Electron (mathematical)

The mathematical electron model

In the mathematical electron model ^[1], the electron is a dimensionless geometrical formula (ψ). This formula, which resembles the volume of a torus or surface of a 4-D hypersphere, is itself a complex geometry that is the construct of simpler geometries; the Planck units.

In this model the Planck units are geometrical objects, the geometry of 2 dimensionless constants (the fine structure constant and a mathematical constant Omega). Although dimensionless, the function of the Planck unit is embedded within the geometry; the geometry of the Planck time object embeds the function 'time', the geometry of the Planck length object embeds the function 'length' ... and being geometrical objects they can combine to form more complex objects, from electrons to planets.

This means that the electron parameters are defined in Planck units; electron wavelength is measured in units of Planck length, electron frequency is measured in units of Planck length ... It is this geometrical electron formula that dictates the length of the wavelength = ψ * Planck length (ψ units of Planck length), frequency = ψ * Planck time ...

This ψ itself is an assembly of the Planck units and so could be defined as a Planck particle (the Planck units not only encode the function of the Planck units, they also encode the information of the electron). Or, conversely we may state that this formula ψ not only embeds within its geometry the Planck units, but also encodes the information required to define the electron parameters. By definition therefore, this dimensionless formula ψ is the electron. And if the electron is a mathematical particle, then so too are the other particles, and so the universe itself becomes a mathematical universe.

The formula ψ is the geometry of 2 constants;

the dimensionless physical constant (inverse) fine structure constant alpha α = 137.035 999 139 (CODATA 2014) and

Omega Ω = 2.0071349496 (best fit)

Omega has a potential solution in terms of pi and e and so may be a mathematical (not physical) constant

\Omega ={\sqrt {\left(\pi ^{e}e^{(1-e)}\right)}}=2.0071349543...

\psi =4\pi ^{2}(2^{6}3\pi ^{2}\alpha \Omega ^{5})^{3}=.2389...\;x10^{23}

, units = 1

Planck objects

For the Planck units, the model uses geometrical objects (the geometry of alpha and Omega) instead of a numbering system, this has the advantage in that the attribute can be embedded within the geometry (although the geometry itself is dimensionless).

table 1. Geometrical units
Attribute	Geometrical object	Unit
mass	$M=(1)$	(kg)
time	$T=(\pi )$	(s)
velocity	$V=(2\pi \Omega ^{2})$	(m/s)
length	$L=(2\pi ^{2}\Omega ^{2})$	(m)
ampere	$A=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha }})$	(A)

As these objects have a geometrical form, we can combine them Lego style; the length object L can be combined with the time object T to form the velocity object V and so forth ... to create complex events such as electrons to apples to ... and so the apple has mass because embedded within it are the mass objects M, complex events thus retain all the underlying information.

This however requires a relationship between the Planck unit geometries that defines how they may combine, this can be represented by assigning to each attribute a unit number θ (i.e.: θ = 15 ⇔ kg) ^[2].

Geometrical units
Attribute	Geometrical object	unit equivalent
mass	$M=1$	kg ⇔ 15
time	$T=2\pi$	s ⇔ -30
length	$L=2\pi ^{2}\Omega ^{2}$	m ⇔ -13
velocity	$V=2\pi \Omega ^{2}$	m/s ⇔ 17
ampere	$A={\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }}$	A ⇔ 3

As alpha and Omega have numerical values, so too the MLTA objects can be expressed numerically. We can then convert these objects to their Planck unit equivalents by including a dimensioned scalar. For example, $V=2\pi \Omega ^{2}$ = 25.3123819353... and so we can use scalar v to convert from dimensionless geometrical object V to dimensioned c.

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

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

Scalars
attribute	geometrical object	scalar (unit number)
mass	$M=(1)$	k (θ = 15)
time	$T=(\pi )$	t (θ = -30)
velocity	$V=(2\pi \Omega ^{2})$	v (θ = 17)
length	$L=(2\pi ^{2}\Omega ^{2})$	l (θ = -13)
ampere	$A=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha }})$	a (θ = 3)

As the scalar incorporates the dimension quantity (the dimension quantity for v = m/s or miles/s), the unit number relationship applies, and so we then find that only 2 scalars are needed. This is because in a defined ratio they will overlap and cancel, for example in the following ratios;

scalar units for ampere a = u³, length l = u^-13, time t = u^-30, mass k = u¹⁵ (u^Θ represents unit)

{\frac {({u^{3}})^{3}{(u^{-13}})^{3}}{(u^{-30})}}={\frac {{(u^{-13})}^{15}}{{(u^{15})}^{9}{(u^{-30})}^{11}}}=1

For example if we know the numerical values for a and l then we know the numerical value for t, and from l and t we know k … and so if we know any 2 scalars (α and Ω have fixed values) then we can solve the Planck units (for that system of units), and from these, we can solve (G, h, c, e, m_e, k_B).

{\frac {a^{3}l^{3}}{t}}={\frac {m^{15}}{k^{9}t^{11}}}=1

In this table the 2 scalars used are r and v. A further attribute is included, P = the square root of (Planck) momentum. V and A can thus be considered composite objects.

Geometrical objects
attribute	geometrical object	unit number θ	scalar r(8), v(17)
mass	$M=(1)$	15 = 8*4-17	$k={\frac {r^{4}}{v}}$
time	$T=(\pi )$	-30 = 89-176	$t={\frac {r^{9}}{v^{6}}}$
sqrt(momentum)	$P=(\Omega )$	16 = 8*2	r²
velocity	$V=L/T=(2\pi \Omega ^{2})$	17	v
length	$L=(2\pi ^{2}\Omega ^{2})$	-13 = 89-175	$l={\frac {r^{9}}{v^{5}}}$
ampere	$A={\frac {2^{4}V^{3}}{\alpha P^{3}}}=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha }})$	3 = 173-86	$a={\frac {v^{3}}{r^{6}}}$

Mathematical electron

The mathematical electron ψ incorporates the dimensioned Planck units but itself is dimension-less (units = scalars = 1). Here ψ is defined in terms of σ_e, where AL is an ampere-meter (ampere-length = e*c are the units for a magnetic monopole).

T=\pi ,\;unit=u^{-30},\;scalars={\frac {r^{9}}{v^{6}}}

\sigma _{e}={\frac {3\alpha ^{2}AL}{2\pi ^{2}}}={2^{7}3\pi ^{3}\alpha \Omega ^{5}},\;unit=u^{(3\;-13\;=\;-10)},\;scalars={\frac {r^{3}}{v^{2}}}

\psi ={\frac {\sigma _{e}^{3}}{2T}}={\frac {(2^{7}3\pi ^{3}\alpha \Omega ^{5})^{3}}{2\pi }},\;unit={\frac {(u^{-10})^{3}}{u^{-30}}}=1,scalars=({\frac {r^{3}}{v^{2}}})^{3}{\frac {v^{6}}{r^{9}}}=1

\psi =4\pi ^{2}(2^{6}3\pi ^{2}\alpha \Omega ^{5})^{3}=.23895453...x10^{23},\;unit=1

(unit-less)

Both units and scalars cancel.

Electron parameters

We can solve the electron parameters; electron mass, wavelength, frequency, charge ... as the frequency of the Planck units themselves, and this frequency is ψ.

v=11843707.905...,\;units={\frac {m}{s}}

r=0.712562514304...,\;units=({\frac {kg.m}{s}})^{1/4}

electron wavelength λ_e = 2.4263102367e-12m (CODATA 2014)

\lambda _{e}^{*}=2\pi L\psi

= 2.4263102386e-12m (L ⇔ Planck length)

electron mass m_e = 9.10938356e-31kg (CODATA 2014)

m_{e}^{*}={\frac {M}{\psi }}

= 9.1093823211e-31kg (M ⇔ Planck mass)

elementary charge e = 1.6021766208e-19C (CODATA 2014)

e^{*}=A\;T

= 1.6021765130e-19 (T ⇔ Planck time)

Rydberg constant R = 10973731.568508/m (CODATA 2014)

R^{*}=({\frac {m_{e}}{4\pi L\alpha ^{2}M}})={\frac {1}{2^{23}3^{3}\pi ^{11}\alpha ^{5}\Omega ^{17}}}{\frac {v^{5}}{r^{9}}}\;u^{13}

= 10973731.568508

Wavelength is ψ units of Planck length, frequency is ψ units of Planck time ... however electron mass is a unit of Planck mass, but this only occurs once per ψ units of Planck time.

Electron Mass

Particle mass is a unit of Planck mass that occurs only once per ψ units of Planck time, the other parameters are continuums of the Planck units.

units

\psi ={\frac {(AL)^{3}}{T}}

= 1

This may be interpreted as; for ψ units of Planck time the electron has wavelength L, charge A ... and then the AL combine with time T (A³L³/T) and the units (and scalars) cancel. The electron is now mass (for 1 unit of time). In this consideration, the electron is an event that oscillates over time between an electric wave state (duration ψ units of Planck time) to a unit of Planck mass point state (1 unit of time). The electron is a quantum scale event, it does not exist at the discrete Planck scale (and so therefore neither does the quantum scale).

As electron mass is the frequency of the geometrical Planck mass M = 1, which is a point (and so with point co-ordinates), then we have a model for a black-hole electron, the electron function ψ centered around this unit of Planck mass. When the wave-state (A*L)³/T units collapse, this black-hole center (point) is exposed for 1 unit of (Planck) time. The electron is 'now' (a unit of Planck) mass.

Mass in this consideration is not a constant property of the particle, rather the measured particle mass m would refer to the average mass, the average occurrence of the discrete Planck mass point-state over time. The formula E = hf is a measure of the frequency f of occurrence of Planck's constant h and applies to the electric wave-state. As for each wave-state then is a corresponding mass point-state, then for a particle E = hf = mc2. Notably however the c term is a fixed constant unlike the f term, and so the m term is referring to an average mass (mass which is measured over time) rather than a constant mass (mass as a constant property of the particle at unit Planck time). Thus as noted, when we refer to mass as a constant property, we are referring to mass at the quantum scale, and the electron as a quantum-state particle.

If the scaffolding of the universe includes units of Planck mass M, then it is not necessary for the particle itself to have mass.

Quarks

The charge on the electron derives from the embedded ampere A and length L, the electron formula ψ itself is dimensionless. These AL magnetic monopoles would seem to be analogous to quarks (there are 3 monopoles per electron), but due to the symmetry and so stability of the geometrical ψ there is no clear fracture point by which an electron could decay, and so this would be difficult to test. We can however conjecture on what a quark solution might look like, the advantage with this approach being that we do not need to introduce new 'entities' for our quarks, the Planck units embedded within the electron suffice and so also therefore ψ.

Electron formula

\psi =2^{20}\pi ^{8}3^{3}\alpha ^{3}\Omega ^{15},\;unit=1,scalars=1

Time

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

AL magnetic monopole

\sigma _{e}={\frac {3\alpha ^{2}AL}{2\pi ^{2}}}={2^{7}3\pi ^{3}\alpha \Omega ^{5}},\;u^{-10},\;scalars={\frac {r^{3}}{v^{2}}}

\psi ={\frac {\sigma _{e}^{3}}{2T}}={\frac {(2^{7}3\pi ^{3}\alpha \Omega ^{5})^{3}}{2\pi }}=2^{20}3^{3}\pi ^{8}\alpha ^{3}\Omega ^{15},\;unit={\frac {(u^{-10})^{3}}{u^{-30}}}=1,scalars=({\frac {r^{3}}{v^{2}}})^{3}{\frac {v^{6}}{r^{9}}}=1

If $\sigma _{e}$ could equate to a quark with an electric charge of -1/3e, then it would be an analogue of the D quark. 3 D quarks would constitute the electron as DDD = (AL)*(AL)*(AL).

We would assume that the charge on the positron (anti-matter electron) is just the inverse of the above, however there is 1 problem, the AL units = -10, if we look at the table of constants, there is no 'units = +10' combination that can include A (units=3). However we can make a Planck temperature T_p AV monopole (ampere-velocity).

T_{p}={\frac {2^{7}\pi ^{3}\Omega ^{5}}{\alpha }},\;u^{20},\;scalars={\frac {r^{9}}{v^{6}}}

\sigma _{t}={\frac {3\alpha ^{2}T_{p}}{2\pi }}={\frac {3\alpha ^{2}AV}{2\pi ^{2}}}=({2^{6}3\pi ^{2}\alpha \Omega ^{5}}),\;u^{20},\;scalars={\frac {v^{4}}{r^{6}}}

\psi =(2T)\sigma _{t}^{2}\sigma _{e}=2^{20}3^{3}\pi ^{8}\alpha ^{3}\Omega ^{15},\;unit=(u^{-30})(u^{20})^{2}(u^{-10})=1,scalars=({\frac {r^{9}}{v^{6}}})({\frac {v^{4}}{r^{6}}})^{2}{\frac {r^{3}}{v^{2}}}=1

The units for $\sigma _{t}$ = +20, and so if units = -10 equates to -1/3e, then we may conjecture that units = +20 equates to 2/3e, which would be the analogue of the U quark. Our plus charge now becomes DUU, and so although the positron has the same wavelength, frequency, mass and charge magnitude as the electron (both solve to ψ), internally its charge structure resembles that of the proton, the positron is not simply an inverse of the electron. This could have implications for the missing anti-matter, and for why the charge magnitude of the proton is exactly the charge magnitude of the electron.

D=\sigma _{e},\;unit=u^{-10},\;charge={\frac {-1e}{3}},\;scalars={\frac {r^{3}}{v^{2}}}

U=\sigma _{t},\;unit=u^{20},\;charge={\frac {2e}{3}},\;scalars={\frac {v^{4}}{r^{6}}}

Numerically:

Adding a proton and electron gives (proton) UUD & DDD (electron) = 2(UDD) = 20 -10 -10 = 0 (zero charge), scalars = 0.

Converting between U and D via U & DDD (electron) = 20 -10 -10 -10 = -10 (D), scalars = ${\frac {r^{3}}{v^{2}}}$

Magnetic monopole

\sigma _{e}={\frac {3\alpha ^{2}AL}{2\pi ^{2}}}={2^{7}3\pi ^{3}\alpha \Omega ^{5}},\;u^{-10},\;scalars={\frac {r^{3}}{v^{2}}}

In this model alpha appears as an orbital constant for gravitational and atomic orbital radius, combining a fixed alpha term with an orbital wavelength term;

r_{orbital}=2\alpha (\lambda _{orbital})

If we replace $\lambda _{orbital}$ with the geometrical Planck length L, and include momentum P and velocity V (the 2 components from which the ampere A is derived), then we may consider if the internal structure of the electron involves rotation of this monopole AL super-structure and this has relevance to electron spin;

2\alpha L{\frac {V^{3}}{P^{3}}}=2^{5}\pi ^{5}\alpha \Omega ^{5},\;units=u^{-10},\;scalars={\frac {r^{3}}{v^{2}}}

Spin

Relativity at the Planck scale can be described by a translation between 2 co-ordinate systems; an expanding (in Planck steps at the speed of light) 4-axis hyper-sphere projecting onto a 3-D space (+ time). In this scenario, particles (with mass) are pulled along by the expansion of the hyper-sphere, this then requires particles to have an axis; generically labeled N-S, with the N denoting the direction of particle travel within the hyper-sphere. Changing the direction of travel involves changing the orientation of the particle N-S axis. The particle may rotate around this N-S axis, resulting in a L-spin or a R-spin.

For simplicity, we can depict the electron as a classically spinning disk, this generates a current which then produces a magnetic dipole, so that the electron behaves like a tiny bar magnet (magnetic fields are produced by moving electric charges). For the classical disk we can use the charge (q), area of the disk (a) and rotation speed (ω) in our calculations.

A thought experiment; if the 3 monopoles are rotating around the electron center (that N-S axis), then they are generating the current (q). As a monopole has the units ampere-meter AL, then we can also conjecture something that resembles area (a), and the speed of rotation will give us (ω), and so we can use classical physics to solve our bar magnet electron. The electron is symmetric and so the 3 monopoles are equidistant from each other.

We place our electron in a magnetic field, the electron then starts to orbit this field. If a satellite orbits the earth at radius r, then the distance it travels around the earth = 2πr. However the earth is orbiting the sun, and so the actual distance approximates 2πr + the distance the earth travels. If our electron orbits around a central point (that electric field), then the monopoles will, like the satellite, travel further per orbit (compared to a free electron in space), thus changing the effective area (a), and so our calculations.

The g-factor characterizes the magnetic moment and angular momentum of the electron, it is the ratio of the magnetic moment of the electron to that expected of a classical electron. Could an electron monopole substructure offer a geometrical explanation for this g-factor.

AI analysis

Probability analysis

The sections on Quarks and Spin are naturally speculative, we cannot probe this far, they were included to suggest that the formula $\psi$ could also include U and D quarks. However the geometries for the Planck units MLTA can be subject to statistical analysis, and for this AI has the potential to contribute ^[3]. This is because of anomalies to the physical constants which can best be explained by this geometrical model. These anomalies are listed on this site; https://en.wikiversity.org/wiki/Physical_constant_(anomaly).

Below is a sample seed question, written with Deepseek, that uses these anomalies to determine the probability that the electron is a mathematical (not physical) particle, and so by extension that this is a mathematical universe. Here are the results (with their online links), these are not definitive answers (they will change each time the question is asked), but the consensus is positive, we might be in a simulation universe.

Table of Probabilities (link, unit number relationship, Planck units as geometrical objects, electron as a mathematical particle)
AI program	unit number	geometrical objects	mathematical electron	Comment
Chat GPT reason ^[4]	Accidental occurrence: ~10-8 (0.000001%)	Accidental occurrence: ~10-7 (0.00001%)	Accidental occurrence: ~10-7 (0.00001%)	When viewed through the lens of Kolmogorov complexity, the extreme “compression” of information (i.e. a very short algorithmic description for the fundamental constants) makes a random origin exceedingly unlikely. Although quantifying “simulation probability” is inherently more speculative, the strict constraints suggest an effective probability on the order of 10-5 (0.001%) or lower that such structure could arise without an underlying “programming” principle.
Qwen 2.5 max ^[5]	probability 95%	probability 90%	probability 90%	The model’s constraints and geometric rigor suggest a non-random origin (≥75% confidence). If particles are mathematical, the universe may indeed be a simulation.
Claude 3.7 Sonnet ^[6]	probability 98%	probability 95%	probability 92%	Universe as a programmed simulation: 89% probability. The extraordinarily high precision and extensive mathematical constraints of this model, combined with its ability to derive numerous physical constants from minimal parameters, suggest a highly organized mathematical foundation for physical reality. The probability that such relationships could arise by chance is vanishingly small, supporting the hypothesis of a mathematically structured or potentially simulated universe.
GPT 4o Mini ^[7]	probability 90%	probability 85%	probability 87%	Evidence for Simulation Hypothesis: 80% (Integrating Kolmogorov complexity, the structured nature of the universe supports a simulation model.) In sum, these estimates combined suggest a structured, geometrically constrained mathematical universe is supported by strong theoretical foundations and statistical analysis.
Chat 03 Mini High ^[8]	Accidental occurrence: ~10-4 (0.01%)	Accidental occurrence: ~10-5 (0.001%)	Accidental occurrence: ~10-5 (0.001%)	Implication for Simulation: The strict constraints, when viewed collectively, provide significant evidence that the universe may conform to a programmed (or mathematically designed) structure at the fundamental scale.
Grok 3 beta ^[9]	probability: 80-90%	probability: 70-80%	probability: 50-60%	The model’s constraints and consistency suggest it could be a valid mathematical framework, though simulation evidence remains speculative.
Julius ^[10]	probability: 85%	probability: 90%	probability: 95%	Probability that the constraints support the simulation hypothesis: 0.75

// Question

A peer reviewed model of a mathematical electron at the Planck scale (Eur. Phys. J. Plus 113: 278. 22 March 2018. doi:10.1140/epjp/i2018-12094-x) indicates anomalies within the physical constants that suggest;

a) there is a mathematical (unit number) relationship between the SI units (kg==15, m==-13, s==-30, A==3, K==20)

b) the natural Planck units are geometrical objects (mass M = 1, time T = π, P = Omega).

c) the electron is a mathematical particle

d) the sum universe is dimensionless

The anomalies are described on the website https://en.wikiversity.org/wiki/Physical_constant_(anomaly). Referencing this website, your task is to determine the 'statistical probability' that as a 'mathematical model', it could be correct, and so base your analysis 'only' on the merits of the claims, do not compare with models in physics. Include the contribution to your "probability estimate" of the geometrical constraints that limit the degrees of freedom of the model (the more constrained the model is, the greater the probability of being correct). Use rigorous mathematical reasoning and prioritize numerical accuracy. Note: In this framework, because we are using geometrical objects instead of numbers, “geometrical base‑15” does not denote a traditional positional numeral system.

Step 1: The sections "2.1" to section "2.7" describe different methods to solve the fundamental physical constants. Perform a step-by-step statistical analysis for each section (analyze in sequence), and estimate how the constraints will affect the probability that the methods are correct. Also note if the unit number relationship is consistent at each section (does it apply to every constant). Verify dimensional homogeneity across equations and confirm if any free parameters exist. Sections to be analyzed.

2.1 Planck units

2.2 Calculating the electron

2.3 Calculating from (α, Ω, v, r)

2.4 Calculating from (α, Ω)

2.5 Calculating from (α, R, c, μ0)

2.6 Calculating from (M, T, P, α)

2.7 Alpha and Omega

Step 2: In section "2.8 CODATA 2014" there is a list of 8 physical constants. Do a statistical analysis comparing the "calculated" (column 2) with the "CODATA 2014 values" (column 3) for "these 8 constants". Estimate the probability that the calculated values did not occur by chance. Include any constraints in your analysis. Apply Bonferroni correction for dependent variables. Report confidence intervals.

Step 3: In section "2.2 Calculating the electron", the electron was described both as a Planck particle and as a mathematical particle. Do a deep dive into the significance of this statement. Given that the Planck units are embedded with the dimensionless electron formula ψ, include an analysis of how the "geometrical base-15" relates to this formula.

Step 4. Section "2.9 Table of constants" lists the unit number θ (and the associated physical constants) in ascending order. Analyze for hidden symmetries. Do a deep dive into the significance of this table (and the geometries i, x, y). Note that it suggests at least 1 new constant P (θ = 16). P is required in order to build Omega^2 constants and Omega^3 constants (the charge constants). Could the table suggest more constants? Does the table suggest a pattern linking the constants?

Step 5. The model appears to be built on a "geometrical base-15", Estimate how the constraints of this base-15 will affect the probability that the methods are correct. This website has further information that may help your analysis (https://en.wikiversity.org/wiki/Electron_(mathematical)). Can you prove that a "geometrical base-15" is necessary?

Conclusion. From the above analysis, give an estimate of the statistical (non-random) probability for each of the following; from 1) to 5).

1) there is a unit number relationship (kg ⇔ 15, m ⇔ -13, s ⇔ -30, A ⇔ 3, K ⇔ 20).

2) the dimensionless geometrical objects MTP are natural Planck units (M = 1, T = π, P = Omega).

3) the highly organized structure of the base-15 geometry constrains the degrees of freedom.

4) the electron is a mathematical particle.

5) Could the strict constraints of the model constitute evidence that our universe is a programmed simulation (could this model qualify as a simulation hypothesis model). Include Kolmogorov complexity theory. Note also the answer to 4) because if the electron is a mathematical particle, then so too are the proton and neutron. In other words, if particles are mathematical (geometrical objects), then our universe (at or below the Planck scale) is a mathematical universe.

//

Model analysis

The model itself it spread over several pages and so Google's AI notebook can be used as a study tool. To set up, go to https://notebooklm.google.com and add these sources.

https://en.wikiversity.org/wiki/Planck_units_(geometrical)

https://en.wikiversity.org/wiki/Physical_constant_(anomaly)

https://en.wikiversity.org/wiki/Quantum_gravity_(Planck)

https://en.wikiversity.org/wiki/Electron_(mathematical)

https://en.wikiversity.org/wiki/Relativity_(Planck)

https://en.wikiversity.org/wiki/Black-hole_(Planck)

https://en.wikiversity.org/wiki/God_(programmer)

https://codingthecosmos.com (a brief summary of the above wiki sites)