The bare join coordinates do not determine contact coherence

Start with a primitive tag $g$ and close it under the ordered binary join constructor $(T,R)\mapsto T\vee R$, retaining a distinct tag for each finite binary tree. Let $n_T\ge 1$ count its primitive leaves, give it amount $n_T$ of the one primitive label and weight $n_T$, and set ${\mathrm{\Gamma }}_T={\mathbb{R}}^{n_T+1}$ with one energy and $n_T$ work coordinates. Define $J_{T,R}((u,v),(u^{\mathrm{\prime }},v^{\mathrm{\prime }}))=(u+u^{\mathrm{\prime }},v,v^{\mathrm{\prime }})$ in the distinct tagged chart ${\mathrm{\Gamma }}_{T\vee R}$. Since $n_{T\vee R}=n_T+n_R$ and energies range over all reals, this chart has exactly the join coordinate-set formula. Reassociation and permutation use the corresponding energy-preserving permutations of work coordinates. Encode a state by ${\mathsf{J}}_T(x)=n_T{e}_{(T,x)}$; separated pairs and joins have equal mass but different raw supports.

On finite encoded compounds define accessibility as the smallest equivalence relation containing, for every $x\in {\mathrm{\Gamma }}_g$, the pair $2e_{(g,x)}$ and $2e_{(g\vee g,J_{g,g}(x,x))}$. Each generating pair has mass two, so every class at any other mass is a singleton; all comparisons have equal mass. Define ${\sim }_T$ by [F1]'s mutual-accessibility rule for the separated pair and its joined state. Fix $x\in {\mathrm{\Gamma }}_g$ and put $z=J_{g,g}(x,x)$. The generating pair gives $x{\sim }_{T}x$. But testing $z{\sim }_{T}x$ compares $2e_{(g\vee g,z)}+e_{(g,x)}$ with $3e_{((g\vee g)\vee g,J_{g\vee g,g}(z,x))}$. These distinct mass-three compounds lie in distinct singleton classes. Thus $z\mathrm{̸}{\sim }_{T}x$, so C1 fails.

On the same tagged charts and encodings, instead make every pair of equal-mass compounds accessible. The supplier's mutual-accessibility rule now makes every thermal comparison true, so C1 holds with exactly the same coordinate formula. These models test only the implication from bare join coordinates; neither is asserted to satisfy all forward-sector, geometric or contact axioms. Both realize thermal equivalence through an actual compound relation. Their different C1 outcomes show that the coordinate data alone do not force coherence.