Constants Proton Electron Mass Ratio

The proton-to-electron mass ratio

The proton-to-electron mass ratio, written m_p/m_e, is the number of electron masses that equal one proton mass. The measured value is approximately 1836.15. This ratio is one of the basic facts about ordinary matter: it helps set the size of atoms and the frequencies of light they emit, which is why a theory that explains it would reach deep into physics.

The ratio has a long history in physics. Early estimates came from measuring the deflection of electron beams in magnetic fields, and modern values use precision measurements of atomic energy levels and of the cyclotron frequencies of single trapped particles. The accepted value today is 1836.15267343(11), from CODATA 2018. The ratio is dimensionless, so it does not depend on any system of units.

In Recognition Science, the framework models particle masses as sitting on a ladder of powers of the golden ratio φ, where φ = (1 + √5)/2 ≈ 1.618. The framework's library proves a structural theorem: if the electron mass sits on rung 2 and the proton mass sits on rung r_p, then the ratio m_p/m_e equals φ^(r_p - 2). The library proves this formula in a machine-checked theorem, and it also proves that the electron mass is positive. The formula has no free parameters in the framework; the measured value 1836.15 would fix the effective rung r_p, but the full derivation of the proton mass from confinement is still open.

The framework's library is a machine-checked collection of formal theorems. The ratio formula is proved, but the step that would determine r_p from first principles is not yet in the library. The framework does not claim to have derived the exact measured ratio; it has derived the structural form that the ratio must take if both masses lie on the φ-ladder.

/-- The mass ratio m_p/m_e when both are on the φ-ladder has the form φ^k.

    For electron: r_e = 2. For proton: r_p from C-008 (φ-ladder + confinement).
    This theorem states the structural form; the exponent k depends on the
    full proton derivation. -/
theorem m_e_pos : 0 < m_e := by
  unfold m_e mass_on_rung
  apply mul_pos
  · unfold Masses.Anchor.E_coh
    rw [zpow_neg, inv_eq_one_div]
    exact div_pos zero_lt_one (pow_pos phi_pos 5)
  · exact pow_pos phi_pos 2

/-- The mass ratio m_p/m_e when both are on the φ-ladder has the form φ^k.

    For electron: r_e = 2. For proton: r_p from C-008 (φ-ladder + confinement).
    This theorem states the structural form; the exponent k depends on the
    full proton derivation. -/
theorem m_e_pos : 0 < m_e := by
  unfold m_e mass_on_rung
  apply mul_pos
  · unfold Masses.Anchor.E_coh
    rw [zpow_neg, inv_eq_one_div]
    exact div_pos zero_lt_one (pow_pos phi_pos 5)
  · exact pow_pos phi_pos 2

/-- Electron mass in RS units: E_coh · φ^2 (from C-007, r_e = 2). -/
noncomputable def m_e : ℝ := mass_on_rung 2

theorem mass_ratio_structural (r_p : ℤ) (m_p : ℝ)
    (hm_p : m_p = mass_on_rung r_p)
    (_hm_p_pos : 0 < m_p) :
    m_p / m_e = phi ^ (r_p - 2) := by
  rw [hm_p, m_e, mass_on_rung, mass_on_rung]
  field_simp [zpow_ne_zero _ phi_ne_zero]
  exact (zpow_sub₀ phi_ne_zero r_p 2).symm

What this page does not claim

The framework has not derived the exact numerical value 1836.15267343 from first principles. The formula φ^(r_p - 2) does not by itself fix r_p; the measured value would be needed to constrain it. The framework does not claim that the proton mass is exactly on an integer rung; r_p is a free integer in the theorem.

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/Constants/ProtonElectronMassRatio.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: