= Zeroth-law thermometer comparison
Put a contact <thermometer> $C$ in <thermal contact> with $A$, wait for <thermal equilibrium>, and record response $X_A$. Repeat with $B$ under the same operating constraints. If $X_A=X_B$, and the calibrated response is an <injective function> of <temperature> on this range, $C$ occupies the same thermal reference state in both comparisons. The <zeroth law of thermodynamics> then predicts $A\sim B$. Equality of raw numbers from different uncalibrated <thermometers> supplies no such premise.<University Physics Volume 2 by Samuel J. Ling and others>[§1.1, Figure 1.2 and following paragraph: a thermometer measures its own temperature and transfers the equilibrium comparison]{ref}
Use a probe whose participating <heat capacity> is small relative to each sample, or use maintained <heat baths>. Otherwise the matching readings describe the disturbed final states. Also hold the probe's immersion and mechanical conditions fixed: its response can depend on more than <temperature>. Matching readouts only tests agreement within resolution and repeatability, rather than proving exact equality of <temperatures>.
Agreement within a tolerance is not itself an <equivalence relation>. With a $0.10\,{}^\circ\mathrm C$ tolerance, readings $25.00$ and $25.16\,{}^\circ\mathrm C$ each agree with $25.08\,{}^\circ\mathrm C$, yet disagree with each other. This is a limitation of the measurement rule, not a failure of the exact <transitivity> asserted by the <zeroth law of thermodynamics>.
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