 205-/
 206
 207/-- The continuous measure in 3D: the solid angle 4π. -/
 208noncomputable def continuousMeasure3D : ℝ := 4 * Real.pi
 209
 210/-- The discrete measure on a 2D face: the vertex count V = 4. -/
 211def discreteMeasure2DFace : ℕ := faceVertexCount 3
 212
 213/-- Verify the discrete measure equals 4 (the vertex count of a square). -/
 214theorem discreteMeasure_eq_4 : discreteMeasure2DFace = 4 := faceVertexCount_D3
 215
 216/-- The e→μ fractional contribution: 1 / (continuous measure).
 217    This is 1/(4π), the same as in FractionalStepDerivation.lean. -/
 218noncomputable def eMuContribution : ℝ := 1 / continuousMeasure3D
 219
 220/-- The μ→τ normalized contribution: F / (discrete measure).
 221    Each face contributes, normalized by its vertex anchoring points. -/
 222noncomputable def muTauContribution : ℝ :=
 223  (faceCount 3 : ℝ) / (discreteMeasure2DFace : ℝ)
 224
 225/-- The μ→τ contribution equals 3/2. -/
 226theorem muTauContribution_eq : muTauContribution = 3 / 2 := by
 227  unfold muTauContribution discreteMeasure2DFace faceCount faceVertexCount
 228  norm_num
 229
 230/-- **The Duality Theorem**: Both steps follow the same pattern.
 231
 232    e→μ: contribution = (active edges) / (continuous measure) = 1/(4π)
 233    μ→τ: contribution = (face count) / (discrete measure) = F/V = 3/2
 234
 235    The vertex count V is the "discrete solid angle" for faces. -/
 236theorem discrete_continuous_duality :
 237    -- e→μ uses 1/(continuous measure)
 238    eMuContribution = 1 / (4 * Real.pi) ∧

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