CounterexampleConstruction: AI-generatedVerification: AI-adaptedPipeline-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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A controlled nonconcave entropy model with ensemble inequivalence

Statement refuted

For every sequence of finite systems with admissible exact-group microcanonical energy densities tending to an interior value $e$ , there is a fixed positive inverse temperature for which the canonical energy density concentrates at $e$ as system size tends to infinity, without further hypotheses such as entropy concavity.

Example

Fix an energy scale $ε > 0$ . For each $N \geq 1$ and $k = 0, \dots, N$ , set $e_{N, k} = 1 + k / N$ , $E_{N, k} = N ε e_{N, k}$ , and $g_{N, k} : = ⌊ e^{N e_{N, k}^{2}} ⌋ .$ Construct the finite state set ${(k, j) : 0 \leq k \leq N, 1 \leq j \leq g_{N, k}}$ with energy $E_{N, k}$ for each state in group $k$ . Its dimensionless exact-group entropy density tends uniformly on the grid to $s (e) = e^{2}$ on $[1, 2]$ , which is nonconcave. For fixed inverse temperature $β > 0$ , put $b = β ε$ and let $p_{N} (b)$ be the canonical log partition sum per site. Then $p_{N} (b) ⟶ \max_{1 \leq e \leq 2} (e^{2} - b e) = \max (1 - b, 4 - 2 b) .$ For every compact $K \subset (1, 2)$ , the canonical probability that the energy density lies in $K$ tends to zero, while an exact microcanonical group with $e_{N, k_{N}} \to e \in (1, 2)$ is supported at that interior density and has entropy density tending to $e^{2}$ . This is an explicit constructed finite sequence demonstrating that ensemble equivalence does not follow from large $N$ alone.

Facts & Assumptions

Given: The finite group degeneracies, energies, and canonical weighting defined in the statement.

[A1]

$ε > 0$ , $N \geq 1$ , $e_{N, k} = 1 + k / N$ , and $g_{N, k} = ⌊ e^{N e_{N, k}^{2}} ⌋$ .

[A2]

At $b = β ε > 0$ , the group weight is $g_{N, k} e^{- N b e_{N, k}}$ ; an exact microcanonical group is uniform on its $g_{N, k}$ states.

[F1]

The real exponential function and the number $e$ by a power series and The natural logarithm as the inverse of the exponential function: the exponential is positive and $\log$ is its inverse on positive reals.

[F2]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log$ is continuous and increasing on $(0, \infty)$ .

[F3]

Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line: closed bounded intervals are compact. A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value: a continuous real-valued function on a nonempty compact metric space attains its minimum and maximum.

Counterexample

Given: [A1]–[A2] and [F1]–[F3].

1.1A1F1F2

Since $e_{N, k} \geq 1$ , $e^{N e_{N, k}^{2}} \geq e^{N}$ . For $x \geq e^{N}$ , $x - 1 \leq ⌊ x ⌋ \leq x$ , so $e_{N, k}^{2} + \frac{1}{N} \log (1 - e^{- N}) \leq \frac{1}{N} \log g_{N, k} \leq e_{N, k}^{2} .$ The error tends to zero uniformly in $k$ . The grid mesh is $1 / N$ ; for a continuous function on $[1, 2]$ , uniform continuity makes its grid maxima converge to its interval maximum.

2.1A1A2F1F2step 1.1algebra

Let $Z_{N} (b) = \sum_{k = 0}^{N} g_{N, k} e^{- N b e_{N, k}}$ and $m_{N} = \max_{k} [N^{- 1} \log g_{N, k} - b e_{N, k}]$ . The largest summand is $e^{N m_{N}}$ and there are $N + 1$ summands, so $m_{N} \leq N^{- 1} \log Z_{N} (b) \leq m_{N} + N^{- 1} \log (N + 1) .$ By 1.1, $m_{N} \to \max_{e \in [1, 2]} (e^{2} - b e)$ . If $e = θ \cdot 1 + (1 - θ) \cdot 2$ with $0 < θ < 1$ , direct expansion gives $e^{2} - b e = θ (1 - b) + (1 - θ) (4 - 2 b) - θ (1 - θ),$ strictly below the weighted endpoint values. Thus the maximum is at an endpoint and equals $\max (1 - b, 4 - 2 b)$ .

3.1A1A2F1F2F3step 1.1step 2.1

If $K = \emptyset$ , the probability is zero. Otherwise, for compact $K \subset (1, 2)$ strict convexity gives $η : = \min_{e \in K} [\max (1 - b, 4 - 2 b) - (e^{2} - b e)] > 0$ by [F3]. Uniform approximation in 1.1 makes every grid exponent in $K$ at most the maximum minus $η / 2$ for all sufficiently large $N$ . On the other hand, an endpoint group has exponent tending to the maximum. By [F2], $\log (N + 1) / N \leq 2 \sqrt{N + 1} / N \to 0$ , so the polynomial factor is dominated by the exponential gap and the canonical probability of $K$ , at most $(N + 1) e^{- N η / 2 + o (N)}$ , tends to zero.

4.1A1A2step 1.1step 3.1∎

Choose integers $k_{N}$ with $e_{N, k_{N}} \to e \in (1, 2)$ . The microcanonical exact-group law has energy density exactly $e_{N, k_{N}}$ at every state. By 1.1 its entropy density $N^{- 1} \log g_{N, k_{N}}$ tends to $e^{2}$ . Thus, for example at the crossing $b = 3$ , canonical mass escapes every compact neighborhood of any interior target while the microcanonical group remains there. Finally $s (3 / 2) = 9 / 4 < 5 / 2 = (s (1) + s (2)) / 2$ , so $s (e) = e^{2}$ is not concave. The example is a constructed model, not an observation about a material.

A controlled nonconcave entropy model with ensemble inequivalence

Statement refuted

Example

Facts & Assumptions

Counterexample

Depends on

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