{
 "script": "part2_constants",
 "passed": 14,
 "failed": 0,
 "seconds": 0.5,
 "checks": [
  {
   "label": "eq:std-onset",
   "desc": "s=3/10: n_s' = 1, n_s = 13707",
   "status": "PASS",
   "method": "interval arithmetic (Arb) + exact",
   "detail": "computed n_s'=1, n_s=13707, signs certified: True"
  },
  {
   "label": "eq:std-onset",
   "desc": "s=1/2: n_s' = 4, n_s = 25051",
   "status": "PASS",
   "method": "interval arithmetic (Arb) + exact",
   "detail": "computed n_s'=4, n_s=25051, signs certified: True"
  },
  {
   "label": "eq:std-onset",
   "desc": "s=4/5: n_s' = 25, n_s = 93949",
   "status": "PASS",
   "method": "interval arithmetic (Arb) + exact",
   "detail": "computed n_s'=25, n_s=93949, signs certified: True"
  },
  {
   "label": "eq:std-onset",
   "desc": "s=19/20: n_s' = 169, n_s = 493795",
   "status": "PASS",
   "method": "interval arithmetic (Arb) + exact",
   "detail": "computed n_s'=169, n_s=493795, signs certified: True"
  },
  {
   "label": "eq:std-onset",
   "desc": "f_s'' = (1+2c12)/(2(n+1)^2) > 0 (convex); f_s(n)/n -> gamma(3/4 - c12(72(1+s)+1/4)) > gamma/2 for all s in (0,1)",
   "status": "PASS",
   "method": "symbolic",
   "detail": ""
  },
  {
   "label": "eq:std-onset",
   "desc": "f_s(1) < 0 at the four values of s",
   "status": "PASS",
   "method": "interval arithmetic (Arb)",
   "detail": ""
  },
  {
   "label": "prop:early",
   "desc": "beta_s^2/(1-s^2) = sqrt(s) log(1/sqrt s)/(2(1+sqrt s)(1+s))",
   "status": "PASS",
   "method": "symbolic",
   "detail": ""
  },
  {
   "label": "prop:early",
   "desc": "gamma_s <= 1/(256 e) on (0,1) (analytic: x log(1/x) <= 1/e; grid of 1999 points)",
   "status": "PASS",
   "method": "interval arithmetic (Arb)",
   "detail": "max 256 e gamma_s on grid = 0.7099"
  },
  {
   "label": "prop:early",
   "desc": "(4e^-3)^(1/2) <= e^-0.8 (chi-squared tail with x=4)",
   "status": "PASS",
   "method": "interval arithmetic (Arb)",
   "detail": "[0.446260 +/- 3.21e-7] vs [0.449329 +/- 3.60e-8]"
  },
  {
   "label": "prop:early",
   "desc": "inf_lambda e^{-lambda m x}(1-2lambda)^{-m/2} = (x e^{1-x})^{m/2} at lambda=(1-1/x)/2",
   "status": "PASS",
   "method": "symbolic",
   "detail": ""
  },
  {
   "label": "prop:early",
   "desc": "eq:std-Vinf exponent = 2n phi; (sqrt s + s)/(s(1-s)) = 1/(sqrt s (1-sqrt s)); exponent at |z|=beta_s is (n/4)log(1/s); truncation term = s^(1/2) s^(n/4)",
   "status": "PASS",
   "method": "symbolic",
   "detail": ""
  },
  {
   "label": "eq:std-gain",
   "desc": "12-point average of phi minus phi(z0) = r2^2/(2(1-s^2)) = y0^2/(32(1-s^2)) = gamma_s; Phi_max bound (3y0/2)^2/(2(1-s)) = 36(1+s)gamma_s",
   "status": "PASS",
   "method": "symbolic",
   "detail": ""
  },
  {
   "label": "eq:std-average",
   "desc": "c_12 = 2*2^-12/(1-2^-12) = 2 sum_m 2^-12m = 2/4095",
   "status": "PASS",
   "method": "exact rational arithmetic",
   "detail": ""
  },
  {
   "label": "prop:early",
   "desc": "s=1/2: beta_s^2, gamma_s, c_s = gamma_s/4, C_s = 14 e^{gamma_s n_s/4}",
   "status": "INFO",
   "method": "interval arithmetic (Arb)",
   "detail": "beta^2=[0.035888870 +/- 3.64e-10], gamma=[0.00037384240 +/- 3.80e-13], c_s=[9.34606e-5 +/- 1.2e-13], C_s=[145.524 +/- 3.21e-4]"
  },
  {
   "label": "prop:early",
   "desc": "RHS of eq:std-early is below 1 only for n > 4 log(13)/gamma_s, about 2.7e4 at s=1/2",
   "status": "PASS",
   "method": "interval arithmetic (Arb)",
   "detail": "4 log 13/gamma_(1/2) = [27444.18 +/- 1.82e-3]"
  }
 ],
 "data": {
  "3/10": {
   "n_s": 13707,
   "n_s_prime": 1,
   "certified": true
  },
  "1/2": {
   "n_s": 25051,
   "n_s_prime": 4,
   "certified": true
  },
  "4/5": {
   "n_s": 93949,
   "n_s_prime": 25,
   "certified": true
  },
  "19/20": {
   "n_s": 493795,
   "n_s_prime": 169,
   "certified": true
  }
 }
}