J. Rogers, SE Ohio
Abstract
Recent high-precision measurements demonstrate that the fine-structure constant α varies with energy scale, increasing from approximately 1/137 at low energies to 1/127 at the Z boson mass scale. Standard Framework interprets this "running" through vacuum polarization—virtual particle loops screening the bare charge. We propose an alternative interpretation: α's energy dependence reflects cross-term contributions when geometric intensity fields superpose before interaction. Under this framework, what Standard QED requires 12,672 tenth-order Feynman diagrams to calculate emerges naturally from a single equation: F = I₁ × I₂, where intensities I = Σ(nᵢ × gᵢ)/r superpose across multiple geometric projection modes. Recent measurements of hadronic contributions to running (>5σ significance), structural interpretations of α (November 2025), and enhanced sensitivity factors (December 2025) align with predictions of geometric cross-terms becoming visible at shorter distance scales. While this interpretation remains speculative pending direct experimental validation, the convergence of independent observations toward structural rather than fundamental interpretations of α warrants serious theoretical consideration.
1. Introduction: The Energy Dependence Problem
The fine-structure constant α ≈ 1/137.036 is among the most precisely measured quantities in physics. Yet despite a century of investigation, its physical origin remains unexplained within Standard Framework. What has become increasingly clear through precision measurements is that α is not actually constant—it varies with the energy scale at which it is measured.
1.1 Observational Evidence
Contemporary measurements establish clear energy dependence:
• At low energy (Thomson limit): α⁻¹ ≈ 137.036
• At Z boson mass (91.2 GeV): α⁻¹ ≈ 127.916 ± 0.015
• Recent hadronic contribution measurements show >5σ significance
This variation is not measurement error—it reflects genuine physics. The question is: what physics?
1.2 Standard Framework Interpretation
Quantum Electrodynamics (QED) explains the running through vacuum polarization. Virtual electron-positron pairs spontaneously appear near a charged particle, with the virtual electron attracted toward and the virtual positron repelled from the bare charge. This creates a dielectric screening effect: at large distances (low energies) the charge appears partially screened, while at short distances (high energies) probes penetrate closer to the bare charge.
The mathematical implementation requires calculating contributions from virtual particle loops at all orders. The current state-of-the-art calculation for the electron magnetic moment involves 12,672 tenth-order Feynman diagrams. While impressively successful at matching experimental values, this framework treats α as fundamentally mysterious—a parameter to be measured, not derived.
2. The Geometric Intensity Framework
2.1 Core Postulates
We propose a fundamentally different ontology where all interactions emerge from a single law:
F = I₁ × I₂
where intensity I is defined as:
I = (count × geometry) / r
Key elements:
Count: Discrete integer (nucleons, charges, etc.) – not continuous parameters
Geometry: Dimensionless projection ratio from unity substrate
r: Distance in natural (dimensionless) units
What Standard Framework calls distinct forces are different geometric projection ratios:
g_strong ≈ 1.0 (direct projection)
g_EM = √(α/2π) ≈ 0.0303 (lever-arm geometry)
g_weak ≈ 10⁻⁶ (torsion spring)
g_gravity = m_nucleon/m_Planck ≈ 10⁻¹⁹ (sparse mesh)
2.2 The Critical Difference: Intensity Superposition
Standard Framework assumes forces add independently:
F_total = F_EM + F_gravity + F_weak + ...
The geometric framework proposes intensities superpose before interaction:
I_total = I_EM + I_gravity + I_weak + ...
F = I₁_total × I₂_total
Expanding this multiplication reveals cross-terms that do not exist in Standard Framework:
F = (I₁_EM + I₁_grav)(I₂_EM + I₂_grav)
= I₁_EM·I₂_EM + I₁_grav·I₂_grav + 2(I₁_EM·I₂_grav)
The final term—the cross-term between electromagnetic and gravitational intensities—has no analog in Standard Framework, where EM and gravity are strictly independent interactions.
3. Reinterpreting the Running of Alpha
3.1 Standard Terminology Confusion
Standard QED uses confusing terminology. What they call "electric potential" V is actually the integral of the intensity gradient:
V = ∫ I dr
The "electric field intensity" E is then defined as E = -∇V, which equals our I. Standard Framework has the ontology backwards: they treat potential as fundamental and intensity as derived, when the geometric intensity gradient is the actual physical structure and potential is merely a mathematical convenience for calculation.
3.2 Cross-Terms and Energy Scales
At different distance scales (corresponding to different energy probes), different geometric structures contribute to the total intensity field:
Long distance (low energy): Primarily electromagnetic geometry dominates. Measured α ≈ pure g_EM²
Short distance (high energy): Additional geometric structures (weak, strong, hadronic) contribute significantly. Cross-terms I_EM × I_weak, I_EM × I_strong become measurable. Effective coupling appears stronger.
The "running" of α is not the bare charge becoming less screened. It is cross-term contributions from multiple geometric projection modes becoming significant at shorter scales.
3.3 Computational Implications
Standard QED requires calculating vacuum polarization through Feynman diagrams. The tenth-order calculation involves 12,672 diagrams to achieve experimental precision. The geometric framework suggests this massive computational effort is approximating what a single algebraic equation would predict directly:
α_eff(E) = [g_EM + corrections_from_other_geometries(E)]²
The 12,672 Feynman diagrams are computing cross-term contributions that emerge naturally when intensities superpose before interacting.
4. Convergent Evidence from Independent Sources
Several recent developments align with predictions of the geometric intensity framework:
4.1 Hadronic Contributions (2024-2025)
Recent measurements demonstrate "more than 5σ significance of the hadronic contribution to the running of α." Standard Framework interprets this through hadronic vacuum polarization—virtual quark-antiquark pairs affecting the photon propagator.
The geometric framework offers an alternative interpretation: hadrons possess both electromagnetic charge (g_EM contribution) and mass/strong-force structure (g_strong, g_gravity contributions). At energies where hadronic structure becomes relevant, cross-terms I_EM × I_hadronic contribute measurably to the effective coupling. The 5σ significance reflects these cross-terms becoming visible, not virtual particle screening.
4.2 Alpha as Structural Ratio (November 2025)
In November 2025, Sanctuary published "The Fine-Structure Constant in the Bivector Standard Model," proposing that "α may be understood not as a fundamental parameter to be fitted, but as a consequence of a deeper internal structure." Within their geometric algebra framework, α emerges as a structural ratio α = r_e/λ_C between the classical electron radius and reduced Compton wavelength.
While their specific geometric interpretation differs from ours, the convergence is remarkable: independent theoretical work arriving at the conclusion that α is a geometric ratio, not a fundamental constant. Both frameworks reject the Standard Model's treatment of α as an unexplained input requiring 12,672 Feynman diagrams to compute accurately.
4.3 Enhanced Sensitivity to Variations (December 2025)
December 2025 measurements using thorium-229 nuclear transitions demonstrated sensitivity to α variations enhanced by a factor of 6,000 compared to atomic systems. This "huge" enhancement factor arises because nuclear processes probe much shorter distance scales than atomic transitions.
The geometric framework predicts exactly this behavior: at nuclear distance scales (femtometers), strong-force geometric contributions (g_strong ≈ 1.0) become dominant. Cross-terms between electromagnetic and strong-force intensities would create much larger apparent variations in effective coupling than seen in atomic systems where only electromagnetic geometry contributes significantly.
4.4 Scale-Dependent Polarization (2020)
Research on QED vacuum response notes that "the running of the fine structure constant is due to equal components of electric screening (polarization of vacuum) and magnetic anti-screening (magnetization of vacuum)." The geometric framework naturally accounts for both through intensity field geometry: electric and magnetic components are dual aspects of the same geometric projection, with their ratio determining the observed coupling at each scale.
5. Critical Assessment and Falsifiability
5.1 What This Framework Claims
Strong claims:
The running of α is cross-term contributions, not vacuum polarization
12,672 Feynman diagrams approximate single algebraic equation
α is geometric ratio √(g_EM/2π), not unexplained fundamental constant
Modest claims:
Recent measurements are consistent with this interpretation
Independent theoretical work converges on structural rather than fundamental interpretation
Cross-term hypothesis warrants investigation as alternative to vacuum polarization
5.2 What Evidence Would Falsify This
Decisive falsification would come from:
Measurement showing α variation that cannot be expressed as intensity superposition
Demonstration that cross-terms between EM and other geometric modes are exactly zero
Proof that 12,672 Feynman diagrams cannot be reduced to simpler algebraic expression
Supportive validation would come from:
Direct measurement of residual force after subtracting pure Coulomb and gravitational contributions showing non-zero cross-term
Demonstration that intensity superposition equation matches QED predictions with fewer computational steps
Independent derivation of α from geometric first principles matching experimental value
5.3 Current Status: Speculative but Supported
This framework remains speculative. It has not been validated through dedicated experimental programs designed to test cross-term predictions. However, existing measurements intended to probe vacuum polarization show patterns consistent with intensity superposition:
Hadronic contributions at precisely the scales where cross-terms would appear
Enhanced sensitivity factors matching predictions for strong-force geometry contributions
Independent theoretical convergence on structural interpretations
6. Implications for Fundamental Physics
6.1 Unification Programs
If the geometric intensity framework is correct, the century-long quest to unify fundamental forces reflects a category error. Physics has been attempting to unify what are already expressions of a single law (F = I₁ × I₂) operating through different geometric projection ratios. The forces are not separate entities requiring unification—they are different measurement modes of unified geometric intensity.
This would explain why unification programs like string theory, requiring particles at ever-higher energies to work, have produced no testable predictions in 50 years. They seek to unify what is already unified at the level of intensity geometry.
6.2 Naturalness and Fine-Tuning
The hierarchy problem—why the Higgs mass is 125 GeV rather than Planck mass—assumes particle masses are fundamental properties requiring explanation. The geometric framework treats mass as a count × geometric_ratio product. The Higgs mass value is whatever the geometric projection ratio happens to be at that scale. There is no "naturalness" requirement because geometric ratios can be any dimensionless number.
Supersymmetry was invented to solve this non-existent problem. The failure to find SUSY particles at LHC energies may reflect not that SUSY exists at higher energies, but that the problem SUSY was designed to solve is an artifact of treating Jacobian coefficients as fundamental properties.
6.3 Future Colliders
Plans for the Future Circular Collider (€15-20 billion, 100 TeV) assume finding new physics at these energies will resolve foundational questions. The geometric framework suggests different expectations: higher energies probe shorter distances where different geometric structures contribute. New particles found (if any) would represent different geometric projection modes becoming accessible, not evidence of "more fundamental" physics.
Critically: if SUSY particles exist only at 50+ TeV energies, they cannot solve the hierarchy problem, provide dark matter candidates, or enable gauge coupling unification—the three phenomena they were invented to explain. Particles requiring that much energy to manifest are, by definition, irrelevant to low-energy physics.
7. Conclusion
The running of alpha may be evidence that Standard Framework has the ontology inverted. What QED treats as vacuum polarization screening a fundamental bare charge might actually be cross-term contributions when geometric intensity fields superpose before interaction.
This interpretation remains speculative. It requires validation through experiments designed to test cross-term predictions rather than vacuum polarization. However, convergent evidence from hadronic measurements, structural interpretations of α, enhanced sensitivity factors, and scale-dependent polarization studies suggests this alternative deserves serious theoretical investigation.
The stakes are high. If the geometric framework is correct, physics has spent a century calculating virtual particle loops to approximate what a single algebraic equation predicts directly. The quest to unify forces would dissolve into recognizing they were never separate. The hierarchy problem would vanish as a category error. And the path forward would require not bigger colliders seeking higher-energy physics, but deeper investigation of the geometric structure underlying what we already observe.
The critical question is not whether this interpretation is proven, but whether it is empirically testable. The answer appears to be yes—and the data already collected may contain evidence we have been interpreting through the wrong framework.
References
[1] Particle Data Group (2010). "Electroweak model and constraints on new physics." Reviews of Modern Physics. Notes α⁻¹(M_Z) = 127.916 ± 0.015 from running coupling measurements.
[2] Aoyama, T., et al. (2018). "Tenth-order QED contribution to the electron g-2." Physical Review Letters. Calculation involving 12,672 Feynman diagrams to achieve experimental precision for fine-structure constant.
[3] Sanctuary, B. (2025). "The Fine-Structure Constant in the Bivector Standard Model." Mathematics, November 2025. Proposes α as structural ratio r_e/λ_C rather than fundamental parameter. Published November 17, 2025, MDPI.
[4] Kraemer, S., et al. (2025). "Nuclear-clock measurement of thorium-229 transition." Nature Communications, December 2025. Reports factor 6,000 enhanced sensitivity to fine structure constant variations in nuclear systems. Physics World, December 1, 2025.
[5] Delahaye, P., et al. (2020). "QED Response of the Vacuum." Symmetry 2(1). Describes running of α as equal contributions from electric screening and magnetic anti-screening. MDPI, January 2020.
[6] Bronnikov, K.A., Ivashchuk, V.D., Khruschov, V.V. (2025). "The fine-structure constant: a review of measurement results and possible space-time variations." Measurement Techniques 68, 125-134. arXiv:2506.18328, June 2025.
[7] Jegerlehner, F., Nyffeler, A. (2009). "The Muon g-2." Physics Reports. Documents hadronic contribution significance to running of α exceeding 5σ.
[8] Wikipedia contributors. "Vacuum polarization." Wikipedia, accessed January 2026. Describes α_eff(p²) momentum-transfer dependence and self-energy tensor formulation.
[9] Murphy, M.T., et al. (2022). "A limit on variations in the fine-structure constant from spectra of nearby Sun-like stars." Science. Sets 50 ppb limit on α variations in local Milky Way region.
[10] Wilczynska, M.R., et al. (2020). "Four direct measurements of the fine-structure constant 13 billion years ago." Science Advances 6, eaay9672. Probes α at z=7.1, constrains Δα/α = (-2.18 ± 7.27) × 10⁻⁵.
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