"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."
— Inconsistent Constants, Scientific American, 292 p.56 (2005)
Novel aspects:
There are apparent anomalies within the dimensioned physical constants \( G, h, c, e, m_e, k_B, \dots \) which strongly suggest an underlying mathematical structure expressed through geometrical objects (Planck unit analogues) for mass, time, and momentum \( MTP \). This is introduced in detail here geometrical Planck objects. This web-page summarizes 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 required. 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):
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 (independent of any system of units).
From \( MTPLA \) we can derive geometrical SI constant analogues \( G^*, h^*, c^*, e^*, (y_e/g_e)^*, m_e^*, k_B^* \) using standard equations for the Planck unit conversions. No arbitrary free parameters are introduced; the constants emerge from a unique geometric lattice.
This page is from Article 6: Anomalies in the Physical Constants, an 8-article series on The Simulation Universe and the Fine structure constant alpha.
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 resulted in exact values assigned to 4 constants \( h, c, e \), and \( k_B \), but this then obscured any underlying mathematical dependency by forcing independence through metrological convention.
In this "Simulation Hypothesis" framework, 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 favor 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.
Note: This analysis uses CODATA 2014 values because, in that framework, only two dimensioned anchors (\(c\) and \(\mu_0\)) are taken as exact; (in this model only 2 constants can be assigned fixed values as from these 2 values all other constants are subsequently defined by default). 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, \(c\) and \(\mu_0\) were exact, while \(h\), \(e\), and \(k_B\) were adjusted measured constants. After 2019, \(\mu_0\) is no longer exact and is determined through \(\alpha\).
To assign 4 constants exact values is only possible if these constants are independent of each other. It is argued here that this premise is not correct.
The core of the model is the unit-number relationship (θ), a mapping that breaks the presumed independence of SI units by assigning specific integer values to base attributes. 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. a = (inverse)alpha = 137.0599...).
| Object | Geometry | Attribute | Unit number \(\theta\) |
|---|---|---|---|
| \(M\) | \(1\) | mass | \(15\) |
| \(T\) | \(\pi\) | time | \(-30\) |
| \(P\) | \(\Omega\) | \(\sqrt{\text{momentum}}\) | \(16\) |
| \(V\) | \(2\pi P^2 / M = 2\pi \Omega^2\) | velocity | \(17\) |
| \(L\) | \(VT = 2\pi^2 \Omega^2\) | length | \(-13\) |
| \(A\) | \(16 V^3 / (a P^3) = \dfrac{2^7 \pi^3 \Omega^3}{a}\) | charge/current | \(3\) |
| \(K\) | \(A V / (2\pi) = \dfrac{2^7 \pi^3 \Omega^5}{a}\) | temperature | \(20\) |
The internal consistency of this mapping is enforced by the fundamental constraint equation:
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 (requiring 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 \(T = -30\).
Conversion from the dimensionless geometric sector to a conventional unit system requires only two dimensioned scalars. This is because once any two scalars are assigned values (numerical and units), 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).
| Object | Geometry | Attribute | \(\theta\) |
|---|---|---|---|
| \(M\) | \(1 \times (r^4 / v)\) | mass | \(15\) |
| \(T\) | \(\pi \times (r^9 / v^6)\) | time | \(-30\) |
| \(P\) | \(\Omega \times (r^2)\) | \(\sqrt{\text{momentum}}\) | \(16\) |
| Constant | Geometrical Object | Unit number \(\theta\) from \(r, v\) |
|---|---|---|
| Speed of light | \(c^* = V = 2\pi \Omega^2 (v)\) | \(17\) |
| Vacuum permeability | \(\mu_0^* = \dfrac{4 \pi V^2 M}{a L A^2} = \dfrac{a}{2^{11} \pi^5 \Omega^4} (r^7)\) | \(8\cdot7 = 56\) |
| Planck constant | \(h^* = 2\pi M V L = 2^3 \pi^4 \Omega^4 \left( \dfrac{r^{13}}{v^5} \right)\) | \(8\cdot13 - 17\cdot5 = 19\) |
| Gravitational constant | \(G^* = \dfrac{V^2 L}{M} = 2^3 \pi^4 \Omega^6 \left( \dfrac{r^5}{v^2} \right)\) | \(8\cdot5 - 17\cdot2 = 6\) |
| Elementary charge | \(e^* = A T = \left( \dfrac{2^7 \pi^4 \Omega^3}{a} \right) \left( \dfrac{r^3}{v^3} \right)\) | \(8\cdot3 - 17\cdot3 = -27\) |
| Boltzmann constant | \(k_B^* = \dfrac{2\pi V M}{A} = \dfrac{a}{2^5 \pi \Omega} \left( \dfrac{r^{10}}{v^3} \right)\) | \(8\cdot10 - 17\cdot3 = 29\) |
The model defines the electron not as a physical particle, but as a dimensionless mathematical invariant (ψ). This invariant is mechanically constructed from magnetic monopoles (AL) and time (T). The point is not that ψ 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 ψ 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 ψ).
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 2 more constants; the Rydberg constant \(R^*\) and the gyromagnetic ratio \((\y_e/g_e)^*\).
(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.
| Constant | Calculated α⁻¹ | Uncertainty (1σ) |
|---|---|---|
| \(R^* = R\) | \( 137.03599636878(22)\) | \(\pm 0.00000000022\) |
| \(e^* = e\) | \( 137.035994984(59)\) | \(\pm 0.000000059\) |
| \(\lambda_e^* = \lambda_e \) | \( 137.035993139(36)\) | \(\pm 0.000000036\) |
| \(m_e^* = m_e\) | \(137.03599292(46)\) | \(\pm 0.00000046\) |
| \(h^* = h\) | \(137.03599274(90)\) | \(\pm 0.00000090\) |
| \((y_e/g_e)^* = (y_e/g_e)\) | \(137.03599158(39)\) | \(\pm 0.00000039\) |
| \(G^* = G\) | \(136.9905083084\) | — (formula scaling differs) |
| \(k_B^* = k_B \) | \(136.9568994960\) | — (formula scaling differs) |
We can plot the sum of precisions to find the minima over a range of alpha, this can give us an estimate for an optimal alpha based only the constants. We find a minima at 137.035993138. If we use CODATA 2022 values then we find a similiar minima but with less precision (y-axis) and more spread out, indicating that the model handles 2014 better (as only 2 constants are pre-defined).
START = mp.mpf("137.0359900000")
END = mp.mpf("137.0359998000")
loop {...
a1 = (C['R']**7) * (C['two']**260) * (mp.pi**122) * (a**26) * (C['Omega']**155) * (C['mu0']**9) * (C['three']**21) / (C['v']**35)
a2 = (C['e']**7) * (a**10) * (C['v']**21) / ((C['two']**82) * (mp.pi**43) * (C['Omega']**33) * (C['mu0']**3))
a3 = (C['h']**7) * (a**13) * (C['v']**35) / ((C['two']**164) * (mp.pi**93) * (C['mu0']**13) * (C['Omega']**80))
a4 = (C['lambda_e']**7) * (C['v']**35) / ((C['two']**253) * (mp.pi**122) * (C['Omega']**155) * (C['mu0']**9) * (a**12) * (C['three']**21))
a5 = (C['me']**7) * (C['two']**96) * (mp.pi**36) * (C['Omega']**89) * (a**25) * (C['v']**7) * (C['three']**21) / (C['mu0']**4)
a6 = (C['ye']**7 / C['ge']**7) * (C['v']**14) * C['mu0'] / ((C['two']**164) * (mp.pi**72) * (C['Omega']**122) * (C['three']**21) * (a**15))
sum = abs(a1-1) + abs(a2-1) + abs(a3-1) + abs(a4-1) + abs(a5-1) + abs(a6-1))
By reorganizing the constants into dimensionless combinations (where both units and scalars cancel), the SI constants should return the same numerical values as the \(MTP\alpha\) analogues. This is because in this model the \(MTP\alpha\) objects are considered to be natural Planck units (embedded in the 'source code'), and so 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:
Dimensionless combinations; \(\theta\) sums to 0| CODATA 2014 Value | Geometrical Equivalent | Uncertainty (1σ) | Fractional Deviation \(\delta = \frac{\text{CODATA} - \text{Geom}}{\text{CODATA}}\) |
Check \(|\delta| \leq \delta_{\text{total}}\)? |
|---|---|---|---|---|
| \(R_1 = 1.407919747902 \times 10^{64}\) (\(e\)) | \(\displaystyle \frac{2^{103}\pi^{64}\Omega^{75}}{a^{10}} = 1.407919937531 \times 10^{64}\) | \(1.35 \times 10^{-7}\) | \(-1.35 \times 10^{-7}\) | ✓ 1.0 |
| \(R_2 = 3.093571370933 \times 10^{268}\) (\(\lambda_e\)) | \(2^{288}\pi^{157}a^{12}\Omega^{225}3^{21} = 3.093571370731 \times 10^{268}\) | \(3.18 \times 10^{-9}\) | \(+6.54 \times 10^{-11}\) | ✓ 1/49 |
| \(R_3 = 3.066510823237 \times 10^{-301}\) (\(R_\infty\)) | \(\displaystyle \frac{1}{2^{295}\pi^{157}a^{26}\Omega^{225}3^{21}} = 3.066512702950 \times 10^{-301}\) | \(6.13 \times 10^{-7}\) | \(-6.13 \times 10^{-7}\) | ✓ 1 |
| \(R_4 = 1.405363033591 \times 10^{141}\) (\(h\)) | \(\displaystyle \frac{2^{199}\pi^{128}\Omega^{150}}{a^{13}} = 1.405362981160 \times 10^{141}\) | \(9.32 \times 10^{-8}\) | \(+3.73 \times 10^{-8}\) | ✓ 1/2.5 |
| \(R_5 = 6.262971955941 \times 10^{183}\) (\(y_e/g_e\)) | \(2^{178}\pi^{86}a^{15}\Omega^{150}3^{21} = 6.262973025999 \times 10^{183}\) | \(1.76 \times 10^{-7}\) | \(-1.71 \times 10^{-7}\) | ✓ 1 |
| \(R_6 = 4.542849889477 \times 10^{-128}\) (\(m_e\)) | \(\displaystyle \frac{1}{2^{89}\pi^{29}a^{25}\Omega^{75}3^{21}} = 4.542849712333 \times 10^{-128}\) | \(9.33 \times 10^{-8}\) | \(+3.90 \times 10^{-8}\) | ✓ 1/2.4 |
The geometric model relies on a single dimensionless constant, Omega (\(\Omega\)), 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 (there are still unknowns). Here is a qualified \(\Omega\):
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), there appeared a local minima, we then tuned to this minima (source code: best fit; alpha, Omega). The final results were cross-checked with Maple.
The Boltzmann constant kB appears in two distinct measurement contexts in this analysis.
kB = 1.380 648 52(79) e-23 CODATA 2014 (thermal, from macroscopic thermometry)
\(k_B^* = 2\pi\; \cdot2\pi\Omega^2v\; \cdot 1r^4/v\;\cdot \dfrac{a r^6}{2^7 \pi^3 \Omega^3 v^3} = \dfrac{a}{2^5 \pi \Omega} \left( \dfrac{r^{10}}{v^3} \right) = 1.379\;510\;194 \times 10^{-23}\)
unit number = 17 + 15 - 3 = 29
\(k_B\) ** (derived as an electromagnetic constant by using the electron gyromagnetic ratio \(y_e\) = \(\gamma_e / 2\pi\) = 28024951640(170) units = kg-1sA (-15 + (-30) + 3 = -42) and the g-factor \(g_e\) = −2.00231930436092(36)).
\(k_B** = \dfrac{g_e h}{4\pi c \gamma_e m_e}\) = 1.379 510 310(24) × 10-23
unit number = 19 - 17 - (-42) - 15 = 29
rewriting the above formula gives
\(k_B** = (g_e \cdot \;2^3\pi^4 \Omega^4 r^{13}/v^5)\; /\; (4\pi\;\cdot 2\pi\Omega^2v\; \cdot\gamma_e\; \cdot (1r^4/v)/\psi\;) \)
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.
Note: \(k_B^{**}\) is not the thermodynamic Boltzmann constant measured by macroscopic thermometry. It is a model-defined electromagnetic scalar constructed from electron-sector quantities. Its numerical closeness to \(k_B\) is therefore a model result, not an independent CODATA measurement of the thermal Boltzmann constant.
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 (source code: base 15 3M + 2T (1), source code: base 15 3M + 2T (2)).
Here $x$, $y$, and $i$ are intermediate algebraic substitutions used to map the $M, T, P$ geometries to the base-15 unit numbers.
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 table suggests more constants, the example used in this model as a primary constant is the sqrt of momentum P which serves as a link between the mass and charge domains.
| Constant | \(\theta\) | Geometrical object (\(\alpha, \Omega, v, r), \;\; \Omega^n\) | Unit |
|---|---|---|---|
| gyromagnetic ratio | \(-42\) | \(\gamma_e = \dfrac{x^\theta i^3}{y^5} = \dfrac{\pi \Omega^3}{v^2 r}\), n = (-42)+45=3 | \(\dfrac{m^{3/2}}{s^{1/2}\cdot kg^{5/2}} \cdot f_\psi = A \cdot s/kg\) |
| Time (Planck) | \(-30\) | \(T = \dfrac{x^\theta i^2}{y^3} = \dfrac{\pi r^9}{v^6}\), (-30)+30=0 | \(s\) |
| Elementary charge | \(-27\) | \(e^* = \dfrac{2^7 \pi^3}{a} \dfrac{x^\theta i^2}{y^3} = \dfrac{2^7 \pi^4 \Omega^3 r^3}{a v^3}\), (-27)+30=3 | \(A \cdot T\) |
| Length (Planck) | \(-13\) | \(L = 2\pi \dfrac{x^\theta i}{y} = \dfrac{2\pi^2 \Omega^2 r^9}{v^5}\), (-13)+15=2 | \(m\) |
| Ampere | \(3\) | \(A = \dfrac{2^7 \pi^3}{a} x^\theta = \dfrac{2^7 \pi^3 \Omega^3 v^3}{a r^6}\), (3)-0=3 | \(A = m^{3/2}/ (kg^{3/2}\cdot s^{3/2}) \) |
| Gravitational constant | \(6\) | \(G^* = 2^3 \pi^3 x^\theta y = \dfrac{2^3 \pi^4 \Omega^6 r^5}{v^2}\), (6)-0=6 | \(m^3 / (kg \cdot s^2)\) |
| Mass (Planck) | \(15\) | \(M = \dfrac{x^\theta y^2}{i} = \dfrac{r^4}{v}\), 15-15=0 | \(kg\) |
| sqrt(momentum) | \(16\) | \(P = \dfrac{x^\theta y^2}{i} = \Omega r^2\), 16-15=1 | \(\sqrt({kg\cdot m / s})\) |
| Velocity | \(17\) | \(V = 2\pi \dfrac{x^\theta y^2}{i} = 2\pi \Omega^2 v\), 17-15=2 | \(m / s\), 17-15=2 |
| Planck constant | \(19\) | \(h^* = 2^3 \pi^3 \dfrac{x^\theta y^3}{i} = \dfrac{2^3 \pi^4 \Omega^4 r^{13}}{v^5}\), 19-15=4 | \(m^2 kg / s\) |
| Planck temperature | \(20\) | \(T_p^* = \dfrac{2^7 \pi^3}{a} \dfrac{x^\theta y^2}{i} = \dfrac{2^7 \pi^3 \Omega^5 v^4}{a r^6}\), 20-15=5 | \(A \cdot V\) |
| Boltzmann constant | \(29\) | \(k_B^* = \dfrac{a}{2^5 \pi} \dfrac{x^\theta y^4}{i^2} = \dfrac{a r^{10}}{2^5 \pi \Omega v^3}\), 29-30=-1 | \(kg \cdot m / (s \cdot A)\) |
| Vacuum permeability | \(56\) | \(\mu_0^* = \dfrac{a}{2^{11}\pi^4} \dfrac{x^\theta y^7}{i^4} = \dfrac{a r^7}{2^{11}\pi^5 \Omega^4}\), 56-60=-4 | \(kg \cdot m / (s^2 A^2)\) |
Methodological & Statistical Notes:
The decisive question is not whether any single ratio $R_i$ is close to unity, but whether a single compact framework simultaneously recovers all independently measured constants to within their experimental uncertainties, utilizing the minimum possible number of free parameters.
H0: The $k$ observed dimensionless ratios {$R_1, \dots, R_k$} are independent numerical accidents — each $R_i$ deviates from unity for unrelated reasons, with no common generative structure.
The model has only two empirical anchors ($c$ and $\mu_0$) and derives seven quantities — six dimensionless combinations and the fine-structure constant itself — from geometry alone. Under H0, the simultaneous near-unity of all six combinations using only two fixed scalars, while deriving $\alpha$ entirely from the remaining constraints, is an extraordinary coincidence requiring explicit quantification.
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 $\mu_0$.
| Constant | CODATA 2014 $u_{\text{rel}}$ | Bits (null model) | Status in geometric model |
|---|---|---|---|
| $e$ | $6.1 \times 10^{-9}$ | ~27.3 | Derived via $R_1$ |
| $h$ | $1.2 \times 10^{-8}$ | ~26.3 | Derived via $R_4$ |
| $m_e$ | $1.2 \times 10^{-8}$ | ~26.3 | Derived via $R_6$ |
| $\lambda_e$ | $4.5 \times 10^{-10}$ | ~31.1 | Derived via $R_2$ |
| $R_\infty$ | $5.9 \times 10^{-12}$ | ~37.3 | Derived via $R_3$ |
| $\gamma_e/g_e$ | $6.1 \times 10^{-9}$ | ~27.3 | Derived via $R_5$ |
| $a$ | $2.3 \times 10^{-10}$ | ~32.0 | Derived from joint minimization |
| Total — null model | ~208 bits | — | |
Note: $\Omega = \sqrt{\pi^e\, e^{1-e}}$ is a pure mathematical constant (like $\pi$ or $e$ themselves) and is therefore excluded from both the null-model and geometric-model bit budgets. Under either model, $\Omega$ is computable from $\pi$ and $e$ at zero additional storage cost. The 10-digit agreement between the theoretically defined $\Omega$ and the empirically optimised value serves as independent confirmation of the formula, but does not contribute to the information-theoretic accounting.
The geometric model specifies all outputs using:
Total — geometric model: ~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. This establishes that the constants are not informationally independent: a compact generative structure exists that encodes them far more efficiently than a flat table.
Note. The bit counts should be read as an illustrative MDL accounting rather than a formal Bayes-factor calculation. A rigorous MDL result would require an executable encoding of the rule-set (including the allowed exponent search space, the invariant-selection rule, and the $\Omega$ formula), together with a precise residual encoding. The ~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 plausible compression advantage over a flat table of constants.
Table 3.2 reports the six combinations evaluated at the model-derived $\alpha^{-1} = 137.035993138$ against CODATA 2014. All six combinations are of order $\sigma \le 1.0$, with the largest about \(1.3\sigma\). No individual formula was tuned; all six share the same two anchors and the same derived $\alpha$.
Table 5.2 explicitly breaks down the fractional deviations and corresponding statistical significance for each dimensionless combination at this optimized consensus baseline.
| Combination ($R_i$) | Mapping | CODATA 2014 Target Value | Model Prediction | Fractional Deviation ($\delta$) | Statistical Significance ($\sigma$) |
|---|---|---|---|---|---|
| $R_1$ | $e/a$ | $1.6021766208 \times 10^{-19}$ | $1.6021766155 \times 10^{-19}$ | $-3.31 \times 10^{-9}$ | $0.54\,\sigma$ |
| $R_2$ | $\lambda_e$ | $2.4263102367 \times 10^{-12}$ | $2.4263102381 \times 10^{-12}$ | $+5.77 \times 10^{-10}$ | $1.28\,\sigma$ |
| $R_3$ | $R_\infty$ | $10973731.568508$ | $10973731.568501$ | $-6.38 \times 10^{-13}$ | $0.11\,\sigma$ |
| $R_4$ | $h$ | $6.626070040 \times 10^{-34}$ | $6.626070084 \times 10^{-34}$ | $+6.64 \times 10^{-9}$ | $0.55\,\sigma$ |
| $R_5$ | $y_e$ | $1.760859644 \times 10^{11}$ | $1.760859632 \times 10^{11}$ | $-6.81 \times 10^{-9}$ | $1.11\,\sigma$ |
| $R_6$ | $m_e$ | $9.10938356 \times 10^{-31}$ | $9.10938362 \times 10^{-31}$ | $+6.59 \times 10^{-9}$ | $0.55\,\sigma$ |
Because the Standard Model of physics mandates that fundamental constants are independent parameters, a first-order global test can be constructed assuming a diagonal covariance matrix. The correct omnibus test is the generalized chi-squared:
where $\mathbf{d}$ is the vector of deviations $(R_i - 1)$ and $\mathbf{C}$ is the covariance matrix. Because calculating $\alpha$ from the dataset consumes one degree of freedom, this test operates with $k - 1 = 5$ degrees of freedom (if the six invariants are treated as pre-specified and if correlations are neglected, fitting one shared value of \(a\) leaves a nominal \(6-1=5\) residual degrees of freedom). With all six in the range $\sigma \le 1.0$, the resulting reduced $\chi^2$ is near unity. This constitutes a formal statistical failure to reject the geometric model against CODATA 2014 under the assumption of independent variables.
Note. A full global \(\chi^2\) test requires the covariance matrix of the six constructed ratios, including correlations inherited from the CODATA adjustment and the fitted value of \(a\). The diagonal calculation should be treated as a first-order diagnostic, not a formal p-value. Also we have looked at 6 constants independently, we can note that these can be determined in terms of each other; ($\lambda_e, R_\infty,$ and $y_e$ are mathematically linked to $e, h, m_e$), we could redefine as combinations, this would still give 6 sets, hwoever this does shift the degrees of freedom to between 3...5.
Table 3.1 demonstrates that when each geometric formula is independently solved for the $\alpha^{-1}$ that makes it exactly equal its CODATA 2014 counterpart, the six resulting values span a highly constrained range of only $\Delta \alpha^{-1} = 4.79 \times 10^{-6}$. The joint minimization narrows this to a single consensus value: $\alpha^{-1} = 137.035993138$.
It is important to note that this specific value of $\alpha$ represents the optimal best-fit for the six specific high-precision constants selected for this analysis (it is the value of \(a\) that minimizes the chosen joint residual within this geometric model and this selected set of invariants). 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. Reducing the number of constants merely reduces this averaging; indeed table 3.1 uses only 1 constant each.
A pattern emerges whereby the constants with the highest experimental precision show the smallest relative deviations from the model. Constants with large or historically troubled measurements (such as $G$ and the thermal Boltzmann constant) show the largest discrepancies. Because the model is a rigid generative structure—once the scalars $(r, v)$ and $\alpha$ are fixed, everything else is forced—it serves as an anomaly detector for metrology. A constant whose measured value lies far from the model’s prediction is flagged as a candidate where the empirical measurement might be influenced by underestimated systematics, rather than a failure of the underlying fundamental geometry.
The results reported here converge from six independent analytical directions:
Each line of evidence points toward a deep mathematical order underlying the numerical values of the dimensioned physical constants.
At the core of this order are just two dimensionless numbers—$\alpha$ and $\Omega$—together with $\pi$. The entire scaffolding of dimensioned physical constants reduces to these three, plus two SI scaling anchors ($c$ and $\mu_0$). Crucially, this analysis demonstrates that $\alpha$ itself can be recovered from the other constants, meaning the ultimate free input is just $\Omega$ and the two anchors. Because $\Omega = \sqrt{\pi^e e^{1-e}}$ 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."
The model's central mathematical claim is that a framework with only two empirical anchors ($c, \mu_0$) simultaneously recovers six independently measured constants and the fine-structure constant itself. This constitutes an overdetermined system in the strongest sense.
Under the null hypothesis (H0), the proper global test statistic is the generalized chi-squared ($\chi^2_{\text{global}} = \mathbf{d}^\top \mathbf{C}^{-1} \mathbf{d}$). Evaluated at the model-derived $\alpha^{-1} = 137.035993138$, all six combinations satisfy $\sigma \le 1.0$ against CODATA 2014. Treating the variables as independent (as standard metrology dictates), the reduced $\chi^2$ approaches unity, resulting in a formal statistical failure to reject the geometric model at that data vintage.
Furthermore, the MDL compression yields $\Delta \approx 86$ bits of algorithmic reduction, corresponding to a Bayes factor of $\sim 10^{26}$ in favor of the geometric model over the null. The successive prediction pathway makes this overdetermination concrete: multiple quantities are recovered with zero additional free parameters beyond the initial axioms.
Alpha from measured constants. The joint minimisation of residuals across R1 to R6 yields a = 137.035993138 (CODATA 2014 dataset). This value differs from the CODATA 2014 recommended (inverse)alpha = 137.035999139(31) and so lies outside the CODATA uncertainty. This is not a failure of the model; it is the model reporting a consistent internal value of α that minimises the joint residual across all constants simultaneously. The discrepancy may reflect the fact that the CODATA 2014 alpha is determined primarily from the anomalous magnetic moment of the electron, while the model's derived α represents the value that minimises disagreement across the full set of electromagnetic constants. Both are legitimate determinations; they agree at the level of (137.035999139−137.035993138)/137.035999139 = 4.4 x 10^-8 relative, but not compatible at CODATA precision and so it should be interpreted as a model-internal consensus value, not as a replacement CODATA measurement.
Omega validation. The joint (α,Ω) optimisation converges to Ω = 2.007134954516 (CODATA 2014) and 2.0071349551 (CODATA 2022), both consistent with the theoretical value \(\Omega = \sqrt{\pi^e\, e^{1-e}} = 2.0071349543\) to 10 significant figures. Given that the CODATA uncertainties of the individual constants are not propagated into the optimisation, this 10-digit agreement supports using the theoretical Ω with confidence.
Furthermore, 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 \(2.32\times10^{-7}\), an improvement factor of 1.019. The fitted \(\Omega\) differs from \(\Omega_0\) by only \(3.45\times10^{-10}\) relatively. Thus the residual floor is not primarily caused by the fixed mathematical value of \(\Omega\).
When the identical framework is applied to the CODATA 2022 constants, two significant changes occur:
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 (here, $c$ and $\mu_0$), after which the geometric structure rigidly dictates the rest. Forcing additional constants into exactitude inherently over-constrains the system, a tension that manifests mathematically as the 2022 discrepancy.
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.
The 2022 discrepancy could therefore reflect either a limitation of the model or a mismatch between the model's two-anchor assumption and the post-2019 exact-constant convention. This is an important question.
Regardless of the final empirical verdict on the model's physical exactness, several results stand as permanent methodological contributions to theoretical physics and metrology:
In summary: the geometric model recovers six independently measured constants and the fine-structure constant from only two empirical anchors. The joint overdetermination test, the MDL compression, and the alpha-from-constants derivation suggests that the alignment is unlikely to be explained by a naive independent-random-constants model. The 2022 discrepancy defines precise, quantified targets for future experimental resolution, elevating the model from a numerical curiosity to a productive, falsifiable theoretical framework.