W1  Constants (CODATA 2022) and Aspden's inputs
  1    e (C), m_e (kg), c (m/s), hbar (J s)                                   = 1.602176634e-19, 9.1093837139e-31, 299792458.0, 1.054571817e-34 
  2    q^2 = e^2/(4 pi eps0) (J m), the Gaussian e^2 in SI                    = 2.30708e-28 
  3    Aspden: N = 1843, r/d = (3/4)(8/N)^(1/6)                               = 0.302916 

W2  The units, one formula per line
  1    r = hbar/(2 m_e c)                                                     = 1.9308e-13 m
  2    Omega = (c/2)/r                                                        = 7.76344e+20 rad/s
  3    hbar Omega                                                             = 510.999 keV
  4    d = r/(r/d)                                                            = 6.37403e-13 m
  5    n = 1/d^3 (quons per m^3)                                              = 3.86151e+36 m^-3
  6    sigma = n e (continuum charge density)                                 = 6.18681e+17 C/m^3
  7    k = sigma e / eps0 (Gaussian 4 pi sigma q), force per metre            = 1.11951e+10 N/m
  8    m0 = 2k/Omega^2 (from m0 Omega^2 r = k (2r), Aspden 7.5)               = 0.0407813 m_e
  9    m0 c^2                                                                 = 20.8392 keV
  10   b = q^2/(3 m0 c^2) (Aspden 7.4)                                        = 2.30329e-14 m
  11   a = 2 q^2/(3 m_e c^2) (electron Thomson radius)                        = 1.87863e-15 m
  12   N = (b/a)^3                                                            = 1843.0 

W3  Force on one quon when every quon shifts by u relative to the continuum (Gaussian units, per q u)
  1    own cell's continuum, taken as a sphere: -(4 pi/3) sigma               = -4.18879 sigma 
  2    all other quons and their cells: Lorentz cavity field +(4 pi/3) P, P = sigma u = +4.18879 sigma 
  3    sum of the local terms                                                 = 0 
  4    what remains is q times the macroscopic field E of the displaced lattice =  
  5    Ewald check, k -> 0: longitudinal / w_p^2 (macroscopic field only)     = 1.000000 
  6    Ewald check, k -> 0: transverse / w_p^2 (no local restoring force)     = -1.31e-10 
  7    Ewald check: one quon moved alone (average of D over the zone) / w_p^2, expect 1/3 = 0.33333 

W4  Equations of motion: m0 u_E'' = q E,  M u_G'' = -q E,  E = -4 pi P (uniform motion), P = n q (u_E - u_G)
  1    w_p^2 = 4 pi n q^2/m0 (quons alone, continuum held)                    = 361.331 keV (hbar w_p)
  2    Omega_pl^2 = 4 pi n q^2 (1/m0 + 1/M), M = m0                           = 510.999 keV (hbar Omega_pl)
  3    Omega_pl / Omega (Aspden's orbital frequency)                          = 1.0 
  4    so the orbit is the uniform plasma oscillation of the two frames       =  

W5  Response to a wave of frequency w: P = -(Omega^2/(4 pi w^2)) E, eps(w) = 1 - Omega^2/w^2,
    Maxwell for transverse waves: c^2 k^2 = eps w^2 = w^2 - Omega^2
       photon                            eps  propagates?      v_group/c  1 - v_group/c
       radio, 100 MHz           -1.52666e+24           no              -              -
       visible, 500 nm           -4.2466e+10           no              -              -
       X-ray, 10 keV                 -2610.2           no              -              -
       gamma, 100 keV                -25.112           no              -              -
       Cs-137, 661.7 keV             0.40355          yes     0.63525569         0.3647
       1 MeV                         0.73888          yes     0.85958134         0.1404
       35 MeV                       0.999787          yes     0.99989341      0.0001066
       6 GeV                             1.0          yes            1.0       3.627e-9
       31 GeV                            1.0          yes            1.0      1.359e-10
  1    decay length of a blocked wave, c/sqrt(Omega^2 - w^2), visible light   = 3.86159e-13 m
  2      (equal to 2r, the orbit diameter)                                    = 3.86159e-13 m
  3    GRB 090510: speed difference 35 MeV vs 31 GeV, and the limit (Part 1, U14) = 0.000107  vs  3.6e-18  (x 3.0e+13) 

W6  The same dispersion as a photon with rest energy hbar Omega
  1    photon rest energy implied, hbar Omega                                 = 510999.0 eV
  2    PDG limit (Ryutov 2007, solar wind): limit (eV) and ratio              = 1.0e-18   x 5.1e+23 
  2    laboratory, Lakes 1998: limit (eV) and ratio                           = 1.0e-17   x 5.1e+22 
  2    Coulomb's law, Williams, Faller and Hill 1971: limit (eV) and ratio    = 1.0e-14   x 5.1e+19 

W7  Static fields: a slowly varying applied field E_ext is screened by the factor 1 - w_p^2/w_L(k)^2
       k = 0.01 pi/d    w_L^2/w_p^2 = 1.00003   field left = +0.00003 of the applied field
       k = 0.1 pi/d     w_L^2/w_p^2 = 1.00257   field left = +0.00256 of the applied field
       k = 0.5 pi/d     w_L^2/w_p^2 = 1.05229   field left = +0.04970 of the applied field
       k = pi/d (X)     w_L^2/w_p^2 = 1.10424   field left = +0.09440 of the applied field
  1    wavelength 2d/0.01 in metres (fields smoother than this are cancelled to < 0.01%) = 1.27481e-10 m

W8  Aspden's 1982 resonance picture: each quon held by its own spring of frequency Omega
  1    eps_A(w) = 1 + Omega^2/(Omega^2 - w^2)   (spring plus bare vacuum; Part 1 reading A) =  
  2    eps_B(w) = Omega^2/(Omega^2 - w^2)        (spring, quons carry all displacement; reading B) =  
  3    a spring independent of the macroscopic field would need a local force that W3 shows is absent =  

W9  Free charges with no restoring force at all: the plasma frequency needs only charge, mass and density
  1    w_p = sqrt(n q^2/(eps0 m)) (SI); no spring constant appears            =  
  2    ionosphere F layer, electrons n = 1e12 m^-3: f_p                       = 8.97866e+6 Hz  (hbar w = 3.71328e-8 eV) 
  2    Aspden's quons against a still continuum, n = 1/d^3, m = m0: f_p       = 8.73694e+19 Hz  (hbar w = 361331.0 eV) 
  3    radio below about 9 MHz is reflected by the ionosphere (shortwave 'skip'); same physics, same formula =  

W10 The lattice flexing: its own elastic (shear) waves, from the exact lattice sums at long wavelength
  1    w_p d / c (converts lattice units to fractions of c)                   = 1.16717 
  2    shear-wave speeds along [100] cube axis                                = 0.134c x i (unstable), 0.134c x i (unstable) 
  2    shear-wave speeds along [110] face diagonal                            = 0.134c x i (unstable), 0.279c 
  2    shear-wave speeds along [111] body diagonal                            = 0.215c, 0.215c 
  3    stiffness increase needed for c along [111] and [110]: (c/s)^2         = 21.7 and 12.8 
  4    mass density the waves must move, m0/d^3                               = 143.452 g/cm^3

W11 Periodic motion driven by the cosmic background (T = 2.72548 K)
  1    kT                                                                     = 0.000234864 eV
  2    rms speed if a quon were free, sqrt(3kT/m0)                            = 55.125 km/s
  3    orbital speed Aspden already gives every quon                          = 149896.0 km/s
  4    crystal coupling q^2/(a kT), a = (3/4 pi n)^(1/3) (a Coulomb crystal melts near 175) = 1.55054e+7 
  5    thermal amplitude in the lattice, sqrt(kT/(m0 w_p^2)), and as a fraction of d = 5.79761e-17 m   9.09566e-5 
  6    the response to a wave is linear, so it adds to any existing motion and is unchanged by it =  

W12 The lattice as light's reference frame: three choices
    (a) at rest in the cosmic background's frame (this analysis's first choice, Part 1A v1.1; not Aspden's)
  1    Earth's speed through the background's rest frame (Planck 2018)        = 369.82 km/s
  2    light-speed anisotropy for an ether without length contraction, (1/2)(v/c)^2 = 7.60868e-7 
  3    rotating-cavity limit (Herrmann et al. 2009; Nagel et al. 2015)        = about 1e-17 to 1e-18 
  4    ratio                                                                  = 7.6e+10 
    (b) Aspden's: carried along with the Earth, not turning with it (PU ch. 9; Lecture 8; 1983b)
  5    lab's speed through that frame, Omega_E R cos(latitude), Berlin 52.52 N = 282.69 m/s
  6      the same at 40 N (Aspden 1983: 'some 350 m/s at 40oN')               = 355.889 m/s
  7    expected lab-fixed signal at twice the turntable rate, (1/2)(v/c)^2, Berlin = 4.4458e-13 
  8    measured: lab-fixed signals at twice the rotation rate (Herrmann et al. 2009, Berlin) = below 3e-16 
  9    ratio                                                                  = 1481.93 
  10   largest steady flow of the frame past the Berlin lab that this allows, c sqrt(2 x 3e-16) = 7.34339 m/s
    (c) turning with the Earth (a co-rotating disk or sphere of aether)
  11   rotation a ring laser fixed to the ground would then sense             = 0 (no Sagnac effect) 
  12   measured: the Wettzell ring laser senses the Earth's full rotation (main paper 3.3) = to 48 parts per billion 
  13   in between, Stokes's theorem: mean flow past the ground round Berlin's latitude circle = 282.69 m/s
  14     as a multiple of the cavity limit in line 10                         = 38.4959 

W13 Tutorial Note 18 (from PU p. 67): how a backflow at u = c/2 gives light the speed c relative to the moving Earth
  1    momentum balance (61): (1-k) v + k u = 0, so k = v/(v - u); Aspden's u = -c/2 gives k = v/(v + c/2) ~ 2v/c =  
  2    the lattice is thinner by the fraction k, so (62) n^2 = 1 - k and light's speed is c1 = c/n = c/sqrt(1 - k) =  
  3    to first order c1 = c(1 + k/2) = c - v c/(2u): light gains v times c/(2|u|) =  
  4      gain in light's speed per unit of v, c/(2|u|), |u| = c/4             = 2.0 
  4      gain in light's speed per unit of v, c/(2|u|), |u| = c/2 (Aspden: the quons' orbital speed) = 1.0 
  4      gain in light's speed per unit of v, c/(2|u|), |u| = c               = 0.5 
    Earth's orbital speed, 29.78 km/s
  5      fraction of the lattice in the backflow, k                           = 0.000198631 
  6      c1 - c, the gain in light's speed (first order: v)                   = 29778.5 m/s
  7      light's speed relative to the Earth minus c: ahead, behind, across   = -1.47896, 59558.5, 29777.0 m/s 
  8      Michelson-Morley signal t_along/t_across - 1 with this scalar c1, and (1/2)(v/c)^2 = 4.93278e-9  4.93376e-9 
    Earth through the background, 369.82 km/s
  5      fraction of the lattice in the backflow, k                           = 0.0024611 
  6      c1 - c, the gain in light's speed (first order: v)                   = 369592.0 m/s
  7      light's speed relative to the Earth minus c: ahead, behind, across   = -227.822, 739412.0, 369364.0 m/s 
  8      Michelson-Morley signal t_along/t_across - 1 with this scalar c1, and (1/2)(v/c)^2 = 7.58996e-7  7.60868e-7 
  9    a density change speeds light equally in every direction, so it leaves the Michelson-Morley signal intact; =  
       the null result needs light to move at c relative to the Earth in every direction, which Note 18 asserts =  

W14 The postulate that the two frames move at c relative to each other (Part 1A section 1.2; Part 1 U41)
  1    Aspden: Omega (2r) = c (TG 1960 'assumed'; POC 6.6 'a likely candidate'). Write Omega (2r) = s c =  
  2    the force balance m0 Omega^2 = 8 pi q^2/d^3 contains no r: the frequency is fixed, the orbit's size is free =  
  3    POC 7.19-7.26 redone with s: hbar c/q^2 = 144 pi (r/d)/s,  m0/m_e = (4/9)(r/d)^2/s^2,  N = (9/8)^3 s^6 (d/r)^6 =  
  4    so for a given N: 1/alpha = 108 pi (8/N)^(1/6) and d, m0, b, E0 do not depend on s; only r and r/d scale with s =  
  5    s = 0.9999: zero-energy N, odd N, r/d, 1/alpha, d (m), m0/m_e          = 1843.424  1843  0.3028856  137.0359148  6.37403e-13  0.0407813 
  5    s = 1.0: zero-energy N, odd N, r/d, 1/alpha, d (m), m0/m_e             = 1844.538  1843  0.3029159  137.0359148  6.37403e-13  0.0407813 
  5    s = 1.00004: zero-energy N, odd N, r/d, 1/alpha, d (m), m0/m_e         = 1844.984  1843  0.302928  137.0359148  6.37403e-13  0.0407813 
  6    range of s that keeps N = 1843 (the zero-energy N between 1843 and 1845) = 0.99986198 to 1.0000414 
  7      in parts per million                                                 = -138.0 to +41.45 ppm 
  8    outside it N becomes 1841 or 1845 and 1/alpha jumps by (1/6)(2/1843)   = 180.865 ppm
  9    so the postulate cannot move the 0.6 ppm gap: within the range 1/alpha is unchanged, outside it jumps =  
