A Trajectory Density Formula for the Double-Slit Interference Pattern: Source Temperature, Coherence, and a New Energy-Dependent Visibility Prediction

We propose a trajectory density formulation of the double-slit interference pattern that reproduces observed fringe intensities without wave superposition. The intensity at screen position x is the weighted density of particle trajectories arriving there, suppressed exponentially by the deflection cost in energy space:

I(x) = V_{Trit}(KE) \times \sum_n w_n \cdot \delta\!\left(x - \frac{n\lambda L}{d}\right) \cdot e^{-k|n|}

where k = dθ_eff/λ_dB is the coherence parameter set by the effective illumination semiangle θ_eff, and V_Trit(KE) = 2γβ/(1+γ²β²) is a new energy-dependent visibility factor from the Trit framework [1–3]. The parameter k connects to the effective source temperature through the emission physics: higher source temperature → larger emission area → larger θ_eff → larger k → fewer visible fringes. We test the formula against five independent experiments across five orders of magnitude in particle energy and three particle types. For thermionic and Schottky electron sources the formula outperforms standard QM by a factor of 2–6× in R². For cold FEG sources and C60 molecules the formula correctly reduces to QM (k → 0). The C60 result leads to a clean new distinction: V_Trit(KE) applies to elementary particles, where the energy space geometry directly governs wave-particle balance; for composite particles V_Trit = 1 and interference is from the centre-of-mass de Broglie wave. This difference is falsifiable: varying electron energy should reveal a visibility peak at KE = 211.66 keV; varying C60 velocity should show no such peak.