Masses Smverification Muon Rung Minus Electron Rung

The muon-electron gap

The muon is a heavier cousin of the electron, roughly 207 times as massive. In the Recognition Science framework, a discrete record of events, particle masses sit on a ladder whose rungs are powers of the golden ratio phi. The declaration muon_rung_minus_electron_rung proves that the muon sits exactly 11 rungs above the electron on this ladder. This is a formal theorem, checked by a machine, with no free parameters in the derivation.

The mass law in the framework is m(particle) = yardstick(Sector) × phi^(r - 8 + gap(Z)), where the yardstick, rung r, and charge-dependent gap Z come from cube geometry and wallpaper groups. The theorem states r_lepton "mu" - r_lepton "e" = 11. This means the predicted mass ratio between the muon and electron is phi^11, about 199.0. The measured PDG 2024 ratio is about 206.8, an agreement to roughly 4 percent.

This is a structural claim about the framework's internal ladder, not a derivation of the measured masses themselves. The comparison against experimental values is an empirical check, documented as a hypothesis, not a theorem. The framework's library proves the positivity of all fermion masses and the phi-scaling law, but the numerical agreement with PDG values is stated as a hypothesis with documented experimental values, not as a proved result.

What the theorem does not claim: it does not predict the absolute masses of the muon or electron, only their ratio on the ladder. It does not claim that the 11-rung gap is the cause of the observed mass difference; it derives the gap from the framework's geometry and then compares it with experiment. The tauon, the third charged lepton, sits 17 rungs above the electron, giving a predicted ratio of phi^17, about 3571, against the measured 3477.

theorem muon_rung_minus_electron_rung :
    r_lepton "mu" - r_lepton "e" = 11 := by
  simp only [r_lepton, tau, Anchor.E_passive, passive_field_edges,
             cube_edges, active_edges_per_tick, D, wallpaper_groups]
  norm_num

theorem muon_rung_minus_electron_rung :
    r_lepton "mu" - r_lepton "e" = 11 := by
  simp only [r_lepton, tau, Anchor.E_passive, passive_field_edges,
             cube_edges, active_edges_per_tick, D, wallpaper_groups]
  norm_num

theorem pdg_mu_e_ratio_approx : abs (pdg_mu_e_ratio - 206.8) < 1 := by
  simp only [pdg_mu_e_ratio, pdg_muon_MeV, pdg_electron_MeV]
  norm_num

What this page does not claim

This does not derive the absolute masses of the muon or electron. This does not prove the numerical agreement with PDG values; that remains an empirical hypothesis. This does not explain why the mass ladder has this particular rung spacing.

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

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