Foundation Electron Mass From Phi Ladder Electron Muon Rung Gap Eq

The electron-muon gap

The golden ratio φ, the number satisfying φ² = φ + 1, is about 1.618. In Recognition Science, a framework that treats reality as maintaining a discrete record of events, particle masses sit on a ladder: each step multiplies a mass by φ. The electron occupies rung 8 of this lattice, and the muon, the electron's heavier cousin, sits at rung 14. The gap between them is 6 rungs, a fact the machine-checked declaration electron_muon_rung_gap_eq proves by simple arithmetic.

The framework's structural prediction for the mass ratio is φ⁶, which lies between 17.9 and 18.0. The measured electron-to-muon mass ratio is 206.77, a number any physicist can look up. The framework's value is too low by a factor of about 11.5. This is the key empirical test of the phi-ladder idea, and it fails.

In Recognition Science, the library of formal theorems proves the rung gap is 6 and the ratio is φ⁶, but it does not prove that this ratio matches nature. The ledger, the discrete record of recognition events, fixes the structure; it does not fix which rung the muon occupies. The framework's own falsifier states this plainly: a precision measurement placing the ratio outside any integer power of φ between φ¹ and φ¹² by more than about 0.118 on the log scale would refute the ladder. The measured value 206.77 sits between φ¹¹ ≈ 199.0 and φ¹² ≈ 321.8, nearest to φ¹¹ at 3.8% off. The dimensional bridge that would select the correct rung is named as the open follow-on, not a proved result.

What the declaration establishes is internal consistency: given the electron at rung 8 and the muon at rung 14, the gap is 6 and the ratio is φ⁶. It is a theorem about the framework's own definitions, not a statement about the physical world. The electron mass itself is φ³ in coherence-energy units, between 4.22 and 4.24, another structural result with no claim to match the measured 0.511 MeV. The framework offers a coherent picture of mass quantization, but the muon gap is where that picture meets measurement and comes up short.

theorem electron_muon_rung_gap_eq : electron_muon_rung_gap = 6 := by
  native_decide

/-- 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 electron-muon mass ratio is not predicted to match measurement. The framework does not derive the muon's rung from first principles. This theorem does not connect the framework's mass units to kilograms.

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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