Foundation Electron Mass From Phi Ladder Electron Muon Ratio Rs Band

The electron-muon ratio band

The electron to muon mass ratio is one of the most precisely known numbers in physics, about 206.77. In the Recognition Science framework, masses sit on a ladder of powers of the golden ratio phi, roughly 1.618. The framework's machine-checked library of formal theorems defines an electron-muon ratio as phi^6, and a proved theorem places that value between 17.9 and 18.0. That is the entire content of the declaration: a narrow numerical band, derived from a structural assumption about how masses are spaced, not a match to measurement.

The framework models the electron and muon as sitting on rungs of a lattice, with the electron at rung 8 and the muon at rung 14, a gap of 6. The ratio of their masses is then phi raised to that gap, phi^6. The theorem electron_muon_ratio_RS_band proves the band 17.9 < phi^6 < 18.0 using only the definition of phi and its basic algebraic properties. The measured ratio, 206.77, falls between phi^11 and phi^12, with the nearest power, phi^11, about 3.8 percent away. The framework names a falsifier: a precision measurement placing the ratio outside the phi^k ladder for any integer k from 1 to 12 by more than about 0.118 on the log-mass scale would refute the structural claim.

The gap between the predicted band and the measured value is large, roughly a factor of 11.5. The framework does not claim this gap is closed. The docstring states plainly that the phi^6 prediction is too low, and the dimensional bridge that selects the correct rung k is the named follow-on, an open target. What the theorem establishes is internal consistency: given the rung assignment, the ratio is exactly phi^6 and lies in the proved band. The assignment itself is a modeling choice, not a derived fact.

In Recognition Science, the electron mass itself is defined as phi^3, which the library proves lies between 4.22 and 4.24 in coherence-energy units, a scale where the coherence energy is phi^(-5). These are framework-native numbers, not SI values. The page's classical content ends here; the framework turn is one labeled section within a larger encyclopedia entry on the electron-muon mass ratio.

/-- Numerical band: `phi^6 ∈ (17.9, 18.0)`.
`phi^6 = (phi^3)^2 = (2 phi + 1)^2 = 4 phi^2 + 4 phi + 1
  = 4(phi + 1) + 4 phi + 1 = 8 phi + 5`.
With `1.61 < phi < 1.62`, `17.88 < 8 phi + 5 < 17.96`. -/
theorem electron_muon_ratio_RS_band :
    17.9 < electron_muon_ratio_RS ∧ electron_muon_ratio_RS < 18.0 := by
  unfold electron_muon_ratio_RS
  have h1 := phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  have hsq := phi_sq_eq
  have : phi ^ (6 : ℕ) = (phi ^ (3 : ℕ)) ^ 2 := by ring
  rw [this]
  have hcube : phi ^ (3 : ℕ) = phi * (phi + 1) := by
    have : phi ^ (3 : ℕ) = phi * phi ^ 2 := by ring
    rw [this, hsq]
  rw [hcube]
  refine ⟨?_, ?_⟩ <;> nlinarith [hsq]

/-- The electron rung on the full recognition lattice: 8 (the 8-tick
octave boundary, matching T7). -/
def r_electron : ℕ := 8

/-- The muon rung on the full lattice: 14 (= 8 + 6, second generation). -/
def r_muon : ℕ := 14

/-- Numerical band: `phi^6 ∈ (17.9, 18.0)`.
`phi^6 = (phi^3)^2 = (2 phi + 1)^2 = 4 phi^2 + 4 phi + 1
  = 4(phi + 1) + 4 phi + 1 = 8 phi + 5`.
With `1.61 < phi < 1.62`, `17.88 < 8 phi + 5 < 17.96`. -/
theorem electron_muon_ratio_RS_band :
    17.9 < electron_muon_ratio_RS ∧ electron_muon_ratio_RS < 18.0 := by
  unfold electron_muon_ratio_RS
  have h1 := phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  have hsq := phi_sq_eq
  have : phi ^ (6 : ℕ) = (phi ^ (3 : ℕ)) ^ 2 := by ring
  rw [this]
  have hcube : phi ^ (3 : ℕ) = phi * (phi + 1) := by
    have : phi ^ (3 : ℕ) = phi * phi ^ 2 := by ring
    rw [this, hsq]
  rw [hcube]
  refine ⟨?_, ?_⟩ <;> nlinarith [hsq]

/-- Numerical band: `phi^6 ∈ (17.9, 18.0)`.
`phi^6 = (phi^3)^2 = (2 phi + 1)^2 = 4 phi^2 + 4 phi + 1
  = 4(phi + 1) + 4 phi + 1 = 8 phi + 5`.
With `1.61 < phi < 1.62`, `17.88 < 8 phi + 5 < 17.96`. -/
theorem electron_muon_ratio_RS_band :
    17.9 < electron_muon_ratio_RS ∧ electron_muon_ratio_RS < 18.0 := by
  unfold electron_muon_ratio_RS
  have h1 := phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  have hsq := phi_sq_eq
  have : phi ^ (6 : ℕ) = (phi ^ (3 : ℕ)) ^ 2 := by ring
  rw [this]
  have hcube : phi ^ (3 : ℕ) = phi * (phi + 1) := by
    have : phi ^ (3 : ℕ) = phi * phi ^ 2 := by ring
    rw [this, hsq]
  rw [hcube]
  refine ⟨?_, ?_⟩ <;> nlinarith [hsq]

What this page does not claim

The framework does not claim the phi^6 prediction matches the measured ratio of 206.77. The framework does not derive the rung assignments 8 and 14 from first principles; they are modeling choices. The framework does not claim the dimensional bridge that selects the correct k has been found.

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/Foundation/ElectronMassFromPhiLadder.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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