Sparse reconstruction in spin systems II: Ising and other factor of IID measures

The paper’s Theorem 1.6 states exactly the four claims the candidate addresses: full sparse reconstruction (SR) for magnetization and SR for majority under diverging susceptibility; full SR for majority under a unique giant FK cluster; specializations to Zd-tori and Curie–Weiss with the stated thresholds. The paper proves these using general Section 3 tools, a Paley–Zygmund + Lebowitz inequality bound for majority, and standard FK geometry for the giant-cluster regime, with explicit statements of parts (1)–(4) (Theorem 1.6) and their proofs in §7.2 . The candidate gives a different, valid proof strategy: an exact FK/Edwards–Sokal L2 identity for Var(Mn) and Var(E[Mn|FUn]) and a clean Bernoulli(pn) calculation yielding pnχβn→∞ as the right scaling; a cluster-mass decomposition with Paley–Zygmund and Hoeffding for majority; and a direct giant-cluster argument. This matches the theorem’s conclusions (including the Bernoulli(pn) clause in Theorem 1.6) and the Zd and Curie–Weiss corollaries (Theorem 1.6 (3),(4)) . One minor slip in the candidate’s Curie–Weiss algebra is noted below, but it does not affect the final threshold statements.