Mathematical verification: F = hbar^2/(gamma*m*r^3) vs k*e^2/r^2
Repository: https://git.esus.name/esus/spin_paper/

Bohr radius calculation:
Force balance: hbar^2/(m*r^3) = k*e^2/r^2
Solving for r: r = hbar^2/(m*k*e^2)
a0 calculated = 5.29177210255e-11 m
a0 defined    = 5.29177210903e-11 m
ratio         = 0.999999998776072

Hydrogen (Z = 1):
  Z_eff = 1.000000
  radius = 52.918 pm = 5.291772e-11 m
  gamma = 1.000027
  F_centripetal = 8.238504e-08 N
  F_coulomb     = 8.238504e-08 N
  ratio = 0.999999998776072
  agreement = 99.9999998776072%
  deviation = 1.224 ppb

Carbon (Z = 6):
  Z_eff = 5.689918
  radius = 9.300 pm = 9.300260e-12 m
  gamma = 1.000959
  F_centripetal = 1.516219e-05 N
  F_coulomb     = 1.516219e-05 N
  ratio = 0.999999998776072
  agreement = 99.9999998776072%
  deviation = 1.224 ppb

Gold (Z = 79):
  Z_eff = 78.688429
  radius = 0.672 pm = 6.724969e-13 m
  gamma = 1.223838
  F_centripetal = 3.279961e-02 N
  F_coulomb     = 3.279961e-02 N
  ratio = 0.999999998776072
  agreement = 99.9999998776072%
  deviation = 1.224 ppb

Verification across periodic table:
Formula: F = hbar^2/(gamma*m*r^3) vs F = k*e^2/r^2

  Z Elem    Z_eff    gamma         F_ratio     Dev(ppb)
------------------------------------------------------------
  1    H    1.000   1.0000  0.999999998776        1.224
  2   He    1.690   1.0001  0.999999998776        1.224
  3   Li    2.690   1.0002  0.999999998776        1.224
  4   Be    3.690   1.0004  0.999999998776        1.224
  5    B    4.690   1.0007  0.999999998776        1.224
  6    C    5.690   1.0010  0.999999998776        1.224
  7    N    6.690   1.0013  0.999999998776        1.224
  8    O    7.690   1.0017  0.999999998776        1.224
  9    F    8.690   1.0022  0.999999998776        1.224
 10   Ne    9.690   1.0027  0.999999998776        1.224
 11   Na   10.690   1.0032  0.999999998776        1.224
 12   Mg   11.690   1.0038  0.999999998776        1.224
 13   Al   12.690   1.0045  0.999999998776        1.224
 14   Si   13.690   1.0053  0.999999998776        1.224
 15    P   14.690   1.0060  0.999999998776        1.224
 16    S   15.690   1.0069  0.999999998776        1.224
 17   Cl   16.690   1.0078  0.999999998776        1.224
 18   Ar   17.690   1.0087  0.999999998776        1.224
 19    K   18.690   1.0098  0.999999998776        1.224
 20   Ca   19.690   1.0108  0.999999998776        1.224
 26   Fe   25.690   1.0185  0.999999998776        1.224
 47   Ag   46.689   1.0646  0.999999998776        1.224
 79   Au   78.688   1.2238  0.999999998776        1.224
 92    U   91.688   1.3493  0.999999998776        1.224
------------------------------------------------------------

Statistical results:
  Elements calculated: 100
  Mean ratio: 0.999999998776072
  Mean agreement: 99.99999987761%
  Mean deviation: 1.223928 ppb
  Std deviation:  0.000000 ppb
  Min deviation:  1.223928 ppb
  Max deviation:  1.223928 ppb
  Range of deviations: 0.000001 ppb
  Deviation range < 1 ppb: True
  All ratios > 0.999: True
  All ratios < 1.001: True

Unit verification:
  F = hbar^2/(gamma*m*r^3)
  [F] = [J*s]^2 / ([kg][m^3]) = [kg*m*s^-2] = [N]
  F = k*e^2/r^2
  [F] = [N*m^2*C^-2][C^2] / [m^2] = [N]

Key numbers:
  Mean force ratio across 100 elements: 0.999999998776072
  Deviation from unity: 1.224 ppb
  Expected if identical: 0.000 ppb
  CODATA electron mass uncertainty: ~300 ppb
