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A new formal theorem, verified in Lean, shows that for thresholds tending to infinity, almost every positive integer falls below the threshold within 436 log N Collatz steps, strengthening Terence Tao's earlier result with explicit bounds and natural density.
Unicode's transliteration rules (UTS #35) are proven to be Turing-complete by compiling 2-tag systems, showing termination is undecidable. This result affects the ICU library used in many systems.