DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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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Signed comparison of finite coefficient maps

Definition

Let $u, v : X \to R$ be finite-support real coefficient maps with equal total coefficient $\sum_{x} u (x) = \sum_{x} v (x)$ . These signed maps are formal comparison notation, not compounds. Put $d : = v - u$ and define $d^{+} (x) : = \max {d (x), 0}$ and $d^{-} (x) : = \max {- d (x), 0}$ . Then $d^{+}, d^{-}$ are nonnegative finite-support compounds of equal mass, as verified in Spectator cancellation and signed-order algebra ↗. Define

$u ⪯_{s} v ⟺ d^{-} ⪯ d^{+},$

using the compound comparison relation of Comparison and stability for compounds. Thus positive coefficients of $v - u$ remain on the right and negative coefficients move to the left. Signed notation is never a claim that a negative amount of a physical system exists.

Depends on

Dependency tree · 2 levels

Sources

Lieb and Yngvason, The Physics and Mathematics of the Second Law of Thermodynamics, §II.D