Conditional contact calibrator and product comparison

Fix two simple types $i,j$ in the declared family. Give every primitive material its unit amount vector and use the aggregate composition of Tagged scaled systems and separated products for each scaled, separated or joined type $\mathrm{\Gamma }$; its sum is $w_{\mathrm{\Gamma }}>0$ unless the type is empty. Construct [F2]'s encoded state and interpretation maps using the physical tagged scaling maps: a raw coefficient $a>0$ at $(\mathrm{\Gamma },x)$ denotes the preparation $(a\mathrm{/}{w}_{\mathrm{\Gamma }})x$. The map preserves aggregate composition and formal mass, and distinct raw tags remain distinct. The interpretations of ${\mathsf{J}}_{s\mathrm{\Gamma }}({\delta }_{s}x)$ and $s{\mathsf{J}}_{\mathrm{\Gamma }}(x)$ are the same declared scaled preparation by composition of scaling maps; reflexivity gives reversible transport. When factor coefficients combine or split, [A1]'s reversible-splitting axiom supplies reversibility, so the pullback inherits composition, positive scaling and stability from [A1]. Dividing fixed-type encoded maps by $w_{\mathrm{\Gamma }}$ and scaling back preserves and reflects comparisons, so the one-copy forward sectors and all geometry in [A2] are unchanged. Any common sign chosen in the mathematical orientation deduction is used only for proof coordinates $u^{\mathrm{\prime }}=\epsilon U$; the physical energy $U$ retains its fixed SI meaning, units and all independently adopted positive-$S_U$ or calibration qualifications. Since the same sign is applied to each component and join proof coordinate, summed-energy identities are preserved. Thus the actual preparation model supplies the exact formal conventions and all order/geometric hypotheses required by [F1].

By [A3], the physical operation Thermal joins and thermal equivalence supplies T1–T4 with the same summed-energy coordinates and reversible splitting used in [F1]. Its pair and join retain the same aggregate composition by Thermal joins and thermal equivalence, so their encoded weights both equal $w_i+w_j$; no mass-one join is compared to an unweighted mass-two pair.

[A4] supplies the separately stated T5 quantifiers for every ordered simple-system pair, and [A5] supplies the separate C1 nested-join equivalence. Thus every premise of [F1], including its two stronger contact inputs, is met on the whole declared join-closed family.

Apply [F1] to this pair $i,j$. Its first conclusion is states $x_0\prec \prec x_1$ and $y_0\prec \prec y_1$ with $(x_0,y_1)\sim (x_1,y_0)$; its second is comparison for every finite separated product made from scaled copies of these two types. This is precisely the stated pairwise conclusion.