Divisibility Thresholds, Binomial Counts, and Record Growth in Pascal's Triangle

We consider the earliest binomial coefficient divisible by a variable integer, counts of divisible positions in a truncated Pascal triangle, and several associated threshold, equality, and greatest-common-divisor problems. A prime-power characterization of the earliest qualifying row and exact formulas for products of two distinct primes lead to a classification of semiprime binomial-coefficient values and a density-one theorem for the indices at which the earliest divisible coefficient exceeds the modulus. For a moving modulus we give exact prime-power counts, a carry-based finite-state algorithm, and families along which the proportion of divisible positions tends to one, whereas it tends to zero along primes. The running maxima of the earliest-divisibility quotient exceed every fixed power of their argument; a corresponding logarithmic restriction is obtained for successive strict-record indices. The paper also treats central-binomial prime-power thresholds, an interior-binomial gcd, and fixed-modulus examples. Claims involving finite sequence comparisons are distinguished from their proofs for infinitely many integers. Classical congruence, carry, and rowwise counting results are cited throughout.Licensing note: The manuscript is licensed under CC BY 4.0. The OEIS b-file snapshots included in the supplementary archive retain their original CC BY-SA 4.0 license, with attribution and source information provided in the accompanying README.