Two Preliminary Pre-Tests for i.i.d.-Like Digit Structure

This paper introduces two preliminary pre-tests for the local i.i.d.-like structure of mathematical constants and other deterministic digit sequences. The first method is the (Z_n) occupancy test. For a block of length (n) over an alphabet of (n) symbols, (Z_n) counts how many symbols attain the maximum frequency. The paper derives the exact i.i.d. reference distribution through a complete occupancy-profile formula and explains the characteristic non-monotone hump structure of the resulting curves. The second method is Shannon block entropy. Exact finite-block i.i.d. expectations are derived for several matched alphabet and block sizes. Numerical comparisons indicate a stable high-entropy level near (4.19) bits in the base-32 setting for sequences such as (\pi), (e), (\sqrt{2}), (\sqrt{3}), (\sqrt{5}), and several deterministic recursive concatenations. Other structured constructions show markedly lower entropy levels. A central methodological example is the base-32 Champernowne concatenation. Although it is normal in the classical sense, it strongly fails both pre-tests on accessible finite ranges. This demonstrates that normality does not imply finite-range i.i.d.-like behaviour. Conversely, agreement with the (Z_n) reference distribution and convergence of block entropy provide a stronger indication of local i.i.d.-like structure than matching one-symbol frequencies alone, while still constituting neither a proof of independence nor a general criterion for randomness or normality. The paper includes a freely usable and modifiable Python 3 script. The script generates one million decimal places of (\pi), evaluates (Z_{10}) and block entropy across all offsets, writes detailed observed-versus-expected CSV files, calculates chi-square statistics, and produces logarithmic comparison plots. Readers are invited to adapt the code to (\sqrt{2}), Khinchin's constant, the Apéry constant, or other mathematical constants. Artificial intelligence was used as a research and writing assistant for organization, language editing, symbolic and numerical checking, and preparation of plots. The author assessed the results and assumes responsibility for the final content and conclusions.