Constants Proton Electron Mass Ratio Mass Ratio Structural

The structural ratio

The proton-to-electron mass ratio, about 1836.15, is one of the most precisely measured numbers in physics. It tells you how much heavier the proton is than the electron, and it sets the scale of atomic spectra and chemistry. In the Recognition Science framework, this ratio is not a free parameter but a consequence of a ledger, a discrete record of recognition events that the framework uses to derive physical constants. The framework's library, a machine-checked collection of formal statements, proves a structural statement: if both masses sit on the same phi-ladder, then their ratio is exactly phi raised to the difference of their ladder rungs.

The declaration mass_ratio_structural is a formal statement in the framework's library. It says that for any integer r_p, if the proton mass equals the mass assigned to rung r_p, then the ratio m_p / m_e equals phi^(r_p - 2). Here phi is the golden ratio, about 1.618, and the electron sits at rung 2. The proof is a short algebraic manipulation of the definition of mass_on_rung, using the fact that phi is nonzero. It establishes the form of the ratio, not its numerical value. The exponent r_p is not fixed by this declaration; it depends on the full derivation of the proton mass from confinement, which the framework marks as BLOCKED.

Historically, the mass ratio has been known since the early 20th century, with measurements improving from Rutherford's scattering experiments to modern Penning-trap techniques. The framework's contribution is to show that the ratio must be a power of phi, a structural claim that constrains any future derivation. If the proton mass is on the ladder, the ratio cannot be an arbitrary number; it must be phi^k for some integer k. The measured value 1836.15 lies between phi^15 (about 1364) and phi^16 (about 2207), so the effective r_p would be between 17 and 18, but the framework does not derive that exponent yet.

In Recognition Science, the framework models the electron mass as E_coh times phi^2, where E_coh is a coherence energy, and the proton mass follows from the same ladder plus confinement. The declaration mass_ratio_structural is a conditional result: it holds for any r_p, provided the proton mass is on the ladder. It does not claim that the proton is actually on the ladder; that is the open part. The framework's own status note says the full derivation is BLOCKED on the proton mass from confinement. What the declaration does prove is the structural form, a clean and testable prediction that any complete derivation must satisfy.

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 numerical value 1836.15 is not derived; only the structural form phi^(r_p - 2) is proved. The declaration does not prove that the proton is on the phi-ladder; that premise is an assumption in the statement. No claim is made about the fine-structure constant or any other coupling constant.

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

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