Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Regulated generalized entropy monotonicity under explicit balance

Physical scope

A conditional deduction for the finite-dimensional nested-regulator model with faithful reference states and the adopted modular-geometric balance. The finite-regulator inequality is pointwise in the named data. A limiting inequality is concluded only when both endpoint entropy sequences have finite limits in the same fixed scheme. The result does not assert limit existence, scheme independence, continuum quantum field theory, physical evolution, semiclassical backreaction, or an unrestricted generalized second law.

Statement

For every finite-regulator data set satisfying the nested balance postulate, set $D_{i} : = D (ρ_{i} ∥ σ_{i})$ for $i = 1, 2$ . Then

$S_{g e n, 2} - S_{g e n, 1} = k_{B} [D (ρ_{1} ∥ σ_{1}) - D (ρ_{2} ∥ σ_{2})] \geq 0.$

If a sequence of such data sets satisfies the postulate term by term under one fixed scheme, and both endpoint sequences converge to finite real values $S_{g e n, i, n} \to L_{i}$ , then $L_{2} \geq L_{1}$ .

Statement references

Nested finite regulator and gravitational balance premises

Finite relative entropy equals modular energy minus entropy

Aron C. Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices

H. S. Reall, Part III Black Holes

John Watrous, The Theory of Quantum Information

Facts & Assumptions

Given: At one named finite regulator, the ordered spaces, density operators, faithful reference operators, partial-trace consistency, geometric balance, and entropy values from the Postulate. For the limit clause, a sequence of such data sets under one fixed scheme with finite real limits at both cuts.

[F1]

Finite relative entropy equals modular energy minus entropy: for finite-dimensional density operators $ρ, σ$ with $σ$ positive definite, $K_{σ} = - \log σ$ and $s (τ) = - tr (τ \log τ)$ in nats, $D (ρ ∥ σ) = tr ((ρ - σ) K_{σ}) - (s (ρ) - s (σ))$ . For a nonzero finite tensor factorization, partial tracing both density operators over one factor cannot increase $D$ .

[A1]

Nested finite regulator and gravitational balance premises: $ρ_{2} = {Tr}_{E} ρ_{1}$ , $σ_{2} = {Tr}_{E} σ_{1}$ , $a_{2} - a_{1} = Δ_{1} - Δ_{2}$ , and $S_{g e n, i} = k_{B} [a_{i} + s (ρ_{i}) - s (σ_{i})]$ , where $Δ_{i} = tr ((ρ_{i} - σ_{i}) (- \log σ_{i}))$ and $k_{B} > 0$ .

Proof

1.1givenF1A1algebra

For $i = 1, 2$ , put $q_{i} : = s (ρ_{i}) - s (σ_{i})$ and $Δ_{i} : = tr ((ρ_{i} - σ_{i}) (- \log σ_{i}))$ . Each $σ_{i}$ is positive definite, so the faithful-reference hypothesis in [F1] is met and the finite identity gives $D_{i} = Δ_{i} - q_{i}$ .

2.1A1step 1.1algebra

By [A1], $a_{2} - a_{1} = Δ_{1} - Δ_{2}$ and $S_{g e n, i} = k_{B} (a_{i} + q_{i})$ . Substitute the two identities from step 1.1 and expand: $S_{g e n, 2} - S_{g e n, 1} = k_{B} [(Δ_{1} - Δ_{2}) + (q_{2} - q_{1})] = k_{B} [(D_{1} + q_{1}) - (D_{2} + q_{2}) + q_{2} - q_{1}] = k_{B} (D_{1} - D_{2})$ . This is an exact finite-dimensional equality.

3.1givenF1A1step 2.1algebra

The factorization and consistency in [A1] identify $(ρ_{2}, σ_{2})$ as the partial traces over $H_{E}$ of $(ρ_{1}, σ_{1})$ . Apply the data-processing inequality in [F1] to obtain $D_{2} \leq D_{1}$ ; multiplication by $k_{B} > 0$ and step 2.1 give $S_{g e n, 2} - S_{g e n, 1} \geq 0$ .

4.1A1step 3.1

For each regulator index $n$ , the finite-regulator argument in step 3.1 gives $S_{g e n, 2, n} - S_{g e n, 1, n} \geq 0$ .

5.1givenstep 4.1algebra∎

Let $ε > 0$ . The assumed finite convergence supplies $N$ such that for all $n \geq N$ , $∣ S_{g e n, 2, n} - L_{2} ∣ < ε / 2$ and $∣ S_{g e n, 1, n} - L_{1} ∣ < ε / 2$ . For any such $n$ , step 4.1 and algebra give $L_{2} - L_{1} = (L_{2} - S_{g e n, 2, n}) + (S_{g e n, 2, n} - S_{g e n, 1, n}) + (S_{g e n, 1, n} - L_{1}) > - ε$ . If $L_{2} - L_{1} < 0$ , choosing $ε = - (L_{2} - L_{1})$ contradicts this strict inequality. Thus $L_{2} - L_{1} \geq 0$ .

Remarks

The balance identity is an adopted premise. The proof uses finite-dimensional partial-trace data processing and real-sequence convergence only; it does not establish that the balance premise holds in a continuum horizon model or that either limit exists.

Regulated generalized entropy monotonicity under explicit balance

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources