Conditional weak entropy principle across mixing channels

Fix an admitted transition $(\mathrm{\Gamma },x)\mathcal{R}({\mathrm{\Gamma }}^{\mathrm{\prime }},y)$. [F1] supplies finite baseline state values on both endpoint types, and [F2] supplies the surjective product maps, positive scaling bijections, zero unit and baseline compatibility required by the exact decorated-chain definition, and identifies its cost convention for this pair. Use the composition-preserving type-weighted representation from [F1]'s construction: interpret each encoded coefficient by its amount divided by the positive type weight, keeping the raw scaled and joined tags distinct. The network compares equal formal masses, and this interpretation preserves its admitted endpoint relation and each finite baseline entropy sum; hence each physical finite-chain witness and its coefficient-map image have the same real edge costs. Reversible regrouping of factors uses the adopted reversible-splitting axiom in [F2], so it changes neither the represented endpoints nor their baseline values. The cost $F$ used by [F3] is therefore this exact witness infimum, not one inferred from coordinate equality. Applying [F3] to this exact transition gives ${\sigma }_{{\mathrm{\Gamma }}^{\mathrm{\prime }}}(y)-{\sigma }_{\mathrm{\Gamma }}(x)\ge F(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\mathrm{\prime }})\mathrm{.}$ If the edge exists, the one-edge chain makes $F(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\mathrm{\prime }})<+\mathrm{\infty }$, so this inequality has an ordinary real right side under the stated no-sinks hypothesis.

The network’s no-sinks condition and [A1] match the hypotheses of [F4]; its cost is the same $F$ as in step 1.1 by [F2] and [F3]. Thus [F4] supplies one offset function $B$, common to all types, with additivity, positive-scale extensivity and $B(\mathrm{\Gamma })-B({\mathrm{\Gamma }}^{\mathrm{\prime }})\le F(\mathrm{\Gamma },{\mathrm{\Gamma }}^{\mathrm{\prime }})$ for every pair.

Subtract the bound in step 2.1 from that in step 1.1: $[{\sigma }_{{\mathrm{\Gamma }}^{\mathrm{\prime }}}(y)+B({\mathrm{\Gamma }}^{\mathrm{\prime }})]-[{\sigma }_{\mathrm{\Gamma }}(x)+B(\mathrm{\Gamma })]\ge 0.$ By the definition of $S$ in the Statement, this is $S_{{\mathrm{\Gamma }}^{\mathrm{\prime }}}(y)\ge S_{\mathrm{\Gamma }}(x)$. The same $B$ works for every admitted transition, proving the conditional weak entropy principle.