No two phases at weak coupling

Theorem 13.2 · thm:dichotomy · page 31

What is happening

Suppose the weak-coupling theory did break the ℤ₂ symmetry: a bounded order parameter B localized in O, with |ω(B)| ≥ δ > 0 in each phase. Translate it far away by a, and clustering applies to the pair.

The uniform gap gives the truncated correlator a bound that holds uniformly in the cutoff, |ω_n(B τ_a(B)) − ω_n(B) ω_n(τ_a(B))| ≤ 2‖B‖² e^−m₁R_a/2, and it survives the limit. Push R_a out and the correlations are forced to zero — but a broken phase needs them to persist at δ². Beyond the crossing the two requirements cannot both hold, which is the contradiction that proves the theorem.

Theorem 13.2 is stated as the impossibility of (B1)–(B4) holding together, conditional on the same imports as Theorem 12.2 — so it inherits U_GJS. It excludes a two-phase structure of that specific form; it is not a uniqueness theorem for ω_∞ among all locally normal ground states, which the manuscript explicitly does not claim.