Electron (mathematical)

The mathematical electron model

In the mathematical electron model [1], the electron is a dimensionless geometrical formula (ψ). This formula ψ, 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 alpha 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 galaxies.

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 time ... It is this geometrical electron formula ψ that dictates the magnitude of the electron parameters; length of the wavelength = ψ * Planck length (ψ units of Planck length), frequency = ψ * Planck time ...

This ψ thus not only embeds the Planck units required for the electron parameters, it also dictates the magnitude of these parameters, and so technically it is the electron. This suggests there is no physical electron (only physical parameters), and if the electron is therefore 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

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

ψ = 4 π 2 ( 2 6 3 π 2 α Ω 5 ) 3 = 0.238954531 x 10 23 {\displaystyle \psi =4\pi ^{2}(2^{6}3\pi ^{2}\alpha \Omega ^{5})^{3}=0.238954531\;x10^{23}} , units = 1

### Planck objects

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Main resource: Planck units (geometrical)

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 ) {\displaystyle M=(1)} | (kg)
time  |  T = ( π ) {\displaystyle T=(\pi )} | (s)
velocity  |  V = ( 2 π Ω 2 ) {\displaystyle V=(2\pi \Omega ^{2})} | (m/s)
length  |  L = ( 2 π 2 Ω 2 ) {\displaystyle L=(2\pi ^{2}\Omega ^{2})} | (m)
ampere  |  A = ( 2 7 π 3 Ω 3 α ) {\displaystyle 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 number (θ)

|
|

mass  |  M = 1 {\displaystyle M=1} | kg ⇔ 15
time  |  T = 2 π {\displaystyle T=2\pi } | s ⇔ -30
length  |  L = 2 π 2 Ω 2 {\displaystyle L=2\pi ^{2}\Omega ^{2}} | m ⇔ -13
velocity  |  V = 2 π Ω 2 {\displaystyle V=2\pi \Omega ^{2}} | m/s ⇔ 17
ampere  |  A = 2 6 π 3 Ω 3 α {\displaystyle A={\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }}} | A ⇔ 3

As alpha and Omega can be assigned numerical values (α = 137.035999139, Ω = 2.0071349496), 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 π Ω 2 {\displaystyle 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 vSI = 11843707.905 m/s gives c = V*vSI = 25.3123819 * 11843707.905 m/s = 299792458 m/s (SI units)

scalar vimp = 7359.3232155 miles/s gives c = V*vimp = 186282 miles/s (imperial units)
Scalars  attribute  | geometrical object  | scalar (unit number)

|
|

mass  |  M = ( 1 ) {\displaystyle M=(1)} | k (θ = 15)
time  |  T = ( π ) {\displaystyle T=(\pi )} | t (θ = -30)
velocity  |  V = ( 2 π Ω 2 ) {\displaystyle V=(2\pi \Omega ^{2})} | v (θ = 17)
length  |  L = ( 2 π 2 Ω 2 ) {\displaystyle L=(2\pi ^{2}\Omega ^{2})} | l (θ = -13)
ampere  |  A = ( 2 7 π 3 Ω 3 α ) {\displaystyle 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 (θ) 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 (uΘ represents unit)

( u 3 ) 3 ( u − 13 ) 3 ( u − 30 ) = ( u − 13 ) 15 ( u 15 ) 9 ( u − 30 ) 11 = 1 {\displaystyle {\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, me, kB).

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

In this table the 2 scalars used are r (θ = 8) and v (θ = 17). 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 ) {\displaystyle M=(1)} | 15 = 8*4-17  |  k = r 4 v {\displaystyle k={\frac {r^{4}}{v}}}
time  |  T = ( π ) {\displaystyle T=(\pi )} | -30 = 8*9-17*6  |  t = r 9 v 6 {\displaystyle t={\frac {r^{9}}{v^{6}}}}
sqrt(momentum) |  P = ( Ω ) {\displaystyle P=(\Omega )} | 16 = 8*2  | r2
velocity  |  V = L / T = ( 2 π Ω 2 ) {\displaystyle V=L/T=(2\pi \Omega ^{2})} | 17  | v
length  |  L = ( 2 π 2 Ω 2 ) {\displaystyle L=(2\pi ^{2}\Omega ^{2})} | -13 = 8*9-17*5  |  l = r 9 v 5 {\displaystyle l={\frac {r^{9}}{v^{5}}}}
ampere  |  A = 2 4 V 3 α P 3 = ( 2 7 π 3 Ω 3 α ) {\displaystyle A={\frac {2^{4}V^{3}}{\alpha P^{3}}}=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha }})} | 3 = 17*3-8*6  |  a = v 3 r 6 {\displaystyle a={\frac {v^{3}}{r^{6}}}}

Further information: Planck units (geometrical) § Scalars

### Mathematical electron

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

T = π , u n i t = u − 30 , s c a l a r s = r 9 v 6 {\displaystyle T=\pi ,\;unit=u^{-30},\;scalars={\frac {r^{9}}{v^{6}}}}

σ e = 3 α 2 A L 2 π 2 = 2 7 3 π 3 α Ω 5 , u n i t = u ( 3 − 13 = − 10 ) , s c a l a r s = r 3 v 2 {\displaystyle \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}}}}

ψ = σ e 3 2 T = ( 2 7 3 π 3 α Ω 5 ) 3 2 π , u n i t = ( u − 10 ) 3 u − 30 = 1 , s c a l a r s = ( r 3 v 2 ) 3 v 6 r 9 = 1 {\displaystyle \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}

ψ = 4 π 2 ( 2 6 3 π 2 α Ω 5 ) 3 = .23895453 . . . x 10 23 , u n i t = 1 {\displaystyle \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

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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... , u n i t s = m s {\displaystyle v=11843707.905...,\;units={\frac {m}{s}}}
r = 0.712562514304... , u n i t s = ( k g . m s ) 1 / 4 {\displaystyle r=0.712562514304...,\;units=({\frac {kg.m}{s}})^{1/4}}

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

λ e ∗ = 2 π L ψ {\displaystyle \lambda _{e}^{*}=2\pi L\psi } = 2.4263102386e-12m (L ⇔  Planck length)

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

m e ∗ = M ψ {\displaystyle m_{e}^{*}={\frac {M}{\psi }}} = 9.1093823211e-31kg (M ⇔  Planck mass)

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

e ∗ = A T {\displaystyle e^{*}=A\;T} = 1.6021765130e-19 (T ⇔  Planck time)

Rydberg constant R = 10973731.568508/m (CODATA 2014)

R ∗ = ( m e 4 π L α 2 M ) = 1 2 23 3 3 π 11 α 5 Ω 17 v 5 r 9 u 13 {\displaystyle 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

From the above formulas, we see that wavelength is ψ units of Planck length, frequency is ψ units of Planck time ... however the electron mass is only 1 unit of Planck mass.

##### Electron Mass

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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  ψ = ( A L ) 3 T {\displaystyle \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 Planck 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 Planck 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 there 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 the frequency term, it 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 a particle itself to have mass, what we define as electron mass would be the absence of electron.

##### Quarks

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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.

Electron formula

ψ = 2 20 π 8 3 3 α 3 Ω 15 , u n i t = 1 , s c a l a r s = 1 {\displaystyle \psi =2^{20}\pi ^{8}3^{3}\alpha ^{3}\Omega ^{15},\;unit=1,scalars=1}

Time

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

AL magnetic monopole

σ e = 3 α 2 A L 2 π 2 = 2 7 3 π 3 α Ω 5 , u − 10 , s c a l a r s = r 3 v 2 {\displaystyle \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}}}}

ψ = σ e 3 2 T = ( 2 7 3 π 3 α Ω 5 ) 3 2 π = 2 20 3 3 π 8 α 3 Ω 15 , u n i t = ( u − 10 ) 3 u − 30 = 1 , s c a l a r s = ( r 3 v 2 ) 3 v 6 r 9 = 1 {\displaystyle \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  σ e {\displaystyle \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 of these 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 (A; θ=3, L; θ=-13) units = -10, and if we look at the table of constants, there is no 'units = +10' combination that can include A. We cannot make an inverse electron. However we can make a Planck temperature Tp AV monopole (ampere-velocity).

T p = 2 7 π 3 Ω 5 α , u 20 , s c a l a r s = r 9 v 6 {\displaystyle T_{p}={\frac {2^{7}\pi ^{3}\Omega ^{5}}{\alpha }},\;u^{20},\;scalars={\frac {r^{9}}{v^{6}}}}

σ t = 3 α 2 T p 2 π = 3 α 2 A V 2 π 2 = ( 2 6 3 π 2 α Ω 5 ) , u 20 , s c a l a r s = v 4 r 6 {\displaystyle \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}}}}

ψ = ( 2 T ) σ t 2 σ e = 2 20 3 3 π 8 α 3 Ω 15 , u n i t = ( u − 30 ) ( u 20 ) 2 ( u − 10 ) = 1 , s c a l a r s = ( r 9 v 6 ) ( v 4 r 6 ) 2 r 3 v 2 = 1 {\displaystyle \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  σ t {\displaystyle \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 = σ e , u n i t = u − 10 , c h a r g e = − 1 e 3 , s c a l a r s = r 3 v 2 {\displaystyle D=\sigma _{e},\;unit=u^{-10},\;charge={\frac {-1e}{3}},\;scalars={\frac {r^{3}}{v^{2}}}}

U = σ t , u n i t = u 20 , c h a r g e = 2 e 3 , s c a l a r s = v 4 r 6 {\displaystyle 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 =  r 3 v 2 {\displaystyle {\frac {r^{3}}{v^{2}}}}

##### Magnetic monopole

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Further information: Quantum_gravity_(Planck)

σ e = 3 α 2 A L 2 π 2 = 2 7 3 π 3 α Ω 5 , u − 10 , s c a l a r s = r 3 v 2 {\displaystyle \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 o r b i t a l = 2 α ( λ o r b i t a l ) {\displaystyle r_{orbital}=2\alpha (\lambda _{orbital})}

If we replace  λ o r b i t a l {\displaystyle \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 α L V 3 P 3 = 2 5 π 5 α Ω 5 , u n i t s = u − 10 , s c a l a r s = r 3 v 2 {\displaystyle 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

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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 magnetic monopole quarks 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 (L a length term), 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

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The sections on Quarks and Spin are naturally speculative, we cannot probe this far, they were included to suggest that the formula  ψ {\displaystyle \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. 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 are answers to a seed question written with Deepseek [3] 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.

Table 11. Table of Probabilities (05.2025)  AI program  | unit number relation  | Planck units as geometrical objects  | underlying base 15 geometry  | mathematical electron  | Comment

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Chat GPT [4] | Accidental occurrence: ~1.6%  | Accidental occurrence: ~0.0004%  | Accidental occurrence: ~0  | Accidental occurrence: ~0.026%  | Taken at face value each sub-claim is vanishingly unlikely to be a “random accident,” so from this internal consistency alone one would assign essentially unity confidence that some deep mathematical structure is at work. Caveat: this “integer-coincidence” probability model is only as good as the assumption that each exponent match is an independent uniform pick in [–30…+30]. But even under that very conservative assumption, the combined odds against pure chance exceed 1 in 10²⁶.
Qwen [5] | probability 98%  | probability 95%  | probability 90%  | probability 94%  | The model’s strict geometrical constraints, flawless unit consistency, and agreement with CODATA strongly suggest non-random validity. Its potential as a simulation hypothesis framework stems from its finite, computable base-15 structure.
Claude 3.7 Sonnet [6][7] | probability 90-95%  | probability 85-95%  | probability 92-97%  | probability 80-90%  | From a Kolmogorov complexity perspective, the model represents a potentially minimal description of physical reality using mathematical objects. If the electron, proton, and neutron are all mathematical particles, then the universe at the Planck scale could indeed be described as a mathematical universe. The high degree of constraint and mathematical elegance in the model, combined with its apparent ability to accurately produce physical constants, suggests a non-random structure to physical reality that is captured by this mathematical framework.
Grok 3 [8][9] | probability: 95%  | probability: 90%  | probability: 85%  | probability: 80%  | Strict constraints and mathematical nature (Kolmogorov complexity: simpler model = more likely programmed) suggest support, but not proof. If particles are mathematical, the universe may be too.
Deepseek [10] | probability: 99%  | probability: 95%  | probability: 90%  | probability: 85%  | The model's rigid constraints and predictive accuracy make it statistically improbable to arise by chance. Geometrical base-15 and dimensionless particles strongly suggest a mathematical universe, aligning with speculative simulation hypothesis.

### Geometrically coded universe

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* Simulation_hypothesis_(Planck): A geometrical Planck scale simulation universe
* Electron_(mathematical): Mathematical electron from Planck units
* Planck_units_(geometrical): Planck units as geometrical forms
* Physical_constant_(anomaly): Anomalies in the physical constants
* Quantum_gravity_(Planck): Gravity at the Planck scale
* Fine-structure_constant_(spiral): Quantization via pi
* Relativity_(Planck): 4-axis hypersphere as origin of motion
* Black-hole_(Planck): CMB and Planck units
* Sqrt_Planck_momentum: Link between charge and mass

### External links

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* Programming at the Planck scale using geometrical objects -Malcolm Macleod's website
* Simulation Argument -Nick Bostrom's website
* Our Mathematical Universe: My Quest for the Ultimate Nature of Reality -Max Tegmark (Book)
* Dirac-Kerr-Newman black-hole electron -Alexander Burinskii (article)
* The Matrix, (1999)
* Pythagoras "all is number" \- Stanford University
* Simulation Hypothesis
* Mathematical universe hypothesis
* Philosophy of mathematics
* Philosophy of physics
* Platonism

### References

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1. ↑ Macleod, M.J. "Programming Planck units from a mathematical electron; a Simulation Hypothesis". Eur. Phys. J. Plus 113: 278. 22 March 2018. doi:10.1140/epjp/i2018-12094-x.
2. ↑ Macleod, Malcolm J. (22 March 2018). "Programming Planck units from a mathematical electron; a Simulation Hypothesis". Eur. Phys. J. Plus 113: 278. doi:10.1140/epjp/i2018-12094-x.
3. ↑ https://codingthecosmos.com/ai-answers-programmer-god-simulation-hypothesis/anomalies-constants.html Physical constants Q&A with podcast
4. ↑ https://chatgpt.com/share/68259395-31d0-8012-954e-c52230a813fd Chat anomalies analysis
5. ↑ https://chat.qwen.ai/s/f33ec1b1-899b-4fa5-9429-67e527b78433 Qwen anomalies analysis
6. ↑ https://claude.ai/public/artifacts/797a802b-fa21-4ee7-9ae9-9dbdf5d7b29a Claude anomalies analysis
7. ↑ https://codingthecosmos.com/ai_pdf/Claude-anomalies-05-2025.pdf Claude anomalies analysis pdf
8. ↑ https://x.com/i/grok/share/JzeRFDlj721zNnm1Jo2vbETk3 Grok anomalies analysis
9. ↑ https://codingthecosmos.com/ai_pdf/Grok-anomalies-05-2025.pd Grok anomalies analysis pdf
10. ↑ https://codingthecosmos.com/ai_pdf/Deepseek-anomalies-05-2025.pdf Deepseek anomalies analysis