Constants Proton Electron Mass Ratio M E

The electron mass rung

The proton-to-electron mass ratio is the number you get when you divide the mass of a proton by the mass of an electron. The measured value is about 1836.15, a figure that has puzzled physicists for a century because no simple theoretical reason explains why the proton should be that much heavier. The Recognition Science framework approaches the puzzle by placing both masses on a single ladder, a discrete set of allowed masses indexed by whole numbers. Each rung of the ladder is a power of the golden ratio, the number approximately 1.618 that solves the equation r² = r + 1.

Within the framework, the declaration m_e fixes the electron's mass as the mass on rung 2, meaning the electron sits at the second step of this ladder. The framework's library, a machine-checked collection of formal theorems, proves that this mass is positive, so the ladder does not produce a negative or zero electron. It also proves a structural statement: if the proton sits on any rung r_p, then the ratio of the proton mass to the electron mass equals the golden ratio raised to the power (r_p minus 2). In other words, the ratio is determined entirely by the spacing between the two rungs, with no free parameters.

This statement is conditional. It does not claim to know which rung the proton occupies. The full derivation of the proton mass, which would fix r_p, is blocked on a separate problem about quark masses and confinement. The ratio formula is a theorem about the ladder's structure, but the specific value 1836.15 is not derived from it. Instead, the measured value constrains the effective rung: if you know the ratio, you can infer what r_p must be, but the framework does not yet produce that rung from first principles.

What the declaration does establish is a clean structural claim: the proton-to-electron mass ratio, if both masses live on the ladder, must be a power of the golden ratio. That is a sharp, testable prediction. The measured ratio 1836.15 is close to phi raised to the power 15.5, which is about 1836.0, but the framework does not yet derive the half-integer exponent. The gap between the structural theorem and the measured value marks the open problem, not a completed derivation.

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

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 does not derive the measured value 1836.15; it only constrains the proton's rung once the ratio is known. The declaration does not prove that the proton actually sits on a rung; that is a premise of the ladder model. The theorem does not say the ratio is exactly a power of phi; it says the ratio equals phi to a power if both masses are on the ladder.

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.

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