Constants Proton Electron Mass Ratio M E Pos

The electron's positive mass

The proton-to-electron mass ratio, about 1836.15, is one of the plainest numbers in physics: it says a proton is that many times heavier than an electron. A full theory of particle masses would derive this ratio from deeper principles rather than measuring it. The Recognition Science framework attempts such a derivation, and its first step is a formal, machine-checked proof that the electron mass is positive.

In the framework, masses sit on a ladder of powers of the golden ratio φ ≈ 1.618, a structure that emerges from the framework's core cost function. The electron occupies rung 2, meaning its mass is written as E_coh · φ², where E_coh is a coherence energy scale. The theorem m_e_pos (a declaration in the framework's machine-checked library of formal theorems) proves that this quantity is strictly greater than zero: 0 < m_e. The proof is short and rests on two facts: the coherence energy is positive, and any positive power of φ is positive.

Why does positivity matter? A mass that could be zero or negative would break the ladder's logic. The ratio m_p/m_e = φ^(r_p - 2) only makes sense if both masses are nonzero, so the positivity theorem is a necessary foundation. The framework also proves a structural formula: if the proton sits on rung r_p, then the ratio equals φ raised to (r_p - 2). This is a theorem about the ladder's form, not a numerical prediction. The measured value 1836.15 would constrain r_p once the proton's rung is derived, but that derivation is blocked on a separate problem: deriving the proton mass from confinement.

The theorem does not claim the electron mass has a specific numerical value. It does not derive the proton-to-electron ratio, and it does not say what r_p is. It only establishes positivity, a modest but rigorously checked foundation stone. In the framework's own terms, the full ratio derivation remains an open target, awaiting the proton mass derivation.

/-- 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 theorem does not assign a numerical value to the electron mass. It does not derive the proton-to-electron mass ratio. It does not specify the proton's rung r_p.

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)

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