Removing the Large-M Assumption from FD Diversity Bounds
Testing whether diversity holds for M < 9/(2p(1-p)) in D > 1 for finite-difference Schrodinger operators with Bernoulli potentials.
Diversity Probability by Dimension (Below Threshold M)
Effect of Sample Size N (M=2, D=2)
Dimension Comparison
Summary: Below-Threshold Results
D
Mean Prob.
Std
Min Prob.
Max Prob.
1
0.953
0.029
0.950
1.000
2
0.898
0.074
0.860
1.000
3
0.753
0.041
0.700
0.790
Diversity holds with substantial probability (>0.70) even well below the theoretical threshold M_thresh = 28. In D=1, the probability exceeds 0.95. Increasing sample size N dramatically compensates for small grid sizes. These results strongly support the conjecture that the large-M assumption is a proof artifact.