COUNTING INDEPENDENT SETS IN AMENABLE GRAPHS

The paper proves an exact computational threshold for additive approximation of the hard‑core free energy on free almost‑transitive amenable group actions: existence of an FPTAA below λc(Δ) and nonexistence above it (unless NP = RP). This is stated as Theorem 8.5 and derived via Proposition 8.3 (algorithm under SSM using SAW trees and an invariant order) and Proposition 8.4 (hardness reduction from finite graphs by replicating across a group action) . The candidate solution outlines the same two directions: a telescoping chain‑rule/SAW‑tree additive scheme under SSM below λc(Δ), and the same replication hardness above λc(Δ). Its representation of fG via one‑site “information” terms aligns with the paper’s randomized/deterministic order formula (Theorem 7.5/7.6) and uses uniform correlation decay below the tree threshold consistent with the paper’s SSM machinery (Theorem 5.7 and related discussion) . A minor difference in presentation (additive control via a Lipschitz bound vs the paper’s multiplicative sandwich) does not affect correctness. Hence both are correct and substantially the same proof.