6. Physical Constant Anomalies as Evidence of a Mathematical Universe

Kolmogorov Complexity and Statistical Significance Report

Malcolm J. Macleod
based on the Mathematical Electron model (theprogrammergod.com)

1. Introduction

“The physical 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.”

Novel aspects:

1. Builds on a base-15 unit-number geometry.
2. Replaces independent SI base-unit assumptions with unit-number relationships: \(kg(15)\), \(m(-13)\), \(s(-30)\), and \(A(3)\).
3. Assigns three primary geometrical Planck-like objects: \(M=1\), \(T=\pi\), and \(P=\sqrt{\pi^e e^{1-e}}\).
4. Derives geometrical analogues \(G^*\), \(h^*\), \(c^*\), \(q_e^*\), \((y_e/g_e)^*\), \(m_e^*\), and \(k_B^*\) from the unit-number lattice and the two SI translation scalars.
5. Derives a model-internal inverse fine-structure constant, \(a=\alpha^{-1}\simeq137.035993138\), from the measured constants.
6. Reduces the electron sector to a dimensionless mathematical invariant \(\psi\), from which \(m_e\), \(\lambda_e\), \(R_\infty\), and \(y_e/g_e\) are linked.

There are apparent anomalies within the dimensioned constants \(G, h, c, e, m_e, \gamma_e, k_B\) which suggest an underlying mathematical structure expressed through the geometrical objects (Planck unit analogues) for mass, time, and (sqrt)momentum \(MTP\) [1]. This paper summarises the significance of these anomalies and to what extent they constitute evidence of a mathematical universe. The anomalies do not lie in dimensioned SI numbers by themselves, since such numbers depend on unit conventions and so a numberless (geometrical approach) is applied. Geometrical because the physical attribute (of mass, space, time, …) may be embedded within the geometry itself. The claim is that after a specified two‑scalar translation from the geometric \(MTP\) sector to the traditional SI, multiple dimensionless combinations and reconstructed constants align with CODATA 2014 values more closely than expected under a simple coincidence model.

The model uses two specific dimensionless constants: the physical fine‑structure constant \(\alpha\) and a mathematical constant \(\Omega\). Omega itself is a function of the mathematical constants \(\pi\) and \(e = 2.718281828459\dots\) (Euler’s number): \[ \Omega = \sqrt{\pi^e e^{1-e}} = 2.0071349543249462\dots \] From mass \(M = 1\), time \(T = \pi\), momentum \(P = \Omega\), and \(\alpha\) (where alpha is the only physical constant required), we can construct \(LVA\) (length, velocity, charge). These geometrical \(MTPLVA\) objects are proposed as natural Planck units (being independent of any system of units).

From \(MTPLA\) we can derive geometrical analogues \(G^*, h^*, c^*, e^*, (y_e/g_e)^*, m_e^*, k_B^*\) of the SI constants using standard equations for the Planck unit conversions. No arbitrary free parameters are introduced; the constants emerge from a unique geometric lattice.

This webpage is from Article 6: Anomalies in the Physical Constants, an 8-article series on The Simulation Universe and the Fine structure constant alpha.

2. Contextual Framework: From Arbitrary Constants to Geometric Order

Physical constants form the fundamental scaffolding upon which the laws of physics are constructed. Conventionally, these values are treated as independent, arbitrary “givens” discovered through observation. The 2019 SI redefinition fixed exact numerical values for \(h\), \(e\), \(k_B\), and \(N_A\), while \(c\) was already exact. In the pre-2019 SI used by CODATA 2014, only \(c\) and \(\mu_0\) were exact, while \(h\), \(e\), and \(k_B\) were adjusted measured constants. We can only assign fixed values if the constants are independent of each other; this certified the independence of their associated units, but then also obscured any underlying mathematical dependency by forcing independence through this metrological convention. This change is important for the present comparison, because in this geometric model we can only independently assign two SI anchors, the model then treats the remaining constants as outputs of the lattice.

In this “Simulation Hypothesis” framework [2], the scaffolding of mass, space, and time emerges from a unified, overdetermined mathematical framework. If the universe is a code‑driven structure, then the traditional independence of dimensions must break down in favour of a fundamental relationship between units. The following report quantifies the validity of a geometrical unit‑number approach, viewing the physical constants as outputs of a compact, generative constraint system rather than a collection of random parameters.

3. Theoretical Foundation: The Unit‑Number Mapping (\(\theta\)) and Lattice Constraints

The core of the model is the unit‑number relationship (\(\theta\)), a mapping that breaks the presumed independence of SI units by assigning specific integer values to base attributes [5]. This allows physical dimensions to be treated as geometrical objects whose attributes are embedded within their underlying geometry (for example the attribute of ‘length’ is embedded within the geometry of the \(L\) object). The following geometries are Planck unit analogues. Note, throughout this article series we use the popular form of alpha; \[ a\equiv\alpha^{-1} = 137.03599\dots \]

The internal consistency of this mapping is enforced by the fundamental constraint equation: \[ 3M + 2T = -15 \] In the language of lattice geometry, this represents a “null transformation” or a “closed loop in exponent space.” Multiplying any physical expression by this specific combination leaves the net unit‑number unchanged, creating a discrete family of equivalent mathematical representations. The value of \(15\) is not arbitrary; it is the closure length of the model’s dimensionless loop on the unit lattice. This “guide‑rail” functions as the unique stable integer structure that satisfies the model’s requirements for dimensional homogeneity, ensuring that the base‑15 geometry is a structural necessity rather than a numerological preference.

An exhaustive integer‑space search over admissible unit‑number assignments, subject to dimensional homogeneity, a dimensionless electron invariant \(\psi\), and quark bookkeeping constraints (a valid quark substructure with \(D = AL\) and \(U = AV\), such that the electron triplet \(DDD = T\)), collapses onto a unique equivalence class characterized by the base‑15 rail, selecting the canonical representative \(M = 15\) and \(T = -30\).

3.1 The Scalar Translation System

We can convert from the dimensionless geometric sector to a conventional unit system using dimensioned scalars. Only two scalars are required to be assigned values (numerical and units), this is because the unit‑number relationship constraints force all others — the system is exactly determined. In this model the two scalars \(r\) (\(\theta = 8\)) and \(v\) (\(\theta = 17\)) are chosen as they are derivable from the two SI constants that were assigned exact values in CODATA 2014; \(c = 299792458\) and \(\mu_0 = 4\pi/(10^7)\). Because \(\alpha\) and \(\Omega\) are fixed (dimensionless numbers), the scalars must be unit specific; in this analysis \(v\) and \(r\) have SI unit system conversion values (Section 4.1).

Using the formulas that convert from Planck units we can construct geometrical analogues; \(r, v\) are the dimensioned components (table 3).

3.2 The Mathematical Electron: the \(\psi\) Invariant

The model defines the electron not as a physical particle, but as a dimensionless mathematical invariant (\(\psi\)). This invariant is mechanically constructed from magnetic monopoles (\(AL\)) and time (\(T\)). The point is not that \(\psi\) is itself a measured electron property, but that it is a pure number whose internal bookkeeping is claimed to encode the electron construction — the information required to reproduce the physical electron. Because all units and scalars cancel perfectly within the \(\psi\) formula, the electron observables (mass, charge, wavelength) are revealed as mere frequencies of the Planck unit objects (all the information required to generate the physical parameters of the electron is embedded within this dimensionless geometrical formula \(\psi\)).

where \(\sigma_e\) has the units for a magnetic monopole (ampere‑meter) and the \(\psi\) units = scalars = \(1\). Because the magnetic monopole \(AL\) has a unit number of \(-10\) (where \(A=3\) and \(L=-13\)), the numerator \((AL)^3\) yields \(-30\), which perfectly cancels the time denominator \(T=-30\), rendering \(\psi\) completely dimensionless. The electron formula resembles the volume of a torus or surface area of a 4‑axis hypersphere \((4\pi^2(AL)^3)\) and can be divided into 3 magnetic monopoles \((AL)^3\), suggesting a ‘quark’ model for the electron with \(D = AL\) constituents forming a \(DDD\) triplet. With \(\psi\) we can now add four more constants: electron mass, Compton wavelength, the Rydberg constant \(R^*\) and the gyromagnetic ratio \((y_e/g_e)^*\).

3.3 Alpha, the Fine Structure Constant

(A) Algebraic consistency check (non‑statistical): The standard relation \(\alpha^{-1} = \dfrac{2h}{\mu_0 e^2 c}\), when the numerical constants are replaced by their geometrical analogues \((h^*, \mu_0^*, e^*, c^*)\), collapses exactly to return \(a\). Both units and scalars cancel:

This is a deterministic pass/fail test of internal model consistency.

4. Analysis

4.1 Dimensioned Constants

Via the \(MTP\) objects we defined geometrical constants \(G^*, h^*, c^*, e^*, (y_e/g_e)^*, \lambda_e, m_e^*, k_B^*\) in terms of \(\pi, \alpha, \Omega\) and 2 SI adjusted scalars (see Table 3). As Omega has been assigned a fixed value, our floating value is alpha. Here we determine the value of alpha required to solve the CODATA mean for each of the following geometrical constants using the CODATA 2014 values:

CODATA 2014 values

1. \(e = 1.6021766208(98) \times 10^{-19}\)
2. \(h = 6.626070040(81) \times 10^{-34}\)
3. \(\lambda_e = 2.4263102367(11) \times 10^{-12}\)
4. \(R = 10973731.568508(65)\)
5. \(y_e = \gamma_e/2\pi = 28024951640(170),\; g_e = 2.00231930436182(52)\)
6. \(m_e = 9.10938356(11) \times 10^{-31}\)
7. \(c = 299792458\) (exact)
8. \(\mu_0 = 4 \pi/10^7\) (exact)
9. \(\alpha^{-1} = 137.035999139(31)\)
10. \(G = 6.67408(31) \times 10^{-11}\)
11. \(k_B = 1.38064852(79) \times 10^{-23}\)

We can plot the sum of precisions of the 6 most precise constants to find the minima over a range of alpha (fig. 1); this can give us an estimate for an optimal alpha based only on these constants. We find a minimum at \(137.035993138\). Using this value gives us solutions to the individual constants (via scalars \(r, v\)). Precision lies between 7-8 digits depending on the proximity to the optimised alpha (table 5).

If we use CODATA 2022 values then we find a similar minimum but with less precision (y‑axis) and more spread out, indicating that the model handles 2014 better (as only 2 constants are pre‑defined).

4.2 Dimensionless Combinations

By reorganizing the constants into unit-free, scalar-free combinations, both the model unit numbers and the two SI translation scalars cancel. These combinations are not generally dimensionless in ordinary SI units; rather, they are dimensionless in the unit-number geometry, with \(\theta=0\). This distinction is important because in combinations where both units and scalars cancel, the SI constants should return the same numerical values as the \(MTP\alpha\) analogues. If the \(MTP\alpha\) objects are natural Planck units (embedded in the universe ‘source code’), then in dimensionless combinations the SI constants will shed their 2 SI scalar components (here we are using \(r, v\)), and reduce to the underlying embedded \(MTP\) objects (e.g.: \(c = V v = 299792458\) m/s and thus sans scalar \(v\), \(c = V\)).

Here we consider six independent combinations of CODATA constants (dimensionless according to the unit‑number model; \(\theta = 0\), dimensioned according to the SI system). Each combination comprises the 2 fixed constants \(c, \mu_0\) and 1 high‑precision constant. If the geometric model correctly describes nature, there are no dimensioned components to include in the calculations and so each (CODATA / Geometrical) should closely approach unity at the model’s residual scale (sans scalars the CODATA combination is the Geometrical combination):

The uncertainty column (Table 7) is not a pure CODATA experimental uncertainty. It includes the finite model-fit envelope of the derived alpha. For a geometrical expression \(G_i(a)\propto a^{s_i}\), the contribution from the fitted-alpha range is \[ \delta_{a,i}=|s_i|\frac{\Delta a}{a}, \] where \[ \Delta a=a_{\max}-a_{\min}=4.78878\times10^{-6} \] is the span of the six independently implied alpha values. The total diagnostic uncertainty is then estimated as \[ \delta_{\rm total,i} = \sqrt{ \delta_{{\rm CODATA},i}^{\,2} + \delta_{a,i}^{\,2} }. \] Thus the table tests whether each invariant lies inside the model’s self-consistent fitted-alpha envelope, not whether it lies inside the much smaller CODATA uncertainty of each individual measured constant.

We can likewise take 1 of our combinations and use it to define the other constants. This demonstrates the flexibility of the model and also serves to cross-check the unit number relationship. For example Table 6. \(R_3\) uses constants expressed in terms of \(c\), \(\mu_0\), \(R\), and \(\alpha\). We first look for combinations in which the unit numbers are equal, and then add dimensionless numbers as required. For example;

We then replace the geometrical object with the numerical SI (\(c\), \(\mu_0\), \(R\)) \[ {(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 \alpha_{inv}^{13} {R}^2} \]

4.3 Omega

The geometric model relies on a single dimensionless constant, \(\Omega\), which governs the scaling of all Planck‑scale objects. Theoretically, \(\Omega\) arises from the requirement that \(\Omega^2\) have integer exponents in the mass–space domain, while \(\Omega^3\) accommodates plus‑or‑minus solutions in the electromagnetic sector. The complete model is predicated on reduction to \(\pi, \Omega, e\) where possible (at present there are still unknowns). Here is a qualified \(\Omega\): \[ \Omega = \sqrt{\pi^e\, e^{1-e}} = 2.0071349543249\dots \] Note: Here \(e\) is Euler’s number, not the elementary charge.

If we use the same range of 6 constants and free float both alpha and Omega, then we can investigate optimal solutions using sum precision as the reference. We ran a broad range (alpha: \(137.035990\) to \(137.060000\); Omega: \(2.00710\) to \(2.00717\)); a local minimum appeared, we then tuned to this minimum. The final results were cross‑checked with Maple.

CODATA 2014 (Maple): \[ a = 137.03599289524 \dots,\qquad \Omega = 2.007134954516 \dots \] CODATA 2022 (Maple): \[ a = 137.0359921141 \dots,\qquad \Omega = 2.0071349551 \dots \] There is a slight divergence between 2014 and 2022, but again there is also a markedly reduced precision (y‑axis) using the 2022 data set (see fig. 1). For CODATA 2014, fixing \(\Omega=\Omega_0\) gives an RMS log residual of \(2.36\times10^{-7}\), while allowing \(\Omega\) to float improves this only to \(2.32\times10^{-7}\), an improvement factor of 1.019. The fitted value of \(\Omega\) differs from \(\Omega_0\) by only \(3.45\times10^{-10}\) relatively. The fixed mathematical value of \(\Omega\) is not the dominant source of the residual floor, thus we may offer that the theoretical Omega may be used with confidence.

4.4 Note on Boltzmann Constant Variant

The Boltzmann constant \(k_B\) appears in two distinct measurement contexts in this analysis.

• \(k_B = 1.380\,648\,52(79) \times 10^{-23}\) (CODATA 2014, thermal, from macroscopic thermometry)
• \(k_B^* = 2\pi \cdot 2\pi\Omega^2 v \cdot 1r^4/v \cdot \frac{a r^6}{2^7 \pi^3 \Omega^3 v^3} = \frac{a}{2^5 \pi \Omega} \left( \frac{r^{10}}{v^3} \right) = 1.379\,510\,194 \times 10^{-23}\)
  unit number = \(17 + 15 - 3 = 29\)

We may propose \(k_B^{**}\), derived as an electromagnetic constant by using the electron gyromagnetic ratio \(y_e = \gamma_e/2\pi = 28\,024\,951\,640(170)\) units = \(\text{kg}^{-1}\,\text{s}\,\text{A}\) (\(-15 + (-30) + 3 = -42\)) and the \(g\)-factor \(g_e = -2.00231930436182(52) \). This \(k_B^{**}\) closely matches the geometrical \(k_B^*\) : \[ k_B^{**} = \frac{g_e h}{4\pi c y_e m_e} = 1.379\,510\,310(24) \times 10^{-23},\;\theta = 19 - 17 - (-42) - 15 = 29 \] Rewriting the above formula gives \[ k_B^{**} = \left(g_e \cdot 2^3\pi^4 \Omega^4 r^{13}/v^5\right) \big/ \left(4\pi \cdot 2\pi\Omega^2 v \cdot y_e \cdot (1r^4/v)/\psi\right) \] The thermal \(k_B\) discrepancy may indicate that the present electromagnetic‑geometric mapping is not capturing the thermodynamic temperature sector, or that \(k_B^*\) corresponds to a different electromagnetic scale rather than the macroscopic thermodynamic Boltzmann constant.

5. The Base‑15 Constraint: Structural Uniqueness, Not a Probabilistic Event

The base‑15 guide‑rail \(3M + 2T = -15\) arises from an exhaustive integer‑space search over admissible unit‑number assignments, subject to three simultaneous requirements:

1. dimensional homogeneity across all standard physics equations;
2. the dimensionless status of the electron invariant \(\psi\) (units = 1, scalars = 1); and
3. the existence of a valid quark substructure satisfying \(DDD = T\).

Under this constraint bundle, the search collapses to a single equivalence class. This is qualitatively different from the numerical coincidence tests above: it is a uniqueness result — an algebraic fact rather than a probabilistic one and so it is structural evidence that base‑15 geometry is a necessary consequence of the model’s building blocks, not a tunable parameter.

5.1 Unit Number Operations

Unit‑less combination: \[ f_\psi = \sqrt{\frac{m^{15}}{kg^9 s^{11}}}; \quad \text{units } (-13)\cdot15 - (15\cdot9 + (-30)\cdot11) = 0, \quad \text{scalars } \left(\frac{r^9}{v^5}\right)^{15} \big/ \left(\frac{r^4}{v}\right)^9 \left(\frac{r^9}{v^6}\right)^{11} = 1 \] Example: \[ \text{length units}\quad L^4 = \left(\frac{r^9}{v^5}\right)^4 = \left(\frac{kg^9 m^9}{s^9}\right) \left(\frac{s^{20}}{m^{20}}\right) = \frac{kg^9 s^{11}}{m^{11}} = m^4 \frac{kg^9 s^{11}}{m^{15}} = \frac{m^4}{f_\psi} = m^4 \]

Here \(x\), \(y\), and \(i\) are intermediate algebraic substitutions used to map the \(M,T,P\) geometries to the base‑15 unit numbers: \[ i = \pi^2 \Omega^{15}; \quad \theta = 0, \quad \text{units} = f_\psi \quad (\text{dimensionless}) \] \[ x = \frac{\Omega v}{r^2}; \quad \theta = 17 - 16 = 1, \quad \text{units} = \sqrt{\frac{m}{kg \cdot s}} \] \[ y = M^2 T = \frac{\pi r^{17}}{v^8}; \quad \theta = 30 + (-30) = 0, \quad \text{units} = kg^2 s \]

Note: The constants with unit numbers \(\theta\) in the series \(\theta = 15n\) have no \(\Omega\). All other constants have an Omega value relative to \(\theta + 15n\). The example used in this model as a primary constant is the square root of momentum \(P\), which serves as a link between the mass and charge domains.

6. Statistical Significance and Algorithmic Evidence

6.1 The Correct Null Hypothesis: An Overdetermined System

The decisive question is not whether any single ratio \(R_i\) is close to unity, but whether a single compact framework can organize all six scalar-free combinations within one internally derived alpha envelope, using the same two SI anchors and the same value of \(a=\alpha^{-1}\).

H\(_0\): The observed scalar-free alignments are independent numerical accidents: each \(R_i\) deviates from unity for unrelated reasons, with no common generative structure.

The model has only two empirical SI anchors, \(c\) and \(\mu_0\), and derives six scalar-free combinations together with a model-internal value of \(a=\alpha^{-1}\). Under \(H_0\), the simultaneous organization of all six combinations by the same lattice, the same electron invariant \(\psi\), and the same fitted \(a\) requires explicit explanation.

6.2 Algorithmic Compression and Minimum Description Length (MDL)

The Kolmogorov complexity \(K(x)\) of a data set is the length of the shortest program that outputs it. The Minimum Description Length (MDL) principle offers a practical proxy: a model is preferred if it achieves the same descriptive accuracy with a shorter instruction set. Under the null hypothesis, each numerical agreement must be treated as an independent, unexplained fact.

The information content of specifying a constant to its relative precision \(u_{\text{rel}}\) is approximately \(-\log_2(u_{\text{rel}})\) bits. Under the null (independence) model, each constant must be recorded separately. Under the geometric model, the same constants are recovered from a shared rule‑set plus only two empirical scalars anchored to \(c\) and \(\mu_0\).

The geometric model specifies all outputs using:

• Rule‑set (\(\theta\)-algebra, base‑15 constraint, six geometric expressions, \(\Omega\) definition): \(\sim 80\) bits
• Scalars \(r, v\) (fixed by \(c\) and \(\mu_0\), both exact in CODATA 2014): \(\sim 17\) bits
• Residuals (the \(\sim 10^{-7}\) deviations): \(\sim 25\) bits

Total — geometric model: \(\sim 122\) bits.

The compression \(\Delta \approx 208 - 122 = 86\) bits corresponds to a Bayes factor of \(2^{86} \approx 10^{26}\) in favor of the geometric model over the null. The bit counts should be read as an illustrative MDL accounting rather than a formal Bayes‑factor calculation, the \(\sim\)80‑bit estimate for the rule‑set is a lower bound; a formal encoding of the six geometric expressions and their exponent structures would likely require more bits, which would reduce the compression advantage accordingly. Nevertheless, the present estimate shows that the model does have a distinctly significant compression advantage over a flat table of constants.

6.3 The Six Dimensionless Combinations: Fit-Envelope Consistency Test

Table 11 reports the six scalar-free combinations evaluated at the model-derived value \(a=\alpha^{-1}=137.035993138\) against CODATA 2014. The uncertainty column is the model-fit envelope uncertainty, including the allowed spread in the derived value of \(a\) implied by the six independent alpha solutions.

For a geometrical expression \(G_i(a)\propto a^{s_i}\), a finite uncertainty in \(a\) contributes \(\delta_{a,i}=|s_i|\frac{\Delta a}{a}\). Here \(\Delta a = a_{\max}-a_{\min}=4.78878\times10^{-6}\), \(\frac{\Delta a}{a}=3.49454\times10^{-8}\). Total diagnostic uncertainty \(\delta_{\rm total,i}=\sqrt{\delta_{{\rm CODATA},i}^{\,2}+\delta_{a,i}^{\,2}}\).

All six residuals lie inside the fitted-alpha envelope. The envelope statistic \(Q_{\rm env} = \sum_{i=1}^{6} (\delta_i/\delta_{\rm total,i})^2 \approx 0.71\). This small value indicates that the six residuals are comfortably contained within the fitted-alpha envelope. However, it should not be interpreted as a formal chi-squared statistic. A pure CODATA experimental chi-squared test would be much stricter and produce very large pulls for some rows, especially \(R_\infty\). The present table answers a different question: whether the six scalar-free invariants are mutually consistent with the model’s internally derived value of \(a\).

Statistical note. A formal global test would require the covariance matrix of the six constructed ratios, including correlations inherited from the CODATA adjustment and from the fitted value of \(a\). Moreover, \(\lambda_e\), \(R_\infty\), \(m_e\), and \(y_e/g_e\) are not fully independent in either CODATA adjustment theory or in the present \(\psi\)-based electron construction. Therefore the effective number of independent residual degrees of freedom is smaller than six. The fit-envelope calculation should be read as a consistency diagnostic, not as a formal global \(p\)-value.

6.4 The Alpha Derivation and Future Tuning

Table 4 demonstrates that when each geometric formula is independently solved for the value of alpha that makes it exactly equal its CODATA 2014 counterpart, the six resulting values span a constrained range of only \(\Delta a=4.79\times10^{-6}\). The joint minimization narrows this to a single consensus value, \(a_{\rm fit}=137.035993138\). This value is the optimal best-fit for the six high-precision constants selected for the present analysis. It is the model-internal consensus value, not a replacement for the CODATA adjustment of \(\alpha\). Because the framework operates as an overdetermined geometric lattice, the inclusion of future, additional high‑precision constants into the joint minimization protocol will serve to further constrain and tune this value of \(\alpha\), continually increasing the statistical confidence of the global fit.

6.5 Anomaly Detection

A pattern emerges whereby constants with the highest experimental precision and closest connection to the electromagnetic/electron sector show the smallest relative deviations from the model. Constants with large or historically difficult measurements, such as \(G\) and the thermal Boltzmann constant, show larger discrepancies. Because the model is a rigid generative structure, once the scalars \(r\), \(v\), and the value of \(a\) are fixed, all remaining predictions are forced. In this sense the model can be used as an anomaly detector for metrology and for the model itself. A constant whose measured value lies far from the model’s prediction is flagged either as a possible metrological/systematic outlier or as a point where the present geometric model must be revised.

7. Conclusion: Synthesis of the Overdetermined Framework

The results reported here converge from seven analytical directions:

1. The \(\theta\) unit-number bookkeeping and base-15 structural uniqueness.
2. The geometrical Planck-object construction, with \(M\), \(T\), and \(P\) derived from \(\pi\), \(e\), and \(\Omega\).
3. The dimensionless electron invariant \(\psi\), which links \(m_e\), \(\lambda_e\), \(R_\infty\), and \(y_e/g_e\) through one geometric construction.
4. The derivation of \(a=\alpha^{-1}\) from the measured constants via joint residual minimization.
5. The robustness of the theoretical \(\Omega\) under the joint \((a,\Omega)\) optimization.
6. Minimum Description Length (MDL) algorithmic compression.
7. The fit-envelope consistency test across six scalar-free combinations.

Together, these lines of evidence point toward a shared mathematical structure underlying the numerical values of the dimensioned physical constants. At the core of this order are two dimensionless numbers, \(\alpha\) and \(\Omega\), together with \(\pi\). The scaffolding of the dimensioned physical constants then reduces to these dimensionless quantities plus two SI scaling anchors, \(c\) and \(\mu_0\). Crucially, the analysis demonstrates that \(a\) itself can be recovered from the other constants as a model-internal consensus value. Since \(\Omega=\sqrt{\pi^e e^{1-e}}\) is a pure mathematical constant, this realizes, in precise mathematical form, Max Planck’s vision of “natural units that retain their meaning for all times and all cultures.”

7.1 The Joint Consistency Argument

The model’s central mathematical claim is that a framework with only two empirical anchors, \(c\) and \(\mu_0\), organizes six scalar-free combinations of measured constants together with a single internally derived value of the inverse fine-structure constant. This constitutes an overdetermined consistency test. The appropriate diagnostic is the fit-envelope test; using that uncertainty, all six scalar-free combinations lie inside the model’s self-consistent alpha envelope. The envelope statistic \(Q_{\rm env}\approx0.71\) shows that the six residuals are mutually compatible with the same model-derived value of \(a\). This statistic should not be interpreted as a formal global \(p\)-value. A pure CODATA \(\chi^2\) test would be much stricter. The present result is best understood as a fit-envelope consistency test.

7.2 The Alpha and Omega Determinations

Alpha from measured constants. The joint minimization yields \(a=137.035993138\) for CODATA 2014. This differs from the CODATA 2014 recommended value \(\alpha^{-1}=137.035999139(31)\) and therefore lies outside the CODATA uncertainty. It is not treated as a replacement CODATA measurement; it is the model’s internal consensus value that minimizes the joint residual across the selected electromagnetic and electron-sector constants. The two values agree at the \(4.4\times10^{-8}\) relative level.

Omega validation. The joint \((a,\Omega)\) optimization converges near the theoretical value \(\Omega_0=\sqrt{\pi^e e^{1-e}}=2.0071349543\ldots\). For CODATA 2014, fixing \(\Omega=\Omega_0\) gives an RMS log residual of \(2.36\times10^{-7}\); allowing \(\Omega\) to float improves this only to \(2.32\times10^{-7}\). The fitted \(\Omega\) differs from \(\Omega_0\) by only \(3.45\times10^{-10}\) relatively. The residual floor is not primarily caused by the fixed mathematical value of \(\Omega\).

7.3 The Electron as the Central Geometric Object

The most physically consequential part of the model is that the electron sector appears to be generated by a single dimensionless invariant, \(\psi\). The invariant \(\psi=\frac{\sigma_e^3}{2T}=4\pi^2(2^6\cdot3\cdot\pi^2\cdot a\cdot\Omega^5)^3\) is dimensionless: both unit-number bookkeeping and SI translation scalars cancel exactly. The electron mass, Compton wavelength, Rydberg constant, and gyromagnetic combination are not independent; they are linked by the same \(\psi\)-based construction. Therefore, repeated agreement among these constants should not be counted as fully independent statistical evidence. Instead, their shared residual pattern tests whether one electron invariant can simultaneously organize the electron-sector constants. If the \(\psi\) construction is physically meaningful, the electron is not a primitive empirical input but a derived geometric object. The proposed magnetic-monopole decomposition of \(\psi\) (\(DDD\) structure) is a model-internal geometric interpretation, not an established particle-physics result. Nevertheless, it gives the framework a concrete internal mechanism. In this sense, the electron is the central falsifiable object of the model.

7.4 The CODATA 2022 Result: Refutation or Precision Instrument?

When the identical framework is applied to the CODATA 2022 constants, two significant changes occur:

• The joint minimum in the \(\alpha\) scan becomes broader and shallower, indicating a reduction in discriminating power.
• The model‑derived \(\alpha^{-1}\) shifts from \(137.035993138\) (2014 dataset) to \(137.0359921141\) (2022 dataset).

These observations are highly consistent with the metrological shifts introduced by the 2019 SI redefinition. By fixing \(e, h,\) and \(k_B\) to exact numerical values, the 2019 convention removed the empirical variation that previously allowed this model to locate a sharp global minimum in residual space. The geometric model fundamentally permits only two constants to be assigned exact values; forcing additional constants into exactitude inherently over‑constrains the system. Distinguishing whether this represents a genuine limitation of the geometric lattice or a mathematically induced artifact of the 2019 SI convention requires future independent, high‑precision measurements of \(\alpha\) that do not rely on the post‑2019 exact‑constant network.

7.5 Permanent Methodological Contributions

Regardless of the final empirical verdict, several results stand as permanent methodological contributions:

• Electron invariant: The dimensionless \(\psi\) construction offers a single geometric origin for \(m_e\), \(\lambda_e\), \(R_\infty\), and \(y_e/g_e\).
• Alpha from constants: The demonstration that \(\alpha\) can be derived by minimizing a joint residual across dimensionless combinations of measured constants is a methodology applicable independently of the MLTA framework.
• Omega confirmation: The 10‑digit agreement between the theoretically motivated \(\Omega\) and the empirically optimized minimum provides independent mathematical support for the \(\Omega\) hypothesis.
• Base‑15 uniqueness: The algebraic uniqueness proof (Section 5) is a pure mathematical result, independent of any empirical data. The base‑15 structure is the unique solution to the constraint system.
• MDL compression: The estimated \(\sim86\)-bit compression of the CODATA 2014 constant set suggests that the model captures shared structure not present in a flat empirical table.
• Joint consistency framework: The fit-envelope test provides a template for evaluating unified-constants models without relying only on isolated numerical coincidences.

In summary: the geometric model organizes six scalar-free combinations of measured constants together with a model-internal value of the fine-structure constant using only two empirical SI anchors. The alpha-from-constants derivation, the \(\Omega\) robustness test, the electron invariant \(\psi\), and the fit-envelope consistency test together suggest that the observed alignment is unlikely to be explained by a naive independent-random-constants model. At the same time, the statistical interpretation should be understood as diagnostic rather than final: a formal assessment requires covariance propagation, objective-function sensitivity tests, and out-of-sample prediction. The CODATA 2022 discrepancy therefore defines precise, quantified targets for future experimental and theoretical resolution, elevating the model from a numerical curiosity to a productive, falsifiable framework.

Note

This webpage has been transcribed from Article 6. Article 6 is a statistical analysis of the Mathematical Electron model [1]. The simulation programs used for this article are listed here;
source code: best fit; alpha, Omega (3.3 Omega).
source code: base 15 3M + 2T (1) or source code: base 15 3M + 2T (2) (5. The Base‑15 Constraint).