The satisfiability threshold conjecture: a limiting random k-SAT threshold for every k >= 3

Claims Theorem 1.1: for every fixed $k\ge3$ there is $\alpha_k\in(0,\infty)$ with $\Pr(F_{n,\lfloor cn\rfloor}\in\mathrm{SAT})\to1$ for $c<\alpha_k$ and $\to0$ for $c>\alpha_k$; it says nothing at $c=\alpha_k$ and does not identify $\alpha_k$ for small $k$. Companions claim hitting-time variance $\Theta_k(n)$ for $k\ge4$ and $\Theta(n)$ for $k=3$, and that $\alpha_3$ is a computable real. Priority: the release credits Gaia Carenini's ECCC TR26-229 (made public 5 October 2026) with first resolving the existence conjecture; this is an independent second proof of the same result, not a contradiction. Carenini's paper states that none of its mathematical arguments was generated by a large language model; it proves the existence via a scaling window of width $O(n^{1/2+1/k})$ and the Abbe-Montanari criterion.

Claims Theorem 1.1: for every fixed $k\ge3$ there is $\alpha_k\in(0,\infty)$ with $\Pr(F_{n,\lfloor cn\rfloor}\in\mathrm{SAT})\to1$ for $c<\alpha_k$ and $\to0$ for $c>\alpha_k$; it says nothing at $c=\alpha_k$ and does not identify $\alpha_k$ for small $k$. Companions claim hitting-time variance $\Theta_k(n)$ for $k\ge4$ and $\Theta(n)$ for $k=3$, and that $\alpha_3$ is a computable real. Priority: the release credits Gaia Carenini's ECCC TR26-229 (made public 5 October 2026) with first resolving the existence conjecture; this is an independent second proof of the same result, not a contradiction. Carenini's paper states that none of its mathematical arguments was generated by a large language model; it proves the existence via a scaling window of width $O(n^{1/2+1/k})$ and the Abbe-Montanari criterion.