LemmaStatement: 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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Scaling invariance of thermal equivalence

Statement

Under T1–T3 and the equal-state reversible split in Thermal joins and conditional contact hypotheses, if $x \sim_{T} y$ then $s x \sim_{T} t y$ for every $s, t > 0$ .

Facts & Assumptions

Given: The contact and scaling assumptions for tagged states and their joins.

[A1]

Thermal joins and conditional contact hypotheses: each equal-state scaled pair is reversibly equivalent to its join, and thermal equivalence is transitive.

Proof

1.1A1given

Let $x$ be any typed state and $s > 0$ . The state $z = (1 + s) x$ in the scaled type has the equal-state reversible split with fractions $1 / (1 + s)$ and $s / (1 + s)$ ; its two components are exactly $x$ and $s x$ . Thus $(x, s x)$ is reversible with its join, which by definition is $x \sim_{T} s x$ . The same argument gives $y \sim_{T} t y$ .

2.1A1step 1.1given∎

Since $x \sim_{T} y$ , symmetry gives $y \sim_{T} x$ . Transitivity with $x \sim_{T} s x$ yields $y \sim_{T} s x$ ; symmetry then gives $s x \sim_{T} y$ . Transitivity with $y \sim_{T} t y$ now yields $s x \sim_{T} t y$ .

Depends on

Thermal joins and conditional contact hypotheses

Scaling invariance of thermal equivalence

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources