U01  Orbit radius r = hbar/(2 m_e c)   [POC 7.1; input taken from the electron, not derived]
  1     r = hbar/(2 m_e c)                                                                1.9307963e-13 m
  2     Dirac zitterbewegung amplitude hbar/(2 m_e c)  (same expression)                  1.9307963e-13 m
  3     range of QED vacuum polarization (Uehling potential ~ exp(-2 m_e c x/hbar)): hbar/2m_e c    1.9307963e-13 m

U02  Orbital angular frequency Omega = (c/2)/r   [POC 7.1, speed c/2 per frame, relative speed c]
  1     Omega                                                                             7.7634407e+20 rad/s
  2     hbar Omega / (m_e c^2)  (= 1: the electron Compton angular frequency)                       1.0 
  3     Dirac zitterbewegung angular frequency 2 m_e c^2/hbar, divided by Omega                     2.0 

U03  Lattice spacing d   [POC 7.24 with N = 1843; PU 156]
  1     r/d = (3/4)(8/1843)^(1/6)                                                          0.3029158782 
  2     d = r/(r/d)                                                                       6.3740348e-13 m
  3     d/a  (Aspden: 108 pi = 339.29; holds when 1/alpha = 144 pi r/d)                       339.29222 
  4     108 pi                                                                                339.29201 
  5     d/r_e (= 72 pi = 226.19)                                                              226.19481 
  6     Aspden 2001 (Fifth Question): cube side 6.374e-11 cm -> in m                          6.374e-13 m

U04  Quon mass from the force balance   [POC 7.5-7.6, 7.25]
  1     m0 from force balance m0 Omega^2 r = 4 pi sigma q (2r)   [m_e units]              0.04078132126 
  2     m0 from (7.25) m0/m_e = (4/9)(r/d)^2                    [m_e units]               0.04078134634 
  3     ratio (identity: should be 1)                                                    0.999999385229 
  4     m0 c^2                                                                                20.839225 keV

U05  Quon charge radius b and the Thomson-type mass   [POC 7.2, 7.4, 7.15, 7.26]
  1     b = d (d/r)^2/(96 pi)                                                             2.3032933e-14 m
  2     b/d  (POC: 0.035)                                                                   0.036135562 
  3     m0 from (7.4) q^2/(3 b c^2), divided by m0 from (7.25)  (identity: 1)            0.999999385229 
  4     quon Thomson energy 2q^2/3b                                                           41.678425 keV
  5     (b/a)^3 = (9/8)^3 (d/r)^6 with his r/d  (1843 by construction)                      1843.003399 
  6     (9/8)^3 (d/r)^6 with the pure-lattice r/d  (POC: 1844.53)                             1844.5379 
  7     f = 1/2 (Aspden): N before rule, odd N, 1/alpha, ppm vs measured                                1844.538  1843  137.0359148  -0.61538 ppm
  7     f = 1 (full Thomson mass): N before rule, odd N, 1/alpha, ppm vs measured                       15254.2  15253  136.262937  -5641.3 ppm
  7     f = 5/4 (sphere plus fluid added mass): N before rule, odd N, 1/alpha, ppm vs measured                  30583.46  30583  135.6683728  -9980.1 ppm

U06  Photon unit: 27 quons, moment of inertia 36 m0 d^2 = 4 h r/(c pi)   [POC 7.19-7.24; PWE 6.59-6.61]
  1     36 m0 d^2 / (4 h r/(c pi))  (identity once r/d fixes m0: 1)                                 1.0 
  2     1/alpha = 144 pi (r/d) with N = 1843 (PU 157: 137.035915)                         137.035914847 
  3     measured 1/alpha (CODATA 2022)                                                    137.035999177 
  4     difference, ppm                                                                         -0.6154 ppm
  5     photon-unit size 3d                                                               1.9122104e-12 m

U07  Number and mass density of the quon lattice   [PWE p.117: 3.87e30 per cc, 144 g/cc]
  1     quons per m^3 = 1/d^3                                                                 3.862e+36 m^-3
  2     per cm^3                                                                              3.862e+30 cm^-3
  3     mass density m0/d^3                                                                       143.5 g/cm^3
  4     continuum charge density sigma = e/d^3                                                6.187e+17 C/m^3

U08  Energy per lattice cell, the 'virtual muon pair'   [POC 7.3, 7.27-7.29; PU 159-160]
  1     E0 from (7.29)  [m_e c^2 units]  (PU 160: 412.665816)                               412.6658162 
  2     E0 from (7.3) with b, d above, divided by (7.29)  (identity: 1)                    0.9999993852 
  3     two muon masses  [m_e units]                                                        413.5365654 
  4     E0 / (2 m_mu) - 1, ppm                                                                  -2105.6 ppm
  5     E0                                                                                      210.872 MeV

U09  Energy density of the aether compared with the observed dark-energy density
  1     E0/d^3                                                                                1.305e+26 J/m^3
  2     dark-energy density Omega_L 3 H0^2 c^2/(8 pi G)  (Planck 2018: H0 67.4, Omega_L 0.6847)        5.251e-10 J/m^3
  3     ratio                                                                                  2.48e+35 
  4     for scale: (200 MeV)^4/(hbar c)^3, a QCD-scale vacuum estimate                         3.34e+34 J/m^3

U10  Natural frequency of a quon displaced against the continuum
  1     natural frequency sqrt(k (1/m0 + 1/M_G)) with M_G = m0, divided by Omega (expect 1)   0.999999692614 
  2     hbar x natural frequency                                                              510.99879 keV
  3     if the continuum did not move (M_G infinite): sqrt(k/m0) x hbar                         361.331 keV

U11  Static polarization of the quon lattice   [POC ch. 6: displacement current = quon displacement]
  1     static P / (eps0 E)  (=1: quon displacement alone gives exactly the vacuum D = eps0 E)              1.0 
  2     the same with M_G = 10 m0 (the identity does not depend on the inertias)                    1.0 

U12  Frequency response if the quons alone carry the displacement current (single displacement)
  B     reading B, photon 0.002 keV: 1 - v_group/c                                            2.298e-11 
  B     reading B, photon 1.0 keV: 1 - v_group/c                                               5.744e-6 
  B     reading B, photon 10.0 keV: 1 - v_group/c                                             0.0005744 
  B     reading B, photon 100.0 keV: 1 - v_group/c                                              0.05689 
  B     reading B, photon 300.0 keV: 1 - v_group/c                                               0.4695 
  B     reading B: eps_r < 0 (no propagating wave) for every photon energy above                510.999 keV
  A     reading A: stop band from (keV)                                                         510.999 keV
  A                          to (keV)                                                           722.662 keV
  A     reading A, photon 100.0 keV: v_group/c                                                0.9704904 
  A     reading A, photon 1.0e+3 keV: v_group/c                                                1.010933 
  A     reading A, photon 6.0e+6 keV: v_group/c                                                1.414214 
  B     reading B: 1/e decay length of a 661.7 keV (Cs-137) wave                              2.453e-13 m
  A     reading A: 1/e decay length of a 661.7 keV wave (inside the stop band)                  6.1e-13 m

U13  Lab speed of GeV photons: Alvaeger et al. 1964, photons > 6 GeV, c' = (2.9977 +/- 0.0004)e8 m/s
  1     measured / c                                                                           0.999925 
  2     reading A prediction / c                                                              1.4142136 
  3     reading A excluded by (sigma)                                                            3105.0 sigma

U14  GRB 090510: keV-MeV and 31 GeV photons arrive within 0.859 s after crossing z = 0.903
  1     lookback (light-travel) time to z = 0.903, Planck 2018 flat LCDM                          7.535 Gyr
  2     allowed |delta v|/c ~ 0.859 s / lookback time (approximate; LIV papers weight by (1+z))         3.61e-18 
  3     reading B: |v(10 keV) - v(300 keV)|/c                                                    0.4689 
  4       arrival spread this would cause, in years                                             3.53e+9 yr
  5       excess over the limit                                                                 1.3e+17 
  6     reading A: |v(10 keV) - v(31 GeV)|/c                                                     0.4145 
  7       excess over the limit                                                                1.15e+17 

U15  What a plain oscillator would need to pass GRB 090510 (resonance far above 31 GeV)
  1     required resonance energy                                                               2.0e+10 GeV
  2       in Planck energies                                                                    1.64e-9 
  3       in electron rest energies (Aspden: exactly 1)                                        3.91e+13 
  4       orbit radius r that would give it, r = hbar c/(2 E_res)                              4.94e-27 m

U16  Lattice discreteness
  1     shortest wave a lattice of spacing d carries: 2d                                      1.275e-12 m
  2       photon energy with that wavelength                                                     0.9726 MeV
  3       highest frequency of a nearest-neighbour lattice wave, hbar 2c/d                       0.6192 MeV
  4     LHAASO 1.4 PeV photon wavelength                                                       8.86e-22 m
  5       2d / that wavelength                                                                  1.44e+9 
  6     lattice-wave speed deficit (kd)^2/24 for 500 nm light                                  2.67e-12 
  7       [100]-[111] direction difference (kd)^2/36 for 500 nm light                          1.78e-12 
  6     lattice-wave speed deficit (kd)^2/24 for 10 keV X-ray                                   4.35e-5 
  7       [100]-[111] direction difference (kd)^2/36 for 10 keV X-ray                            2.9e-5 
  6     lattice-wave speed deficit (kd)^2/24 for 100 keV gamma                                  0.00435 
  7       [100]-[111] direction difference (kd)^2/36 for 100 keV gamma                           0.0029 

U17  Semiconductor light holes   [Lecture 1 appendix: m0/m_e = 0.04078; Ge holes 0.04 and 0.3]
  1     InSb  light hole 0.015 (Ioffe NSM): ratio to Aspden's m0/m_e                              0.368 
  1     Ge    light hole 0.043 (Ioffe NSM): ratio to Aspden's m0/m_e                               1.05 
  1     Ge    light hole 0.044 (Van Zeghbroeck): ratio to Aspden's m0/m_e                          1.08 
  1     GaAs  light hole 0.082 (Van Zeghbroeck): ratio to Aspden's m0/m_e                          2.01 
  1     Si    light hole 0.16 (Van Zeghbroeck / NSM): ratio to Aspden's m0/m_e                     3.92 
  2     spread of light-hole masses across these crystals (max/min)                                10.7 

U18  The 1969 version: r/d = 0.30292 from a 'priming energy' y = 0.000456 e^2/d   [PWE 6.57-6.61, 6.74-6.75]
  1     y = 2 (m0 c^2/8)/5063 in units of e^2/d  (PWE prints 0.000456)                        0.0004555 
  2     1/alpha = 144 pi x 0.30292  (PWE 6.61: 137.038)                                       137.03778 
  3       half-width from rounding r/d to 5 digits, ppm                                            16.5 ppm
  4       vs July 1969 value 137.03602(21) (CODATA 1969), ppm                                      12.8 ppm

U19  The 2003 k term: 8 pi (r/d)^2 = 2.31368 - (1 - 2/3k)(d/r)^4/4608 pi   [POC 7.17-7.18; absent from PU 1980]
  1     F = 1 - 2/3k at which the zero-energy r/d equals the N = 1843 value (exact lattice sum)         0.922413 
  2       k at that point (exact lattice sum; POC: 8.34)                                          8.593 
  1     F = 1 - 2/3k at which the zero-energy r/d equals the N = 1843 value (Aspden's 2.31368)         0.920134 
  2       k at that point (Aspden's 2.31368; POC: 8.34)                                           8.347 
  3     upper bound: displaced charge stays in the cell, kb <= d/2 -> k <= d/(2b)                 13.84 
  4       F at that bound                                                                        0.9518 

U20  The 0.6 ppm is one number: every identity among the units is exact once his own alpha is used
  1     his 1/alpha divided by measured 1/alpha                                          0.999999384616 
  2     m0 (force balance, measured e^2) / m0 (7.25)   [U04]                             0.999999385229 
  3     m0 (force balance, his alpha) / m0 (7.25)                                                   1.0 
  4     b from (7.4) with his alpha / b from (7.15)                                                 1.0 
  5     E0 from (7.3) with his alpha / E0 from (7.29)                                               1.0 

U21  His printed figures re-checked
  1     N with r/d = 0.302874 (POC: 'approximately 1844.53')                                    1844.53 
  2     1/alpha = 108 pi (8/1843)^(1/6) (PU 157: 137.035915)                                137.0359148 
  3     E0 = (3/4 pi)(108 pi)^3 1843^(-4/3) (PU 160: 412.665816)                            412.6658162 
  4     lattice term: his hand sum 2.31368 vs exact Ewald 2.3136987, difference                 1.87e-5 
  5       1/alpha with his hand sum and with the exact sum, after the integer rule          137.0359148 
  6       (exact sum)                                                                       137.0359148 
  7     b/d (POC p.109 prints 0.035)                                                           0.036136 
  8     k b/d with k = 8.34: POC says 'more than 0.29d'                                          0.3014 

U22  What correction would close the gap (final 1/alpha depends only on N and two relations)
  1     required change to the photon-unit relation, eps (1/alpha moves by 1.5 eps)             0.41026 ppm
  2     required change to the size relation N = (9/8)^3 (d/r)^6, delta (1/alpha moves by delta/6)           3.6923 ppm
  3       equivalently the N that would give measured alpha                                   1842.9932 
  4     required change to the orbit radius r = hbar/2 m_e c (enters N as its cube)              1.2308 ppm
  5     finite-size factor F = 0.93: odd N, 1/alpha  (lattice details only pick the integer)                  1843  137.035914847
  5     finite-size factor F = 0.95: odd N, 1/alpha  (lattice details only pick the integer)                  1843  137.035914847
  5     finite-size factor F = 1.0: odd N, 1/alpha  (lattice details only pick the integer)                  1843  137.035914847
  5     finite-size factor F = 1.02: odd N, 1/alpha  (lattice details only pick the integer)                  1843  137.035914847
  6     today's uncertainty of 1/alpha, ppm (a real fix must land within this)                 0.000153 ppm

U23  The restoring coefficient 4 pi sigma q (Gaussian)
  1     his coefficient (unbounded medium, Maxwell displacement current): 1/alpha           137.0359148 
  2     electrostatic sphere coefficient 4 pi/3 instead: odd N, 1/alpha                                 67  79.363639
  3     cylinder (axis normal to the orbit plane) 2 pi instead: odd N, 1/alpha                          229  96.995752

U24  The photon unit's moment of inertia 36 m0 d^2: what it leaves out
  1     spin rate Omega/4; speed of the 12 quons at distance d, as a fraction of c              0.41266 
  2     speed of the 12 quons at distance d sqrt 2                                              0.58358 
  3     angular momentum increase if mass rises with speed (Aspden accepts this, PU ch. 4)            18.69 %
  4     speed-dependent mass: change to I, then odd N and 1/alpha                                       18.69 %   N 3083   1/alpha 162.63672
  4     quons spin with the unit (field mass out to d/2): change to I, then odd N and 1/alpha                  1.257 %   N 1913   1/alpha 138.76332
  4     quons spin with the unit (uniform-mass spheres): change to I, then odd N and 1/alpha                  0.03917 %   N 1845   1/alpha 137.09166
  4     continuum co-rotates: change to I, then odd N and 1/alpha                                       112.5 %   N 17699   1/alpha 291.16259
  5     the change actually needed, eps                                                        4.103e-5 %

U25  The 2003 k computed from Aspden's own description (displaced charge spread through the cell)
  1     mean of d/R over the cube (POC 7.12 prints 2.380077382)                             2.380077364 
  2     k = d/(b x mean of d/R outside the void)                                                 11.665 
  3     F = 1 - 2/3k                                                                           0.942849 
  4     N before rule, odd N, 1/alpha with the integer rule                                             1843.405  1843  137.0359148
  5     pure lattice with this k (no integer rule): 1/alpha, ppm                                        137.0308995   -37.21 ppm
  6     pure lattice with Aspden's F = 1, for comparison: 1/alpha, ppm                                  137.0168651   -139.6 ppm

U26  Small numbers of about the right size (shown to make the point that matching is not fixing)
  1     eps = alpha^3 = 0.3886 ppm: 1/alpha, residual ppm, residual in today's sigma                    137.035994724   -0.0325 ppm   -212.0 sigma
  1     eps = 1/N^2 = 0.2944 ppm: 1/alpha, residual ppm, residual in today's sigma                      137.035975364   -0.174 ppm   -1134.0 sigma
  1     eps = (alpha/2 pi)^2 = 1.349 ppm: 1/alpha, residual ppm, residual in today's sigma                  137.036192113   1.41 ppm   9187.0 sigma
