What question did this study set out to answer?

The research aims to demonstrate the independent derivation of proton charge radius and vacuum energy density by two separate theoretical frameworks and validate these findings through experimental results.

April 19, 2026Open Access

Independent Convergence on the Proton Charge Radius and Vacuum Energy Density

Key Points

The research aims to demonstrate the independent derivation of proton charge radius and vacuum energy density by two separate theoretical frameworks and validate these findings through experimental results.
Derived proton charge radius from holographic ratios and topology frameworks.
Validated the proton charge radius via muonic hydrogen measurements.
Matched vacuum energy density with cosmological observations using geometric approaches.
Compared results from both theoretical frameworks to establish convergence.
Proton charge radius confirmed at r_p ≈ 0.841 fm, resolving the proton radius puzzle.
Vacuum energy density estimated at ρ_Λ ≈ 5.88 × 10⁻²⁷ kg/m³, matching observations within 2%.
Demonstrated convergence between two independent theoretical methods, supporting Planck-scale geometry theories.

Abstract

Two dimensionless ratios — the proton charge radius rₚ expressed in Planck units, and the dark energy density ρ_Λ expressed as a fraction of the Planck energy density — have been derived independently by two distinct Planck-scale vacuum geometry frameworks and confirmed by experiment. The proton charge radius formula rₚ = 4ℓP (mP/mₚ) was derived by Haramein (2012) from holographic surface-to-volume ratios of Planck Spherical Units, and independently by the Unified Foam Field Theory (UFFT) from the topology of the truncated octahedron; these are algebraically identical expressions. The result rₚ ≈ 0. 841 fm was confirmed experimentally by the muonic hydrogen measurement of Antognini et al. (2013), resolving the proton radius puzzle at 7σ from the Standard Model prediction. For vacuum energy density, Haramein's generalized holographic weighting (2019) and the UFFT Euler characteristic correction (Paper #53) both match the observed cosmological constant ρ_Λ ≈ 5. 88 × 10⁻²⁷ kg/m³ to within 2% from different geometric arguments. Independent convergence of this kind — two unrelated frameworks reaching identical formulas, confirmed by experiment — constitutes evidence not merely for the specific numerical results but for the class of theory: Planck-scale discrete vacuum geometry determines nuclear and cosmological observables. We document both convergences precisely, identify the exact formulas, and discuss the implications.

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