CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Interacting marginals are not free Gibbs laws

Statement refuted

A finite marginal of an interacting Gibbs chain always equals the free Gibbs law obtained by retaining only interaction terms wholly inside the observed sites.

Facts & Assumptions

Given: A three-site zero-field Ising chain with spins $s_{1}, s_{2}, s_{3} \in {- 1, + 1}$ , open-chain Hamiltonian $H = - J (s_{1} s_{2} + s_{2} s_{3})$ , $β > 0$ , and $J \neq 0$ .

[A1]

Put $K = β J$ and define $\sinh K : = (e^{K} - e^{- K}) / 2$ , $\cosh K : = (e^{K} + e^{- K}) / 2$ , and $\tanh K : = \sinh K / \cosh K = (e^{K} - e^{- K}) / (e^{K} + e^{- K})$ . The denominator is positive by positivity of the exponential; since $K \neq 0$ , strict increase gives $e^{K} \neq e^{- K}$ , so $\tanh K \neq 0$ (The real exponential function and the number $e$ by a power series, The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , The exponential function is strictly increasing).

[A2]

The specification uses conditional laws given the exterior; it does not identify them with unconditional finite marginals (Lattice Gibbs kernels and DLR states).

[F1]

The finite Gibbs weights are proportional to $e^{- β H}$ and have positive finite normalizer (Finite Gibbs relative entropy and free-energy minimum).

Counterexample

1.1givenA1F1algebra

With $K$ , $\sinh K$ , $\cosh K$ , and $\tanh K$ as in [A1], summing over the middle spin gives $Z = 2 (2 \cosh K)^{2}$ and $E (s_{1} s_{3}) = \frac{2 (2 \sinh K)^{2}}{2 (2 \cosh K)^{2}} = \tanh^{2} K > 0$ . Thus the endpoint marginal is correlated.

2.1A2F1step 1.1algebra∎

The only interaction terms are on bonds ${1, 2}$ and ${2, 3}$ , so no term is wholly supported on the retained endpoint set ${1, 3}$ . The claimed retained-term free law therefore has zero energy and assigns independent uniform endpoint spins, with correlation zero. This contradicts step 1.1, refuting the statement.

Interacting marginals are not free Gibbs laws

Statement refuted

Facts & Assumptions

Counterexample

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