Live Test Here (replicated HTML from the original Python code by LLM - not 100% match for percisions sub-ppt level)

From the Pythagoreans’ hymn to numbers to Feynman’s “1/137” physics has long carried the suspicion that a single ratio ties disparate phenomena together. Sommerfeld introduced the fine-structure constant as a universal coupling , Eddington dared that it should be a pure number, Born framed it as the hidden governor of atomic detail, and Dirac argued that dimensionless combinations like $\alpha$ must be explained by structure rather than units . The riddle endured, acquiring almost mythic overtones—an Ariadne’s thread promised but never found.

In quantum physics, the fine-structure constant $\alpha$ appears almost everywhere, yet its origin remains arguably the field’s most stubborn mystery. After a century of attempts, no first-principles derivation has predicted its value even at the percent level; the rare multi-decimal matches have come from numerology or ad-hoc parameter tuning rather than a physical explanation.

In this paper, I take that challenge literally. I show that $\alpha$ is emergent and parameter-free: its value follows from a purely geometric, gauge-invariant construction rooted in the Relator postulate $R, \omega = c$ (luminal internal evolution on $\mathbb{C}$ orthogonal to spatial winding in $\mathbb{R}^3$). No measured dimensional constants are invoked—no $e$, no $c$, no $\hbar$—and no fitted numbers appear. A closed root condition fixes $\alpha$ by locking a Coulombic shell functional $\mathcal{D}_C$ to the vector-inductive sector through a universal map:

$$ C_{\log} \equiv \frac{\pi^2}{\mathcal{D}_C} \zeta (1 + \zeta) = \frac{1}{3}, \quad \zeta = \frac{K}{2\pi^2} \Lambda, $$

so that the electromagnetic coupling is set by geometry alone. The construction yields rigid, dimensionless ratios between the Coulomb and $\Lambda$-channel sectors. These geometric invariants, not empirical inputs, pin down $\alpha$.

The same mechanism unifies how "time" flows for quantum phases . In a companion analysis, the electron $g$-factor appears as an evolution-rate shift of the phase clock induced by the large $\mathcal{D}$ functional on the matching shell—precisely analogous to time dilation in general relativity, whether momentum or gravity-induced, now for the Coulomb field predicted by the Relator . Thus, the Relator framework does more than produce a number; it provides a single geometric origin for coupling and for evolution-rate renormalization, turning the century-old riddle of $\alpha$ into a calculable constant and opening a concrete path toward band-like stability structures for leptons within a background-free, gauge-invariant setting .

Our closed pipeline predicts an emergent value:

$$ \alpha_{\rm pred} = 0.007297352564326 $$

agreeing with CODATA 2022 $\alpha = 7.2973525643(11) \times 10^{-3}$ to $4.47$ ppt ($z = 0.03 \sigma$), thereby reproducing all certain published digits and predicting subsequent ones.

The numerical outcome—as shown—emerges from a deliberately minimalist formal and computational pathway. While a small background risk of bias toward overfitting can never be fully excluded, the relations employed here are grounded in physically meaningful structure and rigorous mathematics rather than ad hoc symbol-play. In principle, the final equation for $\alpha$ can be compressed into a more compact form, but such a reduction strips away its physical content—which I do not advocate.

This repository contains the code to compute the fine-structure constant $\alpha$ as an emergent and parameter-free invariant within Relator theory. The framework solves a closed root equation that couples the Coulombic (scalar) and inductive (vector) channels in a unified C ⊕ R3 geometry. This computation does not rely on any measured constants like $e$, $c$, or $\hbar$, and produces a result matching the CODATA 2022 $\alpha$ value with an extraordinary precision.

The method relies on a geometric and gauge-invariant construction that predicts $\alpha$ without any empirical inputs. It is based on a unique approach to coupling quantum electrodynamics and relativistic mechanics using Relator geometry.

For more details, see the full paper: Alpha Paper.

• Precision: Sub-ppb accuracy matching the CODATA 2022 value of $\alpha$.
• No Fitted Constants: The value of $\alpha$ is emergent from geometry, with no empirical tuning.
• Pure Geometry: Derived from Relator theory and geometry, offering a novel approach to the fine-structure constant.
• Self-Contained: No need for external constants such as $e$, $c$, or $\hbar$.

This framework implements the closed root equation for $\alpha$, coupling the Coulombic and inductive (Λ) sectors. The main steps include:

1. Coulombic Base Calculation: Using the scalar channel to compute the baseline for $\alpha$.
2. Inductive Channel Correction: Incorporating the effects of the vector channel via a logarithmic correction.
3. Geometric Closure: Ensuring the correctness of the derived $\alpha$ by locking the scalar and vector sectors using geometric relations.
4. Convergence and Precision: Iterating to convergence, with built-in checks for numerical stability and precision.

To run this code, you will need Python 3.x and the following libraries:

• mpmath (for arbitrary precision arithmetic)
• numpy (for mathematical operations)

Install dependencies using pip:

pip install mpmath numpy

After installing the necessary libraries, you can run the code using:

python "Full Alpha Geometry Calculation.py"

The script will compute the emergent $\alpha$ and output the results to the console.

alpha_emergent     = 0.007297352564332633809798
alpha_em^-1        = 137.0359991769773
Λ_eff (final)      = 0.6916840202847290215451
K (spectral)       = 0.002231538916531970186409
P^(IR)_χ(ℓ0)       = 0.08577919258455560110975
∆Λ_OUT (η0)        = -0.01396715806205758007151
∆Λ^(UV→IR)         = 0.05448534958655209065325
C_log(α_em)        = 0.333333333333333333   (∆ vs 1/3 = -2.71498346735e-30)
[Context] α_ref    = 0.007297352564311  →  ∆α(ppb) = 0.00296461074167

This work is based on the paper Emergent Fine-Structure Constant from Relator Geometry by M.Pajuhaan.

Pajuhaan, M. (2025). Alpha. Zenodo. https://doi.org/10.5281/zenodo.16951008

Key Insight: This paper introduces the Relator principle (Rω = c), which leads to the natural emergence of Special Relativity (SR) and General Relativity (GR) without invoking spacetime curvature. It shows how both time dilation and the energy-momentum relation come directly from quantum phase evolution, unifying quantum mechanics with relativity.

Impact:

• Revolutionizes our understanding of SR and GR by deriving them from intrinsic quantum-phase dynamics.
• Establishes quantum evolution as the foundation of spacetime behavior, bypassing traditional spacetime transformations.

Key Insight: Entanglement arises geometrically in the Relator framework, where internal and external frequencies ω_C and ω_R3 interact. This paper links quantum entanglement to internal quantum dynamics, resolving the measurement problem by removing the need for wavefunction collapse or hidden variables.

Impact:

• Provides a concrete, geometric foundation for understanding entanglement.
• Explains the physical origin of quantum nonlocality via intrinsic relational frequencies.

Key Insight: Measurement in quantum mechanics is redefined as a geometric bifurcation within the Relator framework. The paper explains how quantum measurement does not collapse the wavefunction but causes a deterministic restructuring of the system’s relational geometry, which governs quantum entanglement and locality.

Impact:

• Resolves quantum nonlocality and wavefunction collapse ambiguities.
• Provides a deterministic and geometric explanation for quantum measurement and entanglement, avoiding the pitfalls of traditional interpretations like Copenhagen.

Key Insight: The g-factor of the electron is derived analytically in the Relator framework without quantum electrodynamics (QED). The paper connects relativistic time dilation and Coulomb interactions to the observed value of the g-factor with high precision, predicting the value without the need for perturbative QED expansions.

Impact:

• Achieves ppt precision with the experimental g-factor using geometric principles.
• Demonstrates a direct, QED-independent path to calculating the g-factor.