Mixed phases in feedback Ising models

The paper derives the zero-temperature discontinuous ODE ṁ = −m + sgn(h − g(m)) with regions R±, explicit flows, and the trapping/super-(in)stability conditions (Eqs. (9)–(19)) and identifies C0: h = g(m) as the MP locus. It then specializes to the linear feedback f(m)=1+γm, giving g(m)=−m−(3/2)γ m^2, endpoints h±, the vertex at m=−1/(3γ) and stability split for γ>1/3, exactly as used by the model solution. For Maxwell points, the paper formulates ΔĤ = ∫(g−h) du and proves uniqueness between two stable branches, then computes, for the linear case, h* between m± as −γ/2 for γ∈[0,1] and h* between m+=1 and the stable MP m1=−(γ+1)/(2γ) (valid only for γ>1), with h* = −(1/8)((1+γ)(−1+3γ)/γ), and concludes stable MPs are ground states only when γ>1. All of these match the model’s Part A–C. The model’s use of Filippov sliding to write ṁ = h′/g′ along C0 is consistent with the paper’s trapping condition, and its short proof of Maxwell uniqueness via ΔĤ′(h)=−(mU−mL)<0 refines the paper’s uniqueness statement. No substantive discrepancies were found. Key points: zero-T dynamics and trapping criteria (Eqs. (15)–(19) in the paper) align with the model’s Part A; the linear FIM geometry and stability (vertex and γ-thresholds) match (Fig. 2 and text); and the Maxwell constructions and ground-state regimes coincide, including the conditions γ∈[0,1] for the m± jump and γ>1 for the m+=1 vs MP jump. Citations: zero-T dynamics and equilibria C±, C0, R±, and trapping criteria ; definition of g and general setup ; linear FIM geometry and stability split at γ=1/3 ; Maxwell construction, uniqueness, and linear-case formulas (including m1 and h*) .