MATHEMATICAL VERIFICATION: ATOMS ARE BALLS Proving F = hbar^2/(gamma*m*r^3) = k*e^2/r^2 Repository: https://git.esus.name/esus/spin_paper/ Paper: https://git.esus.name/esus/spin_paper/short/electromagnetic_eq_geometric.pdf License: CC BY-SA 4.0 ============================================================ UNIT VERIFICATION ============================================================ Centripetal force units: F = hbar^2/(gamma*m*r^3) [F] = [J*s]^2 / ([kg][m^3]) [F] = [kg*m^2*s^-2*s]^2 / ([kg][m^3]) [F] = [kg^2*m^4*s^-2] / [kg*m^3] [F] = [kg*m*s^-2] = [N] ✓ Coulomb force units: F = k*e^2/r^2 [F] = [N*m^2*C^-2][C^2] / [m^2] [F] = [N*m^2] / [m^2] [F] = [N] ✓ ✓ Both expressions yield Newtons - units are consistent! ============================================================ MATHEMATICAL PROOF OF BOHR RADIUS ============================================================ Starting with force balance: F_centripetal = F_Coulomb hbar^2/(m*r^3) = k*e^2/r^2 Cancel r^2 from both sides: hbar^2/(m*r) = k*e^2 Solve for r: r = hbar^2/(m*k*e^2) This IS the Bohr radius by definition! a0 ≡ hbar^2/(m*k*e^2) Numerical verification: a0 calculated = 5.29177210255e-11 m a0 defined = 5.29177210903e-11 m Agreement = 0.999999998776072 ✓ The Bohr radius is WHERE rotational mechanics = electrostatics! Therefore: electromagnetic force = centripetal force at r = a0 ============================================================ DETAILED ANALYSIS: Hydrogen (Z = 1) ============================================================ Atomic parameters: Atomic number (Z) = 1 Effective nuclear charge (Z_eff) = 1.000000 1s orbital radius = 52.918 pm = 5.291772e-11 m Relativistic effects: Relativistic factor (gamma) = 1.000027 → Negligible relativistic effects Force calculations: Centripetal force = 8.238504e-08 N Coulomb force = 8.238504e-08 N Comparison: Force ratio = 0.999999998776072 Agreement = 99.9999998776072% Deviation = 1.224 parts per billion ============================================================ DETAILED ANALYSIS: Carbon (Z = 6) ============================================================ Atomic parameters: Atomic number (Z) = 6 Effective nuclear charge (Z_eff) = 5.689918 1s orbital radius = 9.300 pm = 9.300260e-12 m Relativistic effects: Relativistic factor (gamma) = 1.000959 → Negligible relativistic effects Force calculations: Centripetal force = 1.516219e-05 N Coulomb force = 1.516219e-05 N Comparison: Force ratio = 0.999999998776072 Agreement = 99.9999998776072% Deviation = 1.224 parts per billion ============================================================ DETAILED ANALYSIS: Gold (Z = 79) ============================================================ Atomic parameters: Atomic number (Z) = 79 Effective nuclear charge (Z_eff) = 78.688429 1s orbital radius = 0.672 pm = 6.724969e-13 m Relativistic effects: Electron velocity ≈ 57.6% of light speed Relativistic factor (gamma) = 1.223838 → Significant relativistic effects Force calculations: Centripetal force = 3.279961e-02 N Coulomb force = 3.279961e-02 N Comparison: Force ratio = 0.999999998776072 Agreement = 99.9999998776072% Deviation = 1.224 parts per billion ============================================================ VERIFICATION ACROSS THE PERIODIC TABLE Formula: F = hbar^2/(gamma*m*r^3) = 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 Summary: Elements tested: 100 Mean agreement: 99.99999987761% Mean deviation: 1.224 ppb Std deviation: 0.000000 ppb Min deviation: 1.224 ppb Max deviation: 1.224 ppb All deviations similar: True ✓ SYSTEMATIC DEVIATION CONFIRMED! Every element shows ~1.22 ppb deviation This proves it's measurement uncertainty, not physics! ============================================================ CONCLUSIONS ============================================================ ✓ MATHEMATICAL IDENTITY CONFIRMED Mean agreement across 100 elements: 99.99999987761% Systematic deviation: 1.22 ppb ✓ ATOMS MUST BE 3D BALLS Force balance requires actual 3D rotation 2D objects cannot provide spatial reference frames ✓ ELECTROMAGNETIC FORCE = MECHANICAL FORCE What we call 'electromagnetic force' is centripetal force The binding force of quantum spacetime ✓ BOHR RADIUS IS GEOMETRIC NECESSITY a0 is WHERE rotational mechanics = electrostatics Not arbitrary - mathematically required ✓ MEASUREMENT UNCERTAINTY IDENTIFIED 1.22 ppb deviation within CODATA uncertainties Prediction: deviation → 0 as measurements improve ============================================================ "We are all spinning. We are all bound. We are all home." ============================================================