# Electron (mathematical)

Mathematical Electron

The mathematical electron model [1] can be used in Simulation Hypothesis (the premise that the Universe is a Computer Simulation) programs to derive physical structures (the physical constants; G, h, c, e, me, kB, ... ) from a single mathematical object - a dimensionless electron formula fe (see Mathematical Universe), such that the physical structures are indistinguishable from their respective mathematical objects. fe is derived from the geometry of 2 dimensionless physical constants, the fine structure constant α and Omega Ω, and so fe is also a natural (dimensionless) physical constant independent of any system of units.

${\displaystyle f_{e}=4\pi ^{2}(2^{6}3\pi ^{2}\alpha \Omega ^{5})^{3}=.23895453...x10^{23},\;units=1}$

## Geometrical objects

From fe, the Planck units; mass ${\displaystyle M}$ , length ${\displaystyle L}$ , time ${\displaystyle T}$ , and ampere ${\displaystyle A}$  are defined as dimensionless geometrical objects in terms of (α, Ω). Being independent of any numerical system and of any system of units, these MLTA units qualify as "natural units";

...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...

...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"... -Max Planck [2][3]
${\displaystyle M=(1)}$
${\displaystyle T=(2\pi )}$
${\displaystyle P=(\Omega )}$
${\displaystyle V=(2\pi \Omega ^{2})}$
${\displaystyle L=(2\pi ^{2}\Omega ^{2})}$
${\displaystyle A=({\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }})}$

## Unit u

The SI units (kg, m, s, A, k) are replaced by mathematical relationships between the geometrical objects.

${\displaystyle u^{3}\;}$  (ampere)
${\displaystyle u^{-13}\;}$  (length)
${\displaystyle u^{15}\;}$  (mass)
${\displaystyle u^{16}\;}$  ( sqrt of momentum)
${\displaystyle u^{17}\;}$  (velocity)
${\displaystyle u^{-30}\;}$  (time)

## Scalars

To translate from geometrical objects to a numerical system requires scalars (ktpvpa) that can be assigned numerical values.

Geometrical units
Unit Geometrical object Scalar
mass ${\displaystyle M=1}$  ${\displaystyle k,\;unit=u^{15}}$
time ${\displaystyle T=2\pi }$  ${\displaystyle t,\;unit=u^{-30}}$
momentum (sqrt of) ${\displaystyle P=\Omega }$  ${\displaystyle p,\;unit=u^{16}}$
velocity ${\displaystyle V=2\pi \Omega ^{2}}$  ${\displaystyle v,\;unit=u^{17}}$
length ${\displaystyle L=2\pi ^{2}\Omega ^{2}}$  ${\displaystyle l,\;unit=u^{-13}}$
ampere ${\displaystyle A={\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }}}$  ${\displaystyle a,\;unit=u^{3}}$

### Scalar relationships

The following un groups cancel, as such only 2 (associated) scalars are actually required, for example, if we know a and l then we know k and t (as in the following examples). AL as an ampere-meter (ampere-length) are the units for a magnetic monopole.

${\displaystyle {\frac {a^{3}l^{3}}{t}}({\frac {{u^{3}}^{3}{u^{-13}}^{3}}{u^{-30}}})={\frac {l^{15}}{k^{9}t^{11}}}({\frac {{u^{-13}}^{15}}{{u^{15}}^{9}{u^{-30}}^{11}}})=\;...\;=1}$

#### MT to LPVA

In this example units LPVA are derived from MT. The formulas for MT;

${\displaystyle M=(1)k,\;unit=u^{15}}$
${\displaystyle T=(2\pi )t,\;unit=u^{-30}}$

Replacing scalars pvla with kt

${\displaystyle P=(\Omega )\;{\frac {k^{12/15}}{t^{2/15}}},\;unit=u^{12/15*15-2/15*(-30)=16}}$
${\displaystyle V={\frac {2\pi P^{2}}{M}}=(2\pi \Omega ^{2})\;{\frac {k^{9/15}}{t^{4/15}}},\;unit=u^{9/15*15-4/15*(-30)=17}}$
${\displaystyle L={\frac {TV}{2}}=(2\pi ^{2}\Omega ^{2})\;k^{9/15}t^{11/15},\;unit=u^{9/15*15+11/15*(-30)=-13}}$
${\displaystyle A={\frac {8V^{3}}{\alpha P^{3}}}=\left({\frac {64\pi ^{3}\Omega ^{3}}{\alpha }}\right)\;{\frac {1}{k^{3/5}t^{2/5}}},\;unit=u^{9/15*(-15)+6/15*30=3}}$

#### PV to MTLA

In this example units MLTA are derived from PV. The formulas for PV;

${\displaystyle P=(\Omega )p,\;unit=u^{16}}$
${\displaystyle V=(2\pi \Omega ^{2})v,\;unit=u^{17}}$

Replacing scalars klta with pv

${\displaystyle M={\frac {2\pi P^{2}}{V}}=(1){\frac {p^{2}}{v}},\;unit=u^{16*2-17=15}}$
${\displaystyle T^{2}=(2\pi \Omega )^{15}{\frac {P^{9}}{2\pi V^{12}}}}$
${\displaystyle T=(2\pi ){\frac {p^{9/2}}{v^{6}}},\;unit=u^{16*9/2-17*6=-30}}$
${\displaystyle L={\frac {TV}{2}}=(2\pi ^{2}\Omega ^{2}){\frac {p^{9/2}}{v^{5}}},\;unit=u^{16*9/2-17*5=-13}}$
${\displaystyle A={\frac {8V^{3}}{\alpha P^{3}}}=({\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }}){\frac {v^{3}}{p^{3}}},\;unit=u^{17*3-16*3=3}}$

### Physical constants

In this example, to maintain integer exponents, scalar p is defined in terms of a scalar r.

${\displaystyle r={\sqrt {p}}={\sqrt {\Omega }},\;unit\;u^{16/2=8}}$

As α and Ω have fixed values, 2 scalars are also needed to solve the physical constants. The Planck units are known with a low precision, conversely 2 of the CODATA 2014 physical constants have been assigned exact numerical values; c and permeability of vacuum μ0. Thus scalars r and v were chosen as they can be derived directly from V = c and μ0.

${\displaystyle v={\frac {c}{2\pi \Omega ^{2}}}=11843707.9...,\;units=m/s}$
${\displaystyle r^{7}={\frac {2^{11}\pi ^{5}\Omega ^{4}\mu _{0}}{\alpha }};\;r=.712562514...,\;units=({\frac {kg.m}{s}})^{1/4}}$
Physical constants; geometrical vs experimental (CODATA)
Constant As Planck units Geometrical object Calculated (r, v, Ω, α*) CODATA 2014 [4]
Speed of light V ${\displaystyle c^{*}=(2\pi \Omega ^{2})v,\;u^{17}}$  c* = 299 792 458, unit = u17 c = 299 792 458 (exact)
Fine structure constant α* = 137.035 999 139 (mean) α = 137.035 999 139(31)
Rydberg constant ${\displaystyle R^{*}=({\frac {m_{e}}{4\pi L\alpha ^{2}M}})}$  ${\displaystyle R^{*}={\frac {1}{2^{23}3^{3}\pi ^{11}\alpha ^{5}\Omega ^{17}}}{\frac {v^{5}}{r^{9}}},\;u^{13}}$  R* = 10 973 731.568 508, unit = u13 R = 10 973 731.568 508(65)
Vacuum permeability ${\displaystyle \mu _{0}^{*}={\frac {\pi V^{2}M}{\alpha LA^{2}}}}$  ${\displaystyle \mu _{0}^{*}={\frac {\alpha }{2^{11}\pi ^{5}\Omega ^{4}}}r^{7},\;u^{17*2+15+13-6=7*8=56}}$  μ0* = 4π/10^7, unit = u56 μ0 = 4π/10^7 (exact)
Planck constant ${\displaystyle h^{*}=2\pi MVL}$  ${\displaystyle h^{*}=2^{3}\pi ^{4}\Omega ^{4}{\frac {r^{13}}{v^{5}}},\;u^{15+17-13=8*13-17*5=19}}$  h* = 6.626 069 134 e-34, unit = u19 h = 6.626 070 040(81) e-34
Gravitational constant ${\displaystyle G^{*}={\frac {V^{2}L}{M}}}$  ${\displaystyle G^{*}=2^{3}\pi ^{4}\Omega ^{6}{\frac {r^{5}}{v^{2}}},\;u^{34-13-15=8*5-17*2=6}}$  G* = 6.672 497 192 29 e11, unit = u6 G = 6.674 08(31) e-11
Elementary charge ${\displaystyle e^{*}=AT}$  ${\displaystyle e^{*}={\frac {2^{7}\pi ^{4}\Omega ^{3}}{\alpha }}{\frac {r^{3}}{v^{3}}},\;u^{3-30=3*8-17*3=-27}}$  e* = 1.602 176 511 30 e-19, unit = u-19 e = 1.602 176 620 8(98) e-19
Boltzmann constant ${\displaystyle k_{B}^{*}={\frac {\pi VM}{A}}}$  ${\displaystyle k_{B}^{*}={\frac {\alpha }{2^{5}\pi \Omega }}{\frac {r^{10}}{v^{3}}},\;u^{17+15-3=10*8-17*3=29}}$  kB* = 1.379 510 147 52 e-23, unit = u29 kB = 1.380 648 52(79) e-23
Electron mass ${\displaystyle m_{e}^{*}={\frac {M}{f_{e}}},\;u^{15}}$  me* = 9.109 382 312 56 e-31, unit = u15 me = 9.109 383 56(11) e-31
Classical electron radius ${\displaystyle \lambda _{e}^{*}=2\pi Lf_{e},\;u^{-13}}$  λe* = 2.426 310 2366 e-12, unit = u-13 λe = 2.426 310 236 7(11) e-12
Planck temperature ${\displaystyle T_{p}^{*}={\frac {AV}{\pi }}}$  ${\displaystyle T_{p}^{*}={\frac {2^{7}\pi ^{3}\Omega ^{5}}{\alpha }}{\frac {v^{4}}{r^{6}}},\;u^{3+17=17*4-6*8=20}}$  Tp* = 1.418 145 219 e32, unit = u20 Tp = 1.416 784(16) e32
Planck mass M ${\displaystyle m_{P}^{*}=(1){\frac {r^{4}}{v}},\;u^{15}}$  mP* = .217 672 817 580 e-7, unit = u15 mP = .217 647 0(51) e-7
Planck length L ${\displaystyle l_{p}^{*}=(2\pi ^{2}\Omega ^{2}){\frac {r^{9}}{v^{5}}},\;u^{-13}}$  lp* = .161 603 660 096 e-34, unit = u-13 lp = .161 622 9(38) e-34
Planck time T ${\displaystyle t_{p}^{*}=(2\pi ){\frac {r^{9}}{v^{6}}},\;u^{-30}}$  tp* = 5.390 517 866 e-44, unit = u-30 tp = 5.391 247(60) e-44
Ampere A ${\displaystyle A^{*}={\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }}{\frac {v^{3}}{r^{6}}},\;u^{3}}$  A^* = 0.148 610 6299 e25, unit = u3
Von Klitzing constant ${\displaystyle R_{K}^{*}=({\frac {h}{e^{2}}})^{*}}$  RK* = 25812.807 455 59, unit = u73 RK = 25812.807 455 5(59)
Gyromagnetic ratio ${\displaystyle \gamma _{e}/2\pi ={\frac {gl_{p}^{*}m_{P}^{*}}{2k_{B}^{*}m_{e}^{*}}},\;unit=u^{-13-29=3-30-15=-42}}$  γe/2π* = 28024.953 55, unit = u-42 γe/2π = 28024.951 64(17)

Note that r, v, Ω, α are dimensionless numbers, it is only when we replace un with the SI unit equivalents (u15 → kg, u-13 → m, u-30 → s, ...) that the geometrical objects (i.e.: c* = 2πΩ2v = 299792458, units = u17) become indistinguishable from their respective physical constants (i.e.: c = 299792458, units = m/s).

### SI Planck unit scalars

${\displaystyle M=(1)k;\;k=m_{P}=.21767281758...\;10^{-7},\;u^{15}\;(kg)}$
${\displaystyle T={2\pi }t;\;t={\frac {t_{p}}{2\pi }}=.17158551284...10^{-43},\;u^{-30}\;(s)}$
${\displaystyle L={2\pi ^{2}\Omega ^{2}}l;\;l={\frac {l_{p}}{2\pi ^{2}\Omega ^{2}}}=.20322086948...10^{-36},\;u^{-13}\;(m)}$
${\displaystyle V={2\pi \Omega ^{2}}v;\;v={\frac {c}{2\pi \Omega ^{2}}}=11843707.90527...,\;u^{17}\;(m/s)}$
${\displaystyle A=({\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }})a;\;a={\frac {A\alpha }{64\pi ^{3}\Omega ^{3}}}=.12691858859...10^{23},\;u^{3}\;(A)}$

Example MLT;

${\displaystyle {\frac {L^{15}}{M^{9}T^{11}}}={\frac {l_{p}^{15}}{m_{P}^{9}t_{p}^{11}}}={\frac {(2\pi ^{2}\Omega ^{2}l)^{15}}{(1k)^{9}(2\pi t)^{11}}}=2^{4}\pi ^{19}\Omega ^{30}}$
${\displaystyle {\frac {l^{15}}{k^{9}t^{11}}}={\frac {(.203...x10^{-36})^{15}}{(.217...x10^{-7})^{9}(.171...x10^{-43})^{11}}}{\frac {u^{-13*15}}{u^{15*9}u^{-30*11}}}=1}$

Example ALT;

${\displaystyle {\frac {A^{3}L^{3}}{T}}={\frac {A_{p}^{3}l_{p}^{3}}{t_{p}}}={\frac {(2^{6}\pi ^{3}\Omega ^{3}a)^{3}(2\pi ^{2}\Omega ^{2}l)^{3}}{(\alpha )^{3}(2\pi t)}}={\frac {2^{20}\pi ^{14}\Omega ^{15}}{\alpha ^{3}}}}$
${\displaystyle {\frac {a^{3}l^{3}}{t}}={\frac {(.126...x10^{23})^{3}(.203...x10^{-36})^{3}}{(.171...x10^{-43})}}{\frac {u^{3*3}u^{-13*3}}{u^{-30}}}=1}$

Example PV;

The geometry Ω15 is common to unit-less ratios.

${\displaystyle {\frac {L^{30}}{M^{18}T^{22}}}={\frac {2^{180}\pi ^{210}\Omega ^{225}P^{135}}{V^{150}}}/{\frac {2^{18}\pi ^{18}P^{36}}{V^{18}}}.{\frac {2^{154}\pi ^{154}\Omega ^{165}P^{99}}{V^{132}}}}$
${\displaystyle {\frac {L^{30}}{M^{18}T^{22}}}={(2^{4}\pi ^{19}\Omega ^{30})}^{2}}$
${\displaystyle {\frac {A^{6}L^{6}}{T^{2}}}={\frac {2^{18}V^{18}}{\alpha ^{6}P^{18}}}.{\frac {2^{36}\pi ^{42}\Omega ^{45}P^{27}}{V^{30}}}/{\frac {2^{14}\pi ^{14}\Omega ^{15}P^{9}}{V^{12}}}}$
${\displaystyle {\frac {A^{6}L^{6}}{T^{2}}}=({\frac {2^{20}\pi ^{14}\Omega ^{15}}{\alpha ^{3}}})^{2}}$

### Electron formula

Although the Planck units MLTA can be expressed in terms of the electron formula fe, this formula is both unit-less and non scalable k0t0v0r0a0u0 = 1. It is therefore a dimensionless physical constant, σe has units for a magnetic monopole, σtp a temperature `monopole'.

${\displaystyle T=(2\pi ){\frac {r^{9}}{v^{6}}},\;u^{-30}}$
${\displaystyle \sigma _{e}={\frac {3\alpha ^{2}AL}{\pi ^{2}}}={2^{7}3\pi ^{3}\alpha \Omega ^{5}}{\frac {r^{3}}{v^{2}}},\;u^{-10}}$
${\displaystyle f_{e}={\frac {\sigma _{e}^{3}}{T}}={\frac {(2^{7}3\pi ^{3}\alpha \Omega ^{5})^{3}}{2\pi }},\;units={\frac {(u^{-10})^{3}}{u^{-30}}}=1}$
${\displaystyle \sigma _{tp}={\frac {3\alpha ^{2}T_{P}}{2\pi }}={2^{6}3\pi ^{2}\alpha \Omega ^{5}}{\frac {v^{4}}{r^{6}}},\;units=u^{20}}$
${\displaystyle f_{e}=T^{2}\sigma _{tp}^{3}=4\pi ^{2}({2^{6}3\pi ^{2}\alpha \Omega ^{5}})^{3},\;units=(u^{-30})^{2}(u^{20})^{3}=1}$

### Fine structure constant

The Sommerfeld fine structure constant alpha is a dimensionless physical constant. The following uses a common formula for (inverse) alpha = 1/137.03599...

${\displaystyle \alpha ={\frac {2h}{\mu _{0}e^{2}c}}}$
${\displaystyle \alpha =2({8\pi ^{4}\Omega ^{4}})/({\frac {\alpha }{2^{11}\pi ^{5}\Omega ^{4}}})({\frac {128\pi ^{4}\Omega ^{3}}{\alpha }})^{2}(2\pi \Omega ^{2})=\alpha }$
${\displaystyle scalars={\frac {r^{13}}{v^{5}}}.{\frac {1}{r^{7}}}.{\frac {v^{6}}{r^{6}}}.{\frac {1}{v}}=1}$
${\displaystyle units={\frac {u^{19}}{u^{56}(u^{-27})^{2}u^{17}}}=1}$

### Omega

The most precise of the experimentally measured constants is the Rydberg R = 10973731.568508(65) 1/m. Here c, μ0, R are combined into a unit-less ratio;

${\displaystyle {\frac {(c^{*})^{35}}{(\mu _{0}^{*})^{9}(R^{*})^{7}}}=(2\pi \Omega ^{2})^{35}/({\frac {\alpha }{2^{11}\pi ^{5}\Omega ^{4}}})^{9}.({\frac {1}{2^{23}3^{3}\pi ^{11}\alpha ^{5}\Omega ^{17}}})^{7}}$
${\displaystyle units={\frac {(u^{17})^{35}}{(u^{56})^{9}(u^{13})^{7}}}=1}$

We can now define Ω using the geometries for (c*, μ0*, R*) and then solve by replacing (c*, μ0*, R*) with the numerical (c, μ0, R) CODATA 2014 values.

${\displaystyle \Omega ^{225}={\frac {(c^{*})^{35}}{2^{295}3^{21}\pi ^{157}(\mu _{0}^{*})^{9}(R^{*})^{7}\alpha ^{26}}},\;units=1}$
${\displaystyle \Omega =2.007\;134\;9496...,\;units=1}$

There is a close natural number for Ω that is a square root implying that Ω can have a plus or a minus solution;

${\displaystyle \Omega ={\sqrt {\left({\frac {\pi ^{e}}{e^{(e-1)}}}\right)}}=2.007\;134\;9543...}$

### G, h, e, me, kB

The physical constants (G, h, e, me, kB) can then be rewritten using the geometrical formulas for (c*, μ0*, R*) and solved using the CODATA 2014 numerical values for (c, μ0, R, α), i.e.:.

${\displaystyle {(h^{*})}^{3}=(2^{3}\pi ^{4}\Omega ^{4}{\frac {r^{13}u^{19}}{v^{5}}})^{3}={\frac {2\pi ^{10}{(\mu _{0}^{*})}^{3}}{3^{6}{(c^{*})}^{5}\alpha ^{13}{(R^{*})}^{2}}},\;unit=u^{57}}$

Physical constants; calculated vs experimental (CODATA)
Constant Geometry Calculated from (R*, c, μ0, α*) CODATA 2014 [5]
Speed of light ${\displaystyle c^{*}=(2\pi \Omega ^{2})v,\;unit=u^{17}}$  c* = 299 792 458, unit = u17
Fine structure constant ${\displaystyle \alpha ^{3}={\frac {8(h^{*})^{3}}{(\mu _{0}^{*})^{3}(e^{*})^{6}(c^{*})^{3}}}=\alpha ^{3},\;unit=1}$  α* = 137.035 999 139
Rydberg constant ${\displaystyle R^{*}={\frac {1}{2^{23}3^{3}\pi ^{11}\alpha ^{5}\Omega ^{17}}}{\frac {v^{5}}{r^{9}}},\;unit=u^{13}}$  R* = 10 973 731.568 508, unit = u13
Vacuum permeability ${\displaystyle \mu _{0}^{*}={\frac {\alpha }{2^{11}\pi ^{5}\Omega ^{4}}}r^{7},\;unit=u^{56}}$  μ0* = 4π/10^7, unit = u56
Planck constant ${\displaystyle {(h^{*})}^{3}={\frac {2\pi ^{10}{(\mu _{0}^{*})}^{3}}{3^{6}{(c^{*})}^{5}\alpha ^{13}{(R^{*})}^{2}}},\;unit=u^{57}}$  h* = 6.626 069 134 e-34, unit = u19 h = 6.626 070 040(81) e-34
Gravitational constant ${\displaystyle {(G^{*})}^{5}={\frac {\pi ^{3}{(\mu _{0}^{*})}}{2^{20}3^{6}\alpha ^{11}{(R_{\infty }^{*})}^{2}}},\;unit=u^{30}}$  G* = 6.672 497 192 29 e11, unit = u6 G = 6.674 08(31) e-11
Elementary charge ${\displaystyle {(e^{*})}^{3}={\frac {4\pi ^{5}}{3^{3}{(c^{*})}^{4}\alpha ^{8}{(R_{\infty }^{*})}}},\;unit=u^{-81}}$  e* = 1.602 176 511 30 e-19, unit = u-19 e = 1.602 176 620 8(98) e-19
Boltzmann constant ${\displaystyle {(k_{B}^{*})}^{3}={\frac {\pi ^{5}{(\mu _{0}^{*})}^{3}}{3^{3}2{(c^{*})}^{4}\alpha ^{5}{(R_{\infty }^{*})}}},\;unit=u^{87}}$  kB* = 1.379 510 147 52 e-23, unit = u29 kB = 1.380 648 52(79) e-23
Electron mass ${\displaystyle {(m_{e}^{*})}^{3}={\frac {16\pi ^{10}{(R_{\infty }^{*})}{(\mu _{0}^{*})}^{3}}{3^{6}{(c^{*})}^{8}\alpha ^{7}}},\;unit=u^{45}}$  me* = 9.109 382 312 56 e-31, unit = u15 me = 9.109 383 56(11) e-31
Planck mass ${\displaystyle (m_{P}^{*})^{15}={\frac {2^{25}\pi ^{13}(\mu _{0}^{*})^{6}}{3^{6}(c^{*})^{5}\alpha ^{16}(R_{\infty }^{*})^{2}}},\;unit=(u^{15})^{15}}$  mP* = .217 672 817 580 e-7, unit = u15 mP = .217 647 0(51) e-7
Planck length ${\displaystyle (l_{p}^{*})^{15}={\frac {\pi ^{22}(\mu _{0}^{*})^{9}}{2^{35}3^{24}\alpha ^{49}(c^{*})^{35}(R_{\infty }^{*})^{8}}},\;unit=(u^{-13})^{15}}$  lp* = .161 603 660 096 e-34, unit = u-13 lp = .161 622 9(38) e-34
Gyromagnetic ratio ${\displaystyle (\gamma _{e}/2\pi )^{3}={\frac {g^{3}3^{3}(c^{*})^{4}}{2^{8}\pi ^{8}\alpha (\mu _{0}^{*})^{3}(R_{\infty }^{*})^{2}}},\;unit=u^{-126}}$  γe/2π* = 28024.953 55, unit = u-42 γe/2π = 28024.951 64(17)

## 2019 SI unit revision

Following the 26th General Conference on Weights and Measures (2019 redefinition of SI base units) are fixed the numerical values of the 4 physical constants (h, c, e, kB). In the context of this model however only 2 base units may be assigned by committee as the rest are then numerically fixed by default (i.e.: derived via the electron itself) and so the revision may lead to unintended consequences where experimentally measured values may differ according to which instruments (means of calibration) are being used;

Physical constants
Constant CODATA 2018 [6]
Speed of light c = 299 792 458 (exact)
Planck constant h = 6.626 070 15 e-34 (exact)
Elementary charge e = 1.602 176 634 e-19 (exact)
Boltzmann constant kB = 1.380 649 e-23 (exact)
Fine structure constant α = 137.035 999 084(21)
Rydberg constant R = 10973 731.568 160(21)
Electron mass me = 9.109 383 7015(28) e-31
Vacuum permeability μ0 = 1.256 637 062 12(19) e-6

For example;

${\displaystyle R^{*}={\frac {4\pi ^{5}}{3^{3}c^{4}\alpha ^{8}e^{3}}}=10973\;729.082\;465}$

${\displaystyle {(m_{e}^{*})}^{3}={\frac {2^{4}\pi ^{10}R\mu _{0}^{3}}{3^{6}c^{8}\alpha ^{7}}},\;m_{e}^{*}=9.109\;382\;3259\;10^{-31}}$

${\displaystyle {(\mu _{0}^{*})}^{3}={\frac {3^{6}h^{3}c^{5}\alpha ^{13}R^{2}}{2\pi ^{10}}},\;\mu _{0}^{*}=1.256\;637\;251\;88\;10^{-6}}$

${\displaystyle {(h^{*})}^{3}={\frac {2\pi ^{10}\mu _{0}^{3}}{3^{6}c^{5}\alpha ^{13}R^{2}}},\;h^{*}=6.626\;069\;149\;10^{-34}}$

${\displaystyle {(e^{*})}^{3}={\frac {4\pi ^{5}}{3^{3}c^{4}\alpha ^{8}R}},\;e^{*}=1.602\;176\;513\;10^{-19}}$

## u as √{length/mass.time}

### u = √{L/M.T}

${\displaystyle u,\;units={\sqrt {\frac {L}{MT}}}={\sqrt {u^{-13-15+30=2}}}=u^{1}}$
${\displaystyle x,\;units={\sqrt {\frac {M^{9}T^{11}}{L^{15}}}}=u^{0}=1}$
${\displaystyle y,\;units=M^{2}T=u^{0}=1}$

Gives;

${\displaystyle u^{3}={\frac {L^{3/2}}{M^{3/2}T^{3/2}}}=A,\;(ampere)}$
${\displaystyle u^{6}(y)=L^{3}/T^{2}M,\;(G)}$
${\displaystyle u^{13}(xy)=1/L,\;(1/l_{p})}$
${\displaystyle u^{15}(xy^{2})=M,\;(m_{P})}$
${\displaystyle u^{17}(xy^{2})=V,\;(c)}$
${\displaystyle u^{19}(xy^{3})=ML^{2}/T,\;(h)}$
${\displaystyle u^{20}(xy^{2})={\frac {L^{5/2}}{M^{3/2}T^{5/2}}}=AV,\;(T_{P})}$
${\displaystyle u^{27}(x^{2}y^{3})={\frac {M^{3/2}{\sqrt {T}}}{L^{3/2}}}=1/AT,\;(1/e)}$
${\displaystyle u^{29}(x^{2}y^{4})={\frac {M^{5/2}{\sqrt {T}}}{\sqrt {L}}}=ML/AT,\;(k_{B})}$
${\displaystyle u^{30}(x^{2}y^{3})=1/T,\;(1/t_{p})}$
${\displaystyle u^{56}(x^{4}y^{7})={\frac {M^{4}T}{L^{2}}}={\frac {ML}{T^{2}A^{2}}},\;(\mu _{0})}$

### β (unit = u)

i (from x) and j (from y).

${\displaystyle R={\sqrt {P}}={\sqrt {\Omega }}r,\;units=u^{8}}$
${\displaystyle \beta ={\frac {V}{R^{2}}}={\frac {2\pi R^{2}}{M}}={\frac {A^{1/3}\alpha ^{1/3}}{2}}\;...,\;unit=u}$
${\displaystyle i={\frac {1}{2\pi {(2\pi \Omega )}^{15}}},\;unit=1}$
${\displaystyle j={\frac {r^{17}}{v^{8}}}=k^{2}t={\frac {k^{8}}{r^{15}}}...,\;unit={\frac {u^{17*8}}{u^{8*17}}}=u^{15*2}u^{-30}...=1}$

For example; the constants solved in terms of (r, v)

${\displaystyle \beta ={\frac {V}{R^{2}}}={\frac {2\pi \Omega ^{2}v}{\Omega r^{2}}},\;u}$
${\displaystyle A=\beta ^{3}({\frac {2^{3}}{\alpha }})={\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }}{\frac {v^{3}}{r^{6}}},\;u^{3}}$
${\displaystyle G={\frac {\beta ^{6}}{2^{3}\pi ^{2}}}(j)=2^{3}\pi ^{4}\Omega ^{6}{\frac {r^{5}}{v^{2}}},\;u^{6}}$
${\displaystyle L^{-1}=4\pi \beta ^{13}(ij)={\frac {1}{2\pi ^{2}\Omega ^{2}}}{\frac {v^{5}}{r^{9}}},\;u^{13}}$
${\displaystyle M=2\pi \beta ^{15}(ij^{2})={\frac {r^{4}}{v}},\;u^{15}}$
${\displaystyle P=\beta ^{16}(ij^{2})=\Omega r^{2},\;u^{16}}$
${\displaystyle V=\beta ^{17}(ij^{2})=2\pi \Omega ^{2}v,\;u^{17}}$
${\displaystyle h=\pi \beta ^{19}(ij^{3})=8\pi ^{4}\Omega ^{4}{\frac {r^{13}}{v^{5}}},\;u^{19}}$
${\displaystyle T_{P}^{*}={\frac {2^{3}\beta ^{20}}{\pi \alpha }}(ij^{2})={\frac {2^{7}\pi ^{3}\Omega ^{5}}{\alpha }}{\frac {v^{4}}{r^{6}}},\;u^{20}}$
${\displaystyle e^{-1}={\frac {\alpha \pi \beta ^{27}(i^{2}j^{3})}{4}}={\frac {\alpha }{128\pi ^{4}\Omega ^{3}}}{\frac {v^{3}}{r^{3}}},\;u^{27}}$
${\displaystyle k_{B}={\frac {\alpha \pi ^{2}\beta ^{29}(i^{2}j^{4})}{4}}={\frac {\alpha }{32\pi \Omega }}{\frac {r^{10}}{v^{3}}},\;u^{29}}$
${\displaystyle T^{-1}=2\pi \beta ^{30}(i^{2}j^{3})={\frac {1}{2\pi }}{\frac {v^{6}}{r^{9}}},\;u^{30}}$
${\displaystyle \mu _{0}^{*}={\frac {\pi ^{3}\alpha \beta ^{56}}{2^{3}}}(i^{4}j^{7})={\frac {\alpha }{2^{11}\pi ^{5}\Omega ^{4}}}r^{7},\;u^{56}}$
${\displaystyle \epsilon _{0}^{*-1}={\frac {\pi ^{3}\alpha \beta ^{90}}{2^{3}}}(i^{6}j^{11})={\frac {\alpha }{2^{9}\pi ^{3}}}v^{2}r^{7},\;u^{90}}$

### limit j

The numerical SI values for j suggest a limit (boundary) to the values the SI constants can have.

${\displaystyle j={\frac {r^{17}}{v^{8}}}=k^{2}t={\frac {k^{17/4}}{v^{15/4}}}=...=.812997...x10^{-59},\;units=1}$

In SI terms unit β has this value;

${\displaystyle a^{1/3}={\frac {v}{r^{2}}}={\frac {1}{t^{2/15}k^{1/5}}}={\frac {\sqrt {v}}{\sqrt {k}}}...=23326079.1...;unit=u}$

The unit-less ratios;

${\displaystyle (AL)^{3}/T=A^{3}T^{-1}/(L^{-1})^{3};\;units={\frac {u^{3}(u^{30}x^{2}y^{3})}{(u^{13}xy)^{3}}}=1/x}$
${\displaystyle T^{2}T_{P}^{3}={\frac {T_{P}^{3}}{(T^{-1})^{2}}};\;units={\frac {(u^{20}xy^{2})^{3}}{(u^{30}x^{2}y^{3})^{2}}}=1/x}$
${\displaystyle {M^{9}(L^{-1})^{15}}/{(T^{-1})^{11}};\;units={\frac {(u^{15}xy^{2})^{9}(u^{13}xy)^{15}}{(u^{30}x^{2}y^{3})^{11}}}=x^{2}}$

### Photon

The energy of a photon (Rydberg formula)

${\displaystyle E={\frac {hc}{2\pi 2\alpha }}{\frac {1}{\alpha \lambda _{orbital}}}({\frac {1}{n_{i}^{2}}}-{\frac {1}{n_{f}^{2}}})}$

re-written in terms of amperes ${\displaystyle A^{2}}$  (see quantum angular momentum)

${\displaystyle {\frac {hc}{2\pi \alpha ^{2}}}={\frac {j^{2}A^{2}}{2^{8}2\pi t_{p}}}}$

## Mathematical Universe

The mathematical universe refers to universe models whose underlying premise is that the physical universe has a mathematical origin, the physical (particle) universe is a construct of the mathematical universe, and as such physical reality is a perceived reality. It can be considered a form of Pythagoreanism or Platonism in that it proposes the existence of mathematical objects; and a form of mathematical monism in that it denies that anything exists except these mathematical objects.

Simulation theory (The Simulation Universe Hypothesis where the universe is a simulated reality, the analogy being a computer game), is a limited mathematical universe model in which the mathematical objects exist only within the framework of (computer) "source code", the Simulated Universe as manipulated data and the question of what lies outside the simulation has no meaning. Universe simulation models are typically associated with digital time.

Physicist Max Tegmark in his book "Our Mathematical Universe: My Quest for the Ultimate Nature of Reality"[7][8] proposed that Our external physical reality is a mathematical structure.[9] That is, the physical universe is not merely described by mathematics, but is mathematics (specifically, a mathematical structure). Mathematical existence equals physical existence, and all structures that exist mathematically exist physically as well. Any "self-aware substructures will subjectively perceive themselves as existing in a physically 'real' world".[10]

Many works of science fiction as well as some forecasts by serious technologists and futurologists predict that enormous amounts of computing power will be available in the future. Let us suppose for a moment that these predictions are correct. One thing that later generations might do with their super-powerful computers is run detailed simulations of their forebears or of people like their forebears. Because their computers would be so powerful, they could run a great many such simulations. Suppose that these simulated people are conscious (as they would be if the simulations were sufficiently fine-grained and if a certain quite widely accepted position in the philosophy of mind is correct). Then it could be the case that the vast majority of minds like ours do not belong to the original race but rather to people simulated by the advanced descendants of an original race. It is then possible to argue that, if this were the case, we would be rational to think that we are likely among the simulated minds rather than among the original biological ones. |Nick Bostrom, Are you living in a computer simulation?, 2003[11]