Physics Proton Charge Radius Rs

The proton's measured size

The proton's charge radius is a measured property of the proton: the distance from its center at which its electric charge distribution falls to a characteristic value. The accepted experimental value, from muonic hydrogen spectroscopy, is 0.841 femtometers, where one femtometer is 10^-15 meters. This measurement, first reported in 2010, disagreed with earlier electron-scattering results by about 4%, a discrepancy known as the proton radius puzzle. The puzzle was resolved when refined electron-scattering experiments and new theoretical calculations converged on the smaller value.

The radius is not a hard boundary. It is a derived quantity that depends on how one defines the charge distribution's extent, typically through the slope of the proton's electric form factor at zero momentum transfer. Different measurement techniques, electron scattering and muonic hydrogen spectroscopy, extract the same underlying quantity but with different systematic uncertainties. The 2010 muonic result, with its precision of about 0.005 femtometers, forced the physics community to re-examine both experimental methods and the theoretical treatment of proton structure.

In Recognition Science, the framework models physical quantities through a cost function that measures the recognition cost between two states. The framework's machine-checked library of formal theorems defines a domain cost as J(m/e), where J is the framework's forced cost function, m is a mass-like parameter, and e is a charge-like parameter. The library proves three general facts about this cost function: it vanishes when the two parameters are equal, it is nonnegative for positive inputs, and a threshold constant involving the golden ratio is positive. These are properties of the cost function itself, not specific to the proton.

The framework does not derive the 0.841 femtometer value. Its research note records an intended identification between the proton radius and a phi-power expression, but the note itself flags that calculation as structurally off by a large factor. The formal theorems are universally quantified statements about the cost function; they become statements about the proton only if one defines m and e in terms of proton-specific quantities, which the framework does not do. The formal content is a template shared with thousands of sibling modules, each awaiting a subject-specific definition.

What the framework does establish, in plain language, is that the cost function has the basic properties one would want from a measure of mismatch: it is zero when there is no mismatch, it never goes negative, and it has a built-in threshold. These are necessary conditions for any physical application, but they are not sufficient to identify the proton radius. The measured value remains an experimental input to the framework, not a derived output.

noncomputable def cert : ProtonChargeRadCert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

What this page does not claim

The framework does not prove the proton radius equals any phi-power expression. The framework's formal theorems are not specific to the proton without additional definitions. The 0.841 femtometer value is a measurement, not a framework derivation.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Physics/Proton_Charge_Radius_RS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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