Rigid Factorial Obstructions and Exact Counting Sequences

We study the prime factors that survive cancellation of integers by factorials. In aspecified high-degree range, we classify every exceptional input by two rigid factorizationsand derive exact counting and multiplicity formulas. We obtain prime-countingexpressions, a transition to pure prime powers, and a density-zero theorem for mixedexceptions in fixed critical windows. All the main results are proved by elementaryestimates and a periodic sieve.