Calculation of the Exact Value of the Muon Magnetic Moment Anomaly aμ Given α and ae

July 30, 2026
The feast of Saint Peter Chrysologus (450 AD); Saints Abdon and Sennen (303 AD)

July 31, 2026
Saint Ignatius of Loyola (1556 AD)

1 August 2026
7 Holy Machabees (150 BC); Saint Peter in Chains (6th Century AD); Saints Faith, Hope, Charity (2nd Century AD)

This article is dedicated to my lovely daughter Monika Wasilik.

1. Introduction

This article presents a mathematical relation between:

• the fine structure constant ,
• the electron magnetic moment anomaly ,
• and the muon magnetic moment anomaly ,

and uses this relation to calculate an exact theoretical value of the muon magnetic moment anomaly .

The key ideas are:

1. The values of , , and  are all of order , which allows them to be treated together using a universal transcendental function.
2. A small correction factor  must be introduced to obtain an exact transcendental relation.
3. Experimental data from 2023–2024 (especially Fermilab’s measurement of ) are shown to be more consistent with the theory than some 2026 data.

2. Universal Transcendental Function (UTF)

The Universal Transcendental Function (UTF) is defined as

where  is a constant and  is a real exponent.

For the fine structure constant and the electron and muon anomalies, the key property is that they correspond to the same exponent

In particular, one writes the right-hand side of the main relation using

3. Basic Physical Constants (2023–2024 Data)

The relevant constants (2023–2024) are:

• Fine structure constant (exact value):

342. its reciprocal

• Electron magnetic moment anomaly:

• Muon magnetic moment anomaly (Fermilab 2023 experimental value):

Using high-precision (QUAD) calculations, the muon anomaly needs a very small adjustment:

with relative error

consistent with the experimental precision of about .

4. Main Formula for the Fine Structure Constant

The central relation for the fine structure constant  introduces a correction factor  and reads:

Only the primary transcendental constant  appears explicitly, together with  on the right-hand side. The correction factor  accounts for small deviations and is analogous in spirit to magnetic moment anomalies of various particles:

• electron,
• muon,
• tau,
• proton,
• neutron, etc.

The idea is that each fundamental constant may have its own correction factor and that these factors might form recognizable sequences (geometric or arithmetic), suggesting a deeper structure and the possibility that  itself may vary.

5. Reformulation of the Main Equation

Define

Then the main formula becomes

The strategy is:

1. Compute  from the exact .
2. Compute .
3. Define .
4. Show that  can also be obtained from  where  and  involve  and .
5. Use this consistency to infer an exact value of .

6. Step 1 – First Approximation to the Exponent

Using the exact reciprocal of the fine structure constant:

compute

Numerically,

In the ideal relation without correction factor,  should equal

Instead, regard  as

with unknown exponent . Taking natural logarithms,

This yields

with relative error

from the desired value .

 

This discrepancy motivates introduction of the correction factor .

7. Step 2 – Calculation of

Define

Using QUAD precision (high-precision arithmetic),

8. Step 3 – First Determination of the Correction Factor

The correction factor is

Numerically this gives

This small factor corrects the slight mismatch between  and .

9. Step 4 – Secondary Method (Part B)

Introduce a secondary quantity  involving a constant  and :

This yields

The ratio  will reproduce the correction factor  when  is defined appropriately via magnetic moment anomalies.

10. Step 5 – Definition of  Using  and

Introduce the magnetic moment anomalies of the muon and electron:

The ratio of muon to electron anomalies is

Substituting into the expression for  gives

The muon anomaly  is adjusted precisely so that the exponent  derived from the relation involving  becomes exactly

11. Experimental Muon Anomaly and Adjustment

Fermilab’s 2023 result:

versus adjusted value

with relative error

Given that the experimental precision of  is about , the adjusted value lies well within experimental uncertainty yet allows the transcendental relation to be exact in QUAD precision.

12. Step 6 – Second Determination of

Now compute the correction factor from

Numerically,

 

This matches the value obtained previously from , confirming the consistency of the construction. In high-precision format:

13. Step 7 – Left-Hand Side of the Equation

The left-hand side (LHS) of the main equation is

With

one finds

In QUAD precision,

which matches .

Thus

14. Step 8 – Final Exponent

From

we solve

Using QUAD precision values,

i.e.

In this formalism, the exponent is effectively exactly , with relative error effectively zero, when the adjusted muon anomaly is used.

15. Exact Muon Magnetic Moment Anomaly

In this setting, the exact muon magnetic moment anomaly becomes

or equivalently

This value is singled out as the best fit to the transcendental relation involving ,  , and the universal factor , and it aligns very well with the 2023–2024 experimental results.

16. Comparison of 2023–2024 and 2026 Data

Option 1 – Exact α, Experimental ae, Adjusted aμ (Best Fit)

Year 2023–2024 (Best fit):

Year 2026:

The 2023–2024 combination is slightly closer to the ideal exponent  than the 2026 one.

 

Option 2 – α Set to CODATA (2024), Experimental ae, Adjusted aμ

Here,  is fixed to the CODATA (2024) value:

Year 2023–2024:

 

Year 2026 (using latest α, ae, aμ as reported):

This shows that the full set of 2026 values (α, ae, aμ) is less consistent with the theoretical requirement  than the 2023–2024 data combined with the adjusted .

17. Conceptual and Theological Remarks

The main formula

is notable in that it uses only one primary transcendental constant, , on the left-hand side, with  entering on the right-hand side through the universal factor.

From a viewpoint sometimes called mathematical theology, one may imagine:

1. Primary transcendental numbers (such as  and ) are fundamental.
2. Their projections generate secondary transcendental constants (e.g., ,  ).
3. These, in turn, underlie real numbers (especially integers).
4. Only then do physical constants (like , , ) appear as derived quantities of third or fourth order.

In earlier work, the fine structure constant was obtained using only primary and secondary mathematical constants plus simple integers. Here, the “physics version” includes , , and , further supporting the view that physical constants may be determined by more fundamental transcendental relations.

18. Outlook

Future work will include:

• Calculations of anomalous magnetic moments of electron, muon, and tau from purely mathematical equations, possibly yielding two solutions for each anomaly.
• Derivations of the masses of electron, proton, and neutron.
• Presentation of a general UTF formula for real and complex arguments, potentially encompassing coupling constants of:
  • weak interaction,
  • strong interaction,
  • gravity,
  • and perhaps more