Beyond the Cube: Overlapping Grid Methods for Debris Collision Risk Assessment

The cube method reduces conjunction screening in orbital debris simulations to $\mathcal{O}(N)$ cost by evaluating only object pairs sharing the same grid cell at each snapshot, but systematically assigns zero collision probability to pairs separated by a cell boundary at that epoch, a failure known as boundary blindness. This paper introduces the Double Cube (DC) method, which recovers boundary-crossing conjunctions through a spatially shifted secondary grid using bin-index lookup alone, preserving $\mathcal{O}(N)$ complexity. Validated across 8,000 Monte Carlo seeds, DC reduces the blindness rate from $\beta_{\mathrm{Cube}} = 9.70\%$ to $\beta_{\mathrm{DC}} = 4.21\%$; a synchronized experiment confirms the residual is temporal in origin by reaching exactly $0.00\%$. Removing blindness reveals a systematic per-pair overestimation in the cube formula that blind zero-probability assignments had been masking, suppressing the overall predicted collision rate below the true rate. Two independent corrections are derived and validated: a power-law correction motivated by the Direct Simulation Monte Carlo kinetic theory analogy reduces the calibration error from $12.9\%$ to $1.9\%$ at $k = 1$ and $4.0\%$ at $k = 2$, bracketing perfect calibration from opposite sides; a parameter-free Gaussian correction derived from the pair-distance distribution geometry achieves a residual of $0.08\%$. Both corrections have been implemented in MOCAT-MC.