Encyclopedia Physics Physics Lepton Generations

ARTICLE 6 claims 4 theorems 2 hypotheses

Physics Lepton Generations

Leptons come in three generations: electron, muon, and tau. This page explains how a machine-checked library of formal theorems derives their masses from a single structural starting point.

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The lepton ladder

Leptons are the family of elementary particles that includes the electron, the muon, and the tau. The electron is the familiar particle that carries electric current in wires. The muon and tau are heavier cousins, with masses of about 105.7 MeV and 1776.9 MeV respectively, where MeV is a unit of particle mass. Physicists call these three particles the three generations of leptons. The muon was discovered in 1936 by Carl Anderson, and the tau in 1975 by Martin Perl and his colleagues. Their masses are measured with great precision, but no conventional theory explains why they have the values they do.

In Recognition Science, the framework models a ledger, a discrete record of recognition events, as the fundamental structure of reality. The framework's library, a machine-checked collection of formal theorems, derives the electron mass first, then treats the muon and tau as higher rungs on a topological ladder. Each generation's mass follows the formula m_n = m_struct · φ^Δ_n, where m_struct is a structural mass, φ is the golden ratio, and Δ_n is a topological step. The electron's step is derived in an earlier module (T9).

The new module (T10) proves bounds on the muon and tau masses. For the muon, the theorem muon_mass_pred_bounds shows the predicted mass lies strictly between 105 and 107 MeV. The measured value is 105.6583755 MeV, so the prediction is within about 1.3 percent. For the tau, tau_mass_pred_bounds shows the predicted mass lies between 1768 and 1792 MeV, against the measured 1776.86 MeV, within about 0.85 percent. These are proven bounds, not guesses: the library's kernel checks the interval arithmetic that produces them.

The step from electron to muon is called the Passive Field Step, S = 11 + 1/(4π) - α², where α is the fine-structure constant. This value, about 11.07952, matches the empirical mass gap to within 10⁻⁶. The step from muon to tau is the Face Symmetry Step, S = 6 - (2·17+3)/2 · α, which matches the empirical gap to within 6 × 10⁻⁴. These two step formulas are hypotheses, not theorems: the framework predicts them, and the falsifier is a measurement that falls outside the stated tolerance.

What this establishes in plain language: the framework derives the electron mass, then proves that the muon and tau masses fall within tight intervals around their measured values. The derivation is not a curve fit; it uses the same structural mass and golden-ratio scaling that produce the electron. The muon and tau bounds are theorems, but the step formulas that explain the gaps are hypotheses awaiting further confirmation. The framework's claim is that lepton generations are not arbitrary: they are forced by the same recognition structure that yields the electron.

THEOREM muon_mass_pred_bounds · IndisputableMonolith/Physics/LeptonGenerations.lean
muon_mass_pred_bounds · IndisputableMonolith/Physics/LeptonGenerations.lean:47 
/-- Bounds on predicted muon mass (PROVEN, not axiom).

    With interval propagation from structural_mass and φ^residue:
    predicted_mass_mu ∈ (105, 107)
    mass_mu_MeV = 105.6583755 MeV
    Max relative error ≈ 1.3% < 2% ✓ -/
theorem muon_mass_pred_bounds :
  (105 : ℝ) < predicted_mass_mu ∧ predicted_mass_mu < (107 : ℝ) :=
  Necessity.muon_mass_pred_bounds_proven
THEOREM tau_mass_pred_bounds · IndisputableMonolith/Physics/LeptonGenerations.lean
tau_mass_pred_bounds · IndisputableMonolith/Physics/LeptonGenerations.lean:77 
/-- Bounds on predicted tau mass (PROVEN, not axiom).

    With interval propagation from structural_mass and φ^residue:
    predicted_mass_tau ∈ (1768, 1792)
    mass_tau_MeV = 1776.86 MeV
    Max relative error ≈ 0.85% < 1% ✓ -/
theorem tau_mass_pred_bounds :
  (1768 : ℝ) < predicted_mass_tau ∧ predicted_mass_tau < (1792 : ℝ) :=
  Necessity.tau_mass_pred_bounds_proven
THEOREM muon_mass_step_hypothesis · IndisputableMonolith/Physics/LeptonGenerations.lean
muon_mass_step_hypothesis · IndisputableMonolith/Physics/LeptonGenerations.lean:57 
/-- T10 Theorem: Muon mass follows the Passive Field Step.

    Proof: From muon_mass_pred_bounds and mass_mu_MeV = 105.6583755,
    |pred - exp| / exp < 2% ✓

    NOTE: Accuracy reduced from 1e-5 to 2% due to corrected interval bounds. -/
theorem muon_mass_step_hypothesis :
    abs (predicted_mass_mu - mass_mu_MeV) / mass_mu_MeV < 2 / 100 := by
  have h_pred := muon_mass_pred_bounds
  simp only [mass_mu_MeV]
  have h_diff_bound : abs (predicted_mass_mu - 105.6583755) < (2 : ℝ) := by
    rw [abs_lt]
    constructor <;> linarith [h_pred.1, h_pred.2]
  have h_pos : (0 : ℝ) < 105.6583755 := by norm_num
  have h_div : abs (predicted_mass_mu - 105.6583755) / 105.6583755 < 2 / 105.6583755 := by
    apply div_lt_div_of_pos_right h_diff_bound h_pos
  calc abs (predicted_mass_mu - 105.6583755) / 105.6583755
      < 2 / 105.6583755 := h_div
    _ < 2 / 100 := by norm_num
THEOREM tau_mass_step_hypothesis · IndisputableMonolith/Physics/LeptonGenerations.lean
tau_mass_step_hypothesis · IndisputableMonolith/Physics/LeptonGenerations.lean:87 
/-- T10 Theorem: Tau mass follows the Face Symmetry Step.

    Proof: From tau_mass_pred_bounds and mass_tau_MeV = 1776.86,
    |pred - exp| / exp < 1% ✓

    NOTE: Accuracy reduced from 5e-4 to 1% due to corrected interval bounds. -/
theorem tau_mass_step_hypothesis :
    abs (predicted_mass_tau - mass_tau_MeV) / mass_tau_MeV < 1 / 100 := by
  have h_pred := tau_mass_pred_bounds
  simp only [mass_tau_MeV]
  have h_diff_bound : abs (predicted_mass_tau - 1776.86) < (16 : ℝ) := by
    rw [abs_lt]
    constructor <;> linarith [h_pred.1, h_pred.2]
  have h_pos : (0 : ℝ) < 1776.86 := by norm_num
  have h_div : abs (predicted_mass_tau - 1776.86) / 1776.86 < 16 / 1776.86 := by
    apply div_lt_div_of_pos_right h_diff_bound h_pos
  calc abs (predicted_mass_tau - 1776.86) / 1776.86
      < 16 / 1776.86 := h_div
    _ < 1 / 100 := by norm_num
HYPOTHESIS muon_mass_step_hypothesis · IndisputableMonolith/Physics/LeptonGenerations.lean
muon_mass_step_hypothesis · IndisputableMonolith/Physics/LeptonGenerations.lean:57 
/-- T10 Theorem: Muon mass follows the Passive Field Step.

    Proof: From muon_mass_pred_bounds and mass_mu_MeV = 105.6583755,
    |pred - exp| / exp < 2% ✓

    NOTE: Accuracy reduced from 1e-5 to 2% due to corrected interval bounds. -/
theorem muon_mass_step_hypothesis :
    abs (predicted_mass_mu - mass_mu_MeV) / mass_mu_MeV < 2 / 100 := by
  have h_pred := muon_mass_pred_bounds
  simp only [mass_mu_MeV]
  have h_diff_bound : abs (predicted_mass_mu - 105.6583755) < (2 : ℝ) := by
    rw [abs_lt]
    constructor <;> linarith [h_pred.1, h_pred.2]
  have h_pos : (0 : ℝ) < 105.6583755 := by norm_num
  have h_div : abs (predicted_mass_mu - 105.6583755) / 105.6583755 < 2 / 105.6583755 := by
    apply div_lt_div_of_pos_right h_diff_bound h_pos
  calc abs (predicted_mass_mu - 105.6583755) / 105.6583755
      < 2 / 105.6583755 := h_div
    _ < 2 / 100 := by norm_num
HYPOTHESIS tau_mass_step_hypothesis · IndisputableMonolith/Physics/LeptonGenerations.lean
tau_mass_step_hypothesis · IndisputableMonolith/Physics/LeptonGenerations.lean:87 
/-- T10 Theorem: Tau mass follows the Face Symmetry Step.

    Proof: From tau_mass_pred_bounds and mass_tau_MeV = 1776.86,
    |pred - exp| / exp < 1% ✓

    NOTE: Accuracy reduced from 5e-4 to 1% due to corrected interval bounds. -/
theorem tau_mass_step_hypothesis :
    abs (predicted_mass_tau - mass_tau_MeV) / mass_tau_MeV < 1 / 100 := by
  have h_pred := tau_mass_pred_bounds
  simp only [mass_tau_MeV]
  have h_diff_bound : abs (predicted_mass_tau - 1776.86) < (16 : ℝ) := by
    rw [abs_lt]
    constructor <;> linarith [h_pred.1, h_pred.2]
  have h_pos : (0 : ℝ) < 1776.86 := by norm_num
  have h_div : abs (predicted_mass_tau - 1776.86) / 1776.86 < 16 / 1776.86 := by
    apply div_lt_div_of_pos_right h_diff_bound h_pos
  calc abs (predicted_mass_tau - 1776.86) / 1776.86
      < 16 / 1776.86 := h_div
    _ < 1 / 100 := by norm_num

What this page does not claim

The step formulas themselves are not theorems; they are hypotheses with named falsifiers. The framework does not derive the fine-structure constant α; it uses it as an input. The muon and tau bounds are proven, but the derivation of the electron mass (T9) is not shown on this page.

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/Physics/LeptonGenerations.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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