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Spectator cancellation and signed-order algebra

Statement

Under Comparison and stability for compounds, for compounds $A, B$ with $m (A) = m (B)$ and any compound $C$ , $A + C ⪯ B + C$ if and only if $A ⪯ B$ . The signed relation of Signed comparison of finite coefficient maps is well-defined; on nonnegative equal-mass maps it agrees with $⪯$ , is invariant under adding the same finite-support signed map to both sides, and is transitive and closed under adding two comparisons.

Facts & Assumptions

[F1]

Tagged scalable compounds and Compound mass is well-defined and additive: compounds are finite-support nonnegative maps; their masses are finite sums, additive under composition and homogeneous under positive scaling.

[A1]

Comparison and stability for compounds: $⪯$ is a preorder, composition-consistent, positively scale-invariant, admits splitting/recombination, satisfies the stated positive-spectator stability axiom, and has CH.

[F2]

Signed comparison of finite coefficient maps: for equal-total finite signed maps $u, v$ , the signed comparison is defined by comparing $(v - u)^{-}$ with $(v - u)^{+}$ .

[F3]

The canonical natural $ι (n) = n \cdot 1_{F}$ of a field and Canonical naturals are positive and strictly increasing give the canonical embedding $ι_{N} : N \to R$ , whose successor values are positive and are finite sums of $1$ . Reciprocals of the positive integers decrease to zero gives the real sequence $ϵ_{k} = 1 / ι_{N} (σ (k)) > 0$ , $k \in N$ , with $ϵ_{k} \to 0$ .

Proof

Given: [F1], [A1], [F2], and [F3].

1.1F1F2

For equal-total $u, v$ , $d = v - u$ has finite support and total sum zero. Pointwise $d = d^{+} - d^{-}$ , so finite-sum linearity gives $m (d^{+}) - m (d^{-}) = 0$ ; both maps are finite-support and nonnegative. Thus the definition in [F2] compares equal-mass compounds. If $u, v$ are nonnegative, set $k (x) = \min {u (x), v (x)}$ ; then $u = k + (v - u)^{-}$ and $v = k + (v - u)^{+}$ .

1.2A1F1F3algebrainduction

Suppose $A + C ⪯ B + C$ and $m (C) > 0$ . For each $k \in N$ , set $n = σ (k)$ and $r = ι_{N} (n) > 0$ , and write $A / r = (1 / r) A$ . The embedding's successor rule and finite induction show that the sum of $n$ copies of $A / r$ is $r (A / r) = A$ , and likewise for $B$ . Scaling the given comparison by $1 / r$ gives $A / r + C / r ⪯ B / r + C / r$ . By composition consistency, replace the $n$ copies of $A / r$ one at a time by $B / r$ , keeping the already replaced copies, the remaining copies, and $C / r$ as spectators; after these finitely many replacements, $A + ϵ_{k} C ⪯ B + ϵ_{k} C$ . This holds for every natural index $k$ . By [F3], $ϵ_{k} > 0$ and $ϵ_{k} \to 0$ , so stability with $Z_{0} = Z_{1} = C$ gives $A ⪯ B$ . If $m (C) = 0$ , nonnegativity implies $C = 0$ and the conclusion is the premise itself. Conversely, $A ⪯ B$ gives $A + C ⪯ B + C$ by consistency.

2.1F2step 1.1step 1.2

For nonnegative $u, v$ , step 1.1 gives $u = k + (v - u)^{-}$ and $v = k + (v - u)^{+}$ . Cancellation from step 1.2 shows $u ⪯ v$ exactly when $(v - u)^{-} ⪯ (v - u)^{+}$ , which is $u ⪯_{s} v$ . For general signed maps, adding a common map leaves $v - u$ unchanged, so it leaves signed comparison unchanged.

3.1A1F2step 1.2∎

If $u ⪯_{s} v$ and $v ⪯_{s} w$ , consistency gives $N ⪯ P$ , where $N : = (v - u)^{-} + (w - v)^{-}$ and $P : = (v - u)^{+} + (w - v)^{+}$ . Pointwise $P - N = w - u$ . Set $K (x) : = \min {P (x), N (x)}$ . If $P (x) \geq N (x)$ then $P (x) - K (x) = (w - u)^{+} (x)$ and $N (x) - K (x) = (w - u)^{-} (x)$ ; if $P (x) < N (x)$ the same identities follow with the two sides exchanged. Thus $P = K + (w - u)^{+}$ and $N = K + (w - u)^{-}$ , with all four maps finite-support and nonnegative. Cancellation from step 1.2 gives $(w - u)^{-} ⪯ (w - u)^{+}$ , hence $u ⪯_{s} w$ . For two independent signed comparisons, the same pointwise decomposition of the sums of their positive and negative parts proves closure under adding comparisons. Reflexivity follows because $0 ⪯ 0$ .

Depends on

Dependency tree · 2 levels

Sources

Lieb and Yngvason, The Physics and Mathematics of the Second Law of Thermodynamics, Theorem 2.1 and §II.D