Construct a first-digit distribution

The two midpoint samples each contain 900 deterministic values between 1 and 10. A separate mode lists every integer from 1 through 999. Choose the spacing rule. This illustrates a mathematical construction, not a randomness or fraud test.

The logarithmic sample is 10^((i+0.5)/900), i=0…899. The ordinary-value sample is 1 + 9×(i+0.5)/900. The integer list contains 1…999. Finite proportions are counted; Benford’s theoretical column is log10(1+1/d).