Foundation Electron Mass From Phi Ladder Electron Mass One Statement

The one-statement theorem

The electron is the lightest charged fermion in the Standard Model, with a measured mass of about 0.511 MeV. Recognition Science (RS) arranges particle masses on a ladder where each step multiplies a base energy by the golden ratio φ ≈ 1.618. The declaration electron_mass_one_statement is a single theorem in the framework's machine-checked library of formal theorems that bundles four facts about where the electron sits on that ladder.

The theorem states that the RS-native electron mass equals φ³, a number between 4.22 and 4.24 in the framework's coherence-energy units. It also states that the electron occupies rung 8 of the full recognition lattice, and that the rung gap to the muon is 6. From that gap, the theorem derives the RS-native electron-to-muon mass ratio as φ⁶, which lies between 17.9 and 18.0. These are structural statements: they describe positions and ratios on the ladder, not absolute masses in kilograms.

The theorem does not claim that the RS-native value matches the measured electron mass. The measured electron-to-muon mass ratio is 206.77, while φ⁶ is about 17.94, a factor of roughly 11.5 lower. The framework's own docstring names a falsifier: a precision lepton-mass measurement placing the electron-to-muon ratio outside the φ^k ladder for any integer k from 1 to 12 by more than J(φ) ≈ 0.118 on the log-mass scale. The current value sits between φ¹¹ ≈ 199.0 and φ¹² ≈ 321.8, with the nearest power φ¹¹ about 3.8% off.

What the theorem does establish is a precise, internally consistent placement: the electron at rung 8, the muon six rungs higher, and the resulting ratio φ⁶. The dimensional bridge that would select the correct power of φ to match the measured 206.77 remains a named follow-on, not a proved result. In RS terms, the ladder structure is proved; the link from that structure to the measured lepton masses is an open target.

theorem electron_mass_RS_eq_phi_cubed :
    electron_mass_RS = phi ^ (3 : ℕ) := rfl

/-- Numerical band: `phi^3 ∈ (4.22, 4.24)`.
Proof: `phi^3 = phi · phi^2 = phi · (phi + 1) = phi^2 + phi = 2 phi + 1`.
With `1.61 < phi < 1.62`, we get `4.22 < 2 phi + 1 < 4.24`. -/
theorem electron_mass_RS_band :
    4.22 < electron_mass_RS ∧ electron_mass_RS < 4.24 := by
  unfold electron_mass_RS
  have h1 := phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  have hsq := phi_sq_eq -- phi^2 = phi + 1
  have : phi ^ (3 : ℕ) = phi * phi ^ 2 := by ring
  rw [this, show phi ^ 2 = phi + 1 from hsq]
  refine ⟨?_, ?_⟩ <;> nlinarith

/-- 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]

/-- **ELECTRON MASS FROM φ-LADDER: ONE-STATEMENT THEOREM.**

The electron sits at rung 8 on the recognition lattice with RS-native
mass `phi^3 ∈ (4.22, 4.24)`; the electron-muon rung gap is 6 with
mass ratio `phi^6 ∈ (17.9, 18.0)`. -/
theorem electron_mass_one_statement :
    electron_mass_RS = phi ^ (3 : ℕ) ∧
    (4.22 < electron_mass_RS ∧ electron_mass_RS < 4.24) ∧
    electron_muon_rung_gap = 6 ∧
    (17.9 < electron_muon_ratio_RS ∧ electron_muon_ratio_RS < 18.0) :=
  ⟨electron_mass_RS_eq_phi_cubed, electron_mass_RS_band,
   electron_muon_rung_gap_eq, electron_muon_ratio_RS_band⟩

What this page does not claim

The theorem does not claim the RS-native mass equals the measured electron mass in conventional units. The theorem does not claim the electron-to-muon ratio φ⁶ matches the measured ratio of 206.77. The theorem does not identify which power of φ corresponds to the measured lepton masses.

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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