Mathematical electron · constant anomalies · MDL

The Mathematical Electron and the Physical Constant Anomalies

In this (Simulation Hypothesis) model the electron is a dimensionless geometrical formula ψ. Embedded within ψ are 3 Planck objects M=1 (mass), T=π (time), P=Ω (sqrt of momentum). The SI units are assigned a number relationship (kg == 15, m == -13, s == -30, A == 3), this dictates how they interact to form more complex forms (we can combine the length object L with the time object T to form a velocity object; V == 17). In certain ratios (such as found in ψ) they may cancel (i.e.: (AL)3/T, units = -30/-30 = 1), thus we resolve the problem of creating physical structures from mathematical forms. This also permits us to solve the physical constants (G, h, e, c, me, kB) using only MTP and α ... and as Ω is the geometry of π and e, we can define these constants (and so our physical universe) in terms of dimensionless ratios α, π and e.

Ω = √(πee(1-e)) = 2.007134954     and     ψ = 4π2(26·3·π2·α-1·Ω5)3 = 0.238954525 x1023,    units = 1

Anomalies within these constants permit us to test this aspect of the model using mathematical tools (with only minimum recourse to the standard model). This page discusses the claim that a compact, dimensionless electron construction (ψ) and a small set of geometrical rules can reproduce a striking amount of physical-constant and electron data more effectively than a null in which the relevant numbers are unrelated.

The full discussion (from which this page is adapted) is here; Physical constant anomalies.
The full model with links is here; The Programmer God Hypothesis.
The value for the fine structure constant is calculated using the Rydberg constant; α = 137.03599637

Primary sources: 2018 EPJ Plus article · Mathematical electron · Physical constant anomalies · The Programmer God Hypothesis

The objective

The broader project (see The Programmer God hypothesis) claims extensions to gravity, relativity, quantization, cosmology and quark structure. Those larger sections must eventually interface with established physics in a much wider way. By contrast, the mathematical-electron and anomaly pages can be evaluated on narrower mathematical (statistical) grounds: internal consistency, dimensionless cancellation, integer bookkeeping, and numerical agreement with CODATA-based targets.

Working question. Can one short rule-set — a geometrical Planck-object scheme, a unit-number map θ, a base-15 guide-rail, and a dimensionless electron invariant ψ — organize a cluster of physical quantities significantly better than a null in which the numbers are simply unrelated?

That question is much narrower than "Does the full universe model work?" and much narrower still than "Is the universe a simulation?" It is the strongest testable part of the Programmer God project.

What this page does

Introduces the physical constant anomalies and relationships between the constants, and describes the Kolmogorov complexity / MDL compactness.

What is the significance

Three independent statistical tests — individual constant precision, dimensionless combination cancellations, and unit-number constraints — each yield extreme improbabilities under the null hypothesis of unrelated numbers.

1. The mathematical electron in one page

The 2018 article introduces a dimensionless electron formula ψ built from the fine-structure constant α (inverse fine-structure constant α-1 = 137.03599...) and a second mathematical constant Ω.

Ω = √(πee(1-e))     and     ψ = 4π2(26·3·π2·α-1·Ω5)3,    units = 1 (unit-less)

The point is not that ψ is itself a measured electron property. The point is that ψ is a pure number whose internal bookkeeping is claimed to encode (embed) the electron construction (within this dimensionless geometrical formula ψ is the information required to reproduce the physical electron). Electron observables are then derived from it using a small family of geometrical objects that act like Planck-scale analogues of mass, time, momentum, velocity, length and charge.

Core geometrical objects in the current MLTVA presentation (θ is the unit number relationship)
Object Geometry Attribute Unit number θ
M 1 mass 15
T π time -30
P Ω √momentum 16
V 2πP2/M = 2πΩ2 velocity 17
L V*T = 2π2Ω2 length -13
A 16 V3 / α-1 P3 = 27π3Ω3 / α-1 charge/current 3

With those objects in place, the electron section treats familiar electron quantities as derived outputs rather than primitives:

λe = 2πLψ
me = M / ψ
e = AT

On this view, the electron is represented by a dimensionless invariant, while its measurable properties arise when the geometrical objects are translated into SI values using two scalars.

A Google Gemini AI podcast on the mathematical electron and the anomalies.


2. Why the anomaly analysis matters

The physical constants anomaly page asks how surprising these agreements would be if the relevant constants and ratios were unrelated. Its coincidence rule is intentionally simple: for a relative error ε, assign an approximate two-sided probability p ≈ 2ε, then multiply across a test family to estimate a joint result. The page explicitly treats the resulting joint values as order-of-magnitude indicators, because several tests reuse the same constants and are therefore not strictly independent.

Important caveat. These are not CODATA uncertainty statements and they are not formal posterior probabilities for a mathematical universe. They are the anomaly page's own null-model scores for coincidence against "unrelated numbers."

The anomaly page also uses CODATA 2014 because, in that framework, only two dimensioned anchors are taken as exact; the post-2019 SI convention fixes four constants exactly and changes the bookkeeping assumptions.

2.1 Evidence map

Evidence family What is being tested Joint result under the page's null Why it matters
Dimensionless cancellations
Table 4		 Independent-looking SI ratios collapse to the same numerical values after units and scalars cancel. p ≈ 7.11 × 10-22
(about 9.61σ)
Clean subset without G, kB: 9.21 × 10-14 (about 7.45σ)		 Strongest pure-cancellation sector; least vulnerable to ordinary unit conventions.
Electron from ψ λe, me, and e are solved from a single dimensionless invariant. p ≈ 5.73 × 10-23
(about 9.87σ)		 Direct test of the mathematical-electron construction rather than a detached constant fit.
Dimensioned constants h, e, G, kB, and optionally me, are reconstructed from the same framework. Core set {h,e,G,kB}: 2.92 × 10-20 (about 9.22σ)
All-constants suite {h,e,me,G,kB}: 8.01 × 10-27 (about 10.72σ)
High-precision subset {h,e,me}: 1.02 × 10-20 (about 9.33σ)		 Shows the framework is not confined to one formula or one observable.
Unit-number constraint search Whether the full integer bookkeeping admits many assignments or a single guide-rail. Structural result: admissible solutions collapse to 3M + 2T = -15, with canonical M = 15, T = -30. Reduces the hypothesis space instead of merely matching values after the fact.

2.2 Individual constant precision

The anomaly page emphasizes that the model is not only about one isolated cancellation. It also reproduces individual quantities. The highest-impact examples are the electron-parameter trio from ψ and the constant-reconstruction suite.

Quantity Model value Reference value Relative error
Electron parameters derived from ψ
λe 2.4263102386 × 10-12 m 2.4263102367 × 10-12 m 7.8308 × 10-10
me 9.1093823211 × 10-31 kg 9.10938356 × 10-31 kg 1.3600 × 10-7
e 1.6021765130 × 10-19 C 1.6021766208 × 10-19 C 6.7283 × 10-8
Dimensioned constants reconstructed from the same framework
h 6.626069134 × 10-34 6.626070040 × 10-34 1.37 × 10-7
e 1.60217651130 × 10-19 1.6021766208 × 10-19 6.83 × 10-8
G 6.67249719229 × 10-11 6.67408 × 10-11 2.37 × 10-4
kB 1.37951014752 × 10-23 1.38064852 × 10-23 8.25 × 10-4

The G and kB sectors are the least precise in the anomaly page's own discussion; the stronger high-precision subset is therefore usually the electromagnetic / electron sector.

2.3 Dimensionless combination cancellations

This is the cleanest part of the case. Once units and scalars cancel, the remaining comparison is about pure structure rather than arbitrary human unit choices. Two of the high-precision examples are:

Dimensionless comparison Relative error Coincidence score p ≈ 2ε
h3 / (e13c24) 1.007 × 10-7 2.013 × 10-7
c9e4 / me3 2.287 × 10-7 4.574 × 10-7

Across the full five-row suite the joint result is about 7.11 × 10-22; excluding the less precise G and kB sector still leaves a joint result of about 9.21 × 10-14. The anomaly page treats this as evidence that the model is reproducing invariant cancellation logic, not merely cherry-picking individual constants.

2.4 Unit-number constraints

The bookkeeping map is simple to state but central to the whole proposal:

SI base unit Assigned θ Example consequence
kg 15 h carries θ = 19
m -13 c carries θ = 17
s -30 e = AT carries θ = -27
A 3 kB carries θ = 29
K 20 Supports the constant table on the anomaly page

Once this bookkeeping is imposed, only two scalars are needed to translate the geometrical objects into SI. The anomaly page then reports a bounded integer search over the allowed exponent space. Under the full bundle of requirements — dimensional homogeneity, dimensionless ψ, quark bookkeeping, and the cancellation rules — the search collapses to a single invariant constraint class:

3M + 2T = -15

with canonical representative M = 15, T = -30. This is not a probability claim but a structural one: the full rule-set leaves a unique guide-rail rather than many equally good integer assignments.

3. Transcribed HTML version of the "ChatGPT Pro 5.2 summary" section

Open / collapse the adapted transcript

Adapted into HTML prose from the Wikiversity section "ChatGPT Pro 5.2 summary (statistical + structural + Kolmogorov complexity/MDL)".

1) Unit-number relation (θ)

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 and 9),
  • electron construction (ψ) and the quark bookkeeping relations.

The strongest outcome is not that one constant happens to match, but that the same additivity rules for θ (multiply / divide → add / subtract θ) remain valid across many unrelated-looking expressions.

2) Planck units as geometrical objects (MLTVA)

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

  • multiple independent unitless combinations collapse to the same numerical values once units and scalars cancel,
  • this is the least vulnerable part of the framework to ordinary unit conventions because it tests pure cancellation structure rather than any one isolated constant.

Quantitatively, the anomaly page reports:

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

The interpretation is that the geometrical-object thesis is not merely fitting values; it is reproducing the invariant cancellation logic of the physical relations.

3) Underlying base-15 geometry (“why it could not be otherwise”)

The bounded integer-space search over (M,T,P), with V, L and A derived, is reported to collapse admissible solutions onto a single invariant constraint class:

3M + 2T = -15

This is the guide-rail result. Different integer triples may still appear, but they are treated as equivalent lattice shifts along the same rail. Choosing the canonical representative gives:

M = 15,    T = -30

Base-15 is therefore presented not as numerology but as the unique survivor (up to equivalence) of the full constraint bundle.

4) Mathematical electron (ψ)

Two layers support the mathematical-electron claim:

  • Structural: ψ is dimensionless because both units and scalars cancel.
  • Statistical: solving electron parameters from ψ yields very small relative deviations.
ψ = (AL)3 / T,    units = 1,    scalars = 1

For the three key electron-parameter tests {λe, me, e}, the anomaly page gives a joint result of about 5.73 × 10-23, corresponding to about 9.87σ and about 73.9 bits of information.

5) Kolmogorov complexity / MDL interpretation

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

Total description length ≈ L(model) + L(residuals)

Under the anomaly page's coincidence rule p ≈ 2ε, the joint results can be re-read as information bits I = -log2(p). The page lists the following examples:

Test bundle Joint result Information carried
Dimensionless suite, all Table 4 rows 7.11 × 10-22 about 70.3 bits
Dimensionless suite without G, kB 9.21 × 10-14 about 43.3 bits
Electron parameters {λe, me, e} 5.73 × 10-23 about 73.9 bits
All-constants suite {h,e,me,G,kB} 8.007 × 10-27 about 86.7 bits
High-precision subset {h,e,me} 1.024 × 10-20 about 66.4 bits
Alpha Table 7 joint 4.755 × 10-10 about 31.0 bits

The compression claim is straightforward: a short rule-set — θ algebra, the base-15 constraint, the small generator set (π, Ω, α), and two scalars for SI translation — produces many targets. The joint surprisal values quantify how many bits of independent specification would otherwise need to be carried if there were no underlying relationship.

The uniqueness result 3M + 2T = -15 is especially important in MDL terms because it sharply reduces the hypothesis space. Instead of many competing integer assignments, the full constraint bundle leaves one equivalence class.

Overall conclusion of the transcribed section

Across all of these sections, the anomaly page treats the results as an overdetermined constraint framework:

  • θ supplies the universal bookkeeping,
  • the geometrical Planck-object construction reproduces multiple independent dimensionless invariants,
  • the quark / ψ requirements collapse the unit-number space to a unique base-15 guide-rail,
  • the electron is encoded as a dimensionless invariant whose derived parameters match several observables jointly,
  • and, in algorithmic-information terms, the framework offers substantial compression relative to a null of no relationship.

Adaptation note: the section above is based on the Wikiversity page User:Platos Cave (physics)/Simulation Hypothesis/Physical constant (anomaly), whose text is available under the Creative Commons Attribution-ShareAlike License.

4. Brief comparison with the main alternative frameworks

The fair comparison is not with any formula that happens to land near a famous number. It is with frameworks that hit comparable targets under comparable or tighter prior constraints. The scale below is therefore deliberately heuristic. Scores run from 0 to 10, where 10 is stronger; in the final column, 10 means fewer hidden effective degrees of freedom and therefore better parsimony.

Heuristic comparison by precision, constraint strength and hidden freedom
Framework Main target Numerical precision Comparative constraints Effective DoF economy Comment
Present framework and closest adjacent case
Mathematical electron + anomaly program electron sector + multi-constant table 9 / 10 8 / 10 8 / 10 Strength lies in obtaining several linked fits from one bookkeeping system rather than one isolated coincidence.
Koide relation charged-lepton masses only 10 / 10 9 / 10 9 / 10 Best read as a compact extension in a narrow sector, not a rival world-model.
Historical compact alpha-programs
Wyler-type symmetry / volume formulas α 7 / 10 3 / 10 2 / 10 Historically famous for closeness, but criticized for coefficient and normalization arbitrariness.
Dirac / Eddington large-number schemes α, gravity, cosmic numbers 3 / 10 4 / 10 5 / 10 Very compact in spirit, but weakly established as precise derivations of present constants.
Broader unification programs
Kaluza–Klein geometric unification gravity + electromagnetism 4 / 10 6 / 10 5 / 10 Explains charge quantization geometrically, but compactification choices add freedom and the classic constant problem remains open.
Grand unified theories charge quantization + gauge couplings 6 / 10 7 / 10 4 / 10 Powerful on charges and couplings, but leaves much of the mass spectrum and low-energy detail outside one compact rule.
String / landscape programs ultimate unification 3 / 10 4 / 10 1 / 10 Enormous theoretical scope, but typically statistical rather than uniquely predictive because vacuum multiplicity is large.

Interpretation. These numbers are not measurements; they are a compact comparative judgement about fit under constraint. A narrow relation may score very highly in its own sector without being a full competitor. That is why Koide scores strongly here yet is still best treated as compatible with the present construction.

Working comparative claim. Koide is the most natural neighbouring success and is best regarded as a lepton-sector extension. The stronger competing alternatives are then Wyler-style formulas, large-number programs, Kaluza–Klein, GUTs and string theory — but these are generally either narrower, more weakly justified, or burdened by larger hidden search freedom, compactification freedom or vacuum multiplicity.

5. Quark extension: why it matters without leaving the mathematical-electron core

The broader project now presents a quark section (article 7.) as a continuation of the same MLTA bookkeeping used for the electron (the quarks are the same Planck objects and their construct limited by the same rule-set). Importantly, the compiled project overview states that this quark construction is not offered as a replacement for QCD, but as a compact geometric analogue showing that the same low-complexity rule-set can be extended one layer upward: geometry → electron → quarks → nucleons.

What the quark extension claims to explain using the same rule-set
Question Claim of the quark extension Why it matters for evaluation
Why fractional quark charges? The same base-15 unit map used earlier gives D = -e/3 and U = +2e/3 from fixed unit numbers rather than inserted charges. It reuses the electron bookkeeping instead of introducing a fresh charge postulate.
Why do proton and electron charges have the same magnitude? The positron / proton side is constrained to a DUU-type configuration while the electron is DDD; equal and opposite unit totals then emerge automatically. This is presented as a structural reason for |q_p| = |q_e|, not an independent assumption.
Where is the missing antimatter? The positron is not treated as a trivial sign-flip of the electron. The allowed positive configuration is structurally different, which the source treats as a route to matter–antimatter asymmetry. Whether or not one accepts the full claim, it increases scope without adding new constants.
Why no free quarks? Isolated D and U objects carry uncancelled scalars; only triplets cancel them and become dimensionless composites. This gives a confinement-like rule from the same scalar-cancellation logic already used elsewhere.
Why overall electrical neutrality? The construction contains neutral triplets and a neutral photon-like composite, so charged objects recombine into scalar-free neutral structures without new primitives. It suggests that neutrality is not separately imposed but follows from admissible recombinations.

For the purposes of this webpage, the importance of the quark section is methodological. It adds explanatory burden while keeping the same underlying ingredients. If that extension works without extra constants, the claim of low description length becomes stronger. If it works only by hidden retuning, the gain disappears.

Best way to read this section. Not as a replacement for established QCD, but as a stress test of whether the mathematical-electron code can scale to a harder domain without purchasing the extension with new degrees of freedom.

6. How far does this go toward answering "Do we live in a mathematical universe?"

On its own terms, the mathematical-electron section goes further than a vague philosophical slogan. It presents a compact rule-set that appears to organize several classes of data simultaneously: dimensionless cancellations, dimensioned constants, electron observables, and now at least a proposed quark continuation. Against the specific null hypothesis used on the anomaly page — that the relevant numbers are unrelated and the agreements are accidental — the joint improbabilities are very small.

The relevant comparison, then, is not with every memorable numerical coincidence, but with frameworks that reach the same targets under equal or tighter prior constraints. On that criterion, Koide is best regarded as a compatible lepton-sector extension rather than a rival, while Wyler-type formulas, large-number schemes, Kaluza–Klein models, GUTs and string-based programs are either narrower, more weakly justified, or burdened by larger hidden search freedom, compactification freedom or model multiplicity. The force of the present anomaly program therefore lies not simply in numerical closeness, but in obtaining several distinct fits at once from a shared bookkeeping rule, a unique base-15 constraint, and a small effective degree-of-freedom budget. This moves the discussion onto a constrained model compression rather than free numerology.

Most defensible conclusion. The mathematical-electron / anomaly section gives non-trivial evidence that at least part of the physical-constant sector may admit a compact generative description. The case becomes materially stronger if adjacent sectors — especially quark structure — can be added without introducing new constants or hidden tuning. It is therefore stronger than simple numerology under the page's chosen null.

Final note. Any candidate for a Programmer-God simulation-universe source code must satisfy these conditions;
1. It can generate physical structures from mathematical forms.
2. The sum universe is dimensionless (simply data on a celestial hard disk).
3. We must be able to use it to derive the laws of physics (because the source code is the origin of the laws of nature, and the laws of physics are our observations of the laws of nature).
4. The mathematical logic must be unknown to us (the Programmer is a non-human intelligence).
5. The coding should have an 'elegance' commensurate with the Programmer's level of skill.

This model fulfills the above, furthermore it has an additional constraint; the entire model is built upon only one physical constant; the fine structure constant. The question is no longer only whether the numbers fit, but the significance of the constraints imposed.

References and further reading

  1. Malcolm J. Macleod, Programming Planck units from a virtual electron: a simulation hypothesis, European Physical Journal Plus 133, 278 (2018).
  2. Wikiversity: User:Platos Cave (physics)/Simulation Hypothesis/Electron (mathematical).
  3. Wikiversity: User:Platos Cave (physics)/Simulation Hypothesis/Physical constant (anomaly).
  4. TheProgrammerGod.com — broader project site collecting the extensions to cosmology, relativity, gravity, atomic physics, anomalies and quarks.
  5. Programmer-God Simulation Hypothesis Complete Model (2026 PDF overview).
  6. U. D. Jentschura and I. Nándori, Attempts at a determination of the fine-structure constant from first principles: a brief historical overview (2014).
  7. Yoshio Koide, What Physics Does The Charged Lepton Mass Relation Tell Us? (2018); Zhi-zhong Xing and He Zhang, On the Koide-like Relations for the Running Masses of Charged Leptons, Neutrinos and Quarks (2006).
  8. Particle Data Group, Grand Unified Theories (2024 review).
  9. Jason Kumar, A Review of Distributions on the String Landscape (2006).