The Ford-Konyagin-Luca conjecture: the prime factors of $p-1$ follow the Poisson-Dirichlet law

Theorem 1.1: for every $k$ and bounded continuous $F$ on $[0,1]^k$, the average of $F(V_1(p),\ldots,V_k(p))$ over primes $3\le p\le x$ tends to $\mathbb E F(L_1,\ldots,L_k)$ with $(L_j)\sim\mathrm{PD}(1)$, unconditionally and with equal weight on each prime. The manuscript notes this gives Granville's fixed-$u$ asymptotic $\#\{p\le x:P^+(p-1)\le x^{1/u}\}\sim\pi(x)\rho(u)$, so a positive proportion of primes have $x^{\varepsilon}$-smooth predecessors for each fixed $\varepsilon>0$. No error terms or uniformity as $u$ grows are claimed. The companion on $\mu(p-1)=1$ (infinitely many primes with $p-1$ squarefree with an even number of prime factors) is linked here, not cataloged, since it cites no posing source.

Theorem 1.1: for every $k$ and bounded continuous $F$ on $[0,1]^k$, the average of $F(V_1(p),\ldots,V_k(p))$ over primes $3\le p\le x$ tends to $\mathbb E F(L_1,\ldots,L_k)$ with $(L_j)\sim\mathrm{PD}(1)$, unconditionally and with equal weight on each prime. The manuscript notes this gives Granville's fixed-$u$ asymptotic $\#\{p\le x:P^+(p-1)\le x^{1/u}\}\sim\pi(x)\rho(u)$, so a positive proportion of primes have $x^{\varepsilon}$-smooth predecessors for each fixed $\varepsilon>0$. No error terms or uniformity as $u$ grows are claimed. The companion on $\mu(p-1)=1$ (infinitely many primes with $p-1$ squarefree with an even number of prime factors) is linked here, not cataloged, since it cites no posing source.