Digital Physics/Simulation Hypothesis/Physical constants (anomalies)

Anomalies within the dimensioned physical constants (G, h, c, e, m_e, k_B) suggest a mathematical relationship between the units (kg ⇔ 15, m ⇔ -13, s ⇔ -30, A ⇔ 3, K ⇔ 20).

A dimensioned physical constant, sometimes denoted a fundamental physical constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. Common examples being the speed of light c, the gravitational constant G, the Planck constant h and the elementary charge e. These constants are usually measured in terms of SI units mass (kilogram), length (meter), time (second), charge (ampere), temperature (Kelvin) ... (kg, m, s, A, K ...).

These constants form the scaffolding around which the theories of physics are erected, and they define the fabric of our universe, but science has no idea why they take the special numerical values that they do, for these constants follow no discernible pattern. The desire to explain the constants has been one of the driving forces behind efforts to develop a complete unified description of nature, or "theory of everything". Physicists have hoped that such a theory would show that each of the constants of nature could have only one logically possible value. It would reveal an underlying order to the seeming arbitrariness of nature ^[1].

Notably a physical universe, as opposed to a mathematical universe (a computer simulation), has as a fundamental premise the concept that the universe scaffolding (of mass, space and time) exists, that somehow mass is, space is, time is ... these dimensions are real, and independent of each other ... we cannot measure distance in kilograms and amperes, or mass using length and temperature. The 2019 redefinition of SI base units resulted in 4 physical constants (h, c, e, k_B) being assigned exact values, and this confirmed the independence of their associated SI units as shown in this table.

2019 redefinition of SI base units

constant		SI units
Speed of light	c	${\displaystyle \frac{m}{s}}$
Planck constant	h	${\displaystyle \frac{kg{m}^2}{s}}$
Elementary charge	e	${\displaystyle C=As}$
Boltzmann constant	kB	${\displaystyle \frac{kg{m}^2}{s^{2}K}}$

However there are anomalies which occur in certain combinations of the fundamental (dimensioned) physical constants (G, h, c, e, m_e, k_B) which suggest a mathematical relationship between the units (kg ⇔ 15, m ⇔ -13, s ⇔ -30, A ⇔ 3, K ⇔ 20).

In order for these physical constants (G, h, c, e, m_e, k_B) to be fundamental, the units must be independent of each other, there cannot be such a unit number relationship ... however these anomalies question this fundamental assumption. Physics has a set of constants defined directly in terms of the units (kg, m, s, A, K), these are called Planck units (Planck mass, Planck length, Planck time ...), and these Planck units are interchangeable with the physical constants.

If we include this unit number relationship (kg ⇔ 15, m ⇔ -13, s ⇔ -30, A ⇔ 3, K ⇔ 20), then we find that we need only these 3 Planck unit analogues (MTP; mass, time, momentum) and the fine structure constant alpha to derive and solve all 6 fundamental physical constants (G, h, c, e, m_e, k_B) consistent with CODATA values. This would then question their status as being fundamental. Furthermore our MTP are themselves constructs of 2 mathematical constants; pi and e, the only physical constant required is alpha, and this may be because its mathematical origin is still unknown ^[2].

${\displaystyle M=1}$

${\displaystyle T=\pi }$

${\displaystyle P=\mathrm{\Omega }=\sqrt{{\pi }^e{e}^{(1-e)}}=2.0071349543249462...}$

Every test listed in the following examples using this unit number relationship (kg ⇔ 15, m ⇔ -13, s ⇔ -30, A ⇔ 3, K ⇔ 20) returns answers consistent with the premise. Furthermore there is only 1 possible number relationship that satisfies all conditions. Statistically therefore, can these anomalies be dismissed as coincidence.

The Planck units are direct measures of the SI units; Planck mass in kg, Planck length in m, Planck time in s ... and so they are analogues to the attributes listed in Table 2.. The SI Planck units have numerical values, however to derive a mathematical relation between these SI units we cannot use numerical values, this is because numerical values are simply dimensionless frequencies of the SI unit itself, 299792458 could refer to the speed of light 299792458m/s or equally to the number of apples in a container (299792458 apples), numbers such as 299792458 carry no unit-specific information, and so the units are treated as independent by default. This therefore requires that to the number 299792458 is added a descriptive (the unit), which could be m/s or apples.

This inherent restriction can be resolved by assigning to each unit a geometrical object for which the geometry embeds the attribute (for example, the geometry of the time object T embeds the function time and so a descriptive unit s = seconds is not required). We may then combine these objects Lego-style to form more complex objects; from electrons to galaxies, while still retaining the underlying attributes (of mass M, wavelength L, frequency T ...). An apple has mass because its 'geometry' includes the geometrical object for mass.

From MTP we can construct (Planck) units for L length, V velocity, A ampere and K Kelvin.

Table 2. MLTVA Geometrical objects

attribute	geometrical object	unit number θ
mass	${\displaystyle M=(1)}$	15
time	${\displaystyle T=(\pi )}$	-30
[| sqrt(momentum)]	${\displaystyle P=(\mathrm{\Omega })}$	16
velocity	${\displaystyle V=\frac{2\pi P^2}{M}=(2\pi {\mathrm{\Omega }}^2)}$	17
length	${\displaystyle L=VT=(2{\pi }^2{\mathrm{\Omega }}^2)}$	-13
ampere	${\displaystyle A=\frac{2^4{V}^3}{\alpha P^3}=(\frac{2^7{\pi }^3{\mathrm{\Omega }}^3}{{\alpha }_{inv}})}$	3
temperature	${\displaystyle K=\frac{AV}{2\pi }=(\frac{2^7{\pi }^3{\mathrm{\Omega }}^5}{{\alpha }_{inv}})}$	20

We can use the text-book formulas to generate analogues of the common physical constants.

Table 3. Physical constant unit numbers

SI constant	geometrical analogue	unit number θ
Speed of light	c* = V	17
Planck constant	${\displaystyle h^{*}=2\pi MVL}$	15+17-13=19
Gravitational constant	${\displaystyle G^{*}=\frac{V^{2}L}{M}}$	34-13-15=6
Elementary charge	${\displaystyle e^{*}=AT}$	3-30=-27
Boltzmann constant	${\displaystyle k_B^{*}=\frac{2\pi VM}{A}}$	17+15-3=29
Vacuum permeability	${\displaystyle {\mu }_0^{*}=\frac{4\pi V^{2}M}{{\alpha }_{inv}L{A}^2}}$	34+15+13-6=56

We are using CODATA 2014 values. This is because only 2 dimensioned physical constants can be assigned exact values, once 2 constants have been assigned values, then all other constants are defined by default. In CODATA 2014 2 constants have exact values; ${\displaystyle c}$ and the vacuum permeability ${\displaystyle {\mu }_0}$. After CODATA 2014, 4 constants were assigned exact values which is problematic in terms of this model.

${\displaystyle c=299792458}$ m/s

${\displaystyle {\mu }_0=4\pi /10^7}$

The exception is alpha, the value used here ${\displaystyle {\alpha }_{inv}}$ = 137.0359963688 is derived from the Rydberg constant.

We can apply the unit number relationship to determine unit-less combinations, for example (A^3 L^3 /T) gives (3*3) + (-13 *3) - (-30) = 0.

If MTP are natural Planck units, then the SI unit-less combinations will be stripped of their terrestrial content and so return the same numerical value as for the MTP combinations. For example;

${\displaystyle \frac{(h^{*}{)}^3}{(e^{*}{)}^{13}(c^{*}{)}^{24}}=\frac{(2\pi MVL{)}^3}{(AT{)}^{13}(V{)}^{24}}=\frac{{\alpha }_{inv}^{13}}{2^{106}{\pi }^{64}({\mathrm{\Omega }}^{15}{)}^5}}$ = 0.228 473 662... 10^-58, units = 1

${\displaystyle \frac{h^3}{e^{13}{c}^{24}}=}$ 0.228 473 639... 10^-58, units = ${\displaystyle \frac{k{g}^3{s}^8}{m^{18}{A}^{13}}}$, units = 1 (15*3-30*8+13*18-3*13 = 0)

The 3 most precisely known CODATA 2014 constants; (${\displaystyle c}$ exact, ${\displaystyle {\mu }_0}$ exact and the Rydberg constant ${\displaystyle R}$ 12-digits are used to calibrate alpha (${\displaystyle \alpha }$ = 137.035996369) in this dimensionless combination (for the derivation of R see Calculating the electron).

${\displaystyle \frac{(c^{*}{)}^{35}}{({\mu }_0^{*}{)}^9(R^{*}{)}^7}=2^{295}{\pi }^{157}{3}^{21}{\alpha }_{inv}^{26}({\mathrm{\Omega }}^{15}{)}^{15}=\frac{c^{35}}{{\mu }_0^9{R}^7}}$

Note: the geometry ${\displaystyle ({\mathrm{\Omega }}^{15}{)}^n}$ (integer n ≥ 0) is common to all ratios where units and scalars cancel (i.e.: only combinations with ${\displaystyle {\mathrm{\Omega }}^0,{\mathrm{\Omega }}^{15},{\mathrm{\Omega }}^{30},{\mathrm{\Omega }}^{45}}$... will be dimensionless). However there is no Planck unit with a ${\displaystyle {\mathrm{\Omega }}^{15}}$ component (all constants are combinations of ${\displaystyle {\mathrm{\Omega }}^2}$ and ${\displaystyle {\mathrm{\Omega }}^3}$), and this suggests there is an underlying geometrical base-15.

Table 4. Dimensionless combinations (α, Ω)

CODATA 2014 (mean)	(α, Ω)	units = 1
${\displaystyle \frac{k_{B}ec}{h}}$ = 1.000 8254	${\displaystyle \frac{(k_B^{*})(e^{*})(c^{*})}{(h^{*})}}$ = 1.0	${\displaystyle \frac{(u^{29})(u^{-27})(u^{17})}{(u^{19})}=1}$
${\displaystyle \frac{h^3}{e^{13}{c}^{24}}=}$ 0.228 473 639... 10-58	${\displaystyle \frac{(h^{*}{)}^3}{(e^{*}{)}^{13}(c^{*}{)}^{24}}=\frac{{\alpha }_{inv}^{13}}{2^{106}{\pi }^{64}({\mathrm{\Omega }}^{15}{)}^5}=}$ 0.228 473 662... 10-58	${\displaystyle \frac{(u^{19}{)}^3}{(u^{-27}{)}^{13}(u^{17}{)}^{24}}=1}$
${\displaystyle \frac{c^9{e}^4}{m_e^3}=}$ 0.170 514 342... 1092	${\displaystyle \frac{(c^{*}{)}^9(e^{*}{)}^4}{(m_e^{*}{)}^3}=2^{97}{\pi }^{49}{3}^9{\alpha }_{inv}^5({\mathrm{\Omega }}^{15}{)}^5=}$ 0.170 514 381... 1092	${\displaystyle \frac{(u^{29})(u^{-27})(u^{17})}{(u^{19})}=1}$
${\displaystyle \frac{k_B}{e^2{m}_e{c}^4}=}$ 73 095 484 786.	${\displaystyle \frac{(k_B^{*})}{(e^{*}{)}^2(m_e^{*})(c^{*}{)}^4}=\frac{3^3{\alpha }_{inv}^6}{2^3{\pi }^5}=}$ 73 035 227 214.	${\displaystyle \frac{(u^{29})}{(u^{-27}{)}^2(u^{15})(u^{17}{)}^4}=1}$
${\displaystyle \frac{h{c}^{2}e{m}_p}{G^2{k}_B}=}$ 3.376 716	${\displaystyle \frac{(h^{*})(c^{*}{)}^2(e^{*})(m_p^{*})}{(G^{*}{)}^2(k_B^{*})}=\frac{2^{11}{\pi }^3}{{\alpha }_{inv}^2}=}$ 3.381 507	${\displaystyle \frac{(u^{19})(u^{17}{)}^2(u^{-27})(u^{15})}{(u^6{)}^2(u^{29})}=1}$

This dimensionless combination approach should therefore apply to any set of units, even extraterrestrial and non-human ones, that in the dimensionless combination the numerical result will revert to the MLTA analogue. This suggests that these MLTVA objects could be candidates for the "natural units" as proposed by Max Planck.

...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 ^[3][4]

Note. 1. Combinations involving only (h, e, c) and (c, e, m_e) exhibit errors in the 8th digit, suggesting that h, e, and m_e have extremely low errors relative to the geometric model.

2. Combinations involving k_B exhibit errors in the 4th digit, identifying k_B as the primary source of the discrepancy in the electromagnetic/thermal sector.

3. The contributions of m_P and G cannot be separated, nevertheless the implication is of low precision for both.

Aim. We treat each Table 4 entry as an independent “coincidence test” and estimate:

1. the probability that a random dimensionless value would land as close as observed, and
2. the joint probability that all Table 4 agreements occur together.

Important note. CODATA uncertainties are not used (and not required here), because the purpose is not a strict measurement-error test but an order-of-magnitude estimate of how unlikely the *overall pattern* is under the null hypothesis of “no relationship”.

Because these are dimensionless quantities (units cancel), we use a conservative “random digits” baseline:

• Let the relative error be:

${\displaystyle \epsilon =\left|\frac{x_{\text{model}}-x_{\text{CODATA}}}{x_{\text{CODATA}}}\right|}$

• Approximate the chance of landing within that relative window (two-sided) as:

${\displaystyle p\approx 2\epsilon }$

• Joint probability across N tests (naive independence):

${\displaystyle p_{\text{joint}}={\displaystyle \prod _{i=1}^N}{p}_i}$

• A “sigma-equivalent” is computed by mapping the two-sided probability ${\displaystyle p}$ to a standard Normal:

${\displaystyle \sigma \approx {\mathrm{\Phi }}^{-1}(1-p/2)}$

Table 4: relative error → probability → sigma-equivalent (CODATA 2014 vs (α, Ω))

Row	Quantity (CODATA 2014 vs (α, Ω))	Relative error ${\displaystyle \epsilon }$	${\displaystyle p\approx 2\epsilon }$	Sigma-equivalent
1	${\displaystyle \frac{k_{B}ec}{h}}$: 1.0008254 vs 1.0	${\displaystyle 8.247\times 10^{-4}}$	${\displaystyle 1.649\times 10^{-3}}$	~3.15σ
2	${\displaystyle \frac{h^3}{e^{13}{c}^{24}}}$: 0.228473639…×10-58 vs 0.228473662…×10-58	${\displaystyle 1.007\times 10^{-7}}$	${\displaystyle 2.013\times 10^{-7}}$	~5.20σ
3	${\displaystyle \frac{c^9{e}^4}{m_e^3}}$: 0.170514342…×1092 vs 0.170514381…×1092	${\displaystyle 2.287\times 10^{-7}}$	${\displaystyle 4.574\times 10^{-7}}$	~5.04σ
4	${\displaystyle \frac{k_B}{e^2{m}_e{c}^4}}$: 73,095,507,858 vs 73,035,227,214	${\displaystyle 8.244\times 10^{-4}}$	${\displaystyle 1.649\times 10^{-3}}$	~3.15σ
5	${\displaystyle \frac{h{c}^{2}e{m}_p}{G^2{k}_B}}$: 3.376716 vs 3.381507	${\displaystyle 1.419\times 10^{-3}}$	${\displaystyle 2.838\times 10^{-3}}$	~2.98σ

Assuming independence between the five Table 4 tests:

${\displaystyle p_{\text{all}}={\displaystyle \prod _{i=1}^5}{p}_i\approx 7.11\times 10^{-22}}$

Normal-equivalent (two-sided) significance:

${\displaystyle {\sigma }_{\text{all}}\approx 9.61\sigma }$

The cleanest high-precision sub-set excludes combinations involving ${\displaystyle G}$ and ${\displaystyle k_B}$, leaving only the two “pure” electromagnetic/mechanical ratios:

• ${\displaystyle \frac{h^3}{e^{13}{c}^{24}}}$
• ${\displaystyle \frac{c^9{e}^4}{m_e^3}}$

Joint probability:

${\displaystyle p_{\text{no }G,k_B}\approx (2.013\times 10^{-7})(4.574\times 10^{-7})=9.21\times 10^{-14}}$

Sigma-equivalent:

${\displaystyle {\sigma }_{\text{no }G,k_B}\approx 7.45\sigma }$

Caveats

1. Dependence: the tests reuse the same constants (h, e, c, etc.), so strict independence is not guaranteed; multiplying ${\displaystyle p_i}$ is therefore an optimistic estimator.

To convert from dimensionless geometrical objects to SI Planck units, we can use scalars. We can assign scalars to each geometry (M ⇔ k, T ⇔ t, L ⇔ l, V ⇔ v, A ⇔ a ... ), however as the scalars also carry the unit designation as well as an associated numerical value, they are dimensioned, and so we can apply the unit number relationship (θ) to them. Using the dimensionless ratios introduced above we find that only 2 scalars are required. For example if we know the numerical value for a and for l then we know the numerical value for t (t = a³l³), and from l and t we know the value for k.

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

This means that once any 2 scalars have been assigned values, the other scalars are then defined by default, consequently the CODATA 2014 values are used here as only 2 constants (c, μ₀) are assigned exact values, following the 2019 redefinition of SI base units 4 constants have been independently assigned exact values which is problematic in terms of this model.

Although we could use the (Planck) scalars for length or time or mass or charge, the 2 scalars used here are r (θ = 8) and v (θ = 17). This is because they can be derived from the 2 constants with exact values; v from c and r from μ₀. We can now calibrate our 2 scalars;

${\displaystyle v=\frac{c}{2\pi {\mathrm{\Omega }}^2}=11843707.905...,units=\frac{m}{s}}$

${\displaystyle r^7=\frac{2^{11}{\pi }^5{\mathrm{\Omega }}^4{\mu }_0}{\alpha };r=0.712562514304...,units=(\frac{kg.m}{s}{)}^{1/4}}$

As the scalars are used to translate between the dimensionless geometrical objects MLTP... and local unit systems such as SI, then the numerical values are unit specific.

For example, we can use scalar v to convert from dimensionless geometrical object V to dimensioned c.

scalar v = 11843707.905 m/s gives c = V*v = 25.3123819 * 11843707.905 m/s = 299792458 m/s (SI units)

scalar v = 7359.3232155 miles/s gives c = V*v = 186282 miles/s (imperial units)

Table 5. MLTVA Geometrical objects

attribute	geometrical object	numerical	unit number θ	scalars
mass	${\displaystyle M=(1)}$	1	15	${\displaystyle \frac{r^4}{v}}$
time	${\displaystyle T=(\pi )}$	3.1415926535...	-30	${\displaystyle \frac{r^9}{v^6}}$
[|sqrt(momentum)]	${\displaystyle P=(\mathrm{\Omega })}$	2.00713495...	16	${\displaystyle r^2}$
velocity	${\displaystyle V=\frac{2\pi P^2}{M}=(2\pi {\mathrm{\Omega }}^2)}$	25.3123819...	17	${\displaystyle v}$
length	${\displaystyle L=VT=(2{\pi }^2{\mathrm{\Omega }}^2)}$	79.5211931...	-13	${\displaystyle \frac{r^9}{v^5}}$
ampere	${\displaystyle A=\frac{2^4{V}^3}{{\alpha }_{inv}{P}^3}=(\frac{2^7{\pi }^3{\mathrm{\Omega }}^3}{{\alpha }_{inv}})}$	234.182607...	3	${\displaystyle \frac{v^3}{r^6}}$
temperature	${\displaystyle K=\frac{AV}{2\pi }=(\frac{2^7{\pi }^3{\mathrm{\Omega }}^5}{{\alpha }_{inv}})}$	943.425875...	20	${\displaystyle \frac{v^4}{r^6}}$

Comparison with the SI constants

Table 6. Comparison θ; SI units and scalars

constant	θ from SI units	MLTVA	θ from r(8), v(17)
c	${\displaystyle \frac{m}{s}}$ (-13+30 = 17)	c* = ${\displaystyle V*v}$	17
h	${\displaystyle \frac{kg{m}^2}{s}}$ (15-26+30=19)	h* = ${\displaystyle 2\pi MVL*\frac{r^{13}}{v^5}}$	8*13-17*5=19
G	${\displaystyle \frac{m^3}{kg{s}^2}}$ (-39-15+60=6)	G* = ${\displaystyle \frac{V^{2}L}{M}*\frac{r^5}{v^2}}$	8*5-17*2=6
e	${\displaystyle C=As}$ (3-30=-27)	e* = ${\displaystyle AT*\frac{r^3}{v^3}}$	8*3-17*3=-27
kB	${\displaystyle \frac{kg{m}^2}{s^{2}K}}$ (15-26+60-20=29)	kB* = ${\displaystyle \frac{2\pi VM}{A}*\frac{r^{10}}{v^3}}$	8*10-17*3=29
μ0	${\displaystyle \frac{kgm}{s^2{A}^2}}$ (15-13+60-6=56)	μ0* = ${\displaystyle \frac{4\pi V^{2}M}{\alpha L{A}^2}*r^7}$	8*7=56

This shows the unit number relationship is consistent regardless of the constants and the system of units used. Furthermore an exhaustive search of the unit-number integer space showed a fundamental constraint 3M + 2T = -15 indicating that this base-15 is the only geometric solution that satisfies all requirements of this model.

Dimensional homogeneity across all physics equations.

The dimensionless status of the electron formula ${\displaystyle \psi }$.

The existence of a valid quark substructure (D, U quarks).

Internal consistency for the electron triplet DDD = T.

The following is one of the most important formulas in physics; it describes the relationship between the fine structure constant and the dimensioned constants.

${\displaystyle {\alpha }_{inv}=\frac{2h}{{\mu }_0{e}^{2}c}}$

However, if we replace the numerical (h, μ₀, e, c) with the geometrical (h, μ₀, e, c), we find that the equation collapses to give alpha;

${\displaystyle \frac{2h}{{\mu }_0{e}^{2}c}=2(2^3{\pi }^4{\mathrm{\Omega }}^4)/(\frac{{\alpha }_{inv}}{2^{11}{\pi }^5{\mathrm{\Omega }}^4})(\frac{2^7{\pi }^4{\mathrm{\Omega }}^3}{{\alpha }_{inv}}{)}^2(2\pi {\mathrm{\Omega }}^2)={\alpha }_{inv}}$

Note also the units and scalars cancel

units = ${\displaystyle \frac{u^{19}}{u^{56}(u^{-27}{)}^2{u}^{17}}=1}$

scalars = ${\displaystyle (\frac{r^{13}}{v^5})(\frac{1}{r^7})(\frac{v^6}{r^6})(\frac{1}{v})=1}$

This is a good test of our model, both of the unit numbers thesis and the geometrical objects thesis, because this equation reduces to

${\displaystyle {\alpha }_{inv}=2(2^3{\pi }^4{\mathrm{\Omega }}^4)/(\frac{{\alpha }_{inv}}{2^{11}{\pi }^5{\mathrm{\Omega }}^4})(\frac{2^7{\pi }^4{\mathrm{\Omega }}^3}{{\alpha }_{inv}}{)}^2(2\pi {\mathrm{\Omega }}^2)}$

${\displaystyle \alpha =\alpha }$

There is no uncertainty of measurement and the formula is well established as a key formula.

Table 7. fine structure constant

CODATA 2014	geometrical (α)
${\displaystyle \sqrt{\frac{2^{11}{\pi }^3{G}^2{k}_B}{h{c}^{2}e{m}_p}}=}$ 137.133 167 47	${\displaystyle \sqrt{\frac{2^{11}{\pi }^3(G^{*}{)}^2(k_B^{*})}{(h^{*})(c^{*}{)}^2(e^{*})(m_p^{*})}}=\alpha }$
${\displaystyle (\frac{2^3{\pi }^5{k}_B}{3^3{e}^2{m}_e{c}^4}{)}^{1/6}=}$ 137.054 833 44	${\displaystyle (\frac{2^3{\pi }^5(k_B^{*})}{3^3(e^{*}{)}^2(m_e^{*})(c^{*}{)}^4}{)}^{1/6}=\alpha }$
${\displaystyle \frac{2^9{\pi }^2{t}_p}{m_P^4{ϵ}_0}=}$ 137.119 576 89	${\displaystyle \frac{2^9{\pi }^2(t_p{)}^{*}}{(m_P^{*}{)}^4({ϵ}_0^{*})}=\alpha }$

This section has two distinct components:

This section has two distinct components:

1. Algebraic consistency check (deterministic; non-statistical): a well-established formula for ${\displaystyle \alpha }$ must remain true when the dimensioned constants are replaced by their geometrical analogues. This tests internal consistency (unit-number + geometrical-object substitution), not probability.
2. Statistical alpha-from-CODATA tests (probabilistic; Table 7): several CODATA-style combinations yield numerical estimates of ${\displaystyle {\alpha }_{inv}}$. These are treated as coincidence tests (like Table 4) and combined via joint probability.

Important: all results below are dimensionless and do not use scalars (r, v).

In this analysis we use:

${\displaystyle {\alpha }_0={\alpha }_{inv}=137.0359963688}$

This value is not the CODATA 2014 recommended ${\displaystyle \alpha }$. It is derived from the Rydberg constant (which is more precise than the CODATA ${\displaystyle \alpha }$), and is the only non-CODATA-2014 input used in this paper.

The following identity is a standard relation between ${\displaystyle \alpha }$ and the constants:

${\displaystyle {\alpha }_{inv}=\frac{2h}{{\mu }_0{e}^{2}c}}$

When replacing the numerical constants (${\displaystyle h,{\mu }_0,e,c}$) by their geometrical analogues (${\displaystyle h^{*},{\mu }_0^{*},e^{*},c^{*}}$), the expression collapses to return ${\displaystyle {\alpha }_{inv}}$ exactly:

${\displaystyle \frac{2h}{{\mu }_0{e}^{2}c}\to {\alpha }_{inv}}$

Because this is an algebraic identity (no measurement uncertainty is required), it is a non-statistical pass/fail test of internal model consistency.

Table 7 lists several CODATA-style combinations that numerically evaluate to ${\displaystyle {\alpha }_{inv}}$. Unlike (A), these are treated as statistical coincidence tests: each formula returns an estimated value ${\displaystyle {\hat{\alpha }}_{inv}}$ which may deviate from the reference ${\displaystyle {\alpha }_0}$.

We do not use CODATA uncertainties. Instead, we measure relative error and convert it into an approximate coincidence probability:

• Absolute deviation:

${\displaystyle {\mathrm{\Delta }}_i={\hat{\alpha }}_i-{\alpha }_0}$

• Relative deviation:

${\displaystyle {\epsilon }_i=\frac{|{\mathrm{\Delta }}_i|}{{\alpha }_0}}$

• Two-sided coincidence probability:

${\displaystyle p_i\approx 2{\epsilon }_i}$

• Joint probability (naive independence):

${\displaystyle p_{\text{joint}}={\displaystyle \prod _{i=1}^N}{p}_i}$

• Sigma-equivalent:

${\displaystyle \sigma \approx {\mathrm{\Phi }}^{-1}(1-p/2)}$

Alpha estimates from CODATA-style formulas (dimensionless; no scalars)

Test (from Table 7)	${\displaystyle {\hat{\alpha }}_{inv}}$	${\displaystyle \mathrm{\Delta }=\hat{\alpha }-{\alpha }_0}$	rel. error ${\displaystyle \epsilon }$ (ppm)	${\displaystyle p_i\approx 2\epsilon }$	equiv. ${\displaystyle {\sigma }_i}$
${\displaystyle \sqrt{\frac{2^{11}{\pi }^3{G}^2{k}_B}{h{c}^{2}e{m}_p}}}$	137.13316747	+0.0971711012	709.09 ppm	1.41818\times10^{-3}	~3.19\sigma
${\displaystyle {\left(\frac{2^3{\pi }^5{k}_B}{3^3{e}^2{m}_e{c}^4}\right)}^{1/6}}$	137.05483344	+0.0188370712	137.46 ppm	2.74922\times10^{-4}	~3.64\sigma
${\displaystyle \frac{2^9{\pi }^2{t}_p}{m_P^4{ϵ}_0}}$	137.11957689	+0.0835805212	609.92 ppm	1.21983\times10^{-3}	~3.23\sigma

Assuming independence between the three Table 7 tests (note: they share constants so this is an optimistic estimator):

${\displaystyle p_{\text{joint}}\approx 4.757\times 10^{-10}}$

${\displaystyle -{\log }_{10}(p_{\text{joint}})\approx 9.323}$

Two-sided Normal sigma-equivalent:

${\displaystyle {\sigma }_{\text{joint}}\approx 6.23\sigma }$

1. Part (A) is a deterministic identity check; it is not a probabilistic event.
2. Part (B) is statistical because the CODATA-style formulas depend on measured constants, and therefore yield slightly different numerical ${\displaystyle \hat{\alpha }}$ values.
3. Relations involving ${\displaystyle G}$ and ${\displaystyle k_B}$ tend to be the least precise; the Table 7 deviations are consistent with that pattern.
4. Because the formulas reuse constants, strict independence is not guaranteed; the joint probability should be treated as an order-of-magnitude indicator.

We can now construct the electron from magnetic monopoles AL and time T (AL units ampere-meter (ampere-length) are the units for a magnetic monopole).

${\displaystyle T=\pi \frac{r^9}{v^6},u^{-30}}$

${\displaystyle {\sigma }_e=\frac{3{\alpha }_{inv}^{2}AL}{2{\pi }^2}=2^{7}3{\pi }^3\alpha {\mathrm{\Omega }}^5\frac{r^3}{v^2},u^{-10}}$

${\displaystyle \psi =\frac{{\sigma }_e^3}{2T}=\frac{(2^{7}3{\pi }^3\alpha {\mathrm{\Omega }}^5{)}^3}{2\pi },units=\frac{(u^{-10}{)}^3}{u^{-30}}=1,scalars=(\frac{r^3}{v^2}{)}^3\frac{v^6}{r^9}=1}$

${\displaystyle \psi =4{\pi }^2(2^{6}3{\pi }^2{\alpha }_{inv}{\mathrm{\Omega }}^5{)}^3=}$ 0.23895452462 e23 (dimensionless)

Both units and scalars cancel (units = scalars = 1), and so ψ (the formula for the electron) is dimensionless. We can solve the electron parameters; electron mass, wavelength, frequency, charge ... as the frequency of the Planck units, and this frequency is ψ. Our results (calculated) agree with CODATA 2014. This means that the formula ψ not only determines the frequency of the Planck units (and so the magnitude or duration of the electron parameters), but it also embeds those Planck units.

In other words, this formula ψ contains all the information needed to make the electron, and so by definition this formula ψ is the electron. However it is dimensionless (units = 1), and this means that the electron is a mathematical particle, not a physical particle. And if the electron is not a physical particle, then it is these electron parameters (wavelength, charge, mass ...), and not the electron itself, that we are measuring. The existence of the electron is inferred, it is not observed.

1. Compton wavelength

λ_e = 2.4263102367 e-12m (CODATA 2014)

λ_e = 2*π*L*ψ = 0.2426310335 e-12m (calculated)

2. Electron mass

m_e = 9.10938356 e-31kg (CODATA 2014)

M = (1*r^4/v) = 0.217672822274 e-7kg (M ⇔ Planck mass)

M/ψ = (1*r^4/v)/(4*pi^2*(2^6*3*π^2*α_{inv}*Omega^5)^3) kg

m_e = M/ψ = 0.910938274224 e-30kg (calculated)

3. Rydberg constant

R = 10973731.568508/m (CODATA 2014)

${\displaystyle R=(\frac{m_e}{4\pi L{\alpha }^{2}M})=\frac{1}{2^{23}{3}^3{\pi }^{11}{\alpha }_{inv}^5{\mathrm{\Omega }}^{17}}\frac{v^5}{r^9}{u}^{13}}$ = 10973731.568508/m (note. this will be exact as the Rydberg constant was used to calibrate alpha).

In summary, we have a dimensionless geometrical mathematical electron formula ψ that resembles the formula for the volume of a torus or surface area of a 4-axis hypersphere (${\displaystyle 4{\pi }^2(AL{)}^3}$), and that includes the information needed to make both the electron parameters and to make the Planck units. It can also be divided into 3 magnetic monopoles ${\displaystyle (AL{)}^3}$ and these suggest a potential 'quark' model for the electron.

The electron is encoded by the dimensionless invariant

${\displaystyle \psi =\frac{{\sigma }_e^3}{2T}}$

with units and scalars cancelling (units = 1, scalars = 1), so ψ is a pure number. :contentReference[oaicite:0]{index=0}

The cancellation of units and scalars in ψ is an algebraic property of the construction (a pass/fail internal-consistency check), not a probabilistic event. :contentReference[oaicite:1]{index=1}

In SI calibration (after solving the Planck objects), the paper reports:

${\displaystyle \psi =4{\pi }^2(2^6\cdot 3\cdot {\pi }^2\cdot \alpha \cdot {\mathrm{\Omega }}^5{)}^3=0.2389545307369\times 10^{23}}$ (dimensionless). :contentReference[oaicite:2]{index=2}

We now treat the reproduced electron parameters as coincidence tests against CODATA 2014 means (ignoring CODATA σ, per the approach used in Table 4). The calculated values are listed explicitly in the “Solving the electron parameters using ψ” section. :contentReference[oaicite:3]{index=3}

For each parameter:

• Relative error:

${\displaystyle \epsilon =\left|\frac{x_{\text{calc}}-x_{\text{CODATA}}}{x_{\text{CODATA}}}\right|}$

• Two-sided coincidence probability:

${\displaystyle p\approx 2\epsilon }$

• Joint probability across N tests (naive independence):

${\displaystyle p_{\text{joint}}={\displaystyle \prod _{i=1}^N}{p}_i}$

• Sigma-equivalent (two-sided Normal):

${\displaystyle \sigma \approx {\mathrm{\Phi }}^{-1}(1-p/2)}$

Electron parameter tests from ψ (CODATA 2014 vs calculated; no CODATA σ)

Parameter	CODATA 2014	calculated (from ψ)	rel. error ${\displaystyle \epsilon }$	${\displaystyle p\approx 2\epsilon }$	equiv. ${\displaystyle \sigma }$
Compton wavelength ${\displaystyle {\lambda }_e}$	2.4263102367×10-12 m	2.4263102386×10-12 m	7.8308×10-10 (0.000783 ppm)	1.5662×10-9	~6.04σ
Electron mass ${\displaystyle m_e}$	9.10938356×10-31 kg	9.1093823211×10-31 kg	1.3600×10-7 (0.1360 ppm)	2.7201×10-7	~5.14σ
Elementary charge ${\displaystyle e}$	1.6021766208×10-19 C	1.6021765130×10-19 C	6.7283×10-8 (0.06728 ppm)	1.3457×10-7	~5.27σ

The CODATA and calculated values above are taken directly from the electron-parameter list in the text. :contentReference[oaicite:4]{index=4}

Treating the three tests as independent “wins” (a strong assumption because constants are reused), the joint probability is:

${\displaystyle p_{\text{joint}}\approx (1.5662\times 10^{-9})(2.7201\times 10^{-7})(1.3457\times 10^{-7})\approx 5.73\times 10^{-23}}$

${\displaystyle -{\log }_{10}(p_{\text{joint}})\approx 22.24}$

Two-sided Normal sigma-equivalent:

${\displaystyle {\sigma }_{\text{joint}}\approx 9.87\sigma }$

1. Dependence: λe, me, e are not strictly independent tests because they share the same underlying constants and definitions; multiplying ${\displaystyle p_i}$ likely overstates the joint surprise.
2. Calibration dependence: the translation to SI uses the model’s calibration choices (e.g. fixing v from c and r from μ0 in CODATA 2014 context), so the statistical claim is “given those anchors, the remaining electron parameters land this close”.

In this section we use the 2 scalars (r, v) to solve the constants independently.

Table 8. Dimensioned constants (α, Ω, v, r)

constant	geometrical object	calculated (α_{inv}, Ω, r, v)	CODATA 2014 (mean)[5]
Planck constant	${\displaystyle h^{*}=2\pi MVL=2^3{\pi }^4{\mathrm{\Omega }}^4\frac{r^{13}}{v^5}}$	6.626069134e-34, u19	6.626070040e-34
Gravitational constant	${\displaystyle G^{*}=\frac{V^{2}L}{M}=2^3{\pi }^4{\mathrm{\Omega }}^6\frac{r^5}{v^2}}$	6.67249719229e11, u6	6.67408e-11
Elementary charge	${\displaystyle e^{*}=AT=\frac{2^7{\pi }^4{\mathrm{\Omega }}^3}{{\alpha }_{inv}}\frac{r^3}{v^3}}$	1.60217651130e-19, u-27	1.6021766208e-19
Boltzmann constant	${\displaystyle k_B^{*}=\frac{2\pi VM}{A}=\frac{{\alpha }_{inv}}{2^5\pi \mathrm{\Omega }}\frac{r^{10}}{v^3}}$	1.37951014752e-23, u29	1.38064852e-23

In this section, we show how to numerically solve the least precise dimensioned physical constants (G, h, e, m_e, k_B ...) in terms of the 3 most precise dimensioned physical constants); speed of light c (exact value), vacuum permeability μ₀ (exact value), Rydberg constant R (12-13 digits) and the dimensionless fine structure constant alpha.

R = 10973731.568508 (θ=13) (12-13 digit precision)

c = 299792458 (θ=17) (exact)

μ₀ = 4π/10⁷ (θ=56) (exact)

We first look for combinations in which the unit numbers are equal, and then add dimensionless numbers as required. For example;

${\displaystyle {(h^{*})}^3=(2^3{\pi }^4{\mathrm{\Omega }}^4\frac{r^{13}{u}^{19}}{v^5}{)}^3=\frac{3^{19}{\pi }^{12}{\mathrm{\Omega }}^{12}{r}^{39}{u}^{57}}{v^{15}},\theta =57}$

${\displaystyle \frac{2{\pi }^{10}{({\mu }_0^{*})}^3}{3^6{(c^{*})}^5{\alpha }^{13}{(R^{*})}^2}=\frac{3^{19}{\pi }^{12}{\mathrm{\Omega }}^{12}{r}^{39}{u}^{57}}{v^{15}},\theta =57}$

We then replace the geometrical with the SI (c, μ₀, R)

${\displaystyle {(h^{*})}^3=\frac{2{\pi }^{10}{{\mu }_0}^3}{3^6{c}^5{\alpha }_{inv}^{13}{R}^2}}$

Table 9. R, c, μ₀, α ... (CODATA 2014 mean)

constant	formula*	θ	Units
Planck constant	${\displaystyle {(h^{*})}^3=\frac{2{\pi }^{10}{{\mu }_0}^3}{3^6{c}^5{\alpha }_{inv}^{13}{R}^2}}$	${\displaystyle \frac{k{g}^3}{A^{6}s}}$, 15*3-3*6+30 = 57	${\displaystyle \frac{kg{m}^2}{s}}$, θ = 15-13*2+30 = 19
Gravitational constant	${\displaystyle {(G^{*})}^5=\frac{{\pi }^3{\mu }_0}{2^{20}{3}^6{\alpha }_{inv}^{11}{R}^2}}$	${\displaystyle \frac{kg{m}^3}{A^2{s}^2}}$, 15-13*3-3*2+30*2 = 30	${\displaystyle \frac{m^3}{kg{s}^2}}$, θ = -13*3-15+30*2 = 6
Elementary charge	${\displaystyle {(e^{*})}^3=\frac{4{\pi }^5}{3^3{c}^4{\alpha }_{inv}^{8}R}}$	${\displaystyle \frac{s^3}{m^3}}$, -30*4+13*3 = -81	${\displaystyle As}$, θ = 3-30 = -27
Boltzmann constant	${\displaystyle {(k_B^{*})}^3=\frac{{\pi }^5{{\mu }_0}^3}{3^{3}2c^4{\alpha }_{inv}^{5}R}}$	${\displaystyle \frac{k{g}^3}{s^2{A}^6}}$, 15*3+30*2-3*6 = 87	${\displaystyle \frac{kg{m}^2}{s^{2}K}}$, θ = 15-26+60-20 = 29
Electron mass	${\displaystyle {(m_e^{*})}^3=\frac{16{\pi }^{10}R{{\mu }_0}^3}{3^6{c}^8{\alpha }_{inv}^7}}$	${\displaystyle \frac{k{g}^3{s}^2}{m^6{A}^6}}$, 15*3-30*2+13*6-3*6 = 45	${\displaystyle kg}$, θ = 15
Planck length	${\displaystyle (l_p^{*}{)}^{15}=\frac{{\pi }^{22}{{\mu }_0}^9}{2^{35}{3}^{24}{\alpha }_{inv}^{49}{c}^{35}{R}^8}}$	${\displaystyle \frac{k{g}^9{s}^{17}}{m^{18}{A}^{18}}}$, 15*9-30*17+13*18-3*18 = -195	${\displaystyle m}$, θ = -13
Planck mass	${\displaystyle (m_P^{*}{)}^{15}=\frac{2^{25}{\pi }^{13}{{\mu }_0}^6}{3^6{c}^5{\alpha }_{inv}^{16}{R}^2}}$	u = ${\displaystyle \frac{k{g}^6{m}^3}{s^7{A}^{12}}}$, 15*6-13*3+30*7-3*12 = 225	${\displaystyle kg}$, θ = 15

Tables 8 and 9 present two different numerical routes to the same goal:

• Table 8: solve dimensioned constants directly from ${\displaystyle (\alpha ,\mathrm{\Omega },v,r)}$.
• Table 9: solve the same constants using the most precise anchors ${\displaystyle (\alpha ,R,c,{\mu }_0)}$ (CODATA 2014 context), then derive the least precise constants (${\displaystyle h,e,m_e,G,k_B}$).

These are two approaches to the same model and can be treated as a single statistical test family. As with Table 4, CODATA uncertainties are not used; the goal is an order-of-magnitude estimate of how unlikely the *overall agreement pattern* is under “no relationship”.

For each predicted constant ${\displaystyle x^{*}}$ compared to CODATA mean ${\displaystyle x}$:

• Relative error:

${\displaystyle \epsilon =\left|\frac{x^{*}-x}{x}\right|}$

• Two-sided coincidence probability:

${\displaystyle p\approx 2\epsilon }$

• Joint probability (naive independence):

${\displaystyle p_{\text{joint}}=\prod _i{p}_i}$

• Sigma-equivalent (two-sided Normal):

${\displaystyle \sigma \approx {\mathrm{\Phi }}^{-1}(1-p/2)}$

Per-constant coincidence estimates

Constant	calculated (model)	CODATA 2014 (mean)	rel. error ${\displaystyle \epsilon }$	${\displaystyle p\approx 2\epsilon }$
${\displaystyle h}$	6.626069134×10-34	6.626070040×10-34	1.37×10-7	2.73×10-7
${\displaystyle e}$	1.60217651130×10-19	1.6021766208×10-19	6.83×10-8	1.37×10-7
${\displaystyle G}$	6.67249719229×10-11	6.67408×10-11	2.37×10-4	4.74×10-4
${\displaystyle k_B}$	1.37951014752×10-23	1.38064852×10-23	8.25×10-4	1.65×10-3

Note: ${\displaystyle m_e}$ is derived in the Table 9 pathway (and in the electron section). When included, it is treated as part of the same test family (see “extended joint” below).

Using the four constants that appear explicitly in Table 8:

${\displaystyle p_{\text{joint}}(h,e,G,k_B)\approx 2.92\times 10^{-20}}$

${\displaystyle -{\log }_{10}(p)\approx 19.53}$

sigma-equivalent:

${\displaystyle \sigma \approx 9.22\sigma }$

Excluding the two least precise sector constants ${\displaystyle (G,k_B)}$ leaves only ${\displaystyle (h,e)}$:

${\displaystyle p_{\text{joint}}(h,e)\approx 3.73\times 10^{-14}}$

${\displaystyle -{\log }_{10}(p)\approx 13.43}$

sigma-equivalent:

${\displaystyle \sigma \approx 7.57\sigma }$

If we include the electron mass test (from the Table 9 pathway / electron calculations), the 5-constant set is:

${\displaystyle \{h,e,m_e,G,k_B\}}$

From the computed results already obtained:

${\displaystyle p_{\text{all}}=8.007456088691929\times 10^{-27}}$

${\displaystyle -{\log }_{10}(p_{\text{all}})=26.096505434242797}$

sigma-equivalent:

${\displaystyle {\sigma }_{\text{all}}\approx 10.72\sigma }$

For the 3-constant high-precision subset:

${\displaystyle \{h,e,m_e\}}$

From the computed results already obtained:

${\displaystyle p_{\text{noGkB}}=1.0237562566087324\times 10^{-20}}$

${\displaystyle -{\log }_{10}(p_{\text{noGkB}})=19.98980343106562}$

sigma-equivalent:

${\displaystyle {\sigma }_{\text{noGkB}}\approx 9.33\sigma }$

1. The “clean” electromagnetic/mechanical subset (${\displaystyle h,e,m_e}$) yields very strong joint coincidence (≈10^-20 scale).
2. Including ${\displaystyle G}$ and especially ${\displaystyle k_B}$ degrades per-test precision, but the overall joint remains extremely small.
3. These joint numbers are not exact; they are order-of-magnitude indicators under the null of “no relationship”.

1. Dependence: these tests share constants and model structure, so strict independence is not guaranteed; multiplying probabilities is therefore an optimistic estimator.
2. Look-elsewhere: if many candidate constructions were tried and only the best retained, a search-space correction would reduce the effective significance.

In this analysis we define per-test coincidence probabilities ${\displaystyle p_i\approx 2{\epsilon }_i}$ and combine them via:

${\displaystyle p_{\text{joint}}=\prod _i{p}_i}$

The “sigma-equivalent” is then obtained by mapping the two-sided probability ${\displaystyle p_{\text{joint}}}$ to a standard Normal tail probability.

Because ${\displaystyle 0<p_i<1}$, adding additional tests (even low-precision ones such as ${\displaystyle G}$ and ${\displaystyle k_B}$) typically makes ${\displaystyle p_{\text{joint}}}$ smaller, and therefore makes the combined sigma larger.

Example (from the computed results):

• with ${\displaystyle \{h,e,m_e\}}$ only:

${\displaystyle p_{\text{noGkB}}=1.0237562566\times 10^{-20}}$

• with ${\displaystyle \{h,e,m_e,G,k_B\}}$:

${\displaystyle p_{\text{all}}=8.0074560887\times 10^{-27}}$

Since ${\displaystyle p_{\text{all}}\ll p_{\text{noGkB}}}$, the corresponding joint sigma is higher when ${\displaystyle G}$ and ${\displaystyle k_B}$ are included.

We can construct a table of constants using these 3 geometries. Setting

${\displaystyle f(x)units=(\frac{L^{15}}{M^9{T}^{11}}{)}^n=1}$

i.e.: unit number θ = (-13*15) - (15*9) - (-30*11) = 0

${\displaystyle i={\pi }^2{\mathrm{\Omega }}^{15}}$, units = ${\displaystyle \sqrt{f(x)}}$ = 1 (unit number = 0, no scalars)

${\displaystyle x=\mathrm{\Omega }\frac{v}{r^2}}$ , units = ${\displaystyle \sqrt{\frac{L}{MT}}}$ = u¹ = u (unit number = -13 -15 +30 = 2/2 = 1, with scalars v, r)

${\displaystyle y=\pi \frac{r^{17}}{v^8}}$ , units = ${\displaystyle M^{2}T}$ = 1, (unit number = 15*2 -30 = 0, with scalars v, r)

Note: The following suggests a numerical boundary to the values the SI constants can have.

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

${\displaystyle \frac{r^{17}}{v^8}=k^{2}t=\frac{k^{17/4}}{v^{15/4}}=...}$ gives a range from 0.812997... x10^-59 to 0.123... x10⁶⁰

Note: 
1. The constants with unit numbers ${\displaystyle \theta }$ in the series ${\displaystyle ({\theta }^{15}{)}^n}$ have no Omega. This further suggests an underlying geometrical base-15.  

Table 10. Table of Constants

Constant	θ	Geometrical object (α, Ω, v, r)	Unit	Calculated	CODATA 2014
Time (Planck)	${\displaystyle -30}$	${\displaystyle T=\frac{x^{\theta }{i}^2}{y^3}=\frac{\pi r^9}{v^6}}$	${\displaystyle T}$	T = 5.390 517 866 e-44	tp = 5.391 247(60) e-44
Elementary charge	${\displaystyle -27}$	${\displaystyle e^{*}=(\frac{2^7{\pi }^3}{\alpha })\frac{x^{\theta }{i}^2}{y^3}=(\frac{2^7{\pi }^3}{\alpha })\frac{\pi {\mathrm{\Omega }}^3{r}^3}{v^3}}$	${\displaystyle \frac{L^{3/2}}{T^{1/2}{M}^{3/2}}=AT}$	e* = 1.602 176 511 30 e-19	e = 1.602 176 620 8(98) e-19
Length (Planck)	${\displaystyle -13}$	${\displaystyle L=(2\pi )\frac{x^{\theta }i}{y}=(2\pi )\frac{\pi {\mathrm{\Omega }}^2{r}^9}{v^5}}$	${\displaystyle L}$	L = 0.161 603 660 096 e-34	lp = 0.161 622 9(38) e-34
Ampere	${\displaystyle 3}$	${\displaystyle A=(\frac{2^7{\pi }^3}{\alpha })x^{\theta }=(\frac{2^7{\pi }^3}{\alpha })\frac{{\mathrm{\Omega }}^3{v}^3}{r^6}}$	${\displaystyle A=\frac{L^{3/2}}{M^{3/2}{T}^{3/2}}}$	A = 0.297 221 e25	e/tp = 0.297 181 e25
Gravitational constant	${\displaystyle 6}$	${\displaystyle G^{*}=(2^3{\pi }^3)x^{\theta }y=(2^3{\pi }^3)\frac{\pi {\mathrm{\Omega }}^6{r}^5}{v^2}}$	${\displaystyle \frac{L^3}{M{T}^2}}$	G* = 6.672 497 192 29 e11	G = 6.674 08(31) e-11
Mass (Planck)	${\displaystyle 15}$	${\displaystyle M=\frac{x^{\theta }{y}^2}{i}=\frac{r^4}{v}}$	${\displaystyle M}$	M = .217 672 817 580 e-7	mP = .217 647 0(51) e-7
Velocity	${\displaystyle 17}$	${\displaystyle V=(2\pi )\frac{x^{\theta }{y}^2}{i}=(2\pi ){\mathrm{\Omega }}^{2}v}$	${\displaystyle V=\frac{L}{T}}$	V = 299 792 458	c = 299 792 458
Planck constant	${\displaystyle 19}$	${\displaystyle h^{*}=(2^3{\pi }^3)\frac{x^{\theta }{y}^3}{i}=(2^3{\pi }^3)\frac{\pi {\mathrm{\Omega }}^4{r}^{13}}{v^5}}$	${\displaystyle \frac{L^{2}M}{T}}$	h* = 6.626 069 134 e-34	h = 6.626 070 040(81) e-34
Planck temperature	${\displaystyle 20}$	${\displaystyle {T_p}^{*}=(\frac{2^7{\pi }^3}{\alpha })\frac{x^{\theta }{y}^2}{i}=(\frac{2^7{\pi }^3}{\alpha })\frac{{\mathrm{\Omega }}^5{v}^4}{r^6}}$	${\displaystyle \frac{L^{5/2}}{M^{3/2}{T}^{5/2}}=AV}$	Tp* = 1.418 145 219 e32	Tp = 1.416 784(16) e32
Boltzmann constant	${\displaystyle 29}$	${\displaystyle {k_B}^{*}=(\frac{\alpha }{2^5\pi })\frac{x^{\theta }{y}^4}{i^2}=(\frac{\alpha }{2^5\pi })\frac{r^{10}}{\mathrm{\Omega }{v}^3}}$	${\displaystyle \frac{M^{5/2}{T}^{1/2}}{L^{1/2}}=\frac{ML}{TA}}$	kB* = 1.379 510 147 52 e-23	kB = 1.380 648 52(79) e-23
Vacuum permeability	${\displaystyle 56}$	${\displaystyle {{\mu }_0}^{*}=(\frac{\alpha }{2^{11}{\pi }^4})\frac{x^{\theta }{y}^7}{i^4}=(\frac{\alpha }{2^{11}{\pi }^4})\frac{r^7}{\pi {\mathrm{\Omega }}^4}}$	${\displaystyle \frac{ML}{T^2{A}^2}}$	μ0* = 4π/10^7	μ0 = 4π/10^7

This section is not about statistical agreement (already analysed earlier). It is about why the parameterisation

${\displaystyle i={\pi }^2{\mathrm{\Omega }}^{15},x=\mathrm{\Omega }\frac{v}{r^2},y=\pi \frac{r^{17}}{v^8}}$

works structurally, and why the repeated appearance of “15” behaves like a fundamental guide-rail rather than a numerical coincidence.

The model separates:

• pure geometry (dimensionless): built from ${\displaystyle \pi }$ and ${\displaystyle \mathrm{\Omega }}$, and
• local unit-system scalars (dimensioned): carried by ${\displaystyle r}$ and ${\displaystyle v}$.

The parameters are chosen to isolate these roles:

• ${\displaystyle i={\pi }^2{\mathrm{\Omega }}^{15}}$ is constructed to be unitless and scalar-free.

 It is the “pure geometric driver” that can appear in every constant without importing any SI/terrestrial scaling.

• ${\displaystyle x=\mathrm{\Omega }\frac{v}{r^2}}$ is a “unit-carrier” with the minimal scalar content needed to introduce a net unit number of one unit step (${\displaystyle u^1}$) while retaining Ω-dependence.

• ${\displaystyle y=\pi \frac{r^{17}}{v^8}}$ is constructed so its net unit-number is zero (units cancel), but it still carries the scalar degrees needed to shift magnitudes between the Planck objects and SI.

In short: ${\displaystyle i}$ carries geometry only; ${\displaystyle x}$ carries a single unit-step; ${\displaystyle y}$ carries scalar degrees with net unit-number zero.

A central requirement is that the “table of constants” be generated in a way that:

1. preserves dimensional homogeneity (correct unit numbers θ),
2. preserves the claim that certain key constructions are dimensionless (units = 1 and scalars = 1),
3. allows a single consistent parameterisation to span many constants with different θ.

This forces the existence of a non-trivial “null transformation” on the (M,T,L,...) lattice: a transformation that changes exponents but leaves the net unit-number unchanged.

The model explicitly identifies one such null combination:

${\displaystyle f(x)\text{units}={\left(\frac{L^{15}}{M^9{T}^{11}}\right)}^n=1}$

because its unit number is exactly zero:

${\displaystyle \theta (f)=(-13\cdot 15)-(15\cdot 9)-(-30\cdot 11)=0.}$

This means ${\displaystyle L^{15}{M}^{-9}{T}^{-11}}$ acts like a closed loop in exponent space: you can multiply any expression by ${\displaystyle f(x{)}^n}$ without changing its unit number. That “loop” creates a discrete family of equivalent representations, and it is precisely here that “15” becomes structural:

• the smallest, stable integer loop that closes under the model’s allowed building blocks introduces the factor 15 on L.

So “15” is not chosen to fit a number: it arises as the closure length of the model’s dimensionless loop on the unit lattice.

A separate constraint is that when units and scalars cancel, the remaining dimensionless structure must come only from the allowed geometrical generators (π and Ω) without introducing new “free” numerical content.

Empirically in the construction, the dimensionless-cancellation ratios consistently leave a residual factor of:

${\displaystyle ({\mathrm{\Omega }}^{15}{)}^n,n\ge 0}$

rather than arbitrary Ω-powers.

This is explained by the fact that the primitive Planck objects in the model use Ω in low powers (primarily Ω^2 and Ω^3). When you form general products/ratios and require:

• unit-number cancellation (θ=0),
• scalar cancellation (r,v cancel),

the remaining Ω-power must land in the additive semigroup generated by {2,3}. The smallest non-trivial common “period” that repeatedly reappears across many such cancellations is 15, because:

• 15 is the smallest positive integer with many decompositions into 2s and 3s (e.g., 15=3+...=2+...),
• and therefore acts like a natural “return point” for Ω-power alignment across many independent cancellations.

Thus ${\displaystyle i={\pi }^2{\mathrm{\Omega }}^{15}}$ acts as a universal dimensionless residue that can appear in every constant without violating the cancellation rules.

The scalar sector is intentionally minimal: only two independent scalars are permitted (r and v), and all other scale-factors are derived from them. Therefore any global parameterisation of constants must:

• introduce exactly two scalar degrees of freedom,
• but still be able to shift θ across many constants.

The choices:

${\displaystyle x=\mathrm{\Omega }\frac{v}{r^2},y=\pi \frac{r^{17}}{v^8}}$

achieve this with a clear separation:

• x changes the unit-number by one unit-step (acts like a “ladder operator” on θ),
• y has θ=0 but carries the scalar degrees required to set magnitudes while preserving unit-number closure.

This is why the table can express constants in the generic form:

${\displaystyle \text{constant}\propto x^{\theta }{i}^p{y}^q}$

with integer p,q chosen so that:

• the correct θ is produced,
• the scalar dependence is consistent,
• and the remaining geometry is only π and Ω.

Within the model, the appearance of base-15 is not a free numerical choice; it is a conditional necessity if all of the following are required simultaneously:

1. a fixed integer unit-number assignment (θ) for SI base units,
2. a non-trivial closed-loop (θ=0) transformation on the exponent lattice,
3. strict cancellation rules for “dimensionless invariants” (units=1 and scalars=1),
4. only two independent scalar degrees of freedom (r,v),
5. and a small generator set for geometry (π and Ω with low Ω-powers in the primitive objects).

Under these constraints, a closure loop like ${\displaystyle L^{15}{M}^{-9}{T}^{-11}}$ (θ=0) forces a corresponding universal dimensionless residue, and the natural stable residue across many cancellations is ${\displaystyle {\mathrm{\Omega }}^{15}}$. In this sense the base-15 geometry functions as a guide-rail: it is the smallest stable closure structure compatible with the model’s restricted building blocks and cancellation requirements.

Because x and y are constructed as the minimal carriers of unit-scaling, their numerical values constrain the allowable magnitudes of all constants generated from:

${\displaystyle x^{\theta }{i}^p{y}^q.}$

Hence relationships such as:

${\displaystyle \frac{v}{r^2}=a^{1/3}=\cdots }$

and

${\displaystyle \frac{r^{17}}{v^8}=\cdots }$

act as natural “range setters”: once two scalar degrees are fixed, every constant’s magnitude is forced into a narrow admissible band consistent with its θ.

This provides a mechanism for keeping dimensioned constants within defined ranges, while still allowing unit-system changes (SI → imperial, etc.) via the scalars.

This conclusion integrates the four pillars already tested plus an algorithmic-information (Kolmogorov complexity / MDL) perspective on *why* the model is non-trivial.

The θ-mapping acts as a single accounting system that must remain consistent across *every* section:

• Dimensionless cancellations (Table 4).
• Dimensioned constant reconstruction (Tables 8/9).
• Electron construction (ψ) and the quark bookkeeping (DDD=T, DUU=±e).

The strongest outcome is not that one constant matches, but that the same θ-additivity rules (multiply/divide → add/subtract θ) remain valid across many unrelated expressions, while still supporting the quark relations.

Treating Planck units as geometrical objects is supported by the “dimensionless sector” results:

• Multiple independent unitless combinations collapse to the same numeric values once units/scalars cancel (Table 4).
• This is the part least vulnerable to “unit conventions” because it tests pure cancellation structure rather than individual constants.

Quantitatively (using the coincidence-probability method p≈2ε and joint multiplication):

• Table 4 (all rows): p_joint ≈ 7.11×10^-22 (≈ 9.61σ equiv; information ≈ 70.3 bits).
• Table 4 excluding (G,kB): p_joint ≈ 9.21×10^-14 (≈ 7.45σ equiv; information ≈ 43.3 bits).

Interpretation: the geometric-object thesis is not just fitting values; it is reproducing the *invariant cancellation logic* of physics relations.

The exhaustive integer-space search (bounded scan of (M,T,P), with V,L,A derived) under the full constraint bundle (dimensional homogeneity + ψ dimensionless + quark structure + DDD=T) collapses admissible solutions onto a single invariant constraint class:

${\displaystyle 3M+2T=-15}$

This is the core “guide-rail” result:

• Different integer triples may appear, but they are equivalent lattice shifts along the same constraint rail.
• Selecting the canonical representative is then a modelling choice (privileging the primary objects), giving:

${\displaystyle M=15,T=-30}$

with the familiar derived unit numbers following at that lattice point.

Hence base-15 is not introduced as a numerological preference; it is the unique survivor (up to equivalence) of the full constraint bundle.

Two layers support the “mathematical electron” claim:

• Structural (non-statistical): ψ is dimensionless because both units and scalars cancel:

${\displaystyle \psi =\frac{(AL{)}^3}{T},\text{units}=1,\text{scalars}=1}$

so ψ is a pure number encoding the electron construction. “Electron properties” (mass, charge, wavelength…) are then derived parameters, while the electron itself is represented by the invariant ψ.

• Statistical (parameter reproduction): using ψ to solve electron parameters yields very small relative deviations. For the three key electron-parameter tests (λe, me, e):
• p_joint ≈ 5.73×10^-23 (≈ 9.87σ equiv; information ≈ 73.9 bits).

This indicates the ψ-construction is not only internally consistent (dimensionless) but externally constrained by multiple electron observables simultaneously.

Kolmogorov complexity K(·) is the length of the shortest program that outputs a dataset. Exact K is uncomputable, but we can compare *upper bounds* using the Minimum Description Length (MDL) principle:

${\displaystyle \text{Total description length}\approx L(\text{model})+L(\text{residuals})}$

• • Baseline (no-relationship null):**

If constants/ratios are unrelated, then each reported agreement to within tolerance ε requires specifying those coincident digits explicitly. The surprisal (information content) of an event with probability p is:

${\displaystyle I=-{\log }_2(p)\text{ bits}}$

Under our coincidence rule p≈2ε, the joint results already computed can be re-read as “how many bits of coincidence” the model is explaining/compressing:

• Table 4 (all rows): p≈7.11×10^-22 ⇒ I≈70.3 bits
• Table 4 without (G,kB): p≈9.21×10^-14 ⇒ I≈43.3 bits
• Electron parameters (λe, me, e): p≈5.73×10^-23 ⇒ I≈73.9 bits
• All-constants suite (h,e,me,G,kB): p≈8.007×10^-27 ⇒ I≈86.7 bits
• No-(G,kB) suite (h,e,me): p≈1.024×10^-20 ⇒ I≈66.4 bits
• Alpha Table 7 joint (3 formulas): p≈4.755×10^-10 ⇒ I≈31.0 bits

• • Compression claim (what this means):**
• The model functions as a compact generator: a short rule-set (θ algebra + base-15 constraint + small generator set (π, Ω, α) plus two scalars for SI translation) produces many targets.
• The joint surprisal values above quantify how many “bits of coincidence” would otherwise have to be carried as independent facts if there were no underlying structure.
• Therefore the evidence is not merely “fits numbers,” but “achieves compression”: it replaces tens of bits of independent specification by a small set of generative constraints.

• • Why base-15 matters in MDL terms:**

The uniqueness result (collapse to 3M+2T=-15) is a strong MDL signal: it drastically reduces the hypothesis space. Instead of many competing integer assignments, the constraint bundle leaves only one equivalence class. That is exactly the type of structural reduction MDL rewards.

Across all sections, the results are best interpreted as an overdetermined constraint framework:

• θ supplies the universal bookkeeping.
• The geometrical Planck-object construction reproduces multiple independent dimensionless invariants with strong joint significance.
• The quark/ψ requirements collapse the unit-number space to a unique base-15 guide-rail (up to equivalent lattice shifts).
• The electron is encoded as a dimensionless invariant ψ, and its derived parameters match multiple observables jointly.
• In algorithmic-information terms, the model provides substantial compression (tens to ~90 bits, depending on the test bundle) relative to a “no relationship” null—supporting the claim that the structure is not simple numerology but a genuine generative constraint system.

The mathematical electron model is an example of a digital physics simulation hypothesis that uses geometrical forms. For AI analysis, the relevant journal articles have been compiled into a single .pdf file ^[6].

• Physics:Simulation_argument_(Programmer_God): Introduction to a Planck scale Programmer God Simulation Hypothesis model
• Physics:Digital_Physics/Simulation_Hypothesis/Electron_(mathematical): Mathematical electron from Planck units
• Physics:Digital_Physics/Simulation_Hypothesis/Gravity_via_Atomic_orbitals: Gravitational orbits emerge from Planck scale n-body rotating atomic orbital pairing
• Physics:Digital_Physics/Simulation_Hypothesis/Planck_unit_scaffolding: CMB and a Planck unit universe scaffolding
• Physics:Digital_Physics/Simulation_Hypothesis/Relativity as Perspective: Relativity as a translation between 2 co-ordinate systems

• The Source Code of God; Programming at the Planck scale

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