Fine-Structure Constant as a Cross-Ratio: A Geometric Reframing of α

In the paper's words

The fine-structure constant alpha is reframed as the cross-ratio of two measurable electron length scales (classical radius / Compton wavelength). Version 1.0 adds the adelically extended cross-ratio on Bruhat-Tits buildings, explicit falsifiability conditions, and synthesis with the QNFO adelic physics program (July 2026).

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Prediction Current Constraint Test Feasibility Projective running of $\alpha$ Consistent with SM at LEP precision Requires ~10× improvement in $\alpha(M_Z)$ measurement Integer-ratio at Planck scale Untestable directly Indirect tests via GUT-scale predictions Gauge coupling relations Consistent with MSSM unification Distinguishable with future collider data

   from 6.3 Current Experimental Constraints

2. The paper is organized as follows. §2 reviews the historical and theoretical context of $\alpha$. §3 introduces projective geometry and the cross-ratio. §4 constructs the cross-ratio from electron length scales. §5 develops the mathematical formalism of projective invariance. §6 explores implications for precision measurement. §7 connects this framework to other formalisms, including the adelic extension (§7.7). §8 discusses falsifiability (§8.4) and open questions.

   from 1. Introduction

3. The cross-ratio is invariant under ALL projective transformations (homographies) of the form: $$x \mapsto \frac{ax + b}{cx + d}, \quad ad - bc \neq 0$$ This invariance is absolute: any configuration of four collinear points has the same cross-ratio in any projective coordinate system [established, projective geometry since Poncelet 1822].

   from 3.1 The Cross-Ratio as Projective Invariant

4. The fact that $r_e \ll \lambda_C$ (by a factor of ~1/137) means that electromagnetic self-energy effects are "small" compared to quantum uncertainty — the electron's charge is weak enough that its self-interaction is a small perturbation on its quantum nature.

   from 4.3 Physical Meaning

5. Specifically: $$\alpha^{-1} \approx 137$$ has tantalizing near-integer properties: $$137.035999084 = 137 + \frac{1}{27.79\ldots}$$ While no exact integer formula has been found, the cross-ratio framework provides geometric motivation for searching: if the relevant length scales arise from a discrete geometric structure (e.g., a Bruhat–Tits building — see §7.7), then their ratio would naturally be a rational number.

   from 6.1 Integer-Ratio Precision

6. In two-dimensional conformal field theory (CFT), the cross-ratio plays a central role: four-point correlation functions are determined up to a function of the cross-ratio of the insertion points [established, Belavin–Polyakov–Zamolodchikov 1984].

   from 7.2 Conformal Field Theory

How it is done

1. Its value governs the strength of electromagnetic interactions across all energy scales accessible to current experiment.

   from 1. Introduction

2. Arnold Sommerfeld introduced $\alpha$ in 1916 to explain the fine structure of hydrogen spectral lines [established].

   from 2.1 Measurement History

3. This approach predicts relationships between couplings but does not uniquely determine $\alpha$ without additional parameters [speculative].

   from 2.2 Theoretical Status

4. By Ostrowski's theorem [established], the only non-trivial absolute values on $\mathbb{Q}$ (up to equivalence) are the real absolute value $|\cdot|_\infty$ and the p-adic absolute values $|\cdot|_p$ for each prime $p$.

   from 7.7 Adelic Extension: The Cross-Ratio on Bruhat–Tits Buildings [v1.0 — NEW]

What would falsify it

1. Per the QNFO Research Integrity Mandate (QNFO-POL-COM-001, §0.0), speculative claims must carry explicit falsifiability conditions.

   from 8.4 Falsifiability Conditions [v1.0 — NEW]

2. The following would disconfirm or constrain the framework:

   from 8.4 Falsifiability Conditions [v1.0 — NEW]

3. This framework would be DISCONFIRMED if the relation $\alpha = r_e/\lambda_C$ were shown to be coincidental rather than structural.

   from 8.4.1 Direct Falsification

Where the claim stops

1. Since then, its value has been measured with extraordinary precision through multiple independent methods: Method Measured $\alpha^{-1}$ Uncertainty Year Electron $g-2$ (Gabrielse) 137.035999084(21) 0.15 ppb 2023 Rubidium atom recoil (Morel) 137.035999206(11) 0.081 ppb 2020 Cesium atom recoil (Parker) 137.035999046(27) 0.20 ppb 2018 CODATA 2018 137.035999084(21) 0.15 ppb 2019 The agreement across independent methods confirms the constancy of $\alpha$ at low energies.

   from 2.1 Measurement History

2. However, the $g-2$ and atom-recoil measurements show a ~5.5$\sigma$ tension when interpreted through the Standard Model, hinting at possible new physics [debated, $g-2$ vs.

   from 2.1 Measurement History

3. However, the derivation lacked physical motivation and relied on arbitrary group-theoretic factors [speculative, no experimental support].

   from 2.2 Theoretical Status

Where to read

1. 1. Introduction251 words · 1 min
2. 2. The Fine-Structure Constant: Historical and Theoretical Context346 words · 2 min
3. 3. Projective Geometry and the Cross-Ratio256 words · 1 min
4. 4. Electron Length Scales and the Cross-Ratio Construction289 words · 1 min
5. 5. Mathematical Formalism: Projective Invariance218 words · 1 min
6. 6. Implications for Precision Measurements249 words · 1 min
7. 7. Connections to Other Formalisms984 words · 4 min
8. 8. Discussion533 words · 2 min
9. 9. Cross-References to QNFO Research Program163 words · 1 min
10. Appendix A: Mathematical Conventions70 words · 1 min