The October 2026 refutations of the Exponential Time Hypothesis can count satisfying assignments in $2^{o(n)}$ time, yet the asymptotic bound does not indicate from which input size—the crossover point—they surpass the best known algorithms. We make Anthropic’s #3‑SAT recursion explicit and lower its crossover points by provable modifications.\ For the clause‑count bound at density one, the crossover drops from roughly $2^{33858}$ clauses to about $2^{420}$ clauses; for general 3‑SAT in the variable dimension $n$, our estimate falls from $2^{2.2\cdot10^{12}}$ to $2^{8.6\cdot10^{9}}$.\ Using integer tables, the recursion counts, for every $j$, the assignments satisfying exactly $j$ clauses in time $2^{O(m/\sqrt{\log m})}$.\ We note that this refutes the Sparse Max‑3‑SAT Hypothesis of Alman and Vassilevska‑Williams, which also follows from the Impagliazzo‑Paturi‑Zane reduction; the corresponding crossover point drops from $2^{68364}$ clauses to about $2^{1016}$ clauses.\ Structurally, we define a rewrite‑and‑merge class containing the recursion and all constructions, and prove that in the main‑term model no grid algorithm can gain for general 3‑SAT below roughly $2^{6000}$ variables. For two balanced homogeneous merges we prove with quantum functionals an exponent of at least $(\log_2 K)^{-3}$ per edge bit, excluding running times of the form $2^{n^{1-c}}$.\ Independently of SAT, we formalise the algorithm as a computation model called Signed Network Sums, which subsumes rank‑one methods and contraction trees. On the complete graph with four nodes, one‑level constructions contract the network in $N^{23/6}$ when $\omega=2$ and in about $N^{3.93}$ on the current (preprint) matrix‑multiplication curve, both below the $N^{4}$ bound of any rank‑one method or contraction tree.\
Review: By exposing the recursion and tightening structural arguments, the paper dramatically lowers the practical crossover thresholds of Anthropic’s subexponential 3‑SAT algorithm, strengthening the case against the Exponential Time Hypothesis.