Physics theoremStatement: AI-adaptedProof: 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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Unitary invariance of finite-sector quantum entropy

Physical scope

The exact closed, finite-dimensional sector model and unitary dynamics of the named NRQM supplier, including its adopted canonical preparation and stated choice assumptions; the result identifies no macroscopic entropy, open-system relaxation, or laboratory behaviour.

Statement

Adopt exactly the state, observable, autonomous-dynamics and canonical-ensemble assumptions, readings, finite-dimensional sector class, and normalized canonical preparation of Unitary evolution preserves entropy and cannot produce the canonical preparation from a different one. The sector $H_{N}^{σ}$ and its self-adjoint energy operator $H_{N}^{σ}$ are those of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable. Model time $t$ is in seconds, $H_{N}^{σ}$ is in joules, and $ℏ = h / (2 π)$ is the reduced Planck constant in $J s$ , as in Dimensions, SI unit conventions and quantity types of the framework. The evolution definition Assigned energy observable, unitary flow and the two pictures of autonomous evolution gives $U (t) = \exp (- i t H_{N}^{σ} / ℏ)$ and $ρ (t) = U (t) ρ (0) U (t)^{*}$ . Both the Axiom of Countable Choice and the Axiom of Choice are assumed (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Define $S_{v N} (ρ) = k_{B} s (ρ)$ as in Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model. Then, for every declared sector preparation $ρ (0)$ and every $t \in R$ :

$S_{v N} (ρ (t)) = S_{v N} (ρ (0))$ ;
the adopted canonical preparation is stationary; and
a preparation different from the canonical one is never mapped to it by this unitary flow.

These are statements about the exact closed finite-sector model. They do not identify $S_{v N}$ with the macroscopic equilibrium entropy $S_{c}$ of Macroscopic and microscopic entropy quantities, derive the canonical preparation, or establish bath relaxation or thermalization.

Facts & Assumptions

Given: The complete model assumptions and conclusions (a)–(d) of the published finite-sector supplier, together with its Countable Choice and Choice assumptions; use the definition $S_{v N} = k_{B} s$ and the entropy-domain distinction stated below.

[F1]

Unitary evolution preserves entropy and cannot produce the canonical preparation from a different one: under the stated finite-dimensional model and postulates, its clauses (a)–(c) prove dimensionless entropy conservation, canonical stationarity, and nonproduction of the canonical preparation from any distinct preparation.

[F2]

Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model: in the declared finite-dimensional sector, $S_{v N} (ρ) = k_{B} s (ρ)$ in $J K^{- 1}$ .

[F3]

Macroscopic and microscopic entropy quantities: $S_{c}$ is a function on the equilibrium macrostate domain $D_{c}$ , whereas $S_{v N}$ is a function on finite-sector density operators; no identification is adopted without an additional bridge.

[F4]

The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable: the declared nonzero sector has a self-adjoint energy operator $H_{N}^{σ}$ assigned as its energy observable.

[F5]

Dimensions, SI unit conventions and quantity types of the framework: model time is in seconds, energy is in joules, and $ℏ = h / (2 π)$ is in $J s$ .

[F6]

Assigned energy observable, unitary flow and the two pictures of autonomous evolution: the closed-model state evolves by $ρ (t) = U (t) ρ (0) U (t)^{*}$ with $U (t) = \exp (- i t H_{N}^{σ} / ℏ)$ .

[F7]

Assigned energy observable, unitary flow and the two pictures of autonomous evolution: $U$ is a strongly continuous unitary group with $U (t)^{*} = U (t)^{- 1} = U (- t)$ .

[F8]

Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model: the supplier defines no entropy for its infinite-dimensional spatial sectors.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) and The Axiom of Choice: assume the Axiom of Countable Choice and the Axiom of Choice, exactly as the published supplier does.

Proof

1.1F1F2F4F5F6A1algebra

Under the finite-sector model [F4], units and time convention [F5], flow [F6], and both choice assumptions [A1], clause (a) of [F1] gives $s (ρ (t)) = s (ρ (0))$ for every declared preparation and every real $t$ . Multiplying by the fixed $k_{B}$ and using [F2] yields $S_{v N} (ρ (t)) = S_{v N} (ρ (0))$ . This proves conclusion 1.

1.2F1F4F5F6A1

Clause (b) of [F1] gives conclusion 2 under the same model and choice assumptions. The canonical preparation is an adopted premise of [F1]; its stationarity does not derive that preparation or a relaxation process.

2.1F3F7F8step 1.1step 1.2algebra∎

If $ρ (t) = ρ_{β}$ for some $t$ , then applying $U (t)^{*}$ on the left and $U (t)$ on the right, using [F7], gives $ρ (0) = U (t)^{*} ρ_{β} U (t)$ . Stationarity in step 1.2 at time $- t$ makes the right-hand side $ρ_{β}$ . Thus $ρ (0) \neq ρ_{β}$ implies $ρ (t) \neq ρ_{β}$ for every real $t$ , which is conclusion 3. By [F3] the invariant in step 1.1 is defined on density operators, while the macroscopic entropy is defined on equilibrium macrostates; [F8] also keeps the von Neumann entropy restricted to finite-dimensional sectors. None of the premises identifies those domains or supplies a bath or macroscopic relaxation law, so no monotonicity claim for $S_{c}$ , bath relaxation, or thermalization follows.

Unitary invariance of finite-sector quantum entropy

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources