Complexity, Randomness, and First-Occurrence Positions: A Unified Theory

This research note develops a series of results connecting the first-occurrence positions of prefixes of a real α inside a Martin-Löf random sequence to the Kolmogorov complexity of those prefixes. A universal complexity-occurrence inequality is proved, yielding consequences for mutual information, early-occurrence thresholds, and the impossibility of compressing high-complexity prefixes into subcritical exponential windows. The note also establishes sharp asymptotics for computable reals and characterizes oracle-use density via effective Hausdorff dimension, forming a partial embedding-capacity theory. A final section outlines conjectural extensions involving packing dimension and embedding exponents.

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