Calibration of an arbitrary family of simple-system entropies

Use the positive type normalization in Forward-sector hypotheses for a simple state domain whenever invoking [F2]; state entropies evaluate represented maps $\mathsf{J}(x)$, and their unit-label entropy density is $S_i(x)\mathrm{/}{w}_{{\mathrm{\Gamma }}_i}$. Apply [F2] to the meter type and choose its entropy $S_m$ so $S_m(z_0)=0$ and $S_m(z_1)=1$. Set the meter reference to $r_m=z_0$, and use [A1] to choose a reference state $r_i$ and one same-type entropy representative for every $i\mathrm{\ne }m$. For each $i\mathrm{\ne }m$, use [F1] to choose calibrator states $a_i,b_i\in {\mathrm{\Gamma }}_i$ and $c_i,d_i\in {\mathrm{\Gamma }}_m$ such that $a_i\prec \prec b_i$, $c_i\prec \prec d_i$, and $(a_i,d_i)\sim (b_i,c_i)$; also choose $r_i\in {\mathrm{\Gamma }}_i$. For the meter label set $a_m=c_m=z_0$ and $b_m=d_m=z_1$; the two pairs $(z_0,z_1)$ and $(z_1,z_0)$ are the same two-copy meter compound up to permutation. Write $S_i^0$ for the selected entropy representative, with $S_m^0=S_m$, and put ${\mathrm{\Delta }}_i=S_i^0(b_i)-S_i^0(a_i)>0$ and ${\mu }_i=S_m(d_i)-S_m(c_i)>0$. Define $S_i(x):=\frac{{\mu }_i}{{\mathrm{\Delta }}_i}(S_i^0(x)-S_i^0(r_i))$. Then $S_i(r_i)=0$ and $S_i(b_i)-S_i(a_i)={\mu }_i$, matching the meter entropy increment across the chosen calibrator endpoints. For $i=m$, ${\mathrm{\Delta }}_m={\mu }_m=1$, so this agrees with the fixed meter normalization.

For a finite separated compound $A$ with factors $(t_{\mathrm{\ell }}{i}_{\mathrm{\ell }},x_{\mathrm{\ell }})$, define $S(A):={\sum }_{\mathrm{\ell }}{t}_{\mathrm{\ell }}{S}_{i_{\mathrm{\ell }}}(x_{\mathrm{\ell }})$. The sum is finite and invariant under reordering; concatenating compounds adds the sums, and scaling every amount by $t>0$ multiplies it by $t$. The explicit-factor convention in the Statement therefore makes $S$ well-defined, additive, and extensive.

Fix a composition with amounts $M_i$ for finitely many labels and let $A,B$ be two states of that same composition. For a state $x$ of type $i$, put $s_i(x):=\frac{S_i(x)-S_i(a_i)}{{\mu }_i}\mathrm{.}$ The same-type entropy representation and the CH for type $i$ show $\mathsf{J}(x){\sim }_s(1-s_i(x))\mathsf{J}(a_i)+s_i(x)\mathsf{J}(b_i)\mathrm{.}$ Indeed, the signed difference has total coefficient zero and $S_i$-value zero, so its positive and negative parts have equal mass and entropy and therefore compare both ways by [F2]. The calibrator relation gives $(\mathsf{J}(b_i)-\mathsf{J}(a_i)){\sim }_s(\mathsf{J}(d_i)-\mathsf{J}(c_i))$ by taking the signed difference of $(a_i,d_i)\sim (b_i,c_i)$ and canceling common terms with [F3]. Add and scale these identities over the finitely many factors of $A$. With $q_i(A):={\sum }_{\mathrm{\ell }:i_{\mathrm{\ell }}=i}{t}_{\mathrm{\ell }}{s}_i(x_{\mathrm{\ell }}),$ the resulting signed map is $A{\sim }_s{\sum }_i{M}_i\mathsf{J}(a_i)+{\sum }_i{q}_i(A)(\mathsf{J}(d_i)-\mathsf{J}(c_i)),$ where the sums range over the finite support of the composition. The same holds for $B$. Since they have the same composition, the reference base cancels, and $B-A$ is signed-equivalent to the finite meter map $D:={\sum }_i(q_i(B)-q_i(A))(\mathsf{J}(d_i)-\mathsf{J}(c_i))\mathrm{.}$ Extend the meter entropy linearly to raw finite signed coefficient maps using density $S_m(z)\mathrm{/}{w}_{{\mathrm{\Gamma }}_m}$ at a meter raw tag $({\mathrm{\Gamma }}_m,z)$; in particular its value on $\mathsf{J}(z)$ is $S_m(z)$. Then $S_m(D)={\sum }_i(q_i(B)-q_i(A)){\mu }_i=S(B)-S(A)\mathrm{.}$ The positive and negative parts of $D$ have equal mass. If $S(A)\le S(B)$, the same-type entropy representation [F2] for finite meter products gives $D^{-}\prec D^{+}$, hence $D$ compares in the forward signed direction. Translation, addition, and ordinary-order compatibility [F3] transfer this to $A\prec B$. Conversely, if $A\prec B$, the same signed equivalences and translation in [F3] give $D^{-}\prec D^{+}$. The meter representation in [F2] then forces $S_m(D)=S(B)-S(A)\ge 0$. Reversing the entropy inequality gives the reverse comparison. Thus $S$ characterizes every same-composition comparison. Each particular comparison uses only finitely many labels; no infinite sum or uncountable product occurs.

The meter normalization fixes the multiplicative unit. Changing the chosen reference state for type $i$ changes only its additive constant; on a finite compound these constants contribute amount-weighted conserved-type offsets. This proves the stated freedom and completes the construction.