc \u2016\u222b t in T..U, Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016\n \u2264 ((Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2)) * (U - T) := hint\n _ = (U - T) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2) := by ring\n\n/-! ## 7. The total right-horizontal bound (DL-5, D-074 \u00a73.4) -/\n\n/-- **The upper right-tail bound** (D-074 \u00a73.4): for n \u2265 16 and\nT \u2208 [\u221an, \u221an+1] (the T_cut window), U = (3/2)\u00b7T,\n\u2016HIntegral (Jlag n) 3 (2\u221an) T\u2016 \u2264 ((\u221an+1)/2)\u00b7(e^{5/2}+1)/2 + 8.38 +\nIIMaj(n)/2 \u2014 the exact deformation identity +\nV_left/H_high/V_right piece bounds, all in the zero-free region. -/\ntheorem hIntegral_Jlag_low_le_dogLegUpper {n : \u2115} {T : \u211d}\n (hn : 16 \u2264 n) (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T\u2016 \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2 := by\n have hnR : (16 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have hsqrt4 : (4 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n have h1 : Real.sqrt 16 \u2264 Real.sqrt (n : \u211d) :=\n Real.sqrt_le_sqrt (by exact_mod_cast hn : (16 : \u211d) \u2264 (n : \u211d))\n have h2 : Real.sqrt 16 = 4 := by\n rw [Real.sqrt_eq_iff_eq_sq (by norm_num) (by norm_num)]; norm_num\n rwa [h2] at h1\n have hT0 : (0 : \u211d) < T := by linarith\n have hTn2 : (n : \u211d) \u2264 T ^ 2 := by\n have h1 : (Real.sqrt (n : \u211d)) ^ 2 \u2264 T ^ 2 :=\n pow_le_pow_left\u2080 (Real.sqrt_nonneg _) hTn 2\n rwa [Real.sq_sqrt (by linarith [Real.sqrt_nonneg (n : \u211d)])] at h1\n have hU0 : (0 : \u211d) < (3 / 2) * T := by linarith [hT0]\n have hTU : T \u2264 (3 / 2) * T := by linarith [hT0]\n have hnU : (n : \u211d) \u2264 (4 / 9) * ((3 / 2) * T) ^ 2 := by\n have h1 : (4 / 9) * ((3 / 2) * T) ^ 2 = T ^ 2 := by\n field_simp\n ring\n rw [h1]\n exact hTn2\n have hXR : (3 : \u211d) \u2264 2 * Real.sqrt (n : \u211d) := by linarith\n have hid := upper_right_horizontal_eq_dogleg hT0 hTU hXR (n := n)\n have hH := hIntegral_Jlag_dogleg_norm_le hU0 hnU hXR (n := n)\n have hVr := vIntegral_Jlag_right_norm_le hn hT0 hTU hTn2 (n := n)\n have hVl := vIntegral_Jlag_three_norm_le hT0 hTU hTn2 (n := n)\n have hlen : (3 / 2) * T - T \u2264 (Real.sqrt (n : \u211d) + 1) / 2 := by linarith [hTn1, hTn]\n have hlen0 : (0 : \u211d) \u2264 (3 / 2) * T - T := by linarith [hT0]\n -- V_left \u2264 ((\u221an+1)/2)\u00b7(e^{5/2}+1)/2\n have hVl2 : \u2016VIntegral (Jlag n) 3 T ((3 / 2) * T)\u2016 \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 := by\n refine hVl.trans ?_\n exact div_le_div_of_nonneg_right\n (mul_le_mul hlen (le_refl _) (by positivity) (by positivity)) (by norm_num)\n -- V_right \u2264 IIMaj/2\n have hVr2 : \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T)\u2016 \u2264\n IIMaj (n : \u211d) / 2 := by\n have hC2 : dirichletC2 \u2264 1 := dirichletC2_le_one\n have hK : (0 : \u211d) < (Real.sqrt (n : \u211d) + 1) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))) := by positivity\n have eA : ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * (1 : \u211d)) =\n (1 / 2) * ((Real.sqrt (n : \u211d) + 1) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)))) := by ring\n have eB : IIMaj (n : \u211d) / 2 =\n (0.57 : \u211d) * ((Real.sqrt (n : \u211d) + 1) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)))) := by\n unfold IIMaj\n ring\n calc \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T)\u2016\n \u2264 ((3 / 2) * T - T) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2) := hVr\n _ \u2264 ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * (1 : \u211d)) := by\n have h1 : ((3 / 2) * T - T) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) :=\n mul_le_mul hlen (le_refl _) (by positivity) (by positivity)\n have h2 : (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 \u2264\n (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * (1 : \u211d) :=\n mul_le_mul_of_nonneg_left hC2 (Real.rpow_nonneg (by norm_num) _)\n exact mul_le_mul h1 h2\n (mul_nonneg (Real.rpow_nonneg (by norm_num) _) dirichletC2_nonneg)\n (by positivity)\n _ \u2264 IIMaj (n : \u211d) / 2 := by\n rw [eA, eB]\n apply mul_le_mul_of_nonneg_right _ hK.le\n norm_num\n -- assemble via the exact identity\n rw [hid]\n calc \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) ((3 / 2) * T) -\n VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T) +\n VIntegral (Jlag n) 3 T ((3 / 2) * T)\u2016\n \u2264 \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) ((3 / 2) * T)\u2016 +\n \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T)\u2016 +\n \u2016VIntegral (Jlag n) 3 T ((3 / 2) * T)\u2016 := by\n have h1 := norm_add_le\n (HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) ((3 / 2) * T) -\n VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T))\n (VIntegral (Jlag n) 3 T ((3 / 2) * T))\n have h2 := norm_sub_le\n (HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) ((3 / 2) * T))\n (VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T))\n linarith [h1, h2]\n _ \u2264 (8.38 : \u211d) + IIMaj (n : \u211d) / 2 +\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 :=\n add_le_add (add_le_add hH hVr2) hVl2\n _ = ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2 := by ring\n\n/-- **The lower right-tail bound by conjugation** (D-074 \u00a73.4): the\nsame bound holds at im = \u2212T (the integrand is conjugation-symmetric,\nJlag_conj; the interval integral maps conj through). -/\ntheorem hIntegral_Jlag_low_neg_norm_le_dogLegUpper {n : \u2115} {T : \u211d}\n (hn : 16 \u2264 n) (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T)\u2016 \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2 := by\n have hT0 : (0 : \u211d) < T := by\n have hnR : (16 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have hsqrt4 : (4 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n have h1 : Real.sqrt 16 \u2264 Real.sqrt (n : \u211d) :=\n Real.sqrt_le_sqrt (by exact_mod_cast hn : (16 : \u211d) \u2264 (n : \u211d))\n have h2 : Real.sqrt 16 = 4 := by\n rw [Real.sqrt_eq_iff_eq_sq (by norm_num) (by norm_num)]; norm_num\n rwa [h2] at h1\n linarith\n have hconj : HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T) =\n (starRingEnd \u2102) (HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T) := by\n unfold HIntegral\n rw [\u2190 intervalIntegral.intervalIntegral_conj]\n apply intervalIntegral.integral_congr\n intro \u03c3 h\u03c3\n show Jlag n ((\u03c3 : \u211d) + (((-T : \u211d)) : \u2102) * Complex.I) =\n (starRingEnd \u2102) (Jlag n ((\u03c3 : \u211d) + T * Complex.I))\n have hid2 : ((\u03c3 : \u211d) + (((-T : \u211d)) : \u2102) * Complex.I) =\n (starRingEnd \u2102) ((\u03c3 : \u211d) + T * Complex.I) := by\n simp [Complex.ext_iff]\n rw [hid2]\n exact Jlag_conj n (by\n intro h\n have him : (((\u03c3 : \u211d) + T * Complex.I)).im = T := by simp\n rw [h] at him\n simp at him\n linarith [hT0])\n rw [hconj, Complex.norm_conj]\n exact hIntegral_Jlag_low_le_dogLegUpper hn hTn hTn1\n\n/-- **THE TWO-SIDE RIGHT-HORIZONTAL BOUND** (the DL capstone for\nii\u2032-6-c): both horizontal right tails bounded by the dog-leg charge. -/\ntheorem dogLegUpper_two_sides {n : \u2115} {T : \u211d}\n (hn : 16 \u2264 n) (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T\u2016 +\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T)\u2016 \u2264\n 2 * (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2) := by\n rw [show (2 : \u211d) * (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 +\n (8.38 : \u211d) + IIMaj (n : \u211d) / 2) =\n (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2) +\n (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2) by ring]\n exact add_le_add (hIntegral_Jlag_low_le_dogLegUpper hn hTn hTn1)\n (hIntegral_Jlag_low_neg_norm_le_dogLegUpper hn hTn hTn1)\n\n/-! ## 8. The direct Dirichlet bound at re = 2\u221an (DL-6) \u2014 the I_I /\nV_right zeta factor WITHOUT certifying dirichletC2 \u2264 0.57.\n\nThe I_I charge in IIMaj is 0.57\u00b72^{\u2212(2\u221an\u22122)} = 2.28\u00b72^{\u22122\u221an} (the\n(6.10)-route constant |\u03b6\u2032/\u03b6(2)|). Proving dirichletC2 \u2264 0.57 directly\nneeds a ~100-prime partial-sum build (the antitone-integral tail is too\nfat). AVOIDED here: the DIRECT Dirichlet-series bound at re = 2\u221an gives\nlog2-leading + a 3^{\u2212(2\u221an\u22122)} tail (dirichletC2 \u2264 1), which is \u2264\n0.57\u00b72^{\u2212(2\u221an\u22122)} for n \u2265 5 \u2014 the IIMaj charge exactly. -/\n\n/-- The Dirichlet majorant at s = 2\u221an (summable since 2\u221an > 1). -/\ndef dirichletCX (n : \u2115) : \u211d :=\n \u2211' m : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((2 * Real.sqrt (n : \u211d) : \u211d) : \u2102) m\u2016\n\n/-- \u2016\u03b6\u2032/\u03b6(2\u221an+it)\u2016 \u2264 dirichletCX n for n \u2265 1 (the LSeries majorant). -/\nlemma norm_logDeriv_zeta_le_dirichletCX {n : \u2115} (hn : 1 \u2264 n) {t : \u211d} :\n \u2016logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n dirichletCX n := by\n have h1 : (1 : \u211d) \u2264 Real.sqrt (n : \u211d) := Real.one_le_sqrt.mpr (by exact_mod_cast hn)\n have hs1 : (1 : \u211d) < (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I).re := by\n simp\n linarith\n have h1' : -logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\n = LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) := by\n have h2 := ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div hs1\n rw [logDeriv, Pi.div_apply, h2]\n ring\n have hsumX : Summable (fun m : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016) := by\n rw [summable_norm_iff]\n apply ArithmeticFunction.LSeriesSummable_vonMangoldt\n rw [show (((2 * Real.sqrt (n : \u211d) : \u211d) : \u2102).re) = 2 * Real.sqrt (n : \u211d) from by simp]\n linarith\n have hsumS : Summable (fun m : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) m\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt hs1\n have hle : \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264 dirichletCX n := by\n show \u2016\u2211' m : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) m\u2016 \u2264 _\n calc \u2016\u2211' m : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) m\u2016\n \u2264 \u2211' m : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) m\u2016 :=\n norm_tsum_le_tsum_norm hsumS\n _ \u2264 \u2211' m : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016 :=\n Summable.tsum_le_tsum (fun m => LSeries.norm_term_le_of_re_le_re _ (by\n simp\n linarith) m) hsumS hsumX\n _ = dirichletCX n := rfl\n calc \u2016logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016\n = \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 := by\n rw [\u2190 h1', norm_neg]\n _ \u2264 dirichletCX n := hle\n\n/-- Termwise tail comparison: for m \u2265 3, the term at 2\u221an is \u2264\n3^{\u2212(2\u221an\u22122)}\u00b7(the term at 2). -/\nlemma norm_term_vonMangoldt_X_le_three_pow {n : \u2115} (hn : 1 \u2264 n) {m : \u2115} (hm : 3 \u2264 m) :\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016 \u2264\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016 := by\n have hXre : ((((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102).re) = 2 * Real.sqrt (n : \u211d) := by simp\n have h2re : ((2 : \u2102).re) = 2 := rfl\n simp only [LSeries.norm_term_eq, hXre, h2re]\n split\n \u00b7 norm_num\n \u00b7 next hm0 =>\n have h\u039b : \u2016(ArithmeticFunction.vonMangoldt m : \u2102)\u2016 =\n ArithmeticFunction.vonMangoldt m :=\n Complex.norm_of_nonneg ArithmeticFunction.vonMangoldt_nonneg\n rw [h\u039b]\n have hm2 : (0 : \u211d) < (m : \u211d) := by exact_mod_cast (by omega : 0 < m)\n have hsqrt2 : (2 : \u211d) \u2264 2 * Real.sqrt (n : \u211d) := by\n have h1 : (1 : \u211d) \u2264 Real.sqrt (n : \u211d) := Real.one_le_sqrt.mpr (by exact_mod_cast hn)\n linarith\n have hexp_nn : (0 : \u211d) \u2264 2 * Real.sqrt (n : \u211d) - 2 := by linarith\n have e : ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 * Real.sqrt (n : \u211d)) =\n (1 / (m : \u211d) ^ (2 * Real.sqrt (n : \u211d) - 2)) *\n (ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 : \u211d)) := by\n rw [show (2 * Real.sqrt (n : \u211d)) = (2 * Real.sqrt (n : \u211d) - 2) + 2 by ring,\n Real.rpow_add hm2]\n field_simp [hm2.ne']\n rw [e]\n have hle : (1 : \u211d) / (m : \u211d) ^ (2 * Real.sqrt (n : \u211d) - 2) \u2264\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) := by\n rw [Real.rpow_neg (by norm_num : (0 : \u211d) \u2264 3), \u2190 one_div]\n apply one_div_le_one_div_of_le (Real.rpow_pos_of_pos (by norm_num : (0 : \u211d) < 3) _)\n exact Real.rpow_le_rpow (by norm_num : (0 : \u211d) \u2264 3)\n (by exact_mod_cast hm : (3 : \u211d) \u2264 (m : \u211d)) hexp_nn\n have hpos : (0 : \u211d) \u2264 ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 : \u211d) :=\n div_nonneg ArithmeticFunction.vonMangoldt_nonneg\n (Real.rpow_nonneg (by positivity : (0 : \u211d) \u2264 (m : \u211d)) _)\n exact mul_le_mul_of_nonneg_right hle hpos\n\n/-- The split: dirichletCX n = log 2\u00b72^{\u22122\u221an} + (the m \u2265 3 tail). -/\nlemma dirichletCX_le {n : \u2115} (hn : 1 \u2264 n) :\n dirichletCX n \u2264 Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 := by\n have hsumX : Summable (fun m : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016) := by\n rw [summable_norm_iff]\n apply ArithmeticFunction.LSeriesSummable_vonMangoldt\n rw [show (((2 * Real.sqrt (n : \u211d) : \u211d) : \u2102).re) = 2 * Real.sqrt (n : \u211d) from by simp]\n have h1 : (1 : \u211d) \u2264 Real.sqrt (n : \u211d) := Real.one_le_sqrt.mpr (by exact_mod_cast hn)\n linarith\n have hsplit := tsum_split_finset hsumX {0, 1, 2}\n have hsum2 : Summable (fun m : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n have hfin : \u2211 i \u2208 ({0, 1, 2} : Finset \u2115),\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) i\u2016 =\n Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n have hXre : ((((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102).re) = 2 * Real.sqrt (n : \u211d) := by simp\n rw [Finset.sum_insert (by norm_num), Finset.sum_insert (by norm_num),\n Finset.sum_singleton]\n simp only [LSeries.norm_term_eq, hXre]\n rw [if_pos (by norm_num), if_neg (by norm_num : (1 : \u2115) \u2260 0),\n if_neg (by norm_num : (2 : \u2115) \u2260 0)]\n simp [ArithmeticFunction.vonMangoldt_apply_one]\n have h\u039b2 : \u2016(ArithmeticFunction.vonMangoldt 2 : \u2102)\u2016 = Real.log 2 := by\n have h1 : ArithmeticFunction.vonMangoldt 2 = Real.log 2 :=\n ArithmeticFunction.vonMangoldt_prime Nat.prime_two\n rw [h1]\n exact Complex.norm_of_nonneg (Real.log_two_gt_d9.le)\n rw [h\u039b2]\n have e : (2 : \u211d) ^ (2 * Real.sqrt (n : \u211d)) = (2 : \u211d) ^ (2 * Real.sqrt (n : \u211d)) := rfl\n rw [show Real.log 2 / (2 : \u211d) ^ (2 * Real.sqrt (n : \u211d)) =\n Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) by\n rw [Real.rpow_neg (by norm_num : (0 : \u211d) \u2264 2)]\n field_simp [Real.rpow_pos_of_pos (by norm_num : (0 : \u211d) < 2)\n (2 * Real.sqrt (n : \u211d)).ne']]\n ring\n have htail : (\u2211' i : \u2115, (if i \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) i\u2016)) \u2264\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 := by\n have hterm : \u2200 m : \u2115, (if m \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016) \u2264\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016 := by\n intro m\n by_cases hm : m \u2208 ({0, 1, 2} : Finset \u2115)\n \u00b7 rw [if_pos hm]\n positivity\n \u00b7 rw [if_neg hm]\n have hm3 : 3 \u2264 m := by\n have h : m \u2209 ({0, 1, 2} : Finset \u2115) := hm\n simp at h\n omega\n exact norm_term_vonMangoldt_X_le_three_pow hn hm3\n have hsum_tail : Summable (fun m : \u2115 => (if m \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016)) := by\n apply Summable.of_norm_bounded hsumX.norm\n intro m\n by_cases hm : m \u2208 ({0, 1, 2} : Finset \u2115) <;> simp [hm]\n calc (\u2211' m : \u2115, (if m \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016))\n \u2264 \u2211' m : \u2115, (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016 :=\n Summable.tsum_le_tsum hterm hsum_tail (hsum2.mul_left _)\n _ = (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 := by\n rw [dirichletC2, tsum_mul_left]\n calc dirichletCX n\n = \u2211 i \u2208 ({0, 1, 2} : Finset \u2115),\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) i\u2016 +\n \u2211' i : \u2115, (if i \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) i\u2016) := hsplit\n _ \u2264 Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 :=\n add_le_add (le_of_eq hfin) htail\n\n/-- log(9/4) \u2265 0.811 (via e^{0.811} \u2265 9/4 with e^{0.8} \u2265 2.2254). -/\nlemma log_nine_quarters_ge : (0.811 : \u211d) \u2264 Real.log (9 / 4) := by\n have h08 : (2.2254 : \u211d) \u2264 Real.exp 0.8 :=\n le_trans (by norm_num [Finset.sum_range_succ])\n (Real.sum_le_exp_of_nonneg (by norm_num : (0 : \u211d) \u2264 0.8) 7)\n have h011 : (1.01106 : \u211d) \u2264 Real.exp 0.011 :=\n le_trans (by norm_num [Finset.sum_range_succ])\n (Real.sum_le_exp_of_nonneg (by norm_num : (0 : \u211d) \u2264 0.011) 4)\n have hprod : (9 / 4 : \u211d) \u2264 Real.exp 0.8 * Real.exp 0.011 := by\n calc (9 / 4 : \u211d) = 2.25 := by norm_num\n _ \u2264 2.2254 * 1.01106 := by norm_num\n _ \u2264 Real.exp 0.8 * Real.exp 0.011 :=\n mul_le_mul h08 h011 (by norm_num : (0 : \u211d) \u2264 1.01106) (by norm_num : (0 : \u211d) \u2264 2.2254)\n have h1 : Real.exp 0.811 = Real.exp 0.8 * Real.exp 0.011 := by\n rw [show (0.811 : \u211d) = 0.8 + 0.011 by norm_num, Real.exp_add]\n rw [\u2190 h1] at hprod\n have h2 := Real.log_le_log (Real.exp_pos _) hprod\n rwa [Real.log_exp] at h2\n\n/-- log(4/9) \u2264 \u22120.811 (the negation). -/\nlemma log_four_ninths_le : Real.log (4 / 9) \u2264 (-0.811 : \u211d) := by\n have h1 := log_nine_quarters_ge\n have h2 : Real.log (4 / 9) = -Real.log (9 / 4) := by\n rw [show ((4 : \u211d) / 9) = (9 / 4)\u207b\u00b9 by norm_num, Real.log_inv]\n rw [h2]\n linarith [h1]\n\n/-- The certified (4/9)^{\u221an} \u2264 0.17631 for \u221an \u2265 2.236. -/\nlemma four_ninths_rpow_sqrt_le {x : \u211d} (hx : (2.236 : \u211d) \u2264 x) :\n (4 / 9 : \u211d) ^ x \u2264 (0.17631 : \u211d) := by\n have hlog49 := log_four_ninths_le\n have hlogneg : Real.log (4 / 9) < 0 :=\n Real.log_neg (by norm_num : (0 : \u211d) < 4 / 9) (by norm_num : (4 / 9 : \u211d) < 1)\n have h1 : (4 / 9 : \u211d) ^ x = Real.exp (Real.log (4 / 9) * x) :=\n Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 4 / 9) x\n rw [h1]\n have h2 : Real.log (4 / 9) * x \u2264 (-1.8135 : \u211d) := by\n have h2a : Real.log (4 / 9) * x \u2264 Real.log (4 / 9) * 2.236 :=\n mul_le_mul_of_nonpos_left hx hlogneg.le\n have h2b : Real.log (4 / 9) * 2.236 \u2264 (-0.811) * 2.236 :=\n mul_le_mul_of_nonneg_right hlog49 (by norm_num)\n calc Real.log (4 / 9) * x \u2264 Real.log (4 / 9) * 2.236 := h2a\n _ \u2264 (-0.811) * 2.236 := h2b\n _ \u2264 -1.8135 := by norm_num\n have h3 : Real.exp (-1.8135) \u2264 (0.17631 : \u211d) := by\n have hsum : (5.6721 : \u211d) \u2264 Real.exp 1.8135 := by\n calc (5.6721 : \u211d) \u2264 \u2211 i \u2208 Finset.range 7, (1.8135 : \u211d) ^ i / (Nat.factorial i : \u211d) := by\n norm_num [Finset.sum_range_succ]\n _ \u2264 Real.exp 1.8135 :=\n Real.sum_le_exp_of_nonneg (by norm_num : (0 : \u211d) \u2264 1.8135) 7\n rw [show Real.exp (-1.8135) = 1 / Real.exp 1.8135 by\n rw [Real.exp_neg, one_div]]\n calc 1 / Real.exp 1.8135 \u2264 1 / 5.6721 :=\n one_div_le_one_div_of_le (by positivity : (0 : \u211d) < 5.6721) hsum\n _ \u2264 0.17631 := by norm_num\n calc Real.exp (x * Real.log (4 / 9)) \u2264 Real.exp (-1.8135) :=\n Real.exp_le_exp.mpr h2\n _ \u2264 0.17631 := h3\n\n/-- The 3^{\u2212(2\u221an\u22122)} = 9\u00b7(4/9)^{\u221an}\u00b72^{\u22122\u221an} identity. -/\nlemma three_pow_neg_two_sqrt_sub_two_eq {n : \u2115} :\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) =\n 9 * ((4 / 9 : \u211d) ^ Real.sqrt (n : \u211d)) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n have h9 : Real.log 9 = 2 * Real.log 3 := by\n rw [show (9 : \u211d) = 3 ^ 2 by norm_num, Real.log_pow]\n push_cast\n ring\n have h2 : (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) =\n Real.exp (-(Real.log 2) * (2 * Real.sqrt (n : \u211d))) := by\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n congr 1\n ring\n have h3 : (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) =\n Real.exp (-(Real.log 3) * (2 * Real.sqrt (n : \u211d) - 2)) := by\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 3)]\n congr 1\n ring\n have h49 : (4 / 9 : \u211d) ^ Real.sqrt (n : \u211d) =\n Real.exp (Real.sqrt (n : \u211d) * Real.log (4 / 9)) :=\n Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 4 / 9) _\n rw [h2, h3, h49]\n have hlog49 : Real.log (4 / 9) = Real.log 2 - Real.log 9 + Real.log 2 := by\n rw [Real.log_div (by norm_num : (4 : \u211d) \u2260 0) (by norm_num : (9 : \u211d) \u2260 0)]\n ring\n rw [hlog49, h9]\n rw [\u2190 Real.exp_add, \u2190 Real.exp_add]\n congr 1\n rw [show (9 : \u211d) = Real.exp (Real.log 9) from (Real.exp_log (by norm_num : (0 : \u211d) < 9)).symm]\n rw [\u2190 Real.exp_add]\n congr 1\n ring\n\n/-- **The I_I zeta charge WITHOUT dirichletC2 \u2264 0.57** (DL-6 capstone):\nfor n \u2265 5, \u2016\u03b6\u2032/\u03b6(2\u221an+it)\u2016 \u2264 0.57\u00b72^{\u2212(2\u221an\u22122)} \u2014 the IIMaj zeta form,\nvia the direct log2-leading + 3^{\u2212(2\u221an\u22122)} tail (dirichletC2 \u2264 1) with\nthe certified (4/9)^{\u221an} \u2264 0.17631 bound at \u221an \u2265 \u221a5. -/\ntheorem norm_logDeriv_zeta_le_iimZeta {n : \u2115} (hn : 5 \u2264 n) {t : \u211d} :\n \u2016logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n (0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) := by\n have hn1 : 1 \u2264 n := by omega\n have h1 := norm_logDeriv_zeta_le_dirichletCX hn1 (t := t)\n have h2 := dirichletCX_le hn1\n have hC2 : dirichletC2 \u2264 1 := dirichletC2_le_one\n have hC2nn : (0 : \u211d) \u2264 dirichletC2 := dirichletC2_nonneg\n have hsqrt5 : (2.236 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n have hnR : (5 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have h1 : Real.sqrt 5 \u2264 Real.sqrt (n : \u211d) := Real.sqrt_le_sqrt hnR\n have h2 : (2.236 : \u211d) \u2264 Real.sqrt 5 := by\n rw [Real.le_sqrt (by norm_num) (by norm_num)]\n norm_num\n linarith\n have h49 := four_ninths_rpow_sqrt_le hsqrt5\n have h3 : dirichletCX n \u2264 Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) := by\n refine h2.trans ?_\n exact add_le_add_left (mul_le_mul_of_nonneg_left hC2\n (Real.rpow_nonneg (by norm_num) _)) _\n have h4 : (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) \u2264\n (1.58679 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n rw [three_pow_neg_two_sqrt_sub_two_eq]\n calc 9 * ((4 / 9 : \u211d) ^ Real.sqrt (n : \u211d)) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d)))\n \u2264 9 * (0.17631 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg (by norm_num) _)\n apply mul_le_mul_of_nonneg_left h49 (by norm_num)\n _ = (1.58679 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by ring\n have h5 : Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) \u2264\n (0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) := by\n have hlog2 : Real.log 2 \u2264 (0.6932 : \u211d) := Real.log_two_lt_d9.le\n have hpos2 : (0 : \u211d) < (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by positivity\n have e : (0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) =\n (2.28 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n rw [show (-(2 * Real.sqrt (n : \u211d) - 2)) = -2 * Real.sqrt (n : \u211d) + 2 by ring,\n Real.rpow_add (by norm_num : (0 : \u211d) < 2), Real.rpow_two]\n ring\n rw [e]\n calc Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))\n \u2264 0.6932 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (1.58679 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n apply add_le_add _ h4\n exact mul_le_mul_of_nonneg_right hlog2 hpos2.le\n _ = (0.6932 + 1.58679 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by ring\n _ \u2264 (2.28 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n apply mul_le_mul_of_nonneg_right _ hpos2.le\n norm_num\n exact h1.trans (h3.trans h5)\n\nend NguyenRH1\n\nend\n",
"structuredPatch": [
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"oldStart": 1083,
"oldLines": 8,
"newStart": 1083,
"newLines": 7,
"lines": [
" _ \u2264 \u2211' m : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))",
" (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016 :=",
" Summable.tsum_le_tsum (fun m => LSeries.norm_term_le_of_re_le_re _ (by",
"- simp",
"- linarith) m) hsumS hsumX",
"+ simp) m) hsumS hsumX",
" _ = dirichletCX n := rfl",
" calc \u2016logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016",
" = \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))"
]
}
],
"userModified": false,
"replaceAll": false
}
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"old_string": " simp only [LSeries.norm_term_eq, hXre]\n rw [if_pos (by norm_num), if_neg (by norm_num : (1 : \u2115) \u2260 0),\n if_neg (by norm_num : (2 : \u2115) \u2260 0)]\n simp [ArithmeticFunction.vonMangoldt_apply_one]\n have h\u039b2 : \u2016(ArithmeticFunction.vonMangoldt 2 : \u2102)\u2016 = Real.log 2 := by\n have h1 : ArithmeticFunction.vonMangoldt 2 = Real.log 2 :=\n ArithmeticFunction.vonMangoldt_prime Nat.prime_two\n rw [h1]\n exact Complex.norm_of_nonneg (Real.log_two_gt_d9.le)\n rw [h\u039b2]",
"new_string": " simp only [LSeries.norm_term_eq, hXre]\n rw [if_pos (by norm_num), if_neg (by norm_num : (1 : \u2115) \u2260 0),\n if_neg (by norm_num : (2 : \u2115) \u2260 0)]\n simp [ArithmeticFunction.vonMangoldt_apply_one]\n have h\u039b2 : \u2016(fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) 2\u2016 = Real.log 2 := by\n show \u2016(ArithmeticFunction.vonMangoldt 2 : \u2102)\u2016 = Real.log 2\n rw [ArithmeticFunction.vonMangoldt_apply_prime Nat.prime_two]\n exact Complex.norm_of_nonneg (by linarith [Real.log_two_gt_d9] : (0 : \u211d) \u2264 Real.log 2)\n rw [h\u039b2]"
}
tool: Edit ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"old_string": " _ \u2264 Real.exp 0.8 * Real.exp 0.011 :=\n mul_le_mul h08 h011 (by norm_num : (0 : \u211d) \u2264 1.01106) (by norm_num : (0 : \u211d) \u2264 2.2254)",
"new_string": " _ \u2264 Real.exp 0.8 * Real.exp 0.011 :=\n mul_le_mul h08 h011 (by norm_num : (0 : \u211d) \u2264 1.01106) (Real.exp_pos _).le"
}
result 2026-08-26 09:29:27Z → 2026-08-26 09:29:27Z
{
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"oldString": " _ \u2264 Real.exp 0.8 * Real.exp 0.011 :=\n mul_le_mul h08 h011 (by norm_num : (0 : \u211d) \u2264 1.01106) (by norm_num : (0 : \u211d) \u2264 2.2254)",
"newString": " _ \u2264 Real.exp 0.8 * Real.exp 0.011 :=\n mul_le_mul h08 h011 (by norm_num : (0 : \u211d) \u2264 1.01106) (Real.exp_pos _).le",
"originalFile": "/-\nNguyenRH1/LiDogLeg.lean \u2014 ii\u2032-6-DL-1 (D-074, session 040): the zero-free\ndog-leg deformation of the Lagarias horizontal right tail.\n\nTHE CONTENT (D-074 \u00a73.1\u20133.2): on the dog-leg rectangle [3, X_R]\u00d7[T, U]\n(T > 0), the Lagarias integrand Jlag n s = (s/(s\u22121))\u207f\u22121 \u00b7 (\u2212\u03b6\u2032/\u03b6(s)) is\nholomorphic UNCONDITIONALLY (s \u2260 1 since im > 0; \u03b6 s \u2260 0 for re s \u2265 3 > 1\nby Mathlib's Euler-product nonvanishing `riemannZeta_ne_zero_of_one_lt_re`\n\u2014 NO perimeter hypothesis, no zero-avoidance at U). Cauchy's theorem\n(`HolomorphicOn.vanishesOnRectangle`) then gives the EXACT deformation\nidentity\n H(3\u2192X_R, T) = H(3\u2192X_R, U) \u2212 V(X_R, T\u2192U) + V(3, T\u2192U)\n(the upper right tail = the high horizontal \u2212 the right drop + the left\nlift), plus the Jlag conjugation identity for the lower-side transfer.\n\nThis is the repair of Finding F-D (D-072): the broken low-path envelope\nis ABANDONED (not bounded) \u2014 the magnitude bound comes later from the\nthree dog-leg pieces, all in the unconditionally zero-free region.\n-/\nimport Mathlib\nimport NguyenRH1.LiSides\nimport NguyenRH1.LiTightR\nimport NguyenRH1.LiBasePoints\n\nopen Zeta23 Complex Filter Topology\nopen scoped Nat\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. Dog-leg analyticity (D-074 \u00a73.1; no perimeter) -/\n\n/-- The dog-leg corners (same real-part/imaginary-part form as rectZ/rectW). -/\ndef dogZ (T : \u211d) : \u2102 := ((3 : \u211d) : \u2102) + (T : \u211d) * Complex.I\ndef dogW (XR U : \u211d) : \u2102 := ((XR : \u211d) : \u2102) + (U : \u211d) * Complex.I\n\nlemma dogZ_re (T : \u211d) : (dogZ T).re = 3 := by simp [dogZ]\nlemma dogZ_im (T : \u211d) : (dogZ T).im = T := by simp [dogZ]\nlemma dogW_re (XR U : \u211d) : (dogW XR U).re = XR := by simp [dogW]\nlemma dogW_im (XR U : \u211d) : (dogW XR U).im = U := by simp [dogW]\n\n/-- Jlag is differentiable at every point with 3 \u2264 re, 0 < im:\n\u03b6 is zero-free for 1 < re (Mathlib `riemannZeta_ne_zero_of_one_lt_re`,\nUNCONDITIONAL \u2014 the Euler-product region) and s \u2260 1 (im \u2260 0).\nNO perimeter hypothesis. -/\ntheorem Jlag_differentiableAt_dogleg {n : \u2115} {s : \u2102}\n (hre : 3 \u2264 s.re) (him : 0 < s.im) :\n DifferentiableAt \u2102 (Jlag n) s := by\n apply Jlag_differentiableAt\n \u00b7 rintro rfl\n simp at him\n \u00b7 exact riemannZeta_ne_zero_of_one_lt_re (by linarith)\n\n/-- Jlag is holomorphic on the whole dog-leg rectangle\n[3, X_R]\u00d7[T, U] (T > 0). -/\ntheorem holomorphicOn_Jlag_dogleg {n : \u2115} {T U XR : \u211d}\n (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n HolomorphicOn (Jlag n) (Rectangle (dogZ T) (dogW XR U)) := by\n intro s hs\n rw [Rectangle, Complex.mem_reProdIm, dogZ_re, dogW_re, dogZ_im, dogW_im] at hs\n rw [Set.uIcc_of_le hXR, Set.uIcc_of_le hTU] at hs\n exact (Jlag_differentiableAt_dogleg hs.1.1 (by linarith [hs.2.1])).differentiableWithinAt\n\n/-- **The dog-leg rectangle integral vanishes** (Cauchy; the region is\nunconditionally zero-free \u2014 the residue sum is EMPTY, no perimeter). -/\ntheorem rectangleIntegral_Jlag_dogleg_eq_zero {n : \u2115} {T U XR : \u211d}\n (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n RectangleIntegral (Jlag n) (dogZ T) (dogW XR U) = 0 :=\n (holomorphicOn_Jlag_dogleg hT hTU hXR).vanishesOnRectangle subset_rfl\n\n/-- The (2\u03c0i)\u207b\u00b9-normalized form (API symmetry with rectInt'_Jlag). -/\ntheorem rectangleIntegral'_Jlag_dogleg_eq_zero {n : \u2115} {T U XR : \u211d}\n (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n RectangleIntegral' (Jlag n) (dogZ T) (dogW XR U) = 0 := by\n rw [RectangleIntegral', rectangleIntegral_Jlag_dogleg_eq_zero hT hTU hXR,\n smul_zero]\n\n/-- **THE EXACT DOG-LEG DEFORMATION IDENTITY** (D-074 \u00a73.2 (\u2605), the\ncapstone for ii\u2032-6-c): the low right-horizontal equals the high\nhorizontal minus the right drop plus the left lift. Orientations: all\nH-integrals left-to-right, all V-integrals upward (the main contour's\ntop side is their reversal, handled at the fold site). -/\ntheorem upper_right_horizontal_eq_dogleg {n : \u2115} {T U XR : \u211d}\n (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n HIntegral (Jlag n) 3 XR T =\n HIntegral (Jlag n) 3 XR U - VIntegral (Jlag n) XR T U +\n VIntegral (Jlag n) 3 T U := by\n have h0 := rectangleIntegral_Jlag_dogleg_eq_zero (n := n) hT hTU hXR\n unfold RectangleIntegral at h0\n rw [dogZ_re, dogZ_im, dogW_re, dogW_im] at h0\n linear_combination h0\n\n/-! ## 2. The conjugation identity (for the lower-side transfer, DL-5) -/\n\n/-- kFun is conjugation-symmetric (real-coefficient rational function). -/\ntheorem kFun_conj (n : \u2115) (s : \u2102) :\n kFun n ((starRingEnd \u2102) s) = (starRingEnd \u2102) (kFun n s) := by\n unfold kFun\n simp [map_sub, map_pow, map_one]\n\n/-- **Jlag is conjugation-symmetric** (from `riemannZeta_conj` +\n`iteratedDeriv_riemannZeta_conj`): Jlag n (s\u0304) = conj (Jlag n s).\nNeeded for the lower-horizontal transfer (DL-5). -/\ntheorem Jlag_conj (n : \u2115) {s : \u2102} (hs : s \u2260 1) :\n Jlag n ((starRingEnd \u2102) s) = (starRingEnd \u2102) (Jlag n s) := by\n have h1 : deriv riemannZeta ((starRingEnd \u2102) s) =\n (starRingEnd \u2102) (deriv riemannZeta s) := by\n have h := iteratedDeriv_riemannZeta_conj 1 hs\n rwa [iteratedDeriv_one] at h\n unfold Jlag\n rw [kFun_conj, _root_.riemannZeta_conj, h1]\n simp only [map_mul, map_neg, map_div\u2080]\n\n/-! ## 3. The left-lift bound (DL-2, D-074 \u00a73.3 V_left) -/\n\n/-- **Kernel on the lift**: \u2016kFun n (3+it)\u2016 \u2264 e^{5/2}+1 for n \u2264 t\u00b2\n(D-074 \u00a76: |(3+it)/(2+it)|\u00b2 = (9+t\u00b2)/(4+t\u00b2) = 1 + 5/(4+t\u00b2), whose n-th\npower is \u2264 e^{5n/(2(4+t\u00b2))} \u2264 e^{5/2}). -/\ntheorem kFun_three_add_it_norm_le_exp_five_halves_add_one {n : \u2115} {t : \u211d}\n (hT : (n : \u211d) \u2264 t ^ 2) :\n \u2016kFun n ((3 : \u211d) + t * Complex.I)\u2016 \u2264 Real.exp (5 / 2) + 1 := by\n rcases Nat.eq_zero_or_pos n with hn | hn\n \u00b7 subst hn\n simp only [kFun, pow_zero, sub_self, norm_zero]\n positivity\n \u00b7 have hnR : (1 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have ht2 : (0 : \u211d) < 4 + t ^ 2 := by positivity\n have hn4t2 : (n : \u211d) \u2264 4 + t ^ 2 := by linarith [hT]\n have hQpos : (0 : \u211d) \u2264 (9 + t ^ 2) / (4 + t ^ 2) := by positivity\n have hQle : (9 + t ^ 2) / (4 + t ^ 2) \u2264 Real.exp (5 / (n : \u211d)) := by\n have h1 : (9 + t ^ 2) / (4 + t ^ 2) = 1 + 5 / (4 + t ^ 2) := by\n field_simp [ht2.ne']\n ring\n rw [h1]\n have h2 : 5 / (4 + t ^ 2) \u2264 5 / (n : \u211d) :=\n (div_le_div_iff_of_pos_left (by norm_num) ht2 (by positivity)).mpr hn4t2\n have h3 := Real.add_one_le_exp (5 / (n : \u211d))\n linarith\n have hQn : ((9 + t ^ 2) / (4 + t ^ 2)) ^ n \u2264 Real.exp 5 := by\n calc ((9 + t ^ 2) / (4 + t ^ 2)) ^ n \u2264 (Real.exp (5 / (n : \u211d))) ^ n :=\n pow_le_pow_left\u2080 hQpos hQle n\n _ = Real.exp 5 := by\n rw [\u2190 Real.exp_nat_mul]\n congr 1\n have hn0 : (n : \u211d) \u2260 0 := by exact_mod_cast hn.ne'\n field_simp [hn0]\n have hns3 : Complex.normSq ((3 : \u211d) + t * Complex.I) = 9 + t ^ 2 := by\n simp [Complex.normSq_apply]\n ring\n have hns2 : Complex.normSq ((2 : \u211d) + t * Complex.I) = 4 + t ^ 2 := by\n simp [Complex.normSq_apply]\n ring\n have hnormA : \u2016((3 : \u211d) + t * Complex.I)\u2016 = Real.sqrt (9 + t ^ 2) := by\n have h1 : \u2016((3 : \u211d) + t * Complex.I)\u2016 =\n Real.sqrt (Complex.normSq ((3 : \u211d) + t * Complex.I)) := by\n rw [Complex.normSq_eq_norm_sq, Real.sqrt_sq (norm_nonneg _)]\n rw [h1, hns3]\n have hnormB : \u2016((2 : \u211d) + t * Complex.I)\u2016 = Real.sqrt (4 + t ^ 2) := by\n have h1 : \u2016((2 : \u211d) + t * Complex.I)\u2016 =\n Real.sqrt (Complex.normSq ((2 : \u211d) + t * Complex.I)) := by\n rw [Complex.normSq_eq_norm_sq, Real.sqrt_sq (norm_nonneg _)]\n rw [h1, hns2]\n have hnorm : \u2016((3 : \u211d) + t * Complex.I) / ((2 : \u211d) + t * Complex.I)\u2016 =\n Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2)) := by\n rw [norm_div, hnormA, hnormB,\n \u2190 Real.sqrt_div (by positivity : (0 : \u211d) \u2264 9 + t ^ 2)]\n have hs1 : ((3 : \u211d) + t * Complex.I) - 1 = (2 : \u211d) + t * Complex.I := by\n push_cast\n ring\n unfold kFun\n rw [hs1]\n calc \u2016(((3 : \u211d) + t * Complex.I) / ((2 : \u211d) + t * Complex.I)) ^ n - 1\u2016\n \u2264 \u2016(((3 : \u211d) + t * Complex.I) / ((2 : \u211d) + t * Complex.I)) ^ n\u2016 + \u2016(1 : \u2102)\u2016 :=\n norm_sub_le _ _\n _ = (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n + 1 := by\n rw [norm_pow, hnorm, norm_one]\n _ \u2264 Real.exp (5 / 2) + 1 := by\n have h4 : (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n \u2264 Real.exp (5 / 2) := by\n have hx0 : (0 : \u211d) \u2264 Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2)) :=\n Real.sqrt_nonneg _\n have hx : (0 : \u211d) \u2264 (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n :=\n pow_nonneg hx0 n\n have h5 : ((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2 \u2264 Real.exp 5 := by\n have e : ((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2 =\n ((9 + t ^ 2) / (4 + t ^ 2)) ^ n := by\n rw [\u2190 pow_mul, mul_comm n 2, pow_mul, Real.sq_sqrt hQpos]\n rw [e]\n exact hQn\n have h8 : (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n =\n Real.sqrt (((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2) := by\n rw [Real.sqrt_sq hx]\n rw [h8]\n calc Real.sqrt (((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2)\n \u2264 Real.sqrt (Real.exp 5) := Real.sqrt_le_sqrt h5\n _ = Real.exp (5 / 2) := (Real.exp_half 5).symm\n linarith [h4]\n\n/-- The Dirichlet majorant at s = 3 (summable since 3 > 1). -/\ndef dirichletC3 : \u211d :=\n \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016\n\n/-- Termwise 3-vs-2 comparison for the \u039b L-series: \u039b(n)/n\u00b3 \u2264 (1/2)\u00b7\u039b(n)/n\u00b2. -/\nlemma norm_term_vonMangoldt_three_le_half (n : \u2115) :\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016 \u2264\n (1 / 2) * \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016 := by\n have h3re : ((3 : \u2102).re) = 3 := rfl\n have h2re : ((2 : \u2102).re) = 2 := rfl\n simp only [LSeries.norm_term_eq, h3re, h2re]\n split\n \u00b7 norm_num\n \u00b7 next hn =>\n rcases eq_or_ne n 1 with rfl | hn1\n \u00b7 simp [ArithmeticFunction.vonMangoldt_apply_one]\n \u00b7 have hn2 : 2 \u2264 n := by omega\n have h\u039b : \u2016(ArithmeticFunction.vonMangoldt n : \u2102)\u2016 =\n ArithmeticFunction.vonMangoldt n :=\n Complex.norm_of_nonneg ArithmeticFunction.vonMangoldt_nonneg\n rw [h\u039b]\n have e : ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (3 : \u211d) =\n (1 / (n : \u211d)) * (ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d)) := by\n rw [show (3 : \u211d) = 2 + 1 by norm_num,\n Real.rpow_add (by positivity : (0 : \u211d) < (n : \u211d)), Real.rpow_one]\n field_simp [show (n : \u211d) \u2260 0 by positivity]\n have h12 : (1 : \u211d) / (n : \u211d) \u2264 1 / 2 :=\n one_div_le_one_div_of_le (by norm_num : (0 : \u211d) < 2)\n (by exact_mod_cast hn2 : (2 : \u211d) \u2264 (n : \u211d))\n have hpos : (0 : \u211d) \u2264 ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d) :=\n div_nonneg ArithmeticFunction.vonMangoldt_nonneg\n (Real.rpow_nonneg (by positivity : (0 : \u211d) \u2264 (n : \u211d)) _)\n calc ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (3 : \u211d)\n = (1 / (n : \u211d)) * (ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d)) := e\n _ \u2264 (1 / 2) * (ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d)) :=\n mul_le_mul_of_nonneg_right h12 hpos\n\n/-- dirichletC3 \u2264 1/2 (the termwise comparison + dirichletC2 \u2264 1) \u2014\nthe certified C_zeta3 = 0.5 of D-074 \u00a74. (A tighter 0.17 build is\npossible via a 6-prime partial sum + antitone-integral tail; the fold\nis indifferent to the constant, so the cheap certified 0.5 is charged.) -/\nlemma dirichletC3_le_half : dirichletC3 \u2264 1 / 2 := by\n have hsum3 : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n have hsum2 : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n calc dirichletC3\n = \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016 := rfl\n _ \u2264 \u2211' n : \u2115, (1 / 2) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016 :=\n Summable.tsum_le_tsum (fun n => norm_term_vonMangoldt_three_le_half n) hsum3\n (hsum2.mul_left (1 / 2))\n _ = (1 / 2) * dirichletC2 := by rw [dirichletC2, tsum_mul_left]\n _ \u2264 1 / 2 := by linarith [dirichletC2_le_one]\n\n/-- \u2016\u03b6\u2032/\u03b6 s\u2016 \u2264 dirichletC3 for re s \u2265 3 (the s = 3 clone of the\nDirichlet-series majorant). -/\nlemma norm_logDeriv_zeta_le_dirichletC3 {s : \u2102} (hs : 3 \u2264 s.re) :\n \u2016logDeriv riemannZeta s\u2016 \u2264 dirichletC3 := by\n have hs1 : (1 : \u211d) < s.re := by linarith\n have h1 : -logDeriv riemannZeta s\n = LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s := by\n have h2 := ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div hs1\n rw [logDeriv, Pi.div_apply, h2]\n ring\n have hsum3 : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n have hsumS : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt hs1\n have hle : \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s\u2016\n \u2264 dirichletC3 := by\n show \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016 \u2264 _\n calc \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016\n \u2264 \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016 :=\n norm_tsum_le_tsum_norm hsumS\n _ \u2264 \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016 :=\n Summable.tsum_le_tsum (fun n => LSeries.norm_term_le_of_re_le_re _ (by\n simp\n exact hs) n) hsumS hsum3\n _ = dirichletC3 := rfl\n calc \u2016logDeriv riemannZeta s\u2016\n = \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s\u2016 := by\n rw [\u2190 h1, norm_neg]\n _ \u2264 dirichletC3 := hle\n\n/-- \u2016\u03b6\u2032/\u03b6 s\u2016 \u2264 1/2 for re s \u2265 3 (the charged C_zeta3 = 0.5). -/\nlemma norm_logDeriv_zeta_le_half {s : \u2102} (hs : 3 \u2264 s.re) :\n \u2016logDeriv riemannZeta s\u2016 \u2264 1 / 2 :=\n (norm_logDeriv_zeta_le_dirichletC3 hs).trans dirichletC3_le_half\n\n/-- **The left-lift bound** (D-074 \u00a73.3 V_left): for 0 < T \u2264 U and\nn \u2264 T\u00b2, \u2016VIntegral (Jlag n) 3 T U\u2016 \u2264 (U \u2212 T)\u00b7(e^{5/2}+1)/2. The kernel\n\u2264 e^{5/2}+1 (the 1+5/(4+t\u00b2) computation), the zeta factor \u2264 1/2\n(certified, no perimeter). -/\ntheorem vIntegral_Jlag_three_norm_le {n : \u2115} {T U : \u211d}\n (hT0 : (0 : \u211d) < T) (hTU : T \u2264 U) (hTn : (n : \u211d) \u2264 T ^ 2) :\n \u2016VIntegral (Jlag n) 3 T U\u2016 \u2264 (U - T) * (Real.exp (5 / 2) + 1) / 2 := by\n have hnorm : \u2016VIntegral (Jlag n) 3 T U\u2016 =\n \u2016\u222b t in T..U, Jlag n ((3 : \u211d) + t * Complex.I)\u2016 := by\n unfold VIntegral\n rw [norm_smul, Complex.norm_I, one_mul]\n rw [hnorm]\n have hbound : \u2200 t \u2208 Set.Ioc T U,\n \u2016Jlag n ((3 : \u211d) + t * Complex.I)\u2016 \u2264 (Real.exp (5 / 2) + 1) * (1 / 2) := by\n intro t ht\n have htT : T < t := ht.1\n have ht2 : (n : \u211d) \u2264 t ^ 2 := by nlinarith [hTn, hT0]\n have hre : (3 : \u211d) \u2264 ((3 : \u211d) + t * Complex.I).re := by simp\n have h1 := kFun_three_add_it_norm_le_exp_five_halves_add_one ht2\n have h2 := norm_logDeriv_zeta_le_half (s := (3 : \u211d) + t * Complex.I) hre\n have hnorm3 : \u2016Jlag n ((3 : \u211d) + t * Complex.I)\u2016 =\n \u2016kFun n ((3 : \u211d) + t * Complex.I)\u2016 *\n \u2016logDeriv riemannZeta ((3 : \u211d) + t * Complex.I)\u2016 := by\n show \u2016kFun n ((3 : \u211d) + t * Complex.I) *\n (-logDeriv riemannZeta ((3 : \u211d) + t * Complex.I))\u2016 = _\n rw [norm_mul, norm_neg]\n rw [hnorm3]\n exact mul_le_mul h1 h2 (norm_nonneg _) (by positivity)\n have hint : \u2016\u222b t in T..U, Jlag n ((3 : \u211d) + t * Complex.I)\u2016 \u2264\n ((Real.exp (5 / 2) + 1) * (1 / 2)) * |U - T| := by\n apply intervalIntegral.norm_integral_le_of_norm_le_const\n intro t ht\n rw [Set.uIoc_of_le hTU] at ht\n exact hbound t ht\n have habs : |U - T| = U - T := abs_of_nonneg (by linarith)\n rw [habs] at hint\n calc \u2016\u222b t in T..U, Jlag n ((3 : \u211d) + t * Complex.I)\u2016\n \u2264 ((Real.exp (5 / 2) + 1) * (1 / 2)) * (U - T) := hint\n _ = (U - T) * (Real.exp (5 / 2) + 1) / 2 := by ring\n\n/-! ## 4. The high-horizontal stepping (DL-3, D-074 \u00a77 + D-073 \u00a74) -/\n\n/-- Termwise \u03c3-vs-2 comparison for the \u039b L-series (the (6.10) halving):\n\u039b(m)/m^\u03c3 \u2264 2^{\u2212(\u03c3\u22122)}\u00b7\u039b(m)/m\u00b2 for \u03c3 \u2265 2. -/\nlemma norm_term_vonMangoldt_le_two_pow {\u03c3 t : \u211d} (h\u03c3 : 2 \u2264 \u03c3) (m : \u2115) :\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) m\u2016 \u2264\n (2 : \u211d) ^ (-(\u03c3 - 2)) * \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016 := by\n have h\u03c3re : ((\u03c3 : \u211d) + t * Complex.I).re = \u03c3 := by simp\n have h2re : ((2 : \u2102).re) = 2 := rfl\n simp only [LSeries.norm_term_eq, h\u03c3re, h2re]\n split\n \u00b7 norm_num\n \u00b7 next hm =>\n have h\u039b : \u2016(ArithmeticFunction.vonMangoldt m : \u2102)\u2016 =\n ArithmeticFunction.vonMangoldt m :=\n Complex.norm_of_nonneg ArithmeticFunction.vonMangoldt_nonneg\n rw [h\u039b]\n rcases eq_or_ne m 1 with rfl | hm1\n \u00b7 simp [ArithmeticFunction.vonMangoldt_apply_one]\n \u00b7 have hm2 : 2 \u2264 m := by omega\n have e : ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ \u03c3 =\n (1 / (m : \u211d) ^ (\u03c3 - 2)) * (ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 : \u211d)) := by\n rw [show (\u03c3 : \u211d) = (\u03c3 - 2) + 2 by ring,\n Real.rpow_add (by positivity : (0 : \u211d) < (m : \u211d))]\n field_simp [show (m : \u211d) \u2260 0 by positivity]\n ring_nf\n rw [e]\n have hle : (1 : \u211d) / (m : \u211d) ^ (\u03c3 - 2) \u2264 (2 : \u211d) ^ (-(\u03c3 - 2)) := by\n rw [Real.rpow_neg (by norm_num : (0 : \u211d) \u2264 2), \u2190 one_div]\n apply one_div_le_one_div_of_le (Real.rpow_pos_of_pos (by norm_num : (0 : \u211d) < 2) _)\n exact Real.rpow_le_rpow (by norm_num : (0 : \u211d) \u2264 2)\n (by exact_mod_cast hm2 : (2 : \u211d) \u2264 (m : \u211d)) (by linarith : (0 : \u211d) \u2264 \u03c3 - 2)\n have hpos : (0 : \u211d) \u2264 ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 : \u211d) :=\n div_nonneg ArithmeticFunction.vonMangoldt_nonneg\n (Real.rpow_nonneg (by positivity : (0 : \u211d) \u2264 (m : \u211d)) _)\n exact mul_le_mul_of_nonneg_right hle hpos\n\n/-- \u2016\u03b6\u2032/\u03b6(\u03c3+it)\u2016 \u2264 2^{\u2212(\u03c3\u22122)}\u00b7dirichletC2 for \u03c3 \u2265 2 (the (6.10)\nhalving in LSeries form; dirichletC2 \u2264 1 makes the tail free). -/\nlemma norm_logDeriv_zeta_le_two_pow {\u03c3 t : \u211d} (h\u03c3 : 2 \u2264 \u03c3) :\n \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264\n (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := by\n have hs1 : (1 : \u211d) < ((\u03c3 : \u211d) + t * Complex.I).re := by simp; linarith\n have h1 : -logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\n = LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) ((\u03c3 : \u211d) + t * Complex.I) := 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 : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n have hsumS : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt hs1\n have hle : \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264 (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := by\n show \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) n\u2016 \u2264 _\n calc \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) n\u2016\n \u2264 \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) n\u2016 :=\n norm_tsum_le_tsum_norm hsumS\n _ \u2264 \u2211' n : \u2115, (2 : \u211d) ^ (-(\u03c3 - 2)) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016 :=\n Summable.tsum_le_tsum (fun n => norm_term_vonMangoldt_le_two_pow h\u03c3 n) hsumS\n (hsum2.mul_left _)\n _ = (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := by rw [dirichletC2, tsum_mul_left]\n calc \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\u2016\n = \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I)\u2016 := by\n rw [\u2190 h1, norm_neg]\n _ \u2264 (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := hle\n\n/-- **Kernel envelope** (D-074 \u00a77, the single-exponential form that\nreplaces the unit-step geometric series): \u2016(1+1/((\u03c3\u22121)+iU))\u207f\u2016 \u2264\ne^{(4/9)\u03c3} for \u03c3 \u2265 3 and n \u2264 (4/9)U\u00b2. From the (6.11) core\n\u20161+1/s\u2016 \u2264 1+(re s+1)/normSq s \u2264 1+\u03c3/U\u00b2. -/\ntheorem one_add_div_pow_norm_le_exp_four_ninths {n : \u2115} {\u03c3 U : \u211d}\n (h\u03c3 : 3 \u2264 \u03c3) (hU : 0 < U) (hnU : (n : \u211d) \u2264 (4 / 9) * U ^ 2) :\n \u2016(1 + 1/(((\u03c3 : \u211d) - 1) + U * Complex.I))\u2016 ^ n \u2264 Real.exp ((4 / 9) * \u03c3) := by\n have hs0 : (((\u03c3 : \u211d) - 1) + U * Complex.I) \u2260 0 := by\n intro h\n have him : (((\u03c3 : \u211d) - 1) + U * Complex.I).im = 0 := by rw [h]; simp\n simp at him\n linarith\n have hle := norm_one_add_one_div_le hs0\n have hre : (((\u03c3 : \u211d) - 1) + U * Complex.I).re + 1 = \u03c3 := by\n simp [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.I_im,\n Complex.sub_re, Complex.one_re]\n have hns : Complex.normSq (((\u03c3 : \u211d) - 1) + U * Complex.I) = (\u03c3 - 1) ^ 2 + U ^ 2 := by\n simp [Complex.normSq_apply]\n ring\n have hstep : ((((\u03c3 : \u211d) - 1) + U * Complex.I).re + 1) /\n Complex.normSq (((\u03c3 : \u211d) - 1) + U * Complex.I) \u2264 \u03c3 / U ^ 2 := by\n rw [hre, hns]\n have hU2 : (0 : \u211d) < U ^ 2 := by positivity\n have h1 : (0 : \u211d) \u2264 (\u03c3 - 1) ^ 2 := sq_nonneg _\n exact (div_le_div_iff_of_pos_left (by linarith : (0 : \u211d) < \u03c3)\n (by positivity) hU2).mpr (by linarith [h1])\n have hstep1 : 1 + ((((\u03c3 : \u211d) - 1) + U * Complex.I).re + 1) /\n Complex.normSq (((\u03c3 : \u211d) - 1) + U * Complex.I) \u2264 1 + \u03c3 / U ^ 2 := by\n linarith [hstep]\n have hpow1 : \u2016(1 + 1/(((\u03c3 : \u211d) - 1) + U * Complex.I))\u2016 ^ n \u2264 (1 + \u03c3 / U ^ 2) ^ n :=\n pow_le_pow_left\u2080 (norm_nonneg _) (hle.trans hstep1) n\n have hpow2 : (1 + \u03c3 / U ^ 2) ^ n \u2264 Real.exp ((n : \u211d) * (\u03c3 / U ^ 2)) := by\n calc (1 + \u03c3 / U ^ 2) ^ n \u2264 (Real.exp (\u03c3 / U ^ 2)) ^ n := by\n have h := Real.add_one_le_exp (\u03c3 / U ^ 2)\n exact pow_le_pow_left\u2080 (by positivity) (by linarith) n\n _ = Real.exp ((n : \u211d) * (\u03c3 / U ^ 2)) := (Real.exp_nat_mul (\u03c3 / U ^ 2) n).symm\n have hexp : (n : \u211d) * (\u03c3 / U ^ 2) \u2264 (4 / 9) * \u03c3 := by\n have hU2 : (0 : \u211d) < U ^ 2 := by positivity\n have h\u03c30 : (0 : \u211d) < \u03c3 := by linarith\n have h1 : (n : \u211d) / U ^ 2 \u2264 4 / 9 := by\n have h2 := div_le_div_of_nonneg_right hnU hU2.le\n rwa [mul_div_cancel_right\u2080 _ hU2.ne'] at h2\n rw [show (n : \u211d) * (\u03c3 / U ^ 2) = ((n : \u211d) / U ^ 2) * \u03c3 by ring]\n exact mul_le_mul_of_nonneg_right h1 h\u03c30.le\n calc \u2016(1 + 1/(((\u03c3 : \u211d) - 1) + U * Complex.I))\u2016 ^ n\n \u2264 Real.exp ((n : \u211d) * (\u03c3 / U ^ 2)) := hpow1.trans hpow2\n _ \u2264 Real.exp ((4 / 9) * \u03c3) := Real.exp_le_exp.mpr hexp\n\n/-! ## 5. The product envelope and the integral assembly (DL-3c/d) -/\n\n/-- **The product envelope** (D-074 \u00a77): for \u03c3 \u2265 3, 0 < t, n \u2264 (4/9)t\u00b2,\n\u2016Jlag n (\u03c3+it)\u2016 \u2264 (4\u00b7e^{\u2212c\u03c3} + 4\u00b72^{\u2212\u03c3})\u00b7dirichletC2, c = log 2 \u2212 4/9.\nThe kernel envelope (e^{(4/9)\u03c3}) times the zeta halving (2^{\u2212(\u03c3\u22122)})\ncollapses to the single exponential 4\u00b7e^{\u2212c\u03c3}: no geometric series. -/\ntheorem Jlag_dogleg_norm_le_envelope {n : \u2115} {\u03c3 t : \u211d}\n (h\u03c3 : 3 \u2264 \u03c3) (ht : 0 < t) (hnU : (n : \u211d) \u2264 (4 / 9) * t ^ 2) :\n \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) := by\n have hw1 : ((\u03c3 : \u211d) + t * Complex.I) - 1 = ((\u03c3 : \u211d) - 1) + t * Complex.I := by\n ring\n have hs1ne : (((\u03c3 : \u211d) - 1) + t * Complex.I) \u2260 0 := by\n intro h\n have him : (((\u03c3 : \u211d) - 1) + t * Complex.I).im = 0 := by rw [h]; simp\n simp at him\n linarith\n have hwid : ((\u03c3 : \u211d) + t * Complex.I) / (((\u03c3 : \u211d) + t * Complex.I) - 1) =\n 1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I) := by\n rw [hw1]\n field_simp [hs1ne]\n ring\n have hker := one_add_div_pow_norm_le_exp_four_ninths h\u03c3 ht hnU\n have hzeta := norm_logDeriv_zeta_le_two_pow (show (2 : \u211d) \u2264 \u03c3 by linarith) (t := t)\n have hk : \u2016kFun n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264 Real.exp ((4 / 9) * \u03c3) + 1 := by\n unfold kFun\n rw [hwid]\n calc \u2016(1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I)) ^ n - 1\u2016\n \u2264 \u2016(1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I)) ^ n\u2016 + \u2016(1 : \u2102)\u2016 := norm_sub_le _ _\n _ = \u2016(1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I))\u2016 ^ n + 1 := by rw [norm_pow, norm_one]\n _ \u2264 Real.exp ((4 / 9) * \u03c3) + 1 := by linarith [hker]\n have hprod : \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264\n (Real.exp ((4 / 9) * \u03c3) + 1) * ((2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2) := by\n have hn3 : \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016 =\n \u2016kFun n ((\u03c3 : \u211d) + t * Complex.I)\u2016 *\n \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\u2016 := by\n show \u2016kFun n ((\u03c3 : \u211d) + t * Complex.I) *\n (-logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I))\u2016 = _\n rw [norm_mul, norm_neg]\n rw [hn3]\n exact mul_le_mul hk hzeta (norm_nonneg _)\n (add_nonneg (Real.exp_nonneg _) (by norm_num))\n have h2b : (2 : \u211d) ^ (-\u03c3) = Real.exp (-(Real.log 2) * \u03c3) := by\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n congr 1\n ring\n have h22 : (2 : \u211d) ^ (2 : \u211d) = 4 := by\n rw [Real.rpow_two]\n norm_num\n have h2a : (2 : \u211d) ^ (-(\u03c3 - 2)) = 4 * (2 : \u211d) ^ (-\u03c3) := by\n rw [show (-(\u03c3 - 2)) = -\u03c3 + 2 by ring, Real.rpow_add (by norm_num : (0 : \u211d) < 2), h2b, h22]\n ring\n have hexp_alg : Real.exp ((4 / 9) * \u03c3) * Real.exp (-(Real.log 2) * \u03c3) =\n Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) := by\n rw [\u2190 Real.exp_add]\n congr 1\n ring\n have halg : (Real.exp ((4 / 9) * \u03c3) + 1) * ((2 : \u211d) ^ (-(\u03c3 - 2))) =\n 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3) := by\n rw [h2a, h2b]\n calc (Real.exp ((4 / 9) * \u03c3) + 1) * (4 * Real.exp (-(Real.log 2) * \u03c3))\n = 4 * (Real.exp ((4 / 9) * \u03c3) * Real.exp (-(Real.log 2) * \u03c3)) +\n 4 * Real.exp (-(Real.log 2) * \u03c3) := by ring\n _ = 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * Real.exp (-(Real.log 2) * \u03c3) := by\n rw [hexp_alg]\n calc \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016\n \u2264 (Real.exp ((4 / 9) * \u03c3) + 1) * ((2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2) := hprod\n _ = dirichletC2 * ((Real.exp ((4 / 9) * \u03c3) + 1) * (2 : \u211d) ^ (-(\u03c3 - 2))) := by ring\n _ = dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) := by\n rw [halg]\n\n/-- The FTC-bound \u222b\u2083^XR e^{\u2212c\u03c3}d\u03c3 \u2264 e^{\u22123c}/c for c > 0. -/\nlemma integral_exp_neg_le {c XR : \u211d} (hc : 0 < c) (_hXR : 3 \u2264 XR) :\n \u222b \u03c3 in (3 : \u211d)..XR, Real.exp (-c * \u03c3) \u2264 Real.exp (-3 * c) / c := by\n have hF : \u2200 x \u2208 Set.uIcc (3 : \u211d) XR,\n HasDerivAt (fun \u03c3 => -Real.exp (-c * \u03c3) / c) (Real.exp (-c * x)) x := by\n intro x hx\n have h2 : HasDerivAt (fun \u03c3 => Real.exp (-c * \u03c3)) (Real.exp (-c * x) * (-c)) x := by\n have h := ((hasDerivAt_id' x).const_mul (-c)).exp\n rwa [mul_one] at h\n have h3 := h2.const_mul (-1 / c)\n rw [show (-1 / c) * (Real.exp (-c * x) * (-c)) = Real.exp (-c * x) by\n field_simp [hc.ne']] at h3\n have h5 : (fun \u03c3 => (-1 / c) * Real.exp (-c * \u03c3)) =\n (fun \u03c3 => -Real.exp (-c * \u03c3) / c) := by\n funext \u03c3\n field_simp [hc.ne']\n rw [h5] at h3\n exact h3\n have hint : IntervalIntegrable (fun \u03c3 => Real.exp (-c * \u03c3)) MeasureTheory.volume 3 XR := by\n apply ContinuousOn.intervalIntegrable\n exact (Real.continuous_exp.comp (continuous_const.mul continuous_id)).continuousOn\n have e : Real.exp (-c * 3) = Real.exp (-3 * c) := by\n congr 1\n ring\n rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF hint, e]\n have h1 : (0 : \u211d) \u2264 Real.exp (-c * XR) := (Real.exp_pos _).le\n have h2 : (0 : \u211d) < c := hc\n rw [show -Real.exp (-c * XR) / c - -Real.exp (-3 * c) / c =\n (Real.exp (-3 * c) - Real.exp (-c * XR)) / c by\n field_simp [h2.ne']\n ring]\n rw [show Real.exp (-3 * c) / c = (Real.exp (-3 * c) - (0 : \u211d)) / c by ring]\n exact div_le_div_of_nonneg_right (by linarith [h1]) h2.le\n\n/-- The certified exponent-decay constant c = log 2 \u2212 4/9 \u2265 0.2487. -/\nlemma cdog_gt : (0.2487 : \u211d) \u2264 Real.log 2 - 4 / 9 := by\n have h := Real.log_two_gt_d9\n norm_num at h\n linarith\n\n/-- The certified e^{4/3} \u2264 3.8073 (the (1\u2212x/16)\u207b\u00b9\u2076 route + d9 e). -/\nlemma exp_four_thirds_le : Real.exp (4 / 3) \u2264 (3.8073 : \u211d) := by\n have he13 : Real.exp (1 / 3) \u2264 (1.4006 : \u211d) :=\n le_trans (exp_le_inv_sub_sixteen (by norm_num) (by norm_num)) (by norm_num)\n rw [show (4 / 3 : \u211d) = 1 + 1 / 3 by norm_num, Real.exp_add]\n calc Real.exp 1 * Real.exp (1 / 3) \u2264 2.7182818286 * 1.4006 :=\n mul_le_mul Real.exp_one_lt_d9.le he13 (Real.exp_pos _).le (by norm_num)\n _ \u2264 3.8073 := by norm_num\n\n/-- **THE HIGH-HORIZONTAL BOUND** (D-074 \u00a77 capstone; D-073 Cor 4.2's\nR(1.5) in single-exponential-envelope form): for 0 < U, n \u2264 (4/9)U\u00b2\n(i.e. U \u2265 (3/2)\u221an) and 3 \u2264 XR, \u2016HIntegral (Jlag n) 3 XR U\u2016 \u2264 8.38\n(charged; the true value \u2264 1.2 at n = 10\u2075, E-034 V5). Charged with\ndirichletC2 \u2264 1 \u2014 the tight R(1.5) = 5.94 would need a dirichletC2 \u2264\n0.57 build (documented as the tightening path; the fold is\nindifferent: \u0394 \u2248 +2.4 per side). -/\ntheorem hIntegral_Jlag_dogleg_norm_le {n : \u2115} {U XR : \u211d}\n (hU : 0 < U) (hnU : (n : \u211d) \u2264 (4 / 9) * U ^ 2) (hXR : 3 \u2264 XR) :\n \u2016HIntegral (Jlag n) 3 XR U\u2016 \u2264 (8.38 : \u211d) := by\n have hc : (0 : \u211d) < Real.log 2 - 4 / 9 := by\n have h := cdog_gt\n linarith\n have hpt : \u2200 \u03c3 \u2208 Set.Ioc 3 XR,\n \u2016Jlag n ((\u03c3 : \u211d) + U * Complex.I)\u2016 \u2264\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) :=\n fun \u03c3 h\u03c3 => Jlag_dogleg_norm_le_envelope (by linarith [h\u03c3.1]) hU hnU\n have hcontA : Continuous (fun \u03c3 : \u211d => Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) :=\n Real.continuous_exp.comp (continuous_const.mul continuous_id)\n have hcontB : Continuous (fun \u03c3 : \u211d => (2 : \u211d) ^ (-\u03c3)) := by\n have h1 : Continuous (fun \u03c3 : \u211d => Real.exp (-(Real.log 2) * \u03c3)) :=\n Real.continuous_exp.comp (continuous_const.mul continuous_id)\n have h2 : (fun \u03c3 : \u211d => (2 : \u211d) ^ (-\u03c3)) =\n (fun \u03c3 : \u211d => Real.exp (-(Real.log 2) * \u03c3)) := by\n funext \u03c3\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n congr 1\n ring\n rw [h2]\n exact h1\n have hIntA : IntervalIntegrable (fun \u03c3 : \u211d => 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3))\n MeasureTheory.volume 3 XR := by\n apply ContinuousOn.intervalIntegrable\n exact (hcontA.const_mul 4).continuousOn\n have hIntB : IntervalIntegrable (fun \u03c3 : \u211d => 4 * (2 : \u211d) ^ (-\u03c3)) MeasureTheory.volume 3 XR := by\n apply ContinuousOn.intervalIntegrable\n exact (hcontB.const_mul 4).continuousOn\n have hIntG : IntervalIntegrable (fun \u03c3 : \u211d =>\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)))\n MeasureTheory.volume 3 XR := by\n have h := (hIntA.add hIntB).const_mul dirichletC2\n convert h using 1\n -- step 1: \u2016H\u2016 \u2264 \u222b envelope\n have h1 : \u2016HIntegral (Jlag n) 3 XR U\u2016 \u2264\n \u222b \u03c3 in (3 : \u211d)..XR,\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) := by\n unfold HIntegral\n apply intervalIntegral.norm_integral_le_of_norm_le hXR _ hIntG\n exact Filter.Eventually.of_forall (fun \u03c3 h\u03c3 => hpt \u03c3 h\u03c3)\n -- step 2: \u222b envelope = dirichletC2\u00b7(4\u222be^{\u2212c\u03c3} + 4\u222b2^{\u2212\u03c3})\n have h2 : \u222b \u03c3 in (3 : \u211d)..XR,\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3))\n = dirichletC2 * ((\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3))) := by\n rw [intervalIntegral.integral_const_mul, intervalIntegral.integral_add hIntA hIntB]\n -- step 3: the two integral bounds\n have h3a : \u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) \u2264\n Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) := by\n rw [intervalIntegral.integral_const_mul]\n have h := integral_exp_neg_le hc hXR\n have he3c : Real.exp (-3 * (Real.log 2 - 4 / 9)) = Real.exp (4 / 3) / 8 := by\n have h3l2 : Real.exp (-(3 * Real.log 2)) = 1 / 8 := by\n have e1 : -(3 * Real.log 2) = Real.log (1 / 8) := by\n rw [show ((1 : \u211d) / 8) = (2 ^ 3 : \u211d)\u207b\u00b9 by norm_num, Real.log_inv, Real.log_pow,\n Nat.cast_ofNat]\n rw [e1, Real.exp_log (by norm_num : (0 : \u211d) < 1 / 8)]\n calc Real.exp (-3 * (Real.log 2 - 4 / 9))\n = Real.exp ((4 / 3) + -(3 * Real.log 2)) := by congr 1; ring\n _ = Real.exp (4 / 3) * Real.exp (-(3 * Real.log 2)) := by rw [Real.exp_add]\n _ = Real.exp (4 / 3) / 8 := by rw [h3l2]; ring\n rw [he3c] at h\n calc 4 * \u222b \u03c3 in (3 : \u211d)..XR, Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)\n \u2264 4 * ((Real.exp (4 / 3) / 8) / (Real.log 2 - 4 / 9)) :=\n mul_le_mul_of_nonneg_left h (by norm_num)\n _ = Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) := by\n field_simp [hc.ne']\n ring\n have h3b : \u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3) \u2264 1 / (2 * Real.log 2) := by\n have h2e : (fun \u03c3 : \u211d => 4 * (2 : \u211d) ^ (-\u03c3)) =\n (fun \u03c3 : \u211d => 4 * Real.exp (-(Real.log 2) * \u03c3)) := by\n funext \u03c3\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n congr 1\n ring\n rw [h2e, intervalIntegral.integral_const_mul]\n have h := integral_exp_neg_le (Real.log_pos (by norm_num : (1 : \u211d) < 2)) hXR\n have h3l2 : Real.exp (-3 * Real.log 2) = 1 / 8 := by\n have e1 : (-3 * Real.log 2) = Real.log (1 / 8) := by\n rw [show ((1 : \u211d) / 8) = (2 ^ 3 : \u211d)\u207b\u00b9 by norm_num, Real.log_inv, Real.log_pow,\n Nat.cast_ofNat]\n ring\n rw [e1, Real.exp_log (by norm_num : (0 : \u211d) < 1 / 8)]\n rw [h3l2] at h\n calc 4 * \u222b \u03c3 in (3 : \u211d)..XR, Real.exp (-(Real.log 2) * \u03c3)\n \u2264 4 * ((1 / 8) / Real.log 2) := by\n apply mul_le_mul_of_nonneg_left h (by norm_num : (0 : \u211d) \u2264 4)\n _ = 1 / (2 * Real.log 2) := by\n field_simp [(Real.log_pos (by norm_num : (1 : \u211d) < 2)).ne']\n norm_num\n -- step 4: the certified assembly\n have hlog2 : (0.6931471803 : \u211d) \u2264 Real.log 2 := Real.log_two_gt_d9.le\n have h48 : Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) \u2264 (7.6545 : \u211d) := by\n have hcge := cdog_gt\n have h1 : Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) \u2264 3.8073 / (2 * 0.2487) :=\n div_le_div\u2080 (by norm_num : (0 : \u211d) \u2264 3.8073) exp_four_thirds_le\n (by norm_num : (0 : \u211d) < 2 * 0.2487) (by linarith [hcge])\n calc Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9))\n \u2264 3.8073 / (2 * 0.2487) := h1\n _ \u2264 7.6545 := by norm_num\n have h72 : 1 / (2 * Real.log 2) \u2264 (0.7214 : \u211d) := by\n have h1 : 1 / (2 * Real.log 2) \u2264 1 / (2 * 0.6931471803) :=\n one_div_le_one_div_of_le (by positivity) (by linarith)\n calc 1 / (2 * Real.log 2) \u2264 1 / (2 * 0.6931471803) := h1\n _ \u2264 0.7214 := by norm_num\n have hC2 : dirichletC2 \u2264 1 := dirichletC2_le_one\n have hsum : dirichletC2 * ((\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3))) \u2264 (8.38 : \u211d) := by\n have hA0 : (0 : \u211d) \u2264 Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) := by positivity\n have hB0 : (0 : \u211d) \u2264 1 / (2 * Real.log 2) := by positivity\n have hIntAB : (0 : \u211d) \u2264 (\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3)) := by\n apply add_nonneg\n \u00b7 apply intervalIntegral.integral_nonneg hXR\n intro \u03c3 h\u03c3\n positivity\n \u00b7 apply intervalIntegral.integral_nonneg hXR\n intro \u03c3 h\u03c3\n positivity\n calc dirichletC2 * ((\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3)))\n \u2264 1 * (Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) + 1 / (2 * Real.log 2)) := by\n apply mul_le_mul hC2 (add_le_add h3a h3b) hIntAB (by norm_num)\n _ = Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) + 1 / (2 * Real.log 2) := by ring\n _ \u2264 7.6545 + 0.7214 := add_le_add h48 h72\n _ \u2264 8.38 := by norm_num\n calc \u2016HIntegral (Jlag n) 3 XR U\u2016\n \u2264 \u222b \u03c3 in (3 : \u211d)..XR,\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) := h1\n _ = dirichletC2 * ((\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3))) := h2\n _ \u2264 8.38 := hsum\n\n/-! ## 6. The far-right vertical drop (DL-4, D-074 \u00a78) -/\n\n/-- **Kernel at the far-right vertical** (D-074 \u00a78): for n \u2265 16 and\nt\u00b2 \u2265 n, \u2016kFun n (2\u221an+it)\u2016 \u2264 e^{\u221an/2}+1. (The ii\u2032-4 lemma\n`kFun_shift_norm_le_exp_sqrt_half_add_one` takes re s \u2265 2\u221an, i.e.\nre w \u2265 2\u221an+1 \u2014 this is the w = 2\u221an variant: |1+1/(w\u22121)| \u2264\n1+(2\u221an)/((2\u221an\u22121)\u00b2+t\u00b2) \u2264 1+1/(2\u221an) since 4\u221an\u22121 \u2264 t\u00b2.) -/\ntheorem kFun_right_vertical_norm_le_exp_sqrt_half_add_one {n : \u2115} {t : \u211d}\n (hn : 16 \u2264 n) (ht : (n : \u211d) \u2264 t ^ 2) :\n \u2016kFun n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n Real.exp (Real.sqrt (n : \u211d) / 2) + 1 := by\n have hnR : (16 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have hsq : (0 : \u211d) < Real.sqrt (n : \u211d) := Real.sqrt_pos.mpr (by linarith)\n have hw1 : (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1 =\n ((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I := by\n push_cast\n ring\n have hs0 : ((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I \u2260 0 := by\n intro h\n have him : (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I).im = 0 := by\n rw [h]; simp\n simp at him\n have ht0 : t = 0 := by\n have h1 : (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I).im = t := by simp\n linarith\n rw [ht0] at ht\n nlinarith [hnR]\n have hle := norm_one_add_one_div_le hs0\n have hre : (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I).re + 1 =\n 2 * Real.sqrt (n : \u211d) := by\n simp [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.I_im,\n Complex.sub_re, Complex.one_re]\n have hns : Complex.normSq (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) =\n (2 * Real.sqrt (n : \u211d) - 1) ^ 2 + t ^ 2 := by\n simp [Complex.normSq_apply]\n ring\n have hkey : (2 * Real.sqrt (n : \u211d)) /\n Complex.normSq (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) \u2264\n 1 / (2 * Real.sqrt (n : \u211d)) := by\n rw [hns]\n have hsqn : Real.sqrt (n : \u211d) ^ 2 = (n : \u211d) := Real.sq_sqrt (by linarith)\n have h1 : 4 * Real.sqrt (n : \u211d) - 1 \u2264 t ^ 2 := by\n have h2 : 4 * Real.sqrt (n : \u211d) \u2264 (n : \u211d) + 1 := by\n have h3 : (4 * Real.sqrt (n : \u211d)) ^ 2 \u2264 ((n : \u211d) + 1) ^ 2 := by\n rw [show (4 * Real.sqrt (n : \u211d)) ^ 2 = 16 * (Real.sqrt (n : \u211d) ^ 2) by ring, hsqn]\n nlinarith [sq_nonneg ((n : \u211d) - 1), hnR]\n nlinarith [hsq]\n nlinarith [ht, h2]\n rw [show (2 * Real.sqrt (n : \u211d) - 1) ^ 2 + t ^ 2 =\n 4 * (Real.sqrt (n : \u211d) ^ 2) - 4 * Real.sqrt (n : \u211d) + 1 + t ^ 2 by ring, hsqn]\n have h4 : (0 : \u211d) < 4 * (n : \u211d) - 4 * Real.sqrt (n : \u211d) + 1 + t ^ 2 := by\n nlinarith [ht, sq_nonneg t, hsq]\n rw [div_le_iff\u2080 h4,\n show (1 / (2 * Real.sqrt (n : \u211d))) * (4 * (n : \u211d) - 4 * Real.sqrt (n : \u211d) + 1 + t ^ 2) =\n (4 * (n : \u211d) - 4 * Real.sqrt (n : \u211d) + 1 + t ^ 2) / (2 * Real.sqrt (n : \u211d)) by\n field_simp [hsq.ne'],\n le_div_iff\u2080 (by linarith [hsq] : (0 : \u211d) < 2 * Real.sqrt (n : \u211d))]\n nlinarith [h1, ht, hsq, hsqn]\n have hstep : 1 + ((((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I).re + 1) /\n Complex.normSq (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) \u2264\n 1 + 1 / (2 * Real.sqrt (n : \u211d)) := by\n rw [hre]\n have h1 : (2 * Real.sqrt (n : \u211d)) /\n Complex.normSq (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) \u2264\n 1 / (2 * Real.sqrt (n : \u211d)) := hkey\n linarith [h1]\n have hpow1 : \u2016(1 + 1/(((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I))\u2016 ^ n \u2264\n (1 + 1 / (2 * Real.sqrt (n : \u211d))) ^ n :=\n pow_le_pow_left\u2080 (norm_nonneg _) (hle.trans hstep) n\n have hpow2 : (1 + 1 / (2 * Real.sqrt (n : \u211d))) ^ n \u2264\n Real.exp (Real.sqrt (n : \u211d) / 2) := by\n calc (1 + 1 / (2 * Real.sqrt (n : \u211d))) ^ n\n \u2264 (Real.exp (1 / (2 * Real.sqrt (n : \u211d)))) ^ n := by\n have h := Real.add_one_le_exp (1 / (2 * Real.sqrt (n : \u211d)))\n exact pow_le_pow_left\u2080 (by positivity) (by linarith) n\n _ = Real.exp ((n : \u211d) * (1 / (2 * Real.sqrt (n : \u211d)))) :=\n (Real.exp_nat_mul _ n).symm\n _ = Real.exp (Real.sqrt (n : \u211d) / 2) := by\n congr 1\n have hsqn : Real.sqrt (n : \u211d) ^ 2 = (n : \u211d) := Real.sq_sqrt (by linarith)\n field_simp [hsq.ne']\n nlinarith [hsqn]\n have hwid : (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) /\n ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1) =\n 1 + 1 / (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) := by\n rw [hw1]\n field_simp [hs0]\n push_cast\n ring\n unfold kFun\n rw [hwid]\n calc \u2016(1 + 1/(((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I)) ^ n - 1\u2016\n \u2264 \u2016(1 + 1/(((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I)) ^ n\u2016 + \u2016(1 : \u2102)\u2016 :=\n norm_sub_le _ _\n _ = \u2016(1 + 1/(((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I))\u2016 ^ n + 1 := by\n rw [norm_pow, norm_one]\n _ \u2264 Real.exp (Real.sqrt (n : \u211d) / 2) + 1 := by linarith [hpow1.trans hpow2]\n\n/-- **The far-right vertical drop** (D-074 \u00a78 V_right): for n \u2265 16,\n0 < T \u2264 U, T\u00b2 \u2265 n, \u2016VIntegral (Jlag n) (2\u221an) T U\u2016 \u2264\n(U\u2212T)\u00b7(e^{\u221an/2}+1)\u00b72^{\u2212(2\u221an\u22122)}\u00b7dirichletC2 \u2014 exponentially tiny. -/\ntheorem vIntegral_Jlag_right_norm_le {n : \u2115} {T U : \u211d}\n (hn : 16 \u2264 n) (hT0 : (0 : \u211d) < T) (hTU : T \u2264 U) (hTn : (n : \u211d) \u2264 T ^ 2) :\n \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T U\u2016 \u2264\n (U - T) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2) := by\n have hnorm : \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T U\u2016 =\n \u2016\u222b t in T..U, Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 := by\n unfold VIntegral\n rw [norm_smul, Complex.norm_I, one_mul]\n rw [hnorm]\n have hbound : \u2200 t \u2208 Set.Ioc T U,\n \u2016Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2) := by\n intro t ht\n have htT : T < t := ht.1\n have ht2 : (n : \u211d) \u2264 t ^ 2 := by nlinarith [hTn, hT0]\n have h1 := kFun_right_vertical_norm_le_exp_sqrt_half_add_one hn ht2\n have h2 := norm_logDeriv_zeta_le_two_pow\n (show (2 : \u211d) \u2264 2 * Real.sqrt (n : \u211d) by\n have hnR : (16 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have hsq : (4 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n have h1 : Real.sqrt 16 \u2264 Real.sqrt (n : \u211d) :=\n Real.sqrt_le_sqrt (by exact_mod_cast hn : (16 : \u211d) \u2264 (n : \u211d))\n have h2 : Real.sqrt 16 = 4 := by\n rw [Real.sqrt_eq_iff_eq_sq (by norm_num) (by norm_num)]; norm_num\n rwa [h2] at h1\n linarith) (t := t)\n have hnorm3 : \u2016Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 =\n \u2016kFun n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 *\n \u2016logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 := by\n show \u2016kFun n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) *\n (-logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I))\u2016 = _\n rw [norm_mul, norm_neg]\n rw [hnorm3]\n exact mul_le_mul h1 h2 (norm_nonneg _)\n (add_nonneg (Real.exp_nonneg _) (by norm_num))\n have hint : \u2016\u222b t in T..U, Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n ((Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2)) * |U - T| := by\n apply intervalIntegral.norm_integral_le_of_norm_le_const\n intro t ht\n rw [Set.uIoc_of_le hTU] at ht\n exact hbound t ht\n have habs : |U - T| = U - T := abs_of_nonneg (by linarith)\n rw [habs] at hint\n calc \u2016\u222b t in T..U, Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016\n \u2264 ((Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2)) * (U - T) := hint\n _ = (U - T) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2) := by ring\n\n/-! ## 7. The total right-horizontal bound (DL-5, D-074 \u00a73.4) -/\n\n/-- **The upper right-tail bound** (D-074 \u00a73.4): for n \u2265 16 and\nT \u2208 [\u221an, \u221an+1] (the T_cut window), U = (3/2)\u00b7T,\n\u2016HIntegral (Jlag n) 3 (2\u221an) T\u2016 \u2264 ((\u221an+1)/2)\u00b7(e^{5/2}+1)/2 + 8.38 +\nIIMaj(n)/2 \u2014 the exact deformation identity +\nV_left/H_high/V_right piece bounds, all in the zero-free region. -/\ntheorem hIntegral_Jlag_low_le_dogLegUpper {n : \u2115} {T : \u211d}\n (hn : 16 \u2264 n) (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T\u2016 \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2 := by\n have hnR : (16 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have hsqrt4 : (4 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n have h1 : Real.sqrt 16 \u2264 Real.sqrt (n : \u211d) :=\n Real.sqrt_le_sqrt (by exact_mod_cast hn : (16 : \u211d) \u2264 (n : \u211d))\n have h2 : Real.sqrt 16 = 4 := by\n rw [Real.sqrt_eq_iff_eq_sq (by norm_num) (by norm_num)]; norm_num\n rwa [h2] at h1\n have hT0 : (0 : \u211d) < T := by linarith\n have hTn2 : (n : \u211d) \u2264 T ^ 2 := by\n have h1 : (Real.sqrt (n : \u211d)) ^ 2 \u2264 T ^ 2 :=\n pow_le_pow_left\u2080 (Real.sqrt_nonneg _) hTn 2\n rwa [Real.sq_sqrt (by linarith [Real.sqrt_nonneg (n : \u211d)])] at h1\n have hU0 : (0 : \u211d) < (3 / 2) * T := by linarith [hT0]\n have hTU : T \u2264 (3 / 2) * T := by linarith [hT0]\n have hnU : (n : \u211d) \u2264 (4 / 9) * ((3 / 2) * T) ^ 2 := by\n have h1 : (4 / 9) * ((3 / 2) * T) ^ 2 = T ^ 2 := by\n field_simp\n ring\n rw [h1]\n exact hTn2\n have hXR : (3 : \u211d) \u2264 2 * Real.sqrt (n : \u211d) := by linarith\n have hid := upper_right_horizontal_eq_dogleg hT0 hTU hXR (n := n)\n have hH := hIntegral_Jlag_dogleg_norm_le hU0 hnU hXR (n := n)\n have hVr := vIntegral_Jlag_right_norm_le hn hT0 hTU hTn2 (n := n)\n have hVl := vIntegral_Jlag_three_norm_le hT0 hTU hTn2 (n := n)\n have hlen : (3 / 2) * T - T \u2264 (Real.sqrt (n : \u211d) + 1) / 2 := by linarith [hTn1, hTn]\n have hlen0 : (0 : \u211d) \u2264 (3 / 2) * T - T := by linarith [hT0]\n -- V_left \u2264 ((\u221an+1)/2)\u00b7(e^{5/2}+1)/2\n have hVl2 : \u2016VIntegral (Jlag n) 3 T ((3 / 2) * T)\u2016 \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 := by\n refine hVl.trans ?_\n exact div_le_div_of_nonneg_right\n (mul_le_mul hlen (le_refl _) (by positivity) (by positivity)) (by norm_num)\n -- V_right \u2264 IIMaj/2\n have hVr2 : \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T)\u2016 \u2264\n IIMaj (n : \u211d) / 2 := by\n have hC2 : dirichletC2 \u2264 1 := dirichletC2_le_one\n have hK : (0 : \u211d) < (Real.sqrt (n : \u211d) + 1) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))) := by positivity\n have eA : ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * (1 : \u211d)) =\n (1 / 2) * ((Real.sqrt (n : \u211d) + 1) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)))) := by ring\n have eB : IIMaj (n : \u211d) / 2 =\n (0.57 : \u211d) * ((Real.sqrt (n : \u211d) + 1) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)))) := by\n unfold IIMaj\n ring\n calc \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T)\u2016\n \u2264 ((3 / 2) * T - T) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2) := hVr\n _ \u2264 ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * (1 : \u211d)) := by\n have h1 : ((3 / 2) * T - T) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) :=\n mul_le_mul hlen (le_refl _) (by positivity) (by positivity)\n have h2 : (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 \u2264\n (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * (1 : \u211d) :=\n mul_le_mul_of_nonneg_left hC2 (Real.rpow_nonneg (by norm_num) _)\n exact mul_le_mul h1 h2\n (mul_nonneg (Real.rpow_nonneg (by norm_num) _) dirichletC2_nonneg)\n (by positivity)\n _ \u2264 IIMaj (n : \u211d) / 2 := by\n rw [eA, eB]\n apply mul_le_mul_of_nonneg_right _ hK.le\n norm_num\n -- assemble via the exact identity\n rw [hid]\n calc \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) ((3 / 2) * T) -\n VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T) +\n VIntegral (Jlag n) 3 T ((3 / 2) * T)\u2016\n \u2264 \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) ((3 / 2) * T)\u2016 +\n \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T)\u2016 +\n \u2016VIntegral (Jlag n) 3 T ((3 / 2) * T)\u2016 := by\n have h1 := norm_add_le\n (HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) ((3 / 2) * T) -\n VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T))\n (VIntegral (Jlag n) 3 T ((3 / 2) * T))\n have h2 := norm_sub_le\n (HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) ((3 / 2) * T))\n (VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T))\n linarith [h1, h2]\n _ \u2264 (8.38 : \u211d) + IIMaj (n : \u211d) / 2 +\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 :=\n add_le_add (add_le_add hH hVr2) hVl2\n _ = ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2 := by ring\n\n/-- **The lower right-tail bound by conjugation** (D-074 \u00a73.4): the\nsame bound holds at im = \u2212T (the integrand is conjugation-symmetric,\nJlag_conj; the interval integral maps conj through). -/\ntheorem hIntegral_Jlag_low_neg_norm_le_dogLegUpper {n : \u2115} {T : \u211d}\n (hn : 16 \u2264 n) (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T)\u2016 \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2 := by\n have hT0 : (0 : \u211d) < T := by\n have hnR : (16 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have hsqrt4 : (4 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n have h1 : Real.sqrt 16 \u2264 Real.sqrt (n : \u211d) :=\n Real.sqrt_le_sqrt (by exact_mod_cast hn : (16 : \u211d) \u2264 (n : \u211d))\n have h2 : Real.sqrt 16 = 4 := by\n rw [Real.sqrt_eq_iff_eq_sq (by norm_num) (by norm_num)]; norm_num\n rwa [h2] at h1\n linarith\n have hconj : HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T) =\n (starRingEnd \u2102) (HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T) := by\n unfold HIntegral\n rw [\u2190 intervalIntegral.intervalIntegral_conj]\n apply intervalIntegral.integral_congr\n intro \u03c3 h\u03c3\n show Jlag n ((\u03c3 : \u211d) + (((-T : \u211d)) : \u2102) * Complex.I) =\n (starRingEnd \u2102) (Jlag n ((\u03c3 : \u211d) + T * Complex.I))\n have hid2 : ((\u03c3 : \u211d) + (((-T : \u211d)) : \u2102) * Complex.I) =\n (starRingEnd \u2102) ((\u03c3 : \u211d) + T * Complex.I) := by\n simp [Complex.ext_iff]\n rw [hid2]\n exact Jlag_conj n (by\n intro h\n have him : (((\u03c3 : \u211d) + T * Complex.I)).im = T := by simp\n rw [h] at him\n simp at him\n linarith [hT0])\n rw [hconj, Complex.norm_conj]\n exact hIntegral_Jlag_low_le_dogLegUpper hn hTn hTn1\n\n/-- **THE TWO-SIDE RIGHT-HORIZONTAL BOUND** (the DL capstone for\nii\u2032-6-c): both horizontal right tails bounded by the dog-leg charge. -/\ntheorem dogLegUpper_two_sides {n : \u2115} {T : \u211d}\n (hn : 16 \u2264 n) (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T\u2016 +\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T)\u2016 \u2264\n 2 * (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2) := by\n rw [show (2 : \u211d) * (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 +\n (8.38 : \u211d) + IIMaj (n : \u211d) / 2) =\n (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2) +\n (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2) by ring]\n exact add_le_add (hIntegral_Jlag_low_le_dogLegUpper hn hTn hTn1)\n (hIntegral_Jlag_low_neg_norm_le_dogLegUpper hn hTn hTn1)\n\n/-! ## 8. The direct Dirichlet bound at re = 2\u221an (DL-6) \u2014 the I_I /\nV_right zeta factor WITHOUT certifying dirichletC2 \u2264 0.57.\n\nThe I_I charge in IIMaj is 0.57\u00b72^{\u2212(2\u221an\u22122)} = 2.28\u00b72^{\u22122\u221an} (the\n(6.10)-route constant |\u03b6\u2032/\u03b6(2)|). Proving dirichletC2 \u2264 0.57 directly\nneeds a ~100-prime partial-sum build (the antitone-integral tail is too\nfat). AVOIDED here: the DIRECT Dirichlet-series bound at re = 2\u221an gives\nlog2-leading + a 3^{\u2212(2\u221an\u22122)} tail (dirichletC2 \u2264 1), which is \u2264\n0.57\u00b72^{\u2212(2\u221an\u22122)} for n \u2265 5 \u2014 the IIMaj charge exactly. -/\n\n/-- The Dirichlet majorant at s = 2\u221an (summable since 2\u221an > 1). -/\ndef dirichletCX (n : \u2115) : \u211d :=\n \u2211' m : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((2 * Real.sqrt (n : \u211d) : \u211d) : \u2102) m\u2016\n\n/-- \u2016\u03b6\u2032/\u03b6(2\u221an+it)\u2016 \u2264 dirichletCX n for n \u2265 1 (the LSeries majorant). -/\nlemma norm_logDeriv_zeta_le_dirichletCX {n : \u2115} (hn : 1 \u2264 n) {t : \u211d} :\n \u2016logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n dirichletCX n := by\n have h1 : (1 : \u211d) \u2264 Real.sqrt (n : \u211d) := Real.one_le_sqrt.mpr (by exact_mod_cast hn)\n have hs1 : (1 : \u211d) < (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I).re := by\n simp\n linarith\n have h1' : -logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\n = LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) := by\n have h2 := ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div hs1\n rw [logDeriv, Pi.div_apply, h2]\n ring\n have hsumX : Summable (fun m : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016) := by\n rw [summable_norm_iff]\n apply ArithmeticFunction.LSeriesSummable_vonMangoldt\n rw [show (((2 * Real.sqrt (n : \u211d) : \u211d) : \u2102).re) = 2 * Real.sqrt (n : \u211d) from by simp]\n linarith\n have hsumS : Summable (fun m : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) m\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt hs1\n have hle : \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264 dirichletCX n := by\n show \u2016\u2211' m : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) m\u2016 \u2264 _\n calc \u2016\u2211' m : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) m\u2016\n \u2264 \u2211' m : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) m\u2016 :=\n norm_tsum_le_tsum_norm hsumS\n _ \u2264 \u2211' m : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016 :=\n Summable.tsum_le_tsum (fun m => LSeries.norm_term_le_of_re_le_re _ (by\n simp) m) hsumS hsumX\n _ = dirichletCX n := rfl\n calc \u2016logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016\n = \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 := by\n rw [\u2190 h1', norm_neg]\n _ \u2264 dirichletCX n := hle\n\n/-- Termwise tail comparison: for m \u2265 3, the term at 2\u221an is \u2264\n3^{\u2212(2\u221an\u22122)}\u00b7(the term at 2). -/\nlemma norm_term_vonMangoldt_X_le_three_pow {n : \u2115} (hn : 1 \u2264 n) {m : \u2115} (hm : 3 \u2264 m) :\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016 \u2264\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016 := by\n have hXre : ((((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102).re) = 2 * Real.sqrt (n : \u211d) := by simp\n have h2re : ((2 : \u2102).re) = 2 := rfl\n simp only [LSeries.norm_term_eq, hXre, h2re]\n split\n \u00b7 norm_num\n \u00b7 next hm0 =>\n have h\u039b : \u2016(ArithmeticFunction.vonMangoldt m : \u2102)\u2016 =\n ArithmeticFunction.vonMangoldt m :=\n Complex.norm_of_nonneg ArithmeticFunction.vonMangoldt_nonneg\n rw [h\u039b]\n have hm2 : (0 : \u211d) < (m : \u211d) := by exact_mod_cast (by omega : 0 < m)\n have hsqrt2 : (2 : \u211d) \u2264 2 * Real.sqrt (n : \u211d) := by\n have h1 : (1 : \u211d) \u2264 Real.sqrt (n : \u211d) := Real.one_le_sqrt.mpr (by exact_mod_cast hn)\n linarith\n have hexp_nn : (0 : \u211d) \u2264 2 * Real.sqrt (n : \u211d) - 2 := by linarith\n have e : ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 * Real.sqrt (n : \u211d)) =\n (1 / (m : \u211d) ^ (2 * Real.sqrt (n : \u211d) - 2)) *\n (ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 : \u211d)) := by\n rw [show (2 * Real.sqrt (n : \u211d)) = (2 * Real.sqrt (n : \u211d) - 2) + 2 by ring,\n Real.rpow_add hm2]\n field_simp [hm2.ne']\n rw [e]\n have hle : (1 : \u211d) / (m : \u211d) ^ (2 * Real.sqrt (n : \u211d) - 2) \u2264\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) := by\n rw [Real.rpow_neg (by norm_num : (0 : \u211d) \u2264 3), \u2190 one_div]\n apply one_div_le_one_div_of_le (Real.rpow_pos_of_pos (by norm_num : (0 : \u211d) < 3) _)\n exact Real.rpow_le_rpow (by norm_num : (0 : \u211d) \u2264 3)\n (by exact_mod_cast hm : (3 : \u211d) \u2264 (m : \u211d)) hexp_nn\n have hpos : (0 : \u211d) \u2264 ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 : \u211d) :=\n div_nonneg ArithmeticFunction.vonMangoldt_nonneg\n (Real.rpow_nonneg (by positivity : (0 : \u211d) \u2264 (m : \u211d)) _)\n exact mul_le_mul_of_nonneg_right hle hpos\n\n/-- The split: dirichletCX n = log 2\u00b72^{\u22122\u221an} + (the m \u2265 3 tail). -/\nlemma dirichletCX_le {n : \u2115} (hn : 1 \u2264 n) :\n dirichletCX n \u2264 Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 := by\n have hsumX : Summable (fun m : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016) := by\n rw [summable_norm_iff]\n apply ArithmeticFunction.LSeriesSummable_vonMangoldt\n rw [show (((2 * Real.sqrt (n : \u211d) : \u211d) : \u2102).re) = 2 * Real.sqrt (n : \u211d) from by simp]\n have h1 : (1 : \u211d) \u2264 Real.sqrt (n : \u211d) := Real.one_le_sqrt.mpr (by exact_mod_cast hn)\n linarith\n have hsplit := tsum_split_finset hsumX {0, 1, 2}\n have hsum2 : Summable (fun m : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n have hfin : \u2211 i \u2208 ({0, 1, 2} : Finset \u2115),\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) i\u2016 =\n Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n have hXre : ((((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102).re) = 2 * Real.sqrt (n : \u211d) := by simp\n rw [Finset.sum_insert (by norm_num), Finset.sum_insert (by norm_num),\n Finset.sum_singleton]\n simp only [LSeries.norm_term_eq, hXre]\n rw [if_pos (by norm_num), if_neg (by norm_num : (1 : \u2115) \u2260 0),\n if_neg (by norm_num : (2 : \u2115) \u2260 0)]\n simp [ArithmeticFunction.vonMangoldt_apply_one]\n have h\u039b2 : \u2016(fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) 2\u2016 = Real.log 2 := by\n show \u2016(ArithmeticFunction.vonMangoldt 2 : \u2102)\u2016 = Real.log 2\n rw [ArithmeticFunction.vonMangoldt_apply_prime Nat.prime_two]\n exact Complex.norm_of_nonneg (by linarith [Real.log_two_gt_d9] : (0 : \u211d) \u2264 Real.log 2)\n rw [h\u039b2]\n have e : (2 : \u211d) ^ (2 * Real.sqrt (n : \u211d)) = (2 : \u211d) ^ (2 * Real.sqrt (n : \u211d)) := rfl\n rw [show Real.log 2 / (2 : \u211d) ^ (2 * Real.sqrt (n : \u211d)) =\n Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) by\n rw [Real.rpow_neg (by norm_num : (0 : \u211d) \u2264 2)]\n field_simp [Real.rpow_pos_of_pos (by norm_num : (0 : \u211d) < 2)\n (2 * Real.sqrt (n : \u211d)).ne']]\n ring\n have htail : (\u2211' i : \u2115, (if i \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) i\u2016)) \u2264\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 := by\n have hterm : \u2200 m : \u2115, (if m \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016) \u2264\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016 := by\n intro m\n by_cases hm : m \u2208 ({0, 1, 2} : Finset \u2115)\n \u00b7 rw [if_pos hm]\n positivity\n \u00b7 rw [if_neg hm]\n have hm3 : 3 \u2264 m := by\n have h : m \u2209 ({0, 1, 2} : Finset \u2115) := hm\n simp at h\n omega\n exact norm_term_vonMangoldt_X_le_three_pow hn hm3\n have hsum_tail : Summable (fun m : \u2115 => (if m \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016)) := by\n apply Summable.of_norm_bounded hsumX.norm\n intro m\n by_cases hm : m \u2208 ({0, 1, 2} : Finset \u2115) <;> simp [hm]\n calc (\u2211' m : \u2115, (if m \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016))\n \u2264 \u2211' m : \u2115, (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016 :=\n Summable.tsum_le_tsum hterm hsum_tail (hsum2.mul_left _)\n _ = (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 := by\n rw [dirichletC2, tsum_mul_left]\n calc dirichletCX n\n = \u2211 i \u2208 ({0, 1, 2} : Finset \u2115),\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) i\u2016 +\n \u2211' i : \u2115, (if i \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) i\u2016) := hsplit\n _ \u2264 Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 :=\n add_le_add (le_of_eq hfin) htail\n\n/-- log(9/4) \u2265 0.811 (via e^{0.811} \u2265 9/4 with e^{0.8} \u2265 2.2254). -/\nlemma log_nine_quarters_ge : (0.811 : \u211d) \u2264 Real.log (9 / 4) := by\n have h08 : (2.2254 : \u211d) \u2264 Real.exp 0.8 :=\n le_trans (by norm_num [Finset.sum_range_succ])\n (Real.sum_le_exp_of_nonneg (by norm_num : (0 : \u211d) \u2264 0.8) 7)\n have h011 : (1.01106 : \u211d) \u2264 Real.exp 0.011 :=\n le_trans (by norm_num [Finset.sum_range_succ])\n (Real.sum_le_exp_of_nonneg (by norm_num : (0 : \u211d) \u2264 0.011) 4)\n have hprod : (9 / 4 : \u211d) \u2264 Real.exp 0.8 * Real.exp 0.011 := by\n calc (9 / 4 : \u211d) = 2.25 := by norm_num\n _ \u2264 2.2254 * 1.01106 := by norm_num\n _ \u2264 Real.exp 0.8 * Real.exp 0.011 :=\n mul_le_mul h08 h011 (by norm_num : (0 : \u211d) \u2264 1.01106) (by norm_num : (0 : \u211d) \u2264 2.2254)\n have h1 : Real.exp 0.811 = Real.exp 0.8 * Real.exp 0.011 := by\n rw [show (0.811 : \u211d) = 0.8 + 0.011 by norm_num, Real.exp_add]\n rw [\u2190 h1] at hprod\n have h2 := Real.log_le_log (Real.exp_pos _) hprod\n rwa [Real.log_exp] at h2\n\n/-- log(4/9) \u2264 \u22120.811 (the negation). -/\nlemma log_four_ninths_le : Real.log (4 / 9) \u2264 (-0.811 : \u211d) := by\n have h1 := log_nine_quarters_ge\n have h2 : Real.log (4 / 9) = -Real.log (9 / 4) := by\n rw [show ((4 : \u211d) / 9) = (9 / 4)\u207b\u00b9 by norm_num, Real.log_inv]\n rw [h2]\n linarith [h1]\n\n/-- The certified (4/9)^{\u221an} \u2264 0.17631 for \u221an \u2265 2.236. -/\nlemma four_ninths_rpow_sqrt_le {x : \u211d} (hx : (2.236 : \u211d) \u2264 x) :\n (4 / 9 : \u211d) ^ x \u2264 (0.17631 : \u211d) := by\n have hlog49 := log_four_ninths_le\n have hlogneg : Real.log (4 / 9) < 0 :=\n Real.log_neg (by norm_num : (0 : \u211d) < 4 / 9) (by norm_num : (4 / 9 : \u211d) < 1)\n have h1 : (4 / 9 : \u211d) ^ x = Real.exp (Real.log (4 / 9) * x) :=\n Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 4 / 9) x\n rw [h1]\n have h2 : Real.log (4 / 9) * x \u2264 (-1.8135 : \u211d) := by\n have h2a : Real.log (4 / 9) * x \u2264 Real.log (4 / 9) * 2.236 :=\n mul_le_mul_of_nonpos_left hx hlogneg.le\n have h2b : Real.log (4 / 9) * 2.236 \u2264 (-0.811) * 2.236 :=\n mul_le_mul_of_nonneg_right hlog49 (by norm_num)\n calc Real.log (4 / 9) * x \u2264 Real.log (4 / 9) * 2.236 := h2a\n _ \u2264 (-0.811) * 2.236 := h2b\n _ \u2264 -1.8135 := by norm_num\n have h3 : Real.exp (-1.8135) \u2264 (0.17631 : \u211d) := by\n have hsum : (5.6721 : \u211d) \u2264 Real.exp 1.8135 := by\n calc (5.6721 : \u211d) \u2264 \u2211 i \u2208 Finset.range 7, (1.8135 : \u211d) ^ i / (Nat.factorial i : \u211d) := by\n norm_num [Finset.sum_range_succ]\n _ \u2264 Real.exp 1.8135 :=\n Real.sum_le_exp_of_nonneg (by norm_num : (0 : \u211d) \u2264 1.8135) 7\n rw [show Real.exp (-1.8135) = 1 / Real.exp 1.8135 by\n rw [Real.exp_neg, one_div]]\n calc 1 / Real.exp 1.8135 \u2264 1 / 5.6721 :=\n one_div_le_one_div_of_le (by positivity : (0 : \u211d) < 5.6721) hsum\n _ \u2264 0.17631 := by norm_num\n calc Real.exp (x * Real.log (4 / 9)) \u2264 Real.exp (-1.8135) :=\n Real.exp_le_exp.mpr h2\n _ \u2264 0.17631 := h3\n\n/-- The 3^{\u2212(2\u221an\u22122)} = 9\u00b7(4/9)^{\u221an}\u00b72^{\u22122\u221an} identity. -/\nlemma three_pow_neg_two_sqrt_sub_two_eq {n : \u2115} :\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) =\n 9 * ((4 / 9 : \u211d) ^ Real.sqrt (n : \u211d)) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n have h9 : Real.log 9 = 2 * Real.log 3 := by\n rw [show (9 : \u211d) = 3 ^ 2 by norm_num, Real.log_pow]\n push_cast\n ring\n have h2 : (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) =\n Real.exp (-(Real.log 2) * (2 * Real.sqrt (n : \u211d))) := by\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n congr 1\n ring\n have h3 : (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) =\n Real.exp (-(Real.log 3) * (2 * Real.sqrt (n : \u211d) - 2)) := by\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 3)]\n congr 1\n ring\n have h49 : (4 / 9 : \u211d) ^ Real.sqrt (n : \u211d) =\n Real.exp (Real.sqrt (n : \u211d) * Real.log (4 / 9)) :=\n Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 4 / 9) _\n rw [h2, h3, h49]\n have hlog49 : Real.log (4 / 9) = Real.log 2 - Real.log 9 + Real.log 2 := by\n rw [Real.log_div (by norm_num : (4 : \u211d) \u2260 0) (by norm_num : (9 : \u211d) \u2260 0)]\n ring\n rw [hlog49, h9]\n rw [\u2190 Real.exp_add, \u2190 Real.exp_add]\n congr 1\n rw [show (9 : \u211d) = Real.exp (Real.log 9) from (Real.exp_log (by norm_num : (0 : \u211d) < 9)).symm]\n rw [\u2190 Real.exp_add]\n congr 1\n ring\n\n/-- **The I_I zeta charge WITHOUT dirichletC2 \u2264 0.57** (DL-6 capstone):\nfor n \u2265 5, \u2016\u03b6\u2032/\u03b6(2\u221an+it)\u2016 \u2264 0.57\u00b72^{\u2212(2\u221an\u22122)} \u2014 the IIMaj zeta form,\nvia the direct log2-leading + 3^{\u2212(2\u221an\u22122)} tail (dirichletC2 \u2264 1) with\nthe certified (4/9)^{\u221an} \u2264 0.17631 bound at \u221an \u2265 \u221a5. -/\ntheorem norm_logDeriv_zeta_le_iimZeta {n : \u2115} (hn : 5 \u2264 n) {t : \u211d} :\n \u2016logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n (0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) := by\n have hn1 : 1 \u2264 n := by omega\n have h1 := norm_logDeriv_zeta_le_dirichletCX hn1 (t := t)\n have h2 := dirichletCX_le hn1\n have hC2 : dirichletC2 \u2264 1 := dirichletC2_le_one\n have hC2nn : (0 : \u211d) \u2264 dirichletC2 := dirichletC2_nonneg\n have hsqrt5 : (2.236 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n have hnR : (5 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have h1 : Real.sqrt 5 \u2264 Real.sqrt (n : \u211d) := Real.sqrt_le_sqrt hnR\n have h2 : (2.236 : \u211d) \u2264 Real.sqrt 5 := by\n rw [Real.le_sqrt (by norm_num) (by norm_num)]\n norm_num\n linarith\n have h49 := four_ninths_rpow_sqrt_le hsqrt5\n have h3 : dirichletCX n \u2264 Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) := by\n refine h2.trans ?_\n exact add_le_add_left (mul_le_mul_of_nonneg_left hC2\n (Real.rpow_nonneg (by norm_num) _)) _\n have h4 : (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) \u2264\n (1.58679 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n rw [three_pow_neg_two_sqrt_sub_two_eq]\n calc 9 * ((4 / 9 : \u211d) ^ Real.sqrt (n : \u211d)) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d)))\n \u2264 9 * (0.17631 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg (by norm_num) _)\n apply mul_le_mul_of_nonneg_left h49 (by norm_num)\n _ = (1.58679 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by ring\n have h5 : Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) \u2264\n (0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) := by\n have hlog2 : Real.log 2 \u2264 (0.6932 : \u211d) := Real.log_two_lt_d9.le\n have hpos2 : (0 : \u211d) < (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by positivity\n have e : (0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) =\n (2.28 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n rw [show (-(2 * Real.sqrt (n : \u211d) - 2)) = -2 * Real.sqrt (n : \u211d) + 2 by ring,\n Real.rpow_add (by norm_num : (0 : \u211d) < 2), Real.rpow_two]\n ring\n rw [e]\n calc Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))\n \u2264 0.6932 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (1.58679 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n apply add_le_add _ h4\n exact mul_le_mul_of_nonneg_right hlog2 hpos2.le\n _ = (0.6932 + 1.58679 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by ring\n _ \u2264 (2.28 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n apply mul_le_mul_of_nonneg_right _ hpos2.le\n norm_num\n exact h1.trans (h3.trans h5)\n\nend NguyenRH1\n\nend\n",
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"oldStart": 1227,
"oldLines": 7,
"newStart": 1227,
"newLines": 7,
"lines": [
" calc (9 / 4 : \u211d) = 2.25 := by norm_num",
" _ \u2264 2.2254 * 1.01106 := by norm_num",
" _ \u2264 Real.exp 0.8 * Real.exp 0.011 :=",
"- mul_le_mul h08 h011 (by norm_num : (0 : \u211d) \u2264 1.01106) (by norm_num : (0 : \u211d) \u2264 2.2254)",
"+ mul_le_mul h08 h011 (by norm_num : (0 : \u211d) \u2264 1.01106) (Real.exp_pos _).le",
" have h1 : Real.exp 0.811 = Real.exp 0.8 * Real.exp 0.011 := by",
" rw [show (0.811 : \u211d) = 0.8 + 0.011 by norm_num, Real.exp_add]",
" rw [\u2190 h1] at hprod"
]
}
],
"userModified": false,
"replaceAll": false
}
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"old_string": " rw [h1] at hprod\n have h2 := Real.log_le_log (Real.exp_pos _) hprod\n rwa [Real.log_exp] at h2",
"new_string": " rw [\u2190 h1] at hprod\n have h2 := Real.log_le_log (by norm_num : (0 : \u211d) < 9 / 4) hprod\n rwa [Real.log_exp] at h2"
}
tool: Edit ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"old_string": " calc Real.exp (x * Real.log (4 / 9)) \u2264 Real.exp (-1.8135) :=\n Real.exp_le_exp.mpr h2\n _ \u2264 0.17631 := h3",
"new_string": " calc Real.exp (Real.log (4 / 9) * x) \u2264 Real.exp (-1.8135) :=\n Real.exp_le_exp.mpr h2\n _ \u2264 0.17631 := h3"
}
result 2026-08-26 09:29:27Z → 2026-08-26 09:29:27Z
{
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"oldString": " calc Real.exp (x * Real.log (4 / 9)) \u2264 Real.exp (-1.8135) :=\n Real.exp_le_exp.mpr h2\n _ \u2264 0.17631 := h3",
"newString": " calc Real.exp (Real.log (4 / 9) * x) \u2264 Real.exp (-1.8135) :=\n Real.exp_le_exp.mpr h2\n _ \u2264 0.17631 := h3",
"originalFile": "/-\nNguyenRH1/LiDogLeg.lean \u2014 ii\u2032-6-DL-1 (D-074, session 040): the zero-free\ndog-leg deformation of the Lagarias horizontal right tail.\n\nTHE CONTENT (D-074 \u00a73.1\u20133.2): on the dog-leg rectangle [3, X_R]\u00d7[T, U]\n(T > 0), the Lagarias integrand Jlag n s = (s/(s\u22121))\u207f\u22121 \u00b7 (\u2212\u03b6\u2032/\u03b6(s)) is\nholomorphic UNCONDITIONALLY (s \u2260 1 since im > 0; \u03b6 s \u2260 0 for re s \u2265 3 > 1\nby Mathlib's Euler-product nonvanishing `riemannZeta_ne_zero_of_one_lt_re`\n\u2014 NO perimeter hypothesis, no zero-avoidance at U). Cauchy's theorem\n(`HolomorphicOn.vanishesOnRectangle`) then gives the EXACT deformation\nidentity\n H(3\u2192X_R, T) = H(3\u2192X_R, U) \u2212 V(X_R, T\u2192U) + V(3, T\u2192U)\n(the upper right tail = the high horizontal \u2212 the right drop + the left\nlift), plus the Jlag conjugation identity for the lower-side transfer.\n\nThis is the repair of Finding F-D (D-072): the broken low-path envelope\nis ABANDONED (not bounded) \u2014 the magnitude bound comes later from the\nthree dog-leg pieces, all in the unconditionally zero-free region.\n-/\nimport Mathlib\nimport NguyenRH1.LiSides\nimport NguyenRH1.LiTightR\nimport NguyenRH1.LiBasePoints\n\nopen Zeta23 Complex Filter Topology\nopen scoped Nat\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. Dog-leg analyticity (D-074 \u00a73.1; no perimeter) -/\n\n/-- The dog-leg corners (same real-part/imaginary-part form as rectZ/rectW). -/\ndef dogZ (T : \u211d) : \u2102 := ((3 : \u211d) : \u2102) + (T : \u211d) * Complex.I\ndef dogW (XR U : \u211d) : \u2102 := ((XR : \u211d) : \u2102) + (U : \u211d) * Complex.I\n\nlemma dogZ_re (T : \u211d) : (dogZ T).re = 3 := by simp [dogZ]\nlemma dogZ_im (T : \u211d) : (dogZ T).im = T := by simp [dogZ]\nlemma dogW_re (XR U : \u211d) : (dogW XR U).re = XR := by simp [dogW]\nlemma dogW_im (XR U : \u211d) : (dogW XR U).im = U := by simp [dogW]\n\n/-- Jlag is differentiable at every point with 3 \u2264 re, 0 < im:\n\u03b6 is zero-free for 1 < re (Mathlib `riemannZeta_ne_zero_of_one_lt_re`,\nUNCONDITIONAL \u2014 the Euler-product region) and s \u2260 1 (im \u2260 0).\nNO perimeter hypothesis. -/\ntheorem Jlag_differentiableAt_dogleg {n : \u2115} {s : \u2102}\n (hre : 3 \u2264 s.re) (him : 0 < s.im) :\n DifferentiableAt \u2102 (Jlag n) s := by\n apply Jlag_differentiableAt\n \u00b7 rintro rfl\n simp at him\n \u00b7 exact riemannZeta_ne_zero_of_one_lt_re (by linarith)\n\n/-- Jlag is holomorphic on the whole dog-leg rectangle\n[3, X_R]\u00d7[T, U] (T > 0). -/\ntheorem holomorphicOn_Jlag_dogleg {n : \u2115} {T U XR : \u211d}\n (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n HolomorphicOn (Jlag n) (Rectangle (dogZ T) (dogW XR U)) := by\n intro s hs\n rw [Rectangle, Complex.mem_reProdIm, dogZ_re, dogW_re, dogZ_im, dogW_im] at hs\n rw [Set.uIcc_of_le hXR, Set.uIcc_of_le hTU] at hs\n exact (Jlag_differentiableAt_dogleg hs.1.1 (by linarith [hs.2.1])).differentiableWithinAt\n\n/-- **The dog-leg rectangle integral vanishes** (Cauchy; the region is\nunconditionally zero-free \u2014 the residue sum is EMPTY, no perimeter). -/\ntheorem rectangleIntegral_Jlag_dogleg_eq_zero {n : \u2115} {T U XR : \u211d}\n (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n RectangleIntegral (Jlag n) (dogZ T) (dogW XR U) = 0 :=\n (holomorphicOn_Jlag_dogleg hT hTU hXR).vanishesOnRectangle subset_rfl\n\n/-- The (2\u03c0i)\u207b\u00b9-normalized form (API symmetry with rectInt'_Jlag). -/\ntheorem rectangleIntegral'_Jlag_dogleg_eq_zero {n : \u2115} {T U XR : \u211d}\n (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n RectangleIntegral' (Jlag n) (dogZ T) (dogW XR U) = 0 := by\n rw [RectangleIntegral', rectangleIntegral_Jlag_dogleg_eq_zero hT hTU hXR,\n smul_zero]\n\n/-- **THE EXACT DOG-LEG DEFORMATION IDENTITY** (D-074 \u00a73.2 (\u2605), the\ncapstone for ii\u2032-6-c): the low right-horizontal equals the high\nhorizontal minus the right drop plus the left lift. Orientations: all\nH-integrals left-to-right, all V-integrals upward (the main contour's\ntop side is their reversal, handled at the fold site). -/\ntheorem upper_right_horizontal_eq_dogleg {n : \u2115} {T U XR : \u211d}\n (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n HIntegral (Jlag n) 3 XR T =\n HIntegral (Jlag n) 3 XR U - VIntegral (Jlag n) XR T U +\n VIntegral (Jlag n) 3 T U := by\n have h0 := rectangleIntegral_Jlag_dogleg_eq_zero (n := n) hT hTU hXR\n unfold RectangleIntegral at h0\n rw [dogZ_re, dogZ_im, dogW_re, dogW_im] at h0\n linear_combination h0\n\n/-! ## 2. The conjugation identity (for the lower-side transfer, DL-5) -/\n\n/-- kFun is conjugation-symmetric (real-coefficient rational function). -/\ntheorem kFun_conj (n : \u2115) (s : \u2102) :\n kFun n ((starRingEnd \u2102) s) = (starRingEnd \u2102) (kFun n s) := by\n unfold kFun\n simp [map_sub, map_pow, map_one]\n\n/-- **Jlag is conjugation-symmetric** (from `riemannZeta_conj` +\n`iteratedDeriv_riemannZeta_conj`): Jlag n (s\u0304) = conj (Jlag n s).\nNeeded for the lower-horizontal transfer (DL-5). -/\ntheorem Jlag_conj (n : \u2115) {s : \u2102} (hs : s \u2260 1) :\n Jlag n ((starRingEnd \u2102) s) = (starRingEnd \u2102) (Jlag n s) := by\n have h1 : deriv riemannZeta ((starRingEnd \u2102) s) =\n (starRingEnd \u2102) (deriv riemannZeta s) := by\n have h := iteratedDeriv_riemannZeta_conj 1 hs\n rwa [iteratedDeriv_one] at h\n unfold Jlag\n rw [kFun_conj, _root_.riemannZeta_conj, h1]\n simp only [map_mul, map_neg, map_div\u2080]\n\n/-! ## 3. The left-lift bound (DL-2, D-074 \u00a73.3 V_left) -/\n\n/-- **Kernel on the lift**: \u2016kFun n (3+it)\u2016 \u2264 e^{5/2}+1 for n \u2264 t\u00b2\n(D-074 \u00a76: |(3+it)/(2+it)|\u00b2 = (9+t\u00b2)/(4+t\u00b2) = 1 + 5/(4+t\u00b2), whose n-th\npower is \u2264 e^{5n/(2(4+t\u00b2))} \u2264 e^{5/2}). -/\ntheorem kFun_three_add_it_norm_le_exp_five_halves_add_one {n : \u2115} {t : \u211d}\n (hT : (n : \u211d) \u2264 t ^ 2) :\n \u2016kFun n ((3 : \u211d) + t * Complex.I)\u2016 \u2264 Real.exp (5 / 2) + 1 := by\n rcases Nat.eq_zero_or_pos n with hn | hn\n \u00b7 subst hn\n simp only [kFun, pow_zero, sub_self, norm_zero]\n positivity\n \u00b7 have hnR : (1 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have ht2 : (0 : \u211d) < 4 + t ^ 2 := by positivity\n have hn4t2 : (n : \u211d) \u2264 4 + t ^ 2 := by linarith [hT]\n have hQpos : (0 : \u211d) \u2264 (9 + t ^ 2) / (4 + t ^ 2) := by positivity\n have hQle : (9 + t ^ 2) / (4 + t ^ 2) \u2264 Real.exp (5 / (n : \u211d)) := by\n have h1 : (9 + t ^ 2) / (4 + t ^ 2) = 1 + 5 / (4 + t ^ 2) := by\n field_simp [ht2.ne']\n ring\n rw [h1]\n have h2 : 5 / (4 + t ^ 2) \u2264 5 / (n : \u211d) :=\n (div_le_div_iff_of_pos_left (by norm_num) ht2 (by positivity)).mpr hn4t2\n have h3 := Real.add_one_le_exp (5 / (n : \u211d))\n linarith\n have hQn : ((9 + t ^ 2) / (4 + t ^ 2)) ^ n \u2264 Real.exp 5 := by\n calc ((9 + t ^ 2) / (4 + t ^ 2)) ^ n \u2264 (Real.exp (5 / (n : \u211d))) ^ n :=\n pow_le_pow_left\u2080 hQpos hQle n\n _ = Real.exp 5 := by\n rw [\u2190 Real.exp_nat_mul]\n congr 1\n have hn0 : (n : \u211d) \u2260 0 := by exact_mod_cast hn.ne'\n field_simp [hn0]\n have hns3 : Complex.normSq ((3 : \u211d) + t * Complex.I) = 9 + t ^ 2 := by\n simp [Complex.normSq_apply]\n ring\n have hns2 : Complex.normSq ((2 : \u211d) + t * Complex.I) = 4 + t ^ 2 := by\n simp [Complex.normSq_apply]\n ring\n have hnormA : \u2016((3 : \u211d) + t * Complex.I)\u2016 = Real.sqrt (9 + t ^ 2) := by\n have h1 : \u2016((3 : \u211d) + t * Complex.I)\u2016 =\n Real.sqrt (Complex.normSq ((3 : \u211d) + t * Complex.I)) := by\n rw [Complex.normSq_eq_norm_sq, Real.sqrt_sq (norm_nonneg _)]\n rw [h1, hns3]\n have hnormB : \u2016((2 : \u211d) + t * Complex.I)\u2016 = Real.sqrt (4 + t ^ 2) := by\n have h1 : \u2016((2 : \u211d) + t * Complex.I)\u2016 =\n Real.sqrt (Complex.normSq ((2 : \u211d) + t * Complex.I)) := by\n rw [Complex.normSq_eq_norm_sq, Real.sqrt_sq (norm_nonneg _)]\n rw [h1, hns2]\n have hnorm : \u2016((3 : \u211d) + t * Complex.I) / ((2 : \u211d) + t * Complex.I)\u2016 =\n Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2)) := by\n rw [norm_div, hnormA, hnormB,\n \u2190 Real.sqrt_div (by positivity : (0 : \u211d) \u2264 9 + t ^ 2)]\n have hs1 : ((3 : \u211d) + t * Complex.I) - 1 = (2 : \u211d) + t * Complex.I := by\n push_cast\n ring\n unfold kFun\n rw [hs1]\n calc \u2016(((3 : \u211d) + t * Complex.I) / ((2 : \u211d) + t * Complex.I)) ^ n - 1\u2016\n \u2264 \u2016(((3 : \u211d) + t * Complex.I) / ((2 : \u211d) + t * Complex.I)) ^ n\u2016 + \u2016(1 : \u2102)\u2016 :=\n norm_sub_le _ _\n _ = (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n + 1 := by\n rw [norm_pow, hnorm, norm_one]\n _ \u2264 Real.exp (5 / 2) + 1 := by\n have h4 : (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n \u2264 Real.exp (5 / 2) := by\n have hx0 : (0 : \u211d) \u2264 Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2)) :=\n Real.sqrt_nonneg _\n have hx : (0 : \u211d) \u2264 (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n :=\n pow_nonneg hx0 n\n have h5 : ((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2 \u2264 Real.exp 5 := by\n have e : ((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2 =\n ((9 + t ^ 2) / (4 + t ^ 2)) ^ n := by\n rw [\u2190 pow_mul, mul_comm n 2, pow_mul, Real.sq_sqrt hQpos]\n rw [e]\n exact hQn\n have h8 : (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n =\n Real.sqrt (((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2) := by\n rw [Real.sqrt_sq hx]\n rw [h8]\n calc Real.sqrt (((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2)\n \u2264 Real.sqrt (Real.exp 5) := Real.sqrt_le_sqrt h5\n _ = Real.exp (5 / 2) := (Real.exp_half 5).symm\n linarith [h4]\n\n/-- The Dirichlet majorant at s = 3 (summable since 3 > 1). -/\ndef dirichletC3 : \u211d :=\n \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016\n\n/-- Termwise 3-vs-2 comparison for the \u039b L-series: \u039b(n)/n\u00b3 \u2264 (1/2)\u00b7\u039b(n)/n\u00b2. -/\nlemma norm_term_vonMangoldt_three_le_half (n : \u2115) :\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016 \u2264\n (1 / 2) * \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016 := by\n have h3re : ((3 : \u2102).re) = 3 := rfl\n have h2re : ((2 : \u2102).re) = 2 := rfl\n simp only [LSeries.norm_term_eq, h3re, h2re]\n split\n \u00b7 norm_num\n \u00b7 next hn =>\n rcases eq_or_ne n 1 with rfl | hn1\n \u00b7 simp [ArithmeticFunction.vonMangoldt_apply_one]\n \u00b7 have hn2 : 2 \u2264 n := by omega\n have h\u039b : \u2016(ArithmeticFunction.vonMangoldt n : \u2102)\u2016 =\n ArithmeticFunction.vonMangoldt n :=\n Complex.norm_of_nonneg ArithmeticFunction.vonMangoldt_nonneg\n rw [h\u039b]\n have e : ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (3 : \u211d) =\n (1 / (n : \u211d)) * (ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d)) := by\n rw [show (3 : \u211d) = 2 + 1 by norm_num,\n Real.rpow_add (by positivity : (0 : \u211d) < (n : \u211d)), Real.rpow_one]\n field_simp [show (n : \u211d) \u2260 0 by positivity]\n have h12 : (1 : \u211d) / (n : \u211d) \u2264 1 / 2 :=\n one_div_le_one_div_of_le (by norm_num : (0 : \u211d) < 2)\n (by exact_mod_cast hn2 : (2 : \u211d) \u2264 (n : \u211d))\n have hpos : (0 : \u211d) \u2264 ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d) :=\n div_nonneg ArithmeticFunction.vonMangoldt_nonneg\n (Real.rpow_nonneg (by positivity : (0 : \u211d) \u2264 (n : \u211d)) _)\n calc ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (3 : \u211d)\n = (1 / (n : \u211d)) * (ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d)) := e\n _ \u2264 (1 / 2) * (ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d)) :=\n mul_le_mul_of_nonneg_right h12 hpos\n\n/-- dirichletC3 \u2264 1/2 (the termwise comparison + dirichletC2 \u2264 1) \u2014\nthe certified C_zeta3 = 0.5 of D-074 \u00a74. (A tighter 0.17 build is\npossible via a 6-prime partial sum + antitone-integral tail; the fold\nis indifferent to the constant, so the cheap certified 0.5 is charged.) -/\nlemma dirichletC3_le_half : dirichletC3 \u2264 1 / 2 := by\n have hsum3 : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n have hsum2 : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n calc dirichletC3\n = \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016 := rfl\n _ \u2264 \u2211' n : \u2115, (1 / 2) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016 :=\n Summable.tsum_le_tsum (fun n => norm_term_vonMangoldt_three_le_half n) hsum3\n (hsum2.mul_left (1 / 2))\n _ = (1 / 2) * dirichletC2 := by rw [dirichletC2, tsum_mul_left]\n _ \u2264 1 / 2 := by linarith [dirichletC2_le_one]\n\n/-- \u2016\u03b6\u2032/\u03b6 s\u2016 \u2264 dirichletC3 for re s \u2265 3 (the s = 3 clone of the\nDirichlet-series majorant). -/\nlemma norm_logDeriv_zeta_le_dirichletC3 {s : \u2102} (hs : 3 \u2264 s.re) :\n \u2016logDeriv riemannZeta s\u2016 \u2264 dirichletC3 := by\n have hs1 : (1 : \u211d) < s.re := by linarith\n have h1 : -logDeriv riemannZeta s\n = LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s := by\n have h2 := ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div hs1\n rw [logDeriv, Pi.div_apply, h2]\n ring\n have hsum3 : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n have hsumS : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt hs1\n have hle : \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s\u2016\n \u2264 dirichletC3 := by\n show \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016 \u2264 _\n calc \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016\n \u2264 \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016 :=\n norm_tsum_le_tsum_norm hsumS\n _ \u2264 \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016 :=\n Summable.tsum_le_tsum (fun n => LSeries.norm_term_le_of_re_le_re _ (by\n simp\n exact hs) n) hsumS hsum3\n _ = dirichletC3 := rfl\n calc \u2016logDeriv riemannZeta s\u2016\n = \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s\u2016 := by\n rw [\u2190 h1, norm_neg]\n _ \u2264 dirichletC3 := hle\n\n/-- \u2016\u03b6\u2032/\u03b6 s\u2016 \u2264 1/2 for re s \u2265 3 (the charged C_zeta3 = 0.5). -/\nlemma norm_logDeriv_zeta_le_half {s : \u2102} (hs : 3 \u2264 s.re) :\n \u2016logDeriv riemannZeta s\u2016 \u2264 1 / 2 :=\n (norm_logDeriv_zeta_le_dirichletC3 hs).trans dirichletC3_le_half\n\n/-- **The left-lift bound** (D-074 \u00a73.3 V_left): for 0 < T \u2264 U and\nn \u2264 T\u00b2, \u2016VIntegral (Jlag n) 3 T U\u2016 \u2264 (U \u2212 T)\u00b7(e^{5/2}+1)/2. The kernel\n\u2264 e^{5/2}+1 (the 1+5/(4+t\u00b2) computation), the zeta factor \u2264 1/2\n(certified, no perimeter). -/\ntheorem vIntegral_Jlag_three_norm_le {n : \u2115} {T U : \u211d}\n (hT0 : (0 : \u211d) < T) (hTU : T \u2264 U) (hTn : (n : \u211d) \u2264 T ^ 2) :\n \u2016VIntegral (Jlag n) 3 T U\u2016 \u2264 (U - T) * (Real.exp (5 / 2) + 1) / 2 := by\n have hnorm : \u2016VIntegral (Jlag n) 3 T U\u2016 =\n \u2016\u222b t in T..U, Jlag n ((3 : \u211d) + t * Complex.I)\u2016 := by\n unfold VIntegral\n rw [norm_smul, Complex.norm_I, one_mul]\n rw [hnorm]\n have hbound : \u2200 t \u2208 Set.Ioc T U,\n \u2016Jlag n ((3 : \u211d) + t * Complex.I)\u2016 \u2264 (Real.exp (5 / 2) + 1) * (1 / 2) := by\n intro t ht\n have htT : T < t := ht.1\n have ht2 : (n : \u211d) \u2264 t ^ 2 := by nlinarith [hTn, hT0]\n have hre : (3 : \u211d) \u2264 ((3 : \u211d) + t * Complex.I).re := by simp\n have h1 := kFun_three_add_it_norm_le_exp_five_halves_add_one ht2\n have h2 := norm_logDeriv_zeta_le_half (s := (3 : \u211d) + t * Complex.I) hre\n have hnorm3 : \u2016Jlag n ((3 : \u211d) + t * Complex.I)\u2016 =\n \u2016kFun n ((3 : \u211d) + t * Complex.I)\u2016 *\n \u2016logDeriv riemannZeta ((3 : \u211d) + t * Complex.I)\u2016 := by\n show \u2016kFun n ((3 : \u211d) + t * Complex.I) *\n (-logDeriv riemannZeta ((3 : \u211d) + t * Complex.I))\u2016 = _\n rw [norm_mul, norm_neg]\n rw [hnorm3]\n exact mul_le_mul h1 h2 (norm_nonneg _) (by positivity)\n have hint : \u2016\u222b t in T..U, Jlag n ((3 : \u211d) + t * Complex.I)\u2016 \u2264\n ((Real.exp (5 / 2) + 1) * (1 / 2)) * |U - T| := by\n apply intervalIntegral.norm_integral_le_of_norm_le_const\n intro t ht\n rw [Set.uIoc_of_le hTU] at ht\n exact hbound t ht\n have habs : |U - T| = U - T := abs_of_nonneg (by linarith)\n rw [habs] at hint\n calc \u2016\u222b t in T..U, Jlag n ((3 : \u211d) + t * Complex.I)\u2016\n \u2264 ((Real.exp (5 / 2) + 1) * (1 / 2)) * (U - T) := hint\n _ = (U - T) * (Real.exp (5 / 2) + 1) / 2 := by ring\n\n/-! ## 4. The high-horizontal stepping (DL-3, D-074 \u00a77 + D-073 \u00a74) -/\n\n/-- Termwise \u03c3-vs-2 comparison for the \u039b L-series (the (6.10) halving):\n\u039b(m)/m^\u03c3 \u2264 2^{\u2212(\u03c3\u22122)}\u00b7\u039b(m)/m\u00b2 for \u03c3 \u2265 2. -/\nlemma norm_term_vonMangoldt_le_two_pow {\u03c3 t : \u211d} (h\u03c3 : 2 \u2264 \u03c3) (m : \u2115) :\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) m\u2016 \u2264\n (2 : \u211d) ^ (-(\u03c3 - 2)) * \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016 := by\n have h\u03c3re : ((\u03c3 : \u211d) + t * Complex.I).re = \u03c3 := by simp\n have h2re : ((2 : \u2102).re) = 2 := rfl\n simp only [LSeries.norm_term_eq, h\u03c3re, h2re]\n split\n \u00b7 norm_num\n \u00b7 next hm =>\n have h\u039b : \u2016(ArithmeticFunction.vonMangoldt m : \u2102)\u2016 =\n ArithmeticFunction.vonMangoldt m :=\n Complex.norm_of_nonneg ArithmeticFunction.vonMangoldt_nonneg\n rw [h\u039b]\n rcases eq_or_ne m 1 with rfl | hm1\n \u00b7 simp [ArithmeticFunction.vonMangoldt_apply_one]\n \u00b7 have hm2 : 2 \u2264 m := by omega\n have e : ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ \u03c3 =\n (1 / (m : \u211d) ^ (\u03c3 - 2)) * (ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 : \u211d)) := by\n rw [show (\u03c3 : \u211d) = (\u03c3 - 2) + 2 by ring,\n Real.rpow_add (by positivity : (0 : \u211d) < (m : \u211d))]\n field_simp [show (m : \u211d) \u2260 0 by positivity]\n ring_nf\n rw [e]\n have hle : (1 : \u211d) / (m : \u211d) ^ (\u03c3 - 2) \u2264 (2 : \u211d) ^ (-(\u03c3 - 2)) := by\n rw [Real.rpow_neg (by norm_num : (0 : \u211d) \u2264 2), \u2190 one_div]\n apply one_div_le_one_div_of_le (Real.rpow_pos_of_pos (by norm_num : (0 : \u211d) < 2) _)\n exact Real.rpow_le_rpow (by norm_num : (0 : \u211d) \u2264 2)\n (by exact_mod_cast hm2 : (2 : \u211d) \u2264 (m : \u211d)) (by linarith : (0 : \u211d) \u2264 \u03c3 - 2)\n have hpos : (0 : \u211d) \u2264 ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 : \u211d) :=\n div_nonneg ArithmeticFunction.vonMangoldt_nonneg\n (Real.rpow_nonneg (by positivity : (0 : \u211d) \u2264 (m : \u211d)) _)\n exact mul_le_mul_of_nonneg_right hle hpos\n\n/-- \u2016\u03b6\u2032/\u03b6(\u03c3+it)\u2016 \u2264 2^{\u2212(\u03c3\u22122)}\u00b7dirichletC2 for \u03c3 \u2265 2 (the (6.10)\nhalving in LSeries form; dirichletC2 \u2264 1 makes the tail free). -/\nlemma norm_logDeriv_zeta_le_two_pow {\u03c3 t : \u211d} (h\u03c3 : 2 \u2264 \u03c3) :\n \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264\n (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := by\n have hs1 : (1 : \u211d) < ((\u03c3 : \u211d) + t * Complex.I).re := by simp; linarith\n have h1 : -logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\n = LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) ((\u03c3 : \u211d) + t * Complex.I) := 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 : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n have hsumS : Summable (fun n : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) n\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt hs1\n have hle : \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264 (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := by\n show \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) n\u2016 \u2264 _\n calc \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) n\u2016\n \u2264 \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I) n\u2016 :=\n norm_tsum_le_tsum_norm hsumS\n _ \u2264 \u2211' n : \u2115, (2 : \u211d) ^ (-(\u03c3 - 2)) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016 :=\n Summable.tsum_le_tsum (fun n => norm_term_vonMangoldt_le_two_pow h\u03c3 n) hsumS\n (hsum2.mul_left _)\n _ = (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := by rw [dirichletC2, tsum_mul_left]\n calc \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\u2016\n = \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((\u03c3 : \u211d) + t * Complex.I)\u2016 := by\n rw [\u2190 h1, norm_neg]\n _ \u2264 (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := hle\n\n/-- **Kernel envelope** (D-074 \u00a77, the single-exponential form that\nreplaces the unit-step geometric series): \u2016(1+1/((\u03c3\u22121)+iU))\u207f\u2016 \u2264\ne^{(4/9)\u03c3} for \u03c3 \u2265 3 and n \u2264 (4/9)U\u00b2. From the (6.11) core\n\u20161+1/s\u2016 \u2264 1+(re s+1)/normSq s \u2264 1+\u03c3/U\u00b2. -/\ntheorem one_add_div_pow_norm_le_exp_four_ninths {n : \u2115} {\u03c3 U : \u211d}\n (h\u03c3 : 3 \u2264 \u03c3) (hU : 0 < U) (hnU : (n : \u211d) \u2264 (4 / 9) * U ^ 2) :\n \u2016(1 + 1/(((\u03c3 : \u211d) - 1) + U * Complex.I))\u2016 ^ n \u2264 Real.exp ((4 / 9) * \u03c3) := by\n have hs0 : (((\u03c3 : \u211d) - 1) + U * Complex.I) \u2260 0 := by\n intro h\n have him : (((\u03c3 : \u211d) - 1) + U * Complex.I).im = 0 := by rw [h]; simp\n simp at him\n linarith\n have hle := norm_one_add_one_div_le hs0\n have hre : (((\u03c3 : \u211d) - 1) + U * Complex.I).re + 1 = \u03c3 := by\n simp [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.I_im,\n Complex.sub_re, Complex.one_re]\n have hns : Complex.normSq (((\u03c3 : \u211d) - 1) + U * Complex.I) = (\u03c3 - 1) ^ 2 + U ^ 2 := by\n simp [Complex.normSq_apply]\n ring\n have hstep : ((((\u03c3 : \u211d) - 1) + U * Complex.I).re + 1) /\n Complex.normSq (((\u03c3 : \u211d) - 1) + U * Complex.I) \u2264 \u03c3 / U ^ 2 := by\n rw [hre, hns]\n have hU2 : (0 : \u211d) < U ^ 2 := by positivity\n have h1 : (0 : \u211d) \u2264 (\u03c3 - 1) ^ 2 := sq_nonneg _\n exact (div_le_div_iff_of_pos_left (by linarith : (0 : \u211d) < \u03c3)\n (by positivity) hU2).mpr (by linarith [h1])\n have hstep1 : 1 + ((((\u03c3 : \u211d) - 1) + U * Complex.I).re + 1) /\n Complex.normSq (((\u03c3 : \u211d) - 1) + U * Complex.I) \u2264 1 + \u03c3 / U ^ 2 := by\n linarith [hstep]\n have hpow1 : \u2016(1 + 1/(((\u03c3 : \u211d) - 1) + U * Complex.I))\u2016 ^ n \u2264 (1 + \u03c3 / U ^ 2) ^ n :=\n pow_le_pow_left\u2080 (norm_nonneg _) (hle.trans hstep1) n\n have hpow2 : (1 + \u03c3 / U ^ 2) ^ n \u2264 Real.exp ((n : \u211d) * (\u03c3 / U ^ 2)) := by\n calc (1 + \u03c3 / U ^ 2) ^ n \u2264 (Real.exp (\u03c3 / U ^ 2)) ^ n := by\n have h := Real.add_one_le_exp (\u03c3 / U ^ 2)\n exact pow_le_pow_left\u2080 (by positivity) (by linarith) n\n _ = Real.exp ((n : \u211d) * (\u03c3 / U ^ 2)) := (Real.exp_nat_mul (\u03c3 / U ^ 2) n).symm\n have hexp : (n : \u211d) * (\u03c3 / U ^ 2) \u2264 (4 / 9) * \u03c3 := by\n have hU2 : (0 : \u211d) < U ^ 2 := by positivity\n have h\u03c30 : (0 : \u211d) < \u03c3 := by linarith\n have h1 : (n : \u211d) / U ^ 2 \u2264 4 / 9 := by\n have h2 := div_le_div_of_nonneg_right hnU hU2.le\n rwa [mul_div_cancel_right\u2080 _ hU2.ne'] at h2\n rw [show (n : \u211d) * (\u03c3 / U ^ 2) = ((n : \u211d) / U ^ 2) * \u03c3 by ring]\n exact mul_le_mul_of_nonneg_right h1 h\u03c30.le\n calc \u2016(1 + 1/(((\u03c3 : \u211d) - 1) + U * Complex.I))\u2016 ^ n\n \u2264 Real.exp ((n : \u211d) * (\u03c3 / U ^ 2)) := hpow1.trans hpow2\n _ \u2264 Real.exp ((4 / 9) * \u03c3) := Real.exp_le_exp.mpr hexp\n\n/-! ## 5. The product envelope and the integral assembly (DL-3c/d) -/\n\n/-- **The product envelope** (D-074 \u00a77): for \u03c3 \u2265 3, 0 < t, n \u2264 (4/9)t\u00b2,\n\u2016Jlag n (\u03c3+it)\u2016 \u2264 (4\u00b7e^{\u2212c\u03c3} + 4\u00b72^{\u2212\u03c3})\u00b7dirichletC2, c = log 2 \u2212 4/9.\nThe kernel envelope (e^{(4/9)\u03c3}) times the zeta halving (2^{\u2212(\u03c3\u22122)})\ncollapses to the single exponential 4\u00b7e^{\u2212c\u03c3}: no geometric series. -/\ntheorem Jlag_dogleg_norm_le_envelope {n : \u2115} {\u03c3 t : \u211d}\n (h\u03c3 : 3 \u2264 \u03c3) (ht : 0 < t) (hnU : (n : \u211d) \u2264 (4 / 9) * t ^ 2) :\n \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) := by\n have hw1 : ((\u03c3 : \u211d) + t * Complex.I) - 1 = ((\u03c3 : \u211d) - 1) + t * Complex.I := by\n ring\n have hs1ne : (((\u03c3 : \u211d) - 1) + t * Complex.I) \u2260 0 := by\n intro h\n have him : (((\u03c3 : \u211d) - 1) + t * Complex.I).im = 0 := by rw [h]; simp\n simp at him\n linarith\n have hwid : ((\u03c3 : \u211d) + t * Complex.I) / (((\u03c3 : \u211d) + t * Complex.I) - 1) =\n 1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I) := by\n rw [hw1]\n field_simp [hs1ne]\n ring\n have hker := one_add_div_pow_norm_le_exp_four_ninths h\u03c3 ht hnU\n have hzeta := norm_logDeriv_zeta_le_two_pow (show (2 : \u211d) \u2264 \u03c3 by linarith) (t := t)\n have hk : \u2016kFun n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264 Real.exp ((4 / 9) * \u03c3) + 1 := by\n unfold kFun\n rw [hwid]\n calc \u2016(1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I)) ^ n - 1\u2016\n \u2264 \u2016(1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I)) ^ n\u2016 + \u2016(1 : \u2102)\u2016 := norm_sub_le _ _\n _ = \u2016(1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I))\u2016 ^ n + 1 := by rw [norm_pow, norm_one]\n _ \u2264 Real.exp ((4 / 9) * \u03c3) + 1 := by linarith [hker]\n have hprod : \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264\n (Real.exp ((4 / 9) * \u03c3) + 1) * ((2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2) := by\n have hn3 : \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016 =\n \u2016kFun n ((\u03c3 : \u211d) + t * Complex.I)\u2016 *\n \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\u2016 := by\n show \u2016kFun n ((\u03c3 : \u211d) + t * Complex.I) *\n (-logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I))\u2016 = _\n rw [norm_mul, norm_neg]\n rw [hn3]\n exact mul_le_mul hk hzeta (norm_nonneg _)\n (add_nonneg (Real.exp_nonneg _) (by norm_num))\n have h2b : (2 : \u211d) ^ (-\u03c3) = Real.exp (-(Real.log 2) * \u03c3) := by\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n congr 1\n ring\n have h22 : (2 : \u211d) ^ (2 : \u211d) = 4 := by\n rw [Real.rpow_two]\n norm_num\n have h2a : (2 : \u211d) ^ (-(\u03c3 - 2)) = 4 * (2 : \u211d) ^ (-\u03c3) := by\n rw [show (-(\u03c3 - 2)) = -\u03c3 + 2 by ring, Real.rpow_add (by norm_num : (0 : \u211d) < 2), h2b, h22]\n ring\n have hexp_alg : Real.exp ((4 / 9) * \u03c3) * Real.exp (-(Real.log 2) * \u03c3) =\n Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) := by\n rw [\u2190 Real.exp_add]\n congr 1\n ring\n have halg : (Real.exp ((4 / 9) * \u03c3) + 1) * ((2 : \u211d) ^ (-(\u03c3 - 2))) =\n 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3) := by\n rw [h2a, h2b]\n calc (Real.exp ((4 / 9) * \u03c3) + 1) * (4 * Real.exp (-(Real.log 2) * \u03c3))\n = 4 * (Real.exp ((4 / 9) * \u03c3) * Real.exp (-(Real.log 2) * \u03c3)) +\n 4 * Real.exp (-(Real.log 2) * \u03c3) := by ring\n _ = 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * Real.exp (-(Real.log 2) * \u03c3) := by\n rw [hexp_alg]\n calc \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016\n \u2264 (Real.exp ((4 / 9) * \u03c3) + 1) * ((2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2) := hprod\n _ = dirichletC2 * ((Real.exp ((4 / 9) * \u03c3) + 1) * (2 : \u211d) ^ (-(\u03c3 - 2))) := by ring\n _ = dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) := by\n rw [halg]\n\n/-- The FTC-bound \u222b\u2083^XR e^{\u2212c\u03c3}d\u03c3 \u2264 e^{\u22123c}/c for c > 0. -/\nlemma integral_exp_neg_le {c XR : \u211d} (hc : 0 < c) (_hXR : 3 \u2264 XR) :\n \u222b \u03c3 in (3 : \u211d)..XR, Real.exp (-c * \u03c3) \u2264 Real.exp (-3 * c) / c := by\n have hF : \u2200 x \u2208 Set.uIcc (3 : \u211d) XR,\n HasDerivAt (fun \u03c3 => -Real.exp (-c * \u03c3) / c) (Real.exp (-c * x)) x := by\n intro x hx\n have h2 : HasDerivAt (fun \u03c3 => Real.exp (-c * \u03c3)) (Real.exp (-c * x) * (-c)) x := by\n have h := ((hasDerivAt_id' x).const_mul (-c)).exp\n rwa [mul_one] at h\n have h3 := h2.const_mul (-1 / c)\n rw [show (-1 / c) * (Real.exp (-c * x) * (-c)) = Real.exp (-c * x) by\n field_simp [hc.ne']] at h3\n have h5 : (fun \u03c3 => (-1 / c) * Real.exp (-c * \u03c3)) =\n (fun \u03c3 => -Real.exp (-c * \u03c3) / c) := by\n funext \u03c3\n field_simp [hc.ne']\n rw [h5] at h3\n exact h3\n have hint : IntervalIntegrable (fun \u03c3 => Real.exp (-c * \u03c3)) MeasureTheory.volume 3 XR := by\n apply ContinuousOn.intervalIntegrable\n exact (Real.continuous_exp.comp (continuous_const.mul continuous_id)).continuousOn\n have e : Real.exp (-c * 3) = Real.exp (-3 * c) := by\n congr 1\n ring\n rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF hint, e]\n have h1 : (0 : \u211d) \u2264 Real.exp (-c * XR) := (Real.exp_pos _).le\n have h2 : (0 : \u211d) < c := hc\n rw [show -Real.exp (-c * XR) / c - -Real.exp (-3 * c) / c =\n (Real.exp (-3 * c) - Real.exp (-c * XR)) / c by\n field_simp [h2.ne']\n ring]\n rw [show Real.exp (-3 * c) / c = (Real.exp (-3 * c) - (0 : \u211d)) / c by ring]\n exact div_le_div_of_nonneg_right (by linarith [h1]) h2.le\n\n/-- The certified exponent-decay constant c = log 2 \u2212 4/9 \u2265 0.2487. -/\nlemma cdog_gt : (0.2487 : \u211d) \u2264 Real.log 2 - 4 / 9 := by\n have h := Real.log_two_gt_d9\n norm_num at h\n linarith\n\n/-- The certified e^{4/3} \u2264 3.8073 (the (1\u2212x/16)\u207b\u00b9\u2076 route + d9 e). -/\nlemma exp_four_thirds_le : Real.exp (4 / 3) \u2264 (3.8073 : \u211d) := by\n have he13 : Real.exp (1 / 3) \u2264 (1.4006 : \u211d) :=\n le_trans (exp_le_inv_sub_sixteen (by norm_num) (by norm_num)) (by norm_num)\n rw [show (4 / 3 : \u211d) = 1 + 1 / 3 by norm_num, Real.exp_add]\n calc Real.exp 1 * Real.exp (1 / 3) \u2264 2.7182818286 * 1.4006 :=\n mul_le_mul Real.exp_one_lt_d9.le he13 (Real.exp_pos _).le (by norm_num)\n _ \u2264 3.8073 := by norm_num\n\n/-- **THE HIGH-HORIZONTAL BOUND** (D-074 \u00a77 capstone; D-073 Cor 4.2's\nR(1.5) in single-exponential-envelope form): for 0 < U, n \u2264 (4/9)U\u00b2\n(i.e. U \u2265 (3/2)\u221an) and 3 \u2264 XR, \u2016HIntegral (Jlag n) 3 XR U\u2016 \u2264 8.38\n(charged; the true value \u2264 1.2 at n = 10\u2075, E-034 V5). Charged with\ndirichletC2 \u2264 1 \u2014 the tight R(1.5) = 5.94 would need a dirichletC2 \u2264\n0.57 build (documented as the tightening path; the fold is\nindifferent: \u0394 \u2248 +2.4 per side). -/\ntheorem hIntegral_Jlag_dogleg_norm_le {n : \u2115} {U XR : \u211d}\n (hU : 0 < U) (hnU : (n : \u211d) \u2264 (4 / 9) * U ^ 2) (hXR : 3 \u2264 XR) :\n \u2016HIntegral (Jlag n) 3 XR U\u2016 \u2264 (8.38 : \u211d) := by\n have hc : (0 : \u211d) < Real.log 2 - 4 / 9 := by\n have h := cdog_gt\n linarith\n have hpt : \u2200 \u03c3 \u2208 Set.Ioc 3 XR,\n \u2016Jlag n ((\u03c3 : \u211d) + U * Complex.I)\u2016 \u2264\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) :=\n fun \u03c3 h\u03c3 => Jlag_dogleg_norm_le_envelope (by linarith [h\u03c3.1]) hU hnU\n have hcontA : Continuous (fun \u03c3 : \u211d => Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) :=\n Real.continuous_exp.comp (continuous_const.mul continuous_id)\n have hcontB : Continuous (fun \u03c3 : \u211d => (2 : \u211d) ^ (-\u03c3)) := by\n have h1 : Continuous (fun \u03c3 : \u211d => Real.exp (-(Real.log 2) * \u03c3)) :=\n Real.continuous_exp.comp (continuous_const.mul continuous_id)\n have h2 : (fun \u03c3 : \u211d => (2 : \u211d) ^ (-\u03c3)) =\n (fun \u03c3 : \u211d => Real.exp (-(Real.log 2) * \u03c3)) := by\n funext \u03c3\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n congr 1\n ring\n rw [h2]\n exact h1\n have hIntA : IntervalIntegrable (fun \u03c3 : \u211d => 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3))\n MeasureTheory.volume 3 XR := by\n apply ContinuousOn.intervalIntegrable\n exact (hcontA.const_mul 4).continuousOn\n have hIntB : IntervalIntegrable (fun \u03c3 : \u211d => 4 * (2 : \u211d) ^ (-\u03c3)) MeasureTheory.volume 3 XR := by\n apply ContinuousOn.intervalIntegrable\n exact (hcontB.const_mul 4).continuousOn\n have hIntG : IntervalIntegrable (fun \u03c3 : \u211d =>\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)))\n MeasureTheory.volume 3 XR := by\n have h := (hIntA.add hIntB).const_mul dirichletC2\n convert h using 1\n -- step 1: \u2016H\u2016 \u2264 \u222b envelope\n have h1 : \u2016HIntegral (Jlag n) 3 XR U\u2016 \u2264\n \u222b \u03c3 in (3 : \u211d)..XR,\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) := by\n unfold HIntegral\n apply intervalIntegral.norm_integral_le_of_norm_le hXR _ hIntG\n exact Filter.Eventually.of_forall (fun \u03c3 h\u03c3 => hpt \u03c3 h\u03c3)\n -- step 2: \u222b envelope = dirichletC2\u00b7(4\u222be^{\u2212c\u03c3} + 4\u222b2^{\u2212\u03c3})\n have h2 : \u222b \u03c3 in (3 : \u211d)..XR,\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3))\n = dirichletC2 * ((\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3))) := by\n rw [intervalIntegral.integral_const_mul, intervalIntegral.integral_add hIntA hIntB]\n -- step 3: the two integral bounds\n have h3a : \u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) \u2264\n Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) := by\n rw [intervalIntegral.integral_const_mul]\n have h := integral_exp_neg_le hc hXR\n have he3c : Real.exp (-3 * (Real.log 2 - 4 / 9)) = Real.exp (4 / 3) / 8 := by\n have h3l2 : Real.exp (-(3 * Real.log 2)) = 1 / 8 := by\n have e1 : -(3 * Real.log 2) = Real.log (1 / 8) := by\n rw [show ((1 : \u211d) / 8) = (2 ^ 3 : \u211d)\u207b\u00b9 by norm_num, Real.log_inv, Real.log_pow,\n Nat.cast_ofNat]\n rw [e1, Real.exp_log (by norm_num : (0 : \u211d) < 1 / 8)]\n calc Real.exp (-3 * (Real.log 2 - 4 / 9))\n = Real.exp ((4 / 3) + -(3 * Real.log 2)) := by congr 1; ring\n _ = Real.exp (4 / 3) * Real.exp (-(3 * Real.log 2)) := by rw [Real.exp_add]\n _ = Real.exp (4 / 3) / 8 := by rw [h3l2]; ring\n rw [he3c] at h\n calc 4 * \u222b \u03c3 in (3 : \u211d)..XR, Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)\n \u2264 4 * ((Real.exp (4 / 3) / 8) / (Real.log 2 - 4 / 9)) :=\n mul_le_mul_of_nonneg_left h (by norm_num)\n _ = Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) := by\n field_simp [hc.ne']\n ring\n have h3b : \u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3) \u2264 1 / (2 * Real.log 2) := by\n have h2e : (fun \u03c3 : \u211d => 4 * (2 : \u211d) ^ (-\u03c3)) =\n (fun \u03c3 : \u211d => 4 * Real.exp (-(Real.log 2) * \u03c3)) := by\n funext \u03c3\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n congr 1\n ring\n rw [h2e, intervalIntegral.integral_const_mul]\n have h := integral_exp_neg_le (Real.log_pos (by norm_num : (1 : \u211d) < 2)) hXR\n have h3l2 : Real.exp (-3 * Real.log 2) = 1 / 8 := by\n have e1 : (-3 * Real.log 2) = Real.log (1 / 8) := by\n rw [show ((1 : \u211d) / 8) = (2 ^ 3 : \u211d)\u207b\u00b9 by norm_num, Real.log_inv, Real.log_pow,\n Nat.cast_ofNat]\n ring\n rw [e1, Real.exp_log (by norm_num : (0 : \u211d) < 1 / 8)]\n rw [h3l2] at h\n calc 4 * \u222b \u03c3 in (3 : \u211d)..XR, Real.exp (-(Real.log 2) * \u03c3)\n \u2264 4 * ((1 / 8) / Real.log 2) := by\n apply mul_le_mul_of_nonneg_left h (by norm_num : (0 : \u211d) \u2264 4)\n _ = 1 / (2 * Real.log 2) := by\n field_simp [(Real.log_pos (by norm_num : (1 : \u211d) < 2)).ne']\n norm_num\n -- step 4: the certified assembly\n have hlog2 : (0.6931471803 : \u211d) \u2264 Real.log 2 := Real.log_two_gt_d9.le\n have h48 : Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) \u2264 (7.6545 : \u211d) := by\n have hcge := cdog_gt\n have h1 : Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) \u2264 3.8073 / (2 * 0.2487) :=\n div_le_div\u2080 (by norm_num : (0 : \u211d) \u2264 3.8073) exp_four_thirds_le\n (by norm_num : (0 : \u211d) < 2 * 0.2487) (by linarith [hcge])\n calc Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9))\n \u2264 3.8073 / (2 * 0.2487) := h1\n _ \u2264 7.6545 := by norm_num\n have h72 : 1 / (2 * Real.log 2) \u2264 (0.7214 : \u211d) := by\n have h1 : 1 / (2 * Real.log 2) \u2264 1 / (2 * 0.6931471803) :=\n one_div_le_one_div_of_le (by positivity) (by linarith)\n calc 1 / (2 * Real.log 2) \u2264 1 / (2 * 0.6931471803) := h1\n _ \u2264 0.7214 := by norm_num\n have hC2 : dirichletC2 \u2264 1 := dirichletC2_le_one\n have hsum : dirichletC2 * ((\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3))) \u2264 (8.38 : \u211d) := by\n have hA0 : (0 : \u211d) \u2264 Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) := by positivity\n have hB0 : (0 : \u211d) \u2264 1 / (2 * Real.log 2) := by positivity\n have hIntAB : (0 : \u211d) \u2264 (\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3)) := by\n apply add_nonneg\n \u00b7 apply intervalIntegral.integral_nonneg hXR\n intro \u03c3 h\u03c3\n positivity\n \u00b7 apply intervalIntegral.integral_nonneg hXR\n intro \u03c3 h\u03c3\n positivity\n calc dirichletC2 * ((\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3)))\n \u2264 1 * (Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) + 1 / (2 * Real.log 2)) := by\n apply mul_le_mul hC2 (add_le_add h3a h3b) hIntAB (by norm_num)\n _ = Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) + 1 / (2 * Real.log 2) := by ring\n _ \u2264 7.6545 + 0.7214 := add_le_add h48 h72\n _ \u2264 8.38 := by norm_num\n calc \u2016HIntegral (Jlag n) 3 XR U\u2016\n \u2264 \u222b \u03c3 in (3 : \u211d)..XR,\n dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) := h1\n _ = dirichletC2 * ((\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) +\n (\u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3))) := h2\n _ \u2264 8.38 := hsum\n\n/-! ## 6. The far-right vertical drop (DL-4, D-074 \u00a78) -/\n\n/-- **Kernel at the far-right vertical** (D-074 \u00a78): for n \u2265 16 and\nt\u00b2 \u2265 n, \u2016kFun n (2\u221an+it)\u2016 \u2264 e^{\u221an/2}+1. (The ii\u2032-4 lemma\n`kFun_shift_norm_le_exp_sqrt_half_add_one` takes re s \u2265 2\u221an, i.e.\nre w \u2265 2\u221an+1 \u2014 this is the w = 2\u221an variant: |1+1/(w\u22121)| \u2264\n1+(2\u221an)/((2\u221an\u22121)\u00b2+t\u00b2) \u2264 1+1/(2\u221an) since 4\u221an\u22121 \u2264 t\u00b2.) -/\ntheorem kFun_right_vertical_norm_le_exp_sqrt_half_add_one {n : \u2115} {t : \u211d}\n (hn : 16 \u2264 n) (ht : (n : \u211d) \u2264 t ^ 2) :\n \u2016kFun n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n Real.exp (Real.sqrt (n : \u211d) / 2) + 1 := by\n have hnR : (16 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have hsq : (0 : \u211d) < Real.sqrt (n : \u211d) := Real.sqrt_pos.mpr (by linarith)\n have hw1 : (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1 =\n ((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I := by\n push_cast\n ring\n have hs0 : ((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I \u2260 0 := by\n intro h\n have him : (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I).im = 0 := by\n rw [h]; simp\n simp at him\n have ht0 : t = 0 := by\n have h1 : (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I).im = t := by simp\n linarith\n rw [ht0] at ht\n nlinarith [hnR]\n have hle := norm_one_add_one_div_le hs0\n have hre : (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I).re + 1 =\n 2 * Real.sqrt (n : \u211d) := by\n simp [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.I_im,\n Complex.sub_re, Complex.one_re]\n have hns : Complex.normSq (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) =\n (2 * Real.sqrt (n : \u211d) - 1) ^ 2 + t ^ 2 := by\n simp [Complex.normSq_apply]\n ring\n have hkey : (2 * Real.sqrt (n : \u211d)) /\n Complex.normSq (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) \u2264\n 1 / (2 * Real.sqrt (n : \u211d)) := by\n rw [hns]\n have hsqn : Real.sqrt (n : \u211d) ^ 2 = (n : \u211d) := Real.sq_sqrt (by linarith)\n have h1 : 4 * Real.sqrt (n : \u211d) - 1 \u2264 t ^ 2 := by\n have h2 : 4 * Real.sqrt (n : \u211d) \u2264 (n : \u211d) + 1 := by\n have h3 : (4 * Real.sqrt (n : \u211d)) ^ 2 \u2264 ((n : \u211d) + 1) ^ 2 := by\n rw [show (4 * Real.sqrt (n : \u211d)) ^ 2 = 16 * (Real.sqrt (n : \u211d) ^ 2) by ring, hsqn]\n nlinarith [sq_nonneg ((n : \u211d) - 1), hnR]\n nlinarith [hsq]\n nlinarith [ht, h2]\n rw [show (2 * Real.sqrt (n : \u211d) - 1) ^ 2 + t ^ 2 =\n 4 * (Real.sqrt (n : \u211d) ^ 2) - 4 * Real.sqrt (n : \u211d) + 1 + t ^ 2 by ring, hsqn]\n have h4 : (0 : \u211d) < 4 * (n : \u211d) - 4 * Real.sqrt (n : \u211d) + 1 + t ^ 2 := by\n nlinarith [ht, sq_nonneg t, hsq]\n rw [div_le_iff\u2080 h4,\n show (1 / (2 * Real.sqrt (n : \u211d))) * (4 * (n : \u211d) - 4 * Real.sqrt (n : \u211d) + 1 + t ^ 2) =\n (4 * (n : \u211d) - 4 * Real.sqrt (n : \u211d) + 1 + t ^ 2) / (2 * Real.sqrt (n : \u211d)) by\n field_simp [hsq.ne'],\n le_div_iff\u2080 (by linarith [hsq] : (0 : \u211d) < 2 * Real.sqrt (n : \u211d))]\n nlinarith [h1, ht, hsq, hsqn]\n have hstep : 1 + ((((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I).re + 1) /\n Complex.normSq (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) \u2264\n 1 + 1 / (2 * Real.sqrt (n : \u211d)) := by\n rw [hre]\n have h1 : (2 * Real.sqrt (n : \u211d)) /\n Complex.normSq (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) \u2264\n 1 / (2 * Real.sqrt (n : \u211d)) := hkey\n linarith [h1]\n have hpow1 : \u2016(1 + 1/(((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I))\u2016 ^ n \u2264\n (1 + 1 / (2 * Real.sqrt (n : \u211d))) ^ n :=\n pow_le_pow_left\u2080 (norm_nonneg _) (hle.trans hstep) n\n have hpow2 : (1 + 1 / (2 * Real.sqrt (n : \u211d))) ^ n \u2264\n Real.exp (Real.sqrt (n : \u211d) / 2) := by\n calc (1 + 1 / (2 * Real.sqrt (n : \u211d))) ^ n\n \u2264 (Real.exp (1 / (2 * Real.sqrt (n : \u211d)))) ^ n := by\n have h := Real.add_one_le_exp (1 / (2 * Real.sqrt (n : \u211d)))\n exact pow_le_pow_left\u2080 (by positivity) (by linarith) n\n _ = Real.exp ((n : \u211d) * (1 / (2 * Real.sqrt (n : \u211d)))) :=\n (Real.exp_nat_mul _ n).symm\n _ = Real.exp (Real.sqrt (n : \u211d) / 2) := by\n congr 1\n have hsqn : Real.sqrt (n : \u211d) ^ 2 = (n : \u211d) := Real.sq_sqrt (by linarith)\n field_simp [hsq.ne']\n nlinarith [hsqn]\n have hwid : (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) /\n ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1) =\n 1 + 1 / (((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I) := by\n rw [hw1]\n field_simp [hs0]\n push_cast\n ring\n unfold kFun\n rw [hwid]\n calc \u2016(1 + 1/(((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I)) ^ n - 1\u2016\n \u2264 \u2016(1 + 1/(((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I)) ^ n\u2016 + \u2016(1 : \u2102)\u2016 :=\n norm_sub_le _ _\n _ = \u2016(1 + 1/(((2 * Real.sqrt (n : \u211d) - 1) : \u211d) + t * Complex.I))\u2016 ^ n + 1 := by\n rw [norm_pow, norm_one]\n _ \u2264 Real.exp (Real.sqrt (n : \u211d) / 2) + 1 := by linarith [hpow1.trans hpow2]\n\n/-- **The far-right vertical drop** (D-074 \u00a78 V_right): for n \u2265 16,\n0 < T \u2264 U, T\u00b2 \u2265 n, \u2016VIntegral (Jlag n) (2\u221an) T U\u2016 \u2264\n(U\u2212T)\u00b7(e^{\u221an/2}+1)\u00b72^{\u2212(2\u221an\u22122)}\u00b7dirichletC2 \u2014 exponentially tiny. -/\ntheorem vIntegral_Jlag_right_norm_le {n : \u2115} {T U : \u211d}\n (hn : 16 \u2264 n) (hT0 : (0 : \u211d) < T) (hTU : T \u2264 U) (hTn : (n : \u211d) \u2264 T ^ 2) :\n \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T U\u2016 \u2264\n (U - T) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2) := by\n have hnorm : \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T U\u2016 =\n \u2016\u222b t in T..U, Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 := by\n unfold VIntegral\n rw [norm_smul, Complex.norm_I, one_mul]\n rw [hnorm]\n have hbound : \u2200 t \u2208 Set.Ioc T U,\n \u2016Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2) := by\n intro t ht\n have htT : T < t := ht.1\n have ht2 : (n : \u211d) \u2264 t ^ 2 := by nlinarith [hTn, hT0]\n have h1 := kFun_right_vertical_norm_le_exp_sqrt_half_add_one hn ht2\n have h2 := norm_logDeriv_zeta_le_two_pow\n (show (2 : \u211d) \u2264 2 * Real.sqrt (n : \u211d) by\n have hnR : (16 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have hsq : (4 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n have h1 : Real.sqrt 16 \u2264 Real.sqrt (n : \u211d) :=\n Real.sqrt_le_sqrt (by exact_mod_cast hn : (16 : \u211d) \u2264 (n : \u211d))\n have h2 : Real.sqrt 16 = 4 := by\n rw [Real.sqrt_eq_iff_eq_sq (by norm_num) (by norm_num)]; norm_num\n rwa [h2] at h1\n linarith) (t := t)\n have hnorm3 : \u2016Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 =\n \u2016kFun n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 *\n \u2016logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 := by\n show \u2016kFun n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) *\n (-logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I))\u2016 = _\n rw [norm_mul, norm_neg]\n rw [hnorm3]\n exact mul_le_mul h1 h2 (norm_nonneg _)\n (add_nonneg (Real.exp_nonneg _) (by norm_num))\n have hint : \u2016\u222b t in T..U, Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n ((Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2)) * |U - T| := by\n apply intervalIntegral.norm_integral_le_of_norm_le_const\n intro t ht\n rw [Set.uIoc_of_le hTU] at ht\n exact hbound t ht\n have habs : |U - T| = U - T := abs_of_nonneg (by linarith)\n rw [habs] at hint\n calc \u2016\u222b t in T..U, Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016\n \u2264 ((Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2)) * (U - T) := hint\n _ = (U - T) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2) := by ring\n\n/-! ## 7. The total right-horizontal bound (DL-5, D-074 \u00a73.4) -/\n\n/-- **The upper right-tail bound** (D-074 \u00a73.4): for n \u2265 16 and\nT \u2208 [\u221an, \u221an+1] (the T_cut window), U = (3/2)\u00b7T,\n\u2016HIntegral (Jlag n) 3 (2\u221an) T\u2016 \u2264 ((\u221an+1)/2)\u00b7(e^{5/2}+1)/2 + 8.38 +\nIIMaj(n)/2 \u2014 the exact deformation identity +\nV_left/H_high/V_right piece bounds, all in the zero-free region. -/\ntheorem hIntegral_Jlag_low_le_dogLegUpper {n : \u2115} {T : \u211d}\n (hn : 16 \u2264 n) (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T\u2016 \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2 := by\n have hnR : (16 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have hsqrt4 : (4 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n have h1 : Real.sqrt 16 \u2264 Real.sqrt (n : \u211d) :=\n Real.sqrt_le_sqrt (by exact_mod_cast hn : (16 : \u211d) \u2264 (n : \u211d))\n have h2 : Real.sqrt 16 = 4 := by\n rw [Real.sqrt_eq_iff_eq_sq (by norm_num) (by norm_num)]; norm_num\n rwa [h2] at h1\n have hT0 : (0 : \u211d) < T := by linarith\n have hTn2 : (n : \u211d) \u2264 T ^ 2 := by\n have h1 : (Real.sqrt (n : \u211d)) ^ 2 \u2264 T ^ 2 :=\n pow_le_pow_left\u2080 (Real.sqrt_nonneg _) hTn 2\n rwa [Real.sq_sqrt (by linarith [Real.sqrt_nonneg (n : \u211d)])] at h1\n have hU0 : (0 : \u211d) < (3 / 2) * T := by linarith [hT0]\n have hTU : T \u2264 (3 / 2) * T := by linarith [hT0]\n have hnU : (n : \u211d) \u2264 (4 / 9) * ((3 / 2) * T) ^ 2 := by\n have h1 : (4 / 9) * ((3 / 2) * T) ^ 2 = T ^ 2 := by\n field_simp\n ring\n rw [h1]\n exact hTn2\n have hXR : (3 : \u211d) \u2264 2 * Real.sqrt (n : \u211d) := by linarith\n have hid := upper_right_horizontal_eq_dogleg hT0 hTU hXR (n := n)\n have hH := hIntegral_Jlag_dogleg_norm_le hU0 hnU hXR (n := n)\n have hVr := vIntegral_Jlag_right_norm_le hn hT0 hTU hTn2 (n := n)\n have hVl := vIntegral_Jlag_three_norm_le hT0 hTU hTn2 (n := n)\n have hlen : (3 / 2) * T - T \u2264 (Real.sqrt (n : \u211d) + 1) / 2 := by linarith [hTn1, hTn]\n have hlen0 : (0 : \u211d) \u2264 (3 / 2) * T - T := by linarith [hT0]\n -- V_left \u2264 ((\u221an+1)/2)\u00b7(e^{5/2}+1)/2\n have hVl2 : \u2016VIntegral (Jlag n) 3 T ((3 / 2) * T)\u2016 \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 := by\n refine hVl.trans ?_\n exact div_le_div_of_nonneg_right\n (mul_le_mul hlen (le_refl _) (by positivity) (by positivity)) (by norm_num)\n -- V_right \u2264 IIMaj/2\n have hVr2 : \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T)\u2016 \u2264\n IIMaj (n : \u211d) / 2 := by\n have hC2 : dirichletC2 \u2264 1 := dirichletC2_le_one\n have hK : (0 : \u211d) < (Real.sqrt (n : \u211d) + 1) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))) := by positivity\n have eA : ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * (1 : \u211d)) =\n (1 / 2) * ((Real.sqrt (n : \u211d) + 1) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)))) := by ring\n have eB : IIMaj (n : \u211d) / 2 =\n (0.57 : \u211d) * ((Real.sqrt (n : \u211d) + 1) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)))) := by\n unfold IIMaj\n ring\n calc \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T)\u2016\n \u2264 ((3 / 2) * T - T) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2) := hVr\n _ \u2264 ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) *\n ((2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * (1 : \u211d)) := by\n have h1 : ((3 / 2) * T - T) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) :=\n mul_le_mul hlen (le_refl _) (by positivity) (by positivity)\n have h2 : (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 \u2264\n (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * (1 : \u211d) :=\n mul_le_mul_of_nonneg_left hC2 (Real.rpow_nonneg (by norm_num) _)\n exact mul_le_mul h1 h2\n (mul_nonneg (Real.rpow_nonneg (by norm_num) _) dirichletC2_nonneg)\n (by positivity)\n _ \u2264 IIMaj (n : \u211d) / 2 := by\n rw [eA, eB]\n apply mul_le_mul_of_nonneg_right _ hK.le\n norm_num\n -- assemble via the exact identity\n rw [hid]\n calc \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) ((3 / 2) * T) -\n VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T) +\n VIntegral (Jlag n) 3 T ((3 / 2) * T)\u2016\n \u2264 \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) ((3 / 2) * T)\u2016 +\n \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T)\u2016 +\n \u2016VIntegral (Jlag n) 3 T ((3 / 2) * T)\u2016 := by\n have h1 := norm_add_le\n (HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) ((3 / 2) * T) -\n VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T))\n (VIntegral (Jlag n) 3 T ((3 / 2) * T))\n have h2 := norm_sub_le\n (HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) ((3 / 2) * T))\n (VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) T ((3 / 2) * T))\n linarith [h1, h2]\n _ \u2264 (8.38 : \u211d) + IIMaj (n : \u211d) / 2 +\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 :=\n add_le_add (add_le_add hH hVr2) hVl2\n _ = ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2 := by ring\n\n/-- **The lower right-tail bound by conjugation** (D-074 \u00a73.4): the\nsame bound holds at im = \u2212T (the integrand is conjugation-symmetric,\nJlag_conj; the interval integral maps conj through). -/\ntheorem hIntegral_Jlag_low_neg_norm_le_dogLegUpper {n : \u2115} {T : \u211d}\n (hn : 16 \u2264 n) (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T)\u2016 \u2264\n ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2 := by\n have hT0 : (0 : \u211d) < T := by\n have hnR : (16 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have hsqrt4 : (4 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n have h1 : Real.sqrt 16 \u2264 Real.sqrt (n : \u211d) :=\n Real.sqrt_le_sqrt (by exact_mod_cast hn : (16 : \u211d) \u2264 (n : \u211d))\n have h2 : Real.sqrt 16 = 4 := by\n rw [Real.sqrt_eq_iff_eq_sq (by norm_num) (by norm_num)]; norm_num\n rwa [h2] at h1\n linarith\n have hconj : HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T) =\n (starRingEnd \u2102) (HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T) := by\n unfold HIntegral\n rw [\u2190 intervalIntegral.intervalIntegral_conj]\n apply intervalIntegral.integral_congr\n intro \u03c3 h\u03c3\n show Jlag n ((\u03c3 : \u211d) + (((-T : \u211d)) : \u2102) * Complex.I) =\n (starRingEnd \u2102) (Jlag n ((\u03c3 : \u211d) + T * Complex.I))\n have hid2 : ((\u03c3 : \u211d) + (((-T : \u211d)) : \u2102) * Complex.I) =\n (starRingEnd \u2102) ((\u03c3 : \u211d) + T * Complex.I) := by\n simp [Complex.ext_iff]\n rw [hid2]\n exact Jlag_conj n (by\n intro h\n have him : (((\u03c3 : \u211d) + T * Complex.I)).im = T := by simp\n rw [h] at him\n simp at him\n linarith [hT0])\n rw [hconj, Complex.norm_conj]\n exact hIntegral_Jlag_low_le_dogLegUpper hn hTn hTn1\n\n/-- **THE TWO-SIDE RIGHT-HORIZONTAL BOUND** (the DL capstone for\nii\u2032-6-c): both horizontal right tails bounded by the dog-leg charge. -/\ntheorem dogLegUpper_two_sides {n : \u2115} {T : \u211d}\n (hn : 16 \u2264 n) (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T\u2016 +\n \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T)\u2016 \u2264\n 2 * (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2) := by\n rw [show (2 : \u211d) * (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 +\n (8.38 : \u211d) + IIMaj (n : \u211d) / 2) =\n (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2) +\n (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n IIMaj (n : \u211d) / 2) by ring]\n exact add_le_add (hIntegral_Jlag_low_le_dogLegUpper hn hTn hTn1)\n (hIntegral_Jlag_low_neg_norm_le_dogLegUpper hn hTn hTn1)\n\n/-! ## 8. The direct Dirichlet bound at re = 2\u221an (DL-6) \u2014 the I_I /\nV_right zeta factor WITHOUT certifying dirichletC2 \u2264 0.57.\n\nThe I_I charge in IIMaj is 0.57\u00b72^{\u2212(2\u221an\u22122)} = 2.28\u00b72^{\u22122\u221an} (the\n(6.10)-route constant |\u03b6\u2032/\u03b6(2)|). Proving dirichletC2 \u2264 0.57 directly\nneeds a ~100-prime partial-sum build (the antitone-integral tail is too\nfat). AVOIDED here: the DIRECT Dirichlet-series bound at re = 2\u221an gives\nlog2-leading + a 3^{\u2212(2\u221an\u22122)} tail (dirichletC2 \u2264 1), which is \u2264\n0.57\u00b72^{\u2212(2\u221an\u22122)} for n \u2265 5 \u2014 the IIMaj charge exactly. -/\n\n/-- The Dirichlet majorant at s = 2\u221an (summable since 2\u221an > 1). -/\ndef dirichletCX (n : \u2115) : \u211d :=\n \u2211' m : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n ((2 * Real.sqrt (n : \u211d) : \u211d) : \u2102) m\u2016\n\n/-- \u2016\u03b6\u2032/\u03b6(2\u221an+it)\u2016 \u2264 dirichletCX n for n \u2265 1 (the LSeries majorant). -/\nlemma norm_logDeriv_zeta_le_dirichletCX {n : \u2115} (hn : 1 \u2264 n) {t : \u211d} :\n \u2016logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n dirichletCX n := by\n have h1 : (1 : \u211d) \u2264 Real.sqrt (n : \u211d) := Real.one_le_sqrt.mpr (by exact_mod_cast hn)\n have hs1 : (1 : \u211d) < (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I).re := by\n simp\n linarith\n have h1' : -logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\n = LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) := by\n have h2 := ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div hs1\n rw [logDeriv, Pi.div_apply, h2]\n ring\n have hsumX : Summable (fun m : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016) := by\n rw [summable_norm_iff]\n apply ArithmeticFunction.LSeriesSummable_vonMangoldt\n rw [show (((2 * Real.sqrt (n : \u211d) : \u211d) : \u2102).re) = 2 * Real.sqrt (n : \u211d) from by simp]\n linarith\n have hsumS : Summable (fun m : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) m\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt hs1\n have hle : \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264 dirichletCX n := by\n show \u2016\u2211' m : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) m\u2016 \u2264 _\n calc \u2016\u2211' m : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) m\u2016\n \u2264 \u2211' m : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) m\u2016 :=\n norm_tsum_le_tsum_norm hsumS\n _ \u2264 \u2211' m : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016 :=\n Summable.tsum_le_tsum (fun m => LSeries.norm_term_le_of_re_le_re _ (by\n simp) m) hsumS hsumX\n _ = dirichletCX n := rfl\n calc \u2016logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016\n = \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 := by\n rw [\u2190 h1', norm_neg]\n _ \u2264 dirichletCX n := hle\n\n/-- Termwise tail comparison: for m \u2265 3, the term at 2\u221an is \u2264\n3^{\u2212(2\u221an\u22122)}\u00b7(the term at 2). -/\nlemma norm_term_vonMangoldt_X_le_three_pow {n : \u2115} (hn : 1 \u2264 n) {m : \u2115} (hm : 3 \u2264 m) :\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016 \u2264\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016 := by\n have hXre : ((((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102).re) = 2 * Real.sqrt (n : \u211d) := by simp\n have h2re : ((2 : \u2102).re) = 2 := rfl\n simp only [LSeries.norm_term_eq, hXre, h2re]\n split\n \u00b7 norm_num\n \u00b7 next hm0 =>\n have h\u039b : \u2016(ArithmeticFunction.vonMangoldt m : \u2102)\u2016 =\n ArithmeticFunction.vonMangoldt m :=\n Complex.norm_of_nonneg ArithmeticFunction.vonMangoldt_nonneg\n rw [h\u039b]\n have hm2 : (0 : \u211d) < (m : \u211d) := by exact_mod_cast (by omega : 0 < m)\n have hsqrt2 : (2 : \u211d) \u2264 2 * Real.sqrt (n : \u211d) := by\n have h1 : (1 : \u211d) \u2264 Real.sqrt (n : \u211d) := Real.one_le_sqrt.mpr (by exact_mod_cast hn)\n linarith\n have hexp_nn : (0 : \u211d) \u2264 2 * Real.sqrt (n : \u211d) - 2 := by linarith\n have e : ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 * Real.sqrt (n : \u211d)) =\n (1 / (m : \u211d) ^ (2 * Real.sqrt (n : \u211d) - 2)) *\n (ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 : \u211d)) := by\n rw [show (2 * Real.sqrt (n : \u211d)) = (2 * Real.sqrt (n : \u211d) - 2) + 2 by ring,\n Real.rpow_add hm2]\n field_simp [hm2.ne']\n rw [e]\n have hle : (1 : \u211d) / (m : \u211d) ^ (2 * Real.sqrt (n : \u211d) - 2) \u2264\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) := by\n rw [Real.rpow_neg (by norm_num : (0 : \u211d) \u2264 3), \u2190 one_div]\n apply one_div_le_one_div_of_le (Real.rpow_pos_of_pos (by norm_num : (0 : \u211d) < 3) _)\n exact Real.rpow_le_rpow (by norm_num : (0 : \u211d) \u2264 3)\n (by exact_mod_cast hm : (3 : \u211d) \u2264 (m : \u211d)) hexp_nn\n have hpos : (0 : \u211d) \u2264 ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 : \u211d) :=\n div_nonneg ArithmeticFunction.vonMangoldt_nonneg\n (Real.rpow_nonneg (by positivity : (0 : \u211d) \u2264 (m : \u211d)) _)\n exact mul_le_mul_of_nonneg_right hle hpos\n\n/-- The split: dirichletCX n = log 2\u00b72^{\u22122\u221an} + (the m \u2265 3 tail). -/\nlemma dirichletCX_le {n : \u2115} (hn : 1 \u2264 n) :\n dirichletCX n \u2264 Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 := by\n have hsumX : Summable (fun m : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016) := by\n rw [summable_norm_iff]\n apply ArithmeticFunction.LSeriesSummable_vonMangoldt\n rw [show (((2 * Real.sqrt (n : \u211d) : \u211d) : \u2102).re) = 2 * Real.sqrt (n : \u211d) from by simp]\n have h1 : (1 : \u211d) \u2264 Real.sqrt (n : \u211d) := Real.one_le_sqrt.mpr (by exact_mod_cast hn)\n linarith\n have hsplit := tsum_split_finset hsumX {0, 1, 2}\n have hsum2 : Summable (fun m : \u2115 =>\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016) := by\n rw [summable_norm_iff]\n exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n have hfin : \u2211 i \u2208 ({0, 1, 2} : Finset \u2115),\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) i\u2016 =\n Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n have hXre : ((((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102).re) = 2 * Real.sqrt (n : \u211d) := by simp\n rw [Finset.sum_insert (by norm_num), Finset.sum_insert (by norm_num),\n Finset.sum_singleton]\n simp only [LSeries.norm_term_eq, hXre]\n rw [if_pos (by norm_num), if_neg (by norm_num : (1 : \u2115) \u2260 0),\n if_neg (by norm_num : (2 : \u2115) \u2260 0)]\n simp [ArithmeticFunction.vonMangoldt_apply_one]\n have h\u039b2 : \u2016(fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) 2\u2016 = Real.log 2 := by\n show \u2016(ArithmeticFunction.vonMangoldt 2 : \u2102)\u2016 = Real.log 2\n rw [ArithmeticFunction.vonMangoldt_apply_prime Nat.prime_two]\n exact Complex.norm_of_nonneg (by linarith [Real.log_two_gt_d9] : (0 : \u211d) \u2264 Real.log 2)\n rw [h\u039b2]\n have e : (2 : \u211d) ^ (2 * Real.sqrt (n : \u211d)) = (2 : \u211d) ^ (2 * Real.sqrt (n : \u211d)) := rfl\n rw [show Real.log 2 / (2 : \u211d) ^ (2 * Real.sqrt (n : \u211d)) =\n Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) by\n rw [Real.rpow_neg (by norm_num : (0 : \u211d) \u2264 2)]\n field_simp [Real.rpow_pos_of_pos (by norm_num : (0 : \u211d) < 2)\n (2 * Real.sqrt (n : \u211d)).ne']]\n ring\n have htail : (\u2211' i : \u2115, (if i \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) i\u2016)) \u2264\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 := by\n have hterm : \u2200 m : \u2115, (if m \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016) \u2264\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016 := by\n intro m\n by_cases hm : m \u2208 ({0, 1, 2} : Finset \u2115)\n \u00b7 rw [if_pos hm]\n positivity\n \u00b7 rw [if_neg hm]\n have hm3 : 3 \u2264 m := by\n have h : m \u2209 ({0, 1, 2} : Finset \u2115) := hm\n simp at h\n omega\n exact norm_term_vonMangoldt_X_le_three_pow hn hm3\n have hsum_tail : Summable (fun m : \u2115 => (if m \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016)) := by\n apply Summable.of_norm_bounded hsumX.norm\n intro m\n by_cases hm : m \u2208 ({0, 1, 2} : Finset \u2115) <;> simp [hm]\n calc (\u2211' m : \u2115, (if m \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) m\u2016))\n \u2264 \u2211' m : \u2115, (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) *\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016 :=\n Summable.tsum_le_tsum hterm hsum_tail (hsum2.mul_left _)\n _ = (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 := by\n rw [dirichletC2, tsum_mul_left]\n calc dirichletCX n\n = \u2211 i \u2208 ({0, 1, 2} : Finset \u2115),\n \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) i\u2016 +\n \u2211' i : \u2115, (if i \u2208 ({0, 1, 2} : Finset \u2115) then 0\n else \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n (((2 * Real.sqrt (n : \u211d)) : \u211d) : \u2102) i\u2016) := hsplit\n _ \u2264 Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) * dirichletC2 :=\n add_le_add (le_of_eq hfin) htail\n\n/-- log(9/4) \u2265 0.811 (via e^{0.811} \u2265 9/4 with e^{0.8} \u2265 2.2254). -/\nlemma log_nine_quarters_ge : (0.811 : \u211d) \u2264 Real.log (9 / 4) := by\n have h08 : (2.2254 : \u211d) \u2264 Real.exp 0.8 :=\n le_trans (by norm_num [Finset.sum_range_succ])\n (Real.sum_le_exp_of_nonneg (by norm_num : (0 : \u211d) \u2264 0.8) 7)\n have h011 : (1.01106 : \u211d) \u2264 Real.exp 0.011 :=\n le_trans (by norm_num [Finset.sum_range_succ])\n (Real.sum_le_exp_of_nonneg (by norm_num : (0 : \u211d) \u2264 0.011) 4)\n have hprod : (9 / 4 : \u211d) \u2264 Real.exp 0.8 * Real.exp 0.011 := by\n calc (9 / 4 : \u211d) = 2.25 := by norm_num\n _ \u2264 2.2254 * 1.01106 := by norm_num\n _ \u2264 Real.exp 0.8 * Real.exp 0.011 :=\n mul_le_mul h08 h011 (by norm_num : (0 : \u211d) \u2264 1.01106) (Real.exp_pos _).le\n have h1 : Real.exp 0.811 = Real.exp 0.8 * Real.exp 0.011 := by\n rw [show (0.811 : \u211d) = 0.8 + 0.011 by norm_num, Real.exp_add]\n rw [\u2190 h1] at hprod\n have h2 := Real.log_le_log (Real.exp_pos _) hprod\n rwa [Real.log_exp] at h2\n\n/-- log(4/9) \u2264 \u22120.811 (the negation). -/\nlemma log_four_ninths_le : Real.log (4 / 9) \u2264 (-0.811 : \u211d) := by\n have h1 := log_nine_quarters_ge\n have h2 : Real.log (4 / 9) = -Real.log (9 / 4) := by\n rw [show ((4 : \u211d) / 9) = (9 / 4)\u207b\u00b9 by norm_num, Real.log_inv]\n rw [h2]\n linarith [h1]\n\n/-- The certified (4/9)^{\u221an} \u2264 0.17631 for \u221an \u2265 2.236. -/\nlemma four_ninths_rpow_sqrt_le {x : \u211d} (hx : (2.236 : \u211d) \u2264 x) :\n (4 / 9 : \u211d) ^ x \u2264 (0.17631 : \u211d) := by\n have hlog49 := log_four_ninths_le\n have hlogneg : Real.log (4 / 9) < 0 :=\n Real.log_neg (by norm_num : (0 : \u211d) < 4 / 9) (by norm_num : (4 / 9 : \u211d) < 1)\n have h1 : (4 / 9 : \u211d) ^ x = Real.exp (Real.log (4 / 9) * x) :=\n Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 4 / 9) x\n rw [h1]\n have h2 : Real.log (4 / 9) * x \u2264 (-1.8135 : \u211d) := by\n have h2a : Real.log (4 / 9) * x \u2264 Real.log (4 / 9) * 2.236 :=\n mul_le_mul_of_nonpos_left hx hlogneg.le\n have h2b : Real.log (4 / 9) * 2.236 \u2264 (-0.811) * 2.236 :=\n mul_le_mul_of_nonneg_right hlog49 (by norm_num)\n calc Real.log (4 / 9) * x \u2264 Real.log (4 / 9) * 2.236 := h2a\n _ \u2264 (-0.811) * 2.236 := h2b\n _ \u2264 -1.8135 := by norm_num\n have h3 : Real.exp (-1.8135) \u2264 (0.17631 : \u211d) := by\n have hsum : (5.6721 : \u211d) \u2264 Real.exp 1.8135 := by\n calc (5.6721 : \u211d) \u2264 \u2211 i \u2208 Finset.range 7, (1.8135 : \u211d) ^ i / (Nat.factorial i : \u211d) := by\n norm_num [Finset.sum_range_succ]\n _ \u2264 Real.exp 1.8135 :=\n Real.sum_le_exp_of_nonneg (by norm_num : (0 : \u211d) \u2264 1.8135) 7\n rw [show Real.exp (-1.8135) = 1 / Real.exp 1.8135 by\n rw [Real.exp_neg, one_div]]\n calc 1 / Real.exp 1.8135 \u2264 1 / 5.6721 :=\n one_div_le_one_div_of_le (by positivity : (0 : \u211d) < 5.6721) hsum\n _ \u2264 0.17631 := by norm_num\n calc Real.exp (x * Real.log (4 / 9)) \u2264 Real.exp (-1.8135) :=\n Real.exp_le_exp.mpr h2\n _ \u2264 0.17631 := h3\n\n/-- The 3^{\u2212(2\u221an\u22122)} = 9\u00b7(4/9)^{\u221an}\u00b72^{\u22122\u221an} identity. -/\nlemma three_pow_neg_two_sqrt_sub_two_eq {n : \u2115} :\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) =\n 9 * ((4 / 9 : \u211d) ^ Real.sqrt (n : \u211d)) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n have h9 : Real.log 9 = 2 * Real.log 3 := by\n rw [show (9 : \u211d) = 3 ^ 2 by norm_num, Real.log_pow]\n push_cast\n ring\n have h2 : (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) =\n Real.exp (-(Real.log 2) * (2 * Real.sqrt (n : \u211d))) := by\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n congr 1\n ring\n have h3 : (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) =\n Real.exp (-(Real.log 3) * (2 * Real.sqrt (n : \u211d) - 2)) := by\n rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 3)]\n congr 1\n ring\n have h49 : (4 / 9 : \u211d) ^ Real.sqrt (n : \u211d) =\n Real.exp (Real.sqrt (n : \u211d) * Real.log (4 / 9)) :=\n Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 4 / 9) _\n rw [h2, h3, h49]\n have hlog49 : Real.log (4 / 9) = Real.log 2 - Real.log 9 + Real.log 2 := by\n rw [Real.log_div (by norm_num : (4 : \u211d) \u2260 0) (by norm_num : (9 : \u211d) \u2260 0)]\n ring\n rw [hlog49, h9]\n rw [\u2190 Real.exp_add, \u2190 Real.exp_add]\n congr 1\n rw [show (9 : \u211d) = Real.exp (Real.log 9) from (Real.exp_log (by norm_num : (0 : \u211d) < 9)).symm]\n rw [\u2190 Real.exp_add]\n congr 1\n ring\n\n/-- **The I_I zeta charge WITHOUT dirichletC2 \u2264 0.57** (DL-6 capstone):\nfor n \u2265 5, \u2016\u03b6\u2032/\u03b6(2\u221an+it)\u2016 \u2264 0.57\u00b72^{\u2212(2\u221an\u22122)} \u2014 the IIMaj zeta form,\nvia the direct log2-leading + 3^{\u2212(2\u221an\u22122)} tail (dirichletC2 \u2264 1) with\nthe certified (4/9)^{\u221an} \u2264 0.17631 bound at \u221an \u2265 \u221a5. -/\ntheorem norm_logDeriv_zeta_le_iimZeta {n : \u2115} (hn : 5 \u2264 n) {t : \u211d} :\n \u2016logDeriv riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n (0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) := by\n have hn1 : 1 \u2264 n := by omega\n have h1 := norm_logDeriv_zeta_le_dirichletCX hn1 (t := t)\n have h2 := dirichletCX_le hn1\n have hC2 : dirichletC2 \u2264 1 := dirichletC2_le_one\n have hC2nn : (0 : \u211d) \u2264 dirichletC2 := dirichletC2_nonneg\n have hsqrt5 : (2.236 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n have hnR : (5 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n have h1 : Real.sqrt 5 \u2264 Real.sqrt (n : \u211d) := Real.sqrt_le_sqrt hnR\n have h2 : (2.236 : \u211d) \u2264 Real.sqrt 5 := by\n rw [Real.le_sqrt (by norm_num) (by norm_num)]\n norm_num\n linarith\n have h49 := four_ninths_rpow_sqrt_le hsqrt5\n have h3 : dirichletCX n \u2264 Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) := by\n refine h2.trans ?_\n exact add_le_add_left (mul_le_mul_of_nonneg_left hC2\n (Real.rpow_nonneg (by norm_num) _)) _\n have h4 : (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) \u2264\n (1.58679 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n rw [three_pow_neg_two_sqrt_sub_two_eq]\n calc 9 * ((4 / 9 : \u211d) ^ Real.sqrt (n : \u211d)) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d)))\n \u2264 9 * (0.17631 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg (by norm_num) _)\n apply mul_le_mul_of_nonneg_left h49 (by norm_num)\n _ = (1.58679 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by ring\n have h5 : Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) \u2264\n (0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) := by\n have hlog2 : Real.log 2 \u2264 (0.6932 : \u211d) := Real.log_two_lt_d9.le\n have hpos2 : (0 : \u211d) < (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by positivity\n have e : (0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) =\n (2.28 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n rw [show (-(2 * Real.sqrt (n : \u211d) - 2)) = -2 * Real.sqrt (n : \u211d) + 2 by ring,\n Real.rpow_add (by norm_num : (0 : \u211d) < 2), Real.rpow_two]\n ring\n rw [e]\n calc Real.log 2 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (3 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))\n \u2264 0.6932 * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) +\n (1.58679 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n apply add_le_add _ h4\n exact mul_le_mul_of_nonneg_right hlog2 hpos2.le\n _ = (0.6932 + 1.58679 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by ring\n _ \u2264 (2.28 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d))) := by\n apply mul_le_mul_of_nonneg_right _ hpos2.le\n norm_num\n exact h1.trans (h3.trans h5)\n\nend NguyenRH1\n\nend\n",
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raw content_json (database row)
[{"type":"thinking","thinking":"I see the issues now:\n\n1. **Line 1087 (`linarith` inside the by-block at 1085-1087)**: the `by simp; linarith` for `(by ... : ((2√↑n : ℂ)) ≤ (((2√↑n : ℝ) + t·I).re)` — hmm — `LSeries.norm_term_le_of_re_le_re _ (by simp; linarith) m` — the second arg is the re-comparison `s.re ≤ s'.re` — with s := (2√n : ℂ) and s' := σ+tI: needs `(2√n : ℂ).re ≤ ((2√n : ℝ) + t·I).re` — ((2√n:ℂ).re) = 2√n and ((σ+tI).re) = σ = 2√n — equal ✓ — the by simp gives the RHS .re = 2√n?? then linarith closes 2√n ≤ 2√n ✓ — hmm but the error is at 1087:12 \"No goals to be solved\" — meaning `simp` already closed the goal (the re-equality is rfl-simplifiable: ((2√↑n : ℂ)).re ≤ (((2√↑n:ℝ)+↑t·I).re) — simp reduces both sides' .re to 2√↑n: goal becomes 2√↑n ≤ 2√↑n — simp's le_refl closes it ✓ — then linarith has no goals ✗. Fix: `by simp` alone.\n\n2. **Line 1119 (the e in X_le_three_pow)**: `rw [e]` failed? — no, 1119:68 was in hΛ's region... — actually looking at the previous error output: 1119:68 \"unsolved goals\" with context ending at hΛ — hmm, the context showed `hΛ : ‖↑(Λ m)‖ = Λ m` present — and 1087:12 \"No goals\" — hmm these are at DIFFERENT spots than I thought. The 1119:68 error — line 1119 col 68 is inside `e`'s statement (line 1117-1119) — the `field_simp [hm2.ne']` at 1122 — field_simp closed, no ring after ✓ — hmm. Actually 1119:68 might be the `rw [show (2·√↑n) = (2·√↑n−2)+2 by ring, Real.rpow_add hm2]` — the `by ring` proves the exponent identity ✓ — hmm. Wait, maybe 1119:68 is `have e : ... := by rw [...]; field_simp [hm2.ne']` — the field_simp at 1122 — no.\n\nLet me just apply the known fixes and recompile to see what remains. The 1119:68 error with context ending at hΛ — the only tactic between hΛ (1108-1110) and the end of the lemma is at 1111 (`rw [hΛ]`) — \"unsolved goals\" after rw [hΛ]?? — the goal after `simp only [norm_term_eq, hXre, h2re]; split; next hm0` is `‖(fun n => (↑(Λ n)):ℂ) m‖ / ↑m ^ (2·√↑n) ≤ 3^(...)·(‖(fun n => (↑(Λ n)):ℂ) m‖ / ↑m ^ 2)` — rw [hΛ] rewrites `‖(ArithmeticFunction.vonMangoldt m : ℂ)‖` — but the goal has `‖(fun n => (↑(Λ m)):ℂ) m‖` — beta-redex — hΛ's LHS is `‖(ArithmeticFunction.vonMangoldt m : ℂ)‖` — the beta-redex `(fun n => (↑(Λ n)):ℂ) m` vs `(↑(Λ m):ℂ)` — NOT syntactically equal → rw [hΛ] fails OR rewrites nothing... — \"unsolved goals\" — hmm no, rw would say \"pattern not found\". The 1119:68 is at `e`'s have (1117-1122) — probably the field_simp's behavior. Let me just add `show` to beta-reduce before rw [hΛ], and fix the known issues, then recompile.\n\nActually — looking more carefully at the error list: 1119:68 was in the FIRST error batch with the context `n, hn, m, hm, hXre, h2re, hm0, hΛ` — that's exactly the state BEFORE `rw [hΛ]` at line 1111... — no wait, hΛ is PRESENT in the context, so it's AFTER hΛ's proof. The unsolved goals at 1119:68 — col 68 — line 1119 is `(ArithmeticFunction.vonMangoldt m / (m : ℝ) ^ (2 : ℝ)) := by` (e's RHS continuation) — the `field_simp [hm2.ne']` at 1122 — hmm col 68 of line 1119 doesn't align with 1122. The errors might be misaligned due to my earlier edits. Let me just recompile and look at the current first error precisely.","signature":""},{"type":"text","text":"Applying the known fixes and recompiling to see the precise remaining errors."},{"type":"tool_use","id":"Edit_434","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":" Summable.tsum_le_tsum (fun m => LSeries.norm_term_le_of_re_le_re _ (by\n simp\n linarith) m) hsumS hsumX","new_string":" Summable.tsum_le_tsum (fun m => LSeries.norm_term_le_of_re_le_re _ (by\n simp) m) hsumS hsumX"},"extra_content":null},{"type":"tool_use","id":"Edit_435","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":" simp only [LSeries.norm_term_eq, hXre]\n rw [if_pos (by norm_num), if_neg (by norm_num : (1 : ℕ) ≠ 0),\n if_neg (by norm_num : (2 : ℕ) ≠ 0)]\n simp [ArithmeticFunction.vonMangoldt_apply_one]\n have hΛ2 : ‖(ArithmeticFunction.vonMangoldt 2 : ℂ)‖ = Real.log 2 := by\n have h1 : ArithmeticFunction.vonMangoldt 2 = Real.log 2 :=\n ArithmeticFunction.vonMangoldt_prime Nat.prime_two\n rw [h1]\n exact Complex.norm_of_nonneg (Real.log_two_gt_d9.le)\n rw [hΛ2]","new_string":" simp only [LSeries.norm_term_eq, hXre]\n rw [if_pos (by norm_num), if_neg (by norm_num : (1 : ℕ) ≠ 0),\n if_neg (by norm_num : (2 : ℕ) ≠ 0)]\n simp [ArithmeticFunction.vonMangoldt_apply_one]\n have hΛ2 : ‖(fun n => (ArithmeticFunction.vonMangoldt n : ℂ)) 2‖ = Real.log 2 := by\n show ‖(ArithmeticFunction.vonMangoldt 2 : ℂ)‖ = Real.log 2\n rw [ArithmeticFunction.vonMangoldt_apply_prime Nat.prime_two]\n exact Complex.norm_of_nonneg (by linarith [Real.log_two_gt_d9] : (0 : ℝ) ≤ Real.log 2)\n rw [hΛ2]"},"extra_content":null},{"type":"tool_use","id":"Edit_436","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":" _ ≤ Real.exp 0.8 * Real.exp 0.011 :=\n mul_le_mul h08 h011 (by norm_num : (0 : ℝ) ≤ 1.01106) (by norm_num : (0 : ℝ) ≤ 2.2254)","new_string":" _ ≤ Real.exp 0.8 * Real.exp 0.011 :=\n mul_le_mul h08 h011 (by norm_num : (0 : ℝ) ≤ 1.01106) (Real.exp_pos _).le"},"extra_content":null},{"type":"tool_use","id":"Edit_437","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":" rw [h1] at hprod\n have h2 := Real.log_le_log (Real.exp_pos _) hprod\n rwa [Real.log_exp] at h2","new_string":" rw [← h1] at hprod\n have h2 := Real.log_le_log (by norm_num : (0 : ℝ) < 9 / 4) hprod\n rwa [Real.log_exp] at h2"},"extra_content":null},{"type":"tool_use","id":"Edit_438","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":" calc Real.exp (x * Real.log (4 / 9)) ≤ Real.exp (-1.8135) :=\n Real.exp_le_exp.mpr h2\n _ ≤ 0.17631 := h3","new_string":" calc Real.exp (Real.log (4 / 9) * x) ≤ Real.exp (-1.8135) :=\n Real.exp_le_exp.mpr h2\n _ ≤ 0.17631 := h3"},"extra_content":null}]