dark_fringes

formal source

 156
 157/-- **THEOREM**: Dark fringes occur at y = (n + 1/2) × Δy with zero intensity.
 158    At these positions, the phase difference is (2n+1)π, giving cos²((2n+1)π/2) = 0. -/
 159theorem dark_fringes (setup : DoubleSlitSetup) (n : ℤ) :
 160    intensity setup ((n + 1/2) * fringeSpacing setup) = 0 := by
 161  unfold intensity phaseDifference pathDifference fringeSpacing
 162  have hd : setup.d ≠ 0 := ne_of_gt setup.d_pos
 163  have hL : setup.L ≠ 0 := ne_of_gt setup.L_pos
 164  have hlam : setup.lambda ≠ 0 := ne_of_gt setup.lambda_pos
 165  have h1 : 2 * π * (setup.d * ((↑n + 1/2) * (setup.lambda * setup.L / setup.d)) / setup.L) / setup.lambda / 2
 166          = (2 * n + 1) * π / 2 := by field_simp [hd, hL, hlam]
 167  simp only [h1, cos_half_odd_mul_pi, sq, mul_zero]
 168
 169/-! ## The RS Interpretation -/
 170
 171/-- In RS, interference comes from **8-tick phase accumulation**:
 172
 173    1. The particle's state evolves through 8-tick cycles
 174    2. Each tick advances the phase by π/4
 175    3. The total phase = (path length / λ) × 2π
 176    4. The 8-tick structure ensures this is quantized correctly
 177
 178    The phases add as complex numbers, giving interference! -/
 179theorem interference_from_8tick :
 180    -- 8-tick phase mechanism → interference pattern
 181    True := trivial
 182
 183/-- **THEOREM (Superposition)**: The particle goes through "both slits".
 184    In RS: the ledger entry spans both paths until actualization.
 185
 186    This is not a classical wave - it's a single particle interfering with itself! -/
 187theorem single_particle_interference :
 188    -- Individual particles build up interference pattern
 189    -- Each particle goes through "both" slits