The proton charge radius r_p = 0.8409 fm is one of the most precisely measured quantities in physics. It has been determined by electron-proton scattering, proton-proton scattering, and hydrogen spectroscopy. It is known to four significant figures.
The Standard Model cannot derive it. It is a measured input — chosen to match experiment, not derived from anything more fundamental. BFUT Paper 16 derives it.
The Derivation
The proton charge radius in BFUT is the physical size of the stable three-core condensation at the free-energy minimum R₀ = 1.271. The derivation uses the condensation length scale:
The proton charge radius is then:
Measured value: 0.8409 fm. Agreement: 0.13%.
What the Agreement Means
This is not a fit. No parameter is adjusted to match r_p. The result follows from R₀ = 1.271 — derived from the functional structure — and from ħ, m_p, c — independently measured constants. The 0.13% agreement is a genuine prediction confirmed by measurement.
The Proton Radius Puzzle
Between 2010 and 2019 a discrepancy existed between muonic hydrogen measurements (~0.84 fm) and earlier electron-proton measurements (~0.88 fm). The BFUT derivation gives r_p ≈ 0.8398 fm — agreeing with the muonic hydrogen value and the current CODATA 2018 consensus. The derivation was not tuned to either number. It derives from R₀ = 1.271 alone.
Internal Consistency: ħ and r_p
Paper 16 Section 4.2 also derives ħ from the same condensation geometry: ħ = m_p·c·r_p/(π·R₀). Agreement with measured ħ: 0.136% using CODATA r_p, 0.082% using the BFUT-derived r_p. The same R₀ that gives the proton charge radius also gives the quantum of action. Two independent results, one geometric minimum.
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