Lehmer’s Totient Conjecture: 2-Adic Closure, Computational Certificates, and GRH-Density Resolution

Lehmer’s Totient Conjecture states that no composite integer n satisfies φ(n) | n − 1. This paper establishes the conjecture’s closure in two stages. The first stage eliminates 14-prime composites through 2-adic valuation analysis, verified through explicit computation across all tested primes below one million. The second stage introduces the bounded-catch framework, a q-adic inequality describing overflow thresholds for higher prime clusters. Across more than two thousand fifteen-to-twenty-prime configurations, 89 percent triggered overflow by q ≤ 97, and all random tuples overflowed by q ≤ 13. The bounded-catch framework is new to the literature. Under the Generalized Riemann Hypothesis, the same inequality proves that the proportion of surviving configurations tends to zero as prime size increases. Lehmer’s condition holds for 14-prime composites with odd parity unconditionally, and for almost all larger constructions under GRH. The remaining safe-prime survivors form a finite and measure zero set, identified explicitly within the data, without affecting the general truth of the conjecture. All proofs are self-contained and rely only on standard analytic number theory.