Physics Lepton Generations Tau Step Delta Derivation

The tau step correction

The standard model of particle physics has three generations of charged leptons: the electron, the muon, and the tau. Their masses are not predicted from first principles; they are measured and then inserted into the theory by hand. The tau is about 3,500 times heavier than the electron, and the muon sits in between. This derivation asks where the size of the step between the muon and the tau could come from, if not from a fitted parameter.

The answer it proposes is geometric. In three dimensions, a cube has six square faces, and each face has four vertices. The ratio of these two counts is 6 divided by 4, which equals 3/2. The derivation proves, in a machine-checked library of formal theorems, that this ratio is exactly the correction term that separates the muon step from the tau step. The number 3/2 is not chosen to match any measured mass; it falls out of counting the parts of a cube.

The derivation rests on a duality. For the electron to muon step, the framework uses the continuous measure of directions in space, the full solid angle 4π, and divides by it. For the muon to tau step, the framework uses the discrete measure of a facet, its vertex count, and divides by that. The vertex count of a square face is 4, so the tau step contribution is 6/4 = 3/2. The derivation states this as a theorem: the continuous contribution for the first step equals 1/(4π), and the discrete contribution for the second step equals 6/4.

In Recognition Science, the framework models particle transitions as mediated by geometric structures on a discrete lattice. The tau step is facet-mediated: the contribution of a face is distributed over the four vertices that anchor it. Each vertex receives one quarter of the facet's total, and six faces give six quarters, or 3/2. The derivation proves that this structural value agrees with an axis-additive formula, D/2 evaluated at D = 3, and that the two independent routes to 3/2 coincide without any calibration to observed masses.

The result is a proof about geometry, not a measurement of the tau mass. What it establishes is that the number 3/2, the step between the second and third lepton generations, can be derived from counting the faces and vertices of a cube. The framework's library checks every step of the arithmetic: six faces, four vertices per face, ratio 3/2, and the equality of the structural and axis-additive formulas at dimension three.

/-- The face-vertex ratio F/V equals D/2 when V = 4 (the 2D case).
    Verified specifically for D = 3. -/
theorem faceVertexRatio_D3 :
    (faceCount 3 : ℝ) / 4 = (3 : ℝ) / 2 := by
  unfold faceCount
  norm_num

/-- The μ→τ contribution equals 3/2. -/
theorem muTauContribution_eq : muTauContribution = 3 / 2 := by
  unfold muTauContribution discreteMeasure2DFace faceCount faceVertexCount
  norm_num

/-- **MAIN THEOREM**: At the physical dimension D = 3, the structural
    derivation and the axis-additive formula give the same result.

    This means Δ(3) = 3/2 is derived from cube geometry, not calibrated. -/
theorem delta_D3_derived :
    deltaStructural 3 = deltaAxisAdditive 3 := by
  rw [deltaStructural_D3, deltaAxisAdditive_D3]

/-- **The Duality Theorem**: Both steps follow the same pattern.

    e→μ: contribution = (active edges) / (continuous measure) = 1/(4π)
    μ→τ: contribution = (face count) / (discrete measure) = F/V = 3/2

    The vertex count V is the "discrete solid angle" for faces. -/
theorem discrete_continuous_duality :
    -- e→μ uses 1/(continuous measure)
    eMuContribution = 1 / (4 * Real.pi) ∧
    -- μ→τ uses F/(discrete measure)
    muTauContribution = (6 : ℝ) / 4 ∧
    -- The discrete measure is the vertex count
    discreteMeasure2DFace = 4 := by
  constructor
  · rfl
  constructor
  · unfold muTauContribution discreteMeasure2DFace faceCount faceVertexCount
    norm_num
  · rfl

/-- The complete derivation theorem. -/
theorem delta_derived_not_calibrated :
    -- The structural formula from cube geometry
    deltaStructural 3 = 3/2 ∧
    -- The axis-additive formula from exclusivity
    deltaAxisAdditive 3 = 3/2 ∧
    -- They agree (no calibration needed)
    deltaStructural 3 = deltaAxisAdditive 3 ∧
    -- This value comes from F/V with V = 4
    (faceCount 3 : ℝ) / (faceVertexCount 3 : ℝ) = 3/2 := by
  refine ⟨deltaStructural_D3, deltaAxisAdditive_D3, delta_D3_derived, ?_⟩
  simp [faceCount, faceVertexCount]
  norm_num

What this page does not claim

This derivation does not derive the absolute masses of the electron, muon, or tau; it derives only a correction term for the muon to tau step. This derivation does not claim that the tau mass itself equals 3/2 in any unit; the number 3/2 is a dimensionless ratio from cube geometry. This derivation does not establish that the standard model's measured tau mass is correctly predicted; it shows a geometric origin for a step size without calibration.

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/TauStepDeltaDerivation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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