ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Two minimizers in a reduced coexistence model

Physical scope

A dimensionless coarse-grained order-parameter example illustrating a possible failure of a singleton variational-minimizer assumption; it is not a continuum Gibbs-potential calculation.

Example

Consider the reduced coarse-grained variational model with scalar dimensionless order parameter $m \in [- 1, 1]$ and dimensionless free-energy function $g (m) = (m^{2} - 1)^{2}$ . Its global minimizers are $m = - 1$ and $m = 1$ . This model shows why uniqueness must be stated as a parameter-specific assumption. It does not identify these two minimizers with Gibbs measures of a continuum pair potential.

Facts & Assumptions

[F1]

Selected grand-variational phase uniqueness: the selected continuum conclusion is licensed only when the relevant minimizer set on the finite-intensity stationary-law domain is a singleton.

[A1]

Adopt the stated toy variational function $g (m) = (m^{2} - 1)^{2}$ on the closed interval $[- 1, 1]$ ; $m$ and $g$ are dimensionless.

Verification

Given: The domain $[- 1, 1]$ , the explicit function $g$ , and the limited illustrative scope.

1.1givenA1algebra

For every $m \in [- 1, 1]$ , $g (m) = (1 - m^{2})^{2} \geq 0$ . Equality holds exactly when $m^{2} = 1$ , namely at $m = - 1$ or $m = 1$ .

2.1givenF1step 1.1∎

The two equality points are distinct, so this scalar model's global-minimizer set is ${- 1, 1}$ and is not a singleton. This illustrates the need for a separate uniqueness assumption in a variational model. The domain in [F1] consists of stationary configuration laws, whereas this domain is $[- 1, 1]$ ; no map identifying their minimizers has been specified. Thus this calculation neither instantiates nor refutes the continuum assumption [F1].

Depends on

Selected grand-variational phase uniqueness

Two minimizers in a reduced coexistence model

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels