{
 "script": "part3b_firstorder",
 "passed": 16,
 "failed": 0,
 "seconds": 145.7,
 "checks": [
  {
   "label": "(G1)-(G3)",
   "desc": "Gaussian sample seed 106: R* and the next nonreal modulus, real zeros in |w|<0.95, theta*",
   "status": "INFO",
   "method": "numerical (floating point)",
   "detail": "R*=0.74016, next=0.85023, real moduli=[], theta*=2.4601, T*=1.82537"
  },
  {
   "label": "rem:benchmark",
   "desc": "generic tau_c, W, rho* for 1+bw+2w^2 agree with the closed forms; tau_c + W = a/2 = 1",
   "status": "PASS",
   "method": "numerical (floating point)",
   "detail": "nu=0.221465, tau_c=-0.259049, W=1.25905"
  },
  {
   "label": "thm:collision(c)",
   "desc": "benchmark q = 1 + bw + 2w^2, theta* = 1.1: certified bracket (width <= 2e-12) for Lambda_n; exactly two nonreal zeros (certified count) at its lower end, all real at its upper end; the Newton double zero lies in the bracket",
   "status": "PASS",
   "method": "interval arithmetic (Arb)",
   "detail": "certified for n = [50, 100, 101, 110, 150, 200, 207, 250, 300, 350, 400, 408, 800, 1600]"
  },
  {
   "label": "thm:firstorder-intro",
   "desc": "benchmark q = 1 + bw + 2w^2, theta* = 1.1: n(Lambda_n - T*) - (tau_c - W cos Psi(c_n^nat)) -> 0, consistent with O(n^-1 d_n^-1/3) (scaled error bounded for n >= 800, |diff| < 0.01 at n = 3200)",
   "status": "PASS",
   "method": "numerical (floating point)",
   "detail": "max n|diff| d_n^(1/3): 0.67 for n>=800 (exact c_n: 0.68), 0.96 over all n; n x diff: [(50, 0.45), (100, 0.15), (101, 1.88), (110, 4.61), (150, -0.27), (200, -0.88), (207, 0.75), (250, 2.15), (300, 0.47), (350, 0.36), (400, 0.42), (408, -3.18), (800, 0.02), (1600, -1.77), (3200, 0.58)]; tau_c=-0.25905, W=1.25905, nu=0.22146"
  },
  {
   "label": "thm:firstorder-intro",
   "desc": "benchmark q = 1 + bw + 2w^2, theta* = 1.1: place of the double zero, n(z_n - z*) - (formula) -> 0, consistent with O(n^-1 d_n^-2/3)",
   "status": "PASS",
   "method": "numerical (floating point)",
   "detail": "max n|diff| d_n^(2/3) for n>=800: 0.30; n x diff: [(50, -0.03), (100, 0.04), (101, -0.57), (110, -3.05), (150, 0.1), (200, 0.09), (207, -0.19), (250, -0.77), (300, -0.17), (350, -0.27), (400, -0.31), (408, -1.37), (800, 0.06), (1600, -0.21), (3200, -0.29)]"
  },
  {
   "label": "thm:firstorder-intro",
   "desc": "benchmark q = 1 + bw + 2w^2, theta* = 1.1: n(Lambda_n - T*) in the limit interval [tau_c - W, tau_c + W] (up to 0.1, n >= 800)",
   "status": "PASS",
   "method": "numerical (floating point)",
   "detail": "[-1.5181, 1.0000], observed range over all n [-1.4390, 1.0017]"
  },
  {
   "label": "thm:collision(c)",
   "desc": "rational q = (1+2w)(1+4w/3+20w^2/9)/(1-5w/6): certified bracket (width <= 2e-12) for Lambda_n; exactly two nonreal zeros (certified count) at its lower end, all real at its upper end; the Newton double zero lies in the bracket",
   "status": "PASS",
   "method": "interval arithmetic (Arb)",
   "detail": "certified for n = [50, 100, 150, 200, 250, 300, 350, 400, 800, 1600]"
  },
  {
   "label": "thm:firstorder-intro",
   "desc": "rational q = (1+2w)(1+4w/3+20w^2/9)/(1-5w/6): n(Lambda_n - T*) - (tau_c - W cos Psi(c_n^nat)) -> 0, consistent with O(n^-1 d_n^-1/3) (scaled error bounded for n >= 800, |diff| < 0.01 at n = 3200)",
   "status": "PASS",
   "method": "numerical (floating point)",
   "detail": "max n|diff| d_n^(1/3): 1.45 for n>=800 (exact c_n: 1.34), 4.16 over all n; n x diff: [(50, 0.55), (100, -1.43), (150, -9.27), (200, 1.87), (250, 0.45), (300, -1.62), (350, -16.09), (400, 1.73), (800, 1.5), (1600, 1.1), (3200, 0.32)]; tau_c=-1.41762, W=1.38889, nu=0.77491"
  },
  {
   "label": "thm:firstorder-intro",
   "desc": "rational q = (1+2w)(1+4w/3+20w^2/9)/(1-5w/6): place of the double zero, n(z_n - z*) - (formula) -> 0, consistent with O(n^-1 d_n^-2/3)",
   "status": "PASS",
   "method": "numerical (floating point)",
   "detail": "max n|diff| d_n^(2/3) for n>=800: 0.21; n x diff: [(50, -0.1), (100, -0.6), (150, 3.59), (200, -0.14), (250, -0.13), (300, -0.62), (350, 15.26), (400, -0.13), (800, -0.1), (1600, -0.06), (3200, -0.16)]"
  },
  {
   "label": "thm:firstorder-intro",
   "desc": "rational q = (1+2w)(1+4w/3+20w^2/9)/(1-5w/6): n(Lambda_n - T*) in the limit interval [tau_c - W, tau_c + W] (up to 0.1, n >= 800)",
   "status": "PASS",
   "method": "numerical (floating point)",
   "detail": "[-2.8065, -0.0287], observed range over all n [-2.6131, -0.0252]"
  },
  {
   "label": "thm:collision(c)",
   "desc": "Gaussian power series sample (seed 106, degree 4000): certified bracket (width <= 2e-12) for Lambda_n; exactly two nonreal zeros (certified count) at its lower end, all real at its upper end; the Newton double zero lies in the bracket",
   "status": "PASS",
   "method": "interval arithmetic (Arb)",
   "detail": "certified for n = [100, 150, 200, 250, 300, 350, 400, 800, 1600]"
  },
  {
   "label": "thm:firstorder-intro",
   "desc": "Gaussian power series sample (seed 106, degree 4000): n(Lambda_n - T*) - (tau_c - W cos Psi(c_n^nat)) -> 0, consistent with O(n^-1 d_n^-1/3) (scaled error bounded for n >= 800, |diff| < 0.01 at n = 3200)",
   "status": "PASS",
   "method": "numerical (floating point)",
   "detail": "max n|diff| d_n^(1/3): 28.97 for n>=800 (exact c_n: 28.96), 45.57 over all n; n x diff: [(100, -41.94), (150, -53.91), (200, -69.65), (250, -125.15), (300, 14.81), (350, 0.91), (400, -5.22), (800, -28.02), (1600, -25.11), (3200, -20.88)]; tau_c=1.08857, W=2.29962, nu=0.93879"
  },
  {
   "label": "thm:firstorder-intro",
   "desc": "Gaussian power series sample (seed 106, degree 4000): place of the double zero, n(z_n - z*) - (formula) -> 0, consistent with O(n^-1 d_n^-2/3)",
   "status": "PASS",
   "method": "numerical (floating point)",
   "detail": "max n|diff| d_n^(2/3) for n>=800: 3.19; n x diff: [(100, -0.7), (150, 1.73), (200, 6.85), (250, 49.6), (300, 14.72), (350, 7.96), (400, 5.87), (800, 1.89), (1600, 2.16), (3200, 2.51)]"
  },
  {
   "label": "thm:firstorder-intro",
   "desc": "Gaussian power series sample (seed 106, degree 4000): n(Lambda_n - T*) in the limit interval [tau_c - W, tau_c + W] (up to 0.1, n >= 800)",
   "status": "PASS",
   "method": "numerical (floating point)",
   "detail": "[-1.2110, 3.3882], observed range over all n [-0.9410, 3.3596]"
  },
  {
   "label": "tab:benchmark",
   "desc": "rows n=100,101,207,408 of Table tab:benchmark (dist(c_n,piZ), n(Lambda_n-T*), tau_c - W cos Psi_n with the exact c_n) reproduced to the printed digits",
   "status": "PASS",
   "method": "interval arithmetic (Arb)",
   "detail": "n=100: dist 0.6714 (0.671), n(L-2) 0.4538974 (0.4539), pred 0.4500746 (0.45007), n*diff 0.382; n=101: dist 0.0243 (0.024), n(L-2) -1.2411197 (-1.24112), pred -1.2477479 (-1.24775), n*diff 0.669; n=207: dist 1.5151 (1.515), n(L-2) 1.0017067 (1.00171), pred 0.9980490 (0.99805), n*diff 0.757; n=408: dist 0.0095 (0.009), n(L-2) -1.3758216 (-1.37582), pred -1.3722778 (-1.37228), n*diff -1.446"
  },
  {
   "label": "rem:benchmark",
   "desc": "Lambda_n <= a sqrt(n/(n-1)) for every tested n; at n=207, n(Lambda_n - 2) = 1.0017 below the bound 1.0036",
   "status": "PASS",
   "method": "interval arithmetic (Arb)",
   "detail": "n=50: 0.71703 <= 1.01525; n=100: 0.45390 <= 1.00756; n=101: -1.24112 <= 1.00749; n=110: -1.43903 <= 1.00687; n=150: 0.08368 <= 1.00503; n=200: -0.46046 <= 1.00377; n=207: 1.00171 <= 1.00364; n=250: -1.29451 <= 1.00301; n=300: -0.35768 <= 1.00251; n=350: 0.14968 <= 1.00215; n=400: 0.49762 <= 1.00188; n=408: -1.37582 <= 1.00184; n=800: 0.34030 <= 1.00094; n=1600: -1.06468 <= 1.00047"
  },
  {
   "label": "rem:sharp",
   "desc": "benchmark scaled errors n|n(Lambda_n-T*) - (tau_c - W cos Psi(c_n))| near the cusps: 1.62 at n=110, 1.45 at n=408",
   "status": "PASS",
   "method": "interval arithmetic (Arb)",
   "detail": "n=110: 1.6162 (dist(c_n,piZ) = 0.00275), n=408: 1.4459 (dist(c_n,piZ) = 0.00946)"
  }
 ],
 "data": {
  "benchmark q = 1 + bw + 2w^2, theta* = 1.1": [
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    "bound": 1.015254455221075
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    "bound": 1.007563051842415
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   {
    "n": 110,
    "method": "bisection (32 steps)",
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    "hi": "1.98691795262522",
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    "bound": 1.006870248975985
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   {
    "n": 150,
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    "lo": "2.00055784015262",
    "hi": "2.00055784015963",
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    "bound": 1.003637836435993
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    "bound": 1.0030100351264637
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   {
    "n": 300,
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    "lo": "1.99880772051855",
    "hi": "1.99880772052556",
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    "bound": 1.0025069647600242
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   {
    "n": 350,
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   {
    "n": 400,
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    "hi": "2.00124404239695",
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   {
    "n": 408,
    "method": "bisection (31 steps)",
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    "hi": "1.99662788835829",
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    "bound": 1.001841997929542
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   {
    "n": 800,
    "method": "Newton + certified bracket",
    "lo": "2.00042537284567",
    "hi": "2.00042537284767",
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   {
    "n": 1600,
    "method": "Newton + certified bracket",
    "lo": "1.99933457233286",
    "hi": "1.99933457233486",
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    "n_dz": -0.05236668680896209,
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    "bound": 1.0004689942742147
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   {
    "n": 3200,
    "method": "Newton only (not certified)",
    "lo": "2.00025238599558",
    "hi": "2.00025238599558",
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    "bound": 1.0002344360518503
   }
  ],
  "rational q = (1+2w)(1+4w/3+20w^2/9)/(1-5w/6)": [
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    "n": 50,
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    "bound": null
   },
   {
    "n": 100,
    "method": "bisection (33 steps)",
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    "hi": "2.21828419195239",
    "two_nonreal_before": true,
    "dist_cn": 0.8750732580795025,
    "dist_cnat": 0.8753175569745512,
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    "n_dz": 0.18480169374513916,
    "pred_place": 0.1907843502523052,
    "bound": null
   },
   {
    "n": 150,
    "method": "bisection (32 steps)",
    "lo": "2.20722429055253",
    "hi": "2.20722429055994",
    "two_nonreal_before": true,
    "dist_cn": 0.07633571809458593,
    "dist_cnat": 0.07540085675775167,
    "n_dLambda": -2.2496897498983355,
    "pred_exact": -2.183050916304612,
    "pred_nat": -2.1878902141383425,
    "newton_in_bracket": true,
    "n_dz": 0.6299412630758094,
    "pred_place": 0.605981555970742,
    "bound": null
   },
   {
    "n": 200,
    "method": "bisection (32 steps)",
    "lo": "2.21946640119873",
    "hi": "2.21946640120429",
    "two_nonreal_before": true,
    "dist_cn": 0.7247846814524004,
    "dist_cnat": 0.7245158434590478,
    "n_dLambda": -0.5511642041418995,
    "pred_exact": -0.5601318369419448,
    "pred_nat": -0.5604951218609064,
    "newton_in_bracket": true,
    "n_dz": 1.4466446296088766,
    "pred_place": 1.4473306341976084,
    "bound": null
   },
   {
    "n": 250,
    "method": "bisection (31 steps)",
    "lo": "2.22210856836329",
    "hi": "2.22210856837219",
    "two_nonreal_before": true,
    "dist_cn": 1.5252594646351303,
    "dist_cnat": 1.5244325436758475,
    "n_dLambda": -0.028413463620312024,
    "pred_exact": -0.03017588515038977,
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}