Finite-family calibration by a common meter

Let $I$ be a finite nonempty set of system types, $X_i$ a nonempty state set with chosen reference $r_i\in X_i$ for each $i\in I$, and $M$ a common meter state set with distinct $z_0,z_1\in M$. For each $i$, let ${⪯}_i$ be a comparison relation on equal-mass compounds over the Cartesian product $X_i\times M$ (The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$), defined as in Comparison and stability for compounds and satisfying its six hypotheses: preorder, composition consistency, positive scaling, splitting and recombination, stability, and CH. Write ${⪯}_{s,i}$ for the signed comparison supplied by Signed comparison of finite coefficient maps, and ${\sim }_{s,i}$ for mutual signed comparison. Assume $(r_i,z_0){⪯}_i(r_i,z_1)$ and not $(r_i,z_1){⪯}_i(r_i,z_0)$. Let $S_i$ be the normalized representation supplied by Additive extensive representation of compound comparison for this product comparison, and put ${\sigma }_i(x):=S_i(x,z_0)$. In the product comparison, the unit product-state point mass satisfies

$e_{(x,z_0)}{\sim }_{s,i}(1-{\sigma }_i(x))e_{(r_i,z_0)}+{\sigma }_i(x)e_{(r_i,z_1)},$

by Reference entropy supremum is finite and attained. These product-state point masses and product comparisons are separate from the shared state-and-meter compounds and comparison defined next.

Assume a common-meter compatibility interface as follows. Let $\mathcal{Y}$ be a set of raw tags containing three pairwise disjoint families $\{a_{i,x}:i\in I,x\in X_i\}$, $\{{\rho }_i:i\in I\}$, and $\{{\zeta }_0,{\zeta }_1\}$, with ${\zeta }_0\mathrm{\ne }{\zeta }_1$. Compounds over $\mathcal{Y}$ are finite-support nonnegative coefficient maps as in Tagged scalable compounds. Let ${⪯}_{\mathcal{Y}}$ be a relation on equal-mass compounds over $\mathcal{Y}$ satisfying all six hypotheses of Comparison and stability for compounds; write ${⪯}_{s,\mathcal{Y}}$ for its signed comparison from Signed comparison of finite coefficient maps and ${\sim }_{s,\mathcal{Y}}$ for mutual signed comparison. The common relation is a separate relation from each product comparison ${⪯}_i$.

Let $e_{{\rho }_i}$ and $e_{{\zeta }_j}$ be the unit point-mass compounds on their indicated raw tags, and define $u_{i,x}:=e_{a_{i,x}}+e_{{\zeta }_0}$. Thus $u_{i,x}$ is an explicit mass-two state-and-fixed-meter compound, not a unit tag. Require the exact signed substitution

$u_{i,x}{\sim }_{s,\mathcal{Y}}{e}_{{\rho }_i}+(1-{\sigma }_i(x))e_{{\zeta }_0}+{\sigma }_i(x)e_{{\zeta }_1}(i\in I,x\in X_i),$

and require the common meter point masses to satisfy $e_{{\zeta }_0}{⪯}_{\mathcal{Y}}{e}_{{\zeta }_1}$ and not $e_{{\zeta }_1}{⪯}_{\mathcal{Y}}{e}_{{\zeta }_0}$. Both sides of each substitution have total coefficient two, even when ${\sigma }_i(x)$ lies outside $[0,1]$. Finite substitutions compose and common terms cancel by Spectator cancellation and signed-order algebra under the stated common comparison hypotheses; neither property is inferred from the separate product comparisons or from numerical meter values.

For positive coefficients $t_{ik}$ on a finite collection of type-$i$ states $x_{ik}$, define $T_i:={\sum }_k{t}_{ik}$ and $T:={\sum }_i{T}_i$. Put $C_0:={\sum }_{i,k}{t}_{ik}{e}_{a_{i,x_{ik}}}$ and $C:={\sum }_{i,k}{t}_{ik}{u}_{i,x_{ik}}=C_0+T{e}_{{\zeta }_0}$. Thus $C_0$ is the state compound of mass $T$ and $C$ is its fixed-meter encoding of mass $2T$. Empty collections give $T=0$ and $C_0=C=0$. Define

$\mathrm{\Sigma }(C):={\sum }_{i\in I}{\sum }_k{t}_{ik}{\sigma }_i(x_{ik})\mathrm{.}$

For two such compounds $C,D$ with the same type-total vector $(T_i{)}_{i\in I}$, the common comparison satisfies $C{⪯}_{\mathcal{Y}}D$ exactly when $\mathrm{\Sigma }(C)\le \mathrm{\Sigma }(D)$. Equivalently, the same order criterion holds for the state compounds $C_0,D_0$ after canceling their common $T{e}_{{\zeta }_0}$. The index family is finite; no arbitrary-index choice or infinite decomposition is asserted.

The compatibility interface records the compositional assumption needed to move from separate product systems to one shared meter scale. Without it, independent representations need not share a multiplicative calibration. This theorem makes no statement for infinitely many types.