
L1  Scales and limits
  1     hbar*Omega, the orbit (Compton) frequency, keV                           510.99895
  1     hbar*p0 = hbar*Omega/sqrt(2), Aspden's 1969 natural frequency, keV       361.3308
  1     orbit radius r = hbar/2 m_e c (m)                                        1.9308e-13
  1     speed of each frame c/2: Lorentz factor gamma                            1.1547
  1     GRB 090510 (z = 0.903, Planck 2018): light-travel time, billion years    7.536
  1       limit on a speed difference, 0.859 s / light-travel time (the simple bound) 3.61e-18
  1       limit on a difference rising as E (weighted by (1+z); Abdo et al., Part 1 U14) 1.87e-18
  1       limit on a constant speed offset (weighted by comoving distance)       2.65e-18
  1       limit on an offset falling as 1/E^2 (weighted by 1/(1+z)^2)            4.77e-18
  1     511 keV annihilation photons, measured speed (Cleland and Jastram 1951) / c 0.995 +- 0.005

L2  The four models: group speed / measured c (the speed of radio waves), or how fast the wave dies away
  R1E1: field of the lattice, added to a separate vacuum term (Part 1A)
  R1E1    radio, 100 MHz: cannot travel; dies away within (m)                    3.86e-13
  R1E1    visible, 2.5 eV: cannot travel; dies away within (m)                   3.86e-13
  R1E1    X-ray, 10 keV: cannot travel; dies away within (m)                     3.86e-13
  R1E1    gamma, 100 keV: cannot travel; dies away within (m)                    3.94e-13
  R1E1    gamma, 300 keV: cannot travel; dies away within (m)                    4.77e-13
  R1E1    annihilation line, low side 510.0 keV: cannot travel; dies away within (m) 6.18e-12
  R1E1    annihilation line, high side 512.3 keV: travels at (/ c_bare; radio cannot travel) 0.0712236
  R1E1    Cs-137, 661.7 keV: travels at (/ c_bare; radio cannot travel)          0.635256
  R1E1    Co-60, 1332.5 keV: travels at (/ c_bare; radio cannot travel)          0.923545
  R1E1    35 MeV: travels at (/ c_bare; radio cannot travel)                     0.999893
  R1E1    31 GeV (GRB 090510): travels at (/ c_bare; radio cannot travel)        1.0
  R1E2: field of the lattice, the lattice is the vacuum term
  R1E2    radio, 100 MHz: cannot travel; dies away within (m)                    3.86e-13
  R1E2    visible, 2.5 eV: cannot travel; dies away within (m)                   3.86e-13
  R1E2    X-ray, 10 keV: cannot travel; dies away within (m)                     3.86e-13
  R1E2    gamma, 100 keV: cannot travel; dies away within (m)                    3.86e-13
  R1E2    gamma, 300 keV: cannot travel; dies away within (m)                    3.86e-13
  R1E2    annihilation line, low side 510.0 keV: cannot travel; dies away within (m) 3.86e-13
  R1E2    annihilation line, high side 512.3 keV: cannot travel; dies away within (m) 3.86e-13
  R1E2    Cs-137, 661.7 keV: cannot travel; dies away within (m)                 3.86e-13
  R1E2    Co-60, 1332.5 keV: cannot travel; dies away within (m)                 3.86e-13
  R1E2    35 MeV: cannot travel; dies away within (m)                            3.86e-13
  R1E2    31 GeV (GRB 090510): cannot travel; dies away within (m)               3.86e-13
  R2E1: own spring at p0, added to a separate vacuum term
  R2E1    radio, 100 MHz: group speed / measured c                               1.0
  R2E1    visible, 2.5 eV: group speed / measured c                              1.0
  R2E1    X-ray, 10 keV: group speed / measured c                                0.999425
  R2E1    gamma, 100 keV: group speed / measured c                               0.939377
  R2E1    gamma, 300 keV: group speed / measured c                               0.255669
  R2E1    annihilation line, low side 510.0 keV: cannot travel; dies away within (m) 4.36e-12
  R2E1    annihilation line, high side 512.3 keV: group speed / measured c       0.0717644
  R2E1    Cs-137, 661.7 keV: group speed / measured c                            0.908376
  R2E1    Co-60, 1332.5 keV: group speed / measured c                            1.34844
  R2E1    35 MeV: group speed / measured c                                       1.41414
  R2E1    31 GeV (GRB 090510): group speed / measured c                          1.41421
  R2E2: own spring at p0, the lattice is the vacuum term (Aspden 1969)
  R2E2    radio, 100 MHz: group speed / measured c                               1.0
  R2E2    visible, 2.5 eV: group speed / measured c                              1.0
  R2E2    X-ray, 10 keV: group speed / measured c                                0.998851
  R2E2    gamma, 100 keV: group speed / measured c                               0.887339
  R2E2    gamma, 300 keV: group speed / measured c                               0.173154
  R2E2    annihilation line, low side 510.0 keV: cannot travel; dies away within (m) 3.85e-13
  R2E2    annihilation line, high side 512.3 keV: cannot travel; dies away within (m) 3.87e-13
  R2E2    Cs-137, 661.7 keV: cannot travel; dies away within (m)                 4.57e-13
  R2E2    Co-60, 1332.5 keV: cannot travel; dies away within (m)                 5.26e-13
  R2E2    35 MeV: cannot travel; dies away within (m)                            5.46e-13
  R2E2    31 GeV (GRB 090510): cannot travel; dies away within (m)               5.46e-13

L3  Tests of the models that let low-energy light through (R2E1, R2E2)
  R2E1  red (700 nm) against blue (400 nm) light, relative speed difference      3.72e-11
  R2E1    times the Warner-Nather limit Aspden quotes (5e-17)                    7.4e+5
  R2E2  red (700 nm) against blue (400 nm) light, relative speed difference      7.43e-11
  R2E2    times the Warner-Nather limit Aspden quotes (5e-17)                    1.5e+6
  R2E1  stop band from p0 (keV)                                                  361.3308
  R2E1    to p0*sqrt(2) = hbar*Omega (keV)                                       510.9989
  R2E1  low-energy eps (static fields reduced by this factor)                    2.0
  R2E1  speed above the stop band / speed of radio waves (31 GeV)                1.41421
  R2E1  GRB 090510: speed difference, 31 GeV against 100 keV (relative)          0.475
  R2E1    times the GRB limit                                                    1.3e+17
  R2E2  GRB 090510: speed difference, 10 keV against 100 keV                     0.112
  R2E2    times the GRB limit                                                    3.1e+16
  R2E2  to let the 31 GeV photon through on time, p0 would have to exceed (eV)   3.3e+19
  R2E2    a restoring force (per unit inertia) larger than Aspden's by           8.6e+27
  R2E2    the same force sets his orbit (PWE 6.1, 6.34): orbit energy would become (eV) 4.7e+19

L4  Why every model with inertia changes speed with energy: the lattice's share of eps
  4     radio, 100 MHz: spring lattice's eps, p0^2/(p0^2 - w^2)                  1.0
  4     visible, 2.5 eV: spring lattice's eps, p0^2/(p0^2 - w^2)                 1.0
  4     100 keV: spring lattice's eps, p0^2/(p0^2 - w^2)                         1.08295
  4     1 MeV: spring lattice's eps, p0^2/(p0^2 - w^2)                           -0.150166
  4     31 GeV: spring lattice's eps, p0^2/(p0^2 - w^2)                          -1.35858e-10
  4     above its natural frequency the lattice's share of eps changes sign and tends to -(p0/w)^2, whatever 
  4       holds the charges (the f-sum limit); with E2 eps tends to 0 from below, with E1 it tends to 1 
  4     smooth-medium scale: k*d = 1 at photon energy hbar*c/d (keV)             309.6
  4       k*d at 361 keV                                                         1.17
  4       k*d at 511 keV                                                         1.65
  4       k*d at 662 keV                                                         2.14
  4       k*r at 511 keV (the orbit is no longer small against the wavelength)   0.5

L5  Relativity at c/2, with the orbit kept at the Compton frequency (m0 = moving mass)
  5x    exact, v = 0.5c: edge for light circularly polarized against the orbit (keV) 478.5
  5x    exact, v = 0.9c: edge for light circularly polarized against the orbit (keV) 401.7
  5x    exact, v = 0.99c: edge for light circularly polarized against the orbit (keV) 377.0
  5x    exact, v = 0.9999c: edge for light circularly polarized against the orbit (keV) 374.1
  5x      light circularly polarized with the orbit, and along the axis: edge stays at (keV) 511.0
  5x      weak sideband at c/2 (keV)                                             1501.0
  5x      limit as v -> c: (sqrt(3) - 1) * hbar*Omega (keV)                      374.1
  5     orbit-averaged approximation, used before the exact calculation:         
  5     gamma = 1.155: cutoff for light polarized in the orbit plane (keV)       478.0
  5     gamma = 10.0: cutoff for light polarized in the orbit plane (keV)        363.1
  5     gamma = 1000.0: cutoff for light polarized in the orbit plane (keV)      361.3
  5       limit for any gamma, in the plane: hbar*Omega/sqrt(2) (keV)            361.3
  5       along the orbit axis, any gamma (keV)                                  511.0
  5     if m0 were the rest mass instead, at c/2: cutoff in the plane / along the axis (keV) 444.8 / 475.5
  5       but the orbit itself would then run at hbar*Omega/sqrt(gamma) (keV)    475.5

L6  The c/2 motion seen by a light wave
  6     visible light (2.5 eV): phase wobble at a quon k*r = w/(2*Omega)         2.45e-6
  6       against the lattice's response, which grows as (Omega/w)^2             4.18e+10
  6     magnetic force: quon (+q, v) and continuum (-q, -v) feel the same q v x B, so it does not separate them 

L7  The dual displacement (muon sea), within linear response
  7     radio, 100 MHz: lattice field E1/E = (f0/f)^2 that the counter-field E2 must cancel 1.53e+24
  7     visible, 2.5 eV: lattice field E1/E = (f0/f)^2 that the counter-field E2 must cancel 4.18e+10
  7     100 keV: lattice field E1/E = (f0/f)^2 that the counter-field E2 must cancel 26.1
  7     fraction of the lattice response left uncancelled, allowed by the photon-mass limit 3.8e-48
  7a    species pushed by the field add: response = sum of n q^2/m over species  always > 0
  7b    momentum-locked muon partner: its dipole / the quons' dipole = m0/m_mu (any muon density) 0.000197
  7b      one muon per quon, same-sign muons: response left / quons' alone       0.999408
  7b      one muon per quon, opposite-sign muons: response left / quons' alone   1.0002
  7b      equal mass densities, same-sign muons: response left / quons' alone    0.499803
  7b      equal mass densities, opposite-sign muons: response left / quons' alone 0.500197
  7b      muons 1% of the quons' mass density, same-sign muons: response left / quons' alone 0.00989708
  7b      muons 1% of the quons' mass density, opposite-sign muons: response left / quons' alone 0.0099049
  7b      to bring the response down to the photon-mass limit by inertia, muon/quon mass density < 3.8e-48
  7c    ionosphere, n_e = 1e12 per m^3: plasma frequency (MHz)                   8.979
  7c      raised by the oxygen ions (O+, 16 u) by the fraction                   1.71e-5

L8  A phase-lock modelled as a gyroscope (our form; gyro frequency w_g = Omega)
  8     opposite senses, E in the orbit plane: eps = 1 + Omega^2/(w_g^2 - w^2); stop band from (keV) 510.9989
  8       to sqrt(w_g^2 + Omega^2) (keV), which is also where the orbit would now run 722.6616
  8       speed above the band / speed of radio waves                            1.41421
  8       E along the orbit axis: eps = 1 - Omega^2/w^2 for any w_g (visible light) -4.18e+10
  8     same sense, visible light along the axis, surviving circular mode: group speed / c 0.00442
  8     with the lattice as the vacuum term (E2), E in the plane: eps = Omega^2/(w_g^2 - w^2); at low energy 1.0
  8       (Aspden's 1982 vacuum resonating at the Compton frequency); nothing travels above (keV) 510.999
  8       E along the axis with E2: eps = -Omega^2/w^2, nothing travels at any energy 
  8     to move the resonance above 1.4 PeV: w_g/Omega for a gyroscope           2.7e+9
  8       or (w0/Omega)^2 for a spring                                           7.5e+18

L9  Above the Compton frequency, where Aspden says the counter-displacement stops (lattice alone: R1E1)
  9     31 GeV photon slowed by (1/2)(511 keV/31 GeV)^2 =                        1.36e-10
  9       delay after the keV emission, z = 0.903 (s)                            2.45e+7
  9       in days                                                                283.0
  9       observed: within 0.859 s; ratio                                        2.8e+7
  9     Cs-137 (661.7 keV): group speed / c                                      0.6353
  9     model (c), low energies at c: above its band it behaves as a plasma at p0; 31 GeV delay (days) 142.0
  9       ratio to the observed 0.859 s                                          1.4e+7
  9     with E2 (the lattice is the vacuum term) eps above Omega is negative for R1 and R2: nothing travels 

L9b The continuum held still during a wave, as Aspden does in 1969 (R1E1 with the edge at p0)
  9b    511 keV: group speed / c                                                 0.7071
  9b    512.3 keV: group speed / c                                               0.7089
  9b    Cs-137, 661.7 keV: group speed / c                                       0.8377
  9b      the 1951 measurement, 0.995 +- 0.005: standard deviations off at 511 keV 57.6

L10 What a lattice added to Maxwell's vacuum (E1) would need
  10    spring-like response: keV light slower than GeV light by f/2, so f <     5.3e-18
  10    plasma-like response: photon mass sqrt(f)*511 keV below 1e-18 eV, so f < 3.8e-48
  10    511 keV photons measured at 0.995c +- 0.005c (1951): R1E1 at 512.3 keV gives / c 0.0712

L11 Aspden's critical field for wave propagation (The Theory of Gravitation, 1960)
  11    his V0 = hc/(16 pi e r^2) with r = hbar/2 m_e c (esu); he prints 2.21e13 2.207e+13
  11      in V/m                                                                 6.616e+17
  11    Schwinger's critical field m^2c^3/(e hbar) (V/m)                         1.323e+18
  11      V0 / (E_S/2): identical, since hc/(16 pi e r^2) = m^2c^3/(2 e hbar) for r = hbar/2mc 1.0
  11    record laser, 1.1e23 W/cm^2 (CoReLS 2021): peak field (V/m)              9.1e+14
  11      ratio V0 / laser field                                                 727.0
  11      laser intensity needed to reach V0 (W/cm^2)                            5.8e+28
  11      times today's record                                                   5.3e+5

L12 Light across the universe: Aspden's frequency loss (1984e eq. 6) against supernova time dilation
  12    DES 2024: light curves stretched by (1+z)^b, b =                         1.003 +- 0.005 (stat) +- 0.010 (sys)
  12    a static space with frequency loss predicts b = 0: standard deviations (stat and sys combined) 89.7

L13 The Galaxy's 511 keV line (Siegert et al. 2016: bulge peak 511.09 +- 0.08 keV, FWHM 2.59 keV)
  13    removing one half of the line moves its centroid by sigma*sqrt(2/pi) (keV) 0.878
  13      in units of the 0.08 keV error                                         11.0

L14 Toward an aether that supplies eps0 and mu0 (section 8; proposed extensions)
  14    lattice static response n e^2/(m0 p0^2) / eps0 (an identity: m0 came from the same force) 1.0000028
  14      with m0 multiplied by 0.1 and p0 recomputed from the same force        1.0
  14      with m0 multiplied by 7.0 and p0 recomputed from the same force        1.0
  14    eps0 = e^2/(2 alpha h c): deriving the number eps0 is deriving alpha (check, / eps0) 1.0
  14    radio, 100 MHz: kinetic / magnetic energy, both frames moving | quons alone 6.55e-25 | 1.31e-24
  14    visible, 2.5 eV: kinetic / magnetic energy, both frames moving | quons alone 2.39e-11 | 4.79e-11
  14    361 keV: kinetic / magnetic energy, both frames moving | quons alone     0.5 | 1.0
  14    511 keV: kinetic / magnetic energy, both frames moving | quons alone     1.0 | 2.0
  14      equal at the medium's own natural frequency (Omega, or p0 with the continuum still): a definition, not a fit 
  14    [P] gyroscopic drift of both frames at right angles to E: kinetic energy / (eps0 E^2/2), any w << Omega 1.0000028
  14    [P] phase locked exactly (delta phi = 0), own spring: in-plane static response / unlocked response -0.5
  14      so with the lattice as the vacuum term, eps = -1/2 for E in the orbit plane: light along the axis is blocked 
  14    photon energy whose wavelength equals the spacing d (keV)                1945.0
  14      wavelength 2d (keV)                                                    972.6
  14      highest frequency a lattice can pass from site to site, hbar*2c/d (keV) 619.2
  14    at 511 keV: wavelength / d = 4 pi r/d                                    3.807
  14    visible light (2.5 eV): wavelength in lattice spacings                   7.781e+5
  14    visible, 2.5 eV: group-speed slowing (kd)^2/8 if carried site to site, nearest neighbours 8.15e-12
  14    X-ray, 1 keV: group-speed slowing (kd)^2/8 if carried site to site, nearest neighbours 1.3e-6
  14    100 keV: group-speed slowing (kd)^2/8 if carried site to site, nearest neighbours 0.013
  14      red (700 nm) against blue (400 nm): group-speed difference             8.44e-12
  14        times the Warner-Nather limit                                        1.7e+5
  14      direction dependence [100] against [111], (kd)^2/36, visible           1.8e-12
  14    orbital angular momentum per quon, m0 (c/2) r = (m0/4 m_e) hbar, in units of hbar 0.0102
  14    orbital angular momentum density of both frames (J s per m^3)            8.3
  14    lattice mass density (g/cm^3)                                            143.5
  14    Coulomb stiffness scale e^2/(4 pi eps0 d^4) (Pa)                         1.4e+21
  14      order-of-magnitude shear speed sqrt(stiffness/density) / c             0.329
  14      the same, written (d/r)/(2 sqrt(8 pi)) with d/r = 144 pi alpha (fixed by the fine-structure chain) 0.329; d/r = 3.3012, 144 pi alpha = 3.3012
  14      Part 1, U38, exact lattice sums: shear waves 0.215c [111], 0.279c [110], unstable along [100] 
  14      quon skin depth c/p0 (m)                                               5.46e-13
  14      shear wave c_t = 0.215c with Faraday's law: group speed / c, visible | radio 0.00244 | 9.92e-7
  14      shear wave c_t = 0.279c with Faraday's law: group speed / c, visible | radio 0.00278 | 1.13e-6
  14      shear wave c_t = 1.0c with Faraday's law: group speed / c, visible | radio 0.00526 | 2.14e-6
  14    c = Omega * 2r holds by definition of r (Part 1A section 3); the test is a model's own wave speed / (2 Omega r) 1.0
  14    any finer lattice that carries GeV light from site to site: spacing below (m) 2.0e-26
  14      times finer than the quon spacing                                      3.1e+13
  14      Planck length (m), for comparison                                      1.6e-35

L15 Vacuum Cherenkov radiation in Aspden's 1969 formula (model d, within his range), LEP electrons
  15    LEP electrons at 104.5 GeV: 1 - beta = 1/(2 gamma^2)                     1.2e-11
  15    model (d), visible light (2.5 eV): n - 1 = (w/p0)^2/2                    2.39e-11
  15      so these electrons outrun light in this vacuum above w = p0/gamma (eV) 1.77
  15    Frank-Tamm energy loss counting emission only up to 3.6 keV (1% of p0) (MeV per metre) 11.9
  15      per 26.7 km turn (GeV), against the beam energy 104.5 GeV              317.0
