Constants Proton Electron Mass Ratio Proton Electron Ratio From Ladder

The ladder ratio

The proton-to-electron mass ratio, written m_p/m_e, is one of the most precisely measured numbers in physics, with a value near 1836.15. It asks a simple question: why is the proton, the particle in every atomic nucleus, so much heavier than the electron that orbits it? The question matters because the ratio sets the size of atoms and the energies of chemical bonds; if it were very different, chemistry and life as we know it would not exist.

The standard model of particle physics does not predict this ratio. It accepts the electron mass and the proton mass as free inputs, measured from experiment. The framework called Recognition Science (RS) instead tries to derive both masses from one shared structure, a ladder of allowed masses. In RS, a ledger, a discrete record of recognition events, forces a special scaling number, the golden ratio φ, approximately 1.618. The framework models particle masses as sitting on a ladder where each rung is a power of φ. The electron sits on rung 2; the proton sits on some higher rung, call it r_p.

Within the framework, the declaration proton_electron_ratio_from_ladder proves a structural theorem: if the proton mass equals the ladder value at rung r_p, and the electron mass equals the ladder value at rung 2, then the ratio m_p/m_e equals φ raised to the power (r_p - 2). The proof is a short algebraic identity, checked by a machine. The theorem does not say what r_p is. It states the form of the ratio, not its numerical value. The measured value 1836.15 would pin down an effective r_p, but the full derivation of the proton's rung, which the framework links to quark confinement, remains blocked.

In Recognition Science, this theorem is a structural step, not a finished prediction. It shows that if both masses fall on the φ-ladder, then their ratio must be a power of φ, with no free parameters in the formula. The framework's library proves this conditional statement. What it does not prove is that the proton actually sits on the ladder, nor which rung it occupies. Those claims depend on further derivations that the pack marks as not yet complete.

/-- 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

/-- **C-009 Status**: The ratio m_p/m_e is determined by the φ-ladder.

    Once C-007 and C-008 give m_e and m_p, the ratio m_p/m_e = φ^(r_p - 2).
    No free parameters. The measured value 1836.15 constrains the effective r_p.

    Full derivation: BLOCKED on C-008 (proton mass from confinement). -/
theorem proton_electron_ratio_from_ladder (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) :=
  mass_ratio_structural r_p m_p hm_p hm_p_pos

What this page does not claim

The theorem does not state the numerical value of the proton-to-electron mass ratio. The theorem does not prove that the proton actually sits on the φ-ladder. The full derivation of the proton mass from confinement is not yet complete.

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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