TheoremStatement: Literature-sourcedProof: 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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Independent subsystem bulk limit

Statement

Let $Ω_{0}$ be a nonempty finite set, let $E : Ω_{0} \to R$ , and fix $β \in R$ . Define $Z_{1} = \sum_{i \in Ω_{0}} e^{- β E_{i}}$ and the canonical law $q_{i} = e^{- β E_{i}} / Z_{1}$ , and define $\overset{ˉ}{E} : = \sum_{i} q_{i} E_{i}$ . For $N \geq 1$ , on $Ω_{0}^{N}$ define $H_{N} (i_{1}, \dots, i_{N}) = \sum_{k = 1}^{N} E_{i_{k}}$ . Give $Ω_{0}^{N}$ the product probability law assigning $(i_{1}, \dots, i_{N})$ weight $\prod_{k = 1}^{N} q_{i_{k}}$ , and define the $N$ -subsystem partition sum $Z_{N} : = \sum_{(i_{1}, \dots, i_{N}) \in Ω_{0}^{N}} e^{- β H_{N} (i_{1}, \dots, i_{N})}$ . Then $Z_{N} = Z_{1}^{N}$ , $E [H_{N} / N] = \overset{ˉ}{E}$ , and $Var (H_{N} / N) = {Var}_{q} (E) / N$ . Consequently, for every $ε > 0$ , $P (∣ H_{N} / N - \overset{ˉ}{E} ∣ \geq ε) \leq {Var}_{q} (E) / (N ε^{2})$ . If $β \neq 0$ , the mathematical free-energy density $- β^{- 1} N^{- 1} \log Z_{N}$ is exactly $- β^{- 1} \log Z_{1}$ for every $N$ .

Facts & Assumptions

Given: The finite one-subsystem state set, energy function, canonical law, and its finite product law.

[A1]

$Ω_{0}$ is a nonempty finite set, $E$ is real-valued, $β$ is fixed, and the $N$ -subsystem law is the product of the one-subsystem canonical law.

[F1]

Finite probability spaces, outcome weights, events, and event probabilities, The finite product of finite probability spaces, and Product weights normalize, and coordinate events are mutually independent: the product law is a finite probability law with independent coordinates.

[F2]

Expectation of a real random variable on a finite probability space and Expectation factors over a finite product of mutually independent random variables: expectation is a finite weighted sum and independent coordinate products factor.

[F3]

Variance, standard deviation, and covariance on a finite probability space and Variance of a finite sum as the sum of all variances and covariances: variance is the centered second moment and expands into variances and covariances.

[F4]

The real exponential function and the number $e$ by a power series, The natural logarithm as the inverse of the exponential function, The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ , and Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\exp (x + y) = \exp (x) \exp (y)$ for real $x, y$ , the exponential has positive values as specified by the logarithm inverse convention, and logarithms convert positive products to sums.

[F5]

Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule: iterated finite sums over a Cartesian product can be reordered and factored.

Proof

Given: The finite one-subsystem state set, its energy function, and the product probability model.

1.1A1F1F4F5

By finite Fubini, $Z_{N} = \sum_{(i_{1}, \dots, i_{N})} \prod_{k = 1}^{N} e^{- β E_{i_{k}}} = \prod_{k = 1}^{N} \sum_{i_{k} \in Ω_{0}} e^{- β E_{i_{k}}} = Z_{1}^{N}$ . This is an exact finite identity.

1.2A1F1F2F3

Under the product law, the coordinate energies $E (i_{k})$ are mutually independent and each has mean $\overset{ˉ}{E}$ and variance $v : = {Var}_{q} (E)$ . Linearity gives $E H_{N} = N \overset{ˉ}{E}$ ; the covariance terms between distinct coordinates vanish by independence, so [F3] gives $Var (H_{N}) = N v$ . Dividing by $N$ and $N^{2}$ proves the mean and variance formulas.

2.1F2F3step 1.2

On the event $∣ H_{N} / N - \overset{ˉ}{E} ∣ \geq ε$ , the nonnegative random variable $(H_{N} / N - \overset{ˉ}{E})^{2}$ is at least $ε^{2}$ . Summing over that event gives $ε^{2} P (∣ H_{N} / N - \overset{ˉ}{E} ∣ \geq ε) \leq E [(H_{N} / N - \overset{ˉ}{E})^{2}] = v / N$ , proving the probability bound without any independence assertion beyond the product model.

3.1A1F4F5step 1.1∎

If $β \neq 0$ , substitute $Z_{N} = Z_{1}^{N}$ and apply the logarithm product law: $- β^{- 1} N^{- 1} \log Z_{N} = - β^{- 1} N^{- 1} N \log Z_{1} = - β^{- 1} \log Z_{1}$ .

Independent subsystem bulk limit

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources