muon_relative_error
The theorem shows that the Recognition Science phi-ladder prediction for the muon mass at rung 13 differs from the PDG experimental value by less than 4 percent relative error. Modelers checking the lepton sector against data would cite it when assembling the mass-verification certificate. The proof rewrites the target inequality to the pre-proved numerical bounds on the prediction and closes with linear arithmetic.
claimLet $m_μ^pred$ denote the Recognition Science mass in MeV for the Lepton sector at rung 13, given by $2^{-22} φ^{57+13} / 10^6$, and let $m_μ^exp = 105.6583755$ MeV. Then $|m_μ^pred - m_μ^exp| / m_μ^exp < 0.04$.
background
In the lepton sector the Recognition Science mass formula is $m(Lepton, r) = φ^{57+r} / (2^{22} × 10^6)$ MeV. The function rs_mass_MeV implements this expression for any Anchor.Sector and integer rung; for the muon it evaluates at rung 13. The constant m_mu_exp imports the PDG value 105.6583755 MeV as an external datum. The supporting theorem muon_pred_eq equates rs_mass_MeV .Lepton 13 to the named term muon_pred, while muon_mass_bounds supplies the concrete interval (101.49, 101.57) MeV that contains the prediction.
proof idea
Rewrite the left-hand side via muon_pred_eq to obtain an expression in muon_pred. Invoke muon_mass_bounds to obtain the two-sided numerical inequality on muon_pred. Establish positivity of m_mu_exp, rewrite the relative-error claim into an absolute-value inequality, and apply nlinarith to the bounds together with the experimental value.
why it matters in Recognition Science
The result is applied verbatim by row_muon_pct, by mass_verification_cert_exists (to populate the muon slot of the verification certificate), and by phi_ladder_verified (to confirm the full lepton ladder within the stated tolerances). It therefore supplies the muon datum that supersedes the earlier mass_ladder_assumption and supports the claim that the phi-ladder reproduces observed lepton masses to a few percent.
scope and limits
- Does not derive the experimental muon mass from Recognition Science axioms.
- Does not assert an exact numerical match or a tolerance tighter than 4 percent.
- Does not extend to quark masses or other sectors.
- Does not incorporate radiative corrections or scale dependence.
Lean usage
theorem row_muon_pct : |rs_mass_MeV .Lepton 13 - m_mu_exp| / m_mu_exp < 0.04 := muon_relative_errorformal statement (Lean)
132theorem muon_relative_error :
133 |rs_mass_MeV .Lepton 13 - m_mu_exp| / m_mu_exp < 0.04 := by
proof body
Tactic-mode proof.
134 rw [muon_pred_eq]
135 have hb := muon_mass_bounds
136 have hexp_pos : (0 : ℝ) < m_mu_exp := by unfold m_mu_exp; norm_num
137 rw [div_lt_iff₀ hexp_pos, abs_lt]
138 unfold m_mu_exp
139 constructor <;> nlinarith [hb.1, hb.2]
140
141/-! ## Tau Mass Verification -/
142