Manuscript Oai276-A@1 · Oai276-A.tex · section · 215068af

Gate 2: the thermal extension

Stage 1: audit the five external proof interfaces › Gate 2: the thermal extension · lines 650–694

Oai276-A@1#eq:audit-thermal

The source's thm:thermal concerns matrices $X, Y$ with $Tr X X^{†} = Tr Y Y^{†} = 1$ and coefficients

α, β, η, δ \geq 0, α + β + η + δ = 1, α δ - β η = K^{2}, K \geq 0.

Its state has the form

T = (α X X^{†} + β Y Y^{†} K Y X^{†} K X Y^{†} η X X^{†} + δ Y Y^{†}) .

Writing $U_{0} = α + β$ , $U_{1} = η + δ$ , and $u = U_{0} U_{1} - K^{2}$ , the asserted bound is

S (T) \geq (α + η) S (X X^{†}) + (β + δ) S (Y Y^{†}) + g_{2} (u) .

The source obtains it from zero-block factorizations SourceThermal276.

The relevant algebra is

u = β U_{1} + η U_{0} .

For $u > 0$ , the two coefficient endpoints used in the source have $(β, η) = (u / U_{1}, 0)$ and $(0, u / U_{0})$ , with weights $β U_{1} / u$ and $η U_{0} / u$ . The essential audit is that both endpoint bounds retain the same scalar term $g_{2} (u)$ . Otherwise entropy concavity would not give the advertised thermal statement. The cases $u = 0$ , vanished row sums, and zero mixture weights must be handled separately.

Nothing in this coefficient condition identifies a Hawking occupation, a bath spectrum, or a temporal radiation history.

Displayed formulas in this passage

Oai276-A@1#eq:audit-thermal · line 669
$S (T) \geq (α + η) S (X X^{†}) + (β + δ) S (Y Y^{†}) + g_{2} (u) .$
unlabeled · line 654
$α, β, η, δ \geq 0, α + β + η + δ = 1, α δ - β η = K^{2}, K \geq 0.$
unlabeled · line 660
$T = (α X X^{†} + β Y Y^{†} K Y X^{†} K X Y^{†} η X X^{†} + δ Y Y^{†}) .$
unlabeled · line 679
$u = β U_{1} + η U_{0} .$