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.

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

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 as numerical values. 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	In Planck units	Geometrical object	Calculated (r, v, Ω, α*)	CODATA 2014 [5]
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). If this mathematical relationship can therefore be identified within the SI units themselves, then we have a strong argument for a mathematical Planck level.

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 f_e, this formula is both unit-less and non scalable k⁰t⁰v⁰r⁰a⁰u⁰ = 1. It is therefore a dimensionless physical constant, σ_e has units for a magnetic monopole, σ_tp 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. Note here I use common usage alpha = 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, m_e, k_B

As geometrical objects, the physical constants (G, h, e, m_e, k_B) can be defined 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 [6]
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)

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;

${\displaystyle (AL)^{3}/T=A^{3}T^{-1}/(L^{-1})^{3};\;units={\frac {u^{3}(u^{30}x^{2}y^{3})}{(u^{13}xy)^{3}}}=1/x}$

${\displaystyle T^{2}T_{P}^{3}={\frac {T_{P}^{3}}{(T^{-1})^{2}}};\;units={\frac {(u^{20}xy^{2})^{3}}{(u^{30}x^{2}y^{3})^{2}}}=1/x}$