outputs/claim5_stdout.txt

== structural lemma: median of 3 is gamma-far only if >= 2 of 3 are ==
  4,000,000 random triples + 41^4 grid triples: 0 + 0 violations -> lemma holds: True

== worst-case median-of-three loss vs n and d_gamma ==
  n*L/d_gamma over the whole grid: min=0.2970  max=0.3468  (bounded constant => L = O(d_gamma/n))
  d_gamma=8:
    n=      8 (=     1 d)  L=3.327844e-01  n*L/d=0.3328
    n=     16 (=     2 d)  L=1.698144e-01  n*L/d=0.3396
    n=     40 (=     5 d)  L=6.877544e-02  n*L/d=0.3439
    n=     80 (=    10 d)  L=3.453178e-02  n*L/d=0.3453
    n=    400 (=    50 d)  L=6.929587e-03  n*L/d=0.3465
    n=    800 (=   100 d)  L=3.466252e-03  n*L/d=0.3466
    n=   4000 (=   500 d)  L=6.934833e-04  n*L/d=0.3467
    n=   8000 (=  1000 d)  L=3.467560e-04  n*L/d=0.3468
  d_gamma=64:
    n=     64 (=     1 d)  L=3.449562e-01  n*L/d=0.3450
    n=    128 (=     2 d)  L=1.729306e-01  n*L/d=0.3459
    n=    320 (=     5 d)  L=6.928132e-02  n*L/d=0.3464
    n=    640 (=    10 d)  L=3.465886e-02  n*L/d=0.3466
    n=   3200 (=    50 d)  L=6.934692e-03  n*L/d=0.3467
    n=   6400 (=   100 d)  L=3.467528e-03  n*L/d=0.3468
    n=  32000 (=   500 d)  L=6.935346e-04  n*L/d=0.3468
    n=  64000 (=  1000 d)  L=3.467691e-04  n*L/d=0.3468

== fitted scaling exponents ==
  d_gamma=   8: d log L / d log n = -0.9954  (predicted -1)   R^2=0.99999
  d_gamma=  32: d log L / d log n = -0.9988  (predicted -1)   R^2=1.00000
  d_gamma= 128: d log L / d log n = -0.9997  (predicted -1)   R^2=1.00000
  n=   5000 fixed:  d log L / d log d_gamma = +1.0000  (predicted +1)   R^2=1.00000
  n=  20000 fixed:  d log L / d log d_gamma = +1.0000  (predicted +1)   R^2=1.00000
  n= 100000 fixed:  d log L / d log d_gamma = +1.0000  (predicted +1)   R^2=1.00000

== sample complexity n(eps) of median-of-three ==
  d_gamma=   8 eps=0.1    n(eps)=      28   n(eps)/(d/eps)=0.3500
  d_gamma=   8 eps=0.05   n(eps)=      56   n(eps)/(d/eps)=0.3500
  d_gamma=   8 eps=0.02   n(eps)=     139   n(eps)/(d/eps)=0.3475
  d_gamma=   8 eps=0.01   n(eps)=     278   n(eps)/(d/eps)=0.3475
  d_gamma=   8 eps=0.005  n(eps)=     555   n(eps)/(d/eps)=0.3469
  d_gamma=  32 eps=0.1    n(eps)=     111   n(eps)/(d/eps)=0.3469
  d_gamma=  32 eps=0.05   n(eps)=     222   n(eps)/(d/eps)=0.3469
  d_gamma=  32 eps=0.02   n(eps)=     555   n(eps)/(d/eps)=0.3469
  d_gamma=  32 eps=0.01   n(eps)=    1110   n(eps)/(d/eps)=0.3469
  d_gamma=  32 eps=0.005  n(eps)=    2219   n(eps)/(d/eps)=0.3467
  d_gamma= 128 eps=0.1    n(eps)=     444   n(eps)/(d/eps)=0.3469
  d_gamma= 128 eps=0.05   n(eps)=     888   n(eps)/(d/eps)=0.3469
  d_gamma= 128 eps=0.02   n(eps)=    2219   n(eps)/(d/eps)=0.3467
  d_gamma= 128 eps=0.01   n(eps)=    4439   n(eps)/(d/eps)=0.3468
  d_gamma= 128 eps=0.005  n(eps)=    8878   n(eps)/(d/eps)=0.3468
  constant C = n(eps)/(d_gamma/eps): mean=0.3473  CV=0.0031 (constant => O(d_gamma/eps), matching the Omega(d_gamma/eps) lower bound)

== Monte-Carlo cross-check (real sampling / real median) ==
  d=   8 n=   40: analytic=0.068775  MC=0.068760 +/- 0.000712  (rel.err 0.02%)   single interpolator MC=0.068005
  d=   8 n=  200: analytic=0.013848  MC=0.013832 +/- 0.000147  (rel.err 0.11%)   single interpolator MC=0.013758
  d=  12 n=  120: analytic=0.034580  MC=0.034519 +/- 0.000304  (rel.err 0.18%)   single interpolator MC=0.034196
  d=  32 n=  160: analytic=0.069209  MC=0.069498 +/- 0.000363  (rel.err 0.42%)   single interpolator MC=0.068602
  d=  32 n=  800: analytic=0.013865  MC=0.013822 +/- 0.000075  (rel.err 0.31%)   single interpolator MC=0.013743
  d= 128 n=  640: analytic=0.069318  MC=0.069216 +/- 0.000182  (rel.err 0.15%)   single interpolator MC=0.068489

== comparison and boundary probes ==
  d=   8 n=   800: median-of-3 = 3.466e-03   single interpolator (its own worst profile) = 3.676e-03   ratio = 1.06x
  d=  32 n=  3200: median-of-3 = 3.467e-03   single interpolator (its own worst profile) = 3.678e-03   ratio = 1.06x
  d= 128 n= 12800: median-of-3 = 3.468e-03   single interpolator (its own worst profile) = 3.679e-03   ratio = 1.06x

  boundary probe: d_gamma = infinity (Theorem 3.10 class)
    median-of-three on the d_gamma = infinity class (n=500, ku=4000000): E[L] = 0.87365  -> the O(d_gamma/n) bound is vacuous there, as it must be

verdict = verified