{
  "file_path": "D:\\Python\\Quantum\\research\\paper\\draft.tex",
  "old_string": "\\appendix\n\\section{Numerical experiments}\n% Tables/figures generated from research/numerics/data/ CSVs.",
  "new_string": "\\appendix\n\\section{Numerical experiments}\n% Tables/figures generated from research/numerics/data/ CSVs.\n\nAll experiments are deterministic scripts with CSV+JSON sidecars in the\nproject archive (\\texttt{research/numerics/}); checks as stated.\n\n\\subsection{Li-coefficient machinery (E-011/E-015/E-021/E-023)}\n\\begin{itemize}\n\\item $\\lambda_n > 0$ verified for $n \\le 3000$ from the arithmetic\nside (Bombieri--Lagarias Thm~2, zero-free; \\textsc{flint} Stieltjes +\nArb $\\zeta$, 1000--1450-digit working precision), cross-validated\nagainst the zero-side computation (agreement $4.8\\times10^{-25}$\nrelative) and $\\lambda_1$ to 102 digits against\n$\\gamma/2$-type closed forms.\n\\item Exact archimedean part: $S_\\infty(n) = \\tfrac12 n\\log n + C_1 n\n- \\tfrac14 - \\tfrac{1}{24n} + o(1/n)$ measured to $n = 5000$\n($c_0 = -1/4$ identified exactly; residuals match $-1/4 - 1/(24n)$ to\nall printed digits), and $|S_f(n)| \\le 13.52$ for $n \\le 5000$\n(max at $n = 3298$) vs the traced worst-case $\\sim 2280$ there \u2014\nthe contour constants are $\\sim 170\\times$ loose, as documented.\n\\item Margin assembly: crossover at $n = 32\\,871$ (traced constants)\n/ $30\\,217$ (measured constants $A = 0.9$, $B_0 = 1.0$); adopted\n$N_0 = 5\\times10^4$ with margin $67\\,705$; margin convex to\n$N_1 \\approx 9\\times10^{24}$.\n\\end{itemize}\n\n\\subsection{Spectral-measure program (E-016--E-025)}\n\\begin{itemize}\n\\item Galerkin realization of Suzuki's Weil-form operator validated\nagainst the zero-side explicit formula to $4.5\\times10^{-6}$;\nzero-location by sign-change + Brent refinement; winding-number\ndiagnostics for off-axis zeros.\n\\item Clark/de~Branges weights $1/K_a(z_k,z_k)$ via Schur-reduced\nbordered Toeplitz solves (validated to $3.6\\times10^{-14}$ against\ndense bordered solves; $h$-refinement $1.1\\times10^{-3}$).\n\\item Valley observable: $\\Delta(a) = 2.16, 2.51, 2.74, 2.87, 2.94$\nfor $a = 4,6,8,10,12$; plateau $K \\to 2a$; floors $\\approx 1.04$;\nmass within $\\pm0.1$ of true zeros $63\\% \\to 72.5\\%$; positional\nlocking $\\approx 0.65/a^2$; residual $\\to Z/(2\\pi)$.\n\\item $\\theta$-independence (E-024): residual spread $\\le 2.2\\%$ over\n$7$ boundary conditions $\\times$ $3$ values of $a$.\n\\item $\\lambda$-flow (E-025): discretization floor documented\n(pencil bottom crosses zero at $a \\approx 5.2$ for $h = 0.01$,\n$O(h)$ artifact); channel response measured at $a=4$\n($\\Delta\\colon 2.16 \\to 6.99$, residual $48 \\to 11$).\n\\end{itemize}"
}

{
  "filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
  "oldString": "lemma norm_term_vonMangoldt_le_two_pow {\u03c3 : \u211d} (h\u03c3 : 2 \u2264 \u03c3) (m : \u2115) :\n    \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (\u03c3 : \u2102) 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 : \u2102).re) = \u03c3 := rfl\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      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) < 2)]\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",
  "newString": "lemma 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)]\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",
  "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\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 : \u211d} (h\u03c3 : 2 \u2264 \u03c3) (m : \u2115) :\n    \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (\u03c3 : \u2102) 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 : \u2102).re) = \u03c3 := rfl\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      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) < 2)]\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    ring\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 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 hstep) 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)) := by\n          rw [\u2190 Real.exp_nat_mul]\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    rw [mul_div_assoc, \u2190 div_le_div_iff_of_pos_right h\u03c30]\n    exact (div_le_div_iff_of_pos_right hU2).mpr hnU\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\nend NguyenRH1\n\nend\n",
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        " ",
        " /-- Termwise \u03c3-vs-2 comparison for the \u039b L-series (the (6.10) halving):",
        " \u039b(m)/m^\u03c3 \u2264 2^{\u2212(\u03c3\u22122)}\u00b7\u039b(m)/m\u00b2 for \u03c3 \u2265 2. -/",
        "-lemma norm_term_vonMangoldt_le_two_pow {\u03c3 : \u211d} (h\u03c3 : 2 \u2264 \u03c3) (m : \u2115) :",
        "-    \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (\u03c3 : \u2102) m\u2016 \u2264",
        "+lemma norm_term_vonMangoldt_le_two_pow {\u03c3 t : \u211d} (h\u03c3 : 2 \u2264 \u03c3) (m : \u2115) :",
        "+    \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))",
        "+        ((\u03c3 : \u211d) + t * Complex.I) m\u2016 \u2264",
        "       (2 : \u211d) ^ (-(\u03c3 - 2)) * \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016 := by",
        "-  have h\u03c3re : ((\u03c3 : \u2102).re) = \u03c3 := rfl",
        "+  have h\u03c3re : ((\u03c3 : \u211d) + t * Complex.I).re = \u03c3 := by simp",
        "   have h2re : ((2 : \u2102).re) = 2 := rfl",
        "   simp only [LSeries.norm_term_eq, h\u03c3re, h2re]",
        "   split"
      ]
    },
    {
      "oldStart": 358,
      "oldLines": 9,
      "newStart": 359,
      "newLines": 10,
      "lines": [
        "         rw [show (\u03c3 : \u211d) = (\u03c3 - 2) + 2 by ring,",
        "           Real.rpow_add (by positivity : (0 : \u211d) < (m : \u211d))]",
        "         field_simp [show (m : \u211d) \u2260 0 by positivity]",
        "+        ring_nf",
        "       rw [e]",
        "       have hle : (1 : \u211d) / (m : \u211d) ^ (\u03c3 - 2) \u2264 (2 : \u211d) ^ (-(\u03c3 - 2)) := by",
        "-        rw [Real.rpow_neg (by norm_num : (0 : \u211d) < 2)]",
        "+        rw [Real.rpow_neg (by norm_num : (0 : \u211d) \u2264 2)]",
        "         apply one_div_le_one_div_of_le (Real.rpow_pos_of_pos (by norm_num : (0 : \u211d) < 2) _)",
        "         exact Real.rpow_le_rpow (by norm_num : (0 : \u211d) \u2264 2)",
        "           (by exact_mod_cast hm2 : (2 : \u211d) \u2264 (m : \u211d)) (by linarith : (0 : \u211d) \u2264 \u03c3 - 2)"
      ]
    }
  ],
  "userModified": false,
  "replaceAll": false
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